Geography Reference
In-Depth Information
Similar scaling shows that the zonal average of the meridional momentum equa-
tion (10.2) can be approximated accurately by geostrophic balance,
f
0
u
=−
∂/∂y
This can be combined with the hydrostatic approximation (10.3) to give the
thermal wind relation
RH
−
1
∂T/∂y
f
0
∂u/∂z
+
=
0
(10.13)
This relationship between zonal-mean wind and potential temperature distribu-
t
ions
imposes a strong constraint on the ageostrophic mean meridional circulation
(v, w). In the absence of a mean meridional circulation the eddy momentum flux
divergence in (10.11) and eddy heat flux divergence in (10.12) would tend sep-
arately to change the mean zonal wind and temperature fields, and hence would
destroy thermal wind balance. The pressure gradient force that results from any
small departure of the mean zonal wind from geostrophic balance will, however,
drive a mean meridional circulation, which adjusts the mean zonal wind and tem-
perature fields so that (10.13) remains valid. In many situations this compensation
allows the mean zonal wind to remain unchanged even in the presence of large eddy
heat and momentum fluxes. The mean meridional circulation thus plays exactly
the same role in the zonal-mean circulation that the secondary divergent circula-
tion plays in the synoptic-
scale
quasi-geostrophic system. In fact, for steady-state
mean flow conditions the (v, w) circulation must just balance the eddy forcing plus
diabatic heating so that the balances in (10.11) and (10.12) are as follows:
Coriolis force (f
0
v)
≈
divergence of eddy momentum fluxes
Adiabatic cooling
≈
diabatic heating plus convergence of eddy heat fluxes
Analysis of observations shows that outside the tropics these balances appear to
be approximately true above the boundary layer. Thus, changes in the zonal- mean
flow arise from small imbalances between the forcing terms and the mean merid-
ional circulation.
The Eulerian mean meridional circulation can be determined in terms of the
forcing from an equation similar to the omega equation of Section 6.4. Before
deriving this equation it is useful to observe that the mean meridional mass cir-
culation is nondivergent in the meridional plane. Thus, it can be represented in
terms of a meridional mass transport stream function, which identically satisfies
the continuity equation (10.8), by letting
∂χ
∂z
;
∂χ
∂y
ρ
0
v
=−
ρ
0
w
=
(10.14)
