Geography Reference
In-Depth Information
9.2.3
Cross-Frontal Circulation
Equations (9.7)-(9.10) form a closed set that can be used to determine the cross-
frontal ageostrophic circulation in terms of the distribution of zonal wind or tem-
perature. Suppose that the large-scale geostrophic flow is acting to intensify the
north-south temperature gradient through deformation as shown in Fig. 9.3. As
the temperature gradient increases, the vertical shear of the zonal wind must also
increase to maintain geostrophic balance. This requires an increase of u g in the
upper troposphere, which must be produced by the Coriolis force associated with
a cross-frontal ageostrophic circulation [see (9.7)]. The structure of this secondary
circulation can be computed by deriving an equation analogous to the omega equa-
tion discussed in Section 6.4.2.
We first differentiate (9.8) with respect to y and use the chain rule to express the
result as
∂b
∂y
N 2
D
Dt
∂v a
∂y
∂b
∂y
∂w
∂y
∂b
∂z
=
Q 2
+
(9.11)
where
∂u g
∂y
∂b
∂x
∂v g
∂y
∂b
∂y
Q 2 =−
(9.12)
is just the y component of the Q vector discussed in Section 6.4.2, but expressed
in the Boussinesq approximation.
Next we differentiate (9.7) with respect to z, again use the chain rule to rearrange
terms, and the thermal wind equation (9.10) to replace ∂u g /∂z by ∂b/∂y on the
right-hand side. The result can then be written as
f ∂u g
∂z
f f
D
Dt
∂v a
∂z
∂u g
∂y
∂w
∂z
∂b
∂y
=
Q 2 +
+
(9.13)
Again, as shown in Section 6.4.2, the geostrophic forcing (given by Q 2 ) tends to
destroy thermal wind balance by changing the temperature gradient and vertical
shear parts of the thermal wind equation in equal but opposite senses. This tendency
of geostrophic advection to destroy geostrophic balance is counteracted by the
cross-frontal secondary circulation.
In this case the secondary circulation is a two-dimensional transverse circulation
in the y, z plane. It can thus be represented in terms of a meridional streamfunction
ψ M defined so that
v a =−
∂ψ M /∂z,
w
=
∂ψ M /∂y
(9.14)
which identically satisfies the continuity equation (9.9). Adding (9.11) and (9.13),
and using the thermal wind balance (9.10) to eliminate the time derivative and
(9.14) to eliminate v a and w, we obtain the Sawyer-Eliassen equation
2 ψ M
∂y 2
F 2 2 ψ M
∂z 2
2S 2 2 ψ M
N s
+
+
∂y∂z =
2Q 2
(9.15)
 
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