704 P.A. Davidson et al.
It appears, therefore, that an Ekman dominated flow is physically realizable provided that F
θ
is not too small. We shall see, in our numerical experiments, that this is exactly what happens. When F
θ
|F
p
| 10
−3
, u
θ
≪ |u
p
|. In the range 10
−3
F
θ
|F
p
| 10
−2
the swirl u
θ
grows to be of order |u
p
|, and for F
θ
|F
p
| 10
−2
we get Ekman pumping, with u
θ
≫ |u
p
|. It is remarkable that the velocity field should be dominated by u
θ
despite the relative weakness of the azimuthal forcing. Finally, we note that, in the arguments above, we have made no assumption about the direction of the poloidal
flow when F
θ
= 0. Our arguments make no distinction between a flow which converges at the surface and one which diverges. We shall see that this is consistent with our numerical experiments.
4. Other possible flow structures So far we have merely shown that our Ekman dominated flow, with F
p
nullified by u
2 θ
r, represents one possible solution of the equations of motion provided F
θ
is not too small. However, there are, in principle,
other structures and scalings for u, and it is important that we identify these. For example, instead of destroying the angular momentum in the boundary layers we could let the swirl build up to a level at which distributed
internal dissipation combats the generation of angular momentum. Indeed, this must represent the flow structure for F
θ
∼0.01|F
p
| where there is no Ekman pumping. To see how such a flow might come about consider 15 and 19 written in the form
u · ∇Ŵ = rF
θ
+ ν
t
∇
2 ∗
Ŵ, 29
Z
rF
θ
dV = −ν
t
Z
∇
2 ∗
Ŵ dV , 30
where 30 now applies to any volume bounded by a streamline closed int the r–z plane. We now look for a solution without Ekman pumping, in which the bulk of the streamlines avoid the boundary layers. From 30
the forcing and diffusive terms must be of similar magnitudes and so, for high Re
t
, 29 demands that u · ∇Ŵ is
of order ν
t
. It follows that Ŵ = Ŵψ + Oν
t
. Note that Ŵ is almost constant along the streamlines. Now 30 says that a steady state is achieved when all of
the angular momentum created by rF
θ
diffuses across the streamlines to the boundary. Consequently, the level of Ŵ will rise in the core of the flow until the right-hand integral in 30 balances the generation term on the
left. All of this happens subject to the boundary condition Ŵ = 0 at the outer surface. This situation is analogous to injecting heat into a prescribed recirculating flow and letting the internal
temperature rise until such time as the diffusion of heat out to the boundary balances the internal generation of heat. Now if the outer boundary condition is one of constant temperature, we would not expect a thermal
boundary layer to develop. Rather, we will have relatively uniform cross-stream gradients in temperature throughout the flow. We would expect the same to be true for angular momentum. That is, there will be a
smooth decline in Ŵ from its elevated value at the centre of the flow to zero at the boundary. Indeed, we may estimate the cross-stream gradients in Ŵ from 30. If Re is large enough then Ŵ ≈ Ŵψ and 30 yields the
estimate
Ŵ
′
ψ ≈
R
r
2
F
θ
dA ν
t
H
r
2
u
p
· dℓ .
31 This may be integrated from the boundary, where Ŵ = ψ = 0, across the streamlines, to give the distribution
of Ŵ.
EUROPEAN JOURNAL OF MECHANICS – BFLUIDS, VOL.
18
, N
◦
4, 1999
The role of Ekman pumping and the dominance of swirl 705
It seems plausible, therefore, that there are at least two structures for Ŵ. In the Ekman dominated flow Ŵ is distributed in such a way that u
2 θ
r eliminates the poloidal forcing. All streamlines are then flushed through the boundary layers by Ekman pumping. The level of swirl is fixed by the balance between the distributed
generation of angular momentum and dissipation of Ŵ in the Ekman layers. In the second option there is no such balance between u
2 θ
and F
p
. Rather, the poloidal flow is dictated, at least in part, by the poloidal forcing and there is no reason to suppose that the streamlines are entrained in the boundary layers. In such a case a
high value of Re
t
requires Ŵ = Ŵψ + Oν
t
. The swirl builds up until such time as the cross-stream diffusion of angular momentum balances the generation of Ŵ by the azimuthal force. The essential difference between
the two structures is the manner in which Ŵ is transported to the boundaries. In one case it is advected to the boundaries, in the other it diffuses.
We shall see in the next section that, provided F
θ
∼0.01|F
p
|, the Ekman dominated structure is preferred. Perhaps this is not surprising. The global integral 30 gives Ŵ ∼ δ
2
F
θ
ν
t
for the Ekman flow but Ŵ ∼ R
2
F
θ
ν
t
for the diffusive case. For the Ekman case, Ŵ ∼ F
59 θ
. See Section 3.2. The Ekman structure leads to a lower kinetic energy, both for the swirling and the poloidal components of the flow. However, for values of F
θ
less
than ∼0.01|F
p
|, the azimuthal torque is not sufficiently large to maintain Ekman pumping and so we might expect the diffusive structure to be seen.
5. Numerical experiments
