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The non-resonant response of fluid in a rapidly rotating sphere undergoing longitudinal libration


We investigate the problem of oscillatory flow of a homogeneous fluid with viscosity ν in a fluid-filled sphere of radius a that rotates rapidly about a fixed axis with angular velocity Ω 0 and that undergoes weak longitudinal libration with amplitude Ω 0 and frequency Ω Ω 0, where is the Poincaré number with ll 1 and Ω is dimensionless frequency with 0< Ω < 2. Three different methods are employed in this investigation: (i) asymptotic analysis at small Ekman numbers E defined as E= ν(a2Ω0)(ii) linear numerical analysis using a spectral method; and (iii) nonlinear direct numerical simulation using a finite-element method. A satisfactory agreement among the three different sets of solutions is achieved when E≤ 10-4. It is shown that the flow amplitude is nearly independent of both the Ekman number E and the libration frequency Ω, always obeying the asymptotic scaling = O even though various spherical inertial modes are excited by longitudinal libration at different libration frequencies Ω. Consequently, resonances do not occur in this system even when Ω is at the characteristic value of an inertial mode. It is also shown that the pressure difference along the axis of rotation is anomalous: this quantity reaches a sharp peak when Ω approaches a characteristic value. In contrast, the pressure difference measured at other places in the sphere, such as in the equatorial plane, and the volume-integrated kinetic energy are nearly independent of both the Ekman number E and the libration frequency Ω. Absence of resonances in a fluid-filled sphere forced by longitudinal libration is explained through the special properties of the analytical solution that satisfies the no-slip boundary condition and is valid for E\ll 1 and ll 1. © 2013 Cambridge University Press.

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