Ambaum M., Thermal Physics of the Atmosphere

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9.7 RADIATIVE–CONVECTIVE EQUILIBRIUM 187

Taking the derivative of this equation with respect to the long-wave optical

depth and substituting the Schwarzschild equations for the long-wave com-

ponents we ﬁnd

2B − L

↓

− L

↑

− 

/

+ dH/dı = 0. (9.61)

Here we have rewritten the derivative of the net upward short-wave compo-

nent in terms of the short-wave heating

; that is,



=−dS

/dz =−

/dı. (9.62)

We can take a further derivative with respect to the optical depth and again

use the Schwarzschild equations for the long-wave components to ﬁnd

I 2

dı

+ L

−

dı









dı

= 0. (9.63)

This equation expresses the temperature lapse rate with optical depth in terms

of the radiative and enthalpy ﬂuxes. It is straightforward to verify that the

temperature lapse rate with geometric height  =−dT/dz is related to dB/dı

dB/dı =−4T

/

. (9.64)

We can now consider the implications of these budget equations for the at-

mospheric proﬁle.

First consider the case where the atmosphere is transparent to short-wave

radiation and there are no enthalpy ﬂuxes, H = 0. There is no convergence

of the short-wave ﬂux, dS

/dz = 0, as there is no black body contribution to

the short-wave ﬂux. The equilibrium condition, Eq. 9.60, then becomes

=−S

= constant = T

, (9.65)

where we have expressed the constant net upward short-wave ﬂux in terms

of the radiation temperature of the planet (T

= 255 K in the case of the

Earth). We conclude that at radiative equilibrium

dı

=−T

. (9.66)

This equation can be integrated with the integration constant set by applying

Eq. 9.61 at the top of the atmosphere: 2B(TOA) = T

. The solution is

I T = T



1 + ı

TOA

− ı



1/4

. (9.67)

188 CH 9 THERMAL RADIATION

The surface temperature T

in this radiative equilibrium model equals

= T



1 + ı

TOA



1/4

. (9.68)

The above set-up gives a more complete description of the greenhouse ef-

fect: in radiative equilibrium the temperature increases going downward

into the atmosphere. The more optically thick the atmosphere is, the larger

this temperature increase is. This explains the high surface temperatures of

Venus: with its high albedo, Venus has a relatively low radiation temperature

≈ 227 K) but the Venusian atmosphere is optically so thick that its surface

temperature can go above 700 K.

If we take the long-wave absorbers to be well mixed, as is the case for

, we can use Eq. 9.44 to express the optical depth coordinate in terms

of pressure. Taking p

the surface pressure, then the optical depth from the

surface to some pressure p is

ı = ı

TOA

+ p)(p

− p)

. (9.69)

Indeed, at the surface p = p

, the optical depth vanishes and at the top of the

atmosphere p = 0 the optical depth becomes ı

TOA

. So in terms of pressure,

the radiative equilibrium proﬁle, Eq. 9.67, becomes

T = T



1 + ı

TOA



1/4

. (9.70)

Figure 9.10 shows a typical proﬁle for radiative equilibrium with Earth-

like parameters. Its lapse rate near the surface is so large it is convec-

tively unstable. We need convective enthalpy ﬂuxes to get more realistic

proﬁles.

Next we will introduce an enthalpy ﬂux, H, while the atmosphere remains

transparent to short-wave radiation. The equilibrium condition, Eq. 9.60,

then becomes

=−S

− H = T

− H. (9.71)

Substituting in Eq. 9.63 we ﬁnd

dı

=−T

+ H −

dı

. (9.72)

For more comprehensive model calculations of radiative-convective equilibrium, see

Manabe, S. & Wetherald, R. T. (1967) J. Atmos. Sci. 24, 241–259; Ramanathan V. &

Coakley, J. A. (1978) Rev. Geoph. Space Phys. 16, 465–489

9.7 RADIATIVE–CONVECTIVE EQUILIBRIUM 189

100

1000

–70

–60 –50 –40 –30 –20 –10

0 10 20 30

p (hPa)

T (ºC)

FIGURE 9.10 Radiative equilibrium proﬁle (thick line) as predicted from Eq. 9.70 using an

long-wave optical depth of ı

TOA

= 3.0. The thin line shows the US Standard Atmosphere,

1976 for comparison.

Integrating this equation between ı and ı

TOA

and using the boundary condi-

tion 2B(TOA) = T

(from Eq. 9.61) we ﬁnd the solution

2T

= T

(1 + ı

TOA

− ı) +

dı

−



TOA

H dı. (9.73)

For H = 0 this equation reduces to Eq. 9.67. The surface temperature T

this model is given by

2T

= T

(1 + ı

TOA

) +

sfc

dı

−



TOA

H dı. (9.74)

Upward enthalpy ﬂuxes act to cool the surface: the second term on the right-

hand side is opposite to the local heating rate at the surface due to the en-

thalpy ﬂuxes. Because enthalpy ﬂuxes tend to be maximal at the surface, this

term tends to be negative. The third term is also always negative for upward

ﬂuxes.

It is instructive to write Eq. 9.72 in terms of a temperature lapse rate by

using Eq. 9.64. The lapse rate is given by

8T



 = T

− H +

dı

. (9.75)

190 CH 9 THERMAL RADIATION

From this it becomes clear how the presence of enthalpy ﬂuxes reduces the

lapse rate. Indeed, the radiative equilibrium proﬁle of Eq. 9.67 is convectively

unstable and the ensuing enthalpy ﬂuxes act to reduce the lapse rate. Strictly

speaking, the enthalpy ﬂuxes act to reduce the combination T

 , but this is

generally equivalent to a reduction in .

To account for the negative lapse rates of the stratosphere we need to

take into account short-wave absorption. The ozone layer is the main con-

tributor to short-wave absorption in the stratosphere. The heating due to the

short-wave absorption,

, follows from the convergence of the net down-

ward shortwave ﬂux as in Eq. 9.62. For simplicity we set the enthalpy ﬂux

H to zero (this is a good approximation in the stratosphere). In this case

the equilibrium equation, Eq. 9.58, and the lapse rate equation, Eq. 9.63,

become

dı





, (9.76a)

dı

=−L

dı









. (9.76b)

The ﬁrst equation expresses how the heating by the shortwave absorption

is compensated by a long-wave cooling. The second equation expresses how

the lapse rate (expressed in terms of dB/dı) is modiﬁed: although the net

downward long-wave ﬂux remains negative, the lapse rate is reduced below

regions of short-wave absorption where the second term on the right-hand

side of Eq. 9.76b is positive.

The lapse rate below a region of short-wave heating is lower than the lapse

rate above a region of short-wave heating. To demonstrate this, take a further

derivative of Eq. 9.76b with respect to the optical depth to ﬁnd

dı

=−





dı









. (9.77)

Now for simplicity assume the short-wave heating is conﬁned between levels

at optical depths ı

and ı

. We choose these bounding levels to be just out-

side the region of short-wave heating itself. Integrating the above equation

between these two levels we ﬁnd

dı

− 2

dı

=−





dz. (9.78)

Here we changed variables in the integral by noting that 

dz = dı. The

right-hand side of this equation is negative. Now using Eq. 9.64 to express

dB/dı in term of lapse rate  , we ﬁnd that the lapse rate at level 1 is

larger than the lapse rate at level 0. Outside this region of short-wave ab-

sorption, the lapse rate is set by Eq. 9.66. Above the region the net upward

9.8 THERMODYNAMICS OF A PHOTON GAS 191

long-wave ﬂux is still T

. This means that the introduction of a single re-

gion of short-wave absorption in a radiative equilibrium model reduces the

lapse rate below that region. This reduced lapse rate effectively deﬁnes the

stratosphere.

9.8 THERMODYNAMICS OF A PHOTON GAS

Thermodynamics can be applied to a gas of photons, particles that mediate

the electromagnetic ﬁeld. In this section we will derive the Stefan–Boltzmann

law and the Wien displacement law from thermodynamic arguments only.

In Chapter 1 we used a kinematic argument to derive how the pressure in a

gas can be written as an average of the momentum transfer in the direction of

a wall over all particles that collide with the wall, see Eq. 1.11. For a photon

gas this equation needs to be reinterpreted: despite being massless, a photon

does have momentum. We can derive an equivalent version of Eq. 1.11 where

we replace M

U by P

, the momentum of the particle in the direction of the

wall. We then ﬁnd for the pressure p

p =

n U P

, (9.79)

with, as before,

n the volumetric particle number density and U the particle

velocity in the direction of the wall. Because the particles on average have

no directional preference we could have used any velocity and momentum

component in the above equation. We can therefore rewrite the pressure as

the average of the contributions from all directions,

p =

n U · P, (9.80)

with U the vector velocity of the particle and P its vector momentum.

A photon moves with the speed of light, c, and has momentum of magni-

tude P, which is related to the energy e of the photon by

e = cP. (9.81)

We can substitute this in the above kinematic expression to ﬁnd the pressure

p of the photon gas,

p =

ncP=

ne. (9.82)

The present derivation is partly based on Adkins, C. J. (1983) Equilibrium thermody-

namics, 3rd edn. Cambridge University Press, Cambridge.

192 CH 9 THERMAL RADIATION

The volumetric energy density

u of the photon gas can be written as

u =

ne. So we arrive at

I p =

u/3, (9.83)

which is the equation of state for a photon gas. The Stefan–Boltzmann law

and the Wien displacement law can be derived from this equation of state.

We use the following expressions for the total entropy S, total energy U,

and pressure p of the equilibrium radiation ﬁeld:

S =

sV, U =

uV, p =

u/3, (9.84)

with

s and

u the volumetric energy densities of entropy and energy. With

these expressions, the ﬁrst law in differential form

dS =

dU + p dV

(9.85)

can be rewritten as

V d

s +

s dV = V

dV. (9.86)

Because

u and, for the same reason,

s are only functions of temperature,

in-

dependent variations in temperature and volume lead to the following equal-

ities:

s =

, d

s =

. (9.87)

In order for both equations to be true, the differential version of the ﬁrst

equation must be equivalent to the second equation. The differential of the

ﬁrst equation is

s =



−



. (9.88)

This equals d

u/T, as in the second Eq. 9.87, if and only if

= 4

. (9.89)

This equation can be integrated to ﬁnd

u = kT

, (9.90)

We can add a constant S

to the total entropy S by adding S

/V to the entropy density

s and still satisfy Eq. 9.87. By the third law, this constant is zero.

9.8 THERMODYNAMICS OF A PHOTON GAS 193

with k an integration constant. As a corollary we ﬁnd a relationship between

entropy density and energy density of the equilibrium radiation ﬁeld,

s =

. (9.91)

The Stefan–Boltzmann equation now follows from the result derived in

Appendix D, which states that the total radiative energy ﬂux J

is related

to the volumetric energy density

u by

uc/4, (9.92)

with c the speed of light. For a photon gas in equilibrium we therefore have

I J

(9.93)

with k as in Eq. 9.90. This is the Stefan–Boltzmann law with J

the total black

body emission and kc/4 the Stefan–Boltzmann constant.

The Wien displacement law follows from a more general thermodynamic

prediction on the possible shape of the spectrum of thermal radiation. We

proceed by performing a thought experiment where we adiabatically com-

press the radiation in a cubic vessel with perfectly reﬂecting walls and

with edge size L. Any single wavemode with wavenumber  will adjust

its wavelength to ﬁt the size of the vessel, see Figure 9.11. We therefore

have

 ∝ L. (9.94)

The total energy of the vessel changes according to the ﬁrst law under adia-

batic conditions,

dU =−p dV. (9.95)

δ L

δλ

FIGURE 9.11 Compressing the vessel of size L leads to a proportional compression of a

wave with wavelength .

194 CH 9 THERMAL RADIATION

If we substitute U =

uV and p =

u/3weﬁnd

u dV + 3V d

u = 0, (9.96)

which integrates to

= constant (9.97)

or, because V = L

and the wavelength  of any mode is proportional to the

edge size L,

u

= constant. (9.98)

From the derivation of the Stefan–Boltzmann law, Eq. 9.90, we know that

is proportional to T

. We conclude that for a single wavemode

T = constant (9.99)

on adiabatic compression.

The ﬁrst law can also be applied to parts of the spectrum. So if we take the

total energy



 V between wavelengths  and  +  then the ﬁrst law

for adiabatic compression reads



 V) =−p



 dV, (9.100)

with p



 =



/3 the partial pressure due to the wavemodes in that

wavelength range. We arrive at the analogous expression of Eq. 9.98,



 = constant. (9.101)

But because the wavelength interval  will compress proportional to L, just

as , we ﬁnd that on adiabatic compression



= constant. (9.102)

Finally, in Section 9.1 we argued that

u can only be a function of  and T.

We have now gathered enough properties of

u to predict some important

properties of the spectrum. We have during the adiabatic compression of the

radiation that for any wavemode 

T = constant, (9.103a)



= constant, (9.103b)



= f (,T). (9.103c)

We have assumed that the radiation remains black during the compression.

9.9 DERIVATION OF THE PLANCK LAW 195

To satisfy these properties for all , we see that



must be of the form



= 

−5

g(T), (9.104)

for some function g. The Planck law, Eq. 9.6, is of this form (remember that

the energy density



is proportional to the energy ﬂux B



). An equivalent

expression of the form of the spectrum is



= T



(T), (9.105)

for some other function g



where g



(x) = g(x)x

−5

. From this form of the

spectral distribution we ﬁnd that the only dependency on the wavelength is

contained in the proportionality to g



(T). The maximum of g



(x) corresponds

to some x

. This then corresponds to some wavelength 

with

I 

= x

/T. (9.106)

This is Wien’s displacement law with x

the displacement constant.

9.9 DERIVATION OF THE PLANCK LAW

One cannot help but be impressed by the thermodynamic arguments leading

to Eq. 9.104, or the equivalent Eq. 9.105. These equations contain the Stefan–

Boltzmann law and the Wien displacement law. However, thermodynamics

alone cannot take us any further. The derivation of the Planck law itself

requires quantum mechanical concepts. We present its derivation here as it is

one of the classic results in physics. This by now canonical derivation is based

on statistical mechanics, which employs the Boltzmann factor introduced in

Section 4.6.

We consider a standing electromagnetic wave of wavelength  in a cubic

vacuum vessel with perfectly reﬂecting walls and with edge size L. The elec-

tromagnetic wave corresponds to a photon with energy, e, which is related

to the wavelength, , as

e = hc/. (9.107)

Introducing the total wavenumber K ,with K = 2, this is written

e =

2

Kc. (9.108)

The wavenumber K is the length of the three-dimensional wave vector K =

For further reading see Kittel C. & Kroemer, H. (1980) Thermal physics 2nd edn. W.

H. Freeman, New York; Baierlein, R. (1999) Thermal physics Cambridge University Press,

Cambridge.

196 CH 9 THERMAL RADIATION

We ﬁrst need to know how many wavevectors of total wavenumber K there

are. In a cubic vessel with reﬂecting walls, the components K

of the wavevec-

tor can only have discrete values such that precisely N

half wavelengths ﬁt

inside the vessel size L; for example, see Figure 9.11. We therefore have

= N

/L. (9.109)

So for each one-dimensional wavenumber /L we can ﬁt one wavenumber

. An elementary volume of the size (/L)

in three-dimensional wavenum-

ber space therefore contains one three-dimensional wavenumber mode. Pho-

tons also have an internal degree of freedom called spin. For a photon the

spin can be either up or down. So each elementary volume in wavenumber

space corresponds to two possible photon modes – each wavenumber mode

has a degeneracy of 2.

The wave vectors with total wavenumber between K and K+dK represent a

three-dimensional spherical shell in wave-vector space with volume 4K

dK.

Because all wavenumber components K

are assumed positive, only 1/8 of this

shell contributes to the total wavenumber volume. An elementary volume of

(/L)

contains two degenerate photon modes so the number of modes g

corresponding to the shell between total wavenumber K and K + dK is

dK =



dK. (9.110)

We now express this as the number of modes g

de with photon energies

between e and e + de. We have

de = g

dK. (9.111)

Using Eq. 9.108 we now ﬁnd that

L

. (9.112)

This function is called the density of states of the energy level.

So what is the probability of ﬁnding N photons at some mode with energy

e? The Boltzmann factor exp (−e/k

T) gives the probability of ﬁnding a

photon in a single mode of energy e, see Section 4.6. The probability P

(n)de

of ﬁnding n photons between energies e and e + de therefore is

(n) = g

exp (−ˇne)

, (9.113)

The argument presented here is not complete in quantum mechanics as we ignore the

ground level of each wave mode. However, this does not change the phase space density

, which is the purpose of the quantization presented here.