Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-29T08:29:47.670Z Has data issue: false hasContentIssue false

Convection in a compressible fluid with infinite Prandtl number

Published online by Cambridge University Press:  19 April 2006

Gary T. Jarvis
Affiliation:
Department of Geodesy and Geophysics, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ Present address: Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7.
Dan P. Mckenzie
Affiliation:
Department of Geodesy and Geophysics, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ

Abstract

An approximate set of equations is derived for a compressible liquid of infinite Prandtl number. These are referred to as the anelastic-liquid equations. The approximation requires the product of absolute temperature and volume coefficient of thermal expansion to be small compared to one. A single parameter defined as the ratio of the depth of the convecting layer, d, to the temperature scale height of the liquid, HT, governs the importance of the non-Boussinesq effects of compressibility, viscous dissipation, variable adiabatic temperature gradients and non-hydrostatic pressure gradients. When d/HT [Lt ] 1 the Boussinesq equations result, but when d/HT is O(1) the non-Boussinesq terms become important. Using a time-dependent numerical model, the anelastic-liquid equations are solved in two dimensions and a systematic investigation of compressible convection is presented in which d/HT is varied from 0·1 to 1·5. Both marginal stability and finite-amplitude convection are studied. For d/HT [les ] 1·0 the effect of density variations is primarily geometric; descending parcels of liquid contract and ascending parcels expand, resulting in an increase in vorticity with depth. When d/HT > 1·0 the density stratification significantly stabilizes the lower regions of the marginal state solutions. At all values of d/HT [ges ] 0·25, an adiabatic temperature gradient proportional to temperature has a noticeable stabilizing effect on the lower regions. For d/HT [ges ] 0·5, marginal solutions are completely stabilized at the bottom of the layer and penetrative convection occurs for a finite range of supercritical Rayleigh numbers. In the finite-amplitude solutions adiabatic heating and cooling produces an isentropic central region. Viscous dissipation acts to redistribute buoyancy sources and intense frictional heating influences flow solutions locally in a time-dependent manner. The ratio of the total viscous heating in the convecting system, ϕ, to the heat flux across the upper surface, Fu, has an upper limit equal to d/HT. This limit is achieved at high Rayleigh numbers, when heating is entirely from below, and, for sufficiently large values of d/HT, Φ/Fu is greater than 1·00.

Type
Research Article
Copyright
© 1980 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

Backus, G. E. 1975 Gross thermodynamics of heat engines in the deep interior of the Earth. Proc. Nat. Acad. Sci. U.S.A. 72, 15551558.Google Scholar
Bénard, M. H. 1901 Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en régime permanent. Anns Chim. Phys. 7e série, XXIII, 62144.Google Scholar
Birch, F. 1952 Elasticity and constitution of the Earth's interior. J. geophys. Res. 57, 227286.Google Scholar
Boussinesq, J. 1903 Théorie analytique de la Chaleur mise en Harmonie avec la Thermodynamique et avec la Théorie mécanique de la Lumière, tome II, pp. 157176. Paris: Gauthier-Villars.
Busse, F. H. 1967 On the stability of two-dimensional convection in a layer heated from below. J. Math. & Phys. 46, 140150.Google Scholar
Busse, F. H. 1971 Stability regions of cellular fluid flow. Proc. IUTAM Symp., Herrenalb, 1969. In Instability of Continuous Systems (ed. H. Leipholz), pp 4147. Springer.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65, 625645.Google Scholar
Froidevaux, C. & Schubert, G. 1975 Plate motion and structure of the continental astheno-sphere: a realistic model of the upper mantle. J. Geophys. Res. 80, 25532564.Google Scholar
Gilbert, F. & Backus, G. E. 1966 Propagator matrices in elastic wave and vibrator problems. Geophys. 31, 326332.Google Scholar
Gough, D. O. 1969 The anelastic approximation for thermal convection. J. Atmos. Sci. 26, 448456.Google Scholar
Graham, E. 1975 Numerical simulation of two-dimensional compressible convection. J. Fluid Mech. 70, 689703.Google Scholar
Griggs, D. T. 1972 The sinking lithosphere and the focal mechanism of deep earthquakes. In The Nature of the Solid Earth (ed. E. C. Robertson), pp. 361384. McGraw-Hill.
Hewitt, J. M., McKenzie, D. P. & Weiss, N. O. 1975 Dissipative heating in convective flows. J. Fluid Mech 68, 721738.Google Scholar
Howard, L. N. 1966 Convection at high Rayleigh numbers. In Proc. 11th Int. Cong. Appl. Mech., Munich, 1964 (ed. H. Görtler), pp. 11091115. Springer.
Jeffreys, H. 1926 On the stability of a layer of fluid heated below. Phil. Mag. VII, 2, 833844.Google Scholar
Jeffreys, H. 1928 Some cases of instability in fluid motion. Proc. Roy. Soc. A 118, 195208.Google Scholar
Jeffreys, H. 1930 The instability of a compressible fluid heated below. Proc. Cam. Phil. Soc. 26, 170172.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.
Low, A. R. 1929 On the criterion for stability of a layer of viscous fluid heated from below. Proc. Roy. Soc. A 125, 180195.Google Scholar
McKenzie, D. P., Roberts, J. M. & Weiss, N. O. 1974 Convection in the Earth's mantle: towards a numerical simulation. J. Fluid Mech. 62, 465538.Google Scholar
McKenzie, D. P. & Weiss, N. O. 1975 Speculations on the thermal and tectonic history of the Earth. Geophys. J. Roy. Astr. Soc. 42, 131174.Google Scholar
Malkus, W. V. R. 1964 Boussinesq equations. Geophys. Fluid Dynamics, Woods Hole Oceanographic Institute Rep. no. 64–46.Google Scholar
Malkus, W. V. R. 1973 Convection at the melting point: a thermal history of the Earth's core. Geophys. Fluid Mech. 4, 267278.Google Scholar
Mihaljan, J. M. 1962 A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid. Astrophys. J. 136, 11261133.Google Scholar
Moore, D. R. 1971 Numerical investigation of astrophysical convection. Ph.D. thesis, University of Cambridge.
Moore, D. R., Peckover, R. S. & Weiss, N. O. 1974 Difference methods for time-dependent two-dimensional convection. Comp. Phys. Commun. 6, 198220.Google Scholar
Moore, D. R. & Weiss, N. O. 1973 Two-dimensional Rayleigh—Bénard convection. J. Fluid Mech. 58, 289312.Google Scholar
Ogura, Y. & Phillips, N. A. 1962 Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173179.Google Scholar
Pellew, A. & Southwell, R. V. 1940 On maintained convective motion in a fluid heated from below. Proc. Roy. Soc. A 176, 312343.Google Scholar
Peltier, W. R. 1972 Penetrative convection in the planetary mantle. Geophys. Fluid Dyn. 5, 4788.Google Scholar
Press, F. 1970 Earth models consistent with geophysical data. Phys. Earth & Planet. Interiors, 3, 322.Google Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. VI, 32, 529546.Google Scholar
Richter, F. M. 1973 Dynamical models for sea floor spreading. Rev. Geophys. & Space Phys. 11, 223287.Google Scholar
Roberts, K. V. & Weiss, N. O. 1966 Convective difference schemes. Math. Comput. 20, 272299.Google Scholar
Romanelli, M. J. 1960 Runge-Kutta methods for the solution of ordinary differential equations. In Mathematical Methods for Digital Computers (ed. A. Ralston & H. S. Wilf), pp. 110120. Wiley.
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129144.Google Scholar
Schmidt, R. J. & Milverton, S. W. 1935 On the instability of a fluid when heated from below. Proc. Roy. Soc. A 152, 586594.Google Scholar
Schmidt, R. J. & Saunders, O. J. 1938 On the motion of a fluid heated from below. Proc. Roy. Soc. A 165, 216228.Google Scholar
Skilbeck, J. M. 1976 The stability of mantle convection. Ph.D. thesis, University of Cambridge.
Spiegel, E. A. 1971 Convection in stars. I. Basic Boussinesq convection. A. Rev. Astron. & Astrophys. 9, 323352.Google Scholar
Spiegel, E. A. & Veronis, G. 1960 On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 442447.Google Scholar
Straus, J. M. 1972 Finite amplitude doubly diffusive convection. J. Fluid Mech. 56, 353374.Google Scholar
Turcotte, D. L., Hsui, A. T., Torrance, K. H. & Schubert, G. 1974 Influence of viscous dissipation on Bénard convection. J. Fluid Mech. 64, 369374.Google Scholar
Turcotte, D. L. & Oxburgh, E. R. 1972 Mantle convection and the new global tectonics. Ann. Rev. Fluid Mech. 4, 3368.Google Scholar