114
DYNAMICAL OCEANOGRAPHY
We now consider the case of the presence of bottom topography. If h b (x, y)=
O(1) we see that the O(1) geostrophic problem no longer gives w 0 ≡ 0.T h e
asymptotic methodology is no longer directly applicable and numerical methods
need to be used. However, in the case h b (x, y)=O(ǫ) asymptotic methods can
still be used. In this case, let
h b (x, y)=ǫη b (x, y).
(5.70)
The boundary conditions at z = −1+ǫη b (x, y) are that both normal and tangential components of the velocity are zero. The tangent vectors and normal at a
particular point is spanned by
t 1 =
⎛
⎝
1
0
ǫη b,x
⎞
⎠ ; t 2 =
⎛
⎝
0
1
ǫη b,y
⎞
⎠ ; n =
⎛
⎝
−ǫη b,x
−ǫη b,y
1
⎞
⎠ ,
(5.71)
where η b,x = ∂η b /∂x (see Fig. 5.3). The conditions
v · t 1 = v · t 2 = v · n =0,
(5.72)
give
u + ǫ
∂η b
∂x
w =0 ,
(5.73a)
v + ǫ
∂η b
∂y
w =0 ,
(5.73b)
w − ǫ(u
∂η b
∂x
+ v
∂η b
∂y
)=0 .
(5.73c)
The vertical velocities w are now O(ǫ) and
w − ǫ(u
0 ∂η b
∂x
+ v
0 ∂η b
∂y
)=0.
(5.74)
If (5.44d) is integrated from ξ =0to ξ = ∞, we find that the correction due to
the bottom topography leads to
lim
ξ→∞
˜
w(x, y, ξ)=ǫ u
0 ·∇η b +
1
2
¯
E
1/2
V ζ
0 .
(5.75)
where u 0 =(u 0 ,v 0 ). This result can be seen as that for the flat bottom plus a part
due to the kinematic boundary condition (v · n =0).
The vertical velocity at the lower boundary of the free surface Ekman layer
follows from vertical integration of the continuity equation (5.63d) with the result
lim
χ→∞
ˆ
w(x, y, χ)= ¯
E
1/2
V ˆ
w
0 (x, y, 0) +
α
2
¯
E
1/2
V ∇.(T ∧ e 3 ),
(5.76)