# Stability Results for Idealized Shear Flows on a Rectangular Periodic Domain

Stability Results for Idealized Shear Flows on a Rectangular Periodic Domain We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain $$[0,2\pi )\times [0,2\pi / \kappa )$$ [ 0 , 2 π ) × [ 0 , 2 π / κ ) for $$\kappa \in \mathbb {R}^+$$ κ ∈ R + , the Euler equations admit a family of stationary solutions given by the vorticity profiles $$\Omega ^*(\mathbf {x})= \Gamma \cos (p_1x_1+ \kappa p_2x_2)$$ Ω ∗ ( x ) = Γ cos ( p 1 x 1 + κ p 2 x 2 ) . We show linear stability for such flows when $$p_2=0$$ p 2 = 0 and $$\kappa \ge |p_1|$$ κ ≥ | p 1 | (equivalently $$p_1=0$$ p 1 = 0 and $$\kappa {|p_2|}\le {1}$$ κ | p 2 | ≤ 1 ). The classical result due to Arnold is that for $$p_1 = 1, p_2 = 0$$ p 1 = 1 , p 2 = 0 and $$\kappa \ge 1$$ κ ≥ 1 the stationary flow is nonlinearly stable via the energy-Casimir method. We show that for $$\kappa \ge |p_1| \ge 2, p_2 = 0$$ κ ≥ | p 1 | ≥ 2 , p 2 = 0 the flow is linearly stable, but one cannot expect a similar nonlinear stability result. Finally we prove nonlinear instability for all steady states satisfying $$p_1^2+\kappa ^2{p_2^2}>\frac{{3(\kappa ^2+1)}}{4(7-4\sqrt{3})}$$ p 1 2 + κ 2 p 2 2 > 3 ( κ 2 + 1 ) 4 ( 7 - 4 3 ) . The modification and application of a structure-preserving Hamiltonian truncation is discussed for the anisotropic case $$\kappa \ne 1$$ κ ≠ 1 . This leads to an explicit Lie-Poisson integrator for the approximate system, which is used to illustrate our analytical results. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

# Stability Results for Idealized Shear Flows on a Rectangular Periodic Domain

, Volume 20 (2) – Jun 22, 2017
12 pages

/lp/springer_journal/stability-results-for-idealized-shear-flows-on-a-rectangular-periodic-fvPmTAC6Pf
Publisher
Springer Journals
Subject
Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics
ISSN
1422-6928
eISSN
1422-6952
D.O.I.
10.1007/s00021-017-0329-2
Publisher site
See Article on Publisher Site

### Abstract

We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain $$[0,2\pi )\times [0,2\pi / \kappa )$$ [ 0 , 2 π ) × [ 0 , 2 π / κ ) for $$\kappa \in \mathbb {R}^+$$ κ ∈ R + , the Euler equations admit a family of stationary solutions given by the vorticity profiles $$\Omega ^*(\mathbf {x})= \Gamma \cos (p_1x_1+ \kappa p_2x_2)$$ Ω ∗ ( x ) = Γ cos ( p 1 x 1 + κ p 2 x 2 ) . We show linear stability for such flows when $$p_2=0$$ p 2 = 0 and $$\kappa \ge |p_1|$$ κ ≥ | p 1 | (equivalently $$p_1=0$$ p 1 = 0 and $$\kappa {|p_2|}\le {1}$$ κ | p 2 | ≤ 1 ). The classical result due to Arnold is that for $$p_1 = 1, p_2 = 0$$ p 1 = 1 , p 2 = 0 and $$\kappa \ge 1$$ κ ≥ 1 the stationary flow is nonlinearly stable via the energy-Casimir method. We show that for $$\kappa \ge |p_1| \ge 2, p_2 = 0$$ κ ≥ | p 1 | ≥ 2 , p 2 = 0 the flow is linearly stable, but one cannot expect a similar nonlinear stability result. Finally we prove nonlinear instability for all steady states satisfying $$p_1^2+\kappa ^2{p_2^2}>\frac{{3(\kappa ^2+1)}}{4(7-4\sqrt{3})}$$ p 1 2 + κ 2 p 2 2 > 3 ( κ 2 + 1 ) 4 ( 7 - 4 3 ) . The modification and application of a structure-preserving Hamiltonian truncation is discussed for the anisotropic case $$\kappa \ne 1$$ κ ≠ 1 . This leads to an explicit Lie-Poisson integrator for the approximate system, which is used to illustrate our analytical results.

### Journal

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: Jun 22, 2017

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