Ambaum M., Thermal Physics of the Atmosphere

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228 APPENDIX C THERMODYNAMIC DIAGRAMS

p = constant

bottom of

tephigram

FIGURE C.1 The dashed rectangle illustrates the location of a typical atmospheric tephi-

gram on a Ts diagram. The solid black lines represent isolines of the pressure. The rotation

is such that the 1000 hPa pressure isoline is quasi horizontal, close to the bottom of the

tephigram. The thin lines represent isolines of temperature and potential temperature,

respectively.

rotation is chosen so as to make the isolines of pressure quasi-horizontal, see

Figure C.2a. The surface pressure is near the bottom of the tephigram and

upper tropospheric pressures are near the top.

The coordinate axes of the tephigram are most easily deﬁned as

x = T

s + c

T, (C.2a)

y = T

s − c

T, (C.2b)

with T

a relevant reference temperature and c

the ideal gas heat capac-

ity. With this transformation, on lines parallel to the x-axis, deﬁned by dy = 0,

the temperature and entropy satisfy

ds = c

dT. (C.3)

But according to Eq. 3.40 this is the differential of s for an ideal gas at constant

p and when the temperature is T

. In other words, isolines of pressure are

horizontal in the tephigram when the temperature is T

. Another advantage

of this mapping is that at ﬁxed temperature the y-coordinate is linear in

ln (p

/p), with p

a reference pressure. So to a good degree of accuracy, the

y-coordinate corresponds to the geometric height of a parcel.

The skewT–logp diagram is strictly speaking only a thermodynamic di-

agram for ideal gases. A skewT–logp diagram is located similarly on a Ts

diagram as a tephigram but it introduces an additional deformation to the

rectangle so as to make the pressure isolines straight and parallel to the

APPENDIX C THERMODYNAMIC DIAGRAMS 229

(a) (b)

(e) (f )

FIGURE C.2 Tephigrams with the bold lines highlighting isolines of (a) pressure, (b) tem-

perature (0

◦

C dashed), (c) potential temperature, (d) equivalent potential temperature,

and (e) saturated water vapour mixing ratio. Panel (f) shows a skewT–logp diagram with

the corresponding isolines.

230 APPENDIX C THERMODYNAMIC DIAGRAMS

horizontal boundaries of the deformed rectangle. The coordinate axes on a

skewT–logp diagram are deﬁned as

x = T/ T

+ ln (p

/p), (C.4a)

y = ln (p

/p), (C.4b)

with T

a ﬁxed reference temperature which determines the skew of different

isolines in the diagram and p

a reference pressure. Lines parallel to the x-axis,

deﬁned by dy = 0, are isolines of pressure. Also, to a good degree of accuracy,

the y-coordinate is proportional to the geometric height of a parcel. We can

now see that for an ideal gas

x dy =

v dp

+ ln (p

/p)

. (C.5)

Any integral over a closed cycle in the xy plane of the second term on the right-

hand side will vanish because it is a function of p alone. Thus the integral

over a closed cycle in the xy plane (that is, on a skewT–logp diagram) will

be proportional to the work performed on an ideal gas parcel (the contour

integral of v dp equals the contour integral of −p dv).

Tephigrams and skewT–logp diagrams look fairly similar, see Fig C.2, and

can be used interchangeably. In tephigrams, entropy isolines are straight and

pressure isolines are curved; in skewT–logp diagrams this is the other way

around.

The isolines of speciﬁc entropy are by deﬁnition the dry adiabats; in a

tephigram the isolines of speciﬁc entropy are normally labelled by their po-

tential temperature, with the potential temperature corresponding to the ac-

tual temperature at the reference level of p

= 1000 hPa. The dry adiabats

are highlighted in Figure C.2c.

We can use Eq. 6.12 to calculate the location of the pseudo-adiabats. These

then produce a set of curved isolines on the tephigram, see Fig-

ure C.2d. Pseudo-adiabats are isolines of saturated pseudo-equivalent poten-

tial temperature 

The pseudo-adiabats are labeled with the wet-bulb po-

tential temperature, 

, which is the temperature where the pseudo-adiabat

crosses the reference pressure level of p

Tephigrams furthermore show isolines of saturated mixing ratio r

, see

Figure C.2e.

Figure C.3 shows a larger scale tephigram with all the isolines labeled.

So a tephigram or a skewT–logp diagram shows isolines of: temperature

T (isotherms), pressure p (isobars), potential temperature  (dry adiabats),

pseudo-equivalent potential temperature 

(pseudo-adiabats), and saturated

water vapour mixing ratio r

. Tephigrams usually show several kinds of ad-

Accurate equations used to construct pseudo-adiabats are given in Bolton, D. (1980)

Mon. Wea. Rev. 108, 1046–1053.

APPENDIX C THERMODYNAMIC DIAGRAMS 231

ditional information, but the above set of ﬁve isolines form the basis of any

tephigram or its variants.

The real power of a tephigram is that it can be used to perform calculations

graphically. As explained in Chapter 6, the calculation of, for example, the

lifting condensation level or the convective available potential energy can be

straightforwardly performed using a tephigram. In this sense a tephigram is

a nomogram, a graphical calculation chart.

0.5

0.2

0.1

1.5

0.001

0.03

0.1

0.01

0.003

200

300

400

500

600

700

800

900

1000

-10

-60

-30

-40

-50

-70

-80

FIGURE C.3 A blank tephigram. Pressures are in hPa, temperatures in

◦

C, and mixing

ratios in g kg

−1

Appendix D

Relationship between energy

density and energy ﬂux

Following the arguments leading to Eq. 1.11, we can deduce that the number

of particles N with positive velocity U hitting the wall per unit time and per

unit area equals

U , (D.1)

with

dU the volumetric number density of particles with x velocity be-

tween U and U +dU . Integrating this equation over positive U we ﬁnd the

total number of particles N hitting the wall per unit time and per unit area,

N =



U >0

U dU . (D.2)

We can write the particle density

in terms of the particle density

particles with vector velocity U if we integrate out the dependence on the

other velocity components, V and W ,



dV dW . (D.3)

We can now write N as

N =



|U |dU dV dW , (D.4)

where the absolute value function allows us to include negative U and the

resulting double counting is compensated by the factor 1/2. The particle

density is a function only of particle speed, say C =|U|, not its direction. We

therefore ﬁnd

N =



∞

|U |

dC, (D.5)

233

Thermal Physics of the Atmosphere Maarten H. P. Ambaum

234 RELATIONSHIP BETWEEN ENERGY DENSITY AND ENERGY FLUX

where |U |

is the average of |U |if the total speed is C. On geometric grounds

it follows that

|U |

= C/2 (D.6)

(this is equivalent to the statement that the average z coordinate of a hemi-

spherical dome on the xy-plane equals z=1/2) and we conclude that

N =



∞

dC =

nC. (D.7)

If the particles carry an average energy e, then the above equation can be

used to relate the volumetric energy

u = e

n to the total energy ﬂux J

.We

ﬁnd

I J

uC. (D.8)

This relationship is used to ﬁnd a relationship between the energy density

and energy ﬂux for the thermal radiation ﬁeld where all the particles, the

photons, have speed c.

Index

absorption, 180

absorptivity, 166

accretion, 151

activation, 136

activation radius, 132

added mass, 80, 123

adiabatic process, 27

afﬁnity of vaporization, 211

aggregation, 151

air

composition of, 11

heat capacity of, 47

parcel, 6

albedo, 176

availability, 223

Avogadro number, 2

barometric distribution, 84

Beer–Lambert law, 180

Bergeron–Findeisen process, 99, 151

Bernoulli equation, 81, 207

black body, 166

boiling point

deﬁnition, 97

elevation, 161

Boltzmann constant, 6

Boltzmann factor, 84

Bowen ratio, 59

Boyle’s law, 9

Brunt–V

ais

a frequency, 79

buoyancy, 77

frequency, 79

capacitance, 150

capillary pressure, 23, 130

carbon dioxide, 11

Celsius, unit, 2

charging work, 20

Charles’s law, 9

chemical potential, 153

Clausius’ postulate, 27

Clausius–Clapeyron equation, 94

cloud radar, 185

collection efﬁciency, 149

colligative properties, 162

collision and coalescence, 148

coloured body, 166

compressibility, 64

continuity equation, 204

convective available potential

energy, 121

convective inhibition, 122

critical supersaturation, 134

Dalton’s law, 11

and water vapour, 97

degrees of freedom, 6

and heat capacity, 47

dewpoint temperature, 101

from Normand’s construction,

119

diabatic, 27

differential, 221

diffusion coefﬁcient, 141

diffusional drop growth, 141

dissipation, 18, 24

viscous, 208

drop growth

by collision and coalescence, 148

by diffusion, 141

dry adiabatic lapse rate, 76

dry static energy, 81

235

Thermal Physics of the Atmosphere Maarten H. P. Ambaum

236 INDEX

efﬁciency, 37, 214

electrostatic energy, 137

emissivity, 166

energy

conversion, 206

density, 22

ﬂux and density, 233

internal, 21, 40

budget equation for, 205

kinetic, 204

potential, 67

budget equation for, 206

enthalpy, 41

and hydrostatic balance, 70

ﬂux, 57

and radiative–convective

equilibrium, 188

for moist air, 104

generalized, 81

in open systems, 53

of fusion, 57, 161

of sublimation, 57

of vaporization, 55

entrainment, 80, 123

entropy, 23

Boltzmann formulation, 27, 36

for moist air, 112

Gibbs formulation, 91

information, 33

of ideal gas, 49

of ideal solution, 158

of liquids and solids, 64

of mixing, 156

of radiation, 193

potential temperature, 60

production, 213

in the climate system, 215

maximum, 219

minimum, 212

pseudo, 113

equation of state, 9

of ideal gas, 8

of photon gas, 192

of saturated air, 110

Van der Waal’s equation, 50

equipartition theorem, 6, 21

derivation, 88

for macroscopic ﬂuctuations, 33

generalized, 89

evaporation, 55, 59

of drops, 145

exergy, 43, 223

exosphere, 5

expansivity, 61

extensive variable, 3

Boltzmann entropy, 29

ohn effect, 77

Fahrenheit, unit, 2

Fick’s law, 141

ﬂow work, 54, 70, 82

ﬂux, 55

diffusive, 141

latent heat, 59

sensible heat, 58

turbulent, 57

free energy

see Helmholtz free energy, 42

free expansion, 27

freezing point depression, 161

friction, 18, 25

gas constant

speciﬁc, 8

universal, 8

Gay–Lussac’s law, 9

general circulation, 176

generalized coordinates, 88

generalized force, 20

geopotential, 67

effective, 69

height, 68

geopotential thickness, 74

Gibbs function, 42

and charge, 137

and surface tension, 127

for coexisting phases, 56, 94

for compound substances, 153

Gibbs’ phase rule, 3

gravity, 67

effective, 69

gravity waves, 80

greenhouse effect, 171, 188

enhanced, 173

runaway, 174

greenhouse gases, 173

grey body, 166

Guoy–Stodola theorem, 215, 225

INDEX 237

haze, 135

heat, 21, 44

uncompensated, 25

heat capacity, 30, 44

of ideal gas, 47

of liquids and solids, 46

of saturated air, 111

of water, 63

heat conduction, 143

Helmholtz free energy, 42, 50

in statistical mechanics, 88

hole clouds, 151

humidity variables, 100

hydrostatic balance, 69

in isentropic coordinates, 83

hydrostatic equation, 70

hypsometric equation, 74

ice crystals, 150

ice–albedo feedback, 175

ideal gas, 1

chemical potential of, 156

entropy of, 49

equation of state, 8

heat capacity of, 47

mixtures, 10, 155

potential temperature of, 61

ideal solution, 131, 157

chemical potential of, 158

implicit function theorem, 221

insolation, 175

absorbed, 176

instability

conditional, 116

parcel, 79

potential/convective, 116

intensity, 177

intensive variable, 3

internal energy, 21, 40

irreversibility, 26

due to precipitation, 211

isentropic coordinates, 82

isentropic process, 27

isotropic radiation, 178

Joule expansion, 27

Joule’s law, 48

Joule, unit, 17

ohler theory, 132

Kelvin effect, 125

Kelvin radius, 129

Kelvin, unit, 2

Kirchhoff’s equation, 56

Kirchhoff’s law, 165

Lagrangian derivative, 203

Langley’s method, 183

lapse rate, 72

adiabatic, 74

of moist air, 109

dry adiabatic, 76

in radiative–convective

equilibrium, 187

moist-adiabatic, 110

pseudo-adiabatic, 111

latent heat, 55

ﬂux, 59

Legendre transform, 41

level of free convection, 115

lifting condensation level, 115

long-wave radiation, 169

Mach number, 91

Maxwell distribution, 86

Maxwell relations, 43, 222

mean ﬁeld method, 53

mean free path, 5

meridional heat transport, 178, 216

mesosphere, 4

Mie scattering, 185

millibar, unit, 3

mixing clouds, 107

mixing ratio, 100

mixtures, 153

moist static energy, 59, 104

molality, 163

molar mass, 2

Montgomery function, 82

nomogram, 2, 13, 231

non-equilibrium processes, 203

Normand’s construction, 119

nucleation

charge enhanced, 137

heterogeneous, 131

homogeneous, 125

open systems, 53

optical depth, 180

238 INDEX

ozone layer, 5

radiative effects of, 190

partition function, 87

Pascal, unit, 3

photon gas, 191

Planck law, 167

derivation, 195

Poisson equations, 62

potential temperature, 60

equivalent, 114

of ideal gas, 61

pseudo-equivalent, 114

virtual, 78

wet-bulb, 115

pressure, 7

as energy density, 22

capillary, 23, 130

charge induced, 138

partial, 11

vapour, 97

pressure broadening, 181

Prigogine theorem, 212

psychrometer, 102

psychrometric chart, 104

psychrometric equation, 102, 106

psychrometric table, 103

radiation, 165

radiative convective equilibrium, 185

radiative transfer, 180

Raoult’s law, 131, 158

Rayleigh radius, 139

Rayleigh scattering, 185

Rayleigh–Jeans law, 199

reciprocal theorem, 222

reciprocity theorem, 222

relative humidity, 101

over a solution, 159

reversibility, 27, 34

Sackur–Tetrode equation, 47, 49

Sandstr

om theorem, 207

saturation ratio, 128, 132, 138

critical, 132

scale height, 72

scattering, 185

Schwarzschild equation, 183, 186

sensible heat ﬂux, 58

short-wave radiation, 169

SI units, 2

simple substance, 3, 39

skewT–logp diagram, 117, 227

solar constant, 170

solutions, 153

soundspeed, 50, 64

speciﬁc humidity, 14, 100

speciﬁc variable, 5

budget equation for, 203

speciﬁc volume, 5

Standard Atmosphere, 72

state variable, 20

statistical mechanics, 84

role of free energy, 42

steam, 93

Stefan–Boltzmann law, 168

thermodynamic derivation, 192

Stirling engine, 36

Stirling’s approximation, 35

stratosphere, 4, 84

radiative effects on, 190

surface tension, 19, 125

temperature, 6

bolometric/radiation, 171

critical, 94

dewpoint, 101

total air temperature, 91

virtual, 14, 71

wet bulb, 101

tephigram, 117, 227

terminal velocity, 149

Teten’s formula, 98

thermal radiation, 165

thermal wavelength, 168

thermalization, 9

thermodynamic

diagram, 117, 227

equilibrium, 33, 36

local equilibrium, 5, 9

potentials, 39, 225

stability, 223

variable, 3

thermodynamics

ﬁrst law of, 21

laws of, 17

of climate, 215

second law of, 23, 208

thermosphere, 4