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Preface The work in this book started twelve years ago when I was studying for a Ph.D degree in the University of Ed...
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i
Preface The work in this book started twelve years ago when I was studying for a Ph.D degree in the University of Edinburgh. At the beginning, I was only attempting to apply the developed meteorological dynamics and classical thermodynamics to study the large-scale processes of energy conversion in the Earth’s atmosphere. As the work went on, it was found that the atmosphere was essentially different from the classical dynamic or thermodynamic systems, and we had no a particular thermodynamic theory to deal with the large-scale inhomogeneous and compressible fluid system in the gravitational field of the Earth. The studies on the particular features of the atmosphere lead to establishment of the new science, called the largescale inhomogeneous thermodynamics here. This new thermodynamics may have many applications. The atmospheric thermodynamics and energetics discussed in this book is one of them. A science, such as the classical dynamics, can be illustrated by different theories based on different definitions or expressions. One may use complicated mathematical symbols to represent simple ideals, or introduce new terminologies to answer questions as: “He was unable to sleep at night because he was suffering from insomnia”. I am more interested in understand the world than creating a new one on paper. Thus, the theories in this book are aimed mainly to reveal the new facts and explain them in the large-scale inhomogeneous thermodynamic systems, but not to make hypothesis for unknown processes and then derive mathematically the theoretical consequences. To find is more difficult than to create sometime. Many facts in front of us may not be seen if we do not intend to find them. They could be found by mind before by eyes. It is always interesting for me to discuss the physical philosophies behind the phenomena, while the mathematics is given in the form of easiest understanding. Some discussions may be challenging to widely accepted theories, but may also be inspiring. It was eight years ago, the author published his first book “Geostrophic Wave Circulations” (1993) in the pen name of Yong L. McHall, introducing a simple and effective method to solve the complete primitive equations in atmospheric dynamics. The previous book was focused on the large-scale dynamic processes in the extratropical troposphere and stratosphere without considering the thermodynamic processes. Since the atmosphere is able to produce kinetic energy by itself, the dynamics cannot be perfect if without the theory of kinetic energy generation. Now, I am glad to forward my new contribution in the first year of the new millenium. The two books tell us that the meteorology is no longer the applications of hydrodynamics, classical thermodynamics and numerical technology, as it has its own theoretical systems powerful to solve successfully many problems which cannot be solved before. Yong Zhu At MIT Cambridge, MA
Contents 1 Introduction
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2 Two classical physical systems 2.1 Introduction . . . . . . . . . . . . . . . . . 2.2 The Newtonian systems . . . . . . . . . . 2.2.1 Principle of friction . . . . . . . . . 2.2.2 Dynamic entropy . . . . . . . . . . 2.3 Simple thermodynamic systems . . . . . . 2.3.1 Mole-number and molecular mass . 2.3.2 Thermodynamic variables . . . . . 2.3.3 Pressure of monatomic gas . . . . 2.4 The first law of thermodynamics . . . . . 2.5 State equation of gases . . . . . . . . . . . 2.6 State equation of ideal gases . . . . . . . . 2.6.1 Ideal-gas equation . . . . . . . . . 2.6.2 More features of ideal gases . . . . 2.6.3 Kelvin temperature . . . . . . . . 2.6.4 Mixing ratio of water vapor . . . . 2.7 Thermodynamic energy law of ideal gases 2.8 Internal energy and heat exchange . . . . 2.9 Polytropic process . . . . . . . . . . . . .
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3 Molecular transport processes 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.2 Diffusion velocity and partial velocities . . . . . . . . 3.2.1 Diffusion element and diffusion velocity . . . 3.2.2 Partial velocities . . . . . . . . . . . . . . . . 3.2.3 Diffusion velocity in non-uniform ideal gases . 3.3 Self-diffusion of ideal gases . . . . . . . . . . . . . . . 3.3.1 Diffusive mass flux . . . . . . . . . . . . . . . 3.3.2 Coefficient of self-diffusion . . . . . . . . . . . 3.4 Viscosity of ideal gases . . . . . . . . . . . . . . . . . 3.4.1 Diffusive momentum flux . . . . . . . . . . . 3.4.2 Momentum conduction . . . . . . . . . . . . 3.4.3 Coefficient of viscosity . . . . . . . . . . . . . 3.4.4 Relation to self-diffusion . . . . . . . . . . . . 3.5 Heat conduction of ideal gases . . . . . . . . . . . . . 3.5.1 Conductive heat flux . . . . . . . . . . . . . . 3.5.2 Heat conductivity . . . . . . . . . . . . . . . 3.5.3 Modified Eucken formula . . . . . . . . . . . 3.5.4 Collisional heat capacity . . . . . . . . . . . . 3.5.5 Comparison with experiments . . . . . . . . . ii
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23 23 24 24 25 27 27 28 30 32 33 36 36 38 40 41 41 44 47
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51 51 52 52 54 56 58 58 60 61 61 62 63 64 64 64 65 66 67 68
CONTENTS
iii
4 Predictability and thermodynamic entropy 4.1 Introduction . . . . . . . . . . . . . . . . . . . . 4.2 Change rate in diffusion processes . . . . . . . . 4.3 Mass conservation law . . . . . . . . . . . . . . 4.3.1 Mass diffusion equation . . . . . . . . . 4.3.2 Mass conservation . . . . . . . . . . . . 4.3.3 Diffusive transport equation . . . . . . . 4.4 Unpredictability in classical thermodynamics . 4.5 Thermodynamic entropy law for uniform states 4.6 Thermodynamic entropy change of non-uniform 4.7 Inadditive and scale-dependent features . . . . 4.8 Thermodynamic entropy balance equation . . . 4.9 Relation to dynamic entropy . . . . . . . . . . 4.10 Calculations for ideal gases . . . . . . . . . . . 5
Newtonian-thermodynamic system 5.1 Introduction . . . . . . . . . . . . . . . . . . 5.2 Field variables . . . . . . . . . . . . . . . . 5.3 Parcel and parcel velocity . . . . . . . . . . 5.4 Mass and heat transport equations . . . . . 5.4.1 Continuity equations . . . . . . . . . 5.4.2 Integrated variations in a system . . 5.4.3 General continuity equation . . . . . 5.4.4 Heat flux equation . . . . . . . . . . 5.4.5 Heat conduction equation . . . . . . 5.5 Inhomogeneous thermodynamic system . . . 5.5.1 Adiabatic and transport processes . 5.5.2 Inhomogeneous thermodynamics . . 5.6 Momentum equation of atmosphere . . . . . 5.6.1 Pressure gradient force . . . . . . . . 5.6.2 Navier-Stokes equation . . . . . . . 5.6.3 Momentum equation of atmosphere 5.7 Shallow water dynamics . . . . . . . . . . . 5.8 Newtonian-thermodynamic system . . . . .
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71 71 72 75 75 76 77 78 79 83 86 89 90 92
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96 96 97 99 102 102 102 103 104 105 106 106 108 108 108 109 111 113 115
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117 117 118 118 119 121 122 124 126 126 128
6 Turbulent entropy and universal principle 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2 Thermodynamic entropy of turbulent system . . 6.2.1 Simple turbulent process . . . . . . . . . . 6.2.2 Thermodynamic entropy changes . . . . . 6.3 Grid thermometers . . . . . . . . . . . . . . . . . 6.4 Turbulent thermodynamic entropy . . . . . . . . 6.5 Turbulent entropy law . . . . . . . . . . . . . . . 6.6 Difference from classical thermodynamic entropy 6.6.1 General discussion . . . . . . . . . . . . . 6.6.2 Example . . . . . . . . . . . . . . . . . . .
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CONTENTS
iv 6.7 6.8
Turbulent entropy and disorderliness . . . Universal principle . . . . . . . . . . . . . 6.8.1 The principle . . . . . . . . . . . . 6.8.2 Applications . . . . . . . . . . . . 6.9 Partition functions . . . . . . . . . . . . . 6.10 Heat capacity and van der Waals equation 6.10.1 Einstein function . . . . . . . . . 6.10.2 van der Waals equation . . . . . .
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130 131 131 134 136 140 140 142
7 Basic conservation laws 7.1 Introduction . . . . . . . . . . . . . . . . . . . 7.2 Parcel and local energy equations . . . . . . . 7.2.1 Mechanic energy equation . . . . . . . 7.2.2 Bernoulli’s equation . . . . . . . . . . 7.2.3 Principle of kinetic energy degradation 7.2.4 Local energy equation . . . . . . . . . 7.3 System energy equation . . . . . . . . . . . . 7.3.1 From kinetic theory of gases . . . . . . 7.3.2 For the whole atmosphere . . . . . . . 7.3.3 For a part of atmosphere . . . . . . . 7.4 Energy conversions . . . . . . . . . . . . . . . 7.4.1 Conversion functions . . . . . . . . . . 7.4.2 Total potential energy and enthalpy . 7.5 Potential enthalpy conservation . . . . . . . .
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143 143 144 144 145 147 148 149 149 150 151 152 152 154 157
8 Thermodynamic and geopotential entropies 8.1 Introduction . . . . . . . . . . . . . . . . . . . 8.2 Thermodynamic entropy variations . . . . . . 8.2.1 General expression . . . . . . . . . . . 8.2.2 Variation tendencies . . . . . . . . . . 8.3 Baroclinic entropy . . . . . . . . . . . . . . . 8.4 Barotropic entropy . . . . . . . . . . . . . . . 8.5 Thermodynamic entropy level . . . . . . . . . 8.6 Static entropy . . . . . . . . . . . . . . . . . . 8.7 Pseudo- reversible process . . . . . . . . . . . 8.8 The reference state . . . . . . . . . . . . . . . 8.9 Thermo-static entropy level . . . . . . . . . . 8.10 Geopotential entropy . . . . . . . . . . . . . . 8.10.1 For dry air parcels . . . . . . . . . . . 8.10.2 For the dry atmosphere . . . . . . . .
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160 160 162 162 163 165 167 168 169 170 171 173 175 175 177
9 Available enthalpy 9.1 Introduction . . . . . . . 9.2 Available enthalpy . . . 9.3 Constraint relationships 9.4 Variational approach . .
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179 179 183 184 186
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CONTENTS
v
9.5 9.6 9.7
The lowest state . . . . . . . . . . . . . . . . . . Maximum available enthalpy . . . . . . . . . . . Approximate approach . . . . . . . . . . . . . . . 9.7.1 The lowest state . . . . . . . . . . . . . . 9.7.2 Maximum available enthalpy . . . . . . . 9.8 Thermodynamic entropy variation . . . . . . . . 9.9 Geopotential entropy variations . . . . . . . . . . 9.10 Discontinuous examples . . . . . . . . . . . . . . 9.10.1 Baroclinic example . . . . . . . . . . . . . 9.10.2 Barotropic example . . . . . . . . . . . . 9.10.3 Thermodynamic and geopotential entropy 9.10.4 Continuous solutions . . . . . . . . . . . . 10 Dry 10.1 10.2 10.3
processes of energy conversion Introduction . . . . . . . . . . . . . . . . . . Dependence on process . . . . . . . . . . . . Sudden warming and cooling . . . . . . . . 10.3.1 Temperature variation . . . . . . . . 10.3.2 Kinetic energy production . . . . . . 10.4 Change of surface pressure . . . . . . . . . . 10.4.1 Surface pressure and static stability 10.4.2 Surface pressure change . . . . . . . 10.4.3 Change of the thickness . . . . . . . 10.5 Change of static stability . . . . . . . . . . 10.5.1 Partition of available enthalpy . . . 10.5.2 Final mean static stability . . . . . . 10.6 Thermo-static entropy level . . . . . . . . . 10.6.1 Change of barotropic entropy . . . . 10.6.2 Change of thermo-static entropy . .
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189 191 193 193 195 197 198 200 200 202 203 206
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207 207 208 211 211 212 215 215 216 217 220 220 221 222 223 225
11 Available moist enthalpy 11.1 Introduction . . . . . . . . . . . . . . . . . . . 11.2 Available moist enthalpy . . . . . . . . . . . . 11.3 Moist potential enthalpy . . . . . . . . . . . . 11.4 Thermodynamic entropy production . . . . . 11.5 Dry reference state . . . . . . . . . . . . . . . 11.6 Moist reference state . . . . . . . . . . . . . . 11.6.1 The isoperimetric problem . . . . . . . 11.6.2 Approximate approach . . . . . . . . . 11.7 Examples of lowest state . . . . . . . . . . . . 11.8 Available moist enthalpy . . . . . . . . . . . . 11.8.1 General and approximate relationships 11.8.2 Examples of available moist enthalpy .
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226 226 229 230 232 233 235 235 237 239 242 242 244
CONTENTS
vi 12 Moist processes of energy conversion 12.1 Introduction . . . . . . . . . . . . . . . . . . . . 12.2 Saturated reference state . . . . . . . . . . . . . 12.2.1 Saturated humidity profile . . . . . . . . 12.2.2 Minimum precipitation . . . . . . . . . 12.2.3 Temperature profile . . . . . . . . . . . 12.3 Effect of baroclinity . . . . . . . . . . . . . . . 12.4 Effect of horizontal humidity gradient . . . . . 12.5 Surface pressure change . . . . . . . . . . . . . 12.6 Available enthalpy of reference state . . . . . . 12.7 Threshold static instability . . . . . . . . . . . 12.8 Equivalent baroclinic and barotropic entropies . 12.9 Equivalent thermo-static entropy level . . . . .
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247 247 249 249 250 252 253 256 259 260 264 265 267
13 Available enthalpy in the atmosphere 13.1 Introduction . . . . . . . . . . . . . . . . . . 13.2 In the Northern Hemisphere . . . . . . . . . 13.2.1 Distributions in winter and summer 13.2.2 Relation to extratropical cyclones . . 13.2.3 Relation to blocking systems . . . . 13.3 In the Southern Hemisphere . . . . . . . . . 13.4 Development of low system . . . . . . . . . 13.5 Baroclinic entropy . . . . . . . . . . . . . . 13.6 Zonal mean distributions . . . . . . . . . . . 13.7 Least thermodynamic entropy production . 13.8 The highest static stabilities . . . . . . . . .
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269 269 273 273 273 276 276 277 279 282 285 287
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14 Available moist enthalpy in the atmosphere 14.1 Introduction . . . . . . . . . . . . . . . . . . . 14.2 Distribution of moist energy sources . . . . . 14.3 Relation to storm tracks . . . . . . . . . . . . 14.4 Tropical and extratropical tropospheres . . . 14.5 Relation to thunderstorms . . . . . . . . . . . 14.6 Relation to precipitation . . . . . . . . . . . . 14.7 Relation to tropical cyclones . . . . . . . . . .
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289 289 290 294 296 299 302 306
15 A case of typhoon recurvature 15.1 Introduction . . . . . . . . . . . 15.2 Typhoon Orchid recurvature . 15.3 Subtropical cyclones . . . . . . 15.4 Threshold surface temperature 15.5 Energy budget . . . . . . . . . 15.6 Self-feeding mechanism . . . . .
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308 308 310 316 317 319 322
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CONTENTS
16 A case of explosive cyclone 16.1 Introduction . . . . . . . . . . . 16.2 Energy steering mechanism . . 16.3 Baroclinic entropy distribution 16.4 Low-level moist jet . . . . . . . 16.5 Self-feeding mechanism . . . . .
vii
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327 327 328 336 338 340
17 States of maximum thermodynamic entropy 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . 17.2 Heat-death ideal gas . . . . . . . . . . . . . . . . 17.3 Heat-death geophysical air mass . . . . . . . . . 17.4 Heat-death atmosphere . . . . . . . . . . . . . . 17.5 Kinetic-death atmosphere . . . . . . . . . . . . . 17.5.1 Isentropic atmosphere . . . . . . . . . . . 17.5.2 Example . . . . . . . . . . . . . . . . . . . 17.5.3 Comparison with heat-death atmosphere . 17.6 Energy conservation constraint . . . . . . . . . . 17.7 Kinetic equilibrium state . . . . . . . . . . . . . . 17.7.1 General expressions . . . . . . . . . . . . 17.7.2 In statically stable atmosphere . . . . . . 17.7.3 In statically unstable atmosphere . . . . . 17.8 Principle of extremal entropy productions . . . .
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343 343 345 347 349 351 351 352 354 356 357 357 358 361 363
18 Energetics of linear disturbance development 18.1 Introduction . . . . . . . . . . . . . . . . . . . . 18.2 Conversion of available enthalpy . . . . . . . . 18.2.1 Method A . . . . . . . . . . . . . . . . . 18.2.2 Method B . . . . . . . . . . . . . . . . . 18.3 Growth of linear disturbances . . . . . . . . . . 18.3.1 Energy constraint equation . . . . . . . 18.3.2 Time-dependent expression . . . . . . . 18.3.3 Alternative expression . . . . . . . . . . 18.3.4 Numerical procedures . . . . . . . . . . 18.4 Eady wave development . . . . . . . . . . . . . 18.4.1 Evaluation equations . . . . . . . . . . . 18.4.2 Examples . . . . . . . . . . . . . . . . . 18.5 Synoptic geostrophic wave development . . . . 18.6 Development of blocking waves . . . . . . . . . 18.7 Wave development in stratosphere . . . . . . .
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366 366 368 368 371 373 373 374 375 376 377 377 379 382 385 388
19 Energetics of moving parcels 19.1 Introduction . . . . . . . . . . . . 19.2 Linear atmosphere . . . . . . . . 19.2.1 The thermal structure . . 19.2.2 Slope of isentropic surface 19.2.3 Slope of isobaric surface .
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391 391 393 393 395 396
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CONTENTS
viii 19.3 External forces on a parcel . . . . . . . 19.3.1 Adiabatic buoyancy oscillations 19.3.2 Horizontal processes . . . . . . 19.4 Slantwise static instability . . . . . . . 19.5 Slantwise lapse rate . . . . . . . . . . . 19.6 Slantwise adiabatic lapse rate . . . . . 19.7 Slantwise circulation instability . . . . 19.8 Height of slantwise convection . . . . . 19.9 Slantwise buoyancy oscillations . . . .
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398 398 400 402 404 406 408 410 411
20 Primary air engine 20.1 Introduction . . . . . . . . . . . . . . . 20.2 Primary air engine . . . . . . . . . . . 20.2.1 Assumed cycle . . . . . . . . . 20.2.2 General parcel energy equation 20.2.3 Relation to external work . . . 20.3 Adiabatic primary air engine . . . . . 20.3.1 Bernoulli’s equation . . . . . . 20.3.2 Extended parcel theory . . . . 20.4 Kinetic energy created on open paths . 20.4.1 On vertical paths . . . . . . . . 20.4.2 On isentropic surfaces . . . . . 20.4.3 On upward sloping paths . . . 20.4.4 On downward sloping paths . .
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414 414 415 415 416 419 419 420 421 422 422 423 426 428
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430 430 431 431 432 434 436 437 438 438 441 443 444 444 446 447 449 449 452 454
21 Dry air engines 21.1 Introduction . . . . . . . . . . . . . . . . . . . 21.2 Joule air engine . . . . . . . . . . . . . . . . 21.2.1 Joule cycle . . . . . . . . . . . . . . . 21.2.2 Condition of doing positive work . . . 21.2.3 Examples of kinetic energy generation 21.2.4 Entropy productions . . . . . . . . . . 21.2.5 Efficiency of Joule engine . . . . . . . 21.3 Energetics of baroclinic waves . . . . . . . . . 21.3.1 The baroclinic waves . . . . . . . . . . 21.3.2 Kinetic energy generation . . . . . . . 21.4 Kinetic energy generation in a system . . . . 21.5 Carnot air engine . . . . . . . . . . . . . . . 21.5.1 Kinetic energy generation . . . . . . . 21.5.2 Efficiency of Carnot engine . . . . . . 21.5.3 Dependence on working substance . . 21.6 Equilibrium air engine . . . . . . . . . . . . . 21.6.1 Equilibrium cycle . . . . . . . . . . . . 21.6.2 Examples . . . . . . . . . . . . . . . . 21.6.3 Entropy productions and efficiency . .
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CONTENTS
22 Wet air engines 22.1 Introduction . . . . . . . . . . . . . . . . . . . 22.2 Primary wet engine . . . . . . . . . . . . . . . 22.2.1 Kinetic energy generation . . . . . . . 22.2.2 Examples . . . . . . . . . . . . . . . . 22.3 Semi- wet Joule engine . . . . . . . . . . . . . 22.3.1 Kinetic energy generation . . . . . . . 22.3.2 Condition of producing kinetic energy 22.3.3 Efficiency . . . . . . . . . . . . . . . . 22.3.4 Thermodynamic entropy production . 22.4 Perfect storm and negative storm . . . . . . . 22.4.1 Perfect storms . . . . . . . . . . . . . 22.4.2 Negative storms . . . . . . . . . . . . 22.5 Development of negative storm . . . . . . . . 22.5.1 Coupling mechanism . . . . . . . . . . 22.5.2 Cross sections of a tropospheric river . 22.5.3 Height of tropical tropopause . . . . . 22.6 Low- and high-level convection . . . . . . . . 22.6.1 Low-level convection . . . . . . . . . . 22.6.2 High-level convection . . . . . . . . . . 22.7 Multiple semi-wet Joule engine . . . . . . . . 22.8 Wet Joule engine . . . . . . . . . . . . . . . . 22.8.1 Kinetic energy generation . . . . . . . 22.8.2 Efficiency . . . . . . . . . . . . . . . . 22.8.3 Thermodynamic entropy production .
ix
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23 Polytropic mixing processes 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 23.2 Lateral entrainment rate . . . . . . . . . . . . . . . 23.3 Heat capacity of mixing . . . . . . . . . . . . . . . 23.4 Polytropic potential temperature . . . . . . . . . . 23.5 Effect of entrainment on dry engines . . . . . . . . 23.5.1 On Joule air engine . . . . . . . . . . . . . 23.5.2 On baroclinic waves . . . . . . . . . . . . . 23.5.3 On equilibrium air engines . . . . . . . . . . 23.6 Moist polytropic mixing processes . . . . . . . . . 23.6.1 Energy equation of moist air . . . . . . . . 23.6.2 Polytropic equivalent potential temperature 23.6.3 Clausius-Clapeyron equation . . . . . . . . 23.7 Effect of entrainment on wet engines . . . . . . . . 23.7.1 On primary wet air engine . . . . . . . . . . 23.7.2 On semi-wet Joule engine . . . . . . . . . . 23.7.3 On multiple semi-wet Joule engine . . . . . 23.7.4 On wet Joule air engine . . . . . . . . . . .
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456 456 458 458 460 461 461 462 464 465 466 466 468 469 469 471 473 475 475 476 478 481 481 484 485
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487 487 489 490 492 494 494 496 497 499 499 500 501 502 502 503 504 506
CONTENTS
x 24 Limitations on frontogenesis 24.1 Introduction . . . . . . . . . . . . . . . . . . . . 24.2 The theoretical model . . . . . . . . . . . . . . 24.2.1 The basic relationships . . . . . . . . . . 24.2.2 Idealized frontal field . . . . . . . . . . . 24.3 Numerical Iteration . . . . . . . . . . . . . . . . 24.4 Limitations by initial field . . . . . . . . . . . . 24.4.1 Initial fields . . . . . . . . . . . . . . . . 24.4.2 Dependence on initial temperature field 24.4.3 Kinetic energy variation . . . . . . . . . 24.5 Baroclinic entropy . . . . . . . . . . . . . . . . 24.6 Available enthalpy . . . . . . . . . . . . . . . . 24.7 Limitations by other factors . . . . . . . . . . . 24.7.1 Scale of atmosphere . . . . . . . . . . . 24.7.2 Baroclinity of background field . . . . . 24.7.3 Latitudinal position of front . . . . . . . 24.8 Geopotential entropy variation . . . . . . . . .
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508 508 509 509 510 512 514 514 515 517 519 519 523 523 523 524 525
25 Grid-scale prediction equations and uncertainties 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 25.2 Scale-dependent data . . . . . . . . . . . . . . . . . 25.3 Subgrid-scale fields . . . . . . . . . . . . . . . . . . 25.4 Diffusive turbulences and negative diffusions . . . . 25.5 Grid-scale prediction equations . . . . . . . . . . . 25.6 Scale-dependent prediction models . . . . . . . . . 25.7 Errors from finite difference schemes . . . . . . . . 25.7.1 Truncation error . . . . . . . . . . . . . . . 25.7.2 Examples . . . . . . . . . . . . . . . . . . . 25.8 Turbulent diffusion and predictability . . . . . . . 25.9 Thermodynamic entropy produced by diffusions . . 25.10Uncertainties in physics . . . . . . . . . . . . . . .
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528 528 531 532 534 536 538 540 540 542 546 549 551
26 Examinations of model results 26.1 Introduction . . . . . . . . . . . . . . . . . . . 26.2 Scientific tests . . . . . . . . . . . . . . . . . . 26.3 Result-dependent models . . . . . . . . . . . 26.4 Thermodynamic entropy balance . . . . . . . 26.5 Partition of thermodynamic entropy change . 26.6 Examination of parameters . . . . . . . . . . 26.7 Features of thermodynamic entropy variation 26.7.1 The entropy change in a system . . . . 26.7.2 Change by dry air exchange . . . . . . 26.7.3 Change by moisture exchange . . . . .
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555 555 556 558 560 562 565 568 568 569 572
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CONTENTS
xi
A Thermopotential energy of gases A.1 Thermopotential energy . . . . . . . . . A.2 Assumed hard-sphere potential . . . . . A.3 Example of reverse sixth-power potential A.4 Comparisons with experiments . . . . . B Thermodynamics of gas expansions B.1 Energy conversions . . . . . . . . . . . B.2 Joule-Thomson effect . . . . . . . . . B.2.1 The new algorithm . . . . . . . B.2.2 Comparisons with experiments B.3 Joule-Thomson coefficient . . . . . . . B.3.1 The new algorithm . . . . . . . B.3.2 Examples . . . . . . . . . . . . B.4 Temperature inversion curves . . . . . B.5 Free expansion . . . . . . . . . . . . .
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574 574 575 577 581
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583 583 584 584 586 588 588 589 591 593
C Derivation of momentum equation
595
D References
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E List of symbols
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Index
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Chapter 1 Introduction In the study of atmospheric thermodynamics, an air parcel, if it is sufficiently small, may be assumed as a classical thermodynamic system (such as a small piece of ideal gas), of which the thermodynamic equilibrium state is uniform. Variations in the thermodynamic state may be illustrated by the state equation and the first law of thermodynamics (Dufour and van Mieghem, 1975; Iribarne and Godson, 1981). When the atmosphere is considered as a single thermodynamic system, additional assumptions and relationships are applied, since the atmosphere manifests the particular features different from those of a classical thermodynamic system. These features are summarized briefly in the following. Firstly, we never see a uniform state of the atmosphere without parcel motions attained by molecular diffusions. The steady states viewed from the datasets over meteorological scales are inhomogeneous, owing to the gravitation and rotation of the Earth and the heat exchanges between the atmosphere and surroundings. Thus, we may not assume a uniform state to represent an equilibrium state for the study of atmospheric dynamics and thermodynamics. The zeroth law of thermodynamics cannot be applied for the inhomogeneous thermodynamic system. The thermodynamic processes, such as energy conversions, must be studied between inhomogeneous equilibrium states of the atmosphere. Unlike in the classical thermodynamic systems, the inhomogeneous equilibrium state attained with conservation of mass and energy may not be unique. The weather predictions are actually looking for one of the equilibrium states according to the initial conditions and physical processes provided. Secondly, the atmosphere in the gravitational field possesses geopotential energy as well as internal energy and kinetic energy. The energy conservation law of the whole atmosphere is different from the first law of classical thermodynamics, as it includes also the mechanic energy equation derived from the atmospheric momentum equation (Brunt, 1944; Starr, 1951; Hess, 1959). According to this law, the mechanic work and parcel kinetic energy may be converted from the geopotential energy and internal energy through the processes inside the atmosphere. Thus, the atmosphere reduces its height of gravity center or increases its static stability when producing kinetic energy in the dry processes. For a provided initial state, the atmosphere possesses multiple barotropic reference states with different static stabilities and energy partitions. A real reference state attained by energy conversion depends on the process as well as initial state. Thirdly, the development of vertical motions in a fluid is produced by the buoyancy force doing mechanic work for fluid parcels. The buoyancy force on air parcels is zero in the statically stable atmosphere, as the downward gravitational force is balanced by the upward pressure gradient force. Thus, the transfer of geopotential energy and internal energy into kinetic energy may take place only in the statically unstable atmosphere if it is barotropic. In other words, the energy conversions in the atmosphere are restricted by the static stability, which depends on not only the 1
2
1. INTRODUCTION
intensity but also the direction of vertical gradient of potential temperature. While, the thermodynamic entropy is independent of the gradient direction. Therefore, the ability of energy conversion in the atmosphere may not be measured by the classical thermodynamic entropy, which depends on magnitude of the gradient only. Fourthly, the disorderliness of atmosphere depends not only on the distribution of molecules at different microscopic energy levels, but also on the distribution of parcels with different potential temperatures. While, the classical thermodynamic entropy is calculated by integrating the entropy for each element of volume or mass, and so depends on the thermodynamic state of each element instead of the element distribution. In the reversible adiabatic processes with conservation of parcel potential temperature, the classical thermodynamic entropy is conserved, but the thermodynamic disorderliness of atmosphere may nevertheless be increased irreversibly by turbulent parcel motions. Thus, a reversible process predicted by the second law of thermodynamics may not be really reversible in the inhomogeneous fluid system. The thermodynamic irreversibility related to the turbulent diffusions may be independent of that related to molecular diffusions. Also, the extremal state with maximum thermodynamic entropy attained through parcel motions are different from the isothermal state attained by molecular diffusions. As the isothermal heat-death state is seldom observed and is not interested for the study of energy conversion in the atmosphere, some particular physical relationship should be used to filter out the dead reference state. Fifthly, the conversion of geopotential energy in the atmosphere is irreversible in isolation with increasing the static stability or the vertical gradients of air density and potential temperature. This reduction of vertical disorderliness resulting from parcel motions is different from the change of horizontal disorderliness. When the statically unstable atmosphere becomes stable by adiabatic parcel convection, the thermodynamic entropy may not change but the process is irreversible. Thus, there are thermodynamically and geopotentially irreversible processes in the atmosphere related to the molecular or turbulent diffusion and geopotential energy conversion respectively. These processes may be independent of each other, and so the thermodynamic entropy is no longer a perfect discriminate parameter for the variation tendency of atmosphere. The adiabatic parcel exchanges in the vertical direction cannot happen in the statically stable atmosphere, though the thermodynamic entropy is conserved. Also, the statically stable atmosphere cannot reduce its static stability by parcel motions, thought the entropy increases in the process. Thus, we need another discriminate parameter together with the thermodynamic entropy to indicate the variation tendency and irreversibility of the geophysical and thermodynamic system. Sixthly, an isolated small piece of gas may reach the uniform equilibrium state with maximum thermodynamic entropy through molecular diffusions. While, the major processes on meteorological scales in the atmosphere are related to parcel motions driven by external Newtonian forces. Due to the gravity effect, the atmosphere tends to lower its gravity center, or increase its vertical disorderliness in mass density and potential temperature. Thus, the energy conversions in an unstable atmosphere drive the geophysical fluid system towards the equilibrium state possessing maximum kinetic energy, minimum geopotential energy and thermody-
3 namic entropy, compared with other equilibrium states. The isothermal heat-death atmosphere with maximum thermodynamic entropy may not be observed. Thus, there is the principle of minimum thermodynamic entropy production, which has been applied implicitly for long for the studies of maximum kinetic energy generation in the atmosphere. However, owing to some other dynamic equilibrium constraints, such as the geostrophic balance and thermal wind balance, the climatological mean state of the open atmosphere is different from the extremal state of minimum thermodynamic entropy. It possesses negative thermodynamic entropy sources, and so the energy conversions in the atmosphere are always possible. These major differences discussed above make the atmosphere a particular thermodynamic system in the gravitational field, which is different from the classical thermodynamic systems studied by the classical thermodynamics. It is necessary to create a new thermodynamics to study the thermodynamic processes in the largescale compressible geophysical fluid. The atmospheric energetics is an application of the new thermodynamics. To introduce the new thermodynamics, we review simply in the next chapter two basic systems studied in the classical physics. A physical system of which the variations are controlled by Newton’s laws may be called the Newtonian system, which can be studied conveniently in the frame composed of time and space coordinates. The study is concentrated on the motion status of the system. The time and distance are two basic variables. Another variable is mass which describes a steady property of the system. Variations of Newtonian system may be predicted by Newton’s laws together with the conservation laws of mass and energy. According to these laws, changes of the motion status in an inertial coordinate frame are caused by external forces referred to as the Newtonian forces in this study. There is a special force called the frictional force, which converts kinetic energy into internal energy of the system irreversibly. This feature allows us to define a state function, called the dynamic entropy in Chapter 2, to indicate the direction of system variation in frictional processes. As variations of Newtonian system may be represented with measurable variables and are predictable in principle, the dynamic entropy is not used in previous studies. Apart from the Newtonian systems, there are the thermodynamic systems, such as the gases, studied by the thermodynamics. The systems are composed of a great number of discrete microscopic particles, like the molecules, with similar features and structures. The particles usually possess kinetic energy of different motion status and potential energy at different levels. The distributions of these particles and their energies manifest a microscopic state of the system, which may not be detected in a macroscopic manner. While, the statistical features of the microscopic state may be represented by measurable thermodynamic variables, which represent the macroscopic thermodynamic state. The measurements of thermodynamic variables depend on the size of instruments in a non-uniform system. The adjective ‘macroscopic’ specifies the scales which are much larger than a particle. When external macroscopic Newtonian forces, such as the gravitational force of the Earth, are ignored, these variables are independent of space coordinates in equilibrium over a certain scale. The classical thermodynamics studies mainly the idealized equilibrium thermodynamic processes and uniform equilibrium states, without considering the nonequilibrium processes between two equilibrium states. In a fluid system without
4
1. INTRODUCTION
parcel motions, changes in the thermodynamic state are produced by molecular diffusions, which are the statistical effects of random molecular motions and collisions on the measurable thermodynamic variables. The classical thermodynamics does not study the causes of the system variations. These two types of physical systems are reviewed simply in Chapter 2. The molecules in a substance possess the kinetic energy associated with translational motions, rotations and vibrations of molecules. They have potential energy also related to the attractive and repulsive intermolecular forces. According to Zhu (2000), the mean kinetic energy of all molecules in a substance is represented by the heat energy, while the integration of intermolecular potentials over the substance is called the thermodynamic potential energy or simply the thermopotential energy. The heat energy and thermopotential energy construct the internal energy of substance. A new approach in calculating the thermopotential energy is discussed in Appendix A. The energy conversions between the thermopotential energy and heat energy or other energies in gas expansions are discussed in Appendix B. The thermopotential energy is also useful for us to derive the state equation of substance in Chapter 2. Since the thermopotential energy is negligibly small compared with the heat energy in the normal conditions of the atmosphere, the air in the atmosphere will be assumed as an ideal gas in this study which possesses the heat energy but not the thermopotential energy. Also, the heat capacities and gas constant will be taken as constants, and the state equation of ideal gases will be used all the time in the study. The temperature means the absolute temperature in this study except indicated particularly. A link between the Newtonian dynamics and classical thermodynamics is provided by the classical kinetic theory of gases. The first law of thermodynamics for ideal gases will be derived from the kinetic theory in Chapter 2. This theory will be applied again in Chapter 3 to study the transport processes in non-uniform gases produced by molecular diffusions. The transport coefficients, such as the heat conductivity, and the coefficients of self-diffusion and viscosity, will be derived from the new transport theory without solving the Boltzmann conservation equation. The effect of inelastic molecular collisions on heat conductivity is studied by the collisional heat capacity introduced, which can be calculated in terms of molecular structure (Zhu, 1999). Although the current theory is independent of the distribution function of molecular energy and intermolecular potentials or forces, the obtained relations between these coefficients agree well with the experiments for the dilute gases with weak or no polar momenta. In the study of transport properties in Chapter 3, we define the instantaneous diffusion velocity for a non-uniform gas according to the additive feature of momentum for a diffusion fluid element. The diffusion velocity depends on the microscale gradients in a fluid, and the transport carried by the velocity is in the downgradient direction of the variable. This feature in heat conduction gives the Clausius statement for the second law of thermodynamics. In other words, the molecular diffusions are controlled by the law. They may also take place in an isobaric system, and are different from the parcel motions dominated by Newton’s second law. As a result of the downgradient transport caused by molecular diffusions, the gradients are destroyed and so an isolated classical thermodynamic system may eventually
5 reach the uniform equilibrium state. Applying the diffusion velocity, we may derive the mass continuity equation for nonequilibrium diffusion processes without parcel motions, which tells that the mass density may be changed by molecular diffusions. Since neither the diffusion velocity nor the related microscale gradients can be detected or evaluated precisely, the mass continuity equation for the diffusion processes cannot be used in practice, and the molecular diffusions included in other thermodynamic equations cannot be evaluated for predicting the system variations. Thus, variations of the three thermodynamic variables are studied by two time-independent equations without including the diffusions in the classical thermodynamics: the state equation and the first law of thermodynamics. These equations are not prediction equations and can only be used for equilibrium states. For a provided initial equilibrium state, we may use these equations to calculate a new equilibrium state if one variable of the new state is given. The new state evaluated with the time-independent relationships has no explicit time relation to the initial state. It may occur before or after the initial state. However, in an isolated system which has no mass and energy exchanges with the surroundings except the mechanic work created as the system changes value, the downgradient transport resulting from molecular diffusions tends to destroy irreversibly the gradients or increase the disorderliness in the variable distribution, and then convert organized molecular motions into random motions. As discussed in Chapter 3, this feature gives a macroscopic description of the second law of thermodynamics. This law may be derived by introducing the thermodynamic entropy, according to the unidirectional transfer of diffusion kinetic energy into internal energy as discussed in Chapter 4. The irreversible energy transfer provides one of the irreversible entropy sources. Since a change of thermodynamic entropy indicates the direction of system variation in the irreversible processes, whether a new equilibrium state evaluated by the classical thermodynamics may really occur or not can be found by calculating the entropy change in the process. Since the diffusion kinetic energy depends on microscale gradients of thermodynamic variables, its variations cannot be evaluated conveniently. For the ideal gases with constant heat capacities, the thermodynamic entropy may be represented as a function of thermodynamic variables. The obtained algorithm may be used conveniently to calculate the entropy changes between two equilibrium states. Except the dissipation of diffusion kinetic energy, another source of thermodynamic entropy is the irreversible downgradient transport including the heat conduction without mass displacement, which may be independent of the kinetic energy dissipation. We shall discuss in Chapter 4 the entropy change in nonequilibrium system, and prove that the entropy increases as a nonequilibrium system becomes equilibrium through the transport processes. This consequence suggests that thermodynamic entropy may also measure the disorderliness of a system or the ability of system change related to the disorderliness. As a system tends to increase its disorderliness, the thermodynamic states with different degrees of disorderliness possess different probabilities of occurrence. So the entropy may also be represented by statistical probabilities in the statistical thermodynamics. To account for the disorderliness related to the distributions of molecules and
6
1. INTRODUCTION
their energies at different levels, the entropy cannot be evaluated by adding the thermodynamic entropy of each molecule (if it can be defined), since the sum of molecular entropy depends on the entropy of individual molecules and is independent of the distribution. When the entropy of each microscopic particle is conserved, the sum of particle entropy is also conserved, but the system disorderliness may nevertheless be changed by rearranging the particles. In this sense, we say that thermodynamic entropy is inadditive. While, the entropy calculated using thermodynamic variables may represent the system disorderliness, since these variables measure the mean features of the elements which include a number of particles. If the particles and their energies in a substance are not changed, the mean features at local places may still be changed by particle exchanges between elements. This means that the change of local mean features may result from the change of particle distribution only. Although the mean thermodynamic variables of the whole system, such as the mean temperature, evaluated with a linear algorithm are conserved, the entropy calculated with a nonlinear algorithm using the thermodynamic variables is changed by molecular exchanges or diffusions. It is discussed in Chapter 2 that the measurements of thermodynamic variables depend on the sample size especially at nonequilibrium states. Thus, thermodynamic entropy as a function of the variables depends on the size too. The calculations over different scales give different amounts of thermodynamic entropy, which are not errors but the entropy on the corresponding scales. An irreversible process may occur in an isolated system only if the thermodynamic entropy on all scales is not destroyed. This scale-dependent feature was not noticed except in the study of Batchelor (1967), since the entropy changes were considered mostly between the uniform equilibrium states in the previous studies of classical thermodynamics. The entropy changes calculated for uniform states are independent of data scale. The irreversible processes related to molecular diffusions or conductions exist also in the classical Newtonian dynamics, and we have introduced in Chapter 2 the dynamic entropy to account for the mechanic irreversibility. The relation to the thermodynamic irreversibility will be discussed also in Chapter 4. Due to the gravitation and rotation of the Earth, a steady thermodynamic state of the atmosphere viewed from the grid-point data over a certain scale manifests vertical stratifications and horizontal gradients in the thermodynamic fields. This non-uniform steady state will be referred to as the inhomogeneous equilibrium state in Chapter 5. The thermodynamic system characterized by inhomogeneous equilibrium state is called the inhomogeneous thermodynamic system. The mechanic and thermodynamic variables of the system vary with space, and so are referred to as the field variables. The inhomogeneous thermodynamic systems may also be studied with the relationships of classical thermodynamics in the time-space coordinates, by expanding the material differentials into partial differentials with respect to the time and space in the moving medium. The expanded first law of thermodynamics is called the heat flux equation, and the mass conservation law gives the continuity equation. The thermodynamics dealing with the inhomogeneous thermodynamic system will be referred to as the inhomogeneous thermodynamics. It is not called the nonequilibrium thermodynamics, because the system studied may not be in the nonequilibrium states related to molecular diffusions.
7 The differential pressure distribution in an inhomogeneous thermodynamic system may produce the pressure gradient force, which pushes the fluid elements or parcels to move from high pressure to low pressure at a constant height. This mass displacement is different from the mass diffusions discussed in Chapter 3, as it is controlled by Newton’s second law on a macroscopic scale. The parcel motions may change not only the motion status but also the thermodynamic state of atmosphere. The meteorology studies the dynamic and thermodynamic variations over meteorological scales, applying all the relationships used for the studies of Newtonian system and classical thermodynamic system. In an incompressible fluid, such as a shallow water, the mechanic and thermodynamic processes may be considered separately, and no mechanic energy is created through thermodynamic processes. But in the compressible atmosphere, the variations in thermodynamic state and motion status interact with each other. Thus, the compressible inhomogeneous thermodynamic system cannot be considered simply as a Newtonian system or a classical thermodynamic system, but is combination of the both called the Newtonian-thermodynamic system in Chapter 5. The equation of parcel motion referred to as the momentum equation will be discussed in the chapter. In a large-scale fluid system with parcel motions, the migratory particles are the macroscopic parcels as well as the microscopic molecules. The distribution of parcels with different temperatures or pressures manifest the disorderliness on a meteorological data scale. In other words, the disorderliness may also be changed by parcel motions as well as molecular diffusions in an inhomogeneous fluid system, or by the migratory elements in different scales at more than one levels. The classical thermodynamic entropy evaluated by adding the entropy of each fluid element, called the parcel or turbulent entity also, may be conserved in the adiabatic process of parcel exchanges without mixing with each other, since it depends on the entropy of parcels and is independent of parcel distribution. Thus, the classical thermodynamic entropy is unable to measure the changes of disorderliness caused by redistribution of parcels, and so there is an idealized process called the reversible adiabatic process in the atmosphere. In fact, this process is not really reversible. We shall introduce in Chapter 6 the turbulent entropy to measure the disorderliness on the data scale and the irreversibility related to parcel redistribution. Like the classical thermodynamic entropy, the turbulent entropy is also scale-dependent, depending on the data scale which is much larger than a parcel. Since the parcel motions are driven by external Newtonian forces including the pressure gradient force, they tend to destroy the pressure gradient in the system irreversibly. This is similar to the effect of molecular diffusions driven by the microscale thermodynamic gradients. Therefore, we may have a new law called the turbulent entropy law, which is similar to the second law of thermodynamic entropy, to account for the changes of system disorderliness caused by parcel motions. This law tells that the processes with increase of turbulent entropy is irreversible, even if the potential temperature or classical thermodynamic entropy is conserved for individual parcels or the whole system. When the classical thermodynamic entropy is conserved in the reversible adiabatic processes, the turbulent entropy may still change, since the processes are not really reversible. The turbulent entropy law is independent of the classical entropy law, as changes of turbulent entropy and clas-
8
1. INTRODUCTION
sical thermodynamic entropy are caused by different processes related to turbulent and molecular diffusions respectively. These processes on different scales may be independent of each other, as they are controlled by different mechanisms. Increase of disorderliness may be a universal feature for a system including a large number of elements with similar properties. This feature, called the the universal principle, will be illustrated mathematically by introducing the universal entropy in Chapter 6. Both the classical entropy law and turbulent entropy law may be examples of the universal principle applied for the classical and inhomogeneous thermodynamic systems respectively. Only in this sense, we may say that the second law of classical thermodynamics may be regarded as an example of turbulent entropy law applied for the simplest thermodynamic systems, of which the disorderliness is represented by distribution of the elements at a single level. We shall derive also from this principle the partition functions studied in the statistical mechanics, without applying the Maxwell-Boltzmann statistics. The obtained partition functions will be used further to derive the expressions of internal energy and isochoric heat capacity of gases. The van der Waals equation of state may also be obtained theoretically from this approach. Energy conversions in a Newtonian-thermodynamic system may take place between parcel kinetic energy and internal energy. Owing to the gravitational force, parcels in the atmosphere possess geopotential energy which depends on the geographic height. Thus, there are also the exchanges between geopotential energy, kinetic energy and internal energy in the atmosphere. The prediction equations discussed in Chapter 5 may be use to derive the energy equations for either a uniform parcel or the whole inhomogeneous system, called the parcel energy equation or system energy equation in Chapter 7. The parcel energy equation in the steady atmosphere is the well known Bernoulli’s equation. The system energy equation tells that the sum of kinetic energy and thermal enthalpy is conserved in the isolated atmosphere, which has no mass and heat exchanges with the exterior. The enthalpy of the whole atmosphere is identical to the total potential energy, defined as the sum of heat energy and geopotential energy in hydrostatic equilibrium. There is a fixed rate of geopotential energy to heat energy in the dry atmosphere, so the conversion of heat energy into kinetic energy is correlated linearly with conversion of geopotential energy. When only the isobaric molecular diffusions are considered and the temperature changes caused by kinetic energy dissipation are ignored, we obtain in Chapter 7 the conservation law of potential enthalpy for the atmosphere. The potential enthalpy of a parcel is defined as the enthalpy which the parcel will possess when it varies adiabatically and reversibly to a reference pressure. This law gives a weaker constraint for the processes of energy conversion discussed, compared with the adiabatic assumption used in the previous studies. It will be found in Chapters 9 and 11 that the reference state of the atmosphere derived from the energy equation without using this law is isothermal without parcel motions. This state, called the heat-death state, possesses maximum thermodynamic entropy and is reached through molecular diffusions in isolation. However, it is not the normal state attained by energy conversions in the atmosphere on a meteorological scale within a few days, and is not the normal predictions interested by meteorologists. The isothermal atmosphere
9 over a limited area in the lower stratosphere is not produced by molecular diffusions in isolation. The conservation law of potential enthalpy can be used to filter out the heat-death reference state for the study of energy conversion without integrating the prediction equations. The energy equations derived in Chapter 7 are independent of time, so they are diagnostic equations but not prediction equations. The diffusion terms which control the direction of energy conversion do not appear in the energy equations applied for equilibrium states. With these equations, the energy conversions are evaluated by choosing, instead of predicting, a reference state for a provided initial state. The assumed reference state may not be reached if the internal thermodynamic entropy decreases in the process. Thus, the energy equations should be used together with thermodynamic entropy laws which provide an important examination for assumed processes. The simple mathematical manipulations in Chapter 8 show that when anisobaric molecular diffusions are filtered out, the thermodynamic entropy increases with reducing three-dimensional inhomogeneity in the potential temperature field, and is maximum in the atmosphere with constant potential temperature called the isentropic atmosphere. Under the effect of gravity, the isentropic state may be attained only in the statically unstable atmosphere through parcel motions. As thermodynamic entropy increases, the disorderliness or randomness of an isolated classical thermodynamic system viewed in all directions increases also. This may not be the case in the large-scale inhomogeneous fluid system affected by the gravitational force. As materials tend to reduce their geopotential energy, the atmosphere tends to reduce its height of gravity center or increase the static stability. Meanwhile, the horizontal gradients such as the baroclinity may be destroyed in the process of geopotential energy conversion. Thus, variations of the disorderliness are anisotropic in the atmosphere. We introduce in Chapter 8 the baroclinic entropy and barotropic entropy to study the features of thermodynamic entropy changes in the horizontal and vertical directions respectively. The sum of baroclinic entropy and barotropic entropy defines the thermodynamic entropy level relative to the isentropic atmosphere, which is the extremal atmosphere with maximum thermodynamic entropy among all the reference states attained by adiabatic parcel exchanges. With the partition of thermodynamic entropy into horizontal and vertical portions, we may discuss conveniently in Chapter 8 the basic conditions for the dry atmosphere to be a kinetic energy source without solving the system energy equation. According to the change of thermodynamic entropy, the barotropic atmosphere may produce kinetic energy only if it is statically unstable, while a baroclinic atmosphere may be the energy source even if it is statically stable. The production of thermodynamic entropy in energy conversions is equivalent to the sum of baroclinic and barotropic entropy changes. Since the entropy changes in the vertical and horizontal directions may have different signs, the total thermodynamic entropy may be conserved in an irreversible process of energy conversion. This irreversible process with conservation of thermodynamic entropy in a whole system will be referred to as the pseudo- reversible process in Chapter 8. It will be illustrated also in the chapter that the reference state depends not only on initial state but also on production of thermodynamic entropy in the process. An extremal example of the reference state
10
1. INTRODUCTION
will be called the lowest state, which is attained through the process with minimum thermodynamic entropy production and possesses the strongest static stability and minimum thermal enthalpy, compared with the other reference states attained from the same initial state. The thermodynamic entropy, which measures the disorderliness of a classical thermodynamic system, depends on the intensities but not directions of the system gradients. The ability of system change depends on the disorderliness, and so may also be measured by thermodynamic entropy. While, changes of large-scale inhomogeneous fluid system, such as the Earth’s atmosphere and oceans, caused by parcel motions, depends on the static stability, since the fluid in the gravity field tends to lower its gravity center. The statically unstable atmosphere is more capable of producing kinetic energy than the stable atmosphere. Thus, the ability of system change related to conversion of geopotential energy depends on not only the intensity but also the vertical direction of potential temperature gradient. It may not be represented fairly by thermodynamic entropy, since the statically stable and unstable barotropic atmospheres may possess an equal amount of thermodynamic entropy. Thus, we introduce in Chapter 8 the static entropy to estimate the ability of energy conversion in the atmosphere associated with vertical gradient of potential temperature. The sum of the baroclinic entropy and static entropy assumes the thermo-static entropy level of the atmosphere, which decreases with the baroclinity and static instability, and so may represent the ability of energy conversion in the geophysical fluid. The irreversibility of pseudo- reversible process is related to conversion of geopotential energy, and may be independent of thermodynamic entropy law. Thus, there are two independent irreversible processes in the atmosphere: One is thermodynamic and the other is geopotential. The change of thermodynamic entropy accounts for the thermodynamic irreversibility but not the geopotential irreversibility. An example is the development of adiabatic convection in the statically unstable and barotropic atmosphere. This process is geopotentially irreversible, though the thermodynamic entropy or potential temperature of each parcel or the system is conserved. Also, the vertical adiabatic convection with conservation of thermodynamic entropy cannot occur in the stable atmosphere, as it raises the gravity center of atmosphere. To provide a full judgment for the variation tendency of the compressible geophysical fluid, a new variable called the geopotential entropy is forwarded in Chapter 8, which measures the geopotential irreversibility of the changes caused by parcel motions in the dry atmosphere. The meteorological processes in the isolated dry atmosphere are possible only when both the thermodynamic entropy and geopotential entropy are not destroyed. The system energy law discussed in Chapter 7 tells that the kinetic energy generated is equivalent to the destruction of enthalpy in the dry atmosphere. The amount of enthalpy which can be converted into kinetic energy in the process permitted by the thermodynamic entropy and geopotential entropy laws is referred to as available enthalpy in Chapter 9. Unlike the available potential energy of Lorenz (1955), the amount of available enthalpy depends on process. For a classical thermodynamic system which is uniform in equilibrium, the amount of enthalpy is determined uniquely for given mass, energy and potential enthalpy at a constant pressure. This, however,
11 is not true for an inhomogeneous thermodynamic system like the atmosphere. The stratified atmosphere may be in variant thermodynamic states possessing different amounts of thermal enthalpy, when the total mass, energy and potential enthalpy are conserved. Among all the states, there exists the limit state called the lowest state in Chapter 8, which possesses minimum enthalpy and maximum kinetic energy. To meet with the geopotential entropy law, the lowest state is evaluated with the variational approach introduced by McHall (1990a) and possesses the maximum geopotential entropy. The maximum available enthalpy may be calculated by taking the lowest state as the reference state. This variational approach is different from that of Dutton and Johnson (1967), as it applies the following constraint relationships for the isoperimetric problem. The entropy equation discussed in Chapter 8 gives the first constraint relationship, as the energy conversion depends on process which may be identified with thermodynamic entropy production. The second constraint is provided by the conservation law of potential enthalpy to filter out the anisobaric molecular diffusions which lead to the heat-death solution of reference state. Moreover, the potential temperature of an isolated system cannot exceed the maximum and minimum values at an initial state. This feature gives the third constraint for the energy conversion as a particular boundary condition. A reference state derived with the new approach manifests the basic features discussed in Chapter 8. For example, it possesses a stronger static stability, if the initial state is more baroclinic and less statically stable. Particularly, the lowest state, which possesses the highest static stability compared with the other reference states, may be attained through a pseudo- reversible process in the isolated baroclinic atmosphere. The new algorithm may be used to evaluate the energy conversions in either the statically stable or unstable atmosphere. It shows that the maximum available enthalpy with respect to the lowest state increseses with the baroclinity and static instability at initial state. Kinetic energy created in the statically unstable atmosphere is far more than in the stable atmosphere, especially when the baroclinity is low. With the new algorithm we may study the energy conversion with respect to a baroclinic reference state. The developments of atmospheric disturbances are local events in most situations, and the new approach may be used for a local region as it does not apply the zonal mean fields. The lowest state derived by the variational approach is continuous in space. If the initial state is discontinuous, the lowest state evaluated with parcel algorithm may also be discontinuous (Margules, 1904). We shall compare the continuous lowest states with the discontinuous for the examples of Margules. Since the continuous lowest state is attained through molecular and turbulent diffusions, the process creates more thermodynamic entropy and less kinetic energy than the parcel exchange process without the diffusions. As the energy conversions in the real atmosphere are highly irreversible, the lowest state may not be reached and the maximum available enthalpy may not be converted entirely. The dependence of energy conversion on process may be studied by considering the thermodynamic entropy production in developments of various weather systems. It will be found in Chapter 10 that energy conversion in a highly irreversible process is characterized by sudden change. This feature may be adopted to study the energetics of explosive or rapid processes in the atmosphere,
12
1. INTRODUCTION
such as the developments of explosive cyclones and stratospheric sudden warmings or coolings. In a highly irreversible process of a major warming, kinetic energy of mean zonal flow is destroyed. The strong turbulent diffusions in the warming events have been confirmed by the wave breaking observed in the stratosphere (McIntyre, 1982; McIntyre and Palmer, 1983). The warming processes cannot be thermodynamically or mechanically isolated, since both the thermodynamic entropy and geopotential entropy are destroyed in the episodes. We discuss also in Chapter 10 the mechanism of energy conversion according to the thermodynamic entropy balance. The conversion of heat energy is correlated to conversion of geopotential energy. As conversion of geopotential energy is realized by downward motion of cold air and upward motion of warm air, the gravity center of the atmosphere decreases or the static stability increases in the process. If the initial state is statically stable, the vertical disorderliness in the potential temperature field decreases, so does the barotropic entropy. Since thermodynamic entropy cannot be destroyed in the isolated atmosphere, the barotropic stable atmosphere cannot produced kinetic energy in isolation. But the baroclinic atmosphere may increase its baroclinic entropy by reducing the baroclinity to compensate the destruction of barotropic entropy, so that the thermodynamic entropy is not reduced in the whole system. However, kinetic energy may be created in the statically unstable atmosphere even if it it is barotropic, since the barotropic entropy increases with the static stability in the process. These consequences are obtained also in Chapter 8 without solving the energy equation. It will be emphasized that reduction of baroclinity merely may not create kinetic energy, unless the static stability increases. The increase of static stability leads to reduction of total enthalpy of the atmosphere, since the change of geopotential energy is correlated to the change of heat energy. Parcel kinetic energy may also be converted from the latent heat of moisture in the atmosphere. The thermal enthalpy plus latent heat is defined as the moist enthalpy in Chapter 11. The moist enthalpy which can be converted into kinetic energy will be referred to as the available moist enthalpy. Applying the variational approach of McHall (1990b, 1991), we may also study the energy conversion in the moist atmosphere with the three constraints similar to those for the dry processes. A reference state attained by moist process depends on the humidity as well as temperature. Since the behaviors of saturated and unsaturated air are different, and it is impossible for us to track down the individual parcels in the three-dimensional atmosphere, we are unable to predict the realistic moist reference state with the time-independent energy equation. Thus, we discuss two particular reference states in Chapter 11. In the first example, changes in the humidity and temperature are independent of each other, and the reference state obtained is dry and barotropic. In the second example, changes of humidity and temperature are considered together according to the change of equivalent potential temperature, and the reference state may be moist. If represented by equivalent potential temperature, the dry and moist reference states are given by a unified expression, and the available moist enthalpy is approximately equal with respect to the dry and moist reference states attained from a provided initial state. Energy conversions in the moist atmosphere depend highly on three-dimensional gradient in the equivalent potential temperature field. As in the dry atmosphere,
13 the energy conversions depend also on process. The turbulent diffusions increase the thermodynamic entropy production but reduce kinetic energy generation, and a highly irreversible process may also occur abruptly in the moist atmosphere. We shall define in Chapter 12 the equivalent baroclinic entropy, equivalent barotropic entropy and equivalent thermo-static entropy to study the ability of energy conversion in the moist atmosphere. Unlike in the dry atmosphere, kinetic energy may be created in the statically stable and barotropic moist atmosphere. The baroclinity may either increase kinetic energy generation or reduce the threshold humidity and static instability for development of moist disturbance. The horizontal gradient in the humidity field has an effect similar to that of baroclinity on energy conversion, as thermodynamic entropy may also be produced by destroying the gradient. It is impossible to convert all the latent heat released into kinetic energy in the barotropic atmosphere, so the atmosphere may possess more heat energy and geopotential energy after development of moist disturbance. When the static stability increases in a dry process of energy conversion, the temperature decreases in the lower troposphere and increases in the upper troposphere. Since the mass density in the lower troposphere is higher than above, the heat energy in the atmosphere is reduced by changing the temperature stratification. According to the state equation, the change of gas volume is not linearly correlated to the change of temperature. The volume increased by the warming in the upper troposphere is more than the volume reduced by the cooling in the lower troposphere, so that the thickness increases in the process. When the top of air column does not change its height, the pressure in the lower part decreases. The pressure reduction in the dry atmosphere will be calculated in Chapter 10. The increase of thickness may be amplified greatly by release of latent heat as shown in Chapter 12. In general, the surface pressure drops lower as more kinetic energy is created in the dry or moist atmosphere. Examples of the strong depressions produced by moist processes are the tropical cyclones and the explosive cyclones developed over extratropical oceans. A successful study must be able to explain the observed features of atmospheric circulations. The global distributions of available dry and moist enthalpy in the atmosphere will be evaluated, in Chapters 13 and 14 respectively, by the new approaches with the semi-daily meteorological data over six years obtained from the European Centre of Medium-Range Weather Forecasts (ECMWF). The results show that there are the basic correlations between the geographic distributions of the major energy sources and storm tracks, thunderstorms and precipitation centers. The tracks of tropical cyclones occur also in the regions where available moist enthalpy exceeds a certain limit. The climatological distribution of baroclinic entropy will be illustrated also. Contribution of static instability on the baroclinic energy sources may be found from the differences between the distributions of available enthalpy and baroclinic entropy. The minimum thermodynamic entropy produced as the dry atmosphere reaches the lowest state after energy conversion will be demonstrated in Chapter 13. The static stability of the six-year mean lowest state are plotted also, which is different from the climatological stratification of temperature. Since the large-scale circulations are constrained by the geostrophic balance, the atmosphere possesses a large amount of available enthalpy and is not in the lowest state.
14
1. INTRODUCTION
To examine in detail the correlations between the energy sources and atmospheric disturbances, we discuss a case of typhoon recurvature and explosive cyclone in Chapters 15 and 16, respectively. It will be found that Typhoon Orchid occurred in October 1991 moved towards the strongest moist energy source close to it. The northward recurvature from a previous westward track took place as a moist energy source to the west was weakened and a new center of high available moist enthalpy developed to the north. According to the mean temperature lapse rate in the tropical and subtropical regions, we may calculate the threshold temperature of saturated air on the surface for development of deep convection. The result agrees with the minimum sea surface temperature in the regions of tropical cyclones obtained from statistical studies. The different features of tropical and extratropical cyclones may be explained by the differences in their energy sources. We shall classify a new category of cyclones called the subtropical cyclones, which develop or live in the quasi-barotropic and equivalent baroclinic atmosphere in the subtropical regions, and possess the intermediate features between those of tropical and extratropical cyclones. According to the energy budget, the tropical cyclones are not supported merely by existing energy sources, because the moving speed estimated from existing energy sources is higher than the real speed observed. In the viscous barotropic atmosphere over oceans, the cyclonic circulation of tropical storm may produce strong convergence of moisture flux near the low center, which includes the effect of evaporation. The concentration of moisture provides a significant amount of energy for maintaining the cyclones over a warm ocean water. This intensification of kinetic energy source produced by the storm itself is referred to as the self-feeding mechanism. Due to this mechanism the tropical storm depend less on existing energy sources compared with the baroclinic cyclones, and so predictions of tropical storm movement are more difficult. When the self-feeding is not strong enough, a tropical cyclone may nevertheless move towards an existing energy source and convert the moist enthalpy along the track into its kinetic energy. This fact makes it possible for us to predict the tropical storm tracks by analysing the energy sources within a limited time period. Since construction of energy source in the moist tropical and subtropical atmospheres is quicker than in the extratropical atmosphere, tropical cyclones may change the direction of motion dramatically towards a newly established energy source, and so the prediction time is relatively short compared with that for extratropical baroclinic cyclones. The case studied in Chapter 16 is an explosive cyclone, called the tropospheric bomb also, developed in the regions of high available enthalpy over the northwestern Pacific in January 1992. Since the humidity was relatively high over the ocean water compared with that over lands, available moist enthalpy was also high around the storm at the early stage. Unlike in the tropical and subtropical regions, the moist energy source overlapped on the dry energy source and depended highly on the baroclinity. The important effect of moisture on the explosive cyclone could be provided by the water vapor flux in the regions. There was a filamentary moisture flux in the lower troposphere off the eastern edge of the cyclone referred to as the tropospheric river. Maximum moisture flux took place near the leading edge of the river. The flux and its convergence can be intensified by the cyclonic circulation.
15 The cyclone moved towards the leading edge of a river to convert the moist enthalpy into kinetic energy. Thus, there is also a self-feeding mechanism for the extratropical cyclones over oceans, which is different from that of tropical cyclones. This mechanism provides an important complement for the traditional theory of baroclinic cyclone development especially in the moist atmosphere. Also, the vertical thermal structure of the moist cyclone is different from that of dry cyclone. However, development of the cyclones in the extratropical atmosphere depends highly on conversion of available enthalpy and increase of baroclinic entropy, and so they are essentially the baroclinic disturbances. An important feature of inhomogeneous thermodynamic system is existence of the multiple reference states for a provided initial state, when the mass, energy and potential enthalpy are conserved. The lowest state is attained by the extremal process with minimum thermodynamic entropy production and maximum geopotential entropy production. It possesses minimum thermal enthalpy and maximum kinetic energy, compared with other reference states attained from the same initial state. We discuss, in Chapter 17, another extremal state in the atmosphere attained by the processes of maximum thermodynamic entropy production. The state is uniform for ideal gases at rest attained by molecular diffusions without the effect of gravity. While, the maximum entropy state, attained by molecular diffusions including kinetic energy dissipation in the gravitational field, is an motionless isothermal state with vertical stratifications in pressure and mass density. The gravitational force affects the profile of mass but not temperature, since the temperature depends on the mean speed of molecules of which the distribution is independent of the gravity. Apart from the molecular diffusions, there are parcel processes which may be more important than the microscopic processes for the circulation changes in a few days. If the anisobaric molecular diffusions are filtered out by the conservation law of potential enthalpy, the atmosphere possesses multi-equilibrium states which are different from the isothermal dead state. The dry and moist reference states including the lowest states attained by energy conversions over a meteorological scale are examples of the equilibrium states. Apart from the lowest state, another extremal example of the equilibrium state is the isentropic atmosphere, attained by the process of maximum thermodynamic entropy production and minimum geopotential entropy production when the potential enthalpy is conserved. This extremal atmosphere will be called the kinetic-death atmosphere, since no thermodynamic variations and energy transfers may be brought about by either the free or forced parcel motions in the isentropic atmosphere without molecular diffusions. The heat-death state of a gas is a true state in physics. While, the atmosphere can hardly become isothermal, because the changes are produced mainly by parcel motions instead of molecular diffusions, and the system is not isolated or closed. The parcel motions are controlled by the Newtonianforces such as the pressure gradient force, gravitational force and Coriolis force in the Earth. The pressure gradient force pushes the parcels move in the downgradient direction, destroying the gradient eventually in the isolated system. But on the rotational Earth the parcel motions are constrained to be quasi- geostrophic balance and hydrostatic equilibrium. The geostrophic circulations may remain the horizontal gradients of thermodynamic variables. When the balance is broken down in disturbance development, the horizontal
16
1. INTRODUCTION
gradients may be destroyed. While, the downward gravitation is independent of the circulation changes. The competition between gravitational force and vertical pressure gradient force causes the Archimedes’ effect or conversion of geopotential energy and correlated heat energy into kinetic energy in the statically or baroclinically unstable atmosphere. As the result, the atmosphere is stratified in its equilibrium state with a high density below and a low density above. Therefore, the energy conversions driven by the gravity increase the static stability or vertical disorderliness of the atmosphere. The geopotential entropy law discussed in Chapter 8 leads the system to the lowest state with maximum geopotential entropy and minimum thermodynamic entropy, compared with other reference state attained in the same atmosphere. This is different from that predicted by thermodynamic entropy law, since the atmosphere affected by the gravitational force is not really isolated. In other words, increase of geopotential entropy is prior to increase of thermodynamic entropy in the geophysical fluid systems. However, the Archimedes’ effect or geopotential entropy law cannot destroy thermodynamic entropy, but may only constrain the entropy production. We shall introduce the principle of extremal entropy productions in Chapter 17, which tells that the energy conversions resulting from parcel motions tend to follow the extremal process with maximum geopotential entropy production and minimum thermodynamic entropy production. According to this principle, the kinetic-death atmosphere is not recorded often unless in the regions of strong convection. In the traditional linear theories of disturbance development, the unstable waves grow exponentially with time at a constant rate. In fact, the growth rate depends on baroclinity of the background field. As available enthalpy is converted into kinetic energy, the baroclinity decreases and the growth rate changes in disturbance development. This interaction between disturbance and background field cannot be studied with the traditional linear theory. The variational approach discussed in Chapter 9 will be applied to study the interaction in Chapter 18, since we may use a baroclinic reference state for the new approach to evaluate the local energy sources not affected by remote fields. The energy constraint equation for the disturbance development will be derived from the interactions between the disturbances and background energy sources. This relationship may be used to illustrate more realistically the development of baroclinic disturbance, when the environmental baroclinity and growth rate change in the process. It will be found in Chapter 18 that the disturbance development under the interaction is significantly different from the wave growth at a constant growth rate. The Eady wave cannot be a good representation of the extratropical disturbances, since the development is too slow compared with the real processes. Also, it was argued by McHall (1993) that the critical baroclinities for the baroclinic instabilities given by previous studies were below the time-mean in the lower troposphere at middle latitudes. The baroclinic disturbances may be considered more likely as the unstable synoptic geostrophic waves (McHall, 1993). The critical baroclinity for the wave instability is slightly stronger than the time-mean baroclinity in the extratropical troposphere. The geostrophic waves in the varying background grow faster than the Eady waves, and have the time scale close to that of observed extratropical cyclones. This new approach allows us to plot the decay phase as well as the development
17 phase of the process. The planetary wave circulations in the atmosphere may also be destabilized by baroclinic instability. According to McHall (1993), developments of blocking systems in the free extratropical troposphere and stratospheric sudden warmings are associated with the nonlinear instability of planetary geostrophic waves in baroclinic regions, as the basic features of the planetary wave development derived from the theory agree with those observed. The current interaction theory based on energy conversion provides additional examinations for the planetary wave instability. The developments of planetary waves in the troposphere and stratosphere will be demonstrated in this chapter. The maximum amplitude and growth rate evaluated in the varying environments are comparable with those in the real processes. The observed dependence of the growth rate on the wave spectrum is confirmed by the study. For example, the long wave components grow slower than the short components, but may get larger amplitudes at the end. The typical large-scale circulations in the extratropical dry atmosphere are the slantwise convection (Green et al., 1966) around the polar front or in the baroclinic waves. The slantwise convection in the dry baroclinic atmosphere will be studied in Chapter 19 with the Newtonian dynamics for air parcels. It is found that the free slantwise convection may occur in the statically stable and baroclinic atmosphere, when the trajectory slope is smaller than the slope of isentropic surface in the dry processes or equivalent isentropic surface in the moist processes. The static stability for slantwise convection may not be measured by the temperature lapse rate studied previously for vertical convection. To find the static stability of the slantwise circulations, we forward in the chapter the algorithms for calculating the environmental temperature lapse rate and adiabatic lapse rate of parcels along a slantwise trajectory, called the slantwise lapse rate and slantwise adiabatic lapse rate respectively. The slantwise static stability depends on the slope of isentropic surface or the baroclinity, and so may also be represented by the baroclinic instability. From the point of slantwise circulation, the baroclinic instability is a kind of static instability. Both of the instabilities may cause conversion of geopotential energy and heat energy into parcel kinetic energy. Moreover, the large-scale horizontal pressure gradient force in the geophysical fluid depends crucially on the gravity effect. This is also found in Chapter 17. Thus, the horizontal processes may nevertheless be related to conversion of geopotential energy in the atmosphere. Unlike the vertical convection, the slantwise convection may form a perturbation pattern with varying amplitude forced by the slantwise static instability and horizontal pressure gradient force. The induced upward propagating slantwise gravity waves may grow up and cause significant weather changes, and so should not be filtered out in a weather forecasting model. The classical hydrodynamics is an application of Newtonian dynamics for fluids, by introducing a particular Newtonian force called the pressure gradient force. This force depends on pressure gradient inside a fluid. It is an external force for a single fluid parcel but may not be external for the whole system, so it cannot change the total energy of the system when creating parcel kinetic energy. Also, the mechanic work created by the gravitational force may also be represented by conversion of
18
1. INTRODUCTION
geopotential energy. Therefore, the parcel kinetic energy produced by the macroscopic Newtonian forces is actually converted from other energies in the system, such as the internal energy and geopotential energy, and the parcel energetics may also be studied with the pure thermodynamics without using the Newton’s laws. We shall introduce the air engine theory in Chapters 20-23 to study the atmospheric dynamics and energetics for the weather systems with typical circulation patterns. We discuss firstly in Chapter 20 the air engine theory for a single parcel. Since parcel motions in the atmosphere may not form closed thermodynamic cycles, we add auxiliary paths to a parcel trajectory in order to complete a simple cycle, called the primary cycle. A parcel working on this cycle will be referred to as the primary air engine. Change of parcel kinetic energy on the open path is evaluated by the mechanic work created over the primary cycle minus the kinetic energy change in the auxiliary paths. The result in the steady atmosphere leads to the general parcel energy equation. The Bernoulli’s equation derived in Chapter 7 gives the example for adiabatic processes. Unlike the classical heat engines, kinetic energy created by air engines may be converted from the geopotential energy as well as heat energy. The examples of parcel kinetic energy generated on a few selected trajectories are demonstrated in the chapter. The results agree with those derived from the classical dynamics in Chapter 19. On vertical paths, kinetic energy can be created only in the statically unstable atmosphere. If parcels move on the isentropic surfaces, the kinetic energy of vertical velocity is conserved, but of horizontal motions may still be changed by horizontal pressure gradient force. This consequence was not made clearly in previous studies. Owing to the horizontal pressure gradient force also, there is no the recognized negative correlation between the signs of kinetic energy generation and trajectory slope with respected to isentropic surface. Parcel kinetic energy are produced in the slantwise upward motions and destroyed in the downward motions. To remain the slantwise vertical circulations across the polar front in the viscous atmosphere, the downward motions on the cold side must start at the height lower than the top height of upward motions on the warm side, and so the tropopause descends discontinuously across the front from the warm side to the cold side. This feature of kinetic energy generation in the slantwise convection may also explain the observed north-east tilt of troughs and ridges in the large-scale baroclinic perturbations. The air engine theory for a single parcel may be applied to study the energy conversion in a weather system or the whole atmosphere, by introducing the equivalent principle in Chapter 21, according to which a cycle completed by more than one parcels can be considered as by one. The system energy equation discussed in Chapter 7 may also be derived from the air engine theory without using the momentum equation. To study the energetics of a weather system, we may design a particular air engine of which the thermodynamic cycle is similar to that of the typical circulation pattern in the system. To give an example, the energetics of large-scale slantwise convection and baroclinic perturbations will be investigated with the Joule air engine, of which the thermodynamic cycle is the Joule cycle studied in the classical thermodynamics. Also, the mean meridional circulations in the atmosphere will be studied by introducing a new reversible air engine, called the equilibrium air engine. The Carnot engine is a particular example of the new reversible heat engine.
19 The traditional linear theories of baroclinic instability provide the necessary conditions but not the sufficient conditions for the wave instability, since the budget of eddy kinetic energy over the wave spectrum is not considered. A unstable wave cannot develop if no eddy kinetic energy is created in the process. The sufficient conditions may be provided by the air engine theory for the baroclinic perturbations as discussed in Chapter 21. The unstable waves may grow only if the eddy kinetic energy created on a wave cycle is greater than that destroyed by viscosity. The critical baroclinity or wavelength for wave development can be derived from the energy budget. The kinetic energy created over a cycle may be applied further to estimate the growth rate. The baroclinic waves may grow when the critical wavelength or baroclinity obtained from a linear theory agrees with that derived from the energetics. It will be found that only the planetary waves may exist stably in the dry and weak baroclinic atmosphere, and the energy generated over a wave cycle may be maximum at a certain wavelength in the viscous atmosphere. The air engine theory may be applied also for the study of moist convection in the atmosphere. Three basic wet air engines will be discussed in Chapter 22 to study the local airmass storms and the super storms developed in the environments with strong vertical wind shears. It is well known that development of sever storms is favored by some conditions, such as the vertical wind shear and low level temperature inversion. These facts may be explained by the theory of wet air engine. We shall find that the intrusion of cold air from the middle troposphere caused by vertical wind shear may increase the energy generation in a storm. Also, net kinetic energy may be created over a convective cycle constrained in the lower troposphere by the inversion, while the convective cycle extending to a higher level may only destroy kinetic energy in the same environments. According to kinetic energy budget of the moist air engines assumed for the storms, we may provide an algorithm for calculating the low limit of sea surface temperature for development of deep convection, such as the tropical storms in a barotropic atmosphere. The surface temperature calculated may be used to check the air engines assumed. A convective system, including the upward and downward convection in mass balance, may not produce net kinetic energy over the whole circulation cycle. The examples are the convective storms forced by the large-scale baroclinic disturbances. These storms will be called the negative storms in Chapter 22, in contrast with the so called positive storms which produce net kinetic energy by themselves. The possible coupling mechanisms between the negative storms and large-scale baroclinic circulations will be discussed also in the chapter. If the Hadley cells in the tropical troposphere are assumed as heat engines, we may calculate the height of tropical tropopause according to the energy balance in the cells. To give the examples, we introduce also in the chapter the equilibrium Hadley engine and extended Hadley engine to study the relatively weak and strong moist Hadley circulations in summer and winter respectively. These engines may also be applied for a single tropical storm. It will be argued that the coupling between tropical barotropic moist convection and extratropical baroclinic slantwise convection is responsible for the seasonal changes in the intensity of Hadley cell and the height of tropical tropopause. Also, the anomalous circulation patterns in the tropical regions may be associated with the extratropical baroclinic circulations.
20
1. INTRODUCTION
For the study of kinetic energy generation in convective motions, entrainment and detrainment, or the mixing between convective clouds and environmental clear air should be considered. Instead of giving a precise algorithm for evaluating the mixing processes, a simple approach is forwarded in Chapter 23 for the study of kinetic energy generation under the effect of entrainment. The entrainment and detrainment will be assumed as polytropic processes by defining the heat capacity of mixing. We shall introduce also the polytropic potential temperature and polytropic equivalent potential temperature, which are the conservative quantities in the dry and moist polytropic convective processes with constant heat capacities of mixing. The static stability of the vertical convection with mixing should be estimated by the vertical gradient of polytropic potential temperature or polytropic equivalent potential temperature of the atmosphere. The previous studies on the effects of entrainment were generally concentrated on the upward convective processes. It will be found that the entrainment or detrainment in the downward convection of a storm may also have an important effect on the energy balance. The large-scale circulations in the rotational atmosphere are constrained by geostrophic balance, so that the equilibrium state of the atmosphere viewed over a meteorological scale may remain horizontal gradients of temperature and mass density. These gradients may be intensified in the process called frontogenesis. Since thermodynamic entropy in a frontal zone decreases in frontogenesis, the produced large-scale temperature gradient cannot exceed a certain limit in an isolated atmosphere. This limit depends on the thermal structures of initial and final states, and is studied in Chapter 24. The results emphasize the importance of using thermodynamic entropy law to examine the processes simulated by simplified models or with unrealistic boundary conditions. For example, some previous model studies showed that an infinitely large temperature contrast over a limited distance could be attained adiabatically from an initial field with a weak baroclinity. This simulated process of frontogenesis may be spurious, since the atmosphere is not infinitely large and the thermodynamic entropy of the isolated atmosphere cannot be reduced infinitely. It is discussed in Chapter 4 that changes of classical thermodynamic system are generally unpredictable with the classical thermodynamics, due to the uncertainty of molecular diffusions, of which the statistical effect does not follow the Newton’s second law. This molecular diffusions may also have an effect on numerical weather predictions. However, they may not be the major error source in general. Unlike in the classical thermodynamics, the primitive prediction equations of the atmosphere, including the state equation, continuity equation, heat flux equation and momentum equation, may still form a complete set for the three independent thermodynamic variables and parcel velocity included, even if the molecular diffusions are ignored entirely. So, the weather changes are predictable with the prediction equations for provided initial and boundary conditions. However, the predictions are not a simple initial data problem. In other words, the prediction errors depend not only on the errors in the initial fields provided, since there are the intrinsic error sources in the prediction equations and the numerical models constructed using these equations. We discuss in Chapter 25 the intrinsic error sources in current numerical models. The variables in the primitive equations are the functions covering continuously each
21 point at the space and time. But in practice, the meteorological fields are represented by discrete grid-point data over a certain scale of time and space. This means that the variables in the equations are scale-dependent, and the differentials and integrations calculated approximately with these date depend on data scale too. The primitive equations including scale-dependent variables are scale-dependent equations, and will be called the grid-scale prediction equations in Chapter 25. Solutions of the scale-dependent equations are also scale-dependent. Prediction errors may then be produced by the differences between the grid-scale prediction equations and the continuous primitive equations. The grid-scale prediction equations predict the circulations over the grid scale. The subgrid-scale circulations can neither be represented by the grid-point data nor predicted by the grid-scale equations. They will be called the meteorological turbulences in Chapter 25, which may be classified into two categories. One is the small- and meso-scale eddies and organized unstable convective systems, which may cascade subgrid-scale perturbation momentum into or produce kinetic energy for grid-scale circulations. These effects will be called the negative diffusions including the negative viscosity on the subgrid scales. The other one is the stable tiny-scale irregular turbulences similar to the classical turbulences studied by classical fluid dynamics. This kind of turbulences, referred to as the diffusive turbulences, cascade kinetic energy from grid-scale circulations into smaller and smaller scales irreversibly. Apart from the negative and positive diffusions, the physical and chemical processes, such as the moist processes and greenhouse effects included in the subgrid-scale circulations, may also have significant effects on the predicted fields after a certain time. In the quantum physics and classical thermodynamics, the effects of uncertain electron and molecule motions on the measurable variables may be illustrated statistically by the probability wave function and the state equation of substance respectively. While, the effects of meteorological turbulences on the grid-scale circulations cannot be illustrated generally by a statistical function with the datasets available, since the subgrid-scale circulations are different from the random motions of fluid parcels. But the meteorological turbulences are treated as a random process in current numerical models with the parameterizations using the grid-point data which do not really include the subgrid-scale information. The effects of subgrid-scale physical or chemical processes are represented also by parameterizations approximately in the numerical models. Thus, the meteorological turbulences form the major error source for the numerical predictions. It is generally difficult to estimate the strength of this error source, as the errors depend essentially on the unknown circulation patterns. This error source may be weakened by increasing data and model resolutions, as more energy conversion and transport processes may be represented by grid-point data. Like the molecular diffusions, the turbulent diffusions may also produce thermodynamic entropy as discussed in Chapter 6. We discuss in Chapter 25 the error source related to the turbulent diffusions in the atmosphere, by estimating the effects of thermodynamic entropy production on the energy conversions and Eady wave development. The results show that the effects of turbulent diffusions on energy conversions in the baroclinic or moist atmosphere may be comparable with the
22
1. INTRODUCTION
effects on kinetic energy dissipation. So, the disturbance developments predicted by a numerical model may be sensitive to the errors in parameterizations of turbulent diffusions, which may change from time to time and from place to place in the atmosphere. Prediction errors may also be produced by the finite difference schemes used for calculating the differentials and integrations in a numerical model. It is illustrated in Chapter 25 that the errors cannot be reduced continuously by increasing the truncation order of a difference scheme without increasing the data and model resolutions. Since the errors produced by deficiency of numerical models cannot be ignored, the study of predictability cannot be completed by estimating the growth of initial data errors only. The parameters included in the parameterizations for the meteorological turbulences and some other subgrid-scale processes are actually the unknown variables which change with time and space. Usually, these subgrid-scale processes depend on each other in the atmosphere, so it is difficult to test a single parameter in a particular process. These parameters are then determined by running the whole model and comparing the model results with the observations. The solutions so obtained may not be unique and may not be all physically meaningful or correct when considered individually. As the numerical models possess several error sources as discussed in Chapter 25, the comparisons of model results with real processes have a limited significance. A similarity between a model result and reality may be achieved functionally but not always principally, and does not mean necessarily that the model and simulated process are physically true. It will be argued in Chapter 26 that the current numerical models are the result-dependent models, as the parameters included are determined functionally by the request of getting ‘realistic’ results. When the real mechanism is not incorporated perfectly in a model, the ‘realistic’ result may be produced artificially. The meteorological turbulences affect largely the thermodynamic entropy balances in the real and model atmospheres. As discussed in some previous chapters, the processes related to frontogenesis or energy conversions are very sensitive to the errors in the entropy balance. The collapse of front and some model instability may be caused by underestimating the thermodynamic entropy production. To suppress the model instability, the entropy production may be exaggerated in a model. Thus, to calculating the thermodynamic entropy budget in the model atmosphere may provide an useful examination for some of the subgrid-scale processes and the parameters related. We study in Chapter 26 the entropy budget equation for the open atmosphere. The basic features of the entropy variations will be discussed. Particularly, we shall reveal the several sources of negative thermodynamic entropy input, which may influence greatly the kinetic energy generation in the atmosphere and are indispensable for maintaining the circulations. As some other relationships, such as the mass, energy and moisture balances in the local or whole atmosphere, may be broken by the intrinsic or technic errors in a model, the balance equations may also be applied to test the parameters in the related processes.
Chapter 2 Two classical physical systems 2.1
Introduction
There are two basic types of physical systems studied in the classical physics. The system of the first type is represented by its kinetic state or motion status, such as the position, velocity and acceleration, measured and studied in a frame of time-space coordinates usually. The kinetic state tells us also the mechanic energy situation including the mechanic potential energy and kinetic energy. This system, called the Newtonian dynamic system here, is studied with Newton’s laws together with mass and energy conservation laws. While, the system of the second type is specified with its macroscopic thermodynamic state represented by thermodynamic variables. This system is called the thermodynamic system, as we are interested mainly in variations of its thermodynamic state which may lead to exchanges of heat energy and mechanic energy in both directions. An example of classical thermodynamic system is ideal gases. Variations in the thermodynamic states of ideal gases are controlled by thermodynamic energy and entropy laws, state equation and mass conservation law. Usually, changes of kinetic state and thermodynamic state may relate to each other, and there may be exchanges between kinetic energy and heat energy in a real physical system. But in idealized situations, the two types of systems may be studied separately, using different sets of physical relationships described above. However, in a physical system of the third kind, such as the Earth’s atmosphere, changes in the kinetic and thermodynamic states cannot be separated at all, and conversion of mechanic energy is correlated essentially to conversion of heat energy. So, we have to adopt all the relationships used for the two basic systems to study the third kind system, which possesses many new features compared with the two systems. To understand these new features, we review simply in this chapter the two types of the simplest systems without discussing in detail the classical dynamics and thermodynamics. The dynamic forces, which may change the motion status of a Newtonian system may be called the Newtonian forces, as they can be applied for the Newton’s laws. A particular example is the frictional force, which always reduces the macroscopic kinetic energy of a system and increases the heat energy. So the energy conversion resulting from friction is unidirectional. In other words, the dynamic processes affected by friction are irreversible if without thermodynamic processes. This feature may be applied to establish the principle of friction for Newtonian systems as discussed in this chapter, which is analogous to the second law of thermodynamics. According to this principle, we introduce a state function, called the dynamic entropy, to illustrate the irreversibility of mechanic processes. The study of classical thermodynamics is based on energy conversions between equilibrium states. There is a particular energy called the internal energy of sub23
2. TWO CLASSICAL PHYSICAL SYSTEMS
24
stance studied in thermodynamics, which depends on microscopic features of a system and is not studied in Newtonian mechanics. When other minor energies are ignored, the internal energy is partitioned, in the current study, into the heat energy related to molecular motions and the thermopotential energy related to intermolecular forces. These energies will be studied in detail in Chapter 6 and in Appendices A and B. In this chapter, we apply the heat energy and thermopotential energy to illustrate the first law of thermodynamics. This energy relationship is then used to derive theoretically the general state equation of substance, which can be simplified to give the ideal-gas equation upon the assumptions provided for ideal gases. The features of ideal gases will be discussed also. Without using the first law of thermodynamics, the energy conservation law of ideal gases may also be derived using the kinetic theory of gases. Changes in the thermodynamic state of gas may be brought about by molecular diffusions. Motions of individual molecules are affected by intermolecular forces as well as macroscopic Newtonian forces. While, the statistical effect of random molecular motions and collisions, called the molecular diffusions, may not be controlled by these forces in a general way. The molecular diffusions may be studied by introducing the diffusion velocity discussed in the next chapter. We extend in this chapter the first law of thermodynamics for equilibrium states to that for nonequilibrium states, in order to account for the irreversible conversion of diffusion kinetic energy into heat energy. However, as the diffusion velocity depends on microscale gradients in the thermodynamic fields, which may not be measured or calculated accurately in practice, the state equation and energy law are usually used for the equilibrium states of thermodynamics systems and do not include the effects of diffusions.
2.2 2.2.1
The Newtonian systems Principle of friction
Variations in the kinetic state of a classical mechanic system are dominated by Newton’s laws together with the principles of mass and energy conservations. A dynamic system is referred to as the Newtonian system if only the variations of kinetic state are studied with Newton’s laws. The study of Newtonian system is carried out in a frame composed of time and space coordinates. The mass, time and distance are the basic variables in Newton’s laws. The simplest Newtonian system is a solid body, which may be viewed as a point mass at a given time and position in the coordinates. The external cause which leads to changes of the motion status is referred to as the external force. The force can be measured by or defined as the rate of momentum variation produced. The classical dynamics deals with the relations between the external forces (or inertial forces) and variations of motion status in the time-space coordinates. In the study of Newtonian system, variations of thermodynamic state are usually considered as a result of mechanic processes. Except in the energy conservation law, the effect of thermodynamic processes on changes of motion status is not discussed. To explain the conversion of kinetic energy to heat energy, Newtonian dynamics
2.2. THE NEWTONIAN SYSTEMS
25
introduced a special force called the frictional force, which is always in the direction opposite to the motion direction. According to Newton’s second law, the frictional force can be evaluated in terms of the deceleration produced. Since frictional force may only destroy kinetic energy and increase heat energy of a system, the resultant exchange between kinetic energy and heat energy is unidirectional. This may be regarded as the principle of friction. The Newton’s laws are time-dependent relationships. All the variables together with the external forces included are measurable on a macroscopic scale. When these relationships form a closed equation set, variations of Newtonian system may be predicted in principle for provided initial conditions. Since the prediction equations include implicitly the principle of friction which gives the restriction on the energy transfer, the solutions do not violate the second law of thermodynamics and the predicted processes may really happen under realistic conditions. Thus, we need not to examine the predictions additionally using the principle of friction. To see more clearly the restriction of friction on variations of Newtonian system, we introduce the concept of dynamic entropy in the next section. Here, we define the reversible and irreversible processes for Newtonian dynamics. A variation of Newtonian system without exchange of kinetic energy and heat energy caused by friction is a reversible process. A Newtonian system without friction may be called the perfect mechanic system. The change of mechanic energy in a reversible process equals the work done by external forces for the system. While, a process with kinetic energy dissipation caused by friction is irreversible. Since kinetic energy dissipation takes place in all Newtonian systems in the real would, we cannot design a permanent working system without energy exchange with environments.
2.2.2
Dynamic entropy
The processes in a frictional dynamic system are irreversible because the energy transfer produced is in the specific direction from kinetic energy to heat energy. We may, at least in principle, find a state function to account for the unidirectional energy conversion. This is discussed in the following. The Newton’s second law for a moving solid body of unit mass gives d2 r =F+F, d2 t where, r indicates the trajectory; t represents the time, and F and F are external and frictional forces, respectively. The external force is along the direction of trajectory, while the frictional force is always in the direction against the motion. Integrating the previous equation from r0 to r1 yields
v1 =
v02
+2
r1 r0
F · dr − WF ,
(2.1)
in which v denotes the moving velocity in the direction of external force, and WF = −
r1 r0
F · dr
(2.2)
2. TWO CLASSICAL PHYSICAL SYSTEMS
26
signifies the work done by frictional force along the trajectory. This relationship manifests the energy conservation in the process, that is k1 = k0 + WF − WF , r1
where WF =
r0
F · dr
(2.3)
is the work done by external force, and k=
v2 2
is the kinetic energy of unit mass called the specific kinetic energy. The energy conservation equation can be rewritten as ∆k = WF − WF .
(2.4)
The kinetic energy is generated by external force doing work and is destroyed by friction. Now, we define a state function as sd = ln
2
r r0
F · dr + v02 . v2
Its variation is calculated from ∆sd = ln
2
r1 r0
F · dr + v02 v12
as the solid body moves from r0 to r1 . Applying (2.1) yields
WF ∆sd = ln 1 − 2 v1
.
Since the frictional force is opposite to the motion direction, we have v · F ≤ 0 and WF ≥ 0. Therefore, the previous equation gives ∆sd ≥ 0 .
(2.5)
Variations of a frictional mechanic system are in the direction with increasing sd , and ∆sd = 0 occurs only in the reversible processes without friction. The state function sd gives an example of dynamic entropy. The existence of an entropy function in any physical system was proved generally by Giles (1964). Equation (2.5) is a mathematical expression of the principle of friction. When a solid body is pushed by a constant external force and moves from r0 to r1 , it will have velocity v1 at r1 . If it returns back from r1 against the same external force, the solid body will possess the speed less than |v0 | as it reaches r0 again under the effect of friction. The dissipated kinetic energy cannot be converted back into kinetic energy without thermodynamic processes. The processes with ∆sd > 0 are then irreversible.
2.3. SIMPLE THERMODYNAMIC SYSTEMS
27
For the simple mechanic system discussed previously, there is little advantage to introduce the dynamic entropy, as motions of a solid body are predictable principally. Introducing the dynamic entropy can neither add more understandings of the Newtonian dynamics, nor make it easier for us to predict the system variations. For a system composed of n independent Newtonian sub-systems, its dynamic entropy may be defined as the sum of dynamic entropy of each sub-systems, that is Sd =
n
Sdj =
j=1
n
mj ln
∆rj
Fj · drj + kj0 kj
j=1
,
where Fj is the external force acting on unit mass of sub-system j; kj is the kinetic energy density and mj is the mass. The dynamic entropy is additive, since it depends on the entropy of each sub-system but not a particular relation between these subsystems. Now, we assume an isotropic system, in which the work done by external force for each sub-system is identical, and the sub-systems have the same mass m and initial kinetic energy k0 . These features may only be statistically meaningful. The dynamic entropy of the isotropic system is evaluated from
Sd = M ln
∆r F
· dr + k0 . k
(2.6)
Here, M = nm is total mass of the system. This equation will be used to study the relation between the dynamic entropy and thermodynamic entropy in Chapter 4.
2.3 2.3.1
Simple thermodynamic systems Mole-number and molecular mass
The substances studied in the classical physics are composed of molecules. The mass of a molecule is referred to as the molecular mass in units of kg/molecule or g/molecule. It is different from the molecular weight ε = 16
m , mo
which is the ratio of molecular mass m to one sixteenth of the mass of an oxygen atom denoted by mo . In fact, this ratio is a number instead of weight. It will be called the mole-number in this study. Obviously, the mole-number of an oxygen atom is 16, or εo = 16. The molecular mass is then evaluated from m=
ε mo . εo
If the measurement of mass is accurate enough to the level of molecular mass, we may find that changes in the mass of a substance are discontinuous, and the features related to the mass may also change discontinuously. The continuous changes of physical variables are a preconceived idea obtained from old sciences dealing with the macroscopic world.
2. TWO CLASSICAL PHYSICAL SYSTEMS
28
The number of molecule in a substance with mass M is calculated from N= or
M , m
M εo . ε mo
N=
For a mole mass MO = ε(M ) ((M ) indicates unit mass such as 1 kg/mole in the mkgs system or 1 g/mole in the cmgs system), we have N=
εo (M ) . mo
It tells that a mole of any substance has an equal number of molecule. This number is called the Avogadro’s number, given by NA = 6.022 × 1023 (molecules/mole, or
NA = 6.022 × 1026 (molecules/mole,
in cmgs)
in mkgs) .
Now, a molecular mass may be written as m=
ε MO = (M ) . NA NA
This equation is useful as the mole-numbers instead of molecular masses can be found easily in the text books.
2.3.2
Thermodynamic variables
In a dissipation mechanic system discussed previously, kinetic energy is converted irreversibly into heat energy by frictional force. But why the winds in the Earth’s atmosphere never stop? From a mechanic point of view, one may say that there is conversion of geopotential energy of air mass into the kinetic energy caused by the gravitational force of the Earth and the pressure gradient force in the atmosphere. However, the conversion of geopotential energy is also irreversible in a mechanic system. Kinetic energy and geopotential energy can be created continuously in the ceaseless atmosphere, because the atmosphere is not only a mechanic system. When the geopotential energy is converted into kinetic energy in the atmosphere, a part of the heat energy is also converted into kinetic energy. The ratio of the heat energy and geopotential energy converted will be studied in Chapter 7. Usually, the Newtonian dynamics studies the changes of mechanic state, and does not discuss how the heat energy is converted into kinetic energy. This energy conversion is studied by the thermodynamic theories of equilibrium and nonequilibrium thermodynamic systems, which are examples of the non-Newtonian systems as their changes cannot be described by Newton’s laws on a macroscopic scale. A simple example of thermodynamic system is a small piece of gas in rest, which includes a number of molecules in random motions (called the Brownian motions also) toward different directions at different speeds. The difference in the speed is
2.3. SIMPLE THERMODYNAMIC SYSTEMS
29
important for remaining the random motions observed in an inertial frame. If the molecules move at an identical speed toward one direction, they may not collide with each other and look in rest from the inertial frame following the movement. The molecules in random motions possess translational kinetic energy. Usually, the atoms in a molecule are not tied up firmly or have their fixed positions relative to the molecular center, so the molecular collisions may cause rotations or vibrations of atoms and molecules in different directions and at various frequencies. Thus, the diatomic and polyatomic molecules may also possess rotational kinetic energy and vibrational kinetic energy at different levels. Moreover, to cause the molecular collisions and remain the fixed volume of a substance, the molecules must possess an attractive intermolecular force. However, the molecules are not pulled together by the attractive force, since there is also a repulsive force between molecules. It is clear now that the intermolecular forces are attractive at a long distance but repulsive at a short distance. These forces are also important for remaining the random molecular motions, against the inelastic collisions caused by transfers between the translational kinetic energy and rotational or vibrational kinetic energy. The different energy levels and motion status of microscopic particles define a microscopic state of the system, which may not be detected with macroscopic methods usually. However, some statistical features of the states may be indicated macroscopically by three basic measurable quantities: the pressure p, temperature T and specific volume α defined as the volume of unit mass. These quantities are called the thermodynamic variables, which define uniquely a macroscopic thermodynamic state of a gas with given mass and constituents in equilibrium. In general, two of the three variables are independent. Thus, we may write p = p(T, V ) ,
or
T = T (p, V ) ,
or
V = V (T, p) .
The changes of thermodynamic state are constrained on a p-α-T surface in the three-dimensional thermodynamic space. A macroscopic thermodynamic state is called the equilibrium state, if no changes in the thermodynamic variables can be found over a certain scale. However, fluctuations of these variables may still exist in a microscale. This implies that the identification of equilibrium state depends on the scale of observations or the size of instruments. Since the thermodynamic variables represent the statistical features of a microscopic state and the statistics depends on the number of samples, the measurements of thermodynamic variables depend also on the instrument size especially at a non-equilibrium state. For a small piece of gas in a laboratory, we assume explicitly or implicitly that, except the pressure force, stress tensors and intermolecular forces inside, the gas is free of external macroscopic Newtonian forces, such as the gravitational force and Coriolis inertial force on the rotating Earth. In this situation, the equilibrium thermodynamic state is independent of time and space (Referring to Chapter 17), and is assumed expediently or practically to be microscopically uniform too, so that the scale-dependent features of thermodynamic variables may be ignored.
2. TWO CLASSICAL PHYSICAL SYSTEMS
30
2.3.3
Pressure of monatomic gas
The pressure of gas is defined as the force asserting by the gas on a surface with unit area in the normal direction. When the external forces are ignored, the pressure in a fluid is equal at each point towards each direction in equilibrium. Thus, the pressure is defined as a scaler. It may not cause motions of mass in a fluid, unless the distribution is non-uniform. The pressure force can be calculated according to the changes of molecular momentum as the molecules encounter a surface of gas container. We consider here only the monatomic gases which possess the translational kinetic energy but not the rotational and vibrational kinetic energies. Let a square container of volume V in rest to be filled with a monatomic gas at a normal condition. If molecules of the gas are assumed as elastic balls with velocity (j = 1, 2, 3, · · · , N ) ,
ˆ + vyj y ˆ + vzj ˆz vj = vxj x
ˆ, y ˆ and z ˆ denote the unit space coordinates, we may derive the relation where x between the pressure and translational kinetic energy of molecules in an equilibrium state. The container has six square surfaces with the normal directions outward. If ˆ , the surface is labeled as +x-surface. When molecule the normal direction towards x j encounters elastically the +x-surface, the impulse on the molecule produced by the collisions within time ∆t may be represented by its momentum change along the x-coordinate, that is fj ∆t = cm(−vxj − vxj ) = −2cmvxj , in which fj is the impulsive force accepted by the molecule in the negative directive ˆ , and c is the number of collisions in ∆t. Since the distance in the x-direction of x covered by the moving molecule is vxj ∆t, the number of collisions is c=
∆t vx , 2l j
where l = V 1/3 is the length of the container. The force on the molecule reads fj = −
mvx2j l
.
According to Newton’s third law, the force acting by the molecule on the +x-surface is equal in magnitude to the force accepted by the molecule but in the opposite direction. If there are N molecules in the gas, the mean force produced by all the molecules on the +x-surface gives F = where vx2j =
N mvx2 , l N 1 v2 N j=1 xj
2.3. SIMPLE THERMODYNAMIC SYSTEMS
31
is the x-component of mean-square speed of molecules. Since N = nV , in which n is number density of the gas, we see F =
V nmvx2 = Anmvx2 , l
where A = l2 is the area of +x-surface. The pressure on the surface is defined as the force over a unit area, that is p = nmvx2 . In general, the rotation and vibration of diatomic and polyatomic molecules may also have an effect on the pressure, especially at very low temperatures or very high pressures. But this effect is ignored here for the study of atmospheric thermodynamics. In this situation, the pressure is proportional to the number density and translational kinetic energy of random molecular motions. The mean molecular velocity has an equal speed in all directions in an equilibrium state, that is v2 . vx2 = vy2 = vz2 = 3 Applying it for the previous equation yields p=
n mv 2 , 3
(2.7)
For a mole gas with volume V , the number density is n=
NA . V
We have now
2NA mv 2 . (2.8) 3 2 It tells that a mole of any gas in a container with fixed volume possesses the same pressure, if the mean molecular kinetic energy is the same. If the container contains I different gases, the pressure of a gas is called the partial pressure given by 2 pi = ni k¯i , 3 with mi 2 v k¯i = 2 in equilibrium. If the mean molecular kinetic energy is defined as pV =
I 1 ni k¯i , k¯ = n i=1
2. TWO CLASSICAL PHYSICAL SYSTEMS
32 the sum of all the partial pressures gives I
pi =
i=1
2 ¯ nk . 3
Comparing with (2.7) finds p=
I
pi .
(2.9)
i=1
This is the Dalton’s law, which tells that the pressure of a gas mixture in equilibrium is the sum of the partial pressures of each gas included. Equation (2.8) represents a state of the gas. It includes pressure and volume but not explicitly the temperature. As a thermodynamic state can be determined by the three independent thermodynamic variables, the temperature must be a function of the mean kinetic energy. This is discussed later on in the chapter.
2.4
The first law of thermodynamics
As discussed earlier, the molecules in a gas move irregularly and collide with each other. They may spin also in the monatomic gases, and rotate and vibrate in the diatomic and polyatomic gases. Since the molecules do not move in one direction at an equal speed, the translational kinetic energy is not identical to the macroscopic kinetic energy of parcel motion. The sum of the translational, rotational and vibrational kinetic energies of each molecule in a substance at rest may be defined as the heat energy. Meanwhile, the intermolecular forces may be represented by intermolecular potentials. The sum of the potential energy of each molecule gives the thermodynamic potential energy of substance, called also the thermopotential energy. The heat energy and thermopotential energy form the major part of internal energy U of substance. The internal energy may change as the substance does work at the cost of its heat energy or thermopotential energy. It may be changed also by energy exchanges with the surroundings. Without mass exchanges with the surroundings, the energy exchanges of a system with zero kinetic energy density are usually the heat exchanges resulting from radiation, heat conduction or latent heating denoted by dQ. So, the energy conservation law of a substance at rest gives dU + dW = dQ ,
(2.10)
where dW is the work done by the substance. This is also called the first law of thermodynamics. This relationship can be supported by physical experiments, such as the earliest experiments of Count Rumford (1798) and Joule (1850) to find the mechanic equivalent of heat. It will be seen in this chapter that this energy law can be derived theoretically from the kinetic theory of ideal gases. The energy law (2.10) is used for thermodynamic equilibrium states, which is the steady uniform state without mass displacement and macroscopic changes in thermodynamic variables. In a non-uniform gas however, there may be mass displacement across the gradient of number density called the mass diffusion. The
2.5. STATE EQUATION OF GASES
33
velocity of mass displacement caused by molecular diffusion is referred to as the diffusion velocity vd , which may be derived from the additive feature of momentum as illustrated in the next chapter. Thus, the molecular assemblies in a non-uniform gas possess the diffusion kinetic energy Kd =
1 2
vd2 dm ,
which changes with time and space. The integration with mass is used since the distribution of diffusion velocity is non-uniform at nonequilibrium states. The first law for nonequilibrium processes is then given by dU + dW + dKd = dQ .
(2.11)
There may be exchanges between the internal energy and mechanic work in an isolated system, for example, in adiabatic expansion and contraction of a gas. While, the transfer of diffusion kinetic energy to internal energy is unidirectional according to the feature of molecular diffusions discussed in the next chapter. Thus, all the diffusion kinetic energy will be converted into internal energy by molecular diffusions as the gas reaches an equilibrium state. The nonequilibrium thermodynamic energy law should be adopted in the nonequilibrium thermodynamics. However, since it is difficult to measure the diffusion velocity and calculate the diffusion kinetic energy, this law is not used in practice. The nonequilibrium energy law is nevertheless theoretically important, as conversion of the diffusion kinetic energy is irreversible in isolation. This feature can be applied to study the thermodynamics entropy law or the second law of thermodynamics in Chapter 4. The energy law for equilibrium states may also be applied for small air parcels in the atmosphere. As a moving parcel possesses kinetic energy and geopotential energy in the gravitational field, the first law of thermodynamics alone may not be enough to illustrate the energy conversions of the parcel. The equation of parcel kinetic energy and geopotential energy obtained from the Newton’s second law will be used also to derive, in Chapter 7, the energy equation of a moving parcel in the atmosphere.
2.5
State equation of gases
The relationship between three thermodynamic variables of a substance at equilibrium states is called the state equation. The exact expression of state equation for real gases, whether it can be used practically or not, has not been found yet. Many state equations of gases and other substances have been constructed theoretically or experimentally, such as the van der Waals equation, Clausius’ equation, BeattieBridgman equation and Dieterici equation (Sears, 1959; Burshtein, 1996). The best state equations of gases used currently are the virial equations, instead of the theoretical equations, obtained by curve-fitting using experimental data. Applications of these equations are given in Appendices A and B. On the theoretical studies then, different state equations have been forwarded by using various assumptions and approximations. The most well known examples are the ideal-gas equation and van der
2. TWO CLASSICAL PHYSICAL SYSTEMS
34
Waals equation. After development of statistical mechanics and quantum statistics, more complicated state equations were derived in terms of the intermolecular forces assumed. The ideal-gas equation and van der Waals equation may be considered as the approximations of these equations. The van der Waals equation of state will be derived in Chapter 6 using the partition function obtained from the maximum entropy principle in classical thermodynamics. Here, we discuss the general expression of the state equation using the thermodynamic energy law. Let a cylinder with a piston at one end to be filled by c mole of a single gas. As the gas expands slowly, it accepts heat from outside in order to keep a constant pressure. After the temperature increases from T0 to T and the volume increases from V0 to V , the heat absorbed in the isobaric process is c∆Qp = c∆Up + ∆Wp , derived from the first law of thermodynamics (2.10). Now, let the gas return back to the original state, and then we input heat to the gas again without moving the piston this time, until the temperature increases to T as before. No mechanic work is produced in the isochoric process, and the heat c∆Qv = c∆Uv is absorbed. Subtracting the isochoric process from the isobaric process yields c(∆Qp − ∆Qv ) = ∆Wp + c(∆Up − ∆Uv ) . Since ∆Up − ∆Uv = U (T, V ) − U (T0 , V0 ) − [U (T, V0 ) − U (T0 , V0 )] = U (T, V ) − U (T, V0 ) , we obtain c(dQp − dQv ) = dWp + c[U (T, V ) − U (T, V0 )] . for an infinitesimal variation. This equation tells that the heat absorbed in the isobaric process is different from that in the isochoric process, because dQp includes the mechanic work dWp and the internal energy difference between the two final states at the same temperature T . The latter is also the internal energy change as the gas changes its volume in an isothermal process. Provide that Cv is the molar heat capacity of a gas at constant volume defined as (2.12) dQv = Cv dT and Cp is the molar heat capacity of a gas at constant pressure given by dQp = Cp dT , the previous equation reads c(Cp − Cv )dT = dWp + c[U (T, V ) − U (T, V0 )] .
2.5. STATE EQUATION OF GASES
35
It shows that the difference between the isobaric and isochoric heat capacities is contributed by the mechanic work created in the isobaric process together with the change of internal energy in an isothermal process. If the piston moves dx in distance, the work done by the external force F is F dx = p0 Adx = p0 dV , where A is the cross-section area of the cylinder. Now, we have T
c T0
R∗ dT = p0
V V0
dV + c[U (T, V ) − U (T, V0 )] ,
(2.13)
in which we have used the Mayer formula R∗ = Cp − Cv .
(2.14)
When Cp and Cv are not constant, R∗ may also not be constant. Changes of the heat capacities and their difference may be very large across the critical point of a gas. The initial condition is represented by (T0 , p0 , V0 ). Since one thermodynamic variable depends on the other two, the initial state at pressure p0 is also represented by (T0 , V0 (p0 )). If the initial state is chosen at any pressure p, it may be given generally by (2.15) V0 = V0 (T0 , p) . Applying it to eliminate p0 in (2.13) gives T
∗
R dT = p
c T0
V V0 (p)
dV + c[U (T, V ) − U (T, V0 (p))] .
Integrating it by parts for the left-hand side yields ∗
∗
cR T = p(V − V0 ) + cR (T0 )T0 + c
T
T T0
where ∆ΠT = U (T, V ) − U (T, V0 ) =
dR∗ dT + c∆ΠT (V ) , dT V ∂U V0
∂V
(2.16)
dV
T
is the change of internal energy at constant temperature. If the heat energy is independent of volume at constant temperature, ∆ΠT gives the change of thermopotential energy. If the substance has volume b at the lowest temperature which can be set to zero for a temperature scale selected if without phase transitions in the process, we have cR∗ T = p(V − b) + c
T 0
T
dR∗ dT + c∆ΠT (V ) . dT
(2.17)
Here b may depend on pressure, and the difference of thermopotential energy is calculated between the volume V and volume b at temperature T . If the substance at the zero point has a hard surface on which the pressure is in equilibrium with the environments, the extremal volume b is constant. Equations (2.16) or (2.17) includes
2. TWO CLASSICAL PHYSICAL SYSTEMS
36
all the three thermodynamic variables which determine uniquely the thermodynamic state in equilibrium. They give the general expressions of the state equation of substance. The parameters R∗ and b and the thermopotential energy in (2.17) have clear physical meanings, though they may not be determined theoretically especially at the zero point. If the true intermolecular potentials can be given analytically, the thermopotential energy of dilute gases may be calculated with the approach of Zhu (2000, see Appendix A also). To provide the initial volume in (2.16) theoretically, we have to know the state equation (2.15) which is independent of (2.16). If we do not have the independent expression of (2.15), the initial volume may be considered as a parameter depending on pressure for provided T0 . The data can be provided by experiments. When a substance changes phase, the change of volume with temperature may be discontinuous, and so the state equation may give several expressions with different parameters for the different phases respectively. For most gases at normal conditions of the atmosphere, this state equation may be simplified to give the ideal-gas equation, with the assumptions discussed in the next section.
2.6 2.6.1
State equation of ideal gases Ideal-gas equation
The simplest state equation of gases was derived using the relationships obtained from physical experiments, and may be applied approximately for dilute gases at normal conditions. This equation is referred to as the ideal-gas equation, since the gases satisfying this equation are called the ideal gases. When the ideal-gas equation is derived from experimental data, the particular features of ideal gases are not made clear on theory. The assumptions of ideal gases proposed afterwards for the study on the kinetic theory of ideal gases were not used for deriving the state equation. The features of ideal gases different from those of real gases can be revealed by the following theoretical approach for deriving the ideal-gas equation. Since the change of internal energy at constant temperature is small at the atmospheric pressure and temperature (Appendix A), we have the first assumption: The internal energy of ideal gases is constant at constant temperature. This means U (T, V ) − U (T, V0 ) = 0 or Π=0. The ideal gases do not possess thermopotential energy. The state equation (2.17) gives T dR∗ dT . T cR∗ T = p(V − b) + c dT 0 The substance without thermopotential energy may not change phase, and so the volume or pressure may change continuously with temperature in all processes. The
2.6. STATE EQUATION OF IDEAL GASES
37
second assumption reads that R∗ is constant for ideal gases. The previous state equation becomes now cR∗ T = p(V − b) . The random molecular motions and inelastic collisions are induced and maintained by intermolecular forces, which are attractive in a long distance but repulsive in a short distance. When these forces are ignored for ideal gases, the third assumption comes as: Molecules of ideal gases are point masses without size and forces acting on each other. Thus, the volume of ideal gases at the zero point disappears. With the three assumptions, the state equation have the simplest form cR∗ T = pV ,
(2.18)
which is just the ideal-gas equation. Here, we do not assume that the pressure of ideal gases vanishes at the zero point. The ideal-gas equation is good enough for studying the meteorological-scale processes in the atmosphere. Since the mass of c mole gas is M = cMO , (2.18) is replaced by M ∗ R T = pV . MO When (2.14) is divided by a mole mass, we gain the Mayer formula R = cp − cv , where R=
R∗ MO
(2.19)
(2.20)
is the specific gas constant, and cp = Cp /MO and cv = Cv /MO are the specific heat capacities at constant pressure and constant volume respectively. We shall use them to represent the heat capacities of the dry air to study the atmospheric thermodynamic. The state equation of ideal gas gives now p = ρRT ,
(2.21)
p α = RT ,
(2.22)
or where ρ = M/V = 1/α measures the gas density. The units of α are different from those of V , so the units of R and R∗ are different as found by comparing (2.22) with (2.18). The units of R∗ is J/(mol.K) and of R is J/(kg.K). With the assumptions of ideal gases, the first law of thermodynamics may be represented in terms of thermodynamic variables. For example, (2.10) gives c(Cp − Cv )dT = p0 dV for c mole ideal gases in the isobaric processes with p = p0 . It tells that the difference between the isobaric and isochoric heat capacities of a substance without thermopotential energy is attributed simply to the mechanic work created in the isobaric process. Integrating the previous equation yields cR∗ (T − T0 ) = p0 (V − V0 ) .
2. TWO CLASSICAL PHYSICAL SYSTEMS
38
As the initial state may be represented generally by (2.15) as discussed earlier, the previous equation is replaced by cR∗ (T − T0 ) = p(V − V0 (p)) . Since we assumed that the volume disappears at the zero point, we gain immediately the ideal-gas equation (2.18). The state equations of real and ideal gases may be derived directly from the energy equations, so they have a clear relation to the energy laws used. For example, the ideal-gas equation (2.18) or (2.22) tells that as an ideal gas at the zero point with pressure p is heated to get temperature T in an isobaric process or isochoric process, the difference in the heat accepted by the gas in the two processes is equivalent to the mechanic work created in the isobaric process. For a real gas, the difference of thermopotential energy in the two final states with temperature T is added also. However, these state equations are independent of the energy equations, because we have used the independent relationship (2.15) which tells that the thermodynamic state of a substance in rest can be determined uniquely by three thermodynamic variables. Equation (2.15) is actually an implicit state equation. When it is used to diminish a variable in the energy equations, we are rewarded by the explicit state equations. Since the state equation (2.15) is unknown, the V0 in (2.16) is also unknown and so is provided by experiments or other assumptions. The ideal-gas equation derived for the particular processes can be applied for any process. For example, we have p1 T2 = p2 T1 in isochoric processes. This is the Gay-Lussac’s law which agrees with the physical experiments in the normal conditions. We may also have the Boyle’s law p1 V1 = p2 V2 for isothermal processes and the Charles’ law T1 V2 = T2 V1 for isobaric processes obtained from physical experiments. In fact, the ideal-gas equation was derived from these laws in earlier studies (e.g., Planck, 1922).
2.6.2
More features of ideal gases
The energy equation (2.10) gives dUv = dQv for an isochoric process with dW = 0. It follows, from (2.12), that dUv = Cv dT , or
∂U ∂T
= Cv . v
2.6. STATE EQUATION OF IDEAL GASES
39
As the internal energy of ideal gases depends explicitly on temperature only, we have ∂U ∂U = =0 ∂V T ∂p T for ideal gases. These equations give
∂Cv ∂V
= T
∂Cv ∂p
=0. T
The isochoric heat capacity is independent of volume and pressure at a constant temperature. Since R∗ is constant, (2.14) shows that the isobaric heat capacity Cp depends explicitly on temperature only. In general, these heat capacities may change with temperature. Since the changes are small in the atmosphere, they will be taken as constants in the study of atmospheric thermodynamic processes. Different ideal gases are distinguished only by their molecular masses or molenumbers. The Avogadro’s law for ideal gases states: Ideal gases with an equal volume possess an equal number of molecules at a provided pressure and temperature. According to this law, a mole of ideal gases a and b satisfy pb Vb pa Va = = R∗ , Ta Tb as they possess an equal number of molecules. It tells immediately that R∗ is identical for all ideal gases. This constant is then called the universal gas constant. The experimental volume gives R∗ ≈ 8.31 J/(mol.K). Reviewing the three assumptions used for the ideal gases finds that the gas assumed for the discussions of gas pressure in Section 2.4 is also an ideal gas. Comparing (2.18) with (2.8) reveals T =
2NA mv 2 3R∗ 2
for monatomic ideal gases in equilibrium. It is proportional to the translational kinetic energy of molecules in random motions. The gas constant of one molecule given by R∗ ζ= NA is also a constant, called the Boltzmann constant, of which the value is ζ = 1.381 × 10−23 J/(molecule.K). The temperature gives T =
mv 2 3ζ
now. Like the pressure, the temperature depends on the translational kinetic energy of molecules only, when the effect of molecular rotation and vibration are ignored. So, this expression may not be applied for the substances at very low temperatures or very high pressures. The pressure of the ideal gas in (2.7) reads p = nζT .
2. TWO CLASSICAL PHYSICAL SYSTEMS
40
This equation may be used to estimate the number density of a gas according to the pressure and temperature. Again, the thermodynamic variables derived from the kinetic theory represents statistically the macroscopic features of microscopic motion status of molecules. The temperature and pressure for a single molecule may be meaningless in physics. The thermodynamic states represented by the statistical variables are regarded to be continuous in space, and the discrete microscopic features are ignored. Since the mean values depend on the number of samples in general, the thermodynamic variables depend on the size of molecule ensemble or the element of volume in a non-uniform system. This scale-dependent feature of thermodynamic variables are not discussed fully in the previous studies, since only the uniform equilibrium states are studied by the classical thermodynamics, while the microscopic structures and fluctuations of the equilibrium states are ignored.
2.6.3
Kelvin temperature
The state equation tells that temperature of ideal gas is proportional to the volume at constant pressure, or proportional to the pressure at constant volume. It is assumed already that T = 0 K as V0 = 0. If we assume again that T1 = 273.15 K at the freezing point of water under the pressure of one atmosphere (atm), the temperature increases 1 K as an ideal gas which has no phase transitions expands each V1 ∆V = 273.15 where V1 is the volume at the freezing temperature. The temperature is then represented by V V (K) . (2.23) = T = 273.15(K) V1 ∆V It is found from experiments that the volume of one mole kilogram gas at the standard condition with T = 273.15 K and p = 1 atm or 101325 Pa is about V1 = 22.4m3 . So we have ∆V ≈ 8.2 × 10−5 m3 , and (2.23) gives T ≈ 1.2194 × 104
V K, m3
where V is the molar volume at 1 atm in units m3 . The temperature so defined is called the Kelvin temperature, and the temperature scale is called the Kelvin scale, since the temperature agrees with that defined by Kelvin according to the efficiency of reversible heat engines as discussed in Chapter 21. The Kelvin temperature may also be given by pressure change at constant volume. For this case, we have to assume that the pressure is zero as T = 0 K. The temperature may then be calculated from T = 273.15(K)
p p (K) , = p1 ∆p
where p1 is the pressure at the freezing temperature, and ∆p = p1 /273.15.
2.7. THERMODYNAMIC ENERGY LAW OF IDEAL GASES
2.6.4
41
Mixing ratio of water vapor
A variable used in meteorology is the mixing ratio of water vapor defined as the rate of the mass of water vapor Mv to the mass of dry air Md in the moist atmosphere, that is Mv . w= Md If the densities of water vapor and dry air are denoted by ρv and ρd respectively, we see ρv . w= ρd When the dry air and water vapor are assumed as ideal gases, we apply the ideal-gas equation (2.21), giving Rd p v , w= Rv pd where Rd and Rv are the specific gas constants of dry air and water vapor respectively, and pd and pv are the partial pressures of dry air and water vapor respectively. According to Dalton’s law (2.9), we have pd = p − pv , in which p is the pressure of moist air. Since pv p usually, we may use pd = p. Also, (2.20) gives Rd ≈ 287 J/(kg.K) and Rv ≈ 461.5 J/(kg.K). Thus, we gain w ≈ 0.622
pv . p
(2.24)
This equation is used frequently in meteorology.
2.7
Thermodynamic energy law of ideal gases
The first law of thermodynamics discussed previously is given directly by the principle of energy conservation. This law for ideal gases may also be derived from the kinetic theory of gases. According to the assumptions applied for ideal gases, a molecule of ideal gas may be considered as a solid body. If the N molecules in an ideal gas of unit mass possess the translational velocity ˆ + vyj y ˆ + vzj ˆz vj = vxj x
(j = 1, 2, 3, · · · , N ) ,
the translational kinetic energy of these molecules is given by Kj =
mj 2 2 2 (v + vyj + vzj ) + Krj + Kvj , 2 xj
in which mj indicates the mass of molecule j, and Krj and Kvj are the rotational and vibrational kinetic energy. The total amount of molecular kinetic energy gives k=
N
Kj .
j=1
This may not be the kinetic energy of gas motion, since the molecules move in different directions at different speeds. Provided that the barycenter of the gas
2. TWO CLASSICAL PHYSICAL SYSTEMS
42
moves at velocity vx , and the molecular velocity relative to the moving barycenter , we have in the x direction is vxj ˆ + vzj zˆ . )ˆ x + vyj y vj = (vx + vxj
Since
N
mj vxj =0,
j=1
there is
k = k∗ + kH ,
(2.25)
where
vx2 2 may be called the ‘ ordered kinetic energy’ or mass kinetic energy, and k∗ =
kH =
N mj j=1
2
2 2 2 (vxj + vyj + vzj ) + Krj + Kvj
may be regarded as the ‘ disordered kinetic energy’, which is also the heat energy or internal energy of ideal gases. The amount of mass kinetic energy depends on choice of the inertial coordinates, while the disordered kinetic energy does not. A thermometer reads the same temperature of a provided sample in any coordinates. When a macroscopic force acts on a gas towards the x direction, we may think that the force acts equally on each molecule in the gas. Thus, we have dkH − Fx dx = −dk∗
(2.26)
derived from (2.4) and (2.25). It is the conservation law of macroscopic mechanic energy of the gas. Although the total amount of molecular kinetic energy is conserved, the mass kinetic energy may be reduced by converting into heat energy. This process of energy conversion is called the molecular dissipation of mass kinetic energy. The ordered kinetic energy may still not be the kinetic energy of gas motion. The displacement of mass may take place inside a rest gas caused by microscale gradients in the pressure or density. The microscale mass displacement may be called the mass diffusion, which is distinguished with the molecular dissipation, as the former may still be an ordered process but on a scale much smaller than that of the gas, and so may cause the mass displacement towards a certain direction within the gas. The mass diffusion may not increase the heat energy immediately until the molecular dissipation takes place afterwards. For a gas at rest, the ordered kinetic energy is the kinetic energy of mass diffusion at diffusion velocity vd , called the diffusion kinetic energy, that is k∗ =
vd · vd = kd . 2
(2.27)
Now, we have dkH − Fx dx = −dkd .
(2.28)
2.7. THERMODYNAMIC ENERGY LAW OF IDEAL GASES
43
The right-hand side gives the dissipation of diffusion kinetic energy. From the theories of classical probability and statistical mechanics, molecular motions tend to be more disordered through collisions, since a more disordered motion status possesses a higher probability of occurrence. Without external influences, the ordered kinetic energy will eventually become disordered kinetic energy through molecular dissipation, and so dkd < 0. Although there are the pressure force and stress tensors inside a gas, this inequality holds nevertheless. This implies that the statistical effect of molecular dissipation and collision is independent of macroscopic Newtonian forces. This is an important difference between molecular diffusion and parcel motion. Equation (2.28) is similar to (2.4). The effect of friction or molecular dissipation always reduces the ordered kinetic energy and increases disordered kinetic energy. For an monatomic ideal gas which possesses translational kinetic energy only, we have 3 v2 = RT . (2.29) kH = 2 2 As the specific heat capacity at constant volume is given by (Reynolds, 1965) cv =
3 R 2
for monatomic ideal gases, we have dkH = cv dT .
(2.30)
It may also be applied for diatomic or polyatomic gases, when the cv includes rotational and vibrational heats. Moreover, the work done by an external force Fx for the gas in a container is represented by Fx dx = −p Adx = −pdα ,
(2.31)
where A denotes the section area perpendicular to dx, and p is the pressure force of the gas acting on surface A. Since the pressure force is identical in magnitude to the external force, but in the reversed direction, a negative sign is added. Now, we have the energy conservation law cv dT + pdα = −dkd for an isolated ideal gas at rest, which has no mass and energy (including heat energy) exchanges with the surroundings. The first term on the left-hand side gives the change of heat energy or internal energy, and the second term measures the mechanic work done by the gas as its volume changes. Adding the specific heat exchange dq between a gas and the exterior yields cv dT + pdα = dq − dkd .
(2.32)
It is the first law of thermodynamics in (2.11) for nonequilibrium processes of ideal gases, called the thermodynamic energy law of ideal gases in this study. With the ideal-gas equation (2.22), the energy law is also represented by cp dT − αdp = dq − dkd .
(2.33)
2. TWO CLASSICAL PHYSICAL SYSTEMS
44
This expression is less accurate than (2.32) when used for real gases. Since the ordered kinetic energy dissipated by molecular diffusion is converted into heat energy as soon as the system reaches an equilibrium state, the diffusion term may be omitted if used for equilibrium states and we have cv dT + pdα = dq
(2.34)
cp dT − αdp = dq .
(2.35)
or In an isolated system with dq = 0, we see cv dT + pdα = 0
(2.36)
or cp dT − αdp = 0 . The last two equations are also called the adiabatic equations of ideal gases. The second term on the left-hand side of (2.34) or (2.36) represents the mechanic work done by the gas against environmental pressure p at the cost of heat energy when αdp > 0. The negative work is created as the gas contracts under the environmental pressure. The mechanic work created by the environment for the gas is then converted into heat energy. Unlike the energy transfer produced by friction in a Newtonian system, exchanges between mechanic energy and heat energy may take place in both directions in a compressible thermodynamic system. Although there are the frictional force and viscous force in Earth’s atmosphere, the winds never die as the energy conversions take place all the time. The generation of kinetic energy in the atmosphere is a major subject of the current study.
2.8
Internal energy and heat exchange
The thermodynamic energy law of ideal gases may be represented by thermodynamic variables. This may not be the case for ideal gases. The internal energy in the energy equation includes the heat energy H and thermopotential energy Π, that is U =H +Π.
(2.37)
The thermopotential energy and heat energy may be converted into each other or to another energy, so the temperature may change in gas expansions even if the internal energy is conserved. The well known processes of gas expansions are the free expansion and Joule-Thomson expansion. The energy exchanges in these processes are discussed in Appendix B. The precise expression of the heat energy at all temperatures and pressures is not clear so far. For the ideal gases, the heat energy can be given theoretically in the quantum mechanics. A different approach in calculating the heat energy derived from the classical thermodynamics will be forwarded in Chapter 6. According to the study of Zhu (Zhu, 2000, referring to Appendix A also), the thermodynamic energy may be calculated according to the intermolecular forces. However, the
2.8. INTERNAL ENERGY AND HEAT EXCHANGE
45
exact expressions of intermolecular forces are unknown in general, so the forces are usually represented by hypothesis in theoretical studies. Thus, we have no a general expression of internal energy for all substances represented by thermodynamic variables. To use the energy law, we have to calculate the change of internal energy usually. The internal energy is a state function, as it is determined uniquely by the thermodynamic state in equilibrium. Since a thermodynamic state can be given by the three thermodynamic variables of which two are independent, the internal energy may be represented as a function of two thermodynamic variables. If we take T and V as the basic variable, the first law (2.10) is rewritten as ∂U ∂U dT + + p dV . dQ = ∂T v ∂V T Here only the mechanic work pdV = dW created as the system changes volume is considered. This work is positive if the gas expands against the environmental pressure, and is negative as the gas is compressed by the environment. For the isochoric processes of a mole substance, the first law gives
Cv =
∂U ∂T
. v
Inserting it into the first law again produces
dQ = Cv dT +
∂U ∂V
+ p dV .
(2.38)
T
Apply this equation for the isobaric processes yields
Cp dTp = Cv dTp + It follows that
∂U ∂V
∂U ∂V
+ p dVp . T
T
∂T = (Cp − Cv ) ∂V
−p .
(2.39)
p
Now, the change of internal energy is given by
∂T dU = Cv dT + (Cp − Cv ) ∂V
− p dV .
(2.40)
p
The change of internal energy may be integrated from an initial equilibrium state to another equilibrium state, if we know the heat capacities and state equation. We may also choose p and V as the basic variable, and have
dQ =
∂U ∂V
It gives
+ p dV + p
Cp dTp =
∂U ∂V
∂U ∂p
+ p dVp p
dp . v
2. TWO CLASSICAL PHYSICAL SYSTEMS
46 in the isobaric processes, and
Cv dTv =
∂U ∂p
dpv v
in the isochoric processes. Solving the last two equations yields
∂U ∂V
and
Thus, we gain
dU = Cp
= Cp p
∂U ∂p
∂T ∂V
= Cv v
∂T ∂p
−p p
. v
∂T ∂V
− p dV + Cv p
∂T ∂p
dp .
(2.41)
v
The change of internal energy may also be solved from this relationship using the state equation. Since the internal energy is a state function and so dU is a perfect differential, we have
∂Cv ∂V
= T
∂(Cp − Cv ) ∂T
v
∂T ∂V
∂p − ∂T p
∂ + (Cp − Cv ) ∂T v
∂T ∂V
(2.42) p v
derived from (2.40) and
∂Cp ∂p
v
∂T ∂V
+ Cp p
∂ ∂p
∂T ∂V
−1= p v
∂Cv ∂V
p
∂T ∂p
+ Cv v
∂ ∂V
∂T ∂p
v p
(2.43) from (2.41). These relationships can be applied to determine the parameters included in a state equation using the experimental data of heat capacities . Certainly, we may also choose T and p as the basic variables to calculate the internal energy. For a mole ideal gases of which the state equation is given by (2.18), we see
∂T ∂V
= p
p . R∗
We have discussed also that the heat capacities of ideal gases depend explicitly on temperature only. So (2.43) gives Cp − Cv = R∗ which is the mayer formula (2.14) used for deriving the state equation earlier. Now, we gain dU = Cv dT from (2.40). The internal energy is T
U = U0 +
Cv dT . T0
This result is consistent with the assumption used for deriving the ideal-gas equation, that the ideal gases possess no thermopotential energy.
2.9. POLYTROPIC PROCESS
47
Unlike the internal energy, the heat exchange depends on process as the work created is not a state function and depends on process also. The heat exchanges in the isobaric and isochoric processes may be evaluated by integrating the isobaric and isochoric heat capacities with temperature respectively. To calculate the heat exchange in isothermal processes of gases, we may define the molar heat capacity at constant temperature according to the volume change, that is dQT = CT dVT . Applying it for (2.38) yields
CT =
∂U ∂V
+p. T
It follows, from (2.39), that
CT = (Cp − Cv )
∂T ∂V
. p
Thus, the heat exchanges are evaluated from
V
∂T (Cp − Cv ) dQT = ∂V V0
dVT .
(2.44)
p
If we know the state equation, the heat exchange may be calculated conveniently. For ideal gases, (∂T /∂V )p = p/R∗ and we have V
dQT =
pdVT . V0
It tells that the heat accepted is converted into the mechanic work created, as the internal energy is unchanged in the isothermal processes of ideal gases. Applying the ideal-gas equation yields dQT = R∗ T ln
V . V0
(2.45)
As the real gases possess thermopotential energy, the heat accepted is converted also to the thermopotential energy. However, if we have the state equation, the heat exchanges in the isothermal processes may be estimated without knowing the change of internal energy or thermopotential energy.
2.9
Polytropic process
Most of the thermodynamic processes in the real world may not be at constant temperature or pressure or volume. The heat capacity in an arbitrary process is called the polytropic heat capacity. The molar polytropic heat capacity may be defined as the heat absorbed by a mole substance, as its temperature increases a unit degree, that is dQ = Cπ dT .
2. TWO CLASSICAL PHYSICAL SYSTEMS
48
The energy equation (2.38) for the polytropic processes gives
Cπ dT = Cv dT +
∂U ∂V
+ p dV . T
It follows, from (2.39), that
∂T ∂V
Cπ dT = Cv dT + (Cp − Cv )
or Cπ = Cv + (Cp − Cv )
∂T ∂V
p
dV dT
dV , p
p=p(T )
,
(2.46)
where p(T ) is the pressure in the process represented as a function of temperature. For example, if the temperature and pressure change with time, that is T = T (t),
p = p(t) ,
as recorded in an experiment, we may get the reverse function t = t(T ) on theory and so the pressure can be given according to the temperature change in the process. For isochoric and isobaric processes, (2.46) gives Cπ = Cv and Cπ = Cp respectively. For isothermal processes, (2.46) gives Cπ = ∞ as dT = 0. In adiabatic processes, we have dQ = 0 and so Cπ = 0 or
Cv + (Cp − Cv )
∂T ∂V
p
dV dT
=0. p=pa (T )
where pa represents the pressure in an adiabatic process. It follows that
Cp = Cv 1 −
∂V ∂T
p
dT dV
p=pa (V )
.
It gives the mayer formula (2.14) for the ideal gases with constant heat capacities. For ideal gases we may use the energy equations (2.35) and (2.34) giving (cp − cπ )dT − αdp = 0 ,
(2.47)
and (cv − cπ )dT + pdα = 0 , respectively, where cπ is the specific heat capacity of polytropic process. These two relationships give dp cp − cπ dα + =0. p cv − cπ α If all the heat capacities cp cv and cπ are constant, this equation is integrated to gain (2.48) pαγ = p0 αγ0 , where γ=
cp − cπ cv − cπ
2.9. POLYTROPIC PROCESS
49
is called the polytropic index. Applying the ideal-gas equation (2.22) for (2.48) yields
p = p0
or α = α0
µ
T T0
T0 T
(2.49)
,
(2.50)
where µ and - are obtained from cπ = cp − µR ,
(2.51)
cπ = cv − -R .
(2.52)
or Now, the heat exchange may be given by dq = (cp − µR)dT , or dq = (cv − -R)dT . We have µ = 0 or - = −1 in isobaric processes, µ = 1 or - = 0 in isochoric processes, µ = - = −∞ in isothermal processes, and µ = cp /R or - = cv /R in adiabatic processes. In the adiabatic processes with cπ = 0, (2.48), (2.49) and (2.50) give pα
cp cv
cp cv
= p0 α0 ,
p = p0 and
T T0
α = α0
cp
T0 T
R
,
cv R
(2.53)
respectively. These equations show that none of the three thermodynamic variables may remain constant in the adiabatic processes of ideal gases. This is different from the polytropic processes. Equation (2.49) may be rewritten as
θπ = T0 = T
p0 p
R cp −cπ
.
(2.54)
The θπ is the temperature at an initial state. It may also be viewed as the polytropic potential temperature with respect to the reference pressure p0 . Since we have dθπ = 0 in a polytropic process, the polytropic potential temperature is conserved in the process. The potential temperature used in meteorology, given by
θ=T
pθ p
κ
,
κ=
R cp
(2.55)
50
2. TWO CLASSICAL PHYSICAL SYSTEMS
is a particular example of the polytropic potential temperature in the adiabatic processes without phase transitions of air. It is the temperature as the parcel pressure changes reversibly and adiabatically to the reference pressure pθ , taken as 1000 hPa usually. This potential temperature is conserved in reversible adiabatic processes with cπ = 0, but not in polytropic processes when cπ = 0. The polytropic processes and polytropic potential temperature will be applied in Chapter 23 for the study of mixing processes in convective processes.
Chapter 3 Molecular transport processes 3.1
Introduction
From the statistical mechanics or kinetic theory of gases, some microscopic features of gases, such as the mean moving speed of molecules, may be represented statistically by macroscopic thermodynamic variables. Equation (2.29) shows that the local mean molecular speed is inhomogeneous in a non-uniform system. The anisotropic molecular motions and molecular collisions may cause transport of mass, momentum and heat across the gradients in related fields, and produce local changes of thermodynamic variables at nonequilibrium states. The transport processes, called the molecular diffusions, are the major causes of thermodynamic variations in a non-uniform system. In the nonequilibrium thermodynamics which deals with the nonequilibrium processes in a classical thermodynamic system, the molecular diffusions are studied by introducing the hypothesis of linear law. However, the linear law fails to give the reasonable predictions for the transport properties. Thus, different theories have been developed to study the transport processes in non-uniform gases. In the classical transport theory for uniform gases, the relation between the coefficient of viscosity η and the coefficient of self-diffusion D assumed D = η/ρ. However, the experimental data showed that the ratio ρD/η was generally in the range between 1.2 and 1.6 for gases at a normal temperature and pressure (Jeans, 1967; Chapman and Cowling, 1970). Also, the classical theory predicted that heat conductivity of ideal gases could be represented by λ = cv η. While the Eucken ratio λ/(cv η) obtained from experiments varied from 1.4 to 2.6 at a normal condition (Jeans, 1967; Kennard, 1938; Sears and Salinger, 1975). It was believed that these large errors were produced by the uniform gas assumption. In fact, the gases studied by the classical theories are not uniform, since there may be gradients in the mean molecular speed and thermodynamic state of a simple gas studied. Several approaches were made either to provide corrections (Eucken, 1913), or to create different theories by solving the Boltzmann conservation equation (Chapman and Cowling, 1970, Wang Chang et al., 1964), the nonequilibrium dynamic equations (Evans and Morriss, 1990) or the nonequilibrium transfer equations (Woods, 1993). The derived algorithms are generally complex and not all the parameters included can be determined theoretically, so that the theoretical calculations of the transport coefficients with good results may be practically available only for dilute noble gases. Most theoretical treatments used the intermolecular potential assumed for hard sphere, such as the Lennard-Jones potential (Jones, 1924), which could be inappropriate for diatomic and polyatomic molecules. The calculations for the dense and moderately dense gases were made mainly by the modified Enskog theory. The results were also not good enough in general (Hanley et al., 1972; Michels and Trappeniers, 1980). The limitations of the classical theories may be referred to the studies in a book of Millat et al., (1996). 51
52
3. MOLECULAR TRANSPORT PROCESSES
Many statistical features of gases may be studied with the classical thermodynamics or kinetic theory of gases without considering the intermolecular forces. The transport processes are also a kind of statistical effect resulting from random molecular motions at nonequilibrium states. Thus, they are studied in this chapter with the diffusion velocity and partial velocities introduced by Zhu (1999) according to the additive feature of momentum. The transport properties derived from this theory show better agreements with experiments, compared with the previous theories which are much more complicated. The effect of inelastic collisions will be represented by introducing the collisional heat capacity, which can be evaluated easily in terms of molecular structure. This study provides an additional approach for estimating or predicting the transport properties, especially when the information of intermolecular forces is not available. It will be found that the diffusion velocity depends on the gradients of thermodynamic variables, and the transport carried by diffusion velocity is in the downgradient directions and destroys the gradients eventually. As the diffusions disappear in a uniform state, an isolated system in equilibrium cannot establish the gradients again by the diffusions, and so the diffusion processes are irreversible. This feature in heat conduction agrees with the Clausius statement of the second law. This means that the second law for the classical thermodynamic systems may be obtained from the current theory of molecular diffusions. Although the molecular diffusions may not be important for the meteorological processes in the atmosphere within a few days compared with the parcel motions, they are substantial for the irreversible processes studied in the classical thermodynamics.
3.2 3.2.1
Diffusion velocity and partial velocities Diffusion element and diffusion velocity
The mean distance between two molecules in dilute gases is much larger than the size of a molecule (about 10 times of the diameter of a molecule) at a normal temperature and pressure. Also, the mean speed of a molecule in a gas is very large, about 102−3 m/s. The molecules in random motions possess different velocities, which keep changing as the molecules collide with each other. The mean number of collisions for a hydrogen molecule is about 1010 per second at 0◦ C and 1 atmosphere. Consider an element of volume in a gas, which contains a large number of molecules, but is small enough so that the molecules in it may move across the boundaries within a short time. This small volume is referred to as a diffusion element. The random molecular motions in the element may cause transport of mass, heat and momentum at a nonequilibrium state. Although molecules in a diffusion element keep exchanging with the exterior, an element possesses certain amounts of mass, momentum and energy at a given time. We may still evaluate the mass density of a diffusion element. The momentum of a diffusion element may be defined by introducing the instantaneous diffusion velocity vd . If the diffusion element has instantaneous mass M , the instantaneous momentum is M vd . When a substance is broken down into pieces
3.2. DIFFUSION VELOCITY AND PARTIAL VELOCITIES
53
by internal forces, such as in explosion of a bomb, the momentum of the substance may be given by summing up the momentum of each piece. This means that the momentum is additive. Thus, the momentum of a diffusion element may be given by the sum of molecular momentum, that is
vd
N
mj =
j=1
N
mj vj ,
j=1
where vj is the velocity of molecule j which possesses mass mj . This equation gives N j=1
vd = N
j=1
mj vj mj
.
The diffusion velocity is also called the mass average velocity (Hirschfelder et al., 1964). The momentum of a bomb is unchanged by explosion, if no external forces act on the bomb or its pieces. Also, the molecules may change their velocities through collisions, but the momentum of a diffusion element is conserved in the collisions. In other words, the momentum is not changed by intermolecular forces and elastic or inelastic molecular collisions in the element. However, the momentum of diffusion element may be changed by external forces acting on the element, or by exchanging molecules inside and outside the element. The diffusion velocity changes as the momentum changes. The word of instantaneous emphasizes the quick changes in both the direction and speed of diffusion velocity. As molecules in a small diffusion element exchange quickly with those in the surroundings, a definition of trajectory for the diffusion element along with the diffusion velocity may not be physically significant. In general, the diffusion velocity depends on the gradients of thermodynamic variables. A non-zero diffusion velocity does not mean necessarily that the diffusion element moves at the velocity. When the mass center moves inside a non-uniform element, the element itself may not move. Also, the diffusion velocity is different from the velocity of a solid body, since its variation may not have a clear relation to external Newtonian forces in a macroscopic scale including the pressure gradient force. The diffusions may also take place at constant pressure, if we put together two pieces of a gas with different temperatures at a given pressure. Although the molecules in each piece move randomly at an equal mean speed towards all directions, the net effect of the random motions is the mass transport from high density to low density, because the molecules across the boundary to the side of low density are more than those to the side of high density in a unit time. This mass displacement caused by molecular diffusion may also take place if without the external forces such as the gravitational force and pressure gradient force, and is different from the mass movement driven by these forces.
3. MOLECULAR TRANSPORT PROCESSES
54
T1
T2
v+x1 ✲
v−x 1 ✛
✛
✲v+x2
v−x 2
p1
p2 ✲ x
Figure 3.1: Diffusion element and partial velocities (after Zhu, 1999)
3.2.2
Partial velocities
The x-component of diffusion velocity derived from the previous equation is given by N+x N−x j=1 m+xj v+xj + j=1 m−xj v−xj vdx = , N j=1 mj in which the signs + and − indicate the variables related to molecular motions in the +x and −x directions respectively. The two terms on the right-hand side will be called the partial velocities which are mass weighted mean velocities. If the molecules in the element are alike, the partial velocities are represented by = v+x
N+x 1 v+xj N j=1
(3.1)
N−x 1 v−xj . N j=1
(3.2)
and = v−x
They may be rewritten as = v+x
N+x v¯+x N
and
N−x v¯−x , N where v¯ denotes mean molecular velocity. Now, the diffusion velocity along xcoordinate reads 1 vdx = (N+x v¯+x + N−x v¯−x ) . N Since v−x (3.3) v¯+x = −¯ = v−x
and N+x = N−x at an equilibrium state, the diffusion velocity is zero and so no mass diffusion takes place in equilibrium. But the diffusion velocity may not vanish at a nonequilibrium state. To study the diffusions at a nonequilibrium state, we assume that there are two uniform pieces of a simple gas standing side by side as shown in Fig.3.1. One has
3.2. DIFFUSION VELOCITY AND PARTIAL VELOCITIES
55
temperatures T1 and pressure p1 , and the other has temperature T2 and pressure p2 . There is no fixed wall between the two pieces. Now we choose a diffusion element over the discontinuous boundary as shown by the small light box in the figure. If in the element there are N+x1 molecules of pieces 1 which have velocity components in the +x direction, and N−x2 molecules of piece 2 which have velocity components in the −x direction, the partial velocities at the boundary are given by = v+x 1
and v−x = 2
N+x1 v¯+x1 N+x1 + N−x1 N−x2 v¯−x1 , N+x1 + N−x1
derived from (3.1) and (3.2). They are related to the motions of different molecules in the element, but not the velocity components of a mass. Their contribution to the diffusion depends not only on mean molecular velocity, but also on the portion of mass. Since each piece in the element is in equilibrium at the very beginning, we have N+x1 = N−x1 ∼
N1 2
and
N2 , 2 where N1 and N2 are the numbers of molecules in the two pieces respectively. Applying these relationships yields N+x2 = N−x2 ∼
N1 v¯+x1 N1 + N2
= v+x 1
and = v−x 2
N2 v¯−x2 . N1 + N2
= −v+x or Moreover, we may use (3.3) to gain v−x 2 2 =− v−x 2
N2 v¯+x2 . N1 + N2
The mean molecular speed included may be assumed as the arithmetic mean speed derived from Maxwellian distribution function at an equilibrium state, that is (Sears and Salinger, 1975) 8ζT , (3.4) v¯2 = πm where ζ is Boltzmann constant. Since R∗ ζ = =R, m NA m where NA is Avogadro’s number; R∗ is the universal gas constant and R is the specific gas constant, we have 8 v¯2 = RT . π
3. MOLECULAR TRANSPORT PROCESSES
56
According to the principle of energy equipartition, the distribution of molecular kinetic energy is identical in each direction at an equilibrium state, that is v¯x2 = v¯y2 = v¯z2 =
8 v¯2 = RT . 3 3π
Now, the partial velocities give √ N1 RT1 =c N1 + N2
(3.5)
√ cN2 RT2 = −c , N1 + N2
(3.6)
v+x 1
and v−x 2
where c = 2(2/3π)1/2 ≈ 0.92. The ratios of number densities included in partial velocities provide the weights of the fluxes carried by partial velocities, and are similar to the Maxwellian-Boltzmann distribution function in the Boltzmann equation (Chapman and Cowling, 1970). The number of molecules is evaluated from N = nV , where V is the volume of gas, and n is the number density of molecules which is evaluated from p (3.7) n= ζT for ideal gases. If the two pieces of gas in the element have an equal volume, we apply pV N= ζT for (3.5) and (3.6) giving =c v+x 1
and v−x 2
√ p1 T2 RT1 p1 T2 + p2 T1
√ p2 T1 RT2 = −c p1 T2 + p2 T1
(3.8)
(3.9)
respectively. The partial velocities have different speeds when the two pieces are not in thermodynamic equilibrium.
3.2.3
Diffusion velocity in non-uniform ideal gases
The local diffusion velocity at the boundary is the sum of the partial velocities, that is + v−x . vdx = v+x 1 2 It follows, from (3.8) and (3.9), that √
c RT1 T2 p1 T2 − p2 T1 . vdx = p1 T2 + p2 T1
(3.10)
3.2. DIFFUSION VELOCITY AND PARTIAL VELOCITIES
57
The diffusion velocity of gases does not vanish if the two pieces are in different thermodynamic states, and may cause transport across the boundary. These partial velocities are sketched in Fig.3.1. The last equation gives, from ideal-gas equation (2.21), that √ √ ρ1 p1 − ρ2 p2 vdx = c (3.11) ρ1 + ρ2 or √ √ √ ρ1 T1 − ρ2 T2 (3.12) vdx = c R ρ1 + ρ2 respectively. For the molecular diffusion at constant pressure, (3.10) gives √
RT1 T2 T2 − T1 . vdx |p = c T1 + T2
(3.13)
If T1 < T2 , we have vdx > 0. The mass moves from the cold side to the warm side, because the density on the cold side is higher than on the warm side. This can be seen more clearly if (3.11) is rewritten as √ √ √ ρ1 − ρ2 . vdx |p = c p ρ1 + ρ2 If the thermodynamic variables vary continuously in the system, the previous two equations may be replaced by
c vdx |p = 4 and vdx |p = −
R ∂T δx T ∂x
c √ ∂ρ δx , RT 4ρ ∂x
where T = (T1 + T2 )/2 is the mean temperature; ρ = (ρ1 + ρ2 )/2 is the mean density and T1 T2 ≈ T 2 . In the classical kinetic theory of gases, δx is represented by the average collision distance in a given direction (Holman, 1980), that is 2 δx = l , 3
(3.14)
where l is called the mean free path of molecules. Now, we have
c vdx |p = l 6 or vdx |p = −
R ∂T T ∂x
c √ ∂ρ l RT . 6ρ ∂x
(3.15)
(3.16)
If the diffusion is caused by pressure gradient only, and the initial temperatures on the two sides are same, we have, from (3.10), √ p1 − p2 vdx |T = c RT p1 + p2
3. MOLECULAR TRANSPORT PROCESSES
58 or
cl √ ∂p , RT 3p ∂x
vdx |T = −
(3.17)
where p = (p1 + p2 )/2 is the mean pressure. The diffusion velocity is from high pressure to low pressure in an isothermal system. We see also, from (3.12), that √ ρ1 − ρ2 vdx |T = c RT ρ1 + ρ2 or vdx |T = −
cl √ ∂ρ . RT 3ρ ∂x
It is also from high density to low density. The speed is higher than at constant pressure compared with (3.16), because the diffusion is accelerated by pressure gradient force. The pressure gradient force will be discussed in Chapter 5. It is noted that the diffusion velocity may not vanish if mass densities on the two sides are identical, since (3.11) and (3.12) give c √ √ vdx |ρ = √ ( p1 − p2 ) 2 ρ and
c√ R( T1 − T2 ) , 2 respectively. As long as the system is inhomogeneous, there may be mass diffusion and so vd = 0. If thermodynamic variables of the system vary continuously, the last two equations may be replaced by
vdx |ρ =
vdx = −
cl √ ∂p RT 6p ∂x
and (3.15). This diffusion velocity is comparable with that in an isobaric system, and is a half of that in an isothermal system as shown by comparing with (3.17).
3.3 3.3.1
Self-diffusion of ideal gases Diffusive mass flux
The statistical effect of random molecular motions and collisions may produce a net transport of mass towards a certain direction in a nonequilibrium system. The diffusive mass flux in the x direction across a surface with unit area normal to the flux direction may be evaluated from the two-flux approach: + ρ2 v−x2 . Fρx = ρ1 v+x1
For the example discussed earlier, we apply (3.8) and (3.9) and gain 3
Fρx
3
p21 T22 − p22 T12 , =c √ RT1 T2 (p1 T2 + p2 T1 )
(3.18)
3.3. SELF-DIFFUSION OF IDEAL GASES
59
where ideal-gas equation (2.21) is used also. This equation can be rewritten, with the ideal-gas equation again, as √
c R 2 Fρx = ρ1 T1 − ρ22 T2 (3.19) ρ1 + ρ2
or
3√ 3√ c ρ12 p1 − ρ22 p2 . (3.20) Fρx = ρ1 + ρ2 The mass diffusion depends on the gradients of all thermodynamic variables. If the system has constant pressure p, the mass diffusion across the boundary gives 3 3 cp 2 2 T2 − T1 (3.21) Fρx |p = √ RT1 T2 (T2 + T1 )
derived from (3.18), or
√ 3 3 c p ρ12 − ρ22 Fρx |p = ρ1 + ρ2
(3.22)
from (3.20). The diffusion in a system without pressure gradient is from the cold side to the warm side or from high density to low density, and is in the same direction as the diffusion velocity. Equation (3.21) shows that the diffusion may also be represented by the gradient of temperature. If the system has constant temperature T , we gain
derived from (3.18), or
p1 − p2 Fρx |T = c √ RT
(3.23)
√ Fρx |T = c RT (ρ1 − ρ2 )
(3.24)
from (3.19). The mass transport is from high pressure to low pressure or from high density to low density. The mass diffusion may also take place in a system with constant density. From (3.20) and (3.19), we have Fρx |ρ =
c√ √ √ ρ ( p1 − p2 ) 2
(3.25)
and
c √ T1 − T2 , (3.26) Fρx |ρ = ρ R 2 respectively. This diffusion in an isochoric system may be called the thermal diffusion, as it results from the temperature or pressure gradient instead of the density gradient and results in changes in the temperature and pressure instead of density distributions. It is along the downgradient of temperature or from high pressure to low pressure. Since 3
3
T22 − T12 =
T2 −
T1
T2 +
T1 T2 + T1
,
we see by comparing with (3.21) that the thermal diffusion is weaker than the diffusion caused by a density gradient, when the temperature gradient is the same as that in the isobaric system.
3. MOLECULAR TRANSPORT PROCESSES
60
Since the mass transport takes place from high density to low density and depends essentially on the gradient of mass density, the diffusion is weakened by the mass displacement produced and disappears eventually as the system reaches an equilibrium state. The temperature and pressure gradients may also not exist after the diffusion vanishes, since it may be eliminated by thermal diffusion afterwards at constant density. All the diffusion kinetic energy is dissipated into heat energy at the uniform equilibrium state. As the gradients cannot be produced again by molecular motions in equilibrium, the mass diffusion, thermal diffusion and kinetic energy dissipation are irreversible in an isolated system. It can be found in the following sections that the heat and momentum transport resulting from molecular diffusions including the effect of molecular collisions are also irreversible in isolation. The downgradient transport implies that occurrence of a more uniform state possesses a higher probability than that of an ordered distribution. The irreversible energy conversion or increase of randomness is a fundamental feature of molecular diffusions, which can be applied to derive the second law of thermodynamics as shown in the next chapter.
3.3.2
Coefficient of self-diffusion
When the thermodynamic variables vary continuously in the system, the differences of thermodynamic variables may be replaced by differentials. The diffusive mass flux at constant pressure, given by (3.22), is then rewritten as 3 √ ∂ρ δx . Fρx |p = − c RT 4 ∂x If δx is replaced by the average collision distance in (3.14), this equation reads Fρx |p = −D|p
∂ρ , ∂x
(3.27)
where
2RT (3.28) 3π is the coefficient of self-diffusion for non-uniform ideal gases at constant pressure. Similarly, the diffusive mass flux in an isothermal system shows, from (3.24), D|p = l
Fρx |T = −D|T where
∂ρ , ∂x
4l 2RT (3.29) D|T = 3 3π is the coefficient of self-diffusion at constant temperature. It is larger than the diffusion coefficient at constant pressure since D|T = 4D|p /3. We may also define the coefficient of thermal diffusion for an isochoric system, as mass diffusion may also take place if the pressure or temperature is not uniform. The diffusion equation (3.26) may be replaced by Fρx |ρ = −D|ρ
∂T , ∂x
3.4. VISCOSITY OF IDEAL GASES
61
where lρ D|ρ = 3
2R 3πT
gives the coefficient of thermal diffusion. The thermal diffusion is from warm place to cold place, and causes fluctuation in the system until the system gradients are destroyed or an equilibrium state is attained.
3.4
Viscosity of ideal gases
3.4.1
Diffusive momentum flux
Molecular diffusion may also cause momentum transport in the system. If the system shown in Fig.3.1 has mass speed w in the z direction, the diffusive momentum flux in the x direction across a surface with unit area normal to the flux direction may also be evaluated from the two-flux approach: + ρ2 w2 v−x2 . Fwx = ρ1 w1 vx1
For the same element assumed previously, we apply (3.8) and (3.9) for it, giving Fwd = √
cT1 T2 −3 −3 p21 T1 2 w1 − p22 T2 2 w2 R(p1 T2 + p2 T1 )
.
This equation is equivalent to Fwd
√
c R 2 = ρ1 w1 T1 − ρ22 w2 T2 ρ1 + ρ2
or Fwd
3 3 c √ √ = ρ12 w1 p1 − ρ22 w2 p2 ρ1 + ρ2
derived by using the state equation (2.21). The last three equations give
cpT1 T2 −3 −3 T1 2 w1 − T2 2 w2 Fwd |p = √ R(T2 + T1 ) or
at constant pressure,
or
√ 3 3 c p 2 2 ρ1 w1 − ρ2 w2 Fwd |p = ρ1 + ρ2 p2 w1 − p22 w2 Fwd |T = c √ 1 RT (p1 + p2 ) √ c RT 2 ρ1 w1 − ρ22 w2 Fwd |T = ρ1 + ρ2
at constant temperature, and Fwd |ρ =
c √ ρ R w1 T1 − w2 T2 2
3. MOLECULAR TRANSPORT PROCESSES
62 or
c√ √ √ ρ (w1 p1 − w2 p2 ) 2 at constant density. These relationships show that momentum diffusion depends on the gradient of mass velocity as well as the gradients of thermodynamic variables. If without the shear of mass velocity, momentum transport may still be produced by diffusive mass transport, as we have Fwd = wFρd . Fwd |ρ =
3.4.2
Momentum conduction
Molecular collisions may cause exchanges of molecular momentum and kinetic energy. In a non-uniform system, molecules on the cold side increase their kinetic energy after collisions, while molecules on the warm side lose kinetic energy. In the assumption of elastic one-dimensional (or head-on) collisions, momentum and kinetic energy of two colliding molecules are exchanged. The processes of momentum and energy transfer resulting from the collisions are called the momentum and heat conductions respectively. To study the momentum and heat conductions, we assume 1) Only binary collisions take place in the system, and 2) the collisions are elastic. As a matter of fact, the collisions are not elastic at least for diatomic and polyatomic molecules, as there may be exchanges between translational kinetic energy and rotational or vibrational kinetic energy of molecules. The correction for inelastic collisions will be made by introducing collisional heat capacity in a later discussion. The conductive momentum flux across the yp-plane with unit area evaluated from the two-flux approach gives + w2 v−x ), Fwc = nc m(w1 v+x 1 2
where nc is the number density of the molecules which collide with other molecules within a mean free path, and the partial velocities are the weighted mean velocities averaged over the molecules in collisions. In the assumption of binary collisions, the numbers of molecules which come from each side of the boundary and collide near the boundary are equal. Thus, we assume N1 = N2 in (3.5) and (3.6), and insert the results into the previous momentum conduction equation yield √
c (3.30) Fwc = nc m R w1 T1 − w2 T2 2 for our previous example. If the momentum conduction is produced by molecular collisions only without the effect of mass diffusion in a thermodynamically equilibrium system, that is T1 = T2 = T , nc may be replaced by n except at a very low pressure. Applying (3.7), we have c √ Fwc = ρ RT (w1 − w2 ) 2 where ρ = nm may be represented by the mean density of the system. Equation (3.10) shows that the diffusion velocity vanishes in equilibrium. However, molecular momentum may still be transferred by collisions. This is similar to the energy transport in a mechanic wave, as the mass is not delivered in the process of energy propagation. However, the physical mechanism of momentum or heat conduction is different from the wave energy propagation in the classical kinetic theory.
3.4. VISCOSITY OF IDEAL GASES
3.4.3
63
Coefficient of viscosity
Except at a very low pressure, (3.30) is rewritten as
c √ Fwc = ρ R w1 T1 − w2 T2 . 2 If the mass speeds and temperatures are compared over the average collision distance δx, this equation gives √ w ∂T c √ ∂w + √ Fwc = − ρ R T . 3 ∂x 2 T ∂x The momentum flux depends on temperature gradient. According to Newton’s second law, the change rate of momentum is a measure of the external force which causes the change. The external force produced by the momentum diffusion or conduction in a fluid system is called the viscosity or viscous force. The viscosity on an element of unit volume is represented by ∂Fw . (3.31) ∂x It tells that the moving speed of a fluid element is reduced in the place with divergence of momentum flux, and is increased in the place with the convergence. If the temperature gradient is ignored as in an isothermal system, the momentum conduction gives ∂w , (3.32) Fwc = −η ∂x where 2ρl 2RT (3.33) η= 3 3π is called the coefficient of viscosity. It may also be called the momentum conductivity. The mean free path derived from the classical kinetic theory gives (Hirschfelder et al., 1964) 1 . l= √ 2nπσ 2 The effective molecular diameter σ in this equation may be evaluated by inserting the mean free path into (3.33), giving Fz = −
2m η= 3πσ 2
RT . 3π
The coefficient of viscosity varies with temperature alone in this expression. However, experiments showed that the viscosity changes with pressure or density at a given temperature, especially when the pressure is very high or very low. Equation (3.32) shows an important effect of molecular diffusions on mass motions, as they destroy the gradient of mass velocity. Under the influence of boundary friction also, the mass velocity will be eliminated eventually and the organized mass motions will be converted into random molecular motions in an isolated system. From the energy point of view, kinetic energy of mass motion is converted irreversibly into heat energy by molecular diffusions.
3. MOLECULAR TRANSPORT PROCESSES
64
3.4.4
Relation to self-diffusion
For a system of constant pressure, the relation between the coefficient of self-diffusion shown in (3.28) and the coefficient of viscosity in (3.33) is given by D=
3η , 2ρ
(3.34)
in which the diffusion coefficient D|p is denoted simply by D. Table 3.1 displays the coefficients of viscosity and self-diffusion obtained by experiments at 0◦ C and 1 atmosphere (Atm). The ratios in the last column of the table are close to unity. The errors, which can be produced by the ideal-gas approximation, are less than those of the classical theory for uniform gases. The new algorithm may also be used to calculate the molecular diffusion in gaseous mixtures. Table 3.1: Comparison of the coefficients of self-diffusion and viscosity, data sources: 1) Chapman and Cowling (1970), 2) Winn (1950), 3) Hutchinson (1949), 4) Jeans (1967), 5) Kaye and Laby (1995), 6) Loeb (1961) and 7) Trautz and Sorg (1931). (after Zhu, 1999)
Gases
3.5 3.5.1
η
D
0 C, 1 Atm
10−6 N.s.m−2
10−6 m2 .s−1
H2 O2 CO CO2 CH4 N2 Ne Ar Kr Xe
8.45 1 19.26 6 16.35 1 13.70 4 10.20 7 16.56 1 29.80 5 21.00 5 23.40 5 21.20 5
131.0 4 18.9 4 19.0 1 10.9 4 20.6 2 18.5 2 45.2 2 15.8 3 7.95 1 4.8 1
◦
ρ kg.m−3 0.090 1.428 1.250 1.964 0.716 1.250 0.901 1.783 3.740 5.860
2Dρ/(3η) 0.930 0.934 0.968 1.042 0.964 0.931 0.911 0.894 0.847 0.885
Heat conduction of ideal gases Conductive heat flux
The heat energy of ideal gases is proportional to the mean kinetic energy of molecules. So, heat flux is also the flux of molecular kinetic energy. The transfer of molecular kinetic energy produced by molecular collisions is called the heat conduction or thermal conduction. The conductive heat flux across the yp-plane with unit area evaluated from the two-flux approach gives + T2 v−x ), Fhc = cπ nc m(T1 v+x 1 2
3.5. HEAT CONDUCTION OF IDEAL GASES
65
where cπ is polytropic heat capacity given by (2.51) or (2.52), and the partial velocities vx1 and vx2 are again the weighted mean velocities averaged over the molecules in collisions. The cπ is used because the transport process is generally polytropic, so both the pressure and density may change simultaneously. This is different from the heat transport in a solid or liquid. For the binary collisions assumed, we may use N1 = N2 for (3.5) and (3.6), and apply the derived partial velocities for the previous heat conduction equation, giving 3 √ 3 c Fhc = cπ nc m R T12 − T22 . 2 The heat conduction is from high temperature to low temperature, and is in the direction opposite to that of molecular diffusion in the same system. Except at a very low pressure, nc may be replaced by n in (3.7). So, the previous equation is rewritten as 3 √ 3 c Fhc = cπ ρ R T12 − T22 , (3.35) 2 where ρ may be given by the mean density of the system. Applying (2.51) for (3.35) yields 3 √ 3 c (3.36) Fhc = ρ(cp − µR) R T12 − T22 . 2 If the experiments for heat conductivities are made at a constant pressure, we have µ = 0 or cπ = cp and so the previous equation gives 3 √ 3 c 2 2 (3.37) Fhc = ρcp R T1 − T2 . 2 The heat conduction is proportional to the gradient of T 3/2 at a given density. Like the mass diffusion, the momentum and heat conduction caused by molecular collisions depend crucially on the gradients of thermodynamic variables, and transport the momentum and heat along the downgradient of the fields. As a result, the transport destroys the gradients and lead to an equilibrium state of the system. Thus, the momentum and heat conductions are also irreversible in an isolated system. The downgradient heat conduction agrees with the second law of thermodynamics described by the Clausius statement (Sears and Salinger, 1975). This implies that the second law is related to the feature of molecular diffusions and can be derived from the current theory of the diffusions. The mass diffusion and momentum conduction show also the downgradient transport in a non-uniform gas, which may also be related to the entropy law in a more general way discussed in the next chapter. This law was illustrated particularly in terms of heat conduction, because heat conduction is most familiar to us and has a straightforward relation to the Kelvin statement of this law in the theory of heat engines.
3.5.2
Heat conductivity
In a non-uniform system, (3.36) may be replaced by Fhc = −λ
∂T , ∂x
(3.38)
66
3. MOLECULAR TRANSPORT PROCESSES
in which
λ = (cp − µR)ρl
2RT 3π
(3.39)
is called the heat conductivity or thermal conductivity of non-uniform gases. Here, the average collision distance given by (3.14) is used. Comparing with the coefficient of viscosity in (3.33) finds 3 (3.40) λ = (cp − µR)η . 2 It gives 3 (3.41) λ = cp η 2 when the heat conduction takes place at constant pressure. This equation may also be obtained by using (3.34) for the expression λ=
1 4
15 − 6
ρD ρD cp − 15 − 10 cv η η η
derived from the classical non-uniform gas theory (Hirschfelder et al., 1964, Eq.(7.623)). Equation (3.40) allows us to calculate the heat conductivity simply from the data of viscosity. The ratios of heat conductivity to the coefficient of viscosity obtained from experiments are not constant for a gas, but change slightly with temperature (Jeans, 1967). For the monatomic ideal gases, we may use cv = 3R/2 for (3.41) and gain 15 5 λ = cv η = Rη . 2 4 These relationships may also be derived from a classical theory of non-uniform gases in terms of the Maxwellian-Boltzmann distribution function and intermolecular forces. Usually, the gradients in thermodynamic variables may produce mass diffusion in the system, so the heat transport includes the effect of molecular diffusion. Although the heat conductivity is not derived using the peculiar velocity defined by other authors (e.g., Woods, 1993), the results are generally good as shown later on in this section.
3.5.3
Modified Eucken formula
In the classical thermodynamics, the degrees of freedom of molecular motions are defined as the dimensions to which molecular kinetic energy is equally partitioned. A monatomic molecule has three translational degrees of freedom, since the molecule may move in three independent directions. Two rotational degrees of freedom are added to the diatomic molecules which have the dumbbell structure, as the rotation about the molecular gravity center has two independent directions. The polyatomic molecules with linear structure have also two rotational degrees of freedom. But nonlinear polyatomic molecules possess three rotational degrees of freedom. The diatomic and polyatomic molecules may also have vibrational degrees of freedom. If a molecular has n atoms, the vibrational degrees of freedom are 3n − 6 for a nonlinear molecule and are 3n − 5 for a linear molecule.
3.5. HEAT CONDUCTION OF IDEAL GASES
67
In the classical thermodynamics also, each translational or rotational degree of freedom contributes R/2 to the heat capacity at constant volume, and a vibrational degree gives R because a vibrational system possesses kinetic and potential energies. In fact, the energies of rotation and vibration are quantized, so contribution of each rotational or vibrational degree may be different from the classical values, especially at low temperatures. If the heat capacity is divided into three components contributed by translational, rotational and vibrational motions of molecules respectively, we have cv = ctr + crot + cvib . Inserting it and (3.34) into (3.41) yields the modified Eucken formula (Chapman and Cowling, 1970) 5 λ = cv η + (crot + cvib )ρD 2 for monatomic gases. We may assume that ctr = 32 R, as a molecule has three translational degrees of freedom. For diatomic and polyatomic molecules, the vibrational heat capacity is calculated from the Einstein function (Einstein, 1907; Loeb, 1961)
cvib = R
νj h ¯
nv νj ¯h 2
ζT
j=1
e ζT νj h ¯
,
(3.42)
(e ζT − 1)2
¯ (= 6.262 × 10−34 J.s) is where νj is the j-th wavenumber of molecular vibration; h Planck’s constant, and nv is the vibrational degree of freedom. Derivation of this equation may be referred to Chapter 6. In the case of degeneracy, one vibrational frequency may be applied for more than one degrees of freedom. The rotational heat capacity may also be calculated theoretically, if we know the moment about each axis of rotation.
3.5.4
Collisional heat capacity
Equation (3.41) is derived from the assumption of elastic collisions. However, the real collisions between diatomic or polyatomic molecules are not elastic, due to the exchanges of molecular translational kinetic energy and rotational or vibrational kinetic energy. The specific heat capacity in (3.41) is the heat which can be transferred by molecular collisions. When the heat transfer is affected by the inelasticity, the heat capacity may be different from that in elastic processes. To provide a correction for inelastic collisions, we define the collisional heat capacity as the heat which can be transferred by molecular collisions along a given direction. In general, the translational kinetic energy in each direction can be transferred by the binary collisions assumed, so we have cˆtr = ctr , whereˆindicates the collisional heat capacity. The rotational kinetic energy can be transferred when the collisions are not in the direction parallel to the axis of rotation
3. MOLECULAR TRANSPORT PROCESSES
68
in a molecule. For diatomic molecules or linear polyatomic molecules, the energy transfer may take place when the axis is in two independent directions perpendicular to the direction of conductive flux. If distribution of the rotation axis in the three independent directions has an equal probability, the collisional heat capacity of rotation is 2 cˆrot = crot . 3 Meanwhile, we have cˆrot = crot for nonlinear polyatomic molecules with a simple structure, since the rotational kinetic energy can be transferred by collisions from all directions. Moreover, we assume that the vibrational kinetic energy may be transferred only by collisions in the direction of vibration. So the collisional heat capacity of vibration is 1 cˆvib = cvib . 3 The heat which can be transferred in a given direction by collisions reads cˆv = cˆtr + cˆrot + cˆvib now. It follows that
cˆvib =
ctr + 23 crot + 13 cvib for linear molecules for nonlinear molecules ctr + crot + 13 cvib
.
Also, we may write cˆp = cˆv + R for ideal gases. The isochoric and isobaric collisional heat capacities for monatomic molecules are given by cˆv = cv and cˆp = cp respectively. These relationships imply that collisions between monatomic molecules are assumed elastic in the current theory.
3.5.5
Comparison with experiments
Although the classical theories for the gases at non-uniform states are very complicated, the results are good only for monatomic gases. The relative errors, defined as − κhexperiment κh , Er = theory κhexperiment are generally large for diatomic and polyatomic gases. The accuracies of the classical theories may be viewed from the study of Reid et al. (1987), which showed that the mean relative error averaged over six or five classical methods was 8.7% for acetylene at 273 K, and 10.1% for carbon dioxide at 200 K.
3.5. HEAT CONDUCTION OF IDEAL GASES
69
Table 3.2: Heat conductivity of monatomic molecules. The data sources are: 1) Kennard (1938), 2) Kannuluik and Carman (1952), 3) Chapman and Cowling (1970) and 4) Keyes (1955). (after Zhu, 1999)
Gas
η
0◦ C, 1 Atm
10−6 N.s.m−2
J.kg−1 .K−1
cv
He Ne Ar Kr Xe
18.65 3 29.80 21.00 23.40 21.20
3122 2 628 2 312 2 149 2 95 2
λexp
λ
14.39 1 4.60 1 1.64 4 0.87 4 0.51 4
14.49 4.60 1.64 0.87 0.50
10−2 W.m−1 .K−1
Er % 0.7 -0.1 0.0 -0.2 -0.6
For monatomic molecules which possess translational kinetic energy only, (3.41) gives 3 (3.43) λ = cˆp η . 2 The heat conductivities evaluated from this equation are compared with experiments in Table 3.2. The results are just as good as those obtained from the classical theory for non-uniform gases. For diatomic and polyatomic molecules, the vibrational heat capacity is calculated with (3.42). The vibrational spectra of diatomic molecules used are taken from Allen and Pritchard (1974), except those for chlorine from Sears and Salinger (1975), and deuterium from Huber and Herzberg (1979). The vibrational spectra of polyatomic molecules are from Shimanouchi (1972). The rotational heat capacity is evaluated with 3 crot = cv − R − cvib 2 using the experimental heat capacity data reported in other literatures. The heat conductivities evaluated for some diatomic and polyatomic molecules with weak or no polar momenta are listed in Table 3.3, which are compared with experiments. The agreements with experimental data are generally good. For real gases, the ratio cp − cv b= R may not be unity. The deviations are very small in general. But for chlorine, the ratio is about 1.07 (Chapman and Cowling, 1970). If we use this ratio, the collisional heat capacity at constant pressure may be represented by cˆp = cˆv + bR . The heat conductivity evaluated for chlorine becomes, then, 0.76×10−2 W.m−1 .K−1 , and the relative error reduces to -0.2%. For the polyatomic molecules with internal degrees of rotational freedom, the collisional heat capacity of rotation may be less than what is assumed previously. The accuracy may be increased by evaluating more precisely the collisional heat capacity for complicated polyatomic molecules.
3. MOLECULAR TRANSPORT PROCESSES
70
Table 3.3: Heat conductivity of diatomic and polyatomic molecules with no or weak polar momenta. The data sources are: 1) Kennard (1938), 2) Eucken (1913), 3) Chapman and Cowling (1970), 4) Johnston and Grilly (1946), 5) Eucken (1911), 6) Senftleben (1964), 7) Misic and Thodos (1966), 8) Kaye and Laby (1995), 9) Jeans (1967), 10) De Rocco and Halford (1958), 11) Vogel (1914), 12) Beeck (1936), 13) Muller (1968), 14) Touloukian et al. (1975), 15) Eucken & Hoffmann (1920) and 16) Touloukian and Makita (1970). (after Zhu, 1999)
Gas
η
0◦ C, 1 Atm
10−6 N.s.m−2
H2 D2 O2 N2 Cl2 CO NO Air CO2 N2 O C2 H2 CH4 C2 H4 C2 H6
8.45 11.92 14 19.26 16.56 12.50 14 16.35 17.94 2 17.30 8 13.70 13.70 8 9.43 11 10.20 9.07 2 8.69 10
cv
cˆv
J.kg−1 .K−1
10125 13 5179 16 653 1 745 1,9 371 15 741 2 693 2 717 628 2 659 2 1335 12 1665 12 1166 12 1392 12
8758 4487 564 645 271 642 600 623 525 532 1001 1595 982 1050
λexp
λ
10−2 W.m−1 .K−1
16.59 4 11.72 13 2.38 2,4,5 2.37 1,2,9 0.77 2 2.31 4 2.38 5 2.37 2 1.46 3 1.47 2 1.89 6 3.15 7 1.75 6 1.77 6
16.40 11.71 2.38 2.34 0.76 2.30 2.36 2.36 1.47 1.48 1.87 3.24 1.75 1.75
Er %
-1.2 -0.1 -0.3 -1.2 -1.1 -0.3 -0.7 -0.2 0.7 0.6 -1.1 3.0 0.2 -1.3
Chapter 4 Predictability and thermodynamic entropy 4.1
Introduction
It is discussed in Chapter 2 that the Newtonian systems may be studied with a complete set of prediction equations in the time-space coordinates, so the variations are predictable in principle. While in the classical thermodynamics, variations of three thermodynamic variables are studied with two thermodynamic relationships: the state equation and the first law of thermodynamics for equilibrium states. For a compressible fluid parcel, such as a gas at rest, another relationship may be provided by mass conservation law, such as the mass diffusion equation discussed in this chapter, which tells that a local mass density may be changed by molecular diffusions in a fluid system without parcel motions. The mass velocity included in the mass diffusion equation is the instantaneous diffusion velocity discussed in the preceding chapter, which is different from the parcel velocity controlled by macroscopic Newtonian forces. It depends on the microscale gradients of non-uniform system, and may not be measured directly and evaluated precisely in a macroscopic scale. The hypothesis of linear law or other diffusion theories used for the previous studies of transport properties may not be applied for the mass diffusion equation to predict either conveniently or successfully the system variations. Thus, molecular diffusions are not studied in the classical thermodynamics which deals with the equilibrium processes only. When the mass diffusion equation is ignored as we do usually, the two thermodynamic equations cannot form a complete set for the three thermodynamic variables. They may be used to evaluate a new equilibrium state for a provided equilibrium initial state, assuming at least one variable of the new state is given already. The two equations used for the equilibrium states are independent of time, and the evaluations of a new equilibrium state are different from predictions. It is discussed in the previous chapters that changes of thermodynamic states caused by molecular diffusions are irreversible in isolation. So, the new state obtained may not really occur due to irreversibility of diffusion processes. To find whether the state evaluated may really occur afterwards, we may introduce a state function, called the thermodynamic entropy, according to the irreversible downgradient transport and energy conversion in the diffusion processes. A microscopic approach for deriving the thermodynamic entropy law was given by Prigogine et al. (1977) using the theory of quantum field. We derive in this chapter the entropy law for ideal gases from the first law of thermodynamics at nonequilibrium states discussed in Chapter 2, which tells that thermodynamic entropy is created by conversion of diffusion kinetic energy into heat energy. A new equilibrium state may 71
72
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
occur afterwards, only if the thermodynamic entropy is not destroyed in isolation. However, the entropy may not be used to predict when and how the new state occurs. However, conversion of mechanic energy into hear energy is not the only source of thermodynamic entropy. From a microscopic point of view, thermodynamic entropy variation is related to the changes in distribution and partition of molecular energies. As the system changes irreversibly in isolation and the ability of system change depends on the system disorderliness, the microscopic thermodynamic states with different degrees of disorderliness possess different probabilities of occurrence. Thus, the entropy variation may also be studied, in the statistical thermodynamics, with the probability theory based on the statistics of molecular distributions against molecular energies. Without using the theory of statistical thermodynamics, we may also prove in this chapter that the entropy increases with the system disorderliness even if no energy conversions take place. We shall provide also the general algorithm for evaluating the entropy change in a non-uniform system, which shows that the entropy of a system depends not only on the entropy of each parcel in the system, but also on the parcel distribution. The entropy of a system may still be changed if the entropy of each parcel in the system is conserved. Thus, thermodynamic entropy is different from many other quantities, and is not entirely transportable and additive. Changes of thermodynamic entropy are caused by molecular diffusions in the classical thermodynamic systems. However, to integrate the diffusions over a nonequilibrium process is not the best way for evaluating the entropy changes, because we are unable to predict the nonequilibrium processes in general. For the ideal gases with constant heat capacities, the state function of thermodynamic entropy may be represented macroscopically in terms of the measurable thermodynamic variables. The entropy change in an irreversible process may be calculated by assuming a reversible process to connect the initial and final equilibrium states provided. It is noted that the expression thermodynamic entropy as a state function for a real gases or another real substances may be different from that of ideal gases, because the state equation is different from that of ideal gases and the heat capacities are not constant. Since the thermodynamic variables represent the statistical features of molecular assemblies and so depend on the size of samples in a non-uniform system, the thermodynamic entropy calculated with these variables depend on the size too. The irreversible energy conversion exists also in the classical mechanic systems, and may be studied by introducing the viscosity or friction to account for the irreversible energy conversion. The dynamic entropy discussed in Chapter 2 may be viewed as an application of the entropy law for Newtonian dynamics. The relation and difference between thermodynamic entropy and dynamic entropy will be discussed in this chapter.
4.2
Change rate in diffusion processes
In a nonequilibrium thermodynamic system, the thermodynamic state varies with time and space. For example, the temperature may be represented by T = T (r, t) ,
(4.1)
4.2. CHANGE RATE IN DIFFUSION PROCESSES
73
where ˆ + zˆ r = xˆ x + yy z indicates the spatial position of a particular point in the temperature field, represented in Cartesian coordinates of which the unit vectors along the axes are denoted ˆ, y ˆ and ˆ by x z. It is discussed in the preceding chapter that the temperature or heat energy can be transported by heat conduction without mass transport accompanied, especially in a solid substance. Thus, the particular point in a variable field may not be an element of volume or mass. As the temperature field moves or changes in heat conduction, the element of volume or mass does not move. For convenience, a particular point in a variable field at a given time and place may be referred to as the field point. A field point driven by molecular diffusion or conduction may change its position continuously with time in the direction of diffusion velocity or conduction velocity, that is r = r(t). The change rate of the field point is given by the derivative of (4.1), that is ∂T ∂T dr dT = + · . dt ∂t ∂r dt We may write ∂T ∂r
∆T ∆r ∆T ∆r = lim ∆r→0 ∆r∆r ∆T ∆r∆r = lim ∆r→0 ∆r ∆r · ∆r ∆T ∆r . = lim ∆r→0 ∆r ∆r =
lim
∆r→0
Here, ∂T ∆T = ∆r→0 ∆r ∂r is the directional derivative of temperature in the direction of r, and lim
r ∆r = ∆r→0 ∆r r lim
is the unit vector along r. Now, we have ∂T r ∂T = . ∂r ∂r r
(4.2)
This expression is independent of coordinates, and is not changed by rotation of coordinates proved by McHall (1993). Moreover, r dr = vgl = vgl dt r
(4.3)
may be referred to as the general velocity, indicating the displacement velocity of the field point. If the field displacement is carried by mass displacement, the general velocity is a mass velocity such as the diffusion velocity. The general velocity may
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
74
also be a conduction velocity of heat or momentum conduction, or a phase velocity or group velocity (The both may be called the propagation velocity) for propagation of perturbation energy in mechanic waves. A conduction velocity or propagation velocity may not be a mass velocity. In a fluid system without macroscopic parcel motions, the variations may be caused by molecular diffusions including molecular collisions. The sum of instantaneous diffusion velocity vd and conduction velocity vc gives the general velocity vgl = vd + vc . It may also be represented by ˆ + vgly y ˆ + vglz ˆz vgl = vglx x in Cartesian coordinates, in which vglx , vgly and vglz are the components in the directions of x, y and z, respectively. If the angles of r to the x, y and z axes are given by αx , αy and αz respectively, the velocity components are vglx = vgl cos αx ,
vgly = vgl cos αy ,
vglz = vgl cos αz .
(4.4)
If the system changes can also be produced by wave propagations, the general velocity may include the propagation velocity. Combining (4.2) and (4.3) gives ∂T dr · ∂r dt
∂T r · r ∂r r 2 ∂T . = vgl ∂r = vgl
The directional derivative included may be expanded as ∂T ∂T ∂T ∂T = cos αx + cos αy + cos αz . ∂r ∂x ∂y ∂z Applying (4.4) for it yields vgl
∂T ∂r
∂T ∂T ∂T vgl cos αx + vgl cos αy + vgl cos αz ∂x ∂y ∂z ∂T ∂T ∂T + vgly + vglz . = vglx ∂x ∂y ∂z =
(4.5)
Now, the rate of temperature change is replaced by ∂T ∂T ∂T ∂T dT = + vglx + vgly + vglz dt ∂t ∂x ∂y ∂z or
∂T dT = + vgl · ∇T , dt ∂t
where ∇=
∂ ∂ ∂ ˆ+ ˆ+ x y zˆ ∂x ∂y ∂z
(4.6)
4.3. MASS CONSERVATION LAW
75
indicates the three dimensional gradient. The left-hand side of (4.6) is a perfect derivative called the material derivative usually, which is the change rate of a field point with unit mass (if it depends on mass) and is evaluated or measured by following the moving field point. The field point may be physically or symbolically independent of mass in some particular situations, such as a packet of waves. The first term on the right-hand side is called the local derivative, which measures the change rate at the local region occupied by the unit mass or specified field point. The other term, called the advection, manifests the transport effect of mass diffusion or conduction or propagation. This equation tells that the change of field variable at a given time and place may be represented by local change plus diffusion or conduction or propagation transport across the region at the time.
4.3 4.3.1
Mass conservation law Mass diffusion equation
Two thermodynamic equations, the state equation and first law of thermodynamics, have been discussed in Chapter 2. To have a complete set of governing equations for the three thermodynamic variables, we need one more equation which can be obtained from mass conservation law. When the pressure decreases, a gas expands through molecular diffusion. The temperature and mass density decrease also in the expansion. Without heat exchange with the surroundings, the process gives an example of adiabatic variation of thermodynamic system. The variation of density produced by diffusion is discussed in the following. If (4.6) is applied for mass density ρ instead of temperature, it gives dρ dt
= =
∂ρ + vd · ∇ρ ∂t ∂ρ + ∇ · (ρvd ) − ρ∇ · vd , ∂t
where ∇·v =
(4.7)
∂vz ∂vx ∂vy + + ∂x ∂y ∂z
denotes the three dimensional divergence. The conduction velocity is ignored in the equation as the contribution to mass displacement is relatively small. Also, ∇ · vd =
∂vdy ∂vdz ∂vdx + + ∂x ∂y ∂z
is a three dimensional divergence of diffusion velocity, which measures the change rate of the volume of diffusion element as discussed later on. The ρvd is a density flux Fρ produced by mass diffusion, and ∇ · (ρvd ) is divergence of the flux which represents the local concentration rate of mass. If there are no internal sources or sinks of mass at the local place, we have ∂ρ + ∇ · (ρvd ) = 0 , ∂t
(4.8)
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
76 and so (4.7) gives
dρ = −ρ∇ · vd . (4.9) dt The last equation may be called the mass diffusion equation, which is the continuity equation of diffusion system without the parcel motions driven by macroscopic Newtonian forces. The relative change rate of density is proportional to the convergence of diffusion velocity. The flux =v is physically meaningful, only if the quantity = can be transported by the velocity v. When the mass is not transported by heat conduction in a solid medium, we have dρ/dt = ∂ρ/∂t = 0, and (4.7) gives ∇ · (ρvhc ) − ρ∇ · vhc = 0 , where vhc denotes the heat conduction velocity. It follows that vhc · ∇ρ = 0 , which may not be true if vhc = 0 and ρ is not constant. This example emphasize that the velocity in the continuity equation must be mass velocity.
4.3.2
Mass conservation
The mass diffusion equation is equivalent to the conservation law of mass. If M = ρV represents the mass in volume V , we have dV dρ dM =ρ +V . dt dt dt It follows, from (4.9), that
dV dM =ρ − V ∇ · vd dt dt
.
(4.10)
From V = δxδyδz, we obtain 1 dδx 1 dδy 1 dδz 1 dV = + + . V dt δx dt δy dt δz dt Since
d(δy) = δvdy , dt
d(δx) = δvdx , dt as δx, δy, δz → 0, we see 1 dV V dt
d(δz) = δvdz dt
δvdy δvdx δvdz + lim + lim δy→0 δy δz→0 δz δx ∂vdy ∂vdz ∂vdx + + = ∂x ∂y ∂z = ∇ · vd . =
lim
δx→0
Therefore, (4.10) gives dM/dt = 0. The velocity divergence in the last equation measures the relative change rate of the volume. When the volume increases, the
4.3. MASS CONSERVATION LAW
77
divergence is positive or ∇ · vd > 0, otherwise ∇ · vd < 0. The negative divergence is also called convergence. Applying the last equation for (4.8) yields 1 dV 1 dρ =− . ρ dt V dt It gives V0 , V which shows the density change in the process of gas expansions. Since ρV = M is the mass of gas, the last equation gives the mass conservation law M = M0 . ρ = ρ0
4.3.3
Diffusive transport equation
Equation (4.8) may be called the diffusive transport equation. It tells that the local variation is produced by divergence of the diffusion flux, if without sources or sinks in a system. As discussed in Chapter 3, the diffusion velocity depends on microscale gradients of thermodynamic variable, and so can hardly be measured precisely in practice. Thus, the classical equilibrium thermodynamics does not discuss either the diffusive transport equation (4.8) or the mass diffusion equation (4.9). The nonequilibrium thermodynamics and hydrodynamics introduced the alternative algorithms to evaluate the diffusion transport based on the hypothesis of linear law (such as the Fickian law for heat transport and Fick’s law for mass diffusion) (Batchelor, 1967; de Groot and Mazur, 1962). When the transport coefficients are constant, (3.27), (3.32) and (3.38) give examples of the linear law for mass, momentum and heat transport, respectively. But in general these flux equations do not show the linear relationships between the fluxes and gradients, as the coefficients are not constant. This can be seen clearly from the flux equations (3.22), (3.30) and (3.36). Applying the diffusive mass flux equation (3.27) for (4.8) yields ∂ρ = D∇2 ρ + ∇D · ∇ρ , ∂t where ∇2 ρ = ∇ · ∇ρ =
∂2ρ ∂2ρ ∂2ρ + + . ∂x2 ∂y 2 ∂z 2
The diffusion coefficient D is not constant as shown by (3.29). Since it is generally difficult to calculate the variation of this coefficient with real data, the coefficient is usually taken as constant. In this case, we gain the mass diffusion equation ∂ρ = D∇2 ρ . ∂t
(4.11)
Use of this equation needs not to know the diffusion velocity. The solution tells that the density decreases while fluctuates with time. However, limitation of this equation is obvious, as it cannot predict the variation of a system with constant gradients. Lorenz (1955) pointed out that the skin friction and surface heating
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
78
evaluated using the hypothesis of linear law have nearly infinite values throughout an infinitesimal depths near the boundaries. Since ∇2 ρ > 0 in a low center of density and ∇2 ρ < 0 in a high center, (4.11) shows that the diffusion destroys the inhomogeneity of the field. This feature of molecular diffusion, discussed also in the preceding chapter, leads to the second law of thermodynamics.
4.4
Unpredictability in classical thermodynamics
The thermodynamic energy law of ideal gases given by (2.32) may be written as the prediction equation dα dq dT +p = + DTd (4.12) cv dt dt dt for a continuous process, where DTd = −
dkd dt
(4.13)
may be called the heat diffusion, which is the rate of heat production resulting from dissipation of diffusion kinetic energy. For ideal gases we may also have cp
dp dq dT −α = + DTd dt dt dt
(4.14)
derived from (2.33). Since dkd < 0 as discussed earlier, (4.13) shows DTd > 0. The molecular dissipation always converts kinetic energy into heat energy. Now, we have three independent relationships for ideal gases: the ideal-gas equation (2.21) for equilibrium state, the mass diffusion equation (4.9) and the energy conservation law (4.12) or (4.14) for nonequilibrium processes. We may also use the transport equation (4.8) which is not independent of the mass diffusion equation. The three equations have five macroscopic variables: three thermodynamic variables T , p, ρ and diabatic heating rate dq/dt together with the dynamic variable vd . The diabatic heating rate may be provided at a laboratory condition, and the diffusion velocity may be evaluated from the transport theory as shown by (3.10)-(3.12). So the three equations can be used to predict the three thermodynamic variables. However, to measure the non-uniform thermodynamic state on a microscale is generally impossible with an accuracy good enough for the predictions. To solve the thermodynamic equations, we may assume that the system varies slowly, so that it is in quasi-equilibrium at any time and the thermodynamic variables are independent of space. The molecular diffusion may be ignored in this situation, and the mass diffusion equation (4.9) or the diffusive transport equation (4.8) gives dρ = 0 or ∂ρ/∂t = 0. This, however, is not true as the density may change from one equilibrium state to another. Thus, the mass conservation equations are ignored in the study of classical thermodynamics. The three thermodynamic variables of ideal gases are then solved from two independent diagnostic equations: the idealgas equation and the thermodynamic energy law (2.34) or (2.35) for equilibrium processes. These equations can be used to evaluate a new equilibrium state, but not to predict the system variations.
4.5. THERMODYNAMIC ENTROPY LAW FOR UNIFORM STATES
79
If the evaluations are carried out between two equilibrium states, the energy equation gives cv ∆Tequ + pequ ∆αequ = q (4.15) or cp ∆Tequ − αequ ∆pequ = q
(4.16)
when the heat capacities are taken as constant in the current study. The subscript ‘equ’ indicates the variables at equilibrium states. For the adiabatic processes of ideal gases, we use the ideal-gas equation (2.22) for (4.12) and (4.14) giving cv
d ln α d ln T +R =0 dt dt
and
d ln p d ln T −R =0, dt dt respectively. If the heat capacities are constant, the solutions are the adiabatic equations of ideal gases R α0 cv (4.17) T = T0 α cp
and
T = T0
p p0
R
cp
.
(4.18)
.
(4.19)
The two equations give another expression
p = p0
α0 α
cp cv
These three equations may also be obtained from (2.49), (2.54) and (2.48) with cπ = 0. To find the solutions of thermodynamic variables, we have to provide not only the initial conditions but also a part of the solutions or to specify a particular process, such as the isobaric process dp = 0, isothermal process dT = 0, isochoric process dα = 0 or adiabatic process dq = 0. The obtained solutions are not the predictions, because they may not tell whether and when the new state may occur. For this reason, the thermodynamic processes are unpredictable in classical thermodynamics.
4.5
Thermodynamic entropy law for uniform states
The state equation and energy conservation equation for equilibrium states used in the classical thermodynamics are diagnostic equations but not prediction equations. A new equilibrium state derived using these equations may occur before or after a provided initial state in time. Since the diffusion process is irreversible in isolation, the new state, which may occur before the initial state, may not take place afterwards. Thus, the two thermodynamic equations may not be enough to give the solutions which are physically significant.
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
80
To know whether the new state may occur after the initial state, we try to integrate the prediction equation (4.12) for ideal gases from an initial time t0 to final time t1 , giving t1 t0
DTd dt =
T1
cv dT +
α1 RT
T0
α0
α
dα − q .
Since DTd > 0 for a diffusion process, both sides of the equation should not be less than zero if the process may really occur. If the both sides are negative, we have DTd < 0. This process cannot happen as a part of heat energy is converted into diffusion kinetic energy at the end of the process. Since the information of molecular diffusion on a microscale is not available, we are unable to integrate the left-hand side of the previous equation. Also, the thermodynamic variables in the integrations on the right-hand side change with time and place in a non-uniform system, and these variations may not be measured or predicted precisely. Except for some particular situations, it is generally impossible for us to integrate the nonequilibrium thermodynamic processes. Thus, the previous equation cannot be used in practice. If (4.12) is rewritten as d ln α dq d ln T DTd = cv +R − T dt dt T dt
(4.20)
for the ideal gases with constant heat capacities, the integration gives t1 DTd t0
T1 α1 dt = cv ln + R ln − T T0 α0
dq . T
(4.21)
The difficulty in calculating the right-hand side occurs only in the heating term now. For an isolated system without heat and mass exchanges with the surroundings, this equation becomes t1 T1 α1 DTd dt = cv ln + R ln . (4.22) T T0 α0 t0 Now, the right-hand side can be calculated according to the initial and final states without considering the variations in the process. If setting ds = d(cv ln T + R ln α) . we gain
(4.23)
t1 DTd
dt dm (4.24) T for an isolated system with mass M . The S is called the thermodynamic entropy, and s is the specific thermodynamic entropy of unit mass. For the open systems, the thermodynamic entropy change is represented generally by ∆S =
M
ds =
t0
dq DTd + . T T
(4.25)
This expression is equivalent to (4.23) only for the ideal gases with constant heat capacities. In many text books, the entropy change is given by ds =
dq . T
(4.26)
4.5. THERMODYNAMIC ENTROPY LAW FOR UNIFORM STATES
81
It is emphasized that this equation may only be applied for reversible processes with DTd = 0. The differential equations, such as the Maxwell equations, derived using this definition of thermodynamic entropy change may also be used for reversible processes only. The left- and right-hand sides of (4.22) give the microscopic and macroscopic descriptions of the entropy changes. This equation shows that the entropy changes of the ideal gases with constant heat capacities may be calculated with the macroscopic thermodynamic variables at the initial and final equilibrium states provided. With the ideal-gas equation (2.22), the entropy variation may also be represented by ds = d(cp ln T − R ln p) .
(4.27)
As in (4.23), ds is a perfect differential for the ideal gases, and so the thermodynamic entropy is a state function. Using the definition of potential temperature for ideal gases given by (2.55), (4.23) and (4.27) can be rewritten as ds = cp d ln θ or
(4.28)
θ2 . θ1 These two equations tell that potential temperature of the ideal gases is also a state function. Since the heat energy may change in the reversible adiabatic process with conservation of thermodynamic entropy, the entropy change may not be found simply by comparing the heat energy between two equilibrium states of a system at different pressures. There are no the simple relations between increases of the entropy and temperature or volume as claimed in some text books (e.g., Young and Freedman, 1996). To compare the heat energy, the system should be justified to a constant reference pressure in order to exclude the change of heat energy in the reversible adiabatic process. Thus, the right-hand side of (4.28) includes potential temperature instead of temperature. According to the energy equation of ideal gases (2.35), the heat exchange at constant pressure may be measured by the change of enthalpy. Since the state equation for real gases is different from ideal-gas equation and the heat capacities of real gases are not constant, the thermodynamic entropy defined for ideal gases may not be a state function again for real gases. If we are looking for a state function to represent the thermodynamic entropy of real gases or real substances, the expression may be different from that of ideal gases. The different thermodynamic entropies for different physical systems are represented by different probability functions in the statistical thermodynamics or statistical mechanics. The thermodynamic entropy of classical thermodynamic system is represented by the Maxwell-Boltzmann statistics. In the quantum world, the entropies of photon gas and electron gas are represented by the Bose-Einstein statistics and Fermi-Dirac statistics, respectively (Sears and Salinger, 1975). Since T > 0 and DTd ≥ 0, the integration on the right-hand side of (4.24) cannot be negative, and (4.24) tells that molecular diffusions are a source of thermodynamic entropy. The entropy increases as ordered kinetic energy is converted into disordered kinetic energy or heat energy. Thus, we have ∆S ≥ 0 for the diffusion processes s2 − s1 = cp ln
82
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
in isolated systems. Since molecular diffusions are irreversible in any substance isolated, the previous consequence is generally true for all substances, and gives the mathematical expression of the thermodynamic entropy law: Thermodynamic entropy cannot be destroyed in an isolated thermodynamic system. A process with ∆S > 0 is irreversible, because the reversed process has ∆S < 0 and so cannot occur in isolation. The processes without molecular diffusions are called the reversible processes since ∆S = 0. Although it is impossible for us to predict the system change in a nonequilibrium process, the entropy law tells us at least the tendency of system variation. A new equilibrium state evaluated may really occur in isolation only if the thermodynamic entropy does not decrease in the process. The thermodynamic irreversibility is associated with the irreversible conversion of diffusion kinetic energy into heat energy and downgradient transport discussed in Chapter 3. Thermodynamic entropy change caused by downgradient transport only without diffusion kinetic energy dissipation will be discussed in the next section. Not just the thermodynamic entropy but also the thermodynamic state or all the thermodynamic variables change irreversibly in a single irreversible process. The value of a thermodynamic variables may either increase or decrease in the process, and only the entropy increases always in an isolated system. Also, the entropy is a state function, but the other thermodynamic variables may be not. So, the irreversibility can be indicated conveniently by the entropy change. While, another single thermodynamic variable (except the potential temperature for ideal gases) may be unable to tell the variation direction of a system. This does not mean that only the thermodynamic entropy changes irreversibly in an irreversible process. In mathematics, changes of a state function depend only on the initial and final states provided and are independent of the processes. Thus, the change of thermodynamic entropy in a continuous process whether reversible or irreversible is the difference of the entropy at the two ends of the process, that is B A
It gives
ds = sB − sA .
ds = 0 . in a continuous cycles of thermodynamic state. The continuous process usually means the reversible process, since an irreversible process is generally discontinuous. It is learned that thermodynamic entropy is conserved in isolated processes. That the entropy of an isolated system is conserved over a cycle may not confirm that the entropy is a state function. For calculating thermodynamic entropy of ideal gases, the thermodynamic state may be represented by the potential temperature only as shown by (4.28). It is discussed in Chapter 2 that ideal gases do not possess thermopotential energy and so have no latent heat. In other words, they may not change the gas phase. The water vapor in the atmosphere may also be considered as an ideal gas unless in the processes of phase transition. When the vapor changes phase, the changes of volume and pressure with temperature are discontinuous, and the ideal-gas equation cannot be applied. In this situation, the thermodynamic state cannot be represented by potential temperature only, and the thermodynamic entropy in a cycle of potential
4.6. THERMODYNAMIC ENTROPY CHANGE OF NON-UNIFORM STATE
83
temperature change may have a net production. The examples can be found in the studies of wet air engines in Chapter 22. It is not proved in this study that the thermodynamic entropy defined as dQ/T is also a state function for the substances other than ideal gases. The proof may be found in the text books of classical thermodynamics, which is based on the two consequences (i.e., Sears and Salinger, 1975): i) All Carnot engines working between two provided heat reservoirs have an equal efficiency represented by the ratio of the net heat accepted to the Kelvin temperature of the warmer reservoir, and ii) the ratio dQ/T vanishes over any reversible cycle as it may be represented by infinite number of Carnot cycles. It will be argued in Chapter 21 that the classical proof is not convincing, since it applied implicitly the assumption that all Carnot engines are reversible. This assumption is not true, as the heat exchanges may take place in nonequilibrium processes, and the entropy changes in the environments should be considered also for an open system. When the working substance is in thermodynamic equilibrium with the reservoirs over a Carnot cycles, different substances working over the same pressure ranges may not produce an equal amount of mechanic work.
4.6
Thermodynamic entropy change of non-uniform state
The thermodynamic entropy law is derived in the preceding section according to the irreversible conversion of diffusion kinetic energy into internal energy. The entropy increases as the internal energy increases in this situation. However, the diffusion kinetic energy is not the only source of thermodynamic entropy in an isolated system, since the irreversible changes may also be produced by the downgradient transport discussed in the preceding chapter while the internal energy is conserved. For example, when a non-uniform gas reaches the uniform equilibrium state by heat conduction or scattering radiation without mass displacement, no diffusion kinetic energy is transfered to internal energy but the downgradient heat conduction is irreversible. It will be found in Chapter 9 that the entropy may also increase as the atmosphere produces kinetic energy. To calculate the entropy for a non-uniform state, we divide the system into J elements of volume. When the mean potential temperature over the element with mass m changes from θ¯ to θ¯ , the change of thermodynamic entropy of the system may be calculated by adding the entropy change of each element, giving ∆S = cp
θ¯j mj ln ¯ . θj j=1 J
(4.29)
The mean potential temperature is used since the potential temperature of an element may not be uniform at a nonequilibrium state. The downgradient diffusive or conductive fluxes of mass, momentum and heat may destroy the thermodynamic gradients eventually as discussed in Chapter 3. According to the Stefan-Boltzmann law (Brunt, 1944; Hess, 1959), the radiation over the whole spectrum is proportional to the fourth power of temperature, and also
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
84
destroys the temperature gradient at the radiation equilibrium. Thus, the heat conduction or scattering radiation leads the system to the thermodynamic equilibrium state viewed over a certain scale, in which the macroscopic thermodynamic state cannot be changed again by molecular diffusions or collisions at constant pressure. It is noted that the microscopic fluctuations of thermodynamic variables may still exist in the equilibrium state, as the system is not microscopically continuous. The number of collisions for a nonequilibrium system becomes equilibrium is studied in the non-uniform gas theory. The molecular motions at an equilibrium state may be considered isotropic, as they do not produce detectable net transport. The equilibrium state of gases, which are free of external Newtonian forces, is uniform in thermodynamic variables. We consider here the entropy change caused by the change of system disorderliness only without dissipation of diffusion kinetic energy. So, we assume that the initial non-uniform state is uniform in mass density but not in temperature. The mean temperature of a single element may be changed by heat conduction, scattering radiation and random molecular exchanges with other elements. While, the volume of whole system does not change in this process, and so the heat energy or mean temperature or potential temperature of the system is conserved. The potential temperature of the equilibrium state may be calculated from J 1 mj θ¯j , θ¯j = M j=1
where M = yields
J
j=1 mj
is mass of the system. Applying it for the previous equation J
∆S = cp M ln M
¯
j=1 mj θj J M ¯mj θj j=1
,
(4.30)
in which Jj=1 denotes the product of J quantities. It may be proved in mathematics (Beckenbach and Bellman, 1965) that J J mj 1 mj θ¯j ≥ θ¯jM M j=1 j=1
(4.31)
for θ¯j > 0. The equality may be applied if all θ¯j are equal. Thus, thermodynamic entropy is created as the non-uniform system reaches the equilibrium state, though no diffusion kinetic energy is dissipated and the heat energy or mean potential temperature of the system is conserved. This entropy change is caused simply by the change of macroscopic disorderliness in the thermodynamic state. Therefore, the entropy may also be regarded as a measure of the disorderliness or randomness. It may also represent the ability of system variation, as the diffusions which are the causes of system variation depend essentially on the disorderliness. At the first glance, the disorderliness is similar to the inhomogeneity. But usually, the disorderliness increases with decreasing the inhomogeneity, and a high inhomogeneity implies a low disorderliness of the system. The disorderliness reaches the highest level as an isolated system becomes equilibrium. In this process, the thermodynamic entropy rises to the maximum when the mass and energy are conserved.
4.6. THERMODYNAMIC ENTROPY CHANGE OF NON-UNIFORM STATE
85
The change of disorderliness may also be applied to explain the entropy change as a system changes between two equilibrium states. When the system is in thermodynamic equilibrium with environments, the reversible adiabatic process of the system does not change the entropy of the system and environments, and so the disorderliness of the system or environments does not change. While in the nonequilibrium processes, such as the isobaric heating or free expansion of a gas, the thermodynamic state changes discontinuously across the system boundary. Thus, we may take the system and surroundings as an expanded isolated system. The changes of disorderliness in the expanded system can be found immediately by calculating the thermodynamic entropy changes. The study of thermodynamic entropy change in a non-uniform system is particularly important as most physical and non-physical systems including the Earth’s atmosphere and oceans are not uniform in an equilibrium state. The entropy of these inhomogeneous systems may be change by redistribution of a conservative quantity or parameter resulting from transport or communications. This is discussed in Chapter 6. The entropy changes between two equilibrium states may also be calculated from (4.30) with J = 1. When the mean potential temperature in this equation is replaced by a statistical parameter related to the features of microscopic particles, the entropy may be applied to study the microscopic processes such as the quantum statistics. The heat conduction is a transport process discussed in the preceding Chapter. The local change rate of thermodynamic entropy caused by heat conduction in a non-uniform system may be calculated from 1 ∂s = − vhc · ∇T , ∂t T
(4.32)
where vhc is called the heat conduction velocity and can be estimated from Fhc = cπ ρT vhc , where Fhc is the conductive heat flux of which the x-component is given by (3.38). The velocity in the x-direction gives
vhcx = −l
2R ∂T . 3πT ∂x
√ It is proportional to the gradient of T and in the downgradient direction. Thus, the entropy produced is positive as shown by (4.32). Analogously, the entropy created by molecular diffusion is evaluated from 1 ∂s = − vd · ∇T . ∂t T
(4.33)
It is also positive since the heat diffusion is also in the downgradient direction of temperature. The thermodynamic entropy is different from other physical parameters, as it depends on the distribution as well as the local quantity. It is not entirely transportable. The downgradient transport increases the entropy. This will be discussed
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
86
again in Chapter 26. The entropy changes in the internal transport processes discussed are caused by destruction of the gradients, or by the local heating and cooling at different temperatures in the process instead of by entropy transport. Thus, the entropy change caused by heat conduction is included in the heat exchange term of (4.25) together with the scattering radiation. While, the entropy change caused by molecular diffusion is in the diffusion term of (4.25). The reason will be given in the next chapter. When the local changes produced are integrated over a closed system, the integrated entropy of the system increases as discussed earlier, even though no transport takes place across the system boundary. If viewed microscopically, molecules in a fluid do not possess an equal amount of kinetic energy for each motion status even if in the equilibrium state, as shown by the Maxwell-Boltzmann distribution function (Sears and Salinger, 1975). The disorderliness of a system may also be represented by the microscopic state or the non-uniform distributions of particles and their energies. Since a thermodynamic system leads to increase its disorderliness, the states of different degrees of microscopic disorderliness possess different probabilities of occurrence. Thus, the entropy may also be studied with the probability theories based on the statistics of parcel distributions and energy partitions. For the classical thermodynamic systems, the thermodynamic probability and the entropy are illustrated by the Maxwell-Boltzmann distribution of particles at different energy levels. While in the quantum mechanics, the entropy is represented by the Bose-Einstein statistics for the systems in which the particles are indistinguishable, or by the Fermi-Dirac statistics for the systems in which the particles follow the Pauli exclusion principle (Sears and Salinger, 1975). It can be found in Chapter 6 that the thermodynamic entropy of ideal gases defined as a thermodynamic probability is consistent with that defined by the classical thermodynamics, for both of them can be applied to derive the same partition functions for the distributions of particles with different energies.
4.7
Inadditive and scale-dependent features
The dynamic entropy discussed in Chapter 2 is additive, as it depends on the dynamic entropy of each sub-system, but not on the relations between different subsystems. If we can define the thermodynamic entropy for a single molecule, the molecular entropy represents the feature of a particle and is independent of other particles. Thus, the total molecular entropy of a system is not changed by redistribution of the particles, if the entropy of each molecule is conserved. In other words, the sum of molecular entropy does not represent the disorderliness of a system. This can be shown by the following relationship I i=1
ln θi =
I
ln θI−i+1 ,
(4.34)
i=1
in which θ denotes the potential temperature of a molecule (if defined) with mass m, and the prime indicates the variable at the final state. This equation tells clearly that the sum of molecular entropy is independent of molecular distribution. Particularly,
4.7. INADDITIVE AND SCALE-DEPENDENT FEATURES
87
the previous equation gives cp Im
I
ln
i=1
θi =0 θi
(4.35)
for θi = θi . This means that the sum of molecular entropy is conserved if molecular potential temperature of each molecule is unchanged, although the disorderliness of the system is changed by redistribution of the molecules at different energy levels. To measure the disorderliness of a system, the entropy is calculated with the thermodynamic variables, which represent the mean features of microscopic molecular thermodynamic state averaged over an element of volume. Thus, the potential temperature in (4.28) is given by the mean value of the element, and we have θ¯j d ln θ¯j = ln ¯ θj for element j. If the element has mass δM , its thermodynamic entropy change is given by δMj δsj = cp δMj d ln θ¯j or
θ¯j δMj δsj = cp δMj ln ¯ θj
(4.36)
for ideal gases. This equation shows clearly the statistical feature of thermodynamic entropy. In general, the molecules do not in an equal energy level even if in the equilibrium state, so the entropy evaluated with the mean potential temperature is different from the sum of molecular entropy since δM ln θ¯ = m
ln θi .
i=1
We may find from the inequalities given in the preceding section that the entropy is greater than the sum of molecular entropy, because the element is assumed uniform for the calculation. This does not mean that the entropy evaluated from (4.36) is an approximation for the sum of molecular entropy, as they are different physical quantities illustrating different properties of the system. The sum of molecular entropy represents only an integrated quantity, while the thermodynamic entropy express the feature of the quantity distribution. Therefore, unlike the dynamic entropy, thermodynamic entropy is not additive and so should not be defined as a sum of the entropy of each molecule. The downgradient transport caused by molecular diffusions may destroy the gradients between the elements in a fluid system, so the disorderliness manifested by the mean thermodynamic variables over each element is increased by particle exchanges across the boundaries of elements. Since the mean potential temperatures of these elements are changed by the particle exchanges, the entropy change of the system evaluated from (4.29) or ∆S = cp
θ¯j δMj ln ¯ θj j=1 J
(4.37)
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
88
is different from zero, even if potential temperatures of single molecules are conserved. However, the total molecular entropy is conserved in the process as shown by (4.35) or Ij J
ln
j=1 i=1
θi =0. θi
Thus, the sum of molecular entropy cannot measure the disorderliness of a system. The system randomness may be illustrated graphically by the molecular features, but the quantitative measurement in the classical thermodynamics depends on the macroscopic variables representing the mean features of molecular assemblies. This is a limitation of the classical thermodynamic entropy represented by thermodynamic variables, as discussed in Chapter 6. The statistical thermodynamics re-defines the thermodynamic entropy in terms of the molecular features, so it is able to reveal the microscopic features of the system. When δM is infinitely small, the sum in (4.37) may be replaced by integration, giving
∆S = M
δS = cp
θ¯ ln ¯ δM . θ M
Here, δM is the mass of fluid element instead of a single molecule. Since the temperature measured in practice is actually the mean temperature in a scale as small as that of thermometer, and the temperature of a single molecule is meaningless, the overbar for the mean value is omitted usually. For the systems with uniform potential temperature in all scales, evaluations of thermodynamic entropy are independent of the element size, since the mean potential temperatures are equal in the elements of different sizes. But for the non-uniform states, the statistical features, such as the mean potential temperature, depends on the number of samples or the size of element, and so does the thermodynamic entropy evaluated. This agrees with the fact that the randomness of a non-uniform state depends on the scale of measurements, and the samples in a certain scale may not reveal the inhomogeneities or gradients in smaller scales. The different values obtained for different sizes are not errors, but are the entropy over the different sizes respectively. In general, the larger the size is, the higher the entropy will be, because the randomness or disorderliness increases with the size on which the mean potential temperature is obtained. If expressed mathematically, we have S2 > S1 , if δV2 > δV1 ,
or
δm2 > δm1 .
So the sum of molecular entropy gives the lower limit of the thermodynamic entropy and the thermodynamic entropy of the uniform state gives the upper limit. The in-additivity of thermodynamic entropy is referring to, in this study, the entropy of the basic elements of which the distribution manifests the system disorderliness. The entropy was also considered to be additive by some authors, for example Gyftopoulos and Beretta (1991), according to (4.29) or (4.37). However, the entropy so evaluated for nonequilibrium states depends on the size of element, and so different results can be obtained when a non-uniform system is divided into
4.8. THERMODYNAMIC ENTROPY BALANCE EQUATION
89
different sets of subsystems. The scale-dependent feature of thermodynamic entropy was ignored, because the statistical features of thermodynamic variables were not emphasized in calculating the entropy for macroscopic thermodynamic systems, and only the entropy changes between equilibrium states were considered in the classical thermodynamics, while the microscale gradients and fluctuations in the equilibrium states were ignored. If the processes between nonequilibrium states are considered, the classical thermodynamic entropy law discussed previously should be stated more generally as: The thermodynamic entropy in all microscales cannot be destroyed in an isolated thermodynamic system. For the gases studied in the classical thermodynamics, of which the disorderliness may only be represented by the distribution of molecules, the entropy changes in different scales may not be independent of each other.
4.8
Thermodynamic entropy balance equation
The heat exchange given by the last integration in (4.21) can be applied for open systems. We discuss in this section only the open systems with heat exchanges but not mass exchanges with the surroundings. The entropy variations of the open systems are given by (4.25). The discussions here give actually the entropy balance equation of the open systems. The balance equation for the open systems with mass exchanges will be discussed in Chapter 26. It is discussed earlier that diffusion term in (4.25) includes the irreversible entropy sources related to diffusion kinetic energy dissipation and molecular diffusion of heat, that is DTd dm , ∆Skd + dShd = M T where 1 ∂kd dt dm ∆Skd = − M ∆t T ∂t is called the kinetic energy dissipation entropy derived from (4.13), and ∆Shd = −
∆t
M
vd · ∇ ln T dt dm
(4.38)
is called the heat diffusion entropy derived from (4.33). The integrations are used because the variables depend on position in the nonequilibrium states. Now, (4.25) is replaced by δq dm , ∆S = ∆Skd + ∆Shd + M T where δ indicates the change at a local position. It is discussed also that the entropy change caused by heat exchange includes also the irreversible component ∆Shc = −
M
∆t
vhc · ∇ ln T dt dm
(4.39)
derived from (4.32). It is produced by heat conduction and called the heat conduction entropy. Resolving this irreversible entropy source from the heat exchange term yields (4.40) ∆S = ∆Si + ∆Sh ,
90
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
where ∆Si = ∆Skd + ∆Shd + ∆Shc
(4.41)
is called the internal thermodynamic entropy production, and
∆Sh =
M
δq dm T
is the entropy change caused by heat exchanges without heat conduction. When mass exchanges take place in an open system, the internal entropy changes may not mean the entropy change inside the system mass. It represents the entropy changes related to irreversible microscale processes which may not be represented directly by thermodynamic variables. Applying (4.28) for it produces
∆Si = cp M
δ ln θ dm − ∆Sh
for the ideal gases with constant heat capacities. Since the kinetic energy dissipation, heat conduction and heat diffusion always create thermodynamic entropy, we have ∆Si ≥ 0 . The variation tendency of an open system may also be indicated by thermodynamic entropy changes. When heat exchanges take place, the system may not be in thermodynamic equilibrium with the environments, and so the exchanges across system boundary are irreversible. If the open system and environment are viewed as a new isolated system, the sum of the entropy changes in the system and environment should not be less than zero, that is ∆Ssys + ∆Senv ≥ 0 , where subscript ‘env’ indicates a variable of environment. In this equation, only the environmental entropy change caused by the heat exchanges is accounted, that is ∆Senv = −
M
δq dm . Tenv
The entropy changes in the system and environment may be calculated separately by assuming different reversible processes for the system and environment respectively. Some examples of calculating the entropy changes for the air engines working in the atmosphere will be given in Chapters 21 and 22. When the heat is conserved, the heat ejected or accepted by the system is equal in amount to that accepted or ejected by the environment. While, the entropy changes produced in the system and environment may be different, as the system and environment have different temperatures.
4.9
Relation to dynamic entropy
If molecules in an ideal gas are viewed as independent Newtonian systems, they form the isotropic system discussed in Section 2.3. The dynamic entropy of the system is
4.9. RELATION TO DYNAMIC ENTROPY
91
represented by (2.6), of which the perfect differential gives
dsd = d ln
∆r Fdr +
k0
k
for unit mass. It follows that Fdr dk . − k ∆r Fdr + k0
dsd =
As discussed in Chapter 2, the friction or inelasticity of molecular collision may cause conversion of translational kinetic energy of molecules into the rotational or vibrational kinetic energy, but not change the total amount of molecular kinetic energy. Thus, we have ∆r
Fdr + k0 = k
and so
Fdr − dk , k where Fdr = dWF is the work done by the external force. From (2.4), we see dsd = 0 or Fdr − dk = 0 . dsd =
The dynamic entropy variation is zero in the ideal gases. Inserting k = kd + kH into the previous equation yields dkH − Fdr = −dkd .
(4.42)
It is also the equation (2.28). If the change of thermodynamic entropy for unit mass is defined as ds = cv
dkH − Fdr , kH
(4.43)
we gain ds = −cv
dkd . kH
Applying (2.30) for it produces ds = −
dkd . T
(4.44)
As dkd ≤ 0, we have ds ≥ 0. Inserting (2.30) and (2.31) into (4.43) gives ds =
cv dT + pdα T
for isolated ideal gases. It follows that ds = d(cv ln T + R ln α) . This equation is the same as (4.23).
(4.45)
92
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
The right-hand side of (4.42) represents the macroscopic effect of molecular diffusions. This equation is similar to (2.4). However, the thermodynamic entropy changes of ideal gases are caused by redistributions of molecules and their momentum and energies, but not by friction of molecules. Thus, the thermodynamic entropy possesses different features from those of dynamic entropy. For example, the dynamic entropy is additive but the thermodynamic entropy is not. However, the redistribution of molecular momentum has a macroscopic effect similar to that of friction in the mechanic processes, that is to convert irreversibly the ordered kinetic energy (parcel and diffusion kinetic energy) into heat energy. Since the macroscopic changes of momentum may be illustrated as an effect of external forces in the mechanic dynamics, we introduce the viscous force in the fluid dynamics to account for the macroscopic kinetic energy dissipation. The frictional force provides a phenomenal link between the thermodynamic entropy and dynamic entropy.
4.10
Calculations for ideal gases
It is generally difficult to detect and predict precisely the molecular diffusions, so we do not use (4.25) to calculate the thermodynamic entropy change in a diffusion process. Since the thermodynamic entropy of ideal gases is a state function, its variations depend on initial and final states only and is independent of process for the initial and final states provided. The processes with different diffusions may lead to different final states. As long as the initial and final states are given, the entropy change integrated along any process gives an equal result. Thus, we may choose a set of successive reversible processes to connect the initial and final states. However, to know whether a process is reversible or not, we may need to calculate the entropy change first. Thus, the reversible processes are chosen empirically as the slow continuous process in which the system is in quasi-equilibrium with the environments. For convenience, these processes are usually selected under special conditions, such as no pressure change or no temperature change. In many cases, more than one reversible processes are applied. In the reversible processes, we have dkd = 0 and (4.25) gives
∆srev =
N δqrevj j=1
Trevj
(4.46)
for the N reversible processes assumed. Here Trevj is the system temperature in the j-th reversible process which is independent of space, and δqrevj is heat exchange of the system with the exterior in the process assumed. This heat exchange may be different from that in the real process. An isolated system may not be isolated again for assumed reversible processes. The previous equation shows that thermodynamic entropy is conserved in reversible adiabatic processes. It can be used conveniently for isothermal processes, if the heat exchange is known. But in general, the temperature changes as a system exchanges heat with the exterior. If the temperature and heat exchanges may be represented by Trevj (t) and δqrevj (t) respectively, the entropy
4.10. CALCULATIONS FOR IDEAL GASES
93
variation can be evaluated by integrating (4.46) with time, that is ∆srev =
N
∆srevj
j=1
with ∆srevj =
tj dqrevj tj−1
Trevj
dt .
However as discussed earlier, the classical thermodynamics does not provide the predictions generally. Since the analytical expression of internal energy of real substances is unknown, we have no the general expression of thermodynamic entropy change represented by thermodynamic variables for all substances. The entropy equations (4.23) and (4.27) are derived for the ideal gases with constant heat capacities. If we know the equilibrium initial and final states, the entropy change for the ideal gases in a continuous process can be calculated easily with (4.23) or (4.27), which is rewritten as T2 α2 + R ln (4.47) ∆s = cv ln T1 α1 or T2 p2 − R ln . (4.48) ∆s = cp ln T1 p1 If a process is discontinuous, we may assume N reversible processes to connect the same initial and final states. The entropy change in these reversible processes is calculated from N Trevj αrevj cv ln + R ln ∆srev = Trevj−1 αrevj−1 j=1 or ∆srev =
N j=1
Trevj prevj cp ln − R ln Trevj−1 prevj−1
The 0 and N of j indicate the initial and final states, respectively. These equations can be used conveniently if the reversible processes are assumed at constant pressure or temperature or volume. The entropy changes evaluated from the last two equations are equivalent to those obtained from the previous two equations for continuous processes. For example, when an ideal gas of unit mass expands adiabatically to the environment with constant pressure pe notably lower than the initial gas pressure p1 , it accepts no heat or dq = 0. The thermodynamic entropy change in the irreversible adiabatic process cannot be evaluated from (4.26), as it is actually given by (4.25) or (4.44). Since the diffusion is unknown, the entropy change may be calculated from (4.47) or (4.48). We may solve firstly the final thermodynamic state, since the pressure of the state is already known as pe . The thermodynamic energy law (4.15) gives (4.49) cv ∆Tequ + pe ∆αequ = 0
94
4. PREDICTABILITY AND THERMODYNAMIC ENTROPY
for the irreversible expansion. Applying the ideal-gas equation (2.22) for it yields pe cv (T2 − T1 ) + RT2 − RT1 = 0 . p1
It follows that
T1 pe cv + R cp p1 The entropy change evaluated from (4.48) gives
T2 =
∆s = cp ln
.
T2 pe − R ln T1 p1
= cp ln
cv pe +κ cp p1
p1 pe
κ
.
(4.50)
It is greater than zero as p1 > pe . We may also assume the following two reversible processes for the calculation: The gas expands reversibly and adiabatically from the initial pressure to pressure pe , then it is heated at the constant pressure to reach the final temperature T2 . In the process of isobaric heating, the gas is in thermodynamic equilibrium with the heat source in the surroundings. The temperature at the end of reversible expansion is κ R pe , κ= T1 = T1 p1 cp obtained from (4.18). The entropy created is zero in the adiabatic process, and is ∆srev = cp ln
T2 T1
in the reversible process of isobaric heating. The result is identical to (4.50) or ∆s = ∆srev . For T1 = 300 K, p1 = 1020 hPa and pe = 1000 hPa, we see T2 = 298.31849 K, T1 = 298.30658 K and ∆s = 0.04 J/(K.kg). The real final temperature T2 is higher than T1 as less mechanic work is created in the irreversible adiabatic expansion than in the reversible adiabatic expansion. The second process assumed is not adiabatic again. The thermodynamic entropy is produced by heat exchange between the gas and environment in the heating process assumed, which is equivalent to the entropy produced by molecular diffusion in the real adiabatic process. The gas accepts the heat cp (T2 − T1 ) in the reversible processes assumed, as it produces more mechanic work than in the real process. The entropy change may also be calculated from (4.28). The potential temperature of the gas is θ1 = T1 at the initial state, but increases to θ2 = T2 at the final state. It is noted that to choose a reversible process for calculating the entropy change may only be used for equilibrium systems. The changes of thermodynamic entropy in non-uniform systems cannot be calculated in this way, but may be calculated by using (4.30). As change of thermodynamic entropy is independent of process for provided initial and final states, the ∆S in (4.40) may be replaced by dsrev for unit mass. This equation may be rewritten, from (4.46), as ∆si =
N δqrev j=1
Trev
−
dq . T
(4.51)
4.10. CALCULATIONS FOR IDEAL GASES
From ∆si ≥ 0, we have
95
N δqrev j=1
Trev
≥
dq . T
(4.52)
It is an important inequality in the classical thermodynamics. The ratio of heat exchange (which does not include the dissipation of ordered kinetic energy) to the system temperature is maximum in reversible processes. In other words, the entropy created by heat exchange is maximum in reversible processes for provided initial and final equilibrium states, as no contribution is made by kinetic energy dissipation. The Earth’s atmosphere is a kind of thermodynamic system. A fundamental feature of the atmospheric variations is irreversibility. The atmosphere never repeats a circulation pattern over any scale in an isolated circumstance. This irreversibility is not simply due to molecular diffusions. The large-scale circulations are also irreversible, although the classical thermodynamic entropy, represented by the potential temperature in meteorology, may be conserved in adiabatic processes. Usually, we do not choose reversible processes for calculating thermodynamic entropy changes in the irreversible processes over a large scale, since the large-scale atmospheric variations are generally slow. The slow adiabatic processes may be considered approximately to be reversible from the point of classical thermodynamics. This implies that the irreversibility of atmospheric circulations may not be interpreted by the classical thermodynamic entropy variations. This irreversibility will be discussed by introducing a new thermodynamic entropy called the turbulent entropy in Chapter 6.
Chapter 5 Newtonian-thermodynamic system 5.1
Introduction
We have discussed in Chapter 2 two types of the simplest physical systems, the Newtonian dynamic system and classical thermodynamic system. Changes in the motion status of Newtonian system are produced by external Newtonian forces. The conversion of kinetic energy into heat energy caused by frictional force is unidirectional, so the kinetic energy lost to friction may not be recovered again in a mechanic system without mechanic energy exchange with the exterior. Meanwhile, changes of thermodynamic state of gases are caused by molecular diffusions, which are the macroscopic effect of microscopic random molecular motions at a nonequilibrium state. Motions of individual molecules may be affected by external forces, while the statistical effect of molecular motions and collisions may not be associated with macroscopic Newtonian forces, and is controlled by the second law of thermodynamics. To study the nonequilibrium thermodynamic processes, the nonequilibrium thermodynamics introduced the generalized forces related to the microscale gradients of thermodynamic variables. The generalized forces may not be applied for Newton’s laws, and so are not regarded as the Newtonian forces. The air in the Earth’s atmosphere will be assumed as an ideal gas in this study, since changes of the heat capacities and thermopotential energy are negligibly small at the normal conditions. The ideal-gas equation and thermodynamic energy law of ideal gases may also be applied to study the atmospheric processes. As in the classical thermodynamics, the molecular diffusions in the atmosphere are also unpredictable. However, they are not as important as the parcel processes for changes of large-scale atmospheric circulations. Thus, the atmospheric dynamics and energetics deal with mainly the processes related to the parcel motions on meteorological scales, which may include the moist effects resulting from phase transitions of water in the atmosphere. Since the atmosphere is compressible, parcel motions in the vertical direction or across the isobaric surfaces may lead to changes in the thermodynamic states. Owing to the thermodynamic processes, exchanges of mechanic energy and internal energy in the atmosphere take place in both directions through expansion and contraction of air parcels, so the parcel kinetic energy lost to boundary friction and fluid viscosity can be recovered again through thermodynamic processes. Therefore, the atmosphere is neither a simple Newtonian system nor a classical thermodynamic system, but is a combination of the both. The changes of kinetic status can be studied by Newton’s laws, and the changes of thermodynamic state are studied by thermodynamic laws. So, the governing equations of atmospheric processes, called the primitive equations, include all the physical relationships adopted for the studies of Newtonian systems and classical thermodynamic systems. The energy equations for the individual air parcels and whole atmosphere may be derived using all the equations as shown in Chapter 7. They include the mechanic energies 96
5.2. FIELD VARIABLES
97
and thermodynamic energies, and so are different from the energy conservation laws used for a mechanic system or classical thermodynamic system. Under the effects of Earth’s gravitation and rotation, the atmosphere, as a thermodynamic system, is different from the classical thermodynamic systems. These differences have not been revealed adequately in the previous studies of atmospheric thermodynamics and energetics. The steady states of rotational atmosphere manifest the inhomogeneities in the vertical and horizontal directions. The inhomogeneous steady states will be referred to as the inhomogeneous equilibrium states. The thermodynamic systems exhibiting the inhomogeneous equilibrium states will be called the inhomogeneous thermodynamic systems. The thermodynamics dealing with these systems is then referred to as the inhomogeneous thermodynamics. It is different from the nonequilibrium thermodynamics, since nonequilibrium thermodynamics deals with mainly the nonequilibrium processes of classical thermodynamic systems of which the equilibrium states are uniform. In the gravitational field, the gradient of pressure or mass density produces a particular internal force in the fluid called the pressure gradient force. This force is also an external force for a fluid parcel. The resultant variations in fluid motions can be studied with Newtonian dynamics. An incompressible fluid like the ocean water is similar to a simple Newtonian system, except that the external forces for parcels may change with thermodynamic state of the system. Motions of incompressible fluid are independent of thermodynamic processes, so the parcel kinetic energy cannot be created inside the fluid if without affected by external mechanic processes. This is the major difference from the processes in a compressible fluid such as the Earth’s atmosphere. The atmospheric processes, in which kinetic energy generation is critical, should not be studied with incompressible approximation.
5.2
Field variables
Due to the gravitational force of the Earth, the atmospheric pressure at a geographic height z indicates also the weight of an steady air column of unit horizontal section area above the height, represented by the hydrostatic equation Z
ρ dz ,
p=g
(5.1)
z
where Z is the top height of the atmosphere. It tells that the local pressure changes with height. The density of a large-scale air mass in equilibrium decreases exponentially with height in the gravitational fields as discussed in Chapter 17. Owing to the input of long wave radiations from the Earth’s surface and the output from the top of atmosphere, the atmosphere exhibits a vertical gradient in the temperature profile also. These vertical gradients cannot be destroyed entirely by molecular diffusions, and do not suggest a nonequilibrium state of the atmosphere. The solar radiation and Earth’s longwave radiation accepted by the atmosphere vary also with the latitude, cloud coverage, surface temperature and albedo, etc.. The temperature, humidity, friction, albedo and evaporation on the surface together with the surface fluxes of sensible and latent heats may change greatly from ocean surfaces to continents. The air masses over oceans and continents or at low and
5.
98
NEWTONIAN-THERMODYNAMIC SYSTEM
high latitudes may be significantly different from each other in their mechanic and thermodynamic features. There are the boundaries with relatively strong contrast of thermodynamic states between the large-scale air masses. Due to rotation of the Earth, the large-scale atmospheric circulations are constrained by geostrophic balance (See Section 5.6), and the ageostrophic component is usually one order smaller than the geostrophic component. Mixing of air masses across the pressure gradient and boundaries resulting from ageostrophic motions is then prevented by the geostrophic balance, so that the boundaries of air masses, like the temperature fronts, may sustain for many days in the geostrophic circulations. Therefore, the atmosphere manifests the large-scale inhomogeneities in the horizontal directions even in the climatological mean fields. The theoretical studies of Paltridge (1975, 1978; Rodgers, 1967; Noda and Tokioka, 1983) show that the climatologically equilibrium state of the open atmosphere with minimum meridional exchanges of heat or thermodynamic entropy manifests the vertical and meridional gradients of thermodynamic variables. To study an inhomogeneous fluid system, we use the time-space coordinates to illustrate the variable distributions and variations or to locate the individual parcels. A thermodynamic variable in the time-space coordinates is a function of time and space, and is called the field variable. In general, the geographic height z of a isobaric surface in (5.1) changes horizontally, and so the pressure depends on the horizontal coordinates also. The variable p represents not only the pressure, but also the vertical position of an air parcel in the pressure coordinate. The parcels at different pressure surfaces possess different amounts of geopotential energy. The variation in geographic height of a parcel may lead to the changes of its thermodynamic state and motion status or the exchanges between its geopotential energy, kinetic energy and internal energy. In an inhomogeneous fluid system with fluid motions, most physical quantities are carried by fluid parcels. Assume that the quantity = is carried by a parcel of unit mass. The change of = in the region occupied by the parcel at a give time is denoted by ∂= δt . ∂t It may be produced by change of the parcel itself d= δt , dt and by replacement of the fluid ∂= δr ∂r across the region. Here, ∂/∂r is the directional derivative along the velocity at the time and place. The negative sign is added, so that the term is positive if the local value is increased by transport. Thus, the local change rate is given by −
∂= ∂t
= =
d= ∂= δr − lim dt ∂r δt→0 δt ∂= d= − vgl . dt ∂r
(5.2)
5.3. PARCEL AND PARCEL VELOCITY
99
Here, dr dt is the general velocity which include the mass velocity, conduction velocity or propagation velocity on all scales. The local change rate is measured at a given place, while the material change rate on the right-hand side of (5.2) is measured by following the general velocity. In a flowing fluid, the local changes are generally different from material changes, unless the fluid is homogeneous so that a particular parcel may not be distinguished from others. The second term on the right-hand side is called the advection produced by fluid motion, conduction or propagation across the gradient. Referring to (4.5) finds ∂= = vgl · ∇= . vgl ∂r The material change rate gives now vgl =
∂= d= = + vgl · ∇= . dt ∂t
(5.3)
It is similar to (4.6) except that the general velocity includes the moving velocity of macroscopic fluid parcels, of which the variations are controlled by Newton’s second law as discussed in the next section. The previous equation may be rewritten as ∂= d= = + ∇ · (=vgl ) − =∇ · vgl . dt ∂t The second term on the right-hand side indicates the gain or loss caused by convergence of = conduction or propagation in the fixed region, which does not include the effect of mass convergence. The last term represents the correction by the volume or density change in the region. For example, when the density decreases resulting from the velocity divergence, a mount of the quantity is removed from the region. If the transport process is independent of mass displacement, the convergence of conduction or propagation velocity in the last term represents a kind of dispersion. For the nondispersive propagation or conduction velocity v which is independent of space, we have ∂= d= = + ∇ · (=v) . dt ∂t This equation may be found in many studies.
5.3
Parcel and parcel velocity
We discussed in Chapter 3 the instantaneous diffusion velocity of a diffusion element. The diffusion element is a small volume of gas. Molecules in a diffusion element keep exchanging with the surroundings. The diffusion velocity represents the statistical effect of molecular motions influenced by molecular collisions, which may not be the moving velocity of a fixed molecular ensemble. It may be changed by exchanging the molecules across the boundary when the effect of external Newtonian forces is
5.
100
NEWTONIAN-THERMODYNAMIC SYSTEM
ignored. We may define mathematically the trajectory of a diffusion element with the instantaneous velocity. However, the definition may not be physically significant, as the constituents of the element keep changing along the trajectory. Although motions of a single molecule are affected by external forces, the macroscopic effect of molecular diffusion may not have a general relation to macroscopic Newtonian forces, or meet with Newton’s second law. The fluid element of which the motion status is controlled by macroscopic Newtonian forces is called the parcel in this study. A parcel is large enough so that the mass and momentum variations caused by molecular diffusions across the boundaries or molecular collisions at the boundaries is negligibly small within a certain time. It is also small enough so that its thermodynamic state is considered to be uniform. Since molecular diffusions exist everywhere in an inhomogeneous fluid, a small parcel is a time-mean feature only. The fluctuations of density and other variables caused by molecular diffusions may be ruled out by making averages over the parcel scale. Motions in a fluid may then be represented by the motions of parcels, and may be studied by using the classical Newtonian dynamics and thermodynamics for individual parcels as in the Lagrangian dynamics. In the Eulerian dynamics, fluid motions are represented by the field of mass velocity covering each point continuously in the three-dimensional space. In general, the total mass velocity includes parcel velocity vp and instantaneous diffusion velocity vd , that is (5.4) v∗ = vp + vd . It is the velocity of mass displacement which may be caused by different mechanisms. The horizontal components of parcel velocity in the atmosphere are called the winds, and can be measured by meteorological instruments. Changes of parcel velocity are caused by Newtonian forces, such as the gravitational fore, pressure gradient force (referring to Section 5.5) and frictional force. While, changes of diffusion velocity follow the second law of thermodynamics and may be independent of macroscopic Newtonian forces. The effect of molecular diffusion on the local changes in a tiny-scale may be different from that of parcel motion. Assuming that an airflow in geostrophic balance is along the isobars in the rotational atmosphere, temperature variations at a local place may be brought about by parcel motions if there is a temperature gradient along isobars. While, the heat conduction resulting from molecular collisions takes place in the direction down to the microscale temperature gradient (seeing Chapter 3). Thus, the heat fluxes resulting from parcel motion and heat conduction may be in different directions. Since the microscale gradient of pressure can not be detected in practice, the identifications of mass diffusion and parcel motion may only be technic in many situations, or depend on the scale resolutions of data and calculation models. If only the mass velocity and conduction velocity are considered, the general velocity is given by vgl = vp + vd + vc , and the material change rate reads ∂= d= = + vp · ∇= + (vd + vc ) · ∇= , dt ∂t
(5.5)
5.3. PARCEL AND PARCEL VELOCITY
101
where (vd + vc ) · ∇= = D1d represents the macroscopic effect of the microscopic processes of molecular diffusions or collisions. When the microscopic processes are ignored, we have ∂= d= = + vp · ∇= . dt ∂t
(5.6)
The material change rates in the thermodynamic equations discussed in Chapter 4 may be expanded with (5.6) or (5.5). The expanded thermodynamic energy law is called the heat flux equation, and the expanded mass conservation law is called the continuity equation. The kinetic energy density of a fluid, called the specific kinetic energy usually, is defined for a moving fluid element in unit mass. This is also true for the kinetic energy of mechanic waves. Thus, only the mass velocities, but not the conduction velocities and propagation velocities, are used for calculating the kinetic energy. When the total mass velocity is resolved into components, the instantaneous kinetic energy is not simply the sum of kinetic energy of each component. This can be seen from m (vp + vd ) · (vp + vd ) 2 vp2 vd2 + + vp · vd , = m 2 2
K∗ =
where m is mass of the fluid element. The last term is different from zero if the parcel velocity and diffusion velocity are not perpendicular to each other. As the parcel is a mean feature, the parcel velocity may be considered as the mean mass velocity over the typical time-scale of parcel denoted by τ , that is 1 vp = τ
v∗ dt .
τ
The instantaneous diffusion velocity is the departure from the mean velocity, and we have vd dt = 0 . τ
Usually, the parcel velocity means the mean velocity in most studies, and so the diffusion velocity is ignored. The mean kinetic energy over the scale of parcel motion is given by vp2 v2 (5.7) K = m+ dm . 2 2 The first term on the right-hand side may be called the parcel kinetic energy, and the last term may be referred to as the diffusion kinetic energy. This equation shows that the effect of molecular diffusion on the kinetic energy cannot be ignored, even if the diffusion is anisotropic in a parcel. As conversion of kinetic energy into heat energy resulting from molecular diffusions is irreversible as discussed in the previous chapters, parcel kinetic energy cannot be conserved in a dynamically isolated system due to molecular diffusions.
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5.4 5.4.1
NEWTONIAN-THERMODYNAMIC SYSTEM
Mass and heat transport equations Continuity equations
If there are parcel motions in a fluid, we may use the total mass velocity v∗ for (4.8) and (4.9) to obtain the mass conservation equations: ∂ρ + ∇ · (ρvp ) + ∇ · (ρvd ) = 0 , ∂t or
dρ = −ρ∇ · vp − ρ∇ · vd . (5.8) dt They are called the continuity equations. The first equation shows that the local change of density is produced by divergences of mass fluxes carried by parcel motion and mass diffusion. The second equation tells that the change of parcel density is produced by divergences of parcel velocity and diffusion velocity. When the effect of diffusion is negligibly small compared with that of parcel motions, the continuity equations are replaced by ∂ρ + ∇ · (ρvp ) = 0 (5.9) ∂t and dρ = −ρ∇ · vp . (5.10) dt The latter is usually written as dα = α∇ · vp , dt
(5.11)
where α is the specific volume of the fluid. Unlike in the study of molecular diffusions in Chapter 4, the continuity equation is still meaningful if molecular diffusions are ignored. This equation together with the heat flux equation, state equation and momentum equation forms a complete set of prediction equations, called the primitive equations, for three thermodynamic variables and parcel velocity. The continuity equation is also used in the nonequilibrium thermodynamics to obtain a complete equation set. The fluids studied by nonequilibrium thermodynamics may not necessarily exhibit parcel motions even at a nonequilibrium state. In this case, the continuity equation should be replaced by the mass diffusion equation (4.9) or diffusive transport equation (4.8).
5.4.2
Integrated variations in a system
If only the transport carried by macroscopic mass velocity or parcel velocity is considered, we multiply (5.6) by mass density ρ and gain ρ
∂= d= =ρ + ρvp · ∇= . dt ∂t
Next, multiplying (5.9) by = gives =
∂ρ + =∇ · (ρvp ) = 0 . ∂t
5.4. MASS AND HEAT TRANSPORT EQUATIONS
103
Adding these two equations together produces
d= ∂(ρ=) dρ ρ = + ∇ · (ρ=vp ) − = ρ∇ · vp + dt ∂t dt
.
(5.12)
When the mass is conserved, the last term disappears according to continuity equation (5.10). Thus, we gain ρ
∂(ρ=) d= = + ∇ · (ρ=vp ) . dt ∂t
(5.13)
The left-hand side is the change of a unit volume. The mass in the moving volume may change. The first term on the right-hand side is the local change of = density in the volume at a given time including the change in mass or density, and the last term is the flux convergence in the element of volume. Integrating it over an atmospheric domain yields 1 g
ps d= A pt
dt
dpdA =
zt ∂(ρ=) A zs
∂t
zt
∇ · (ρ=vp ) dzdA ,
dzdA + A zs
where zs and zt are the heights of the bottom pressure surface ps and top pressure surface pt , respectively. The Gauss equation gives zt zs
∇ · (ρ=vp ) dzdA =
✵
ρ=vn d✵ ,
in which vn is the outward velocity component normal to the outer surface ✵ of the domain integrated in Cartesian coordinates. Thus, we have 1 g
ps d=
dt
A pt
dpdA =
zt ∂(ρ=)
∂t
A zs
dzdA +
✵
ρ=vn d✵ .
(5.14)
For the whole atmosphere or a domain with fixed walls, we see 1 g
ps d= A pt
dt
dpdA =
zt ∂(ρ=) A zs
∂t
dzdA
(5.15)
since vn = 0. The parcel changes integrated over the system may be evaluated by integrating the local changes, as no transport across the boundaries.
5.4.3
General continuity equation
For convenience, the transport process with conservation of the quantity = transported is referred to as the simple transport process. The density of the quantity ρ= may change in the transport, but the total amount m= in mass m is conserved when the mass is conserved. Since there are no the sources or sinks of the quantity in the simple transport process, we have ∂(ρ=) = −∇ · (ρ=v) . ∂t
(5.16)
It tells that the change of local density of = is caused by the flux convergence. The velocity in this equation may be mass velocity or other velocity, which carries
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NEWTONIAN-THERMODYNAMIC SYSTEM
the quantity to form the flux. In other words, the flux may also be produced by the conduction or propagation without mass displacement accompanied. If the = is temperature and the velocity is heat conduction velocity, (5.16) gives the heat conduction equation used widely in physics. It is discussed in Section 4.3 that the velocity in the continuity equation can only be the mass velocity. If the transport is caused by conduction without mass transport, the continuity equation (5.10) cannot be used and (5.12) gives
ρ
dρ d= = −= ρ∇ · v + dt dt
,
where v is the conduction velocity. Adding the first and last terms together yields d(ρ=) = −ρ=∇ · v . dt It tells that the density of = may be changed by dispersion or concentration of the transport velocity, called also the divergence or convergence. This equation may be referred to as the general continuity equation and may be applied for other transport. For mass transport, we set = = 1 and the velocity is a mass velocity. Thus, the continuity equation of mass is an application of this general continuity equation. The discussions above show that the general continuity equation is equivalent to (5.16). It is emphasized that these two equations are only used for simple transport processes.
5.4.4
Heat flux equation
Using the expansion equation (5.5) for the thermodynamic energy law of ideal gases given by (4.12), we obtain the heat flux equation cv
dα dq ∂T = −cv vp · ∇T − p + − cv (vd + vhc ) · ∇T + DTd , ∂t dt dt
where vhc denotes the heat conduction velocity which is not a mass velocity. The first term on the right-hand side is the advection of heat energy caused by parcel motions. The local temperature increases as a flow moves downgradient in the temperature field, or the local mass is replaced by a warmer mass. The second term is the rate of doing mechanic work by a parcel as it changes volume against external pressure force. The third term is the diabatic heating rate including the radiation and latent heating. The fourth term is the heat transport produced by molecular heat diffusion and conduction. The heat diffusion entropy and heat conduction entropy given by (4.38) and (4.39) respectively are derived from this term. The last term is the rate of kinetic energy dissipation resulting from molecular diffusion called the molecular dissipation previously. The heat conduction in the previous equation may be included in the diabatic heating, and the heat diffusion may be added to the last term. Thus, the heat flux equation becomes cv
dα dq ∂T = −cv vp · ∇T − p + + DTd , ∂t dt dt
(5.17)
5.4. MASS AND HEAT TRANSPORT EQUATIONS
105
When the microscopic processes of molecular diffusion and conduction are ignored, the heat flux equation becomes cv
dα dq ∂T = −cv vp · ∇T − p + . ∂t dt dt
(5.18)
dp dq ∂T = −cp vp · ∇T − α + ∂t dt dt
(5.19)
It can be rewritten as cp
derived using (4.14) for ideal gases, where dp/dt may be considered as the vertical velocity in pressure coordinates. The heat flux equation shows only a part of energy transfers in the compressible fluid, and does not account for the energy transfer between heat energy and kinetic energy. The density change in (5.18) and the pressure change in (5.19) provide the physical connections to the kinetic energy transfer. The energy conversions of all kinds in the atmosphere and the energy conservation equation will be studied in Chapter 7.
5.4.5
Heat conduction equation
In the simple transport process caused by heat conduction, the flux equation (5.16) gives ∂(cπ ρT ) + ∇ · (cπ ρT vhc ) = 0 , ∂t in which cπ is a polytropic heat capacity and cπ ρT vhc = Fhc is the conductive heat flux, of which the x-component is given by (3.38). If the flux is represented by Fhc = −λ∇T , where λ is the heat conductivity which is not constant in general as shown by (3.39). we have ∂(cπ ρT ) − ∇ · (λ∇T ) = 0 . ∂t When the heat conductivity, heat capacity and density are taken as constant for a solid substance, we gain the heat conduction equation cv ρ
∂T − λ∇2 T = 0 ∂t
(5.20)
in three dimensions. This equation is similar to mass diffusion equation (4.11). The solution can be found in many text books of mathematical physics. Since ∇2 T is positive at a low center and negative at a high center, the heat conduction destroys the temperature gradient and leads to the isothermal state.
5.
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5.5 5.5.1
NEWTONIAN-THERMODYNAMIC SYSTEM
Inhomogeneous thermodynamic system Adiabatic and transport processes
Adiabatic processes The thermodynamic states of atmosphere are represented by the field variables varying with space at a given time. The parcel and local variations may be produced not only by molecular diffusions, but also by parcel motions. The effect of parcel motions is more important than that of molecular diffusions at least in the dry atmosphere. There are two types of basic processes, adiabatic process and transport process, related to parcel motions and local variations in the atmosphere. In a compressible fluid, the temperature and density of a parcel may change adiabatically as it moves vertically or crosses isobaric surface. If the motion is slow enough, the adiabatic variations may be considered approximately as a thermodynamic reversible process. The parcels are assumed as ideal gases in this study. When ds = 0 in (4.27), we have R . (5.21) d ln T = κd ln p , κ= cp It may be integrated to gain the potential temperature (2.55). Since ds = 0 in reversible adiabatic processes, the potential temperature is conserved as shown by (4.28). It is emphasized that the potential temperature may change in an irreversible adiabatic process. An example of irreversible adiabatic expansion is given in Section 4.10. When the potential temperature is conserved, the temperature may change. To evaluate the change rate as a parcel moves adiabatically in the vertical direction, (5.21) is rewritten as d ln T =κ. d ln p In an adiabatic process of parcel motion, the temperature may be different from environmental temperature. If the variables of a parcel are denoted by´, (5.1) may be replaced by ´ dp (5.22) cp dT´ = α for the parcel. It is assumed usually that the parcel pressure is adjusted to the environment in the movement, so the pressure change in the last equation may be evaluated from hydrostatic equation (5.1), or dp = −gρdz .
(5.23)
In the inhomogeneous atmosphere, the isobaric surfaces may not be horizontal. For a parcel moving on a sloping isobaric surface, the geographic height may change but the pressure may not. So, this equation can only be used in the vertical direction. Inserting (5.23) into (5.22) yields dT = −
gα ´ dz cp α
5.5. INHOMOGENEOUS THERMODYNAMIC SYSTEM
or −
dT = Γd , dz
107
Γd =
gα ´ . cp α
(5.24)
The Γd is called the dry adiabatic lapse rate, which is the change rate of temperature of unsaturated parcels, as the parcels move reversibly and adiabatically in the vertical direction. If the difference between α ´ and α is small, we write Γd ≈
g . cp
(5.25)
The lapse rate is nearly constant. Errors may be produced if this equation is used for the convection with a sloping trajectory, called the slantwise convection, in the atmosphere with strong horizontal gradient of pressure. The adiabatic lapse rate for the slantwise convection will be discussed in Chapter 19. Macroscopic transport processes Apart from the adiabatic processes, the local changes may also be brought about by replacement of air masses. The moving air carrying a physical quantity forms the flux which delivers the quantity from one place to another. A flux of quantity = carried by a flow at a parcel velocity across the plane of unit area perpendicular to the velocity is given by F1 = =vp . The local changes may be produced by convergence of the flux, as shown by continuity equation (5.9). The convergence is evaluated from ∇ · F1 =
∂F1y ∂F1z ∂F1x + + , ∂x ∂y ∂z
which measures the local concentration caused by the flux. To see the adiabatic and transport processes more clearly, the advection of temperature is rewritten as vp · ∇T = ∇ · (T vp ) − T ∇ · vp . Applying continuity equation (5.11) for it yields vp · ∇T = ∇ · (T vp ) −
p dα . R dt
Inserting it into heat flux equation (5.18) gives
cv ∂T = −cv ∇ · (T vp ) + 1 − cv ∂t R
p
dα dq + . dt dt
The first term on the right-hand side is the convergence of heat flux, the second represents the adiabatic expansion or contraction, and the last is the diabatic heating rate. This equation shows that the local changes are produced by the three basic processes, transport process, adiabatic process and heat exchange process.
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5.5.2
NEWTONIAN-THERMODYNAMIC SYSTEM
Inhomogeneous thermodynamics
If the changes caused by molecular diffusions are negligible compared with that resulting from parcel motions, an inhomogeneous fluid system may be in a steady state viewed from a meteorological scale. This steady state, when it is inhomogeneous, will be referred to as the inhomogeneous equilibrium state. The molecular diffusions and produced microscale or tiny-scale variations at an inhomogeneous equilibrium state may not be expressed by the data over a certain scale. A thermodynamic system characterized by the inhomogeneous equilibrium state will be referred to as the inhomogeneous thermodynamic system. The Earth’s atmosphere gives an example of the system. The thermodynamics which studies the inhomogeneous thermodynamic system is referred to as the inhomogeneous thermodynamics. It is not called the nonequilibrium thermodynamics, because the latter deals with the nonequilibrium processes in classical thermodynamic systems. The nonequilibrium states of a classical thermodynamic system are also inhomogeneous, but the equilibrium states are homogeneous. Variations of classical thermodynamic system may be brought about by molecular diffusions only. While, variations of inhomogeneous thermodynamic system depend highly on parcel motions. The mechanic energies, such as the parcel kinetic energy and geopotential energy, are also included in the energy budget equation for individual parcels or the whole system. The inhomogeneous thermodynamic systems possess the properties different from those of classical thermodynamic systems. For example, there is the zeroth law for classical thermodynamic systems which states that when two separate thermodynamic systems are in thermal equilibrium with the third one respectively, the two systems are in thermal equilibrium with each other. This law is no longer available for inhomogeneous thermodynamic systems. The inhomogeneous thermodynamic systems are studied also in the time-space coordinates. In the Lagrangian representation, a parcel may be viewed as a classical thermodynamic system when it is small enough, of which thermodynamic variations may be studied with the classical thermodynamics. In the Eulerian representation then, the thermodynamic state at a given point of the coordinates depend on time only, so a local fluid medium may be considered as an open classical thermodynamic system. The mass and heat exchanges at the point are evaluated with the advection terms, derived by expanding the material derivatives in the thermodynamic equations as discussed earlier.
5.6 5.6.1
Momentum equation of atmosphere Pressure gradient force
To study fluid motions with the classical dynamics, the hydrodynamics has introduced a Newtonian force called the pressure gradient force, which is the net pressure of surroundings acting on the parcel surface in the normal direction. According to Newton’s second law, the external Newtonian force acting on a fluid element may be measured by the resultant change rate of momentum of the element. The mo-
5.6. MOMENTUM EQUATION OF ATMOSPHERE
109
mentum is defined with mass velocity in the classical dynamics. In the quantum physics, the momentum may also be defined with propagation velocity like the light velocity, since a wave packet which propagates at a propagation velocity or group velocity, may be regarded also as a quantum with a certain amount of mass. When the mass of a fluid element is conserved, the change rate of momentum gives the acceleration of fluid motion. The acceleration produced by pressure gradient force is given by dvp p d✵ , =− ρδV dt ✵ where ✵ is the outer surface of a parcel with density ρ and volume δV . The parcel velocity vp instead of general velocity is included in this equation, since variations of the microscopic velocities, such as diffusion velocity and conduction velocity, may be independent of the macroscopic pressure gradient force or Newton’s second law. The environmental pressure acting on the surface is identical in magnitude to the surface pressure of the parcel but in an opposite direction, so it is represented by the parcel pressure p with a negative sign in the equation. From the Gauss equation V
∇p dV =
we obtain
✵
p d✵
dvp =− ∇p dV . ρδV dt V As the volume of parcel tends to zero, we gain dvp = −α lim δV →0 dt
or
∇pdV = −α∇p δV
V
∂p ∂p ∂p dvp ˆ−α y ˆ−α z ˆ = −α x dt ∂x ∂y ∂z
(5.26)
for unit mass. The pressure gradient force is in the downgradient direction of pressure field. Its value is proportional to the intensity of pressure gradient. So, this equation may be derived also in terms of pressure gradient, as shown in Appendix C.
5.6.2
Navier-Stokes equation
Adding the external friction near fixed boundaries F, such as the surface drag, to (5.26), we gain dvp = −α∇p + F . (5.27) dt The derivative on the left-hand side of the previous equation may be expanded as ∂vp dvp = + vp · ∇vp + (vd + vmc ) · ∇vp , dt ∂t where vd and vmc are the diffusion velocity and momentum conduction velocity, and may be derived respectively from the diffusive momentum flux and conductive
5.
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NEWTONIAN-THERMODYNAMIC SYSTEM
momentum flux discussed in Chapter 3. The last term tells that the parcel velocity may be changed by molecular diffusion and momentum conduction. The effects are called the internal friction or viscosity. For the simple transport process caused by momentum conduction without mass displacement, the change of local density of vertical momentum denoted by ρvz may be given by ∂(ρvz ) = −∇ · (ρvz vmc ) ∂t derived from (5.16), where ρvz vmc = Fzc is the conductive flux of z-momentum. Referring to (3.32), the three-dimensional flux may be written as Fzc = −η∇vz , in which η is the coefficient of viscosity. Now, we have ∂(ρvz ) = ∇ · (η∇vz ) . ∂t It follows that
∂vz = ∇η ∇ · vz + η ∇2 vz , ∂t when the local density is unchanged in the transport process, where η = η/ρ is called the kinematic coefficient of viscosity. From (3.33), we see
2l η = 3
2RT . 3π
The local change of three dimensional velocity caused by momentum conduction gives ∂v = ∇η ∇ · v + η ∇2 v (5.28) ∂t with ˆ + ∇ · vy y ˆ + ∇ · vz zˆ ∇ · v = ∇ · vx x and
ˆ + ∇2 vy y ˆ + ∇2 vz zˆ . ∇2 v = ∇2 vx x
The right-hand side of the previous equation gives the expression of viscosity which is similar to the Navier-Stokes viscosity. The velocity equation including this viscosity gives dvp = −α∇p + ∇η ∇ · v + η ∇2 v . dt It is called the Navier-Stokes equation. Apart from the momentum conduction, the viscosity or momentum exchange may also be produced by molecular exchanges resulting from molecular diffusion. This part of viscosity may be evaluated from the diffusive momentum flux discussed in Chapter 3. The viscosity may also be represented by integrating the stress tensors over the surface in the classical hydrodynamics (e.g., Milne-Thomson 1960; Batchelor, 1967). The fluid of which the viscosity may be given approximately by (3.31)
5.6. MOMENTUM EQUATION OF ATMOSPHERE
111
is called the Newton’s fluid. Examples of the steady two-dimensional flows with the balance between pressure gradient force and Newton’s viscosity are the Couette flow and Poiseuille flow (Tritton, 1988). Equation (5.28) may be called the momentum conduction equation which, like the heat conduction equation (5.20) and mass diffusion equation (4.11), is a transport equation. We may also derive the heat diffusion and momentum diffusion equations. These transport processes destroy the system gradients and lead to the uniform state. It is noted that the total momentum of the system may not be changed by the particle momentum exchanges inside the system, if the effects of external forces including the pressure gradient force are ignored. So on theory, the momentum conduction may lead to a uniform velocity of the fluid if without external friction. However, it will be proved in Chapter 7 that the kinetic energy is reduced by the momentum conduction, although the total momentum is conserved in the process of velocity assimilation. Usually, the viscosity and boundary friction are represented together by F in (5.27), and so the derivative on the left-hand side of this equation is replaced by ∂vp dvp = + vp · ∇vp . dt ∂t The momentum equation (5.27) shows that the parcels driven by pressure gradient force move from high pressure to low pressure or from high density to low density at a constant temperature. As a result, the gradients may be destroyed by the mass redistribution. The pressure gradient force depends on thermodynamic states of the parcel and environments, and so may change from time to time and from place to place with the change of thermodynamic states in parcel motions. After the pressure gradient disappears, parcel kinetic energy will be dissipated by internal viscosity and boundary friction. In this process, the molecular diffusions become substantially important. We have learned in Chapter 3 that the diffusive mass flux is also from high density to low density and destroys the gradients. The similarity between the parcel motions and molecular diffusions allows us to extend the classical thermodynamic entropy law to the large-scale inhomogeneous fluid system as discussed in the next Chapter. However, the physical mechanism of molecular diffusions is different from that of parcel motions. The direction of molecular diffusion depends on the gradients of number density of molecules and mean molecular speed. The diffusion follows the second law of thermodynamics and may be independent of Newtonian forces including the gravitational force. An example is the mixing of two gases of different temperatures at a constant pressure. This downgradient mass diffusion may also occur if without the effect of gravity. While, the parcel motions are controlled by Newton’s laws, in which the gravitational force plays an important role. This will be discussed again in the last section of this chapter.
5.6.3
Momentum equation of atmosphere
The moving bodies on the Earth including the air parcels in the atmosphere may remain their velocities in the absolute space, if they are free from external forces.
5.
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NEWTONIAN-THERMODYNAMIC SYSTEM
Since the rotational Earth is not an inertial system with respect to the absolute coordinates, the free motions change their directions when viewed on the Earth. To explain the direction changes in the coordinates of Earth, it is assumed that the moving bodies are affected by an inertial force called the Coriolis force given by (Holton, 1992) Fc = −2Ω × vp , where Ω is the angle acceleration of the Earth with the positive direction to the north. The Coriolis force increases linearly with the speed. Since Ω is positive in the Northern Hemisphere and negative in the Southern Hemisphere, this force is normal to the velocity towards the right if following the moving body in the Northern Hemisphere, and is towards the left in the southern. If viewed in the Cartesian coordinates on the Earth, the Earth’s angle acceleration has the components upward and northward, that is ˆ + Ω sin ϕ zˆ , Ω = Ω cos ϕ y in which ϕ indicates the latitude. The Coriolis force becomes ˆ + 2Ωvx cos ϕ ˆz . x − 2Ωvx sin ϕ y Fc = −2Ω(vz cos ϕ − vy sin ϕ)ˆ The direction of Coriolis force can be found also in this equation. Adding the Coriolis force to the momentum equation (5.27) yields dvp = −α∇p − 2Ω × vp + F . dt When the parcels move from high pressure to low pressure at an increasing speed driven by pressure gradient force in the atmosphere, they are pushed by the Coriolis force towards one side. Since the force increases with the speed, the motion direction keep changing until it is parallel to the isobars if the viscosity is ignored. The steady state, called the geostrophic balance, may be reached as the parcel moves at the horizontal velocity α vg = zˆ × ∇h p f in the statically stable atmosphere. Here, f = 2Ω sin ϕ is the Coriolis parameter, and ∇h indicates a horizontal gradient. It tells that the velocity, called the geostrophic velocity, is perpendicular to the pressure gradient in a geostrophic balance. The velocity components are α ∂p , (5.29) vxg = − f ∂y and α ∂p . vyg = f ∂x The idealized geostrophic balance occur as the air moves along an isobar at a constant latitude in the environment with constant pressure gradient without viscosity. The steady geostrophic motions on the f -plane in the viscous atmosphere have the velocity 1 ˆz × (α∇h p − F) f 1 ˆ×F, = vg − z f
v =
5.7. SHALLOW WATER DYNAMICS
113
in which F depends on the velocity in general. The velocity deflects to the side of low pressure, so that the component of pressure gradient force in the motion direction may be balanced by the viscosity, and the component perpendicular to the velocity may be balanced by the Coriolis force. The departure of velocity from the geostrophic velocity is called the ageostrophic velocity, represented by 1 ˆ×F. va = v − vg = − z f Since the friction is in the opposite direction of the velocity, the ageostrophic component is perpendicular to the real flow towards the low pressure. The previous two equations give v = vg + va . The three velocities form a right triangle. The intensity of ageostrophic component increases with the friction, and so the angle between the real flow and isobars increases also while the flow speed decreases. When the surface friction decreases upward, the wind in the boundary layer turns right or left with increasing height in the Northern or Southern Hemisphere, and the speed increases if the pressure gradient is unchanged. The fluid motions affected by the vertical reduction of viscosity form the Ekman flow (Holton, 1992) in the oceans and atmosphere. Since the friction and viscosity convert the kinetic energy into heat energy, a steady velocity field in the viscous atmosphere must be supported by kinetic energy sources. For the atmosphere in the gravitational field, we may add the gravitational force g (which includes the centrifugal force and is downward in the Cartesian coordinates) to gain the final expression dvp = −α∇p + g − 2Ω × vp + F . dt
(5.30)
It is called the momentum equation of the atmosphere in meteorology, and may also be used to study ocean flows. At the first glance, the gravitational force affect only the vertical motions. This is not true, since the horizontal pressure gradient force depends substantially on the gravity as discussed in the following sections.
5.7
Shallow water dynamics
The shallow water assumption is used frequently for the studies of atmospheric processes. However, the dynamics of shallow water is substantially different from that of the atmosphere. As discussed earlier, there are three basic processes in the atmosphere: the transport process, adiabatic process and diabatic process. The adiabatic process is particularly important, as it leads to exchanges of heat energy and mechanic energy in both directions through the volume change in a compressible fluid. As in the atmosphere, the pressure in a shallow water increases with the depth. But the pressure change alone may not cause the changes of temperature and volume as the water is nearly incompressible. So the shallow water does not possess
5.
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NEWTONIAN-THERMODYNAMIC SYSTEM
the adiabatic processes as in the atmosphere. No mechanic work is created by thermodynamic process when the volume remains constant. Thus, there is no the conversion of internal energy into mechanic energy in water, and the adiabatic processes are purely mechanic characterized by exchanges of the geopotential energy and kinetic energy. The thermodynamic relationships, such as the state equation of gases and the first law of thermodynamics may not be used for the study of incompressible fluid. In other words, the shallow water is a mechanic system dominated by Newton’s laws together with mass and mechanic energy conservation laws. The energy conservation law of shallow water may be derived independently of thermodynamic processes. When ρ is constant, hydrostatic equation (5.1) gives p = gρ(Z − z) . Here, Z = Z(x, y, t) indicates the surface height of the water, and the atmospheric pressure over the surface is ignored for simplicity. Inserting it into (5.30) yields dvh = −g∇h Z + F , dt
(5.31)
where vh is the horizontal component of parcel velocity. The first term on the righthand side shows that the horizontal pressure gradient force in pressure coordinates is proportional to the slope of isobaric surface. The Coriolis force does not appear, as it does not affect the energy conversions discussed in the following. The momentum equation in the vertical direction is not discussed here, since the water is assumed to be hydrostatic equilibrium. The previous equation shows that the horizontal pressure gradient force depends on horizontal gradient of the surface height if without friction. Moreover, the continuity equation (5.11) gives ∂vz = −∇h · vh ∂z for the incompressible fluid. It is considered usually (Gill, 1982; Holton, 1992) that the horizontal velocity in shallow water does not change with height if without friction. In this case, the previous equation may be integrated from the fixed lower boundary at z = 0 up to the water surface, to gain the vertical velocity at the surface: vz|z=Z = −Z∇h · vh . Applying vz|z=Z =
dZ dt
for the previous equation yields ∂Z + vh ∇h Z + Z∇h · vh = 0 . ∂t
(5.32)
The two equations (5.31) and (5.32) form a set of prediction equations for two dynamic variables vh and Z. No thermodynamic processes and the relationships related need to be considered for the shallow water dynamics.
5.8.
NEWTONIAN-THERMODYNAMIC SYSTEM
115
To derive the energy equation of shallow water, (5.31) is multiplied by vh and (5.32) is multiplied by g. Adding the two resultant equations together gives dk dφ + = −∇h · (φvh ) + vh · F , dt dt
(5.33)
where φ = gZ is the specific geopotential energy at the surface height. It tells that changes of the kinetic energy and geopotential energy are caused by convergence of surface geopotential energy flux and kinetic energy dissipation resulting from friction including the surface drag and internal viscosity. Since the frictional force is in the direction against the velocity, we have vh · F < 0 . The kinetic energy destroyed is converted into heat energy eventually by molecular diffusions. However, the heat energy is not included in the energy equation, as the mechanic processes may be independent of thermodynamic processes. So the incompressible fluid may be considered as a mechanic system, which does not convert heat energy into kinetic energy by itself and is driven by external mechanic processes. To keep the motions in a viscous water without changing the surface height at a local region, there must be the flux of surface geopotential energy to the region or the external turbulent forcing on the surface. The positive input of surface geopotential energy may be brought about by the flows come from geographically higher places. The geopotential energy input may then be converted into kinetic energy in the process. Thus, water may flow continuously from a high place to a low place in the gravitational field. The gravitational force of the Moon, run off of inland rivers, precipitation and evaporation over different regions may change the height of sea surface at a local region. Moreover, the winds and turbulences in the atmosphere may produce surface drag or change the surface height also. The ocean waters driven by these external mechanic processes form particular flow patterns, which are studied by the oceanography, oceanophysics and ocean-atmosphere dynamics. If integrated over the whole water with fixed walls, the first term on the righthand side of (5.33) disappears, and so the isolated water without external mechanic forcing will become rest upon the effect of viscosity and boundary friction. The mechanic processes inside the water cannot produce kinetic energy to maintain the circulations.
5.8
Newtonian-thermodynamic system
The Earth’s atmosphere is compressible. The parcel motions in the vertical direction or across the pressure surfaces may cause thermodynamic variations in the atmosphere, through the adiabatic processes which may not occur in an incompressible fluid. The air engine theory developed in Chapters 20-23 reveals that the thermodynamic processes related to parcel motions are the major cause of kinetic energy generation in the atmosphere under the gravity effect. The atmospheric circulations may be maintained thermodynamically without exchanging mechanic energies with the exterior. This is the most important difference between the atmosphere and an incompressible fluid, such as the ocean water.
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NEWTONIAN-THERMODYNAMIC SYSTEM
In some studies, the atmosphere is assumed to be incompressible. An important shortage of using incompressible approximation is excluding the energy generation, as a result of suppressing the thermodynamic processes by this assumption. McHall (1993) pointed out that the incompressible Rossby wave can neither propagate nor follow the quasi- geostrophic balance observed in the real large-scale perturbations. Now, we see also that this classical Rossby wave cannot survive in the viscous atmosphere without external mechanic forcing. The atmospheric processes in which kinetic energy generation is important should not be studied with incompressible approximation. To study the compressible atmosphere, we apply all the physical relationships used for the studies of classical Newtonian systems and thermodynamic systems. These relationships, called the primitive equations, include the Newton’s second law, state equation, continuity equation and heat flux equation. The inhomogeneous fluid system controlled by these equations may be referred to particularly as a Newtonian-thermodynamic system. Variations in the motion status of this system are associated with variations in the thermodynamic state. The momentum equation includes thermodynamic variables, and the thermodynamic equations have dynamic variables. So, these equations form one instead of two complete set if without particular assumptions, such as the incompressible approximation. There is a particular force called the pressure gradient force in the geophysical fluids. Since the fluid pressure depends on the geographic height and acts as a force in all directions, the horizontal component of pressure gradient force is also associated with the gravity, as shown by the first term on the right-hand side of (5.31). In the pressure coordinates, the pressure gradient force is given simply by the gradient of geopotential −∇φ (Holton, 1992). Thus, the vertical gravitational force may change horizontal motions of fluids. The downward gravitational force may also produce upward motions in the compressible atmosphere, when the parcel density is lower than that of the surroundings. This can be seen from the vertical momentum equation or Archimedes’ principle
ρ dvpz =g −1 dt ρ´
derived from (5.30), where ρ´ and ρ are the densities of the parcel and environmental air, respectively. The primitive equations include the variable time and so are prediction equations. In mathematics, the predictions may be obtained by solving these equations for provided initial and boundary conditions. We have discussed in the preceding chapter that variations of classical thermodynamic systems possess the uncertainty to predictions, as we cannot measure and evaluate precisely the molecular diffusions. Apart from the molecular diffusions, there are many other physical and chemical processes in the atmosphere which cannot be represented perfectly by our prediction models and datasets. Also, the current grid-point numerical procedures including the spectral technology have an intrinsic limitation on representing the continuous fields and their variations. Thus, the weather predictions and atmospheric simulations are not a mathematical problem only, and the predictions are acceptable only within a limit time period. The predictability of the atmospheric processes will be discussed in Chapter 25.
Chapter 6 Turbulent entropy and universal principle 6.1
Introduction
The thermodynamic entropy has been mystified to many of us due to its unique macroscopic and microscopic characteristics and wide applications. Like the energy conservation law, the second law of thermodynamics is usually considered as an axiom, which reveals some universal features of the world changes. According to our experiences, the differences, or so called the gradients in a physical or aphysical system, may be weakened or removed by communications. An example of the communications in a thermodynamic system is the mass, heat and momentum transport resulting from molecular diffusions and conductions. A major purpose of the current study is to bring back the clear relations between the entropy law and our previous knowledge and theories. So far, there have been no the general relationship for the changes of inhomogeneities or disorderliness in the various systems, but only the second law of classical thermodynamics has been provided quantitatively in terms of thermodynamic entropy. This law illustrates the basic properties of downgradient molecular transport in a diffusive system, and may be derived using the nonequilibrium energy equation and kinetic theory for ideal gases as shown in Chapter 4. Since the unidirectional transport destroys the gradients irreversibly, the entropy may be used to measure the randomness or disorderliness of the microscopic and macroscopic thermodynamic states. In the large-scale moving fluid, the macroscopic disorderliness in the thermodynamic fields may also be changed by redistribution of fluid parcels with different potential temperatures. The thermodynamic entropy of a non-uniform system may be calculated from the algorithm provided in Chapter 4. In the idealized process of turbulent mixing without molecular diffusions, the system entropy evaluated by adding the entropy of each parcel or fluid element is conserved, but the disorderliness may be changed irreversibly by turbulent motions. To account for the irreversible changes in the disorderliness, we introduce in this chapter the turbulent thermodynamic entropy and the law of turbulent entropy. The turbulent thermodynamic entropy, called simply the turbulent entropy, is also a thermodynamic entropy, since it measures the thermodynamic disorderliness or irreversibility of an inhomogeneous thermodynamic system. However, the turbulent entropy law is independent of the law of classical thermodynamic entropy discussed in Chapter 4, since changes of turbulent entropy are caused by the macroscopic processes of parcel motions which are independent of microscopic molecular diffusions. To the change of turbulent entropy in the idealized turbulent process, 117
118
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
the basic elements manifesting the system disorderliness are the fluid parcels instead of molecules. Thus, the turbulent entropy and classical thermodynamic entropy represent the system disorderliness at different levels and different scales. The gridpoint data used in meteorology may only be used to estimate the turbulent entropy changes which may include the effect of molecular diffusions. The increase of entropy in an isolated system is usually considered as a universal feature of system changes. However, we only have the relationship of entropy increase for the classical thermodynamic systems before. The algorithm of entropy change in non-uniform systems given in this study may be applied also for some other physical and aphysical systems. We introduce in this chapter the universal entropy and the universal principle to manifest the general features of the system changes resulting from communications and exchanges within a system. The classical thermodynamic entropy law and turbulent entropy law are examples of the universal principle. This principle will be applied to derive the partition functions of the microscopic particles at different energy levels without using the concept of thermodynamic probability. These partition functions may then be used to obtain the expressions of internal energy, heat capacity and state equations. The van der Waals equation will be derived in this chapter.
6.2 6.2.1
Thermodynamic entropy of turbulent system Simple turbulent process
Molecular diffusions are the major cause for the change of classical thermodynamic system. While, changes of atmospheric fields viewed in a meteorological scale are produced mainly by parcel motions instead of molecular diffusions. The large-scale circulation changes in the atmosphere are generally irreversible, even if through the reversible adiabatic processes in which the potential temperature or thermodynamic entropy of each fluid parcel is conserved. Apart from the large-scale circulations, there are small-scale turbulent motions in the atmosphere, especially in the boundary layer and near the tropospheric westerly jets such as the clear-air turbulences. According to Pierrehumbert (1991), the chaotic turbulent mixing is a general and basic feature of large-scale circulations in the atmosphere. The turbulent processes may be related to some instabilities (Killworth and McIntyre, 1985; Haynes, 1985; Warn and Gauthier, 1989) or the large-scale deformation field (Welander, 1955). A kind of turbulent mixing in the stratosphere is called particularly the wave breaking (McIntyre, 1982; McIntyre and Palmer, 1983), by which the large-scale potential vorticity perturbations are broken down into turbulent patterns on smaller scales. The turbulences in the atmosphere convert large-scale kinetic energy to smaller scales continuously, and weaken or destroy the horizontal gradients in the atmosphere. The kinetic energy cascade may be considered as the turbulent dissipation of large-scale kinetic energy, caused by diffusive turbulences or turbulent viscosity. To consider the irreversible change of system disorderliness caused by parcel motions instead of molecular diffusions, we assume an idealized turbulent process, called the
6.2. THERMODYNAMIC ENTROPY OF TURBULENT SYSTEM
119
simple turbulent process, in which fluid parcels move slowly and adiabatically and no parcel kinetic energy dissipation and among-parcel molecular exchanges take place. This process may cause rearrangement of the entities or tear an entity into smaller pieces, but may not change the mean potential temperature of the system. The parcels in the simple turbulent process are isolated systems, and called particularly the turbulent entities. They are small enough so that may be considered in their uniform equilibrium states all the time. A close example is the turbulent entity found in the cumulus clouds (Telford et al., 1984). If a system has I turbulent entities, each of which possesses mass mi and potential temperature θi at the beginning, the mean potential temperature of the system reads I
θm =
i=1 mi θi
M
,
(6.1)
where M = Ii=1 mi is total mass of the system. When the molecular diffusions are ignored, an isolated fluid system may reach the turbulent equilibrium state eventually, which is statically stable and horizontally homogeneous viewed over the scale much larger than the size of turbulent entity. The assumption of simple process is not necessary for us to calculate the mean potential temperature or thermodynamic entropy in a system. However, the calculations for this assumed process enable us to see the features of these state functions more clearly. After the system reaches the turbulent equilibrium state by turbulent mixing, the mean potential temperature is unchanged, as potential temperature of each entity is conserved. The position exchanges of turbulent entities may change the disorderliness of the system, and the mixing process is generally irreversible. The changes of disorderliness cannot be represented by variation of classical thermodynamic entropy as discussed in the following.
6.2.2
Thermodynamic entropy changes
For the system of I entities assumed previously, the variation of classical thermodynamic entropy may be calculated by adding the entropy of each entity, giving ∆Sc =
I
mi dsi ,
i=1
in which ds represents the difference between equilibrium states of an entity. When an entity is assumed as an ideal gas, we apply (4.28) for it and gain ∆Sc = cp
I
mi d ln θi ,
(6.2)
i=1
where θi is the mean potential temperature averaged over entity i. This equation is similar to (4.29), except that the overbar is omitted. The mean potential temperature emphasizes the statistical features of thermodynamic variables, especially in a non-uniform state. If the measurement for each entity is available, the entropy may also be evaluated with the local data. Fig.6.1 shows a simple example. When the four entities
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
120
θ1
θ2
y
y
θ3
θ4
θ2
θ4
y
y
❍ ❍✟✟
θ1
θ3
y
y
y
y
Figure 6.1: Evaluation of classical thermodynamic entropy. The block circles indicate thermometers.
exchange their positions, their potential temperatures change from θi to θi . If each entity has unit mass, the entropy variation is evaluated from
θ1 θ θ θ + ln 3 + ln 2 + ln 4 θ1 θ2 θ3 θ4 θ1 θ2 θ3 θ4 = cp ln + ln + ln + ln . θ1 θ2 θ3 θ4
∆Sc = cp ln
(6.3)
The last equation shows that the evaluation with local measurements on a scale not larger than the entity size is identical to the evaluation following individual entities. In general, we have I I θj θ ln i = ln . (6.4) θi i=1,j=i θi i=1 It is similar to (4.34) except that θ represents the mean potential temperature of an entity instead of a molecule. Thus, the entropy calculated with (6.3) depends only on the potential temperature of each entity, and is independent of the entity distribution. If the potential temperatures do not change after position exchanges, the system entropy is conserved though the disorderliness is changed irreversibly by parcel motions. In the theoretical studies, the thermodynamic states are usually represented by continuous functions covering each point in a system, so (6.2) may be replaced by the integration ∆Sc = cp
ρd ln θ δV ,
(6.5)
V
where ρ is mass density of the volume element δV . On theory, the element δV is infinitesimal. But in fact, it cannot be as small as a single molecule. If the thermodynamic entropy can be defined for a single molecule, the sum of molecular entropy
6.3. GRID THERMOMETERS
121
cannot tell the disorderliness of the system, and so loses its physical meanings as discussed in Chapter 4. To account for the changes of disorderliness, we may assume a δV which includes a large number of molecules. In this case, the density and potential temperature in this equation represent the mean values over the element. The turbulent entity may be as small as the element of volume for calculating the thermodynamic entropy. The obtained results depend on the disorderliness of individual entities manifested by the distribution of molecules at different energy levels, but is independent of the disorderliness of the whole fluid system manifested by the entity distribution as shown by (6.4). These two kinds of disorderliness at different levels may be independent of each other, since they are associated with the independent processes of molecular diffusions and parcel motions respectively. The classical thermodynamic entropy evaluated from (6.2) or (6.5) by summing up the entropy of each entity is conserved, as the potential temperature of entity is unchanged in the simple turbulent process without molecular diffusions. Thus, the entropy is unable to tell us the changes of disorderliness caused by parcel exchanges in the reversible adiabatic process. This reversible process is not a purely thermodynamic process, and may not be really reversible due to the unidirectional transport caused by pressure gradient force and gravitational force.
6.3
Grid thermometers
To account for the changes of molecular disorderliness, we apply the thermodynamic variables, which represent the mean features of the fluid element including a large number of molecules, to evaluate the thermodynamic entropy. We do not calculate the mean thermodynamic variables, since the means are made by temperature measurements, which may only read the mean temperature of the element around a thermometer. Analogously, to account for the changes of entity disorderliness in a large-scale fluid system, we may choose a huge element, called the grid, which include a large number of entities to evaluate the thermodynamic entropy. The mean temperature over the grid may be regarded as the temperature measured by a huge thermometer in the size of the grid. The examples of the mean measurements are the grid-point data used in meteorology and oceanography, of which the grid scale is much larger than the size of turbulent entity. A variable measured at a given time and place in the atmosphere or ocean is considered theoretically as the mean over a certain area and time period. So, we may assume that the grid-point temperatures are measured by the huge thermometers at each grid point. The huge thermometer may be called the grid thermometer. Let the atmosphere to be divided into I domains, each of which includes Ji (i = 1, 2, · · · , I) entities. The central position of each domain is a grid point. The potential temperature measured by a grid thermometer at grid point i is the mean potential temperature of Ji entities, that is Ji
Θi = where Mi =
J i
j=1 mij
j=1 mij θij
Mi
,
(6.6)
is total mass of Ji entities. The mean potential temperature
6. TURBULENT ENTROPY AND UNIVERSAL PRINCIPLE
122 of the atmosphere gives
Θm =
Ji I 1 mij θij , M i=1 j=1
where M = Ii=1 Mi is total mass of the atmosphere over I grid points. Applying (6.6) for this equation yields Θm =
I 1 Mi Θi . M i=1
(6.7)
It is similar to (6.1) in the mathematical form. This similarity is based on the assumption given by (6.6). In practice however, the grid-point data may not be the means of all entities around a grid point. Equation (6.7) tells that the mean potential temperature of the simple turbulent system may also be represented by grid-point data. The grid-point data are the only data which we can get from the real measurements for meteorological studies and weather predictions. A uniform field represented by grid-point data may not be uniform on a subgrid scale. The gradients and integrations evaluated with grid-point data give also the smoothed values. The errors depend on grid scale of the data. Thus, the feature of uniform or the gradients and integrations calculated with grid-point data are scale-dependent.
6.4
Turbulent thermodynamic entropy
The classical thermodynamic entropy provides little help for us to understand the turbulent processes in the atmosphere, since the disorderliness of atmosphere can be changed by not only molecular diffusions but also turbulent diffusions. Conservation of classical thermodynamic entropy in the turbulent mixing process does not mean that the process is reversible or the disorderliness is unchanged. For example, as the baroclinic atmosphere becomes barotropic (no temperature gradient on the isobaric surfaces) in development of baroclinic disturbance, the geopotential energy and heat energy are converted into parcel kinetic energy. If the energy conversion is produced by reversible adiabatic parcel motions w
