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  Point vortex models with three-body interactions

+ 1 like - 0 dislike

The classical point vortex model corresponding to the 2D incompressible Euler equation in vorticity form is the system of $N$ ODES

$$\begin{cases} \dot{x}_i(t) = \sum_{1\leq i\neq j\leq N} a_j K(x_i(t),x_j(t)) \\ x_i(0) = x_i^0\end{cases}\tag{1}$$

where $K(x,y) := -\frac{1}{2\pi}(\frac{(x-y)_2}{|x-y|^2}, -\frac{(x-y)_1}{|x-y|^2})$ is the Biot-Savart kernel.

It is evident from the RHS of equation $\ref{a}$ that the point vortices only have binary (i.e. pairwise) interactions. My question is the following: are there physically motivated models generalizing equation \ref{a} which have binary and ternary interactions? By ternary, I have in mind a term of the form

$$\sum_{1\leq i\neq j\neq k\leq N} a_j a_k \tilde{K}(x_i(t),x_j(t),x_k(t)),$$

where $\tilde{K}$ is an $\mathbb{R}^2$-valued map.

asked Feb 21 in Resources and References by Matt Rosenzweig [ no revision ]

I found this:

Point Vortices: Finding Periodic Orbits and their Topological Classification
A dissertation
submitted by
Spencer Ambrose Smith
In partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
August 2012
Advisor: Prof. Bruce Boghosian

On page 34 (43 of 201) or you can search the pdf with control-f to find for the term "savart"

\[v(r)=-\frac{1}{2\pi}\int_{R^2} \frac{r-r'}{|r-r'|^2} X \omega(r')dr'.\]

This is very similar in form to the Biot-Savart law, and indeed in \(R_3\) they are mathematically identical in form.

T@jevanpipitone hank you for your commentary, but I don't see how it answers my question. All you're pointing out is the well-known Biot-Savart law which allows one to recover the velocity field from the vorticity.

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