Ambaum M., Thermal Physics of the Atmosphere

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62 CH 3 GENERAL APPLICATIONS

a diatomic ideal gas (such as air, approximately) we have, from Eqs. 3.28

and 3.29

 =

. (3.85)

A monoatomic gas has  = 2/5. Combining the deﬁnitions of speciﬁc entropy

and potential temperature for an ideal gas we can verify that

I s = c

ln (/T

), (3.86)

with T

a constant temperature that only changes the speciﬁc entropy by an

additive constant. From this equation immediately follows the differential

relation, Eq. 3.80, as applied to an ideal gas.

We can combine the deﬁnition of potential temperature for an ideal gas

with the ideal gas law to ﬁnd

T





, (3.87)

with p

= 

R. From this we can derive the following relationships that

are valid for adiabatic transformations of an ideal gas, sometimes called the

Poisson equations:













. (3.88)

So these relationships quantify how pressure and temperature change on adi-

abatic compression and expansion. For example, an adiabatic compression

of air by a factor of 2 leads to an increase in pressure by a factor of 2.6 and

an increase in temperature by a factor of 1.3. Anyone who has ever used a

bicycle pump will be aware of these effects.

Although in atmospheric science it is common usage to map speciﬁc entropy

onto potential temperature, this is not without its drawbacks. Because of

its widespread use in atmospheric science we will not avoid it in this text.

But there are no sound physical arguments why we should use potential

temperature instead of speciﬁc entropy in any of the applications encountered

here.

Perhaps part of the attraction of using potential temperature is its appar-

ent measurability: it is the temperature a ﬂuid parcel would have if brought

adiabatically to a reference pressure. This is a useful thought experiment

but no one has ever used it as a practical method to measure the potential

temperature of a ﬂuid parcel.

The notion of potential temperature may distract from the more funda-

mental notions of entropy and heat. Derivations using potential temperature

are usually less intuitive than those using entropy. If carelessly applied, they

3.8 POTENTIAL TEMPERATURE 63

give less general results: potential temperature is often read as the ideal gas

potential temperature but, for example, a moist parcel is not an ideal gas.

We are then tempted to introduce such notions as equivalent potential tem-

perature or wet-bulb potential temperature (see Chapter 6), which in fact

are not potential temperatures in the sense of Eq. 3.78. In fact, a proper

deﬁnition of potential temperature for a moist parcel requires the precise

deﬁnition of how the parcel was brought reversibly to the reference pressure.

In other words, there is no unique local deﬁnition of potential temperature.

We can easily avoid any convoluted deﬁnitions and derivations by sticking to

the more fundamental notion of entropy.

There is also a more philosophical drawback. Like speciﬁc entropy, poten-

tial temperature is an intensive quantity; it does not scale with the size of

the system. However, speciﬁc entropy is based on the extensive total en-

tropy. Entropies can be added; subsystems can be joined together and we

can calculate the total entropy of the compound system. No such notion

exists for potential temperature; temperatures cannot be added together.

It appears somewhat illogical to replace a variable based on an extensive

quantity, such as entropy, with a fundamentally intensive quantity such as

temperature.

ROBLEMS

3.1. Legendre transforms. Consider a function f of x: f = f (x). Now deﬁne

a new variable p by

p =

Show that the function g, deﬁned as

g = xp − f

can be seen as a natural function of p (show that dg = x dp). The

function g is called the Legendre transform of function f. In the same

way, show that the function

f deﬁned as

f = xp − g

is a natural function of x. Show that

f is the same as the original f ;in

other words, show that the Legendre transform is its own inverse. Suc-

cessive Legendre transforms of the internal energy deﬁne the different

thermodynamic potentials.

3.2. A water molecule H

O is made up of two hydrogen atoms attached to a

central oxygen atom at an angle of about 105 degrees (

). Suppose

64 CH 3 GENERAL APPLICATIONS

that at room temperature, the vibrational modes are not excited. What

is the expected speciﬁc heat capacity at constant pressure c

for water

vapour? (The observed value is 1.87 × 10

Jkg

−1

3.3. Show that the speciﬁc heat capacity at constant volume for moist air can

be written as

v(moist)

= (1 + q (a

− 1)) c

v(dry)

with q the speciﬁc humidity and a

= c

v(vapour)

v(dry)

≈ 1.96, the ratio

of the heat capacities at constant volume for water vapour and dry air.

Demonstrate the validity of the analogous equation for c

p(moist)

with,

instead of a

, an isobaric heat capacity ratio a

≈ (20/21)a

3.4. The isobaric thermal expansivity and the isothermal compressibility are

deﬁned as



∂v

∂T



,ˇ

=−



∂v

∂p



Show that the relationship between c

and c

can be rewritten as

= c

+ vT˛

/ˇ

Hint: Use the reciprocity relation for partial derivatives, see Appen-

dix A. Find relevant tabulated values for water to show that the iso-

baric and isovolumetric heat capacities of water are typically less than

1% apart.

3.5. Show that the isentropic compressibility ˇ

and the isothermal compress-

ibility ˇ

are related through

= ˇ

+ vT˛

and hence ˇ

/ˇ

= c

Hint: Write the differential of v as a function of p and T and hence

calculate (∂v/∂p)

. What is the physical reason why ˇ

is larger than ˇ

3.6. Show that the speciﬁc entropy of a solid or liquid with speciﬁc heat

capacity c can be approximated as

s = c ln (T/ T

3.7. Use the equation for the speed of sound in an ideal gas, Eq. 3.45, to

explain why the pitch of your voice goes up after breathing in helium

gas. Show that this equation predicts a pitch change of close to one

octave plus a perfect ﬁfth.

3.8 POTENTIAL TEMPERATURE 65

3.8. Show that the speed of sound in humid air C

can be approximated by

= C (1 + 0.26 q),

with C the speed of sound in dry air at the given temperature and q

the speciﬁc humidity. Hint: See Problem 3.3. In light of the above

equation, can we measure temperature in the ﬁeld by measuring the

speed of sound between two points?

3.9. Show that van der Waals’ equation, Eq. 3.51, can be rewritten as

p = RT



1 − b/

−

a





with  the molar mass of the substance, R its speciﬁc gas constant, and

a and b its van der Waals coefﬁcients. Hence show that the second and

third virial coefﬁcients for a van der Waals gas are given by Eq. 3.53.

3.10. Concorde used to cruise at a pressure of around 100 hPa, where the typ-

ical air temperature is around −50

◦

C. If the cabin pressure in Concorde

is 850 hPa, what would the consequences be of adiabatic compression

of the outside air to provide ventilation?

The atmosphere under

gravity

The gravitational potential energy of an air parcel varies considerably over the

depth of the atmosphere when compared to variations in internal energy. This

means that the gravitational potential energy has to be part of the thermody-

namic energy budget when considering vertical structure in the atmosphere.

Hydrostatic balance is the ﬁrst manifestation of the gravitational potential

energy in ﬂuid systems. It reﬂects the fact that the pressure at any point in a

static ﬂuid is determined by the weight of the ﬂuid above it. In this chapter

we give a general thermodynamic derivation of hydrostatic balance.

When the air is in motion, hydrostatic balance is only valid as an approx-

imation. In dynamic terms, hydrostatic balance requires that the vertical ac-

celeration of air parcels is much smaller than the acceleration of gravity. In

thermodynamic terms, hydrostatic balance requires that the kinetic energy

variations are much smaller than potential energy or internal energy vari-

ations. These conditions are essentially equivalent and are usually met in

the atmosphere. Only in strong updrafts or downdrafts is an assumption of

hydrostatic balance expected to be unwarranted.

The rest of the chapter is devoted to exploring what the assumption of

hydrostatic balance implies for the vertical structure of the atmosphere.

The chapter concludes with a discussion of the vertical structure of an

isothermal atmosphere in the context of statistical mechanics. Some key con-

cepts in statistical mechanics are introduced, illustrating the way statistical

mechanics can be used to link the microscopic world of molecules to the

macroscopic world of thermodynamics.

4.1 GEOPOTENTIAL

The acceleration due to gravity is a vector g, which can be written as the

gradient of a geopotential  as

I g =−g

k =−∇, (4.1)

Thermal Physics of the Atmosphere Maarten H. P. Ambaum

68 CH 4 THE ATMOSPHERE UNDER GRAVITY

with

k the unit vector pointing vertically upward and g the magnitude of

the acceleration. The geopotential is in fact used to deﬁne the local vertical:

vertically upward means parallel to the gradient of the geopotential. Because

the vertical direction (spanned by the z-coordinate) is deﬁned to be parallel to

the gradient of , the magnitude of the gravitational acceleration can always

be written as

g =

d

. (4.2)

For simplicity, cartesian geometries with ﬁxed g are often used. In this case

 = gz, (4.3)

which is a good approximation near the Earth’s surface. The energy required

to lift a parcel of mass M against gravity over some small distance dr equals

−Mg · dr. So to move a parcel from some initial location r

to some ﬁnal

location r

requires energy

−



Mg ·dr =



M∇ ·dr = M((r

) − (r

)). (4.4)

This demonstrates that geopotential is the same as the speciﬁc gravitational

potential energy.

Around a spherical gravitating body, such as the Earth, the geopotential is

(see Problem 4.1)

I  = g

a + z

, (4.5)

with a the radius of the body, z the height above the surface of the body,

and g

the acceleration of gravity at the surface of the body. For the Earth,

a = 6371 km and g

= 9.8ms

−2

. For small z it is clear that  = g

Geopotential is often translated into geopotential height Z, which is deﬁned

I Z = /g

. (4.6)

So geopotential height has the units of height but it is in fact a geopoten-

tial. The relationship between Z and z is illustrated in Fig 4.1. The differentials

of Z and z are related by

dZ = g dz. (4.7)

4.2 HYDROSTATIC BALANCE 69

FIGURE 4.1 Isolines of geometric height z, left, and geopotential height Z, right, around the

Earth’s surface, both drawn every 100 (geopotential) km. Near the surface the geometric

and geopotential heights are nearly the same.

For a spherical gravitating mass the geometric height and the geopotential

height transform into each other by

Z =

a + z

and z =

a − Z

. (4.8)

Near the Earth’s surface the geopotential height is very similar to the ge-

ometric height, with the difference increasing for increasing height. The

geopotential height is smaller than the geometric height at any point in the

atmosphere. Geometric height can have any value larger than zero while

geopotential height has a maximum value of a.

The atmosphere is usually considered in a coordinate frame rotating with

the Earth. In this frame of reference, a ﬁxed centrifugal acceleration will add

to the purely gravitational acceleration. This can be taken into account by

deﬁning an effective gravity and an effective geopotential that include this

centrifugal acceleration, see Problem 4.2. The local vertical in the rotating

frame of reference is now deﬁned by the gradient of the effective geopoten-

tial. From now on, ‘effective’ is omitted and we shall simply refer to geopo-

tential or gravitational acceleration.

The main effect of rotation is that the geopotential surfaces are not spheres

but oblate spheroids. The Earth’s surface is, apart from orographic features,

a geopotential surface. It is an oblate spheroid with a polar radius about

1/300 smaller than its equatorial radius. The transformations in Eq. 4.8 are

therefore only approximate for a rotating planet.

4.2 HYDROSTATIC BALANCE

In a ﬂuid at rest the net force on any ﬂuid parcel has to vanish. So in a

gravitational ﬁeld the gravitational force has to be balanced by an internal

force due to a decrease of pressure with height. This is called hydrostatic

balance.

70 CH 4 THE ATMOSPHERE UNDER GRAVITY

We can use thermodynamics to calculate the hydrostatic pressure decrease

with height.

Think of the geopotential ﬁeld as made up of thin layers of a

given geopotential. By symmetry, in hydrostatic balance all thermodynamic

variables are functions of the geopotential alone. We now examine the en-

ergy budget when moving a body of ﬂuid from one geopotential to a nearby

geopotential without putting any heat in. If it was possible to generate energy

through such a process then the atmosphere would be unstable: any small

perturbation of the atmosphere could release internal and potential energy

to create kinetic energy.

We can now view the initial geopotential layer (layer 1) and the ﬁnal

geopotential layer (layer 2) as open systems with matter ﬂowing from one to

the other. From Section 3.5 we know that the change in internal energy ıU

of the whole system on moving the ﬂuid from layer 1 to layer 2 has to equal

the enthalpy gain of layer 2 minus the enthalpy loss of layer 1. The enthalpy

change includes the ﬂow work that is required to move the ﬂuid from layer 1

to layer 2. Because we add no heat to the system, the enthalpy change can

only be due to a pressure change, that is, dH = V dp. We thus ﬁnd

ıU = H

− H



d

d. (4.9)

The change in potential energy, ıP, is

ıP = M (

− 

) =



M d, (4.10)

with M the mass of the displaced ﬂuid. At equilibrium, energy variations

must vanish because otherwise we could generate kinetic energy by some

spontaneous rearrangement of the vertical proﬁle. We thus have

0 = ıU + ıP =





d

+ M



d. (4.11)

Because the two geopotential surfaces were chosen randomly, the integrand

has to vanish. We therefore ﬁnd, after dividing the integrand by V,

d

=−. (4.12)

Writing out the geopotential variations in terms of the gravitational acceler-

ation,

=−g or

=−g

, (4.13)

Ambaum, M. H. P. (2008) Proc. Roy. Soc. A 464, 943–950.

4.2 HYDROSTATIC BALANCE 71

where Eqs. 4.2 and 4.7 have been used. This equation is called the hydrostatic

equation. It is fully general for any ﬂuid system at rest and was ﬁrst derived

(essentially in its integral form, but using a method based on dynamical ar-

guments) by Newton in his Principia. The above thermodynamic derivation

makes no reference to the geometry of the ﬂuid and is therefore more general

than some shorter derivations that are incomplete due to the implicit use of

a ﬂat geometry.

We have assumed that the ﬂuid in our thought experiment is at rest; that

is, there is no kinetic energy. Indeed, if the ﬂuid is in motion, hydrostatic bal-

ance is not strictly valid any longer. However, the atmosphere is usually very

close to hydrostatic balance, despite the motions in the atmosphere. From a

dynamical point of view, as long as vertical accelerations are substantially

smaller than g, the assumption of hydrostatic balance provides an accurate

representation of the atmosphere. From a thermodynamic point of view, as

long as vertical variations in internal or gravitational potential energy are

substantially larger than variations in kinetic energy, hydrostatic balance is

accurate. This depends on the vertical scale considered. For example, the

variation of speciﬁc gravitational potential energy over the depth of the tro-

posphere is about 10

Jkg

−1

. The speciﬁc kinetic energy is V

/2, with V the

ﬂow speed. The variations in speciﬁc kinetic energy in the atmosphere are

much smaller than this variation in gravitational potential energy. However,

the variation of speciﬁc gravitational potential energy over one metre depth

is about 10 J kg

−1

. This is comparable to observed variations in speciﬁc ki-

netic energies, especially in areas of substantial updrafts. So for such scales,

hydrostatic balance is not accurate anymore.

For an ideal gas the hydrostatic relation becomes

=−

d

=−

. (4.14)

This means that if the temperature as a function of height is known then the

pressure as a function of height can be found. Often, the above equation is

reversed:

d

dlnp

=−RT. (4.15)

This equation gives a relationship between geopotential and the (logarithm

of) pressure. This equation is useful because radiosondes normally measure

both pressure and temperature; the above equation is then used to calculate

the geopotential height of the radiosonde.

For moist air, we need to replace the temperature in these equations by the virtual

temperature, see Section 1.3.

72 CH 4 THE ATMOSPHERE UNDER GRAVITY

An interesting application of hydrostatic balance is to assume a constant

temperature T

. The hydrostatic equation then integrates to

p = p

−Z/H

with H =

(4.16)

with p

the pressure at Z = 0. This equation is strictly only valid for geopoten-

tial height Z because g varies with height. A typical value for H, the so-called

scale height, for the atmosphere is 8 km. Constant temperature is obviously

not the most accurate approximation in the atmosphere. Nevertheless, the

pressure still decays with height close to exponentially, with a scale height

fairly close to 8 km.

Although the precise height–pressure relationship depends on the temper-

ature proﬁle it is useful to be aware of approximate values. For illustration,

Table 4.1 shows typical values of height, temperature, pressure, and density

for the 1976 US Standard Atmosphere, a ﬁctional atmosphere with ‘typical’

mid-latitude values of the temperature proﬁle.

Another application is to assume a ﬁxed lapse rate. Lapse rate, usually de-

noted , is the negative derivative of temperature with height or geopotential

TABLE 4.1 Selected values from the US Standard Atmosphere, 1976. Here, the 1013.25 hPa

level provides the reference values for potential temperature  and speciﬁc entropy s.

p (hPa) z (km) T (

◦

C)  (kg m

−3

)  (K) s (J kg

−1

)

1013.25 0.00 15.00 1.225 288.15 0.00

1000 0.11 14.3 1.21 289 1

925 0.76 10.1 1.14 291 9

850 1.5 5.5 1.06 293 17

700 3.0 −4.6 0.91 298 35

600 4.2 −12.3 0.80 303 50

500 5.6 −21.2 0.69 308 68

400 7.2 −31.7 0.58 315 89

300 9.2 −44.6 0.46 324 117

250 10.4 −52.4 0.39 329 134

200 11.8 −56.5 0.32 344 179

150 13.6 −56.5 0.24 374 262

100 16.2 −56.5 0.16 420 378

70 18.5 −56.5 0.11 465 481

50 20.6 −55.9 0.080 513 580

30 23.9 −52.6 0.047 603 741

20 26.6 −50.0 0.031 684 870

10 31.2 −45.5 0.015 852 1089

5 36.0 −33.9 0.0073 1091 1338

2 42.7 −15.3 0.0027 1527 1676

1 48.2 −2.5 0.0013 1954 1924