Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T20:57:05.035Z Has data issue: false hasContentIssue false
  • English
  • Français

Flow instabilities in the wide-gap spherical Couette system

Published online by Cambridge University Press:  05 December 2013

Johannes Wicht
Johannes Wicht*
Affiliation:
Max-Planck-Institut für Sonnensystemforschung, Katlenburg-Lindau, Germany
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The 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.

JFM classification

Boundary Layers: Free shear layers Instability: Instability Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 738 , 10 January 2014 , pp. 184 - 221
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brito, D., Alboussière, T., Cardin, P., Gagnière, N., Jault, D., La Rizza, P., Masson, J.-P., Nataf, H.-C. & Schmitt, D. 2011 Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment. Phys. Rev. E 83 (6), 066310.Google Scholar
Buehler, K. 1990 Symmetries and asymmetric Taylor vortex flow in spherical gaps. Acta Mechanica 81, 3–38.CrossRefGoogle Scholar
Busse, F. H. 1968 Shear flow instabilities in rotating systems. J. Fluid Mech. 33, 577–589.CrossRefGoogle Scholar
Busse, F. H. & Simitev, R. 2004 Inertial convection in rotating fluid spheres. J. Fluid Mech. 498, 23–30.Google Scholar
Cao, H., Russell, C. T., Wicht, J., Christensen, U. R. & Dougherty, M. K. 2012 Saturn’s high degree magnetic moments: evidence for a unique planetary dynamo. Icarus 221, 388–394.Google Scholar
Christensen, U. & Wicht, J. 2007 Numerical dynamo simulations. In Core Dynamics (ed. Olson, P.), Treatise on Geophysics, vol. 8, pp. 245–282. Elsevier.Google Scholar
Christensen, U. R., Aubert, J., Busse, F. H., Cardin, P., Dormy, E., Gibbons, S., Glatzmaier, G. A., Honkura, Y., Jones, C. A., Kono, M., Matsushima, M., Sakuraba, A., Takahashi, F., Tilgner, A., Wicht, J. & Zhang, K. 2001 A numerical dynamo benchmark. Phys. Earth Planet. Inter. 128, 25–34.CrossRefGoogle Scholar
Dormy, E., Cardin, P. & Jault, D. 1998 MHD flow in a slightly differentially rotating spherical shell, with conducting inner core, in a dipolar magnetic field. Earth Planet. Sci. Lett. 158, 15–24.Google Scholar
Egbers, C. & Rath, H. J. 1995 The existence of Taylor vorticies and wide-gap instabilities in spherical Couette flow. Acta Mechanica 111, 125–140.Google Scholar
Figueroa, A., Schaeffer, N., Nataf, H.-C. & Schmitt, D. 2013 Modes and instabilities in magnetized spherical Couette flow. J. Fluid Mech. 716, 445–469.CrossRefGoogle Scholar
Glatzmaier, G. A. 1984 Numerical simulation of stellar convective dynamos. 1. The model and methods. J. Comput. Phys. 55, 461–484.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Breukelen Press.Google Scholar
Guervilly, C. & Cardin, P. 2010 Numerical simulations of dynamos generated in spherical Couette flows. Geophys. Astrophys. Fluid Dyn. 104, 221–248.Google Scholar
Hollerbach, R. 2003 Instabilities of the Stewartson layer. Part 1. The dependence on the sign of Ro . J. Fluid Mech. 492, 289–302.CrossRefGoogle Scholar
Hollerbach, R. 2009 Non-axisymmetric instabilities in magnetic spherical Couette flow. Proc. R. Soc. Lond. A 465, 2003–2013.Google Scholar
Hollerbach, R., Futterer, B., More, T. & Egbers, C. 2004 Instabilities of the Stewartson layer. Part 2. Supercritical mode transitions. Theor. Comput. Fluid Dyn. 18, 197–204.CrossRefGoogle Scholar
Hollerbach, R., Junk, M. & Egbers, C. 2006 Non-axisymmetric instabilities in basic state spherical Couette flow. Fluid Dyn. Res. 38, 257–273.Google Scholar
Kelley, D., Triana, S. A., Zimmerman, D., Tilgner, A. & Lathrop, D. 2007 Inertial waves driven by differential rotation in a planetary geometry. Geophys. Astrophys. Fluid Dyn. 101, 469–487.Google Scholar
Matsui, H., Adams, M., Kelley, D., Triana, S. A., Zimmerman, D., Buffett, B. A. & Lathrop, D. P. 2011 Numerical and experimental investigation of shear-driven inertial oscillations in an Earth-like geometry. Phys. Earth Planet. Inter. 188, 194–202.Google Scholar
Proudman, I. 1956 The almost-rigid rotation of viscous fluid between concentric spheres. J. Fluid Mech. 1, 505–516.Google Scholar
Rieutord, M., Andres Triana, S., Zimmerman, D. S. & Lathrop, D. P. 2012 Excitation of inertial modes in an exerimental spherical Couette flow. Phys. Rev. E 86, 026304.Google Scholar
Rieutord, M. & Valdettaro, L. 1997 Inertial waves in a rotating spherical shell. J. Fluid Mech. 341, 77–99.CrossRefGoogle Scholar
Schaeffer, N. & Cardin, P. 2005 Quasigeostrophic model of the instabilities of the Stewartson layer in flat and depth-varying containers. Phys. Fluids 17 (10), 104111.CrossRefGoogle Scholar
Schaeffer, N. & Cardin, P. 2006 Quasi-geostrophic kinematic dynamos at low magnetic Prandtl number. Earth Planet. Sci. Lett. 245, 595–604.CrossRefGoogle Scholar
Stewartson, K. 1966 On almost rigid rotations. Part 2. J. Fluid Mech. 26, 131–144.Google Scholar
Tagg, R. 1994 The Couette–Taylor problem. Nonlinear Sci. Today 4 (3), 2–25.Google Scholar
Wei, X., Jackson, A. & Hollerbach, R. 2012 Kinematic dynamo action in spherical Couette flow. Geophys. Astrophys. Fluid Dyn. 106, 681–700.CrossRefGoogle Scholar
Wicht, J. 2002 Inner-core conductivity in numerical dynamo simulations. Phys. Earth Planet. Inter. 132, 281–302.Google Scholar
Zhang, K., Earnshaw, P., Liao, X. & Busse, F. H. 2001 On inertial waves in a rotating fluid sphere. J. Fluid Mech. 437, 103–119.Google Scholar
Zimmerman, D. S., Triana, S. A. & Lathrop, D. P. 2011 Bi-stability in turbulent, rotating spherical Couette flow. Phys. Fluids 23 (6), 065104.Google Scholar