Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-23T11:40:38.528Z Has data issue: false hasContentIssue false
  • English
  • Français

Gravito-inertial waves in a differentially rotating spherical shell

Published online by Cambridge University Press:  01 July 2016

G. M. Mirouh ,
C. Baruteau ,
M. Rieutord  and
J. Ballot
G. M. Mirouh*
Affiliation:
Université de Toulouse, UPS-OMP, IRAP, 31400 Toulouse, France CNRS, IRAP, 14 avenue Édouard Belin, 31400 Toulouse, France
C. Baruteau
Affiliation:
Université de Toulouse, UPS-OMP, IRAP, 31400 Toulouse, France CNRS, IRAP, 14 avenue Édouard Belin, 31400 Toulouse, France
M. Rieutord
Affiliation:
Université de Toulouse, UPS-OMP, IRAP, 31400 Toulouse, France CNRS, IRAP, 14 avenue Édouard Belin, 31400 Toulouse, France
J. Ballot
Affiliation:
Université de Toulouse, UPS-OMP, IRAP, 31400 Toulouse, France CNRS, IRAP, 14 avenue Édouard Belin, 31400 Toulouse, France
*
†Email address for correspondence: giovanni.mirouh@irap.omp.eu
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 800 , 10 August 2016 , pp. 213 - 247
Check for updates
Copyright
© 2016 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

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, 1849–1868.Google Scholar
Baruteau, C. & Rieutord, M. 2013 Inertial waves in a differentially rotating spherical shell - I. Free modes of oscillation. J. Fluid Mech. 719, 47–81.Google Scholar
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, 363–365.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.Google Scholar
Dintrans, B., Rieutord, M. & Valdettaro, L. 1999 Gravito-inertial waves in a rotating stratified sphere or spherical shell. J. Fluid Mech. 398, 271–297.Google Scholar
Drazin, P. & Reid, W. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
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, 251–257.Google Scholar
Espinosa Lara, F. & Rieutord, M. 2013 Self-consistent 2D models of fast rotating early-type stars. Astron. Astrophys. 552, A35.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, 845–860.Google Scholar
Fotheringham, P. & Hollerbach, R. 1998 Inertial oscillations in a spherical shell. Geophys. Astrophys. Fluid Dyn. 89, 23–43.CrossRefGoogle Scholar
Friedlander, S. 1982 Turning surface behaviour for internal waves subject to general gravitational fields. Geophys. Astrophys. Fluid Dyn. 21, 189–200.CrossRefGoogle Scholar
Friedlander, S. 1987 Internal waves in a rotating stratified spherical shell: asymptotic solutions. Geophys. J. R. Astron. Soc. 89, 637–655.Google Scholar
Friedlander, S. 1989 Hydromagnetic waves in a differentially rotating, stratified spherical shell. Geophys. Astrophys. Fluid Dyn. 48, 53–67.CrossRefGoogle Scholar
Friedlander, S. & Siegmann, W. 1982a Internal waves in a contained rotating stratified fluid. J. Fluid Mech. 114, 123–156.Google Scholar
Friedlander, S. & Siegmann, W. 1982b Internal waves in a rotating stratified fluid in an arbitrary gravitational field. Geophys. Astrophys. Fluid Dyn. 19, 267–291.Google Scholar
Fuller, J. 2014 Saturn ring seismology: evidence for stable stratification in the deep interior of Saturn. Icarus 242, 283–296.Google Scholar
Gastine, T. & Dintrans, B. 2008 Direct numerical simulations of the 𝜅-mechanism. I. Radial modes in the purely radiative case. Astron. Astrophys. 484, 29–42.Google Scholar
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.CrossRefGoogle Scholar
Goldreich, P. & Schubert, G. 1967 Differential rotation in stars. Astrophys. J. 150, 571.Google Scholar
Goodman, J. & Lackner, C. 2009 Dynamical tides in rotating planets and stars. Astrophys. J. 696, 2054–2067.CrossRefGoogle Scholar
Hypolite, D. & Rieutord, M. 2014 Dynamics of the envelope of a rapidly rotating star or giant planet in gravitational contraction. Astron. Astrophys. 572, A15.Google Scholar
Knobloch, E. & Spruit, H. C. 1983 The molecular weight barrier and angular momentum transport in radiative stellar interiors. Astron. Astrophys. 125, 59–68.Google Scholar
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.Google Scholar
Lignières, F., Califano, F. & Mangeney, A. 1999 Shear layer instability in a highly diffusive stably stratified atmosphere. Astron. Astrophys. 349, 1027–1036.Google Scholar
Maas, L. & Lam, F.-P. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 1–41.CrossRefGoogle Scholar
Maeder, A. 2009 Physics, Formation and Evolution of Rotating Stars. Springer.Google Scholar
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.Google Scholar
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.Google Scholar
Maslowe, S. A. 1986 Critical layers in shear flows. Ann. Rev. Fluid Mech. 18, 405–432.Google Scholar
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.Google Scholar
Morel, P. 1997 CESAM: a code for stellar evolution calculations. Astron. Astrophys. Suppl. Ser. 124, 597–614.CrossRefGoogle Scholar
Ogilvie, G. 2009 Tidal dissipation in rotating fluid bodies: a simplified model. Mon. Not. R. Astron. Soc. 396, 794–806.Google Scholar
Ogilvie, G. I. 2014 Tidal dissipation in stars and giant planets. Annu. Rev. Astron. Astrophys. 52, 171–210.Google Scholar
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.Google Scholar
Rieutord, M. 1987 Linear theory of rotating fluids using spherical harmonics. I. Steady flows. Geophys. Astrophys. Fluid Dyn. 39, 163.CrossRefGoogle Scholar
Rieutord, M. 2006 The dynamics of the radiative envelope of rapidly rotating stars. I. A spherical boussinesq model. Astron. Astrophys. 451, 1025–1036.Google Scholar
Rieutord, M. 2008 The solar dynamo. C. R. Physique 9, 757–765.CrossRefGoogle Scholar
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).Google Scholar
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. 49–73. Springer.Google Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2000 Waves attractors in rotating fluids: a paradigm for ill-posed cauchy problems. Phys. Rev. Lett. 85, 4277–4280.Google 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.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 (2), 026304.Google Scholar
Rieutord, M. & Valdettaro, L. 1997 Inertial waves in a rotating spherical shell. J. Fluid Mech. 341, 77–99.Google Scholar
Rieutord, M. & Valdettaro, L. 2010 Viscous dissipation by tidally forced inertial modes in a rotating spherical shell. J. Fluid Mech. 643, 363–394.Google Scholar
Spruit, H. C. & Knobloch, E. 1984 Baroclinic instability in stars. Astron. Astrophys. 132, 89–96.Google Scholar
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), 16–34.Google Scholar
Unno, W., Osaki, Y., Ando, H., Saio, H. & Shibahashi, H. 1989 Nonradial Oscillations of Stars. University of Tokyo Press.Google Scholar
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, 382–393.Google Scholar
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, 842–852.Google Scholar
Zahn, J.-P. 1974 Rotational instabilities and stellar evolution. IAU Symp. 59: Stellar Instability and Evolution. pp. 185–194. D. Reidel.Google Scholar
Zahn, J.-P. 1992 Circulation and turbulence in rotating stars. Astron. Astrophys. 265, 115.Google Scholar
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. 561–615. North-Holland.Google Scholar
Zhevakin, S. A. 1963 Physical basis of the pulsation theory of variable stars. Annu. Rev. Astron. Astrophys. 1, 367.Google Scholar