354. Vortex line representation. Clebsch variables

Let us continue our brief discussion of behavior of the vorticity field in the Eulerian flow.

(and that’s how vortex lines look like in reality… as if you wouldn’t know )

This time I would really like to derive some equations describing dynamics of vortex lines. For this, it is convenient to use so called Clebsch variables and .

The physical meaning of Clebsch variables is the following: and are two surfaces in space, and their intersection gives the vortex line. Vorticity can be rewritten in terms of Clebsch variables and as

.

Probably, the nicest thing about them is that both variables actually remain Lagrange invariants, if the flow is uncompressible (and that’s exactly the kind of the flow we discussed in the previous post – the reason being that we would actually like to separate sound waves from vortex degrees of freedom and only discuss the latter):

, (1)

so Clebsch variables are actually markers for vortex lines. Namely, we can write

, (2)

where is parameter along the vortex line. Then,

,

where the Jacobian of the mapping is equal to

.

Finally, we are ready to derive equations describing motion of vortex lines. Let us write an equality

,

which trivially follows from the conditions (1) above since and are linearly independent vectors. Using the transformation (2) we find

,

or, in other words,

,

where is the component of velocity perpendicular to the vorticity vector . As we see, any motion along the vortex line does not change its form.

Exercise 1: try to derive equation of motion for vorticity field itself. Answer: it actually has the form

.

Exercise 2 (funnier): check out that the flow described by Clebsch variables actually has zero helicity

.

The latter is topological invariant of the flow – it describes degree of knottiness of vortex lines. What to do if the topology of the flow is non-trivial?