Continuation and stability of rotating waves in the magnetized spherical Couette system: secondary transitions and multistability

Abstract

Rotating waves (RW) bifurcating from the axisymmetric basic magnetized spherical Couette (MSC) flow are computed by means of Newton-Krylov continuation techniques for periodic orbits. In addition, their stability is analysed in the framework of Floquet theory. The inner sphere rotates whilst the outer is kept at rest and the fluid is subjected to an axial magnetic field. For a moderate Reynolds number Re = 10 3 (measuring inner rotation) the effect of increasing the magnetic field strength (measured by the Hartmann number Ha) is addressed in the range Ha ∈ (0, 80) corresponding to the working conditions of the HEDGEHOG experiment at Helmholtz-Zentrum Dresden-Rossendorf. The study reveals several regions of multistability of waves with azimuthal wave number m = 2, 3, 4, and several transitions to quasiperiodic flows, i.e modulated rotating waves (MRW). These nonlinear flows can be classified as the three different instabilities of the radial jet, the return flow and the shear-layer, as found in previous studies. These two flows are continuously linked, and part of the same branch, as the magnetic forcing is increased. Midway between the two instabilities, at a certain critical Ha, the nonaxisymmetric component of the flow is maximum. arXiv:1805.06750v2 [physics.flu-dyn]

Oscillatory flow between concentric spheres driven by an electromagnetic force

“…2013; Kasprzyk et al. 2017; Garcia & Stefani 2018; Kaplan, Nataf & Schaeffer 2018; Ogbonna et al. 2020), it does not imply the use of a magnetizable fluid.…”

“…2017). Even though this problem is simple, the interplay of viscous, inertial and electromagnetic forces gives rise to a wide variety of instabilities (Schrauf 1986; Travnikov, Eckert & Odenbach 2011; Garcia & Stefani 2018; Garcia et al. 2020) and features depending on the electrical conductivity of the fluid and the electrodes, as well as on the magnetic field distribution (Hollerbach 2009; Gissinger et al.…”

Four-Frequency Solution in a Magnetohydrodynamic Couette Flow as a Consequence of Azimuthal Symmetry Breaking

“…Successive Hopf bifurcations [48,49] give rise to quasiperiodic modulated rotating waves (MRWs) and to chaotic turbulent flows, usually following the NRT scenario [48]. In the particular case of the MSC, when the magnetic field is varied, branches of RWs with a m-fold azimuthal symmetry with a prime number m = 2, 3 [34,50] give rise to stable twoand three-tori MRWs [51], and eventually chaotic flows [8], though four-tori MRWs have not yet been found. We show below that MHD four-tori solutions in terms of MRWs can be obtained after successive azimuthal symmetry breaking Hopf bifurcations from a parent branch of RW having m-fold symmetry which is not a prime number, in this case m = 4.…”

Modulated Rotating Waves in the Magnetised Spherical Couette System

“…6. MRW are strongly equatorially asymmetric in the form of a radial jet near the equatorial plane as described for RW corresponding to the radial jet instability [14]. However, for MRW fluid motions start to be noticeable within the cylinder parallel to the rotation axis and tangent to the inner sphere (see meridional sections).…”

“…These are solutions in which a fixed flow pattern with m 1 -fold azimuthal symmetry is rotating at a frequency ω in the azimuthal direction [33]. For the MSC system, RW have been recently computed [14], as a function of Ha, by means of continuation methods for periodic orbits [38]. However, RW can also be computed as fixed points [37,47] of the MSC system written in a reference frame rotating with the same frequency ω:…”