MATHEMATICAL TRIPOS Part III

Tuesday, 12 June, 2012 1:30 pm to 4:30 pm

PAPER 79

ADVANCED TOPICS IN FLUID MECHANICS OF CLIMATE

You may attempt ALL questions, although full marks

can be achieved by good answers to THREE questions.

Completed answers are preferred to fragments.

There are FOUR questions in total.

The questions carry equal weight.

STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS

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Script paper

You may not start to read the questions

printed on the subsequent pages until

instructed to do so by the Invigilator.

2

1 Rossby waves on a zonal jet

Rossby waves can be supported by variations in the rotation rate or bottom

topography. In some cases, these waves can also be supported by a large-scale mean

ﬂow. To show this, consider a steady mean ﬂow, u=U0sin πy

Lin geostrophic balance.

[Assume that the Coriolis parameter f=f0, and the bottom depth, H=H0are constant.]

What is the corresponding free surface elevation, η, in a uniform density ﬂuid?

Start from the shallow water potential vorticity (PV) equation:

D

Dt ζ+f

h= 0.(1)

Let H0be the depth of the ﬂuid at rest, and f0be the Coriolis parameter. Derive the

quasi-geostrophic (QG) equation, including the mean ﬂow given above. Identify the steady

QG potential vorticity qassociated with uand η. By writing q=q′+q, show that the

equation for departures from the mean QG PV can be written

∂q′

∂t +J(ψ′, q′) + u∂q′

∂x +v′∂q

∂y = 0.(2)

Linearize Eq. (2), and assuming that variations in qare on suﬃciently large scale,

derive the dispersion relation associated with Eq. (2) using plane-wave solutions of the

form

ψ′=ˆ

ψei(kx+ly−ωt).(3)

where ψ′is the streamfunction associated with the perturbation velocity (u′, v′). Show

that the wavelength must be < L/2.

Stationary waves have a phase speed that exactly opposes the mean ﬂow so that

ω= 0 in a stationary coordinate frame. Show that waves that are stationary with respect

to the mean ﬂow, uhave a wavenumber |k|=π/L and a group velocity

cg= 2u|k|2

|k|2+k2

d

,(4)

where k2

d≡f2

0/(gH0). Qualitatively describe what this implies for waves generated as a

mean ﬂow passes over a stationary obstacle like a mountain range.

Part III, Paper 79

3

2 The atmospheric ocean

In most locations on the globe, land masses prevent the ocean from ﬂowing

uninterrupted eastward around the globe. An important exception is the Southern Ocean,

where ocean dynamics more closely resemble those in the atmosphere. As a model for the

circulation in the Southern Ocean, consider the following linearised equations

∂u

∂t +f×u=−1

ρ0

∇p−ru+1

ρ0

∂τ

∂z ,(1)

0 = −1

ρ0

∂p

∂z +b, (2)

∂b

∂t =∂B

∂z −αb, (3)

∇ · u= 0,(4)

in the domain shown in Figure 1(a). The coordinates are aligned so that xpoints to the

east, ypoints to the north, and zpoints up so that both gravity and fpoint in the negative

zdirection (consistent with the southern hemisphere). Here, rand αare Rayleigh friction

and Newtonian cooling coeﬃcients, τis the frictional stress, and Bis the buoyancy ﬂux.

Show that in the absence of friction and cooling, the steady-state, depth-integrated

ﬂow follows Sverdrup balance:

βv=∇ × τw

ρ0

,(5)

where the overbar denotes an integral over the full depth of the ocean, τwis the wind

stress, and β=∂f/∂y. State the assumptions used to derive this equation. A possibly

useful vector identity is given at the end of the problem.

While Sverdrup balance is generally a good approximation in the relatively quiescent

ocean gyres, in the Southern Ocean the frictional terms are important in setting the

dynamics and result in a mean meridional circulation called the Deacon cell. Unlike the

atmosphere, forcing in the ocean is concentrated near the sea surface. Suppose that the

wind and buoyancy ﬂux are only felt for depths shallower than some depth, h. Speciﬁcally,

consider the following form for the wind and buoyancy forcing:

τw=(τw

0z+h

h2sin πy

Lˆı,−h < z < 0

0,−H < z < −h(6)

B=(B0z+h

hcos πy

L,−h < z < 0

0,−H < z < −h . (7)

Using these forcing terms, look for steady solutions to Eqns. (1-4) in the zonal

channel sketched in Figure 1(a). For simplicity, assume that the ﬂow is independent of

the x-direction, the bottom depth His constant, and impose conditions of no normal ﬂow

at the walls. Write down the streamfunction associated with ﬂow in the y-zplane and use

arrows to indicate the directions of vand won Figure 1(b), assuming that τw

0and B0are

positive.

Typical values of the various length scales in the Southern Ocean are H≃1km,

h≃100m, L≃1000km, while the scaled wind and buoyancy forcing are typically of

Part III, Paper 79 [TURN OVER

4

z=0

z=-h

z=-H

z

y

xy=0

y=L

z=0

z=-h

z=-H

y=0 y=L

(a)

(b)

Figure 1: (a) Channel geometry (b) Roughly sketch the ﬂow found in part (ii).

comparable magnitude: fτ0/ρ0≃B0. If we further assume that r≃α, show that

the buoyancy forcing is negligible compared to the wind forcing. What extra dynamical

feature, which is not explicitly considered in the model above, is needed to balance the

wind-driven streamfunction? Qualitatively describe what happens to the energy input by

the winds.

[Hint: Vector identity:

∇ × (A×B) = A(∇ · B)−B(∇ · A) + (B· ∇)A−(A· ∇)B.] (8)

Part III, Paper 79

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