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Vortices persist over time, but change their shape.

I am used to a structure that persists being “stable” as in it is robust against small perturbations.

How can one define stability of a Vortex? For example how could one prove that the bathtub vortex is stable? What does it even mean for it to be stable if it’s shape changes?

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    $\begingroup$ It is worth specifying that you are talking about vortices in liquid (or specifically in water) rather than those in atmosphere, superconductor or some other situation. $\endgroup$ Jul 6 at 9:11
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    $\begingroup$ @RogerVadim, Good point. I am talking generally about vortices in fluid where fluid is used in the broader sense of the word. I am being general because in a wide variety of situations vortex's seem to persist in fluids, and from an outside perspective I would expect a general result or lens. $\endgroup$
    – jrudd
    Jul 6 at 13:34
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We can define the stability of a vortex by perturbing it in some way and seeing if that perturbation grows or decays.

For an initially circular vortex ring, we could add a sinusoidal perturbation to its shape and see if the amplitude of the sinusoid grows. It has been shown that vortex rings are unstable to such a sinusoidal perturbation.

For the bathtub vortex you ask about, it is itself the result of an instability of the flow draining down a plug. Water draining from a bath tub could drain without any swirl or vortex, the flow velocity could be entirely radial towards the plughole. Indeed, for low enough $(\lessapprox50)$ Reynolds numbers that is what happens.

However, for higher Reynolds numbers the base flow is unstable to a swirl perturbation and the flow starts to rotate faster and faster, forming the familiar bathtub vortex. More detail can be found in this paper.

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  • $\begingroup$ We can define the stability of a vortex by perturbing it in some way and seeing if that perturbation grows or decays. But what if the perturbation grows but does not lessen the spinning motion. For example, if I perturbed a tornado, and the perturbation caused the tornado to change direction, the perturbation is growing, but I still think of the tornado as stable, since the concentrated vorticity is preserved. $\endgroup$
    – jrudd
    Jul 6 at 13:32
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The basic ingredient of stability is is Kelvin's circulation theorem which says that for inviscid flow $\Gamma=\oint_\gamma {\bf v}\cdot d{\bf r}$ is a constant when the contour $\gamma$ is advected with the flow.

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  • $\begingroup$ This makes sense. And I have seen this result. But it does not seem to say anything about the stability against small perturbations. Unless I am misunderstanding. $\endgroup$
    – jrudd
    Jul 6 at 0:53

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