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Instability of Certain Equilibrium Solutions of the Euler Equations on a Toroidal Domain

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Thursday, May 14, 2015 - 2:00pm

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ECCR 257

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The 2D Euler Equations can be written on a toroidal domain as an infinite dimensional Hamiltonian system. We have studied the stability of the family of stationary solutions cos(mx+ny). We use a Poisson structure-preserving truncation described by Zeitlin (1991) to reduce the full problem into a finite-mode system in Fourier Space, and consider the limit as the truncation goes to infinity. We replicate some results by Li (2000) in this new finite-mode setting, namely the splitting of the linearised problem into ”classes”, most of which are stable. We also show that nearly all the stationary solutions described are unstable. The usefulness of Zeitlin’s truncation, both numerically and analytically, is also demonstrated.