# Point vortex models with three-body interactions

+ 1 like - 0 dislike
74 views

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.

I found this:

https://arxiv.org/pdf/1510.06756.pdf
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
in
Physics
TUFTS UNIVERSITY
August 2012
$v(r)=-\frac{1}{2\pi}\int_{R^2} \frac{r-r'}{|r-r'|^2} X \omega(r')dr'.$
This is very similar in form to the Biot-Savart law, and indeed in $R_3$ they are mathematically identical in form.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.