Hostname: page-component-7bb8b95d7b-495rp Total loading time: 0 Render date: 2024-10-06T14:26:20.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Topographic waves and the evolution of coastal currents

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

An initial-value problem is considered for the oceanographically relevant case of slow flow over obstacles of small height and horizontal scale of order the fluid depth or larger. Previous work on starting flow over obstacles whose contours are closed (Johnson 1984) is extended to flow forced by a source–sink pair to cross a step change in depth bounded by a vertical sidewall. Bottom contours thus end abruptly and the near-periodic solutions of the earlier work are no longer possible. The relevant timescale for the motion is again the topographic vortex-stretching time h/2Ωh0, where h is the fluid depth, Ω the background rotation rate and h0 the step height. This time is taken to be long compared with the inertial period but short compared with the advection time. It is shown that if shallow water lies to the right (looking away from the wall) a wavefront moves outwards exponentially fast leaving behind a flow equivalent to that obtained by replacing the step by a rigid wall. If shallow water lies to the left the wavefront approaches the wall, forming at the wall–step junction an unsteady, exponentially thinning, singular region that transports the whole flux. The relevance of these solutions to experiments and steady solutions for free-surface and two-layer flows in Davey, Gill, Johnson & Linden (1984, 1985) is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 160 , November 1985 , pp. 499 - 509
Copyright
© 1985 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

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Davey, M. K., Gill, A. E., Johnson, E. R. & Linden, P. 1984 Steady flow over a step perpendicular to a vertical boundary. Ocean Modelling 58.Google Scholar
Davey, M. K., Gill, A. E., Johnson, E. R. & Linden, P. 1985 Rossby adjustment over a step. In preparation.
Greenspan, H. P. 1968 The Theory of Rotating Fluids, 1st edn. Cambridge University Press.
Hide, R. 1961 Origin of Jupiter's Great Red Spot. Nature 190, 895–896.Google Scholar
Johnson, E. R. 1984 Starting flow for an obstacle moving transversely in a rapidly rotating fluid. J. Fluid Mech. 149, 71–88.Google Scholar
Leblond, P. H. & Mysak, L. A. 1978 Waves in the Ocean, 1st edn. Elsevier.
Longuet-Higgins, M. S. 1968 On the trapping of waves along a discontinuity of depth in a rotating ocean. J. Fluid Mech. 31, 417–434.Google Scholar
Mysak, L. A. 1969 On the generation of double Kelvin waves. J. Fluid Mech. 37, 417–434.Google Scholar
Rhines, P. B. 1969 Slow oscillations in an ocean of varying depth. Part 1. Abrupt topography. J. Fluid Mech. 37, 161–189.Google Scholar
Rhines, P. B. 1977 The dynamics of unsteady currents. In The Sea (ed. E. D. Goldberg, I. McCave, J. O'Brien & J. Steele), vol. 6, pp. 189–318. Wiley-Interscience.
Willmott, A. J. 1984 Forced double Kelvin waves in a stratified ocean. J. Mar. Res. 42, 319–358.Google Scholar