98
DYNAMICAL OCEANOGRAPHY
5.2.3. Model equations
To summarize the dimensionless equations: if we consider a basin with a bottom topography z = −1+h b (x, y) and an ocean-atmosphere interface given by
z = ǫF η(x, y, t), the equations describing the constant density (barotropic) flow
in the β-plane are (5.7-5.8), i.e.,
ǫ
Du
dt
− v(1 + βǫy)=−
∂p
∂x
+ E H
∂ 2 u
∂x 2 +
∂ 2 u
∂y 2
+ E V
∂ 2 u
∂z 2 , (5.15a)
ǫ
Dv
dt
+ u(1 + βǫy)=−
∂p
∂y
+ E H
∂ 2 v
∂x 2 +
∂ 2 v
∂y 2
+ E V
∂ 2 v
∂z 2 , (5.15b)
0=−
∂p
∂z
(5.15c)
∂w
∂z
+
∂v
∂y
+
∂u
∂x
=0 ,
(5.15d)
with boundary conditions at the upper and layer boundaries given by
z = −1+h b (x, y):n · u = t 1 · u = t 2 · u =0
(5.16a)
z = ǫF η(x, y, t): p = η ; w = ǫF (
∂η
∂t
+ u
∂η
∂x
+ v
∂η
∂y
),
:ˆ ατ
x =
∂u
∂z
;ˆ ατ
y =
∂v
∂z
,
(5.16b)
where (see Fig. 5.3)
Figure 5.3. Sketch of the geometry of the bottom topography with normal n and tangential vectors t1 and t2.
t 1 =
⎛
⎝
1
0
∂h b
∂x
⎞
⎠ ; t 2 =
⎛
⎝
0
1
∂h b
∂y
⎞
⎠ ; n =
⎛
⎝
−
∂h b
∂x
−
∂h b
∂y
1
⎞
⎠ ,
(5.17)
and the parameters are defined as
ǫ =
U
f 0 L
; β =
β 0 L 2
U
;ˆ α =
Dτ 0
ρ 0 A V U