Download Example sheets 3,4 and 5

Examples 3
1 Estimate the Rossby and Ekman numbers for the following geophysical flows,
(a) A cyclone of 1000km in radius and a typical velocity of 10ms−1 .
(b) A tornado or twister of 100 metres in radius and velocities of order 70ms−1 .
(c) Convection in the Earth’s mantle with a typical velocity of 10−9 ms−1 over length scales of 3000
km. Take the viscosity of Mantle rock to be of order 1017 m2 s−1 .
2 Starting from the geostrophic and hydrostatic equations
fv =
1 ∂P
ρ ∂x
fu = −
−ρg =
1 ∂P
ρ ∂y
∂P
∂z
together with conservation of mass for an atmosphere with constant density, derive the TaylorProudman theorem in the form
∂u
= 0.
∂z
Show, by means of a scaling argument for the vertical and horizontal length scales that this is
consistent with
Ω · ∇u = 0
3 Far above a level plain the wind speed is 10 ms−1 towards the east. In the Ekman layer close to
the surface, the eddy viscosity has the value 10 m2 s−1 . At what heights above the surface is the
northward component of the wind speed zero?
4 As a model of the coupling between the atmosphere and the oceans, consider the ocean as a semiinfinite expanse of fluid z < 0, that is rotating with angular velocity (0, 0, Ω). At large depths the
water is stationary, but as a result of the wind the surface of the ocean is moving with velocity
p
(U, 0, 0). Find the flow in the ocean and show that the fluid at depth π ν/Ω is moving in the
opposite direction to the surface. Sketch the hodograph of this flow.
5 Find the horizontal components of the mass flux M
Z 0
M=
ρudz
−∞
of the flow in question 4. Show that the mass flux has magnitude
ν 1/2
ρU
2Ω
in is a direction at 45o to surface flow. Calculate the shear-stress at the surface
∂u ∂v
,
ρν
∂z ∂z
and show that this perpendicular to the direction of the mass flux in the ocean, and hence deduce
that the force of the wind drives a ocean current at 90o to the direct of the applied force.
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Examples 4
1 Calculate the speed of the mean west to east geostrophic flow under which there are exactly 3
stationary Rossby waves around the earth at latitude of 60◦ N.
2 Show that the Rossby wave equation
0
∂
∂v
∂u0
∂
+U
−
= −βv 0
∂t
∂x
∂x
∂y
and the continuity equation
∂u0
∂v 0
+
=0
∂x
∂y
admit oblique wave solutions of the form
u0 = uo exp (iωt − ikx − ily)
v 0 = vo exp (iωt − ikx − ily)
with dispersion relation
ω = kU −
βk
+ l2
k2
3 From question 2, find the phase speeds cx and cy defined as ω/k and ω/l respectively. What is the
westward phase speed relative to the basic flow, cx − U , of Rossby waves moving round latitude 30◦
with latitudinal (y direction) wavelength 3000 km and longitudinal (x direction) wavelength 10000
km? The Earth’s radius is 6370 km.
4 Find the longitudinal wavenumber, k, of Rossby waves with zero phase speed. How does the longitudinal wavelength of such waves vary with the latitudinal wavelength. Calculate the speed of the
mean west to east geostrophic flow under which there are exactly 3 stationary Rossby waves with
latitudinal wavelength 3000 km around the earth at latitude of 60◦ N. Compare with your answer
to question 1.
∂ω
5 For the waves in question 2, find the group velocity cg = ( ∂ω
∂k , ∂l ) and show that this is not parallel
to the wave vector (k, l)
Continued overleaf
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Examples 5
1 (a) Show from Maxwell’s equations that if initial conditions are chosen such that ∇ · B = 0 at t = 0
then ∇ · B = 0 at all times.
(b) Show from the induction equation that if initial conditions are chosen such that ∇ · B = 0 at t = 0
then ∇ · B = 0 at all times.
2 Show, using Maxwell’s equations and the non-relativistic assumptions used in lectures that
|ρe E| |j × B|
and hence that the electrostatic force can be neglected in relation to the Lorentz Force.
3 Estimate the magnetic Reynold’s number for the following geophysical and astrophysical flows
(a) Convection cells on the solar surface with a lengthscale of 106 m and a typical velocity of 1kms−1 .
(The magnetic diffusivity of ionised hydrogen is 107 cm2 s−1 .)
(b) Convection in the Earth’s fluid outer core which has a radius R = 3500km and a typical velocity of
0.04cms−1 . (The magnetic diffusivity of the fluid in the Earth’s core is 2.6m2 s−1 .)
4 Consider the shear flow of infinite extent given by
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u(z) = z e−z , 0, 0 .
This shear flow acts on an initially vertical magnetic field B = (0, 0, B0 ) .
(a) If the fluid is perfectly conducting show that Bz = B0 for all times and that Bx grows linearly with
time. Find the z-dependence of Bx at time t = t1 . Sketch Bx as a function of z at time t = t1 .
(b) If the fluid is not perfectly conducting, but has a diffusivity η, find the steady state for Bx given the
boundary condition Bx → 0 as z → ±∞. Sketch Bx for this steady state.
5 Calculate the current for the magnetic field given by
B = (B0 sin kz, B0 cos kz, 0)
where B0 and k are constants. Hence or otherwise show that B is a force-free field.
6 Find the density as a function of z in an isothermal magnetostatic atmosphere defined between z = 0
and z = 1, where ρ = ρ0 at z = 0 and
B(z) = e−λz , 0, 0 .
[Hint You should consider separately the cases λ = 2g/RT0 and λ 6= 2g/RT0 .]
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