M.Sc. in Computational Science
Fundamentals of
Atmospheric Modelling
Peter Lynch, Met
´
Eireann
Mathematical Computation Laboratory (Opp. Room 30)
Dept. of Maths. Physics, UCD, Belfield.
January–April, 2004.
Lecture 5
Steady Vortical Flows
2
The TaylorProudman Theorem
In deriving the
Shallow Water Equations
, we made the
assumption that the horizontal velocity is independent of
depth.
Although dynamically consistent, this may seem an artificial
limitation.
However, we will show that
rotation acts as a
constraint on the flow
, so that under certain circumstances,
variations in the direction of the spin axis are resisted.
Theorem.
For incompressible, inviscid, hydrostatic, geostrophic flow
on an
f
plane, the velocity is independent of height.
Proof.
We assume the density is constant.
We also ignore varia
tions of the Coriolis parameter
f
.
We assume geostrophic and hydrostatic balance:
f
V
=
1
ρ
k
× ∇
p ,
∂p
∂z
=

gρ .
3
Taking the curl of the geostrophic equation (
f
constant):
∇ ×
(
k
×
V
) = 0
Let us compute the components in Cartesian coordinates:
∇×
(

v, u,
0) =
i
j
k
∂/∂x ∂/∂y ∂/∂z

v
u
0
=

∂u
∂z
,

∂v
∂z
,
∂u
∂x
+
∂v
∂y
= 0
.
Combining this with the continuity equation
∇ ·
V
=
0 we
get
∂u
∂z
=
∂v
∂z
=
∂w
∂z
= 0
.
This result indicates that, under the assumptions of the
theorem, the motion is
quasitwodimensional
, with velocity
independent of depth. If the bottom is flat,
w
≡
0
.
The result surprised G. I. Taylor, who wrote [Taylor, 1923]:
“The idea appears fantastic, but the experiments . . . show
that the motion does . . . approximate to this curious type”.
4