The instability of rotating fluid columns subjected to a weak external Coriolis force

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  • A. Mahalov
    Department of Mechanical Engineering, University of California at Berkeley, Berkeley, California 94720

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Abstract

<jats:p>Instabilities in rotating fluid columns subjected to a weak external Coriolis force are studied. External Coriolis force alters the base flow distorting circular streamlines of the unperturbed columns. The inviscid part of the modified flow (0,r,−2εr sin φ) is an exact solution of Euler equations. Here ε is the strength (nondimensional) of imposed Coriolis force. It is shown that this distortion leads to three-dimensional instabilities. The instability mechanism is generic. It occurs in many flows having circular streamlines distorted by a spectrum of modes cos mφ, sin mφ (m=2 corresponds to elliptical instability). The instabilities occur for wavelengths and frequencies at the intersection points of dispersion curves for the unperturbed columns. Moreover, the instabilities occur at the points of intersection for which modes are coupled with the external Coriolis mode. Here a case is studied where axisymmetric and helical modes are involved in interactions leading to fully three-dimensional flows. It is shown that rotating fluid columns are unstable to disturbances whose axial wavelengths lie in a band, whose width is proportional to the strength ε of imposed Coriolis force. Parametric resonance between a pair of inertial waves (natural modes of oscillation) and the external Coriolis mode is the physical explanation for the instabilities. Numerical results are presented on growth rates of mixed modes and on widths of unstable regions. The growth rates depend linearly on the strength of external Coriolis force. It is shown that viscosity shifts the instability tongues to positive values of ‖ε‖. The results of small-amplitude perturbation analysis are compared with full numerical simulations of the Navier–Stokes equations. Comments are made on competition between centrifugal and parametric instabilities in Taylor–Couette systems subjected to an external Coriolis force studied in Wiener et al. [J. Stat. Phys. 64, 913 (1991)] and Ning et al. [J. Stat. Phys. 64, 927 (1991)].</jats:p>

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