Elliptic instability in the Lagrangian-averaged Euler-Boussinesq-alpha equations

Details Elliptic instability in the Lagrangian-averaged Euler-Boussinesq-alpha equations Elliptic instability in the Lagrangian-averaged Euler-Boussinesq-alpha equations

2004-10-08

arxiv.org/abs/nlin/0410006v2

Phys. Fluids 17 (2005) 054113

Nonlinear Sciences

Chaotic Dynamics

23 pages (sigle spaced w/figure at the end), 9 figures--coarse quality, accepted by Phys. Fluids

Scientific paper

10.1063/1.1897006

We examine the effects of turbulence on elliptic instability of rotating stratified incompressible flows, in the context of the Lagragian-averaged Euler-Boussinesq-alpha, or \laeba, model of turbulence. We find that the \laeba model alters the instability in a variety of ways for fixed Rossby number and Brunt-V\"ais\"al\"a frequency. First, it alters the location of the instability domains in the $(\gamma,\cos\theta)-$parameter plane, where $\theta$ is the angle of incidence the Kelvin wave makes with the axis of rotation and $\gamma$ is the eccentricity of the elliptic flow, as well as the size of the associated Lyapunov exponent. Second, the model shrinks the width of one instability band while simultaneously increasing another. Third, the model introduces bands of unstable eccentric flows when the Kelvin wave is two-dimensional. We introduce two similarity variables--one is a ratio of the Brunt-V\"ais\"al\"a frequency to the model parameter $\Upsilon_0 = 1+\alpha^2\beta^2$, and the other is the ratio of the adjusted inverse Rossby number to the same model parameter. Here, $\alpha$ is the turbulence correlation length, and $\beta$ is the Kelvin wave number. We show that by adjusting the Rossby number and Brunt-V\"ais\"al\"a frequency so that the similarity variables remain constant for a given value of $\Upsilon_0$, turbulence has little effect on elliptic instability for small eccentricities $(\gamma \ll 1)$. For moderate and large eccentricities, however, we see drastic changes of the unstable Arnold tongues due to the \laeba model.

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