Skip to main content

Gravito-inertial waves in a differentially rotating spherical shell

  • G. M. Mirouh (a1) (a2), C. Baruteau (a1) (a2), M. Rieutord (a1) (a2) and J. Ballot (a1) (a2)

The gravito-inertial waves propagating over a shellular baroclinic flow inside a rotating spherical shell are analysed using the Boussinesq approximation. The wave properties are examined by computing paths of characteristics in the non-dissipative limit, and by solving the full dissipative eigenvalue problem using a high-resolution spectral method. Gravito-inertial waves are found to obey a mixed-type second-order operator and to be often focused around short-period attractors of characteristics or trapped in a wedge formed by turning surfaces and boundaries. We also find eigenmodes that show a weak dependence with respect to viscosity and heat diffusion just like truly regular modes. Some axisymmetric modes are found unstable and likely destabilized by baroclinic instabilities. Similarly, some non-axisymmetric modes that meet a critical layer (or corotation resonance) can turn unstable at sufficiently low diffusivities. In all cases, the instability is driven by the differential rotation. For many modes of the spectrum, neat power laws are found for the dependence of the damping rates with diffusion coefficients, but the theoretical explanation for the exponent values remains elusive in general. The eigenvalue spectrum turns out to be very rich and complex, which lets us suppose an even richer and more complex spectrum for rotating stars or planets that own a differential rotation driven by baroclinicity.

Corresponding author
Email address for correspondence:
Hide All
Barker A. J. & Ogilvie G. I. 2010 On internal wave breaking and tidal dissipation near the centre of a solar-type star. Mon. Not. R. Astron. Soc. 404, 18491868.
Baruteau C. & Rieutord M. 2013 Inertial waves in a differentially rotating spherical shell - I. Free modes of oscillation. J. Fluid Mech. 719, 4781.
Carr M. H., Belton M. J. S., Chapman C. R., Davies M. E., Geissler P., Greenberg R., McEwen A. S., Tufts B. R., Greeley R., Sullivan R. et al. 1998 Evidence for a subsurface ocean on Europa. Nature 391, 363365.
Chandrasekhar S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.
Dintrans B., Rieutord M. & Valdettaro L. 1999 Gravito-inertial waves in a rotating stratified sphere or spherical shell. J. Fluid Mech. 398, 271297.
Drazin P. & Reid W. 1981 Hydrodynamic Stability. Cambridge University Press.
Dupret M.-A., Thoul A., Scuflaire R., Daszyńska-Daszkiewicz J., Aerts C., Bourge P.-O., Waelkens C. & Noels A. 2004 Asteroseismology of the 𝛽 Cep star HD 129929. II. Seismic constraints on core overshooting, internal rotation and stellar parameters. Astron. Astrophys. 415, 251257.
Espinosa Lara F. & Rieutord M. 2013 Self-consistent 2D models of fast rotating early-type stars. Astron. Astrophys. 552, A35.
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, 845860.
Fotheringham P. & Hollerbach R. 1998 Inertial oscillations in a spherical shell. Geophys. Astrophys. Fluid Dyn. 89, 2343.
Friedlander S. 1982 Turning surface behaviour for internal waves subject to general gravitational fields. Geophys. Astrophys. Fluid Dyn. 21, 189200.
Friedlander S. 1987 Internal waves in a rotating stratified spherical shell: asymptotic solutions. Geophys. J. R. Astron. Soc. 89, 637655.
Friedlander S. 1989 Hydromagnetic waves in a differentially rotating, stratified spherical shell. Geophys. Astrophys. Fluid Dyn. 48, 5367.
Friedlander S. & Siegmann W. 1982a Internal waves in a contained rotating stratified fluid. J. Fluid Mech. 114, 123156.
Friedlander S. & Siegmann W. 1982b Internal waves in a rotating stratified fluid in an arbitrary gravitational field. Geophys. Astrophys. Fluid Dyn. 19, 267291.
Fuller J. 2014 Saturn ring seismology: evidence for stable stratification in the deep interior of Saturn. Icarus 242, 283296.
Gastine T. & Dintrans B. 2008 Direct numerical simulations of the 𝜅-mechanism. I. Radial modes in the purely radiative case. Astron. Astrophys. 484, 2942.
Gerkema T., Zimmerman J. T. F., Maas L. R. M. & van Haren H. 2008 Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Rev. Geophys. 46, 2006RG000220, 2004.
Goldreich P. & Schubert G. 1967 Differential rotation in stars. Astrophys. J. 150, 571.
Goodman J. & Lackner C. 2009 Dynamical tides in rotating planets and stars. Astrophys. J. 696, 20542067.
Hypolite D. & Rieutord M. 2014 Dynamics of the envelope of a rapidly rotating star or giant planet in gravitational contraction. Astron. Astrophys. 572, A15.
Knobloch E. & Spruit H. C. 1983 The molecular weight barrier and angular momentum transport in radiative stellar interiors. Astron. Astrophys. 125, 5968.
Lainey V., Jacobson R. A., Tajeddine R., Cooper N. J., Murray C., Robert V., Tobie G., Guillot T., Mathis S., Remus F. et al. 2015 New constraints on Saturn’s interior from Cassini astrometric data. ArXiv e-prints.
Lignières F., Califano F. & Mangeney A. 1999 Shear layer instability in a highly diffusive stably stratified atmosphere. Astron. Astrophys. 349, 10271036.
Maas L. & Lam F.-P. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 141.
Maeder A. 2009 Physics, Formation and Evolution of Rotating Stars. Springer.
Marcus P. S., Pei S., Jiang C.-H., Barranco J. A., Hassanzadeh P. & Lecoanet D. 2015 Zombie vortex instability. I. A purely hydrodynamic instability to resurrect the dead zones of protoplanetary disks. Astrophys. J. 808, 87.
Marcus P. S., Pei S., Jiang C.-H. & Hassanzadeh P. 2013 Three-Dimensional Vortices Generated by Self-Replication in Stably Stratified Rotating Shear Flows. Phys. Rev. Lett. 111 (8), 084501.
Maslowe S. A. 1986 Critical layers in shear flows. Ann. Rev. Fluid Mech. 18, 405432.
Mathis S., Neiner C. & Tran Minh N. 2014 Impact of rotation on stochastic excitation of gravity and gravito-inertial waves in stars. Astron. Astrophys. 565, A47.
Morel P. 1997 CESAM: a code for stellar evolution calculations. Astron. Astrophys. Suppl. Ser. 124, 597614.
Ogilvie G. 2009 Tidal dissipation in rotating fluid bodies: a simplified model. Mon. Not. R. Astron. Soc. 396, 794806.
Ogilvie G. I. 2014 Tidal dissipation in stars and giant planets. Annu. Rev. Astron. Astrophys. 52, 171210.
Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P. & Timmes F. 2011 Modules for experiments in stellar astrophysics (MESA). Astrophys. J. Suppl. Ser. 192, 3.
Rieutord M. 1987 Linear theory of rotating fluids using spherical harmonics. I. Steady flows. Geophys. Astrophys. Fluid Dyn. 39, 163.
Rieutord M. 2006 The dynamics of the radiative envelope of rapidly rotating stars. I. A spherical boussinesq model. Astron. Astrophys. 451, 10251036.
Rieutord M. 2008 The solar dynamo. C. R. Physique 9, 757765.
Rieutord M. & Beth A. 2014 Dynamics of the envelope of rapidly rotating stars. I Effects of spin-down of the outer layers. Astron. Astrophys. 1, 1 (to appear).
Rieutord M. & Espinosa Lara F. 2013 Ab initio modelling of steady rotating stars. In SeIsmology for Studies of Stellar Rotation and Convection (ed. Goupil M., Belkacem K., Neiner C., Lignières F. & Green J. J.), Lecture Notes in Physics, vol. 865, pp. 4973. Springer.
Rieutord M., Georgeot B. & Valdettaro L. 2000 Waves attractors in rotating fluids: a paradigm for ill-posed cauchy problems. Phys. Rev. Lett. 85, 42774280.
Rieutord M., Georgeot B. & Valdettaro L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103144.
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 (2), 026304.
Rieutord M. & Valdettaro L. 1997 Inertial waves in a rotating spherical shell. J. Fluid Mech. 341, 7799.
Rieutord M. & Valdettaro L. 2010 Viscous dissipation by tidally forced inertial modes in a rotating spherical shell. J. Fluid Mech. 643, 363394.
Spruit H. C. & Knobloch E. 1984 Baroclinic instability in stars. Astron. Astrophys. 132, 8996.
Swart A., Manders A., Harlander U. & Maas L. 2010 Experimental observation of strong mixing due to internal wave focusing over sloping terrain. Dyn. Atmos. Oceans 50 (1), 1634.
Unno W., Osaki Y., Ando H., Saio H. & Shibahashi H. 1989 Nonradial Oscillations of Stars. University of Tokyo Press.
Valdettaro L., Rieutord M., Braconnier T. & Fraysse V. 2007 Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and arnoldi-chebyshev algorithm. J. Comput. Appl. Maths 205, 382393.
Witte M. G. & Savonije G. J. 1999 The dynamical tide in a rotating 10 M main sequence star. A study of g- and r-mode resonances. Astron. Astrophys. 341, 842852.
Zahn J.-P. 1974 Rotational instabilities and stellar evolution. IAU Symp. 59: Stellar Instability and Evolution. pp. 185194. D. Reidel.
Zahn J.-P. 1992 Circulation and turbulence in rotating stars. Astron. Astrophys. 265, 115.
Zahn J.-P. 1993 Instabilities and turbulence in rotating stars. In Astrophysical Fluid Dynamics – Les Houches 1987 (ed. Zahn J.-P. & Zinn-Justin J.), pp. 561615. North-Holland.
Zhevakin S. A. 1963 Physical basis of the pulsation theory of variable stars. Annu. Rev. Astron. Astrophys. 1, 367.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 1
Total number of PDF views: 90 *
Loading metrics...

Abstract views

Total abstract views: 245 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st January 2018. This data will be updated every 24 hours.