1
Lecture Notes on Fluid Dynamics
(1.63J/2.21J)
by Chiang C. Mei, 2002
chapter
7.2
Vorticity in inviscid rotating fluids
– Taylor -Proudman theorem
Ignoring viscosity, let ζ = ∇ × q and use the identity
|q|
ζ × q = q · ∇q − ∇
2
2
The mommentum equation can be written :
2
∂q × q = − ∇p + ∇ φ − |q|
+ ζ × q + 2Ω
∂t
ρ
2
(7.2.1)
Taking the curl of the above equation:
∂ ζ
× q = ∇ρ × ∇p
+ ζ)
+ ∇ × (2Ω
∂t
ρ2
Using the identity
×B
= A∇
·B
− B∇
·A
+B
· ∇A
−A
· ∇B
∇× A
we get
× q = −q∇ · (2Ω
+ (2Ω
· q + q · ∇(2Ω
− (2Ω
· ∇q
+ ζ)
+ ζ)
+ ζ)∇
+ ζ)
+ ζ)
∇ × (2Ω
= constant and the divergence of curl is zero;
The first term on the right vanishes because Ω
= absolute vorticity
the second vanishes for imcompressible fluids. Let ζa = ζ + 2Ω
∇ρ × ∇p
D ζ
∂ ζ
=
+ q · ∇ζ = ζa · ∇q +
Dt
∂t
ρ2
If the Rossby number is small (slow flow)
= Rossby No. =
and if ρ = constant, then
u
1
2ΩL
ζ
u
∼
1
2Ω
2ΩL
(7.2.2)
2
and
· ∇q = 0
2Ω
(7.2.3)
Th eflow is two dimensional and does not vary along the rotation vector. Let Ω be parallel
to the z axis, then
∂q
=0
(7.2.4)
∂z
This has been proven edperimentally by Taylor, see sketch in Figure 7.2.1.
Theorem 1 Taylor-Proudman theorem : A steady and slow flow in a rotating fluid is twodimensional in the plane perpendicular to the vector of angular velocity.