# Advanced Topics in Fluid Mechanics of Climate - Mathematical Tripos - Paper, Exams for Mathematics. Aliah University

## Mathematics

Description: This is the Past Paper of Mathematical Tripos which includes Combinatorics, Kruska Katona Theorem, Vertex Boundary, Harper’s Inequality, Usual Compression Operator, Construction of Cellular Homology, Intersection Pairing, Differentiable Map etc. Key important points are: Advanced Topics in Fluid Mechanics of Climate, Rossby Waves, Coriolis Parameter, Free Surface Elevation, Water Potential Vorticity, Uniform Density Fluid, Plane-Wave Solutions, Dispersion Relation
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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
Cover sheet None
Treasury Tag
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) + uq
x +vq
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+lyωt).(3)
where ψis 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
df2
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