Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid

Abstract

Inertial modes are the eigenmodes of contained rotating fluids restored by the Coriolis force. When the fluid is incompressible, inviscid and contained in a rigid container, these modes satisfy Poincaré's equation that has the peculiarity of being hyperbolic with boundary conditions. Inertial modes are therefore solutions of an ill-posed boundary-value problem. In this paper we investigate the mathematical side of this problem. We first show that the Poincaré problem can be formulated in the Hilbert space of square-integrable functions, with no hypothesis on the continuity or the differentiability of velocity fields. We observe that with this formulation, the Poincaré operator is bounded and self-adjoint and as such, its spectrum is the union of the point spectrum (the set of eigenvalues) and the continuous spectrum only. When the fluid volume is an ellipsoid, we show that the inertial modes form a complete base of polynomial velocity fields for the square-integrable velocity fields defined over the ellipsoid and meeting the boundary conditions. If the ellipsoid is axisymmetric then the base can be identified with the set of Poincaré modes, first obtained by Bryan (1889) [1], and completed with the geostrophic modes.

Internal Energy Dissipation in Enceladus's Subsurface Ocean From Tides and Libration and the Role of Inertial Waves

“…It is sensible to assume that the response to a body force (per unit volume), f , of frequency ω can be decomposed, at least partially, onto the (infinite) set of inertial modes { u α }: where <·|·> denotes the projection operator over pairs of vector fields. Ivers et al () and Backus and Rieutord () have demonstrated the completeness of the above modal expansion for an inviscid fluid inside a spherical or ellipsoidal container (with no inner core). In analogy with those works, we define the projection operator as .…”

“…where < ·|· > denotes the projection operator over pairs of vector fields. Ivers et al (2014) and Backus and Rieutord (2017) have demonstrated the completeness of the above modal expansion for an inviscid fluid inside a spherical or ellipsoidal container (with no inner core). In analogy with those works, we define the projection operator as < v|w >≡ Re∫  v * · w. A resonance takes place when the factor multiplying one or more u becomes large, which occurs when the distance between the forcing frequency and the eigenvalue approaches zero and/or < f|u > is large.…”

Time-dependent low-latitude core flow and geomagnetic field acceleration pulses

“…A special class of slow, equatorially-symmetric, inertial modes, that we refer to as QG modes (Zhang et al 2001;Busse et al 2005;Maffei et al 2017), have been shown to efficiently describe rotating flow in a sphere at the onset of convection (Zhang & Liao 2004;Zhang et al 2007), and when combined with the geostrophic mode can also describe weakly-nonlinear convection (Zhang & Liao 2004;Liao et al 2012). More generally, geostrophic and inertial modes can be used to describe the transient response of rotating spherical systems to a forcing (Liao & Zhang 2010) and in principle they provide a complete basis for representing flows in a rotating sphere (Cui et al 2014;Ivers et al 2015;Backus & Rieutord 2017). Of particular interest here is that they are well-suited to describing motions at low latitudes.…”

Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle

“…Expansion (10) has proven to be exact for the ellipsoid (Backus & Rieutord 2017;Ivers 2017). From balance (11) it is clear that the axial geostrophic pressure torque vanishes, as the axial Coriolis torque vanishes for any flow u.…”

“…As N is increased more TM with a larger polynomial complexity appear, with one additional TM per two polynomial degrees. This is explained by the introduction of an additional geostrophic basis vector at every second polynomial degree (see Backus & Rieutord (2017), in the sphere and Appendix A in the ellipsoid). The equatorial and meridional sections of the two lowest frequency (and thus largest scale) TM, calculated using B0,QG at N = 7, are presented in Figure 5 for a strongly deformed ellipsoid with equatorial ellipticity = 0.42.…”