366
DYNAMICAL OCEANOGRAPHY
e 3 ∧ f v a = A V
∂ 2 v
∂z 2 − v ·∇v
(15.34b)
The horizontal momentum equation (15.33) can then be written as
e 3 ∧ f v = −
1
ρ 0
∇p + e 3 ∧ f v a
(15.35)
As a next step, we define the depth-integrated horizontal velocity as
V =
0
−H
v dz
(15.36)
and split the total horizontal transport into
V = V a + V s + Hv b ,
(15.37a)
V s =
0
−H
(v g − v b ) dz,
(15.37b)
V a =
0
−H
v a dz.
(15.37c)
Here v b is the bottom horizontal velocity, i.e., v b = v| z=−H , which is independent of z.
To arrive at the depth averaged vorticity equation, we use the fact that the horizontal divergence of V (through vertical integration of the continuity equation) is
zerotogive
∇·(V s + V a + Hv b )=0.
(15.38)
We use Leibnitz’s rule for writing
∇·V s = ∇·
0
−H
(v g − v b ) dz =
=
0
−H
∇·(v g − v b ) dz + ∇H · (v g − v b )| z=−H , (15.39)
and the last term in the right hand side can be neglected if the ageostrophic velocities near the bottom are assumed small.
Using the vector identities (for arbitrary vectors a and b and scalar φ)
∇·(a ∧ b)=b ·∇∧a − a ·∇∧b,
and
∇∧(φa)=φ∇∧a + ∇φ ∧ a,