Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-19T09:45:04.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear Rossby adjustment in a channel: beyond Kelvin waves

Published online by Cambridge University Press:  26 April 2006

Albert J. Hermann ,
Peter B. Rhines  and
E. R. Johnson
Albert J. Hermann
Affiliation:
School of Oceanography, WB-10, University of Washington, Seattle, WA 98195, USA Present address: Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA.
Peter B. Rhines
Affiliation:
School of Oceanography, WB-10, University of Washington, Seattle, WA 98195, USA
E. R. Johnson
Affiliation:
Department of Mathematics, University College London, WC1E 6BT, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Nonlinear advective adjustment of a discontinuity in free-surface height under gravity and rotation is considered, using the method of contour dynamics. After linear wave-adjustment has set up an interior jet and boundary currents in a wide ([Gt ] one Rossby radius) channel, fluid surges down-channel on both walls, rather than only that wall supporting a down-channel Kelvin wave. A wedgelike intrusion of low potential vorticity fluid on this wall, and a noselike intrusion of such fluid on the opposite wall, serve to reverse the sign of relative vorticity in the pre-existing currents. For narrower channels, a coherent boundary-trapped structure of low potential vorticity fluid is ejected at one wall, and shoots ahead of its parent fluid. The initial tendency for the current to concentrate on the ‘right-hand’ wall (the one supporting a down-channel Kelvin wave in the northern hemisphere) is defeated as vorticity advection shifts the maximum to the left-hand side. Ultimately fluid washes downstream everywhere across even wide channels, leaving the linearly adjusted upstream condition as the final state. The time necessary for this to occur grows exponentially with channel width. The width of small-amplitude boundary currents in linear theory is equal to Rossby's deformation radius, yet here we find that the width of the variation in velocity and potential vorticity fields deviates from this scale across a large region of space and time. Comparisons of the contour dynamics solutions, valid for small amplitude, and integration of the shallow-water equations at large amplitude, show great similarity. Boundary friction strongly modifies these results, producing fields more closely resembling the linear wave-adjusted state. Observed features include those suggestive of coastally trapped gravity currents. Analytical results for the evolution of vorticity fronts near boundaries are given in support of the numerical experiments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 205 , August 1989 , pp. 469 - 502
Copyright
© 1989 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

Baines, P. G. 1980 The dynamics of the southerly buster. Austral. Met. Mag. 28, 175–200.Google Scholar
Beardsley, R. C., Dorman, C. E., Friehe, C. A., Rosenfeld, L. K. & Winant, C. D. 1987 Local atmospheric forcing during the coastal ocean dynamics experiment. I. A description of the marine boundary layer and atmospheric conditions over a northern California upwelling region. J. Geophys. Res. 92, 1467–1488.CrossRefGoogle Scholar
Blumen, W. 1972 Geostrophic adjustment. Rev. Geophys. Space Phys. 10, 485–528.Google Scholar
Camerlengo, A. L. & O'Brien, J. J. 1980 Open boundary conditions in rotating fluids. J. Comput. Phys. 35, 12–35.Google Scholar
Csanady, G. T. 1976 Topographic waves in Lake Ontario. J. Phys. Oceanogr. 6, 93–103.Google Scholar
Dorman, C. 1987 Possible role of gravity currents in northern California's coastal summer wind reversals. J. Geophys. Res. 92, 1497–1506.Google Scholar
Gill, A. E. 1976 Adjustment under gravity in a rotating channel. J. Fluid Mech., 77, 603–621.Google Scholar
Gill, A. E. 1977a Coastally trapped waves in the atmosphere. Q. J. R. Met. Soc. 103, 431–440.Google Scholar
Gill, A. E. 1977b The hydraulics of rotating channel flow. J. Fluid Mech. 80, 641–671.Google Scholar
Gill, A. E. 1982 Atmosphere–Ocean Dynamics, ch. 7, 10. Academic.
Gill, A. E., Davey, M. K., Johnson, E. R. & Linden, P. F. 1986 Rossby adjustment over a step. J. Mar. Res. 44, 713–738.Google Scholar
Godfrey, J. S. 1989 A Sverdrup model of the depth-integrated flow of the world ocean allowing for island circulations. Dyn. Atmos. Ocean. in press.Google Scholar
Hermann, A. J., Hickey, B. M., Mass, C. F. & Albright, M. 1989 Orographically-trapped wind events in the Pacific Northwest and their oceanic response. Submitted to J. Geophys. Res.Google Scholar
Johnson, E. R. 1985 Topographic waves and the evolution of coastal currents. J. Fluid Mech. 160, 499–509.Google Scholar
Kawase, M. 1987 Establishment of deep ocean circulation driven by deep-water production. J. Phys. Ocean. 17, 2294–2317.Google Scholar
Killworth, P. D. 1987 A note on van Heijst and Smeed. Ocean Modelling 69, 7.Google Scholar
Lighthill, M. J. 1967 On waves generated in dispersive systems by travelling forcing effects, with applications to the dynamics of rotating fluids. J. Fluid Mech. 27, 725–752.Google Scholar
Lorenz, E. N. & Krishnamurthy, V. 1987 On the nonexistence of a slow manifold. J. Atmos. Sci. 44, 2940–2950.Google Scholar
Mass, C. & Albright, M. 1987 Coastal southerlies and alongshore surges of the west coast of North America: Evidence of topographically trapped response to synoptic forcing. Mon. Weather Rev. 115, 1707–1738.Google Scholar
Middleton, J. F. 1987 Energetics of linear geostrophic adjustment. J. Phys. Ocean. 17, 735–740.Google Scholar
Rossby, C. G. 1937 On the mutual adjustment of pressure and velocity distributions in certain simple current systems. I. J. Mar. Res. 1, 15–28.Google Scholar
Rossby, C. G. 1938 On the mutal adjustment of pressure and velocity distributions in certain simple current systems. II. J. Mar. Res. 2, 239–263.Google Scholar
Sadourney, R. 1975 The dynamics of finite-difference models of the shallow-water equations. J. Atmos. Sci. 32, 680–689.Google Scholar
Saffman, P. G. & Tanveer, S. 1982 The touching pair of equal and opposite uniform vortices. Phys. Fluids 25, 1929–1930.Google Scholar
Stern, M. E. 1985 Lateral wave breaking and ‘shingle’ formation in large-scale shear flow. J. Phys. Ocean. 15, 1274–1283.Google Scholar
Stern, M. E. 1986 On the amplification of convergence's in coastal currents and the formation of ‘squirts’. J. Mar. Res. 44, 403–421.Google Scholar
Stern, M. E. 1987a Horizontal entrainment and detrainment in large-scale eddies. J. Phys. Ocean. 17, 1688–1695.Google Scholar
Stern, M. E. 1987b Large-scale lateral entrainment and detrainment at the edge of a geostrophic shear layer. J. Phys. Ocean. 17, 1680–1687.Google Scholar
Stern, M. E., Whitehead, J. A. & Hua, B. 1982 The intrusion of a density current along the coast in a rotating fluid. J. Fluid Mech. 123, 237–265.Google Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comp. Phys. 30, 96–106.Google Scholar