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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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

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