Geography Reference
In-Depth Information
meridional eddy heat flux, expresses the conversion between zonal-mean and eddy
potential energy.
That the second term on the right in (10.55) and the final term in (10.56) represent
conversion between zonal-mean and eddy energies can be confirmed by performing
analogous operations on the eddy equations (10.51)-(10.53) to obtain equations
for the eddy kinetic and available potential energies:
ρ
0
ρ
0
∂u
∂x
+
u
2
v
2
∂v
∂y
d
dt
+
=+
2
(10.57)
ρ
0
u
v
ρ
0
u
X
+
v
Y
∂u
∂y
−
+
ρ
0
2N
2
2
ρ
0
w
ρ
0
κJ
∂
∂z
N
2
H
∂
∂z
d
dt
∂
∂z
=−
+
ρ
0
N
2
∂
2
∂z∂y
v
(10.58)
∂
∂z
−
If we set w
0, the first term on the right in (10.57) can be rewritten
using the continuity equation (10.54) as
ρ
0
=
0atz
=
ρ
0
w
∂u
∂x
+
∂
∂z
∂v
∂y
∂ (ρ
0
w
)
∂z
=−
=
which is equal to minus the first term on the right in (10.58). Thus, this term
expresses the conversion between eddy kinetic and eddy potential energy for the
Eulerian mean formulation. Similarly, the last term in (10.58) is equal to minus the
last term in (10.56), and thus represents conversion between eddy and zonal-mean
available potential energy.
The Lorenz energy cycle can be expressed compactly by defining zonal-mean
and eddy kinetic and available potential energies:
ρ
0
u
2
2
,
ρ
0
u
2
,
v
2
+
K
≡
K
≡
2
ρ
0
N
2
∂
∂z
2
,
ρ
0
N
2
2
∂
∂z
1
2
1
2
P
≡
P
≡
the energy transformations
ρ
0
w
∂
∂z
,
ρ
0
w
,
P
K
≡
P
·
K
≡
∂
∂z
·
