Geography Reference
In-Depth Information
where from (6.25) the zonal-mean potential vorticity is
ρ
0
N
2
∂
2
∂y
2
1
f
0
f
0
ρ
0
∂
∂z
∂
∂z
q
=
f
0
+
βy
+
+
(10.23)
and the eddy potential vorticity is
∂
2
∂x
2
ρ
0
N
2
∂
2
∂y
2
∂
∂z
1
f
0
f
0
ρ
0
∂
∂z
q
=
+
+
(10.24)
The quantity q
v
on the right-hand side in (10.22) is the divergence of the merid-
ional flux of potential vorticity. According to (10.22), for adiabatic quasi-
geostrophic flow the mean distribution of potential vorticity can be changed only
if there is a nonzero flux of eddy potential vorticity.
Th
e zonal-mean potential vor-
ticity, together with suitable boundary conditions on , completely determines the
distribution of zonal-mean geopotential, and hence the zonal-mean geostrophic
wind and temperature distributions. Thus, eddy-driven mean-flow accelerations
require nonzero potential vorticity fluxes.
It can be shown that the potential vorticity flux is related to the eddy momentum
and heat fluxes. We first note that for quasi-geostrophic motions the eddy horizontal
velocities in the flux terms are geostrophic:
∂
∂x ,
∂
∂y
f
0
v
=
f
0
u
=−
and
Thus,
∂
∂x
2
∂
2
∂x
2
∂
∂x
∂
2
∂x
2
1
f
0
1
2f
0
∂
∂x
v
=
=
=
0
where we have used the fact that a perfect differential in x vanishes when averaged
zonally. Thus,
ρ
0
N
2
v
f
0
∂
2
∂y
2
f
0
ρ
0
∂
∂z
∂
∂z
q
v
=
+
v
We can use the chain rule of differentiation to rewrite the terms on the right in
this expression as
∂
∂x
v
f
0
∂
2
∂y
2
1
f
0
∂
2
∂y
2
=
=
2
=−
∂
∂x
∂
∂y
u
v
∂
∂y
1
f
0
∂
∂y
1
2
∂
∂x
∂
∂y
−
