The Kelvin--Helmholtz instability of momentum sheets in the Euler equations for planar diffeomorphisms
Robert McLachlan and Stephen Marsland
As part of our paper with the above title (to appear in the SIAM Journal of Applied Dynamical Systems) we produced a set of numerical solutions for the Euler equations of momentum sheets. These can be viewed in the paper, but we here provide QuickTime movies of some of the more illustrative cases.
The simulations show 3 symmetric repetitions of a momentum sheet moving under the Euler equations on the full diffeomorphism group (sometimes known as EPDiff). The integrals are approximated by constant-weight quadrature and integrated using the symplectic midpoint rule.
Click on the link to see each movie (and be prepared to wait for them to download).
Gaussian metric, phi=0
Gaussian metric, phi=pi/8
Gaussian metric, phi=pi/2
H^1 metric, phi=pi/8
H^1 metric, phi=pi/2