A barotropic vortex can be generated in a rotating fluid in a number of different ways. One possible way is to place a thin-walled bottomless cylinder in the rotating fluid and then stir the fluid inside this cylinder, either cyclonically or anticycloni- cally. After allowing irregular small-scale motions to vanish and the vortex motion to get established (which typically takes a few rotation periods) the vortex is released by quickly lifting the cylinder out of the fluid. The vortex structure thus created in the otherwise rigidly rotating fluid is referred to as a ‘stirring vortex’. Because these vortices are generated within a solid cylinder with a no-slip wall, the total circulation – and hence the total vorticity – measured in the rotating frame is zero, i.e. stirring vortices are isolated vortex structures:
=
c
vãdr=
A
ωzd A=0. (1.28)
An alternative way of generating vortices is to have the fluid level in the inner cylinder lower than outside it (see Fig. 1.6): the ‘gravitational collapse’ that takes place after lifting the cylinder implies a radial inward motion of the fluid, which by conservation of angular momentum results in a cyclonic swirling motion. After any small-scale and wave-like motions have vanished, the swirling motion takes on the appearance of a columnar vortex. In contrast to the stirring vortices, these
‘gravitational collapse vortices’ have a non-zero net vorticity and are hence not isolated. This technique as well as the generation technique of stirring vortices has been applied successfully by Kloosterziel and van Heijst [18] in their study of the evolution of barotropic vortices in a rotating fluid.
A related generation method has recently been used by Cariteau and Flór [4]:
they placed a solid cylindrical bar in the fluid and after pulling it vertically upwards
Fig. 1.6 Laboratory arrangement for the creation of barotropic vortices
the resulting radial inward motion of the fluid was quickly converted into a cyclonic swirling flow, as in the previous case.
Another vortex generation technique is based on removing some of the rotating fluid from the tank by syphoning through a vertical, perforated tube. Again, the suction- induced radial motion is quickly converted into a cyclonic swirling motion – owing to the principle of conservation of angular momentum. This generation technique has been applied by Trieling et al. [24], who showed that – outside its core – the
‘sink vortex’ has the following azimuthal velocity distribution:
vθ(r)= γ 2πr
1−exp
−r2 L2
, (1.29)
withγ the total circulation of the vortex and L a typical radial length scale. Vortices have also been created in a rotating fluid by translating or rotating vertical flaps through the fluid. Alternatively, buoyancy effects may also lead to vortices in a rotating fluid, as seen, e.g. in experiments with a melting ice cube at the free surface (see, e.g. Whitehead et al. [29] and Cenedese [7]) or by releasing a volume of denser or lighter fluid (see, e.g. Griffiths and Linden [12]).
In all these cases, the vortices are observed to have a columnar structure and
∂vθ
∂z = 0, as follows from the TP theorem, even for larger Ro values. Viscosity is responsible for the occurrence of an Ekman layer at the tank bottom, in which the vortex flow is adjusted to the no-slip condition at the solid bottom. Ekman layers play an important role in the spin-down (or spin-up) of vortices. Kloosterziel and van Heijst [18] have studied the decay of barotropic vortices in a rotating fluid in detail. It was found that this type of vortex, as well as the stirring-induced vor- tex, is characterized by the following radial distributions of vorticity and azimuthal velocity:
ωstir(r)=ω0
1− r2
R2
exp
−r2 R2
, (1.30a)
vstir(r)= ω0r 2 exp
−r2 R2
. (1.30b)
The velocity data in Fig. 1.7a–d have been fitted with (1.30b), which shows a very good correspondence.
Similarly, velocity data of decaying sink-induced vortices turned out to be well fitted (see Kloosterziel and van Heijst [18]; Fig. 1.4) by
ωsink(r)=ω0exp
−r2 R2
, (1.31a)
vsink(r)= ω0R2 2r
1−exp
−r2 R2
. (1.31b)
Note that for large r values (r >> R) this azimuthal velocity distribution agrees with (1.29).
Fig. 1.7 Evolution of collapse-induced vortices in a rotating tank (from [18])
Although vortices with a velocity profile (1.31b) were found to be stable, Carton and McWilliams [6] have shown that those with velocity profile (1.30b) are linearly unstable to m =2 perturbations. It may well be, however, that the instability is not able to develop when the decay (spin-down) associated with the Ekman-layer action is sufficiently fast. In the viscous evolution of stable vortex structures two effects play a simultaneous role: the spin-down due to the Ekman layer, with a timescale
TE = H
(ν)1/2 (1.32)
and the diffusion of vorticity in radial direction, which takes place on a timescale Td = L2
ν , (1.33)
with H the fluid depth and L a measure of the core size of the vortex. For typical valuesν=10−6m2s−1, ∼1 s−1,L∼10−1m, and H =0.2 m one finds
Td∼104s, TE ∼2ã102s. (1.34) Apparently, in these laboratory conditions the effects of radial diffusion take place on a very long timescale and can hence be neglected. For a more extensive dis- cussion of the viscous evolution of barotropic vortices, the reader is referred to [18] and [20].