6Fluid dynamics on a rotating frame

IB Fluid Dynamics



6.3 Geostrophic balance
When we have steady flow, the time derivatives vanish. So we get
u =
y
gh
f
=
y
p
ρf
,
v =
x
gh
f
=
x
p
ρf
.
Hopefully, this reminds us of streamfunctions. The streamlines are places where
h is constant, i.e. the surface is of constant height, i.e. the pressure is constant.
Definition (Shallow water streamfunction). The quantity
ψ =
gh
f
is the shallow water streamfunction.
In general, near a low pressure zone, there is a pressure gradient pushing the
flow towards to the low pressure area. Since flow moves in circles around the
low pressure zone, there is a Coriolis force that balances this force. This is the
geostrophic balance.
L
This is a cyclone. Note that this picture is only valid in the Northern hemisphere.
If we are on the other side of the Earth, cyclones go the other way round.
We now look at the continuity equation, i.e. the conservation of mass.
We consider a horizontal surface D in the water. Then we can compute
d
dt
Z
D
ρh dV =
Z
D
u
H
· n dS,
where u
H
is the horizontal velocity. Applying the divergence theorem, we get
Z
D
t
(ρh) dV =
Z
D
H
· (ρhu
H
) dV,
where
H
=
x
,
y
, 0
.
Since this was an arbitrary surface, we can take the integral away, and we have
the continuity equation
h
t
+
H
· (u
h
h) = 0.
So if there is water flowing into a point (i.e. a vertical line), then the height of
the surface falls, and vice versa.
We can write this out in full. In Cartesian coordinates:
h
t
+
x
(uh) +
y
(uh) = 0.
To simplify the situation, we suppose we have small oscillations, so we have
h = h
0
+ η(x, y, t), where η h
0
, and write
u = (u(x, y), v(x, y)).
Then we can rewrite our equations of motion as
u
t
+ f × u = gη ()
and, ignoring terms like , vη, the continuity equation gives
dη
dt
+ h
0
· u = 0. ()
Taking the curl of the (), we get
ζ
t
+ f · u = 0,
where
ζ = × u.
Note that even though we wrote this as a vector equation, really only the
z
-
component is non-zero. So we can also view
ζ
and
f
as scalars, and get a scalar
equation.
We can express · u in terms of η using (). So we get
t
ζ
η
h
0
f
=
dQ
dt
= 0,
where
Definition (Potential vorticity). The potential vorticity is
Q = ζ
η
h
0
f,
and this is conserved.
Hence given any initial condition, we can compute
Q
(
x, y,
0) =
Q
0
. Then we
have
Q(x, y, t) = Q
0
for all time.
How can we make use of this? We start by taking the divergence of (
) above
to get
t
( · u) f · × u = g
2
η,
and use () to substitute
· u =
1
h
0
η
t
.
We then get
1
h
0
2
η
t
2
f · ζ = g
2
η.
We now use the conservation of potential vorticity, namely
ζ = Q
0
+
η
h
0
f,
to rewrite this as
2
η
t
2
gh
0
2
η + f · fη = h
0
f · Q
0
.
Note that the right hand side is just a constant (in time). So we have a nice
differential equation we can solve.
Example.
Suppose we have fluid with mean depth
h
0
, and we start with the
following scenario:
h
0
η
0
η
0
z
Due to the differences in height, we have higher pressure on the right and lower
pressure on the left.
If there is no rotation, then the final state is a flat surface with no flow.
However, this cannot be the case if there is rotation, since this violates the
conservation of Q. So what happens if there is rotation?
At the beginning, there is no movement. So we have
ζ
(
t
= 0) = 0. Thus we
have
Q
0
=
(
η
0
h
0
f x > 0
η
0
h
0
f x < 0
.
We seek the final steady state such that
η
t
= 0.
We further assume that the final solution is independent of y, so
η
y
= 0.
So η = η(x) is just a function of x. Our equation then says
2
η
x
2
f
2
gh
0
η =
f
g
Q
0
=
f
2
gh
0
η
0
.
It is convenient to define a new variable
R =
gh
0
f
,
which is a length scale. We know
gh
0
is the fastest possible wave speed, and
thus
R
is how far a wave can travel in one rotation period. We rewrite our
equation as
d
2
η
dx
2
1
R
2
η =
1
R
2
η
0
.
Definition (Rossby radius of deformation). The length scale
R =
gh
0
f
is the Rossby radius of deformation.
This is the fundamental length scale to use in rotating systems when gravity
is involved as well.
We now impose our boundary conditions. We require
η ±η
0
as
x ±∞
.
We also require η and
dη
dx
to be continuous at x = 0.
The solution is
η = η
0
(
1 e
x/R
x > 0
(1 e
x/R
) x < 0
.
We can see that this looks quite different form the non-rotating case. It looks
like this:
z
h
0
The horizontal length scale involved is 2R.
We now look at the velocities. Using the steady flow equations, we have
u =
g
f
η
y
= 0
v =
g
f
η
x
= η
0
r
g
h
0
e
−|x|/R
.
So there is still flow in this system, and is a flow in the
y
direction into the
paper. This flow gives Coriolis force to the right, and hence balances the pressure
gradient of the system.
The final state is not one of rest, but one with motion in which the Coriolis
force balances the pressure gradient. This is geostrophic flow.
Going back to our pressure maps, if we have high and low pressure systems,
we can have flows that look like this:
L H
Then the Coriolis force will balance the pressure gradients.
So weather maps describe balanced flows. We can compute the scales here.
In the atmosphere, we have approximately
R
10 · 10
3
10
4
= 10
6
1000 km.
So the scales of cyclones are approximately 1000 km.
On the other hand, in the ocean, we have
R
10 · 10
10
4
= 10
5
= 100 km.
So ocean scales are much smaller than atmospheric scales.