6Fluid dynamics on a rotating frame
IB Fluid Dynamics
6.2 Shallow water equations
We are now going to derive the shallow water equations, where we have some
water shallow relative to its horizontal extent. This is actually quite a good
approximation for the ocean — while the Atlantic is around
4 km
deep, it is
several thousand kilometers across. So the ratio is just like a piece of paper.
Similarly, this is also a good approximation for the atmosphere.
Suppose we have a shallow layer of depth z = h(x, y) with p = p
0
on z = h.
h(x, y, t)
g
1
2
f
We consider motions with horizontal scales
L
much greater than vertical scales
H.
We use the fact that the fluid is incompressible, i.e.
∇ · u
= 0. Writing
u = (u, v, w), we get
∂w
∂z
= −
∂u
∂x
−
∂v
∂y
.
The scales of the terms are
W/H
,
U/L
and
V/L
respectively. Since
H L
, we
know
W U, V
, i.e. most of the movement is horizontal, which makes sense,
since there isn’t much vertical space to move around.
We consider only horizontal velocities, and write
u = (u, v, 0),
and
f = (0, 0, f ).
Then from Euler’s equations, we get
∂u
∂t
− fv = −
1
ρ
∂p
∂x
,
∂v
∂t
+ fu = −
1
ρ
∂p
∂y
,
0 = −
1
ρ
∂p
∂z
− g.
From the last equation, plus the boundary conditions, we know
p = p
0
= gρ(h − z).
This is just the hydrostatic balance. We now put this expression into the
horizontal components to get
∂u
∂t
− fv = −g
∂h
∂x
,
∂v
∂t
+ fu = −g
∂h
∂y
.
Note that the right hand sides are independent of
z
. So the accelerations are
independent of z.
The initial conditions are usually that
u
and
v
are independent of
z
. So we
assume that the velocities always do not depend on z.