Triadic resonances in the wide-gap spherical Couette system

2018 ◽  
Vol 843 ◽  
pp. 211-243 ◽  
Author(s):  
A. Barik ◽  
S. A. Triana ◽  
M. Hoff ◽  
J. Wicht
Keyword(s):  
Differential Rotation ◽  
Numerical Experiments ◽  
Outer Boundary ◽  
Mode Frequency ◽  
Force Balance ◽  
Background Flow ◽  
Concentric Spheres ◽  
Wide Gap ◽  
Spherical Couette ◽  
Axisymmetric Instability

The spherical Couette system, consisting of a viscous fluid between two differentially rotating concentric spheres, is studied using numerical simulations and compared with experiments performed at BTU Cottbus-Senftenberg, Germany. We concentrate on the case where the outer boundary rotates fast enough for the Coriolis force to play an important role in the force balance, and the inner boundary rotates slower or in the opposite direction as compared to the outer boundary. As the magnitude of differential rotation is increased, the system is found to transition through three distinct hydrodynamic regimes. The first regime consists of the emergence of the first non-axisymmetric instability. Thereafter one finds the onset of ‘fast’ equatorially antisymmetric inertial modes, with pairs of inertial modes forming triadic resonances with the first instability. A further increase in the magnitude of differential rotation leads to the flow transitioning to turbulence. Using an artificial excitation, we study how the background flow modifies the inertial mode frequency and structure, thereby causing departures from the eigenmodes of a full sphere and a spherical shell. We investigate triadic resonances of pairs of inertial modes with the fundamental instability. We explore possible onset mechanisms through numerical experiments.

Flow instabilities in the wide-gap spherical Couette system

2013 ◽  
Vol 738 ◽  
pp. 184-221 ◽  
Author(s):  
Johannes Wicht
Keyword(s):  
Shear Layer ◽  
Differential Rotation ◽  
Outer Boundary ◽  
Fluid Flows ◽  
Zonal Flow ◽  
Comprehensive Overview ◽  
Jet Instability ◽  
Rotation Rates ◽  
Boundary Rotation ◽  
Spherical Couette

AbstractThe spherical Couette system is a spherical shell filled with a viscous fluid. Flows are driven by the differential rotation between the inner and the outer boundary that rotate with $\Omega $ and $\Omega + \mathrm{\Delta} \Omega $ about a common axis. This setup has been proposed for second-generation dynamo experiments. We numerically explore the different instabilities emerging for rotation rates up to $\Omega = (1/ 3)\times 1{0}^{7} $, venturing also into the nonlinear regime where oscillatory and chaotic solutions are found. The results provide a comprehensive overview of the possible flow regimes. For low values of $\Omega $ viscosity dominates and an equatorial jet in meridional circulation and zonal flow develops that becomes unstable as the differential rotation is increased beyond a critical value. For intermediate $\Omega $ and an inner boundary rotating slower than the outer one, new double-roll and helical instabilities are found. For large $\Omega $ values Coriolis effects enforce a nearly two-dimensional fundamental flow where a Stewartson shear layer develops at the tangent cylinder. This shear layer is the source of nearly geostrophic non-axisymmetric instabilities that resemble columnar Rossby modes. At first, the instabilities differ significantly depending on whether the inner boundary rotates faster $( \mathrm{\Delta} \Omega \gt 0)$ or slower $( \mathrm{\Delta} \Omega \lt 0)$ than the outer one. For very large outer boundary rotation rates, however, both instabilities once more become comparable. Fast inertial waves similar to those observed in recent spherical Couette experiments prevail for larger $\Omega $ values and $ \mathrm{\Delta} \Omega \lt 0$ in when $ \mathrm{\Delta} \Omega $ and $\Omega $ are of comparable magnitude. For larger differential rotations $ \mathrm{\Delta} \Omega \gg \Omega $, however, the equatorial jet instability always takes over.

Simulation of spiral instabilities in wide-gap spherical Couette flow

2018 ◽  
Vol 50 (2) ◽  
pp. 025507 ◽  
Author(s):  
Suhail Abbas ◽  
Li Yuan ◽  
Abdullah Shah
Keyword(s):  
Couette Flow ◽  
Spherical Couette Flow ◽  
Wide Gap ◽  
Spherical Couette

The nonaxisymmetric instability of the wide-gap spherical Couette flow

1997 ◽  
Vol 9 (4) ◽  
pp. 1197-1199 ◽  
Author(s):  
Keisuke Araki ◽  
Jiro Mizushima ◽  
Shinichiro Yanase
Keyword(s):  
Couette Flow ◽  
Spherical Couette Flow ◽  
Wide Gap ◽  
Spherical Couette

Routes to chaos in wide-gap spherical Couette flow

1999 ◽  
Vol 11 (6) ◽  
pp. 1359-1372 ◽  
Author(s):  
P. Wulf ◽  
C. Egbers ◽  
H. J. Rath
Keyword(s):  
Couette Flow ◽  
Spherical Couette Flow ◽  
Routes To Chaos ◽  
Wide Gap ◽  
Spherical Couette

scholarly journals Bifurcation aspect of polygonal coherence over transitional Reynolds numbers in wide-gap spherical Couette flow

2021 ◽  
Vol 6 (11) ◽  
Author(s):  
Fumitoshi Goto ◽  
Tomoaki Itano ◽  
Masako Sugihara-Seki ◽  
Takahiro Adachi
Keyword(s):  
Couette Flow ◽  
Reynolds Numbers ◽  
Spherical Couette Flow ◽  
Wide Gap ◽  
Spherical Couette

A super-rotating shear layer in magnetohydrodynamic spherical Couette flow

2002 ◽  
Vol 452 ◽  
pp. 263-291 ◽  
Author(s):  
E. DORMY ◽  
D. JAULT ◽  
A. M. SOWARD
Keyword(s):  
Magnetic Field ◽  
Angular Velocity ◽  
Point Source ◽  
Shear Layer ◽  
Outer Boundary ◽  
Outer Sphere ◽  
Free Shear Layer ◽  
Spherical Couette Flow ◽  
Spherical Couette ◽  
Layer Problem

We consider axisymmetric magnetohydrodynamic motion in a spherical shell driven by rotating the inner boundary relative to the stationary outer boundary – spherical Couette flow. The inner solid sphere is rigid with the same electrical conductivity as the surrounding fluid; the outer rigid boundary is an insulator. A force-free dipole magnetic field is maintained by a dipole source at the centre. For strong imposed fields (as measured by the Hartmann number M), the numerical simulations of Dormy et al. (1998) showed that a super-rotating shear layer (with angular velocity about 50% above the angular velocity of the inner core) is attached to the magnetic field line [Cscr ] tangent to the outer boundary at the equatorial plane of symmetry. At large M, we obtain analytically the mainstream solution valid outside all boundary layers by application of Hartmann jump conditions across the inner- and outer-sphere boundary layers. We formulate the large-M boundary layer problem for the free shear layer of width M−1/2 containing [Cscr ] and solve it numerically. The super-rotation can be understood in terms of the nature of the meridional electric current flow in the shear layer, which is fed by the outer-sphere Hartmann layer. Importantly, a large fraction of the current entering the shear layer is tightly focused and effectively released from a point source at the equator triggered by the tangency of the [Cscr ]-line. The current injected by the source follows the [Cscr ]-line closely but spreads laterally due to diffusion. In consequence, a strong azimuthal Lorentz force is produced, which takes opposite signs either side of the [Cscr ]-line; order-unity super-rotation results on the equatorial side. In fact, the point source is the small equatorial Hartmann layer of radial width M−2/3 ([Lt ]M−1/2) and latitudinal extent M−1/3. We construct its analytic solution and so determine an inward displacement width O(M−2/3) of the free shear layer. We compare our numerical solution of the free shear layer problem with our numerical solution of the full governing equations for M in excess of 104. We obtain excellent agreement. Some of our more testing comparisons are significantly improved by incorporating the shear layer displacement caused by the equatorial Hartmann layer.

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