Hostname: page-component-5cf477f64f-54txb Total loading time: 0 Render date: 2025-04-05T15:42:46.730Z Has data issue: false hasContentIssue false
  • English
  • Français

Upper bounds on the energy dissipation in turbulent precession

Published online by Cambridge University Press:  26 April 2006

R. R. Kerswell
R. R. Kerswell
Affiliation:
Department of Mathematics and Statistics, University of Newcastle upon Tyne, NE1 7RU, UK Present address: Department of Mathematics, Bristol University, BS8 1TW, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

Rigorous upper bounds on the viscous dissipation rate are identified for two commonly studied precessing fluid-filled configurations: an oblate spheroid and a long cylinder. The latter represents an interesting new application of the upper-bounding techniques developed by Howard and Busse. A novel ‘background’ method recently introduced by Doering & Constantin is also used to deduce in both instances an upper bound which is independent of the fluid's viscosity and the forcing precession rate. Experimental data provide some evidence that the observed viscous dissipation rate mirrors this behaviour at sufficiently high precessional forcing. Implications are then discussed for the Earth's precessional response.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 321 , 25 August 1996 , pp. 335 - 370
Copyright
© 1996 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

Busse, F. H. 1969a On Howard's upper bound for heat transport by turbulent convection. J. Fluid Mech. 37, 457–477.Google Scholar
Busse, F. H. 1969b Bounds on the transport of mass and momentum by turbulent flow. Z. Angew. Math. Phys. 20, 1–14.Google Scholar
Busse, F. H. 1970 Bounds for turbulent shear flow. J. Fluid Mech. 41, 219–240.Google Scholar
Busse, F. H. 1978 The optimum theory of turbulence. Adv. Appl. Mech. 18, 77–121.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Constantin, P. & Doering, C. R. 1995 Variational bounds on energy-dissipation in incompressible flows: II. channel flow. Phys. Rev. E 51, 3192–3198.Google Scholar
Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69, 1648–1651.Google Scholar
Doering, C. R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49, 4087–4099.Google Scholar
Doering, C. R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows: III. convection. Phys. Rev. E 53, 5957–5981.Google Scholar
Gans, R. F. 1970 On the hydromagnetic precession in a cylinder. J. Fluid Mech. 45, 111–130.Google Scholar
Gebhardt, T., Grossmann, S., Holthaus, M. & Löhden, M. 1995 Rigorous bound on the plane-shear-flow dissipation rate. Phys. Rev. E 51, 360–365.Google Scholar
Hollerbach, R. 1994a Imposing a magnetic field across a nonaxisymmetric shear layer in a rotating spherical shell. Phys. Fluids 6, 2540–2544.Google Scholar
Hollerbach, R. 1994b Magnetohydrodynamic Ekman and Stewartson layers in a rotating spherical shell. Proc. R. Soc. Lond. A 444, 333–346.Google Scholar
Hopf, E. 1941 Ein allgemeiner endlichkeitssatz der hydrodynamik. Mathematische Annalen 117, 764–775.Google Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.Google Scholar
Howard, L. N. 1972 Bounds on flow quantities. Ann. Rev. Fluid Mech. 4, 473–494.Google Scholar
Howard, L. N. 1990 Limits on the transport of heat and momentum by turbulent convection with large scale flow. Stud. Appl. Maths 83, 273–285.Google Scholar
Herley, G. R. & Malkus, W. V. R. 1988 Stability bounds on turbulent Poiseuille flow. J. Fluid Mech. 187, 435–449.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions I. Springer.
Kennett, R. G. 1974 Convectively-driven dynamos. GFD Summer School Fellowship Lectures, Woods Hole, pp. 94–117.
Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72, 107–144.Google Scholar
Kerswell, R. R. 1994 Tidal excitation of hydromagnetic waves and their damping in the Earth. J. Fluid Mech. 274, 219–241.Google Scholar
Kerswell, R. R. 1996 Variational bounds on shear driven turbulence and turbulent Boussinesq convection. Submitted to Physica D.
Kerswell, R. R. & Davey, A. 1996 On the linear instability of elliptic pipe flow. J. Fluid Mech. 316, 307–324.Google Scholar
Kerswell, R. R. & Soward, A. M. 1996 Upper bounds for turbulent Couette flow incorporating the poloidal power constraint. J. Fluid Mech. (submitted).Google Scholar
Kobine, J. J. 1995 Inertial wave dynamics in a rotating and precessing cylinder. J. Fluid Mech. 303, 233–252.Google Scholar
Koper, D. E. 1975 Torque balance and energy budget for the precessionally driven dynamo. Phys. Earth. Planet. Inter. 11, 43–60.Google Scholar
Mahalov, A. 1993 The instability of rotating fluid columns subjected to a weak external Coriolis field. Phys. Fluids 5, 891–900.Google Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 196–212.Google Scholar
Malkus, W. V. R. 1963 Precessional torques as the cause of geomagnetism. J. Geophys. Res. 68, 2871–2886.Google Scholar
Malkus, W. V. R. 1968 Precession of the Earth as the cause of geomagnetism. Science 160, 259–264.Google Scholar
Malkus, W. V. R. 1989 An experimental study of global instabilities due to tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 48, 123–134.Google Scholar
Malkus, W. V. R. 1994 Energy sources for planetary dynamos. In Theory of Solar and Planetary Dynamos. NATO ASI Conference Series, Cambridge University Press.
Malkus, W. V. R. & Smith, L. M. 1989 Upper bounds on functions of the dissipation rate in turbulent shear flow. J. Fluid Mech. 208, 479–507.Google Scholar
Manasseh, R. 1992 Breakdown regimes of inertia waves in a precessing cylinder. J. Fluid Mech. 243, 261–29.Google Scholar
Manasseh, R. 1994 Distortions of inertia waves in a rotating fluid cylinder forced near its fundamental mode resonance. J. Fluid Mech. 265, 345–370.Google Scholar
Payne, L. & Weinberger, H. 1963 An exact stability bound for Navier-Stokes flow in a sphere. In Nonlinear Problems (ed. R. E. Langer). University of Wisconsin Press.
Poincaré, H. 1910 Sur la précession des corps deformable. Bull. Astron. 27, 321.Google Scholar
Proctor, M. R. E. 1979 Necessary conditions for the magnetohydrodynamic dynamo. Geophys. Astrophys. Fluid Dyn. 14, 127–145.Google Scholar
Roberts, P. H. & Gubbins, D. 1987 Origin of the main field: kinematics. In Geomagnetism, vol. 2 (ed. J. A. Jacobs), p. 193. Academic.
Rochester, M. G., Jacobs, J. A., Smylie, D. E. & Chong, K. F. 1975 Can precession power the geomagnetic dynamo? Geophys. J. R. Astron. Soc. 43, 661–678.Google Scholar
Selmi, M. & Herbert, T. 1995 Resonance phenomena in viscous fluids inside partially filled spinning and nutating cylinders. Phys. Fluids 7, 108–120.Google Scholar
Serrin, J. 1959 On the stability of viscous fluid motions. Arch. Rat. Mech. Anal. 3, 1–13.Google Scholar
Smith, L. M. 1991 Turbulent Couette flow profiles that maximise the efficiency function. J. Fluid Mech. 227, 509–525.Google Scholar
Soward, A. M. 1980 Bounds for turbulent convective dynamos. Geophys. Astrophys. Fluid Dyn. 15, 317–341.Google Scholar
Soward, A. M & Jones, C. A. 1983 The linear stability of the flow in the narrow gap between two concentric rotating spheres. Q. J. Mech. Appl. Maths 36, 19–42.Google Scholar
Stacey, F. D. 1973 The coupling of the core to the precession of the Earth. Geophys. J. R. Astron. Soc. 33, 47–55.Google Scholar
Toomre, A. 1996 On the coupling of the Earth's core and the mantle during the 26,000-year precession. In The Earth-Moon System (ed. B. G. Marsden & A. G. W. Cameron), pp. 33–45. Plenum.
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.
Vanyo, J. P. 1984 Earth core motions: experiments with spheroids. Geophys. J. R. Astron. Soc. 77, 173–183.Google Scholar
Vanyo, J. P. 1991 A geodynamo powered by luni-solar precession. Geophys. Astrophys. Fluid Dyn. 59, 209–234.Google Scholar
Vanyo, J. P. & Likins, P. W. 1972 Rigid-body approximations to turbulent motion in a liquid-filled, precessing, spherical cavity. Trans. ASME J. Appl. Mech. 39, 18–24.Google Scholar
Vanyo, J. P., Lu, V. C. & Weyant, T. F. 1975 Dimensionless energy dissipation for precessional flows in the region of Re = 1. Trans. ASME J. Appl. Mech. 42, 881–882.Google Scholar
Vanyo, J. P., Wilde, P., Cardin, P. & Olson, P. 1995 Experiments on precessing flows in the Earth's liquid core. Geophys. J. Intl 121, 136–142.Google Scholar
Wood, W. W. 1966 An oscillatory disturbance of rigidly rotating fluid. Proc. R. Soc. Lond. A 298, 181–212.Google Scholar
Worthing, R. A. 1995 Contributions to the variational theory of convection. PhD thesis, MIT.