Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T06:34:56.325Z Has data issue: false hasContentIssue false
  • English
  • Français

Transcritical rotating flow over topography

Published online by Cambridge University Press:  15 October 2007

J. G. ESLER ,
O. J. RUMP  and
E. R. JOHNSON
J. G. ESLER
Affiliation:
Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, UKgavin@math.ucl.ac.uk
O. J. RUMP
Affiliation:
Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, UKgavin@math.ucl.ac.uk
E. R. JOHNSON
Affiliation:
Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, UKgavin@math.ucl.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The flow of a one-and-a-half layer fluid over a three-dimensional obstacle of non-dimensional height M, relative to the lower layer depth, is investigated in the presence of rotation, the magnitude of which is measured by a non-dimensional parameter B (inverse Burger number). The transcritical regime in which the Froude number F, the ratio of the flow speed to the interfacial gravity wave speed, is close to unity is considered in the shallow-water (small-aspect-ratio) limit. For weakly rotating flow over a small isolated obstacle (M → 0) a similarity theory is developed in which the behaviour is shown to depend on the parameters Γ = (F−1)M−2/3 and ν = B1/2M−1/3. The flow pattern in this regime is determined by a nonlinear equation in which Γ and ν appear explicitly, termed here the ‘rotating transcritical small-disturbance equation’ (rTSD equation, following the analogy with compressible gasdynamics). The rTSD equation is forced by ‘equivalent aerofoil’ boundary conditions specific to each obstacle. Several qualitatively new flow behaviours are exhibited, and the parameter reduction afforded by the theory allows a (Γ, ν) regime diagram describing these behaviours to be constructed numerically. One important result is that, in a supercritical oncoming flow in the presence of sufficient rotation (ν ≳ 2), hydraulic jumps can appear downstream of the obstacle even in the absence of an upstream jump. Rotation is found to have the general effect of increasing the amplitude of any existing downstream hydraulic jumps and reducing the lateral extent and amplitude of upstream jumps. Numerical results are compared with results from a shock-capturing shallow-water model, and the (Γ, ν) regime diagram is found to give good qualitative and quantitative predictions of flow patterns at finite obstacle height (at least for M ≲ 0.4). Results are compared and contrasted with those for a two-dimensional obstacle or ridge, for which rotation also causes hydraulic jumps to form downstream of the obstacle and acts to attenuate upstream jumps.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 590 , 10 November 2007 , pp. 81 - 106
Copyright
Copyright © Cambridge University Press 2007

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

REFERENCES

Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. & Perthame, B. 2004 A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065.CrossRefGoogle Scholar
Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
Baines, P. G. & Davies, P. A. 1980 Laboratory Studies of Topographic Effects in Rotating and/or Stratified Fluids, chap. 8, p. 233. World Meteorological Organization GARP.Google Scholar
Baines, P. G. & Leonard, B. P. 1989 The effects of rotation on flow of a single layer over a ridge. Q. J. R. Met. Soc. 115, 293.CrossRefGoogle Scholar
Benjamin, T. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.CrossRefGoogle Scholar
Bouchut, F., LeSommer, J. Sommer, J. & Zeitlin, V. 2004 Frontal geostrophic adjustment and nonlinear wave phenomena in one-dimensional rotating shallow water. Part II. High resolution numerical simulations. J. Fluid Mech. 514, 35–63.CrossRefGoogle Scholar
Burk, S. D. & Haack, T. 1999 The dynamics of wave clouds upwind of coastal orography. Mon. Weather Rev. 28, 1438–1455.Google Scholar
Chapman, C. J. 2000 High Speed Flow. Cambridge University Press.Google Scholar
Cole, J. D. & Cook, P. 1986 Transonic Aerodynamics. North-Holland.Google Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593–607.CrossRefGoogle Scholar
Engquist, B. & Osher, S. 1980 Stable and entropy satisfying approximations for transonic flow calculations. Math. Comput. 34, 45–75.CrossRefGoogle Scholar
Esler, J. G., Rump, O. J. & Johnson, E. R. 2005 Steady rotating flows over a ridge. Phys. Fluids 17, 116601.CrossRefGoogle Scholar
Esler, J. G., Rump, O. J. & Johnson, E. R. 2007 Nondispersive and weakly dispersive single layer flow over an axisymmetric obstacle: The equivalent aerofoil formulation. J. Fluid Mech. 574, 209–237.CrossRefGoogle Scholar
Goorjian, P. M. & Van Buskirk, R. 1981 Implicit calculations of transonic flows using monotone methods. AIAA Paper 81-331.CrossRefGoogle Scholar
Grimshaw, R. H. J. & Melville, W. K. 1989 On the derivation of the modified Kadomtsev-Petviashvili equation. Stud. Appl. Maths 80, 183–203.CrossRefGoogle Scholar
Grimshaw, R. H. J., Ostrovsky, L. A., Shrira, V. I. & Stepanyants, A., Y. 1998 Long nonlinear surface and internal gravity waves in a rotating ocean. Surveys Geophys. 19, 289–338.CrossRefGoogle Scholar
Grimshaw, R. H. J. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.CrossRefGoogle Scholar
Houghton, D. D. 1969 Effect of rotation on the formation of hydraulic jumps. J. Geophys. Res. 74, 1351–1360.CrossRefGoogle Scholar
Houghton, D. D. & Kasahara, A. 1968 Nonlinear shallow fluid flow over an isolated ridge. Commun. Pure Appl. Maths 21, 1–23.CrossRefGoogle Scholar
Jameson, A. 1979 Acceleration of transonic potential flow calculations on arbitrary meshes by the multiple grid method. AIAA Paper 79-1458.Google Scholar
Jiang, Q. & Smith, R. B. 2000 V-waves, bow shocks, and wakes in supercritical hydrostatic flow. J. Fluid Mech. 406, 27–53.CrossRefGoogle Scholar
Johnson, E. R., Esler, J. G., Rump, O. J., Sommeria, J. & Vilenski, G. G. 2006 Orographically generated nonlinear waves in rotating and nonrotating two-layer flow. Proc. R. Soc. Lond. A 462, 3–20.Google Scholar
Johnson, E. R. & Vilenski, G. G. 2004 Flow patterns and drag in near-critical flow over isolated orography. J. Atmos. Sci. 61, 2909–2918.CrossRefGoogle Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541.Google Scholar
Klemp, J. B., Rotunno, R. & Skamarock, W. C. 1997 On the propagation of internal bores. J. Fluid Mech. 331, 81–106.CrossRefGoogle Scholar
Kuo, A. & Polvani, L. M. 1997 Time-dependent fully nonlinear geostrophic adjustment. J. Phys. Oceanogr. 27, 1614–1634.2.0.CO;2>CrossRefGoogle Scholar
LeVeque, R. J. 2002 Finite Volume methods for Hyperbolic Problems. Cambridge University Press.CrossRefGoogle Scholar
Li, X. F., Dong, C. M., Clemente-Colon, P., Pichel, W. G. & Friedman, K. S. 2004 Synthetic aperture radar observation of the sea surface imprints of up-stream atmospheric solitons generated by flow impeded by an island. J. Geophys. Res. 209, 10.1029.2003JC002168.Google Scholar
Long, R. R. 1954 Some respects of the flow of stratified fluids, II. Experiments with a two-layer system. Tellus 6, 97–115.CrossRefGoogle Scholar
Long, R. R. 1970 Blocking effects in flow over obstacles. Tellus 22, 471–480.CrossRefGoogle Scholar
Lott, F. & Miller, M. J. 1997 A new sub-grid scale orographic drag parameterization: Its formulation and testing. Q. J. R. Met. Soc. 123, 101–127.Google Scholar
Murman, E. M. & Cole, J. D. 1971 Calculations of plane steady transonic flows. AIAA J. 9, 114–121.CrossRefGoogle Scholar
Ostrovsky, L. A. 1978 Nonlinear internal waves in a rotating ocean. Oceanology 18, 181–191.Google Scholar
Roe, P. L. 1981 Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372.CrossRefGoogle Scholar
Rottman, J. W. & Einaudi, F. 1993 Solitary waves in the atmosphere. J. Atmos. Sci. 50, 2116–2136.2.0.CO;2>CrossRefGoogle Scholar
Schär, C. & Smith, R. B. 1993a Shallow water flow past an isolated topography. Part I: Vorticity production and wake formation. J. Atmos. Sci. 50, 1373–1400.2.0.CO;2>CrossRefGoogle Scholar
Schär, C. & Smith, R. B. 1993b Shallow water flow past an isolated topography. Part II: Transition to vortex shedding. J. Atmos. Sci. 50, 1401–1412.2.0.CO;2>CrossRefGoogle Scholar
Shrira, V. I. 1986 On long strongly nonlinear waves in a rotating ocean. Izv. Atmos. Ocean. Phys. 22, 298–305.Google Scholar
Stevenson, R. E. 1980 The 200-mile fishline. In Oceans from Space (ed. Badgley, P. C., Miloy, L. & Childs, L.). Gulf P.Google Scholar
Vilenski, G. G. & Johnson, E. R. 2004 Near critical free-surface rotating flow over topography. Proc. R. Soc. Lond. A 460, 2865–2881.CrossRefGoogle Scholar
Zeitlin, V., Medvedev, S. & Plougonven, R. 2003 Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 1. Theory. J. Fluid Mech. 481, 269–290.CrossRefGoogle Scholar