# 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
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
pru+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 H1km,
h100m, L1000km, 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 ow found in part (ii).
comparable magnitude: fτ00B0. If we further assume that rα, 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