Stratification
181
When (8.23a-b) are both differentiated to z and (8.23c) is used, we find
∂v 0
∂z
= −
∂ρ 0
∂x
,
(8.24a)
∂u 0
∂z
=
∂ρ 0
∂y
(8.24b)
The equations (8.24) are usually referred to as the thermal wind balance and they
Ex. 8.1
describe that in a stratified rotating flow horizontal density gradients are associated
with vertical shear.
Ex. 8.2
The thermal wind balance
The dimensional form of the thermal wind balance is given by
f 0
∂v ∗
∂z
= −
g
ρ 0
∂ρ ∗
∂x ∗
,
f 0
∂u ∗
∂z
=
g
ρ 0
∂ρ ∗
∂y ∗
,
Note that only the vertical shear is given by these equations and not the
velocities itself (see Example 8.1).
◮
Example 8.1: Thermal wind balance
Consider first a situation where the density increases northward, i.e.,
ρ ∗ = ρ 0 (1 + αy ∗ ),
with α>0. According to hydrostatic equilibrium, the vertical pressure increases
faster with depth in the north than in the south, according to
p ∗ = −ρ 0 gz(1 + αy)+p 0 (y),
where p 0 is the surface pressure at z =0. Relative to the mean pressure at particular levels −z 1 and −z 2 , the pressure distribution has the high/low pattern as in
Fig. 8.4. Pressure gradients are balanced by the Coriolis acceleration and hence at
depth the zonal flow is eastward and near the surface the zonal flow is westward.
The vertical shear in the zonal direction is according to the thermal wind balance
given by
∂u ∗
∂z
=
g
f 0 ρ 0
∂ρ ∗
∂y ∗
=
g
f 0
α