Geography Reference
In-Depth Information
ρ
0
N
2
v
ρ
0
u
v
,
∂
2
∂y∂z
K
·
K
≡
P
·
P
≡
∂
∂z
∂u
∂y
and the sources and sinks
ρ
0
N
2
,
ρ
0
N
2
,
κJ
H
∂
∂z
κJ
H
∂
∂z
R
≡
R
≡
ρ
0
u
X
+
v
Y
≡
ρ
0
uX
,
ε
≡
ε
Equations (10.55)-(10.58) can then be expressed in the simple form
=
P
K
+
K
·
K
+
dK/dt
·
ε
(10.59)
=−
P
K
+
P
·
P
+
dP/dt
·
R
(10.60)
=
P
·
K
−
K
·
K
+
dK
/dt
ε
(10.61)
=−
P
·
K
−
P
·
P
+
dP
/dt
R
(10.62)
Here [A
B] designates conversion from energy form A to form B.
Adding (10.59)-(10.62), we obtain an equation for the rate of change of total
energy (kinetic plus available potential):
d
K
·
P
dt
K
+
R
+
ε
+
P
+
=
R
+
ε
+
(10.63)
K
+
Fo
r adiabatic inviscid flow the right side vanishes and the total energy K
+
P
is conserved. In this system the zonal-mean kinetic energy does not include
a contribution from the mean meridional flow because the zonally averaged merid-
ional momentum equation was replaced by the geostrophic approximation. (Like-
wise, use of the hydrostatic approximation means that neither the mean nor the
eddy vertical motion is included in the total kinetic energy.) Thus, the quantities
that are included in the total energy depend on the particular model used. For
any model the definitions of energy must be consistent with the approximations
employed.
In the long-term mean the left side of (10.63) must vanish. Thus, the production
of available potential energy by zonal-mean and eddy diabatic processes must
balance the mean plus eddy kinetic energy dissipation:
P
+
R
=−
ε
+
−
R
ε
(10.64)
Because solar radiative h
ea
ting is a maximum in the tropics, where the temperatures
are high, it is clear that R, the generation of zonal-mean potential energy by the
