Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-22T07:01:49.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Starting flow for an obstacle moving transversely in a rapidly rotating fluid

Published online by Cambridge University Press:  20 April 2006

E. R. Johnson
E. R. Johnson
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT
Get access Rights & Permissions [Opens in a new window]

Abstract

The initial-value problem for Taylor columns is considered for the oceanographically relevant case of slow flow over obstacles of small slope, and horizontal scale of order of the fluid depth or larger. In such flows depth variations cause, in the neighbourhood of the obstacle, a gradient of background potential vorticity. Topographic Rossby waves, forced in the starting flow, propagate across the gradient, cycling the obstacle in times of order of the topographic vortex-stretching time h/2Ωh0, where h is the fluid depth, Ω the background rotation rate and h0 the obstacle height. The linear inertialwave equation and the quasigeostrophic equations are related by considering the case where the inertial period is small compared with the topographic time, which is in turn small compared with the advection time.

When viscosity is present the waves decay to a steady motion in a time of order of the viscous spin-up time. In inviscid flow the waves do not decay. The flow oscillates about steady, irrotational, zero-circulation flow. The drag and lift on the obstacle oscillate with almost constant frequency and amplitude equal to the Coriolis force on a body of fluid whose volume matches that of the obstacle. Particles above a right cylinder move in closed orbits of diameter 1/S, where S is the topographic parameter introduced by Hide (1961). Particles away from the cylinder move irrotationally, but asymmetrically, from upstream to downstream. A shear layer is present at the boundary of the cylinder throughout the motion. The initial vorticity distribution for flow over a paraboloid is everywhere finite. However, regions of positive and negative vorticity interlace ever more closely during the motion, causing at large time a similar shear layer at the column boundary. By this time the vorticity is increasing without bound above the obstacle and neglected nonlinear, horizontal viscous, or vertical effects are important.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 149 , December 1984 , pp. 71 - 88
Copyright
© 1984 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

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Cheng, H. K. 1977 On inertial wave and flow structure at low Rossby number. Z. angew. Math. Phys. 28, 753–769.Google Scholar
Cheng, H. K. & Johnson, E. R. 1982 Inertial waves above an obstacle in an unbounded, rapidly rotating fluid. Proc. R. Soc. Lond. A 383, 71–87.Google Scholar
Grace, S. F. 1927 On the motion of a sphere in a rotating liquid. Proc. R. Soc. Lond. 133, 46–77.Google Scholar
Hickie, B. P. 1972 Taylor columns for small Rossby numbers. In Proc. GFD Summer School, Woods Hole Oceanographic Institute, vol. II, pp. 29–39.
Hide, R. 1961 Origin of Jupiter's Great Red Spot. Nature 190, 895–896.Google Scholar
Hide, R., Ibbetson, A. & Lighthill, M. J. 1968 On slow transverse flow past obstacles in a rapidly rotating fluid. J. Fluid Mech. 32, 251–272.Google Scholar
Huppert, H. E. 1975 Some remarks on the initiation of inertial Taylor columns. J. Fluid Mech. 67, 397–412.Google Scholar
Huppert, H. E. & Bryan, K. 1976 Topographically generated eddies. Deep-Sea Res. 23, 655–679.Google Scholar
Ingersoll, A. P. 1969 Inertial Taylor columns and Jupiter's Great Red Spot. J. Atmos. Sci. 26, 744–752.Google Scholar
Jacobs, S. J. 1964 The Taylor column problem. J. Fluid Mech. 20, 581–591.Google Scholar
James, I. N. 1980 The forces due to geostrophic flow over shallow topography. Geophys. Astrophys. Fluid Dyn. 14, 225.Google Scholar
Johnson, E. R. 1978 Trapped vortices in rotating flow. J. Fluid Mech. 86, 209–224.Google Scholar
Johnson, E. R. 1982 The effects of obstacle shape and viscosity in deep rotating flow over finite-height topography. J. Fluid Mech. 120, 359–383.Google Scholar
Longuet-Higgins, M.S. 1968 Double Kelvin waves with continuous depth profiles. J. Fluid Mech. 34, 49–80.Google Scholar
Mason, P. J. & Sykes, R. I. 1981 A numerical study of rapidly rotating flow over surface mounted obstacles. J. Fluid Mech. 111, 175–195.Google Scholar
Phillips, N. A. 1963 Geostrophic motion. Rev. Geophys. 1, 123–176.Google Scholar
Rhines, P. B. 1969 Slow oscillations in an ocean of varying depth. Part 2. Islands and seamounts. J. Fluid Mech. 37, 191–205.Google Scholar
Smith, R. 1971 The ray paths of topographic Rossby waves. Deep-Sea Res. 18, 477–483.Google Scholar
Stewartson, K. 1953 On the slow motion of an ellipsoid in a rotating fluid. Q. J. Mech. Appl. Maths 6, 141–162.Google Scholar
Stewartson, K. 1966 On almost rigid rotations. Part 2. J. Fluid Mech. 26, 131–144.Google Scholar
Stewartson, K. 1967 On the slow transverse motion of a sphere through a rotating fluid. J. Fluid Mech. 30, 357–369.Google Scholar
Stewartson, K. & Cheng, H. K. 1979 On the structure of inertial waves produced by an obstacle in a deep, rotating container. J. Fluid Mech. 91, 415–432.Google Scholar
Taylor, G. I. 1923 Experiments on the motion of solid bodies in rotating fluids. Proc. R. Soc. Lond. A 104, 213–218.Google Scholar
Watson, G. N. 1944 Theory of Bessel Functions, 2nd ed. Cambridge University Press.