Stratification
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8.5. Exercises on chapter 8
(8.1) Thermal wind balance
An ocean current has a velocity field given by
u ∗ (x ∗ ,y ∗ ,z ∗ )=0; v ∗ (x ∗ ,y ∗ ,z ∗ )=V 0 e
−
(x∗−x 0 ) 2
A
e
z∗/A
a. Determine the density field ρ ∗ (x ∗ ,y ∗ ,z ∗ ) which is in thermal wind balance
with this flow.
b. Sketch the velocity field and the density field of this flow.
(8.2) Velocities from hydrographic data
To determine geostrophic velocities from measurements of temperature T ∗ and
salinity S ∗ one calculates (i) the slope of the isobars, or (ii) the pressure change
along a surface of constant geopotential. Here, the geopotential Φ ∗ is defined
by
Φ ∗ (x ∗ ,y ∗ ,z ∗ )=
z∗
−1
gdz
′
∗
a. Use the hydrostatic and geostrophic balances and derive that
∂Φ ∗
∂x ∗
= −f 0 v ∗ ;
∂Φ ∗
∂y ∗
= f 0 u ∗
where Φ ∗ is the value of the geopotential on an isobaric surface p ∗ = p 0 and
f 0 is the local Coriolis parameter.
Consider now two isobaric surface (with pressures p 1∗,A and p 2∗,A )a ta
station A as in the figure below.
b. Show that the geopotential difference between these two isobaric surfaces
is given by
Φ ∗ (p 1∗,A ) − Φ ∗ (p 2∗,A )=
p 2∗,A
p 1∗,A
α ∗ (S ∗ ,T ∗ ,p ∗ ) dp ∗
where α ∗ =1/ρ ∗ is the specific volume.