Large-Scale Inhomogeneous Thermodynamics
--And applications for atmospheric energetics
Yong Zhu
Contents
1 Introduction2 Two classical physical systems
2.1 Introduction
2.2 The Newtonian systems
2.3 Dynamic entropy
2.4 Simple thermodynamic systems
2.4.1 Mole-number and molecular mass
2.4.2 Thermodynamic variables
2.4.3 Pressure of monatomic gas
2.5 The first law of thermodynamics
2.6 State equation of gases
2.7 State equation of ideal gases
2.7.1 Ideal-gas equation
2.7.2 More features of ideal gases
2.8 Thermodynamic energy law of ideal gases
2.9 Internal energy and heat exchange
2.10 Polytropic process3 Molecular transport properties
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
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 Relation to disorderliness
4.7 Inadditive and scale-dependent features
4.8 For open systems
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 Heat flux equation
5.4.2 Continuity equations
5.4.3 Integrated variations in a system
5.5 Inhomogeneous thermodynamic system
5.5.1 Adiabatic and transport processes
Adiabatic processes
Transport processes
5.5.2 Inhomogeneous thermodynamics
5.6 Momentum equation of atmosphere
5.7 Shallow water dynamics
5.8 Newtonian-thermodynamic system
6 Turbulent entropy and universal principle
6.1 Introduction
6.2 Simple turbulent process
6.3 Thermodynamic entropy of turbulent system
6.4 Grid thermometers
6.5 Turbulent entropy
6.6 Turbulent entropy law
6.7 Difference from classical thermodynamic entropy
6.7.1 General discussion
6.7.2 Example
6.8 Turbulent entropy and disorderliness
6.9 Universal principle
6.9.1 The principle
6.9.2 Applications
6.10 Partition functions
6.11 Heat capacity and van der Waals equation
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 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
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 parcels
8.10.2 For the atmosphere
9 Available enthalpy
9.1 Introduction
9.2 Available enthalpy
9.3 Constraint relationships
9.4 Variational approach
9.5 The lowest state
9.6 Maximum available enthalpy
9.7 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 variations
9.10.4 Continuous solutions
10 Dry processes of energy conversion
10.1 Introduction
10.2 Dependence on process
10.3 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
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
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 instability
12.8 Equivalent baroclinic and barotropic entropies
12.9 Equivalent thermo-static entropy level
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
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 troposphere
14.5 Relation to thunderstorms
14.6 Relation to precipitation
14.7 Relation to tropical cyclones
15 A case of typhoon recurvature
15.1 Introduction
15.2 Typhoon Orchid recurvature
15.3 Subtropical cyclones
15.4 Critical surface temperature
15.5 Energy budget
15.6 Self-feeding mechanism
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
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 Geopotential entropy limitation
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
19 Energetics of 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
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 Adiabatic slantwise oscillations
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
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
21.6 Equilibrium air engine
21.6.1 Equilibrium cycle
21.6.2 Examples
21.6.3 Entropy productions and efficiency
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 convection
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 Convection at low and high levels
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
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
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
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
25.10 Uncertainties in physics
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
Appendices
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
C Derivation of momentum equation
D References
E Index
F List of symbols