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Mean zonal flows induced by weak mechanical forcings in rotating spheroids

Published online by Cambridge University Press:  14 April 2021

David Cébron Open the ORCID record for David Cébron [Opens in a new window] ,
Jérémie Vidal Open the ORCID record for Jérémie Vidal [Opens in a new window] ,
Nathanaël Schaeffer Open the ORCID record for Nathanaël Schaeffer [Opens in a new window] ,
Antonin Borderies  and
Alban Sauret Open the ORCID record for Alban Sauret [Opens in a new window]
David Cébron*
Affiliation:
ISTerre, Université Grenoble Alpes, CNRS, 38000Grenoble, France
Jérémie Vidal
Affiliation:
ISTerre, Université Grenoble Alpes, CNRS, 38000Grenoble, France
Nathanaël Schaeffer
Affiliation:
ISTerre, Université Grenoble Alpes, CNRS, 38000Grenoble, France
Antonin Borderies
Affiliation:
ISTerre, Université Grenoble Alpes, CNRS, 38000Grenoble, France
Alban Sauret
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA93106,USA
*
†Email address for correspondence: [email protected]
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Abstract

The generation of mean flows is a long-standing issue in rotating fluids. Motivated by planetary objects, we consider here a rapidly rotating fluid-filled spheroid, which is subject to weak perturbations of either the boundary (e.g. tides) or the rotation vector (e.g. in direction by precession, or in magnitude by longitudinal librations). Using boundary-layer theory, we determine the mean zonal flows generated by nonlinear interactions within the viscous Ekman layer. These flows are of interest because they survive in the relevant planetary regime of both vanishing forcings and viscous effects. We extend the theory to take into account (i) the combination of spatial and temporal perturbations, providing new mechanically driven zonal flows (e.g. driven by latitudinal librations), and (ii) the spheroidal geometry relevant for planetary bodies. Wherever possible, our analytical predictions are validated with direct numerical simulations. The theoretical solutions are in good quantitative agreement with the simulations, with expected discrepancies (zonal jets) in the presence of inertial waves generated at the critical latitudes (as for precession). Moreover, we find that the mean zonal flows can be strongly affected in spheroids. Guided by planetary applications, we also revisit the scaling laws for the geostrophic shear layers at the critical latitudes, and the influence of a solid inner core.

JFM classification

Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 916 , 10 June 2021 , A39
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© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Aldridge, K.D. & Toomre, A. 1969 Axisymmetric inertial oscillations of a fluid in a rotating spherical container. J. Fluid Mech. 37 (2), 307–323.CrossRefGoogle Scholar
Ascher, U.M., Ruuth, S.J. & Wetton, B.T.R. 1995 Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32 (3), 797–823.CrossRefGoogle Scholar
Aubert, J. 2005 Steady zonal flows in spherical shell dynamos. J. Fluid Mech. 542, 53–67.CrossRefGoogle Scholar
Aubert, J., Jung, S. & Swinney, H.L. 2002 Observations of zonal flow created by potential vorticity mixing in a rotating fluid. Geophys. Res. Lett. 29 (18), 23.CrossRefGoogle Scholar
Brunet, M., Gallet, B. & Cortet, P.-P. 2020 Shortcut to geostrophy in wave-driven rotating turbulence: the quartetic instability. Phys. Rev. Lett. 124 (12), 124501.CrossRefGoogle ScholarPubMed
Burmann, F. & Noir, J. 2018 Effects of bottom topography on the spin-up in a cylinder. Phys. Fluids 30 (10), 106601.CrossRefGoogle Scholar
Busse, F.H. 1968 a Shear flow instabilities in rotating systems. J. Fluid Mech. 33 (3), 577–589.CrossRefGoogle Scholar
Busse, F.H. 1968 b Steady fluid flow in a precessing spheroidal shell. J. Fluid Mech. 33 (4), 739–751.CrossRefGoogle Scholar
Busse, F.H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44 (3), 441–460.CrossRefGoogle Scholar
Busse, F.H. 2010 Mean zonal flows generated by librations of a rotating spherical cavity. J. Fluid Mech. 650, 505–512.CrossRefGoogle Scholar
Caldwell, D.R. & Van Atta, C.W. 1970 Characteristics of Ekman boundary layer instabilities. J. Fluid Mech. 44 (1), 79–95.CrossRefGoogle Scholar
Calkins, M.A., Noir, J., Eldredge, J.D. & Aurnou, J.M. 2010 Axisymmetric simulations of libration-driven fluid dynamics in a spherical shell geometry. Phys. Fluids 22 (8), 086602.CrossRefGoogle Scholar
Cébron, D., Laguerre, R., Noir, J. & Schaeffer, N. 2019 Precessing spherical shells: flows, dissipation, dynamo and the lunar core. Geophys. J. Intl 219 (Supplement 1), S34–S57.CrossRefGoogle Scholar
Cébron, D., Le Bars, M., Moutou, C. & Le Gal, P. 2012 Elliptical instability in terrestrial planets and moons. Astron. Astrophys. 539, A78.CrossRefGoogle Scholar
Cébron, D., Vantieghem, S. & Herreman, W. 2014 Libration-driven multipolar instabilities. J. Fluid Mech. 739, 502–543.CrossRefGoogle Scholar
Chan, K.H., Liao, X. & Zhang, K. 2011 Simulations of fluid motion in spheroidal planetary cores driven by latitudinal libration. Phys. Earth Planet. Inter. 187 (3-4), 404–415.CrossRefGoogle Scholar
Christensen, U.R. 2002 Zonal flow driven by strongly supercritical convection in rotating spherical shells. J. Fluid Mech. 470, 115–133.CrossRefGoogle Scholar
Dudley, M.L. & James, R.W. 1989 Time-dependent kinematic dynamos with stationary flows. Proc. R. Soc. A 425 (1869), 407–429.Google Scholar
Favier, B., Barker, A.J., Baruteau, C. & Ogilvie, G.I. 2014 Non-linear evolution of tidally forced inertial waves in rotating fluid bodies. Mon. Not. R. Astron. Soc. 439 (1), 845–860.CrossRefGoogle Scholar
Favier, B., Grannan, A.M., Le Bars, M. & Aurnou, J.M. 2015 Generation and maintenance of bulk turbulence by libration-driven elliptical instability. Phys. Fluids 27 (6), 066601.CrossRefGoogle Scholar
Fischer, P.F., Loth, F., Lee, S.E., Lee, S. -W., Smith, D.S. & Bassiouny, H.S. 2007 Simulation of high-Reynolds number vascular flows. Comput. Meth. Appl. Mech. Engng 196 (31-32), 3049–3060.CrossRefGoogle Scholar
Godeferd, F.S. & Moisy, F. 2015 Structure and dynamics of rotating turbulence: a review of recent experimental and numerical results. Appl. Mech. Rev. 67 (3), 030802.CrossRefGoogle Scholar
Grannan, A.M., Favier, B., Le Bars, M. & Aurnou, J.M. 2017 Tidally forced turbulence in planetary interiors. Geophys. J. Intl 208 (3), 1690–1703.Google Scholar
Greenspan, H.P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Greenspan, H.P. 1969 On the non-linear interaction of inertial modes. J. Fluid Mech. 36 (2), 257–264.CrossRefGoogle Scholar
Greenspan, H.P. & Howard, L.N. 1963 On a time-dependent motion of a rotating fluid. J. Fluid Mech. 17 (3), 385–404.CrossRefGoogle Scholar
Guervilly, C., Hughes, D.W. & Jones, C.A. 2014 Large-scale vortices in rapidly rotating Rayleigh–Bénard convection. J. Fluid Mech. 758, 407–435.CrossRefGoogle Scholar
Hollerbach, R. & Kerswell, R.R. 1995 Oscillatory internal shear layers in rotating and precessing flows. J. Fluid Mech. 298, 327–339.CrossRefGoogle Scholar
Horimoto, Y., Katayama, A. & Goto, S. 2020 Conical shear-driven parametric instability of steady flow in precessing spheroids. Phys. Rev. Fluids 5 (6), 063901.CrossRefGoogle Scholar
Kerswell, R.R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72 (1-4), 107–144.CrossRefGoogle Scholar
Kerswell, R.R. 1995 On the internal shear layers spawned by the critical regions in oscillatory ekman boundary layers. J. Fluid Mech. 298, 311–325.CrossRefGoogle Scholar
Kerswell, R.R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown. J. Fluid Mech. 382, 283–306.CrossRefGoogle Scholar
Kerswell, R.R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34 (1), 83–113.CrossRefGoogle Scholar
Kida, S. 2011 Steady flow in a rapidly rotating sphere with weak precession. J. Fluid Mech. 680, 150–193.CrossRefGoogle Scholar
Kida, S. 2020 Steady flow in a rapidly rotating spheroid with weak precession: I. Fluid Dyn. Res. 52 (1), 015513.CrossRefGoogle Scholar
Labbé, F., Jault, D. & Gillet, N. 2015 On magnetostrophic inertia-less waves in quasi-geostrophic models of planetary cores. Geophys. Astrophys. Fluid Dyn. 109 (6), 587–610.CrossRefGoogle Scholar
Le Bars, M., Cébron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47, 163–193.CrossRefGoogle Scholar
Le Mouël, J.-L., Narteau, C., Greff-Lefftz, M. & Holschneider, M. 2006 Dissipation at the core-mantle boundary on a small-scale topography. J. Geophys. Res.: Solid Earth 111, B04413.CrossRefGoogle Scholar
Le Reun, T., Favier, B. & Le Bars, M. 2019 Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence. J. Fluid Mech. 879, 296–326.CrossRefGoogle Scholar
Le Reun, T., Gallet, B., Favier, B. & Le Bars, M. 2020 Near-resonant instability of geostrophic modes: beyond Greenspan's theorem. J. Fluid Mech. 900, R2.CrossRefGoogle Scholar
Lemasquerier, D., Grannan, A.M., Vidal, J., Cébron, D., Favier, B., Le Bars, M. & Aurnou, J.M. 2017 Libration-driven flows in ellipsoidal shells. J. Geophys. Res. Planets 122 (9), 1926–1950.CrossRefGoogle Scholar
Lewis, H.R. & Bellan, P.M. 1990 Physical constraints on the coefficients of Fourier expansions in cylindrical coordinates. J. Math. Phys. 31 (11), 2592–2596.CrossRefGoogle Scholar
Lin, Y., Marti, P. & Noir, J. 2015 Shear-driven parametric instability in a precessing sphere. Phys. Fluids 27 (4), 046601.CrossRefGoogle Scholar
Lin, Y. & Noir, J. 2020 Libration-driven inertial waves and mean zonal flows in spherical shells. Geophys. Astrophys. Fluid Dyn. 1–22.CrossRefGoogle Scholar
Livermore, P.W. & Jackson, A. 2005 A comparison of numerical schemes to solve the magnetic induction eigenvalue problem in a spherical geometry. Geophys. Astrophys. Fluid Dyn. 99 (6), 467–480.CrossRefGoogle Scholar
Lorenzani, S. & Tilgner, A. 2001 Fluid instabilities in precessing spheroidal cavities. J. Fluid Mech. 447, 111–128.CrossRefGoogle Scholar
Malkus, W.V.R. 1968 Precession of the Earth as the cause of geomagnetism: experiments lend support to the proposal that precessional torques drive the Earth's dynamo. Science 160 (3825), 259–264.CrossRefGoogle ScholarPubMed
Marti, P., et al. 2014 Full sphere hydrodynamic and dynamo benchmarks. Geophys. J. Intl 197 (1), 119–134.CrossRefGoogle Scholar
Monville, R., Vidal, J., Cébron, D. & Schaeffer, N. 2019 Rotating double-diffusive convection in stably stratified planetary cores. Geophys. J. Intl 219 (Supplement 1), S195–S218.CrossRefGoogle Scholar
Morize, C., Le Bars, M., Le Gal, P. & Tilgner, A. 2010 Experimental determination of zonal winds driven by tides. Phys. Rev. Lett. 104 (21), 214501.CrossRefGoogle ScholarPubMed
Narteau, C., Le Mouël, J.L., Poirier, J.P., Sepúlveda, E. & Shnirman, M. 2001 On a small-scale roughness of the core–mantle boundary. Earth Planet. Sci. Lett. 191 (1-2), 49–60.CrossRefGoogle Scholar
Newell, A.C. 1969 Rossby wave packet interactions. J. Fluid Mech. 35 (2), 255–271.CrossRefGoogle Scholar
Nobili, C., Meunier, P., Favier, B. & Le Bars, M. 2021 Hysteresis and instabilities in a spheroid in precession near the resonance with the tilt-over mode. J. Fluid Mech. 909, A17.CrossRefGoogle Scholar
Noir, J., Brito, D., Aldridge, K. & Cardin, P. 2001 a Experimental evidence of inertial waves in a precessing spheroidal cavity. Geophys. Res. Lett. 28 (19), 3785–3788.CrossRefGoogle Scholar
Noir, J. & Cébron, D. 2013 Precession-driven flows in non-axisymmetric ellipsoids. J. Fluid Mech. 737, 412–439.CrossRefGoogle Scholar
Noir, J., Cébron, D., Le Bars, M., Sauret, A. & Aurnou, J.M. 2012 Experimental study of libration-driven zonal flows in non-axisymmetric containers. Phys. Earth Planet. Inter. 204, 1–10.CrossRefGoogle Scholar
Noir, J., Hemmerlin, F., Wicht, J., Baca, S.M. & Aurnou, J.M. 2009 An experimental and numerical study of librationally driven flow in planetary cores and subsurface oceans. Phys. Earth Planet. Inter. 173 (1-2), 141–152.CrossRefGoogle Scholar
Noir, J., Jault, D. & Cardin, P. 2001 b Numerical study of the motions within a slowly precessing sphere at low Ekman number. J. Fluid Mech. 437, 283–299.CrossRefGoogle Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103–144.CrossRefGoogle Scholar
Roberts, P.H. & Aurnou, J.M. 2012 On the theory of core-mantle coupling. Geophys. Astrophys. Fluid Dyn. 106 (2), 157–230.CrossRefGoogle Scholar
Sauret, A. 2015 Mean zonal flow generated by azimuthal harmonic forcing in a rotating cylinder. Fluid Dyn. Res. 47 (3), 035506.CrossRefGoogle Scholar
Sauret, A., Cébron, D. & Le Bars, M. 2013 Spontaneous generation of inertial waves from boundary turbulence in a librating sphere. J. Fluid Mech. 728, R5.CrossRefGoogle Scholar
Sauret, A., Cébron, D., Le Bars, M. & Le Dizès, S. 2012 Fluid flows in a librating cylinder. Phys. Fluids 24 (2), 026603.CrossRefGoogle Scholar
Sauret, A., Cébron, D., Morize, C. & Le Bars, M. 2010 Experimental and numerical study of mean zonal flows generated by librations of a rotating spherical cavity. J. Fluid Mech. 662, 260–268.CrossRefGoogle Scholar
Sauret, A., Le Bars, M. & Le Gal, P. 2014 Tide-driven shear instability in planetary liquid cores. Geophys. Res. Lett. 41 (17), 6078–6083.CrossRefGoogle Scholar
Sauret, A. & Le Dizès, S. 2013 Libration-induced mean flow in a spherical shell. J. Fluid Mech. 718, 181–209.CrossRefGoogle Scholar
Schaeffer, N. 2013 Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations. Geochem. Geophys. Geosyst. 14 (3), 751–758.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., Jault, D., Nataf, H.-C. & Fournier, A. 2017 Turbulent geodynamo simulations: a leap towards Earth's core. Geophys. J. Intl 211 (1), 1–29.CrossRefGoogle Scholar
Schmitt, D. & Jault, D. 2004 Numerical study of a rotating fluid in a spheroidal container. J. Comput. Phys. 197 (2), 671–685.CrossRefGoogle Scholar
Smith, L.M. & Waleffe, F. 1999 Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence. Phys. Fluids 11 (6), 1608–1622.CrossRefGoogle Scholar
Sous, D., Sommeria, J. & Boyer, D. 2013 Friction law and turbulent properties in a laboratory Ekman boundary layer. Phys. Fluids 25 (4), 046602.CrossRefGoogle Scholar
Stewartson, K. 1966 On almost rigid rotations. Part 2. J. Fluid Mech. 26 (1), 131–144.CrossRefGoogle Scholar
Su, S., Cébron, D., Nataf, H.-C., Cardin, P., Vidal, J., Solazzo, M. & Do, Y. 2020 Acoustic spectra of a gas-filled rotating spheroid. Eur. J. Mech. B Fluids 84, 302–310.CrossRefGoogle Scholar
Suess, S.T. 1971 Viscous flow in a deformable rotating container. J. Fluid Mech. 45, 189–201.CrossRefGoogle Scholar
Vantieghem, S., Cébron, D. & Noir, J. 2015 Latitudinal libration driven flows in triaxial ellipsoids. J. Fluid Mech. 771, 193–228.CrossRefGoogle Scholar
Vidal, J. & Cébron, D. 2017 Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits. J. Fluid Mech. 833, 469–511.CrossRefGoogle Scholar
Vidal, J., Cébron, D., ud Doula, A. & Alecian, E. 2019 Fossil field decay due to nonlinear tides in massive binaries. Astron. Astrophys. 629, A142.CrossRefGoogle Scholar
Vidal, J., Su, S. & Cébron, D. 2020 Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry. J. Fluid Mech. 885, A39.CrossRefGoogle Scholar
Wang, C.-Y. 1970 Cylindrical tank of fluid oscillating about a state of steady rotation. J. Fluid Mech. 41 (3), 581–592.CrossRefGoogle Scholar
Wedemeyer, E.H. 1966 Viscous corrections to Stewartson's stability criterion. BRL Rep. 1325. US Army Ballistic Research Laboratory, Aberdeen Proving Ground, Maryland AD489687.Google Scholar
Williams, J.G., Boggs, D.H., Yoder, C.F., Ratcliff, J.T. & Dickey, J.O. 2001 Lunar rotational dissipation in solid body and molten core. J. Geophys. Res. Planets 106 (E11), 27933–27968.CrossRefGoogle Scholar
Zhang, K., Chan, K.H. & Liao, X. 2012 Asymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration. J. Fluid Mech. 692, 420–445.CrossRefGoogle Scholar
Zhang, K. & Liao, X. 2017 Theory and Modeling of Rotating Fluids: Convection, Inertial Waves and Precession. Cambridge University Press.CrossRefGoogle Scholar