6.13 Exercise 20: Geostrophic Adjustment
155
ξ 2 = −
h
H 2
f
where relative vorticity in the bottom layer is define by ξ = ∂v 2 /∂r . The latter
two equations can be combined to yield an equation of the same form as (6.67). The
solution for the northern hemisphere is:
h(r ) = H 2 exp
r − r o − R
R
v 2 (r ) = −
g H 2 exp
r − r o − R
R
where r o is the initial radius, and the internal Rossby radius of deformation is now
given by:
R =
√ g H 2
| f |
The geostrophic adjustment process is mathematically more difficul to describe
for situations in which both layers have comparable thicknesses and therefore not
included in this book. Generally, geostrophic f ows in the top and bottom layers are
opposite to each other and the ratio of speeds depends on the initial thicknesses of
layers involved. The frontal width is again given by the internal deformation radius:
R =
√ g H ∗
| f |
(6.71)
where the “equivalent” thickness is given by H
∗
= H 1 H 2 /(H 1 + H 2 ). Note that the
numerator in the latter relation is the phase speed of long internal gravity waves for
a two-layer flui (4.27).
6.13 Exercise 20: Geostrophic Adjustment
6.13.1 Aim
The aim of this exercise is to explore the geostrophic adjustment process for an
ocean of two superimposed layers of different densities.
6.13.2 Task Description
The model domain is 200 m deep and has equal sidelengths of 50 km. The surface
patch of lower-density water has a density of 1027 km m
−3
, a thickness of 100 m