MATHEMATICAL TRIPOS
Part III
Thursday, 29 May, 2014
1:30 pm to 4:30 pm
PAPER 70
FLUID DYNAMICS OF THE ENVIRONMENT
Attempt no more than THREE questions.
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
Consider
smallamplitude,
twodimensional
monochromatic
internal
waves
in
a
Boussinesq fluid with constant buoyancy frequency, N 2, where
dρ
N 2
= − g
ρ0
dz ,
and ρ0
is a characteristic value of the background density distribution ρ(z).
In a stationary
ambient fluid, derive the dispersion relation for waves associated with a two
dimensional
vertical displacement perturbation
ζ = Aζ cos(kx + mz − Ωt),
where Aζ , k, m and Ω are all real. Show that statically unstable regions are
predicted to
develop in the flow if Aζ  > 1/m.
Now consider a situation where the ambient fluid velocity varies with height,
such
that U (z)
= sz for z
> 0
where s is a positive real constant.
A two
dimensional wave
packet propagates upwards into the upper halfplane, with horizontal wavenumber
k > 0
and vertical wavenumber m0
< 0 at z = 0.
Using the raytracing equations
(which you
may quote without proof ), show that the intrinsic frequency ω and the
horizontal phase
speed cx
= ω/k measured by a stationary observer remain constant for all time.
Hence
derive an expression for the height zc
of the critical level where cx
= U
(zc ).
Furthermore, show that the ray followed by this wave packet is defined
implicitly
as the solution to the differential equation
dx
ksz
N 2
dz
= tan Θ +
N sin Θ cos2 Θ ,
tan2 Θ(z) =
(ω − ksz)2
− 1.
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