Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-28T17:53:42.510Z Has data issue: false hasContentIssue false

The influence of surface topography on rotating convection

Published online by Cambridge University Press:  26 April 2006

Peter I. Bell
Affiliation:
Department of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK
Andrew M. Soward
Affiliation:
Department of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK Present address: Department of Mathematics, University of Exeter, Laver Building, North Park Road, Exeter, EX4 4QE, UK.

Abstract

Busse's annulus is considered as a model of thermal convection inside the Earth's liquid core. The conventional tilted base and top are modified by azimuthal sinusoidal corrugations so that the effects of surface topography can be investigated. The annulus rotates rapidly about its axis of symmetry with gravity directed radially inwards towards the rotation axis. An unstable radial temperature gradient is maintained and the resulting Boussinesq convection is considered at small Ekman number. Since the corrugations on the boundaries cause the geostrophic contours to be no longer circular, strong geostrophic flows may be driven by buoyancy forces and damped by Ekman suction. When the bumps are sufficiently large, instability of the static state is dominated by steady geostrophic flow with the convection pattern locked to the bumps. As the bump size is decreased, oscillatory geostrophic flow is possible but the preferred mode is modulated on a long azimuthal length scale and propagates as a wave eastwards. This mode only exists in the presence of bumps and is not to be confused with the thermal Rossby waves which are eventually preferred as the bump height tends to zero. Like thermal Rossby waves, the new modes prefer to occupy the longest available radial length scale. In this long-length-scale limit, two finite-amplitude states characterized by uniform geostrophic flows can be determined. The small-amplitude state resembles Or & Busse's (1987) mean flow instability. On losing stability the solution jumps to the more robust large-amplitude state. Eventually, for sufficiently large Rayleigh number and bump height, it becomes unstable to a long-azimuthal-length-scale travelling wave. The ensuing finite-amplitude wave and the mean flow, upon which it rides, are characterized by a geostrophic flow, which is everywhere westward.

Type
Research Article
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

Anufriev, A. P. & Braginsky, S. I. 1975 The effect of a magnetic field on the stream of a rotating liquid at a rough surface. Mag. Gidrod. 4, 6268 (English transl. Magnetohydrodynamics 11, 461–467 (1976)).Google Scholar
Anufriev, A. P. & Braginsky, S. I. 1977 Effect of irregularities of the boundary of the Earth's core on the speed of the fluid and on the magnetic field. III. Geomag. Aeron. 17, 742750 (English transl. pp. 492–496).Google Scholar
Bell, P. I. 1993 The effect of bumps on convection in the Earth's core. PhD dissertation, University of Newcastle upon Tyne.
Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441460.Google Scholar
Busse, F. H. 1994 Convection driven zonal flows and vortices in the major planets. Chaos 4, 123134.Google Scholar
Busse, F. H. & Cuong, P. G. 1977 Convection in rapidly rotating spherical fluid shells. Geophys. Astrophys. Fluid Dyn. 8, 1744.Google Scholar
Busse, F. H. & Hood, L. L. 1982 Differential rotation driven by convection in a rapidly rotating annulus. Geophys. Astrophys. Fluid Dyn. 21, 5974.Google Scholar
Busse, F. H. & Or, A. C. 1986 Convection in a rotating cylindrical annulus: thermal Rossby waves. J. Fluid Mech. 166, 173187.Google Scholar
Busse, F. H. & Wight, J. 1992 A simple dynamo caused by conductivity variations. Geophys. Astrophys. Fluid Dyn. 64, 135144.Google Scholar
Carrigan, C. R. & Busse, F. H. 1983 An experimental and theoretical investigation of the onset of convection in rotating spherical shells. J. Fluid Mech. 126, 287305.Google Scholar
Ewen, S. A. & Soward, A. M. 1994 Phase mixed rotating magnetoconvection and Taylor's condition I. Amplitude equations. Geophys. Astrophys. Fluid Dyn. 77, 209230.Google Scholar
Fearn, D. R. & Proctor, M. R. E. 1992 Magnetostrophic balance in non-axisymmetric, non-standard dynamo models. Geophys. Astrophys. Fluid Dyn. 67, 117128.Google Scholar
Gillman, P. H. 1977 Non-linear dynamics of Boussinesq convection in a deep rotating spherical shell. Geophys. Astrophys. Fluid Dyn. 8, 93135.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press
Gubbins, D. & Richards, M. 1986 Coupling of the core dynamo and mantle: thermal or topographic? Geophys. Res. Lett. 13, 15211524.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer
Hide, R. 1967 Motions of the Earth's core and mantle and variations of the main geomagnetic field. Science 157, 5556.Google Scholar
Hide, R. 1989 Fluctuations in the Earth's rotation and the topography of the core-mantle interface. Phil. Trans. R. Soc. Lond. A 328, 351363.Google Scholar
Hide, R., Clayton, R. W., Hager, B. H., Spieth, M. A. & Voorhies, C. V. 1993 Topographic core–mantle coupling and fluctuations in the Earth's rotation. In Relating Geophysical Structures and Processes, The Jeffreys Volumes ed. (K. Aki & R. Dmowska), vol. 76, pp. 107120. AGU/IUGG.
Holyer, J. Y. 1981 On the collective instability of salt fingers. J. Fluid Mech. 110, 195207.Google Scholar
Jault, D. & Le Mouël, J.-L. 1989 The topographic torque associated with a tangentially geostrophic motion at the core surface and inferences on he flow inside the core. Geophys. Astrophys. Fluid Dyn. 48, 273296.Google Scholar
Jault, D. & Le Mouël, J.-L. 1993 Circulation in the liquid core and coupling with the mantle. Adv. Space Res. 13, 11, 221233.Google Scholar
Kuang, W. & Bloxham, J. 1993 On the effect of boundary topography on flow in the Earth's core. Geophys. Astrophys. Fluid Dyn. 72, 161195.Google Scholar
Moffatt, H. K. & Dillon, R. F. 1976 The correlation between gravitational and geomagnetic fields caused by interaction of the core fluid motion with a bumpy core–mantle interface. Phys. Earth Planet. Inter. 13, 6878.Google Scholar
Munk, W. H. 1950 On the wind-driven ocean circulation. J. Met. 7, 7993.Google Scholar
Or, A. C. 1994 Chaotic transitions of convection rolls in a rapidly rotating annulus. J. Fluid Mech. 261, 119.Google Scholar
Or, A. C. & Busse, F. H. 1987 Convection in a rotating cylindrical annulus. Part 2. Transitions to asymmetric and vascillating flow. J. Fluid Mech. 174, 313326.Google Scholar
Pedlosky, J. 1979 Geophysical Fluid Dynamics. Springer
Roberts, P. H. 1968 On the thermal instability of a rotating fluid sphere containing heat sources. Phil. Trans. R. Soc. Lond. A 263, 93117.Google Scholar
Roberts, P. H. 1988 On topographic core–mantle coupling. Geophys. Astrophys. Fluid Dyn. 44, 181187.Google Scholar
Roberts, P. H. & Soward, A. M. 1992 Dynamo theory. Ann. Rev. Fluid Mech. 24, 459512.Google Scholar
Schnaubelt, M. & Busse, F. H. 1990 Convection in a rotating cylindrical annulus with rigid boundaries. In Nonlinear evolution of spatio–temporal structures in dissipative continuous systems (ed. F. H. Busse & L. Kramer). NATO ASI Series B, vol. 225, pp. 6772. Plenum.
Schnaubelt, M. & Busse, F. H. 1992 Convection in a rotating cylindrical annulus. Part 3. Vacillating and spatially modulated flows. J. Fluid Mech. 245, 155173.Google Scholar
Soward, A. M. 1977 On the finite amplitude thermal instability of a rapidly rotating fluid sphere. Geophys. Astrophys. Fluid Dyn. 9, 1974.Google Scholar
Soward, A. M. 1980 Finite-amplitude thermal convection and geostrophic flow in a rotating magnetic system. J. Fluid Mech. 98, 449471.Google Scholar
Stewartson, K. 1957 On almost rigid rotations. J. Fluid Mech. 3, 1726.Google Scholar
Stommel, H. 1948 The westward intensification of wind-driven ocean currents. Trans. Am. Geophys. Union 99, 202206.Google Scholar
Sun, Z.-P., Schubert, G. & Glatzmaier, G. A. 1994 Numerical simulations of thermal convection in a rapidly rotating spherical shell cooled inhomogeneously from above. Geophys. Astrophys. Fluid Dyn. 75, 199226.Google Scholar
Sverdrup, H. U. 1947 Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific. Proc. Natl Acad. Sci. 33, 318326.Google Scholar
Taylor, J. B. 1963 The magneto-hydrodynamics of a rotating fluid and the Earth's dynamo problem. Proc R. Soc. Lond. A 274, 274283.Google Scholar
Zhang, K. 1992 Spiralling columnar convection in rapidly rotating spherical fluid shells. J. Fluid Mech. 236, 535556.Google Scholar
Zhang, K. & Gubbins, D. 1992 On convection in the Earth's core driven by lateral temperature variations in the lower mantle. Geophys. J. Intl 108, 247255.Google Scholar
Zhang, K. & Gubbins, D. 1993 Convection in a rotating spherical shell with an inhomogenious temperature boundary at infinite Prandtl number. J. Fluid Mech. 250, 209232.Google Scholar