Fowler A. Mathematical Geoscience

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160 3 Oceans and Atmospheres

so that to leading order, mass and momentum conservation equations are

−v =−

¯ρ

∂P

∂x

∂

∂Z

u =−

¯ρ

∂P

∂y

∂

∂Z

, (3.93)

∂u

∂x

∂v

∂y

¯ρ

∂( ¯ρW)

∂Z

=0;

also ∂P/∂Z =O(ε), so that in common with other viscous boundary layers, we can

take P =P(x,y), and equal to the free stream value at the surface.

Using (3.86), and denoting the surface values of the free stream velocity as u

and

,(3.93) can be elegantly solved subject to no slip on the boundary and attainment

of the free stream velocities as Z →∞in the form

u +iv = (u

+iv

)



1 −exp



−

(1 +i)εZ

√



. (3.94)

This solution is known as the Ekman spiral, as the horizontal velocities spiral round

as they approach the free stream velocity.

Of later importance will be the change of W across the Ekman layer. Integration

of (3.93)

from Z = 0toZ =∞(bearing in mind that ¯ρ = 1 +O(ε)) yields the

value of W

,thevalueofW at the edge of the boundary layer:



2ε



∂v

∂x

−

∂u

∂y



. (3.95)

This generation of a vertical velocity by the free stream vorticity is known as Ekman

pumping.

3.4 Poincaré and Kelvin Waves

The geostrophic wind given by Eq. (3.89) is an approximate solution to the govern-

ing equations which is quasi-static, in the sense that the acceleration terms in the

momentum equation are ignored; implicitly, any more rapid transients have died out.

Before we proceed to the higher order approximation which it is necessary to take

in order to determine the perturbed pressure in (3.89), we consider various classes

of wave motion which arise in the model on this shorter transient time scale.

Atmospheric motions are dominated by various kinds of waves. Two particular

sorts of waves which are familiar in ﬂuid mechanics are sound waves and gravity

waves. Sound waves are associated with compressibility; they travel at a speed (the

speed of sound) which depends on density but is independent of wave number: they

are monochromatic. At sea level this speed is about 330 m s

−1

: much faster than

typical wind speeds; as a consequence, we might expect sound waves to be high

3.4 Poincaré and Kelvin Waves 161

frequency phenomena which are not relevant to common atmospheric motions. If

we denote the sound wave speed as c

, then the dispersion relation relating fre-

quency ω to wave speed and wave number k is just ω =kc

. When this is written in

dimensionless units, as above, we have

=gh¯c

, ¯c =



d ¯p

d ¯ρ



1/2

, (3.96)

and the corresponding dimensionless dispersion relation is just

¯c

, (3.97)

where F is the Froude number deﬁned by (3.48). Note from (3.73) that F =ε

3/2

Gravity waves are familiar as the waves which propagate on the surface of the

sea. The ingredients of the theory which describes them are mass conservation

(where horizontal divergence is accommodated by vertical contraction and expan-

sion), acceleration, gravity, pressure gradient, and a vertical stratiﬁcation which, in

the simplest form of the theory, is manifested by the interface between dense un-

derlying ﬂuid (e.g., water) and a lighter overlying ﬂuid (e.g., air). Gravity waves

can be seen propagating at the interface between two incompressible liquids such as

oil and water, and gravity waves will similarly propagate in a continuously stratiﬁed

ﬂuid contained in a vertically conﬁned channel; in this case the waves are less easily

visualised, and they are often called internal waves, or internal gravity waves.

In the sense that the atmosphere consists of a dense troposphere beneath a

light stratosphere, we can expect gravity waves to propagate as undulations in the

tropopause altitude. More generally, gravity waves will propagate as internal waves

in the stratiﬁed atmosphere. Gravity waves can be seen commonly in the atmo-

sphere, because the vertical undulations of the air causes periodic cloud formation

as air rises (and thus cools). Figure 3.2 shows a particular striking example from

Lapland of low lying periodic gravity waves.

For the simple case of an incompressible ﬂuid of depth h, the dispersion re-

lation between frequency and wave number is ω

= gk tanh kh. In the case of a

shallow ﬂuid (such as the atmosphere), the long wave limit kh  1 may be appro-

priate, and then the wave speed is constant, and ω ≈ k

√

gh. This applies to waves

of wavelength larger than 10 km (the waves in Fig. 3.2 are of smaller wavelength).

In dimensionless terms, the dispersion relation becomes

. (3.98)

Comparing (3.98) with (3.97), we see that long gravity waves in the atmosphere

are essentially the same as sound waves. In an incompressible ﬂuid, density is man-

ifested as ﬂuid column depth, and the pressure is proportional to this, so that the

dimensionless ‘sound’ speed is equal to one. For internal waves, the height of the

column need not change, but the common factor is that the height of geopotential

surfaces propagates in both types of wave.

162 3 Oceans and Atmospheres

Fig. 3.2 Periodic gravity waves in Lapland, Northern Finland, October 2004

We can recover gravity waves from the scaled atmospheric model by focussing

on long waves of wave number k ∼O(

√

ε), and time scales of O(ε) (i.e., frequen-

cies ω ∼ O(1/ε)). (Note that then ω/k ∼ 1/ε

3/2

= 1/F , from (3.73) and (3.74),

consistent with (3.97) and (3.98).) We write

t =ετ, (x,y) =(X, Y )/

√

ε, P =

√

(3.99)

(note that P is deﬁned in (3.85)), and retain leading order terms in Eqs. (3.81)

and (3.84), assuming that w ∼ ε. Note that ρ = p

1−α

/θ, and that ∂θ/∂t ≈ 0, so

that

∂ρ

∂t

≈



1 −α



∂p

∂t

. (3.100)

At leading order, mass conservation takes the form



1 −α

¯p



∂Π

∂τ

∂u

∂X

∂v

∂Y

=0; (3.101)

compressibility and stratiﬁcation are manifested by the ﬁrst term in this equation.

3.4 Poincaré and Kelvin Waves 163

At leading order, the momentum equations take the form

∂u

∂τ

−v ≈−

¯ρ

∂Π

∂X

∂v

∂τ

+u ≈−

¯ρ

∂Π

∂Y

(3.102)

We can write these equations in terms of the horizontal divergence Δ =u

the vorticity ζ =v

−u

, and the pressure perturbation Π . We obtain

∂Δ

∂τ

−ζ =−

¯ρ

∇

Π,

∂ζ

∂τ

+Δ = 0, (3.103)

∂Π

∂τ

+¯ρ ¯c

Δ = 0,

where

¯c =



¯p

(1 −α) ¯ρ



1/2

(3.104)

is the dimensionless isentropic sound speed.

These are linear equations, and solutions exist of the form

⎛

⎝

⎞

⎠

=w exp



i(kX +lY +ωτ )



, (3.105)

provided

⎛

⎜

⎝

)

¯ρ

−10 0

−¯ρ ¯c

⎞

⎟

⎠

w =iωw. (3.106)

Solutions to this exist provided either ω =0, or

=1 +





¯c

, (3.107)

and this latter equation is the dispersion relation for gravity waves in a rotating

stratiﬁed atmosphere. These waves are called Poincaré waves.

Another kind of wave can be found by seeking solutions in which v = 0. Such

waves are particularly relevant to propagation in a conﬁned zonal channel (for exam-

ple in the ocean), where the condition v =0 at the north and south boundaries forces

v =0 everywhere. This requires ∂Δ/∂Y =−∂ζ/∂X, and substitution into (3.106)

then shows that we must have l =−ik/ω, and thus solutions are exponential in y,

and

ω =k ¯c; (3.108)

164 3 Oceans and Atmospheres

these waves are called Kelvin waves.Theyareedge waves, because they decay ex-

ponentially away from one or other boundary. Together with the geostrophic mode

ω = 0, Poincaré and Kelvin waves form the complete spectrum of waves for the

ﬂow. The mode ω =0 is associated with low frequency waves which emerge in the

higher order quasi-geostrophic approximation (which is derived in the next section);

these slow waves are called Rossby waves,orplanetary waves.

The constant term in (3.107) arises from rotation and the Coriolis force. In the

high frequency limit, we see that ω ≈ k ¯c (for unidirectional waves), and this is con-

sistent with the long wave limit of gravity wave theory, and the acoustic wave speed

given in (3.97). Gravity waves are essentially long wavelength sound waves, and

Poincaré waves are their modiﬁcation by the effects of rotation. The critical length

scale l/

√

ε above which rotation becomes important is known as the Rossby radius

of deformation. Using (3.76), it is found to be equal to

√

gh/f . For atmospheric mo-

tion, it is of order 3000 km, so that rotation is unimportant for smaller scale gravity

waves, such as those in Fig. 3.2.

3.5 The Quasi-geostrophic Approximation

We now return to the problem of ﬁnding the pressure for the geostrophic approxima-

tion in which (3.89) applies. To do this, we need to carry the approximation to next

order in ε, and this will allow us to deduce the quasi-geostrophic potential vorticity

equation. The equation of mass conservation (3.84) can be written in the form

∂ρ

∂t

+μ

∂(ρu)

∂x

+μ

∂(ρv/μ)

∂y

∂(ρw)

∂z





. (3.109)

Since w =0 at leading order, we put

w =εW. (3.110)

We also deﬁne the perturbed potential temperature Θ by

θ =

θ(z)+ε

Θ; (3.111)

evidently

θ(z) is the time and space-horizontal average of θ correct to O(ε

), and

we can in fact deﬁne it to be the exact such average of θ, without loss of generality.

More generally, we might take

θ =

θ(z,t), but the energy equation then simply im-

plies that

=0. We might have expected

θ to be equal to the wet adiabatic potential

temperature θ

, deﬁned in (3.71), but as we shall see, there is a subtle distinction,

and it is necessary to delineate the difference in the equations. Because the hydro-

static correction in (3.78)isO(ε

), expansion of that equation to O(ε

) yields the

hydrostatic approximation for the perturbation pressure P , deﬁned in (3.85):

Θ =

∂

∂z



¯p

1−α



. (3.112)

3.5 The Quasi-geostrophic Approximation 165

The geostrophic wind approximation (3.86) suggests that we write

P =¯ρψ, (3.113)

where ψ is the geostrophic stream function, thus

u =−

∂ψ

∂y

,v=

∂ψ

∂x

. (3.114)

Bearing in mind that ¯ρ =¯p

1−α

θ, it follows that

Θ =

∂

∂z





∂ψ

∂z

+O(ε), (3.115)

on the assumption that



(z) = O(ε). This relation, together with the geostrophic

wind approximation, gives us the thermal wind equations:

∂u

∂z

=−

∂Θ

∂y

∂v

∂z

∂Θ

∂x

. (3.116)

Next we form an equation for the (vertical) vorticity

ζ =

∂v

∂x

−

∂u

∂y

=∇

ψ (3.117)

by cross differentiating (3.81) (with some care) to eliminate the pressure derivatives.

Using the conservation of mass equation, together with (3.110) and the fact that

ρ =¯ρ(z) +O(ε

), we derive the vorticity equation

Dζ

+β

∂ψ

∂x

¯ρ

∂( ¯ρW)

∂z

, (3.118)

where D/Dt denotes the horizontal material derivative, and the term in β arises

from the variation of sin λ with latitude; β is deﬁned by

β =

Σ cotλ

, (3.119)

and the horizontal material derivative is deﬁned by

∂

∂t

∂

∂x

∂

∂y

∂

∂t

−

∂ψ

∂y

∂

∂x

∂ψ

∂x

∂

∂y

. (3.120)

Next, we consider the energy equation (3.82). Expanding in powers of ε, this can

be written in the form, correct to terms of O(ε

εW

+ε

DΘ

=εW

dθ

+ε

H, (3.121)

166 3 Oceans and Atmospheres

where

H =

∂

∂z



∗

∂

∂z



¯p



1 +

νSt aM(

T,¯p)



(3.122)

is the heating term.

Now we can see the nature of the assumption about the average potential tem-

perature. Bearing in mind that dθ

/dz =O(ε), we see that the ansatz that d

θ/dz=

O(ε) is indeed correct. However, it is generally not the case that

θ =θ

. The ques-

tion then arises how to determine it.

Let us denote the stratiﬁcation function S(z) by

S(z) =



−

dθ



, (3.123)

and note that by observation (and assumption) it is positive and O(1). It is related to

the Brunt–Väisälä frequency N, which is the frequency of small vertical oscillations

in the atmosphere; in fact S ∝ N

. Positive S (and thus real N ) indicates a stably

stratiﬁed atmosphere. If S were to become negative, the atmosphere would become

unstably stratiﬁed and it would overturn. The energy equation is thus

DΘ

=H −WS. (3.124)

In summary, we have the vorticity ζ and potential temperature Θ deﬁned in terms

of the stream function ψ by (3.117) and (3.115). Two separate equations for ζ and

Θ are then (3.118) and (3.124), from which W and S(z) must also be determined,

the latter by averaging the equations.

By an application of Green’s theorem in the plane, we have



DΓ

dS =

∂

∂t



ΓdS−



∂A

Γdψ, (3.125)

where A is any horizontal area at ﬁxed z. In particular, if A is a closed region on the

boundaries of which ψ is constant in space, i.e., there is no ﬂow through ∂A, then

the boundary integral is zero.

Let an overbar denote a space horizontal average

over A. Putting Γ =Θ, it follows that

∂

∂t

=H −

WS, (3.126)

We have in mind that A is the region of zonal mid-latitude ﬂow, bounded to the north by the polar

front, and to the south by the tropical front. We can allow A to be a periodic strip on the sphere

also.

3.5 The Quasi-geostrophic Approximation 167

where W(z)is the horizontal average of W . Applying the same procedure to (3.118),

we have

∂

∂t

¯ρ

∂

∂z

[¯ρ

W ]. (3.127)

According to the Ekman pumping boundary condition (3.95), the value of

W at

z =0is

∗

, (3.128)

where

is the space-averaged vorticity at the surface, and

∗



2ε

. (3.129)

Integrating (3.127), we have (using ¯ρ =1atz =0)

¯ρ

W =



¯ρ

dz +E

∗

, (3.130)

and it follows from this that the stratiﬁcation parameter is deﬁned by the relation

¯ρ



¯ρ

dz +E

∗

H −

. (3.131)

We can go further if we assume that the solutions are stationary (not necessarily

steady), i.e., a well-deﬁned time average exists.

The time averages of the time

derivative terms are zero, and thus it simply follows (since H , S and ¯ρ are functions

only of z) that

H =



WS, ¯ρ



W =



, (3.132)

where



W is the time average of

W , and the constant



is the value of the surface

boundary value of



W at z =0.

The Ekman pumping boundary condition (3.128)

implies that



∗



, (3.133)

where

is the space-averaged vorticity at the surface.

This is what we would generally expect. Unbounded drift of ψ would indicate breakdown of the

perturbation expansion because of the presence of secular terms.

The question arises at this point, why can we not take W

, and thus



, equal to zero? W

the average (scaled) vertical velocity above the planetary boundary layer. If it is not zero, then

apparently there would be a non-zero mass ﬂux into or out of this layer. While that is feasible,

the time average should apparently be zero, unless there is secular growth or decline of the layer

thickness. This would follow from (3.109) were we dealing with the exact horizontal velocities

(u, v). However, the geostrophic stream function prescription in (3.114) is only accurate to O(ε),

and thus if we were to use (3.109) to calculate W = w/ε, there would be an (unknown) O(1)

contribution from the corrections to the horizontal velocities. The point is that we cannot actually

use mass conservation to determine W (and thus we cannot use a natural inference such as



=0).

168 3 Oceans and Atmospheres

The two equations in (3.132) deﬁne S and



W , and in particular we ﬁnd that

¯ρ

∗



. (3.134)

This equation thus deﬁnes the stratiﬁcation function S(z) for a stationary (but

not necessarily steady) atmosphere.

Evidently, the wet adiabatic proﬁle (S =0) is

obtained (in stationary conditions) only if the heating rate H is zero.

We can now use the identity

∂

∂z



K(z)

DΘ





∂

∂z



K(z)

∂ψ

∂z



(3.135)

to show, using (3.124), that

¯ρ

∂

∂z

[¯ρW]=

¯ρ

∂

∂z



¯ρH



−



¯ρ

∂

∂z



¯ρ

∂ψ

∂z



, (3.136)

and therefore (3.118) can be written



∇

ψ +βy +

¯ρ

∂

∂z



¯ρ

∂ψ

∂z



¯ρ

∂

∂z



¯ρH



. (3.137)

This is one form of the quasi-geostrophic potential vorticity equation. It is a single

equation for the geostrophic stream function ψ, providing the stratiﬁcation S is

known. In most treatments of its solutions, the stratiﬁcation parameter S is assumed

known (from measurements), and then the Eq. (3.137) can be considered on its own.

3.5.1 Boundary Conditions

We wish to solve the quasi-geostrophic equation for ψ in a geometric domain con-

sisting of a rectangular channel, representing roughly the mid-latitude cell. It is

simplest to think of ﬁxed boundaries at y =±1, for example, although moving

boundaries (adjoining the Hadley and polar cells) are more appropriate. We sup-

pose the ﬂow is unbounded in the x-direction (the circumference is of O(1/ε), and

thus large). Finally the ﬂow is bounded by an interface at the tropopause, across

which pressure and density are continuous, but temperature gradient is effectively

discontinuous, as a consequence of the different stratospheric thermal régime.

This derivation is somewhat similar to that of Pedlosky (1987); however, he did not provide an

explicit recipe for S(z). See also Question 3.8.

The temperature gradient is in fact continuous; indeed the temperature condition at the

tropopause is a suitably dimensionless version of the ﬂux condition (3.21); but heat conduction

is provided by a singular highest derivative term, so that the energy equation is essentially conduc-

tionless. It is a consequence of this that we may consider the temperature gradient to be discontin-

uous across the tropopause.

3.5 The Quasi-geostrophic Approximation 169

The basic model, (3.1), is one of inviscid ﬂow in a shallow layer with a free

boundary, driven by an imposed poleward temperature gradient due to solar insola-

tion. Consequently, we expect to provide velocity conditions of no ﬂow through the

base; but, as discussed in Sect. 3.3, the planetary viscous boundary layer induces a

non-zero Ekman velocity above it given by (3.95), so that the boundary condition

for (3.118) is in fact

W =E

∗

∇

ψ on z =0. (3.138)

Other conditions of this type are no-ﬂow-through conditions at the side walls

y =±1, and boundedness or periodicity conditions in the x-direction. An initial

condition for the quasi-geostrophic potential vorticity

q =∇

ψ +βy +

¯ρ

∂

∂z



¯ρ

∂ψ

∂z



(3.139)

is sufﬁcient for (3.137), and if q is known, then the periodicity or no-ﬂow-through

conditions in x and y will provide the necessary horizontal boundary conditions

to solve the elliptic (3.139)forψ. However, we also need to specify two vertical

conditions for ψ at the tropopause and surface.

At the tropopause, we expect a kinematic condition and a pressure condition.

We deﬁne the tropopause to be at z = 1 +εη(x,y,t), noting that such a variation

is consistent with observation (the tropopause slopes from perhaps 15 km at the

equator to perhaps 10 km at the poles). The kinematic condition stating that the

tropopause is a material interface then takes the scaled form

Dη

=W at z ≈1, (3.140)

conﬁrming the suggestion that η =O(1). However, just as the planetary boundary

layer induces an Ekman pumping term which modiﬁes the boundary condition on

W at z = 0, so also a (less severe) boundary layer at the troposphere will modify

(3.140)atz =1. We conceive of the stratosphere as a blanket of less dense air which

acts as a brake on the troposphere, and we pose the scaled boundary condition

∂u

∂z

=−γ u

(3.141)

to represent this, where u

=(u, v). In Question 3.4, it is shown that the appropriate

modiﬁcation of (3.140) is then

Dη

=W +Γ ∇

ψ at z ≈1, (3.142)

where

Γ =



2ε



γ +







γ +

√





. (3.143)