Instability of bounded flows with elliptical streamlines

Abstract

In connection with the recent investigations of the instability of unbounded elliptical flows, some methods are discussed for the study of the instability of bounded flows. The stability of a ‘basic flow’ which is two-dimensional and rotating, with elliptical streamlines similar to the elliptical section of an experimentally studied cavity, is investigated in the framework of linear theory (for circular rotation, the flow discussed is stable). The regions of instability for three-dimensional disturbances are found in the plane of the parameters defining the geometry of the system (the height of the ellipsoidal cavity and the degree of ellipticity). It is shown that two types of instability exist, characterized by either monotone or oscillatory growth of the amplitudes of small disturbances.The influence of the Coriolis force field on this instability mechanism is also studied. Rotation of the system as a whole changes the regions of instability in parameter space characterizing the geometry of the cavity and the wavenumbers of unstable disturbances. As a result, the Coriolis force may stabilize or destabilize the basic flow for a given geometry.The instability of rotating density-stratified flow with elliptical streamlines is also considered.

Non-linear tides in a homogeneous rotating planet or star: global modes and elliptical instability

“…We have also analysed the elliptical instability in a rigid ellipsoidal container for global modes with ℓ 5, which extends previous work where this was accomplished for ℓ 3 (Gledzer & Ponomarev 1992a;Kerswell 2002). We find that the freedom of the boundary is unimportant for the instability, and our results are quantitatively very similar to the realistic case with a free surface.…”

“…U ℓ is the subspace of irrotational motions that represent boundary perturbations up to degree ℓ, and a convenient choice of basis for this subspace consist of the gradients of solid ellipsoidal harmonicsthis set is required to study surface gravity modes (hereafter SGMs). V ℓ is the subspace of purely vortical perturbations up to degree ℓ that do not move the boundary, for which a convenient choice of basis can be constructed analytically (Gledzer et al 1977;Lebovitz 1989a;Gledzer & Ponomarev 1992a) -this set is required to study inertial modes (hereafter IMs). The resulting basis depends on the shape of the ellipsoid.…”

“…The widest of these corresponds with the excitation of ℓ = 3, m = 0 and m = 2 IMs (e.g. Gledzer & Ponomarev 1992b). We plot an example of an unstable mode with Ω = 0.3, n = −0.2 and A = 0.1 in Fig.…”

“…The elliptical instability in a rigid ellipsoidal container has previously been studied for modes with ℓ 3 (Gledzer & Ponomarev 1992a;Kerswell 2002). Here, we undertake calculations for modes with ℓ > 3 for the first time.…”

“…A similar construction can be carried out for ℓ > 2, and the set of vectors constructed is equivalent to the basis used by Gledzer & Ponomarev (1992a) for studying the elliptical instability (of IMs) in a rigid container. An equivalent set has also been adopted by Wu & Roberts (2011) (where the basis vectors up to ℓ = 5 are written down explicitly) for studying instabilities of precessional motions in the Earth's core, and by Vantieghem (2014) for studying the free IMs of a triaxial ellipsoid in a rigid container (which was subsequently applied to study latitudinal libration-driven elliptical instability by Vantieghem et al 2015).…”