Libration-driven multipolar instabilities

Abstract

We consider rotating flows in non-axisymmetric enclosures that are driven by libration, i.e. by a small periodic modulation of the rotation rate. Thanks to its simplicity, this model is relevant to various contexts, from industrial containers (with small oscillations of the rotation rate) to fluid layers of terrestial planets (with length-of-day variations). Assuming a multipolar n-fold boundary deformation, we first obtain the two-dimensional basic flow. We then perform a short-wavelength local stability analysis of the basic flow, showing that an instability may occur in three dimensions. We christen it the Libration Driven Multipolar Instability (LDMI). The growth rates of the LDMI are computed by a Floquet analysis in a systematic way, and compared to analytical expressions obtained by perturbation methods.We then focus on the simplest geometry allowing the LDMI, a librating deformed cylinder. To take into account viscous and confinement effects, we perform a global stability analysis, which shows that the LDMI results from a parametric resonance of inertial modes. Performing numerical simulations of this librating cylinder, we confirm that the basic flow is indeed established and report the first numerical evidence of the LDMI. Numerical results, in excellent agreement with the stability results, are used to explore the non-linear regime of the instability (amplitude and viscous dissipation of the driven flow). We finally provide an example of LDMI in a deformed spherical container to show that the instability mechanism is generic. Our results show that the previously studied libration driven elliptical instability simply corresponds to the particular case n = 2 of a wider class of instabilities. Summarizing, this work shows that any oscillating non-axisymmetric container in rotation may excite intermittent, space-filling LDMI flows, and this instability should thus be easy to observe experimentally.

Latitudinal libration driven flows in triaxial ellipsoids

“…This Poisson equation is solved with the algebraic multigrid method BoomerAMG (Henson & Yang 2002). Discussions regarding the performance of this code can be found in Marti et al (2014) and Cébron et al (2014).…”

Experimental study of global-scale turbulence in a librating ellipsoid

“…This method was confirmed in the multipolar stability analysis in a librating deformed cylinder and sphere in Ref. 47. A general formula of the typical growth rate for each calculation of σ inv is given for f around the resonant forcing frequency…”

“…30,47 An analogous equation can be made for a j . Since inertial modes exist within a frequency from [−2, 2], the resonance condition in (14) allows for the existence of elliptical instability in flows from | f | = 0 − 4.…”

“…47 The degree n of multipolar deformation is taken to be n = m lib = 2 for the ellipsoid. This method was confirmed in the multipolar stability analysis in a librating deformed cylinder and sphere in Ref.…”

Inertial Wave Turbulence Driven by Elliptical Instability

“…This instability develops in rotating fluids when the streamlines are deformed into ellipses. Its linear growth, due to the resonance of two inertial waves with the elliptical basic flow, is well described both theoretically [2,[9][10][11] and experimentally [3][4][5]12]. The nonlinear saturation of the elliptical instability remains poorly understood, but it is the relevant regime to describe vortex core breakdown [13], as well as dissipation and magnetic field generation in planetary cores [14].…”