Finite-Difference Presentation of the Coriolis Force for Flows of Rotating Shallow Water

Authors: Karelsky K., Petrosyan A.S., Slavin A.G.

Year: 2009

Issue: 04

Pages: 170-177


In the given work the finite-difference presentation is proposed, that describes the Coriolis force in numerical Godunov-type methods for rotating shallow water flows. The finite-difference schemes are offered for simulating the flows, both on a smooth underlying surface, and for underlying surface of arbitrary profile. The Coriolis force effect is simulated by introducing the fictitious non-stationary boundary. For numerical approximation of source terms, caused by inhomogeneity of underlying surface and Coriolis force effect, the quasi-two-layer model of fluid flow over a stepwise boundary is applied, that takes into account hydrodynamic features. The calculations showing the efficiency of the proposed method are carried out.

Tags: arbitrary underlying surface; Coriolis force; quasi-two-layer method; rotation; shallow water


  1. Bouchut, F., J. Le Sommer, V. Zeitlin (2004) Frontal geostrophic adjustment and nonlinear-wave phenomena  in  one  dimensional  rotating  shallow  Part 2: high-resolution numerical simulations. J. Fluid Mech. 514, pp. 35–63.
  2. Chapman, S., T.G. Cowling (1952) The mathematical theory of non-uniform gases. Cambridge Univ. Press.
  3. Dolzhansky, F.V. (2006) Lectures on geophysical hydrodynamics, Moscow. Institute of Numerical Mathematics RAS, ISBN 5-901854-08-X (in Russian).
  4. Karelsky, K.V, A.S. Petrosyan, A.G. Slavin (2006) Quazi-two-layer model for numerical analysis shallow water flows on step. Russian journal of Numerical Analysis and Mathematical modeling, Vol. 21, No. 6, pp. 539–559.
  5. Karelsky, K.V, A.S. Petrosyan, A.G. Slavin (2007) Numerical simulation of flows of a heavy nonviscous fluid with a free surface in the gravity field over a bed surface with an arbitrary profile. Russian journal of Numerical Analysis and Mathematical modeling, Vol. 22, No. 6, pp. 543–565.
  6. Kolgan, V.P. (1978) Application of smoothing operators in high-accuracy finite-difference schemes. J. Numerical Math. and Math. Physics, Vol. 18, No. 5, pp. 1340 -1345, (In Russian).
  7. Le Sommer, J., S.B. Medvedev, R. Plougonven, V. Zeitlin (2003) Singularity formation during relaxation of jets and fronts toward the state of geostrophic equilibrium. Communications in Nonlinear Science and Numerical Simulation, Vol. 8, pp. 415–442.
  8. LeVeque, R.J. (1998) Balancing source terms and flux gradients in high-resolution Godunov methods:  the  quasi-steady  wave-propagation  algorithm,  Journal  of Computational Physics. 146, pp. 346 –365.
  9. Reznik, G.M, V. Zeitlin, M. Ben Jelloul (2001) Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model. J. Fluid Mech., Vol. 445, pp. 93–120.
  10. Toro, E. F. (1999) Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer–Verlag, Berlin.
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