182
DYNAMICAL OCEANOGRAPHY
which fits with the flow directions in Fig. 8.4. A similar picture can be drawn to
illustrate the vertical shear in meridional direction due to a zonal density gradient.
z = 0
z = -D
y
- z
1
- z
2
y
1
y
2
z
ρ +
ρ -
H
H
L
L
X
u > 0
u < 0
Figure 8.4. Sketch of the mechanism of the thermal wind balance relating the vertical shear in
zonal direction to a meridional density gradient.
◭
The O(ǫ) equations (8.20a-b/d) become
∂u 0
∂t
+ u
0 ∂u 0
∂x
+ v
0 ∂u 0
∂y
− v
1 − βyv
0 = −
∂p 1
∂x
,
(8.26a)
∂v 0
∂t
+ u
0 ∂v 0
∂x
+ v
0 ∂v 0
∂y
+ u
1 + βyu
0 = −
∂p 1
∂y
,
(8.26b)
∂w 1
∂z
+
∂u 1
∂x
+
∂v 1
∂y
=0 .
(8.26c)
The vorticity balance for the vertical component of the vorticity vector ζ 0 =
∂v 0 /∂x − ∂u 0 /∂y can be found by eliminating the pressure p 1 from (8.26a-b)
and with (8.23a-d) and (8.26c) we find
(
∂
∂t
+ u
0 ∂
∂x
+ v
0 ∂
∂y
)(ζ
0 + βy)=
∂w 1
∂z
.
(8.27)
So far we have not used the density balance (8.20e). On the scale L D , stratification must be an important ageostrophic effect. As the mixing of density (heat/