Geography Reference
In-Depth Information
In order to analyze the exchange of energy between mean flow and eddies, we
require a similar set of dynamical equations for the eddy motion. For simplicity
we assume that the eddies satisfy the following linearized set of equations
2
:
∂
∂t
+
u
−
f
0
−
v
=−
∂
∂x
+
∂
∂x
∂u
∂y
X
u
(10.51)
∂
∂t
+
v
+
∂
∂y
+
∂
∂x
f
0
u
=−
Y
u
(10.52)
∂
∂z
∂
∂t
+
∂
∂z
+
κJ
H
∂
∂x
∂
∂y
v
N
2
w
=
u
+
(10.53)
∂
ρ
0
w
∂z
∂u
∂x
+
∂v
∂y
+
1
ρ
0
=
0
(10.54)
where X
and Y
are the zonally varying components of drag due to unresolved
turbulent motions.
We now define a global average
∞
D
L
A
−
1
≡
()dxdydz
0
0
0
where L is the distance around a latitude circle, D is the meridional extent of the
midlatitude β plane, and A designates the total horizontal area of the β plane. Then
for any quantity
∂
∂x
=
0
∂
∂y
=
0,
if vanishes at y
=±
D
∂
∂z
=
0,
if vanishes at z
=
0 and z
→∞
An equation for the evolution of t
h
e mean flow kine
ti
c energy can then be
obtained by multiplying (10.47) by ρ
0
u and (10.48) by ρ
0
ν and adding the results
to get
u
2
2
u
v
∂
∂t
ρ
0
v
∂
∂
∂y
ρ
0
=−
∂y
−
ρ
0
u
+
ρ
0
uX
ρ
0
uu
v
∂y
ρ
0
v
+
∂
ρ
0
∂v
∂
∂y
∂u
∂y
+
=−
∂y
−
+
ρ
0
u
v
ρ
0
uX
2
A similar analysis can be carried out for the nonlinear case.
