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Triadic resonances in the wide-gap spherical Couette system

Published online by Cambridge University Press:  27 March 2018

A. Barik*
Max-Planck-Institut für Sonnensystemforschung, 37077 Göttingen, Germany Fakultät für Physik, Georg-August-Universität Göttingen, 37077 Göttingen, Germany
S. A. Triana
Royal Observatory of Belgium, 1180 Brussels, Belgium
M. Hoff
Lehrstuhl für Aerodynamik und Strömungslehre, Brandenburgische Technische Universität Cottbus-Senftenberg, 03046 Cottbus, Germany
J. Wicht
Max-Planck-Institut für Sonnensystemforschung, 37077 Göttingen, Germany
Email address for correspondence:


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.

JFM Papers
© 2018 Cambridge University Press 

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Baruteau, C. & Rieutord, M. 2013 Inertial waves in a differentially rotating spherical shell. J. Fluid Mech. 719, 4781.Google Scholar
Bellan, P. M. 2008 Fundamentals of Plasma Physics. Cambridge University Press.Google Scholar
Bratukhin, Iu. K. 1961 On the evaluation of the critical Reynolds number for the flow of fluid between two rotating spherical surfaces. Z. Angew. Math. Mech. J. Appl. Math. Mech. 25 (5), 12861299.Google Scholar
Bryan, G. H. 1889 The waves on a rotating liquid spheroid of finite ellipticity. Phil. Trans. R. Soc. Lond. A 180, 187219.Google Scholar
Christensen, U. R. & Wicht, J. 2007 Section 8.08 – Numerical dynamo simulations. In Treatise on Geophysics (ed. Schubert, G.), pp. 245282. Elsevier.Google Scholar
Egbers, C. & Rath, H. J. 1995 The existence of Taylor vortices and wide-gap instabilities in spherical Couette flow. Acta Mechanica 111 (3–4), 125140.Google Scholar
Figueroa, A., Schaeffer, N., Nataf, H.-C. & Schmitt, D. 2013 Modes and instabilities in magnetized spherical Couette flow. J. Fluid Mech. 716, 445469.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Hoff, M., Harlander, U., Egbers, C. & Triana, S. A. 2016a Interagierende Trägheitsmoden in einem differenziell rotierenden Kugelspaltexperiment. In Proceedings der 24. GALA-Fachtagung “Experimentelle Strömungsmechanik” (ed. Egbers, C., Ruck, B., Leder, A. & Dopheide, D.), GALA e.V. (German Association for Laser Anemometry), 12-1–12-8.Google Scholar
Hoff, M., Harlander, U. & Triana, S. A. 2016b Study of turbulence and interacting inertial modes in a differentially rotating spherical shell experiment. Phys. Rev. Fluids 1, 043701.Google Scholar
Hollerbach, R. 2003 Instabilities of the Stewartson layer. Part 1. The dependence on the sign of Ro . J. Fluid Mech. 492, 289302.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 (2), 197204.Google Scholar
Kelley, D. H.2009 Rotating, hydromagnetic laboratory experiment modelling planetary cores. PhD thesis, University of Maryland, College Park, MD.Google Scholar
Kelley, D. H., Triana, S. A., Zimmerman, D. S. & Lathrop, D. P. 2010 Selection of inertial modes in spherical Couette flow. Phys. Rev. E 81, 026311.Google Scholar
Kelley, D. H., Triana, S. A., Zimmerman, D. S., Tilgner, A. & Lathrop, D. P. 2007 Inertial waves driven by differential rotation in a planetary geometry. Geophys. Astrophy. Fluid Dyn. 101 (5–6), 469487.Google Scholar
Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72 (1–4), 107144.Google Scholar
Koch, S., Harlander, U., Egbers, C. & Hollerbach, R. 2013 Inertial waves in a spherical shell induced by librations of the inner sphere: experimental and numerical results. Fluid Dyn. Res. 45 (3), 035504.Google Scholar
Le Bars, M., Cébron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47 (1), 163193.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 (34), 194202; Proceedings of the 12th Symposium of SEDI.Google Scholar
Matsui, H., Heien, E., Aubert, J., Aurnou, J. M., Avery, M., Brown, B., Buffett, B. A., Busse, F., Christensen, U. R., Davies, C. J. et al. 2016 Performance benchmarks for a next generation numerical dynamo model. Geochem. Geophys. Geosyst. 17, 15861607.Google Scholar
McComas, C. H. & Bretherton, F. P. 1977 Resonant interaction of oceanic internal waves. J. Geophys. Res. 82 (9), 13971412.Google Scholar
Munson, B. R. & Joseph, D. D. 1971a Viscous incompressible flow between concentric rotating spheres. Part 1. Basic flow. J. Fluid Mech. 49, 289303.Google Scholar
Munson, B. R. & Joseph, D. D. 1971b Viscous incompressible flow between concentric rotating spheres. Part 2. Hydrodynamic stability. J. Fluid Mech. 49, 305318.Google Scholar
Munson, B. R. & Menguturk, M. 1975 Viscous incompressible flow between concentric rotating spheres. Part 3. Linear stability and experiments. J. Fluid Mech. 69, 705719.Google Scholar
Neiner, C., Floquet, M., Samadi, R., Espinosa Lara, F., Frémat, Y., Mathis, S., Leroy, B., de Batz, B., Rainer, M., Poretti, E. et al. 2012 Stochastic gravito-inertial modes discovered by CoRoT in the hot Be star HD 51452. Astron. Astrophys. 546, A47.Google Scholar
Nikias, C. L. & Raghuveer, M. R. 1987 Bispectrum estimation: a digital signal processing framework. Proc. IEEE 75 (7), 869891.Google Scholar
Pápics, P. I., Briquet, M., Baglin, A., Poretti, E., Aerts, C., Degroote, P., Tkachenko, A., Morel, T., Zima, W., Niemczura, E. et al. 2012 Gravito-inertial and pressure modes detected in the B3 IV CoRoT target HD 43317. Astron. Astrophys. 542, A55.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.Google Scholar
Potter, A. T., Chitre, S. M. & Tout, C. A. 2012 Stellar evolution of massive stars with a radiative 𝛼–𝜔 dynamo. Mon. Not. R. Astron. Soc. 424 (3), 23582370.Google Scholar
Proudman, I. 1956 The almost-rigid rotation of viscous fluid between concentric spheres. J. Fluid Mech. 1, 505516.Google Scholar
Rieutord, M. 1991 Linear theory of rotating fluids using spherical harmonics. Part II. Time-periodic flows. Geophys. Astrophys. Fluid Dyn. 59 (1–4), 185208.Google Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103144.Google Scholar
Rieutord, M., Triana, S. A., Zimmerman, D. S. & Lathrop, D. P. 2012 Excitation of inertial modes in an experimental 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, 7799.Google Scholar
Schaeffer, N. 2013 Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations. Geochem. Geophys. Geosyst. 14 (3), 751758.Google 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.Google Scholar
Schmitt, D., Cardin, P., Rizza, P. L. & Nataf, H.-C. 2013 Magnetocoriolis waves in a spherical Couette flow experiment. Eur. J. Mech. (B/Fluids) 37, 1022.Google Scholar
Sorokin, M. P., Khlebutin, G. N. & Shaidurov, G. F. 1966 Study of the motion of a liquid between two rotating spherical surfaces. J. Appl. Mech. Tech. Phys. 7 (6), 7374.Google Scholar
Spruit, H. C. 2002 Dynamo action by differential rotation in a stably stratified stellar interior. Astron. Astrophys. 381, 923932.Google Scholar
Stewartson, K. 1966 On almost rigid rotations. Part 2. J. Fluid Mech. 26, 131144.Google Scholar
Stewartson, K. & Rickard, J. A. 1969 Pathological oscillations of a rotating fluid. J. Fluid Mech. 35 (4), 759773.Google Scholar
Swami, A., Mendel, J. M. & Nikias, C. L.1998 Higher-Order Spectral Analysis Toolbox. The Mathworks Inc.Google Scholar
Tilgner, A. 1999 Driven inertial oscillations in spherical shells. Phys. Rev. E 59, 17891794.Google Scholar
Tilgner, A. 2007 Zonal wind driven by inertial modes. Phys. Rev. Lett. 99, 194501.Google Scholar
Triana, S. A.2011 Inertial waves in a laboratory model of the Earth’s core. PhD thesis, University of Maryland, College Park, MD.Google Scholar
Vidal, J. & Schaeffer, N. 2015 Quasi-geostrophic modes in the Earth’s fluid core with an outer stably stratified layer. Geophys. J. Intl 202 (3), 21822193.Google Scholar
Wicht, J. 2002 Inner-core conductivity in numerical dynamo simulations. Phys. Earth Planet. Inter. 132 (4), 281302.Google Scholar
Wicht, J. 2014 Flow instabilities in the wide-gap spherical Couette system. J. Fluid Mech. 738, 184221.Google Scholar
Zhang, K., Earnshaw, P., Liao, X. & Busse, F. H. 2001 On inertial waves in a rotating fluid sphere. J. Fluid Mech. 437, 103119.Google Scholar
Zhang, Y. & Pedlosky, J. 2007 Triad instability of planetary Rossby waves. J. Phys. Oceanogr. 37 (8), 21582171.Google Scholar
Zimmerman, D. S.2010 Turbulent shear flow in a rapidly rotating spherical annulus. PhD thesis, University of Maryland, College Park, MD.Google Scholar