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Viscous dissipation by tidally forced inertial modes in a rotating spherical shell

Published online by Cambridge University Press:  15 January 2010

M. RIEUTORD  and
L. VALDETTARO
M. RIEUTORD*
Affiliation:
Laboratoire d'Astrophysique de Toulouse – Tarbes, CNRS et Université de Toulouse, 14 avenue E. Belin, 31400 Toulouse, France
L. VALDETTARO
Affiliation:
MOX – Dipartimento di Matematica, Politecnico di Milano, Piazza L. da Vinci, 32, 20133 Milano, Italy
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the properties of forced inertial modes of a rotating fluid inside a spherical shell. Our forcing is tidal like, but its main property is that it is on the large scales. By numerically solving the linear equations of this problem, including viscosity, we first confirm some analytical results obtained on a two-dimensional model by Ogilvie (J. Fluid Mech., vol. 543, 2005, p. 19); some additional properties of this model are uncovered like the existence of narrow resonances associated with periodic orbits of characteristics. We also note that as the frequency of the forcing varies, the dissipation varies drastically if the Ekman number E is low (as is usually the case). We then investigate the three-dimensional case and compare the results to the foregoing model. The three-dimensional solutions show, like their two-dimensional counterpart, a spiky dissipation curve when the frequency of the forcing is varied; they also display small frequency intervals where the viscous dissipation is independent of viscosity. However, we show that the response of the fluid in these frequency intervals is crucially dominated by the shear layer that is emitted at the critical latitude on the inner sphere. The asymptotic regime, where the dissipation is independent of the viscosity, is reached when an attractor has been excited by this shear layer. This property is not shared by the two-dimensional model where shear layers around attractors are independent of those emitted at the critical latitude. Finally, resonances of the three-dimensional model correspond to some selected least damped eigenmodes. Unlike their two-dimensional counter parts these modes are not associated with simple attractors; instead, they show up in frequency intervals with weakly contracting webs of characteristics. Besides, we show that the inner core is negligible when its relative radius is less than the critical value 0.4E1/5. For these spherical shells, the full sphere solutions give a good approximation of the flows.

JFM classification

Geophysical and Geological Flows: Rotating flows Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 643 , 25 January 2010 , pp. 363 - 394
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Bryan, G. 1889 The waves on a rotating liquid spheroid of finite ellipticity. Phil. Trans. R. Soc. Lond. 180, 187–219.Google Scholar
Dintrans, B. & Rieutord, M. 2000 Oscillations of a rotating star: a non-perturbative theory. A&A 354, 86–98.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.CrossRefGoogle Scholar
Fotheringham, P. & Hollerbach, R. 1998 Inertial oscillations in a spherical shell. Geophys. Astrophys. Fluid Dyn. 89, 23–43.CrossRefGoogle 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, RG2004.CrossRefGoogle Scholar
Giuricin, G., Mardirossian, F. & Mezzetti, M. 1984 Synchronization in eclipsing binary stars. A&A 131, 152–158.Google Scholar
Goodman, J. & Lackner, C. 2009 Dynamical tides in rotating planets and stars. ApJ 696, 2054–2067.CrossRefGoogle Scholar
Greenspan, H. P. 1969 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Hollerbach, R. & Kerswell, R. 1995 Oscillatory internal shear layers in rotating and precessing flows. J. Fluid Mech. 298, 327–339.CrossRefGoogle Scholar
Kerswell, 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. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Lacaze, L., Le Gal, P. & Le Dizès, S. 2005 Elliptical instability of the flow in a rotating shell. Phys. Earth Planet. Inter. 151, 194–205.CrossRefGoogle Scholar
Maas, L. 2001 Waves focusing and ensuing mean flow due to symmetry breaking in rotating fluids. J. Fluid Mech. 437, 13–28.CrossRefGoogle Scholar
Maas, L. & Harlander, U. 2007 Equatorial wave attractors and inertial oscillations. J. Fluid Mech. 570, 47–67.CrossRefGoogle Scholar
Maas, L. & Lam, F.-P. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 1–41.CrossRefGoogle Scholar
Manasseh, R. 1996 Nonlinear behaviour of contained inertia waves. J. Fluid Mech. 315, 151–173.CrossRefGoogle Scholar
Mazeh, T. 2008 Observational evidence for tidal interaction in close binary systems. In EAS Publications Series (ed. Goupil, M.-J. & Zahn, J.-P.), vol. 29, pp. 1–65.CrossRefGoogle Scholar
Ogilvie, G. 2005 Wave attractors and the asymptotic dissipation rate of tidal disturbance. J. Fluid Mech. 543, 19–44.CrossRefGoogle Scholar
Ogilvie, G. 2009 Tidal dissipation in rotating fluid bodies: a simplified model. MNRAS 396, 794–806.CrossRefGoogle Scholar
Ogilvie, G. I. & Lin, D. N. C. 2004 Tidal dissipation in rotating giant planets. ApJ 610, 477–509.CrossRefGoogle 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. 1991 Linear theory of rotating fluids using spherical harmonics. II. Time periodic flows. Geophys. Astrophys. Fluid Dyn. 59, 185–208.CrossRefGoogle Scholar
Rieutord, M. 1997 Une introduction à la dynamique des fluides. Masson.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.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
Rieutord, M. & Valdettaro, L. 1997 Inertial waves in a rotating spherical shell. J. Fluid Mech. 341, 77–99.CrossRefGoogle Scholar
Rieutord, M., Valdettaro, L. & Georgeot, B. 2002 Analysis of singular inertial modes in a spherical shell: the slender toroidal shell model. J. Fluid Mech. 463, 345–360.CrossRefGoogle Scholar
Roberts, P. & Stewartson, K. 1963 On the stability of a Maclaurin spheroid of small viscosity. ApJ 137, 777–790.CrossRefGoogle Scholar
Rocca, A. 1987 Forced oscillations in a rotating star: low frequency gravity mode. A&A 175, 81–90.Google Scholar
Rocca, A. 1989 Tidal effects in rotating close binaries. A&A 213, 114–126.Google Scholar
Savonije, G. J. & Witte, M. G. 2002 Tidal interaction of a rotating 1 M⊙ star with a binary companion. A&A 386, 211–221.Google Scholar
Stewartson, K. 1971 On trapped oscillations of a rotating fluid in a thin spherical shell. Tellus 23, 506–510.CrossRefGoogle Scholar
Tilgner, A. 1999 Driven inertial oscillations in spherical shells. Phys. Rev. D 59, 1789.Google Scholar
Trefethen, L. & Embree, M. 2005 Spectra and Pseudospectra. Princeton University Press.CrossRefGoogle 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. Math. 205, 382–393.CrossRefGoogle Scholar
Witte, M. G. & Savonije, G. J. 1999 a The dynamical tide in a rotating 10M⊙ main sequence star. A study of g- and r-mode resonances. A&A 341, 842–852.Google Scholar
Witte, M. G. & Savonije, G. J. 1999 b Tidal evolution of eccentric orbits in massive binary systems. A study of resonance locking. A&A 350, 129–147.Google Scholar
Witte, M. G. & Savonije, G. J. 2001 Tidal evolution of eccentric orbits in massive binary systems. II. Coupled resonance locking for two rotating main sequence stars. A&A 366, 840–857.Google Scholar
Zahn, J.-P. 1966 Les marées dans une étoile double serrée. Annales d'Astrophysique 29, 313, 489, 565.Google Scholar
Zahn, J.-P. 1975 The dynamical tide in close binaries. A&A 41, 329–344.Google Scholar
Zahn, J.-P. 1977 Tidal friction in close binaries. A&A 57, 383–394.Google Scholar
Zahn, J.-P. 1992 Circulation and turbulence in rotating stars. A&A 265, 115.Google Scholar
Zahn, J.-P. 2008 Tidal dissipation in binary systems. In EAS Publications Series (ed. Goupil, M.-J. & Zahn, J.-P.), vol. 29, pp. 67–90.CrossRefGoogle Scholar
Zhang, K. 1994 On coupling between the Poincaré equation and the heat equation. J. Fluid Mech. 268, 211–229.CrossRefGoogle Scholar
Zhang, K. 1995 On coupling between the Poincaré equation and the heat equation: non-slip boundary condition. J. Fluid Mech. 284, 239–256.CrossRefGoogle Scholar