Geography Reference
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and
2
ρ
0
N
2
ρ
0
N
2
∂
∂z
∂
∂z
∂
∂x
∂
∂z
f
0
ρ
0
∂
∂z
1
ρ
0
∂
∂z
ρ
0
2N
2
∂
∂x
v
=
−
ρ
0
∂
∂z
f
0
ρ
0
∂
∂z
N
2
v
=
Thus,
ρ
0
∂u
v
∂y
∂
∂z
f
0
ρ
0
∂
∂z
q
v
=−
N
2
v
+
(10.25)
so that it is not the momentum flux u
v
or heat flux v
∂
∂z that drives net
changes in the mean-flow distribution, but rather the combination given by the
potential vorticity flux. Under some circumstances the eddy momentum flux and
eddy heat flux may individually be large, but the combination in (10.25) actually
vanishes. This cancellation effect makes the traditional Eulerian mean formulation
a poor framework for analysis of mean-flow forcing by eddies.
Comparing (10.25) and (10.20) we see that the potential vorticity flux is pro-
portional to the divergence of the EP flux vector:
ρ
−
1
0
q
v
=
∇·
F
(10.26)
Thus, the contribution of large-scale motions to the zonal force in (10.17) equals
the meridional flux of quasi-geostrophic potential vorticity. If the motion is adia-
batic and the potential vorticity flux is nonzero, (10.22) shows that the mean-flow
distribution must change in time. Thus, there cannot be complete compensation
between Coriolis torque and zonal force terms in (10.17).
10.3
THE ANGULAR MOMENTUM BUDGET
The previous section used the quasi-geostrophic version of the zonal-mean equa-
tions to show that large-scale eddies play an essential part in the maintenance
of the zonal-mean circulation in the extratropics. In particular, we contrasted the
mean-flow forcing as represented by the conventional Eulerian mean and TEM
formulations. This section expands our consideration of the momentum budget by
considering the overall balance of
angular
momentum for the atmosphere and the
earth combined. Thus, rather than simply considering the balance of momentum
for a given latitude and height in the atmosphere, we must consider the transfer of
angular momentum between the earth and the atmosphere, and the flow of angular
momentum in the atmosphere.
