Fowler A. Mathematical Geoscience

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210 3 Oceans and Atmospheres

and so on. (We retain t as the time variable since it is the same as τ ; note only that

the time derivatives in (3.336) are with respect to constant s.) We suppose that the

leading order solution is the monochromatic wave given by the integrand of (3.333),

thus

(0)

iωt



(I +R)J



2ω

√



+i(I −R)Y



2ω

√



+(cc), (3.337)

where (cc) denotes the complex conjugate. As s → 0, the Bessel functions have

behaviours



2ω

√



∼1 −ω

s, Y



2ω

√



∼



lnω +

ln s +γ



, (3.338)

where γ is Euler’s constant, γ ≈ 0.5772. As a consequence, the right hand side of

(3.336)

is singular as s → 0, and the method of stationary phase dictates that we

choose the straining X so that the solution for η

(1)

is no more singular than η

(0)

s =0. The inhomogeneity causing possible singular terms is in the last term on the

right hand side of (3.336)

, and so this dictates our initial choice for X,

X =U(t)+V(t)lns +···, (3.339)

where

V =−

iωt

(I −R)

+(cc),

U =V −



iωt



I +R +

(I −R)(ln ω +γ)



+(cc)



(3.340)

Note that U and V are real (as indeed they must be).

This determines the straining to leading order, but provides no information on the

amplitude R of the reﬂected wave in terms of the incident amplitude I . In addition,

the choice of X in (3.339) produces further (weaker) singular terms on the right

hand side of (3.336)

, particularly in the ﬁrst and second terms, and these can be

removed by correcting (3.339)to

X =U(t)+V(t)lns +W(t)sln

s +Y(t)sln

s +···, (3.341)

and after some algebra, the suppression of terms proportional to

and

ln s

on the

right hand side of (3.336)

leads to the choice

Y =



V +



,W=



−

V +

U +



, (3.342)

which dictates that we must choose V =0 in order that Y and W be bounded. This

implies that

R =I ; (3.343)

3.12 Notes and References 211

the incoming wave is thus perfectly reﬂected, and (taking I to be real) we have

V =Y =0,U=−4I cos ωt, W =2ω

I cosωt. (3.344)

The straining is thus given by

x ∼s +εI cos ωt



−4 +2ω

s ln



+···, (3.345)

while for small s the surface is given by

(0)

∼4I cos ωt. (3.346)

The position x

of the shoreline is given by x

+εη =0, thus s =0, and hence

≈−4εI cos ωt. (3.347)

A more elaborate theory is necessary to describe the nonlinear ampliﬁcation of the

tsunami wave which occurs in some cases.

3.12 Notes and References

Of the books on geophysical ﬂuid dynamics, that by Pedlosky (1987) is perhaps the

most mathematical, and the present chapter is perhaps most inﬂuenced by his ap-

proach. Another inﬂuential book is that by Gill (1982), which is similar in scope but

less detailed in the mathematical niceties. The books by Houghton (2002) and par-

ticularly Barry and Chorley (1998) are more concerned with weather. Other books

on general meteorology are those by Holton (2004) and Vallis (2006), both com-

prehensive texts, and Andrews (2000), shorter and more like Houghton, and includ-

ing chapters on radiation and stratospheric chemistry. The book edited by Colling

(2001) is a useful primer on ocean circulation. It is an Open University course text.

Ghil and Childress (1987) treat the subject from a dynamical systems perspective.

A more recent book which relates the primitive equations of atmospheric ﬂow

to the problems of numerical weather prediction is that by Kalnay (2003), and a

corresponding book dealing with issues of numerical ocean modelling is that by

Miller (2007). The review paper by Olbers (2001) describes, as it says, a gallery

of mathematical models relevant to climate physics, meteorology and oceanogra-

phy.

Eddy Viscosity Apart from our discussion in Appendix B, Pedlosky (1987,

pp. 181 ff.) gives an account of Reynolds stresses, and discusses the merits of the use

of eddy viscosity as a way of parameterising these. He also discusses the anisotropy

of the eddy viscosity in the atmosphere, and gives estimates for the coefﬁcients ε

and ε

in (3.4) (denoted A

and A

by him).

212 3 Oceans and Atmospheres

Tides A very nice little book on tides is that by Defant (1958), which is short and

to the point. Lamb (1945) has a whole hundred page chapter on tides, unfortunately

rather dated now.

Geopotential Surfaces The choice of a correct coordinate system using geopo-

tential surfaces as the horizontal plane is lucidly described by Gill (1982), although

be careful; his conservation of mass equation 4.12.11 is not a correct deduction from

4.12.9 and 4.12.10.

Quasi-Geostrophic Potential Vorticity Equation The derivation of the quasi-

geostrophic potential vorticity equation provided here largely follows Pedlosky

(1987) in its exposition, up till the point where the stratiﬁcation parameter S is

discussed. At that point in his discussion, Pedlosky declines the challenge of de-

riving it, and simply takes it as a prescribed or measured quantity. Other authors

follow suit, without noting that the stratiﬁcation of the atmosphere must itself be

determined by the solution of the model. The presentation here is not perhaps the

most lucid, but it suggests that from the point of view of perturbation theory, the de-

termination of S follows from an integrability condition from a multiple time scale

expansion of the governing primitive equations; but this is a topic which is worthy

of further investigation.

Two-Phase Flow The discussion following (3.155) on two-phase ﬂow relates to

the well-known ill-posedness of the simplest averaged models; see Fowler (1997),

for instance. Two phase ﬂows exist in a number of different régimes—bubbly, slug,

churn, annular—but it is not known what causes the transition between them. One

suggestion for the bubbly to slug transition is that bubbly ﬂow becomes unstable

to kinematic waves as the bubble volume (void) fraction increases (Matuszkiewicz

et al. 1987). The onset of instability is a harbinger for ill-posedness, but instability

occurs before ill-posedness (Prosperetti and Satrape 1990).

The Global Thermohaline Circulation The idea of the deep ocean circulation as

a conveyor belt is associated with its chief proponent Wally Broecker, see for exam-

ple Broecker (1991). Somewhat unfortunately, the phrase ‘conveyor belt’, together

with the commonly produced cartoon of this, suggests a one-dimensionality of the

motion which is misleading in detail. Broecker’s article paints a more sophisticated

picture, although the basic concept is still very useful.

It is also Broecker’s idea that during ice ages, the circulation can oscillate because

of the interplay of the North Atlantic climate and the quantity of ice sheet ablation.

This idea is attractive, because the response time of the North Atlantic is of the right

magnitude, decades to centuries, for the sudden warmings to occur. Less clear is

what might control the millennial recurrence times.

L. F. Richardson and Weather Prediction If there is an unsung hero of the

present chapter, it would be the appealing ﬁgure of Lewis Fry Richardson, author of

3.13 Exercises 213

the well-known verse describing the essence of the turbulent energy cascade,

pro-

ponent of the mathematical theory of war, and author of an astonishingly precocious

effort at numerical weather prediction published in 1922. Richardson calculated a

weather forecast by hand, some thirty years before the ﬁrst computer weather fore-

cast, and was only thwarted in this endeavour by the inevitable parasitism of gravity

waves in the solution, which wrecks the prediction. Indeed, ﬁltering of gravity waves

is one of the keys to successful modern weather forecasting. Richardson’s attempt

is described in the meticulous book by Lynch (2006).

3.13 Exercises

3.1 The energy equation in the atmosphere is taken to be

ρc

−

=∇. q,

where q is the combined radiative and sensible heat ﬂux. Show how to derive

the equation of global energy balance

(I +P)=q

−q

where q

and q

are the combined radiative and sensible heat ﬂuxes upwards

at sea level and the tropopause, respectively, I =



ρc

Tdzis the internal

enthalpy and P =



ρΦ dz is the potential energy, with Φ being the gravi-

tational potential. You should assume a one-dimensional atmosphere, that the

mass conservation equation

dρ

+ρ∇. u =0

implies

ρdV = 0 for material volume elements dV, and that the pressure p

is related to Φ by

=−ρΦ

where p, ρ and Φ may be taken to be functions of z.

Big whorls have little whorls

That feed on their velocity,

And little whorls have lesser whorls

Andsoontoviscosity.

214 3 Oceans and Atmospheres

3.2 Derive a reference state for a dry atmosphere (no condensation) by using the

equation of state

p =

ρRT

the hydrostatic pressure

∂p

∂z

=−ρg,

and the dry adiabatic temperature equation

ρc

−

=0.

Show that

T =T

−

, ¯p =p

∗

(z),

where

∗

(z) =



1 −



Use the typical values (see Question 2.11) c

/g ≈29 km, M

/R ≈ 3.4,

to show that the pressure can be adequately represented by

¯p =p

exp(−z/H ),

where here the scale height is deﬁned as

H =

≈8.4km.

(A slightly better numerical approximation near the tropopause is obtained if

the scale height is chosen as 7 km.)

3.3 Use the hydrostatic pressure equation

=−

1−α

to show that, for θ = 1 + O(ε) and α relatively small, p ≈ e

−z

. Use this to

show that the conductive heating term



∂

∂z



∗

∂T

∂z



≈

α(5α −1)

−(5α−1)z

assuming that the radiative conductivity is k

∗

, and that

θ =

,ρ=

3.13 Exercises 215

Hence show that for Pe =7 and α =0.29, the heating term is less than 0.02 in

magnitude.

3.4 The Ekman boundary layer equations for the horizontal velocity (u, v) in the

atmospheric boundary layer can be written in the form

−v =−v

∗

+Eu

u =u

∗

+Ev

where (u

∗

) denotes the limiting value of the troposphere velocity as the

Earth’s surface is approached. The vertical (scaled) velocity W satisﬁes the

mass conservation equation

+εW

≈0,

and u

∗

≈0.

Show that U =u +iv satisﬁes

(U −U

∗

and deduce that

U =U

∗

+A exp



−

(1 +i)z

√



where A =A(x,y) is to be chosen.

Show that if U = 0 and W = 0onz = 0, then the value W

∗

of W outside

the boundary layer (i.e., as z/

√

E →∞) is given by

∗

=(v

∗

−u

∗

)



2ε

Now suppose there is an Ekman boundary layer at z = 1, where we pose

the condition U

=−γU. Solve the problem in this case, and show that the

corresponding Ekman pumping term is

W |

z=1

−W |

(z−1)/

√

E→−∞

=Γ(v

∗

−u

∗

where

Γ =



2ε



γ +







γ +

√





3.5 Show that an explicit expression for the atmospheric heating term

H =

∂

∂z



∗

∂T

∂z





1 +

νSt aM(T,p)



216 3 Oceans and Atmospheres

is given by

H =

(4α −1)T

+νSt aM(T , p))

in which you should use the adiabatic approximations that

T(z)=1 −αz,

and

θ =

=1,ρ=

3.6 Consider a planet whose polar axis is at right angles to the direction of the Sun.

If the dimensionless surface temperature T

is proportional to the

-power of

the incident solar radiation, show that

∝cos

1/4

λ,

where λ is the angle of latitude. Hence show, with λ =λ

+Σy, Σ 1, and

=1 +ε

, that

≈1 −s

y −s

where

Σ tanλ

4ε

(4 +3tan

)Σ

32ε

Find typical values of s

and s

for Σ =0.16, λ

, ε =0.2.

3.7 What is wrong with the following argument? By Green’s theorem in the plane,

we have



DΘ

dS =

∂

∂t



ΘdS−



∂A

Θdψ,

where A is any horizontal area. Since Θ ≈∂ψ/∂z,wehave



∂A

Θdψ=

∂

∂z



∂A

ψdψ=

∂

∂z





∂A

=0,

and therefore



DΘ

dS =

∂

∂t



ΘdS.

This is true for any horizontal closed region A, and therefore by shrinking A

to a point, we must have

DΘ

∂Θ

∂t

everywhere. This then implies that Θ =f(ψ).

3.13 Exercises 217

3.8 Suppose that θ satisﬁes the equation

Dθ

+εW

∂θ

∂z

=ε

ΓW +ε

H, (∗)

where Γ and H are constants, W = W(x,y) and the horizontal material

derivative is given by

∂

∂t

−

∂ψ

∂y

∂

∂x

∂ψ

∂x

∂

∂y

where ψ is the geostrophic stream function.

The equation is to be solved in the region V : −L<x<L, −1 <y<1, 0 <

z<1, with the boundary condition θ = 1 +ε

(y) on z = 0, and an initial

condition for θ . We can assume without loss of generality that the average of

over y is zero. (Why?) Assume that ψ =±1ony =±1, and that it is

periodic in x (with period 2L). Comment on the suitability of the initial and

boundary conditions. Does it matter whether W is positive or negative?

If A is any horizontal section of V , show that



Dθ

dS =

∂

∂t



θdS,

and deduce that the equation

Dθ

only has a bounded solution if ¯g(z) = 0, where ¯g is the time average of



gdS.

By expanding θ as θ

+ εθ

+ ε

+··· and assuming that the solution

remains regular, ﬁnd the equations satisﬁed by θ

, i =1, 2, 3, and show that a

solution exists in which θ

=θ

(z); whence also

and θ

=θ

(z), and θ

is given by



Γ +



whence

Dθ



1 −



. (†)

Suppose now that θ

∂ψ

∂z

; show that

[

∂θ

∂z

∂

∂z

(

Dθ

), and deduce that a

solution for θ

can be found in the form θ

(z) +Θ(x,y), where Θ(x,y)

is a particular solution of (†), and show that the secularity constraint at O(ε

)

implies that we can take

=0. Deduce that ψ =zΘ(x, y).

218 3 Oceans and Atmospheres

Suppose now that a diffusion term ε

∂

∂z

is added to the right hand side of

(∗). Show that the preceding discussion still applies, but now Θ represents an

outer solution for θ

away from the boundary z =0. By writing θ

= Θ +χ

and z =εZ, show that χ satisﬁes the approximate boundary layer equation

Dχ

∂χ

∂Z

∂

∂Z

with boundary conditions

χ → 0asZ →∞,

χ = χ

(x, y) =Θ

−Θ on Z =0.

For the particular case of a steady zonal ﬂow in which

∂

∂x

, u = u(y),

W =W(y) and χ



ˆχ

(y)e

ikx

, show that

χ =



ˆχ

(y)e

ikx−αZ

where

α =



+iku



1/2

−

. (‡)

By writing

+iku =(p +iq)

, p>0, and deﬁning the square root in (‡)

as having p>0, show that Reα>0 irrespective of the sign of W . How would

you expect Θ to behave over long time scales in this case?

3.9 The quasi-geostrophic potential vorticity equation is given by



∇

ψ +βy +

¯ρ

∂

∂z



¯ρ

∂ψ

∂z



¯ρ

∂

∂z



¯ρH



, (∗)

and the stratiﬁcation parameter S is determined by

¯ρ

[H −



¯ρ∇

dz +E

∗

∇

, (∗∗)

where the overbars denote a horizontal space average.

In deriving the expression

¯ρH

∗



∇

(†)

for S, where the hat denotes a time and space average for stationary solutions,

we have supposed that S = S(z) is independent of t, although this does not

appear necessary from (∗∗). Show that in fact this assumption is consistent

(i.e., that (†) implies (∗∗)) by using the averaging result

DΓ

∂

∂t

3.13 Exercises 219

to show that

∇

¯ρ

∂

∂z



¯ρ



=0,

and that the boundary condition



∂ψ

∂z



=H −SE

∗

∇

ψ on z =0

implies

=H −SE

∗

∇

ψ on z =0.

Deduce that



¯ρ∇

dz +



¯ρ



=0,

and hence show that, given (∗) and (†), (∗∗) is true if and only if

¯ρH

is constant

(which is indeed the case for (†)).

3.10 Show that the solution A(z) of the Eady model equations

(ikz +σ)





−μ



=0,

where

(ikz +σ)A



−ikA =0onz =0, 1,

can be written in the form

A =α cosh μz +β cosh



μ(1 −z)



providing c =−σ/ik satisﬁes



1 cosh μ −μc sinh μ

coshμ −μ(1 −c)sinh μ 1



=0,

whence

−c +

cothμ

−

=0.

Deduce that

c =−



−coth



−tanh



1/2

[The identity coth μ =

(tanh

+coth

) may be useful.]

3.11 The semi-diurnal M

tides on the Earth are described, neglecting Coriolis

force, by the dimensionless equation

cos

λη

=∇

(η −χ),