Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-23T08:45:48.027Z Has data issue: false hasContentIssue false
  • English
  • Français

The validity of two-dimensional models of a rotating shallow fluid layer

Published online by Cambridge University Press:  12 August 2020

David G. Dritschel Open the ORCID record for David G. Dritschel [Opens in a new window]  and
Mohammad Reza Jalali Open the ORCID record for Mohammad Reza Jalali [Opens in a new window]
David G. Dritschel*
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St AndrewsKY16 9SS, UK
Mohammad Reza Jalali
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St AndrewsKY16 9SS, UK
*
†Email address for correspondence: david.dritschel@st-andrews.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the validity and accuracy of two commonly used two-dimensional models of a rotating, incompressible, shallow fluid layer: the shallow-water (SW) model and the Serre/Green–Naghdi (GN) model. The models differ in just one respect: the SW model imposes the hydrostatic approximation while the GN model does not. Both models assume the horizontal velocity is independent of height throughout the layer. We compare these models with their parent three-dimensional (3-D) model, initialised with a height-independent horizontal velocity, and with a mean height small compared to the domain width. For small to moderate Rossby and Froude numbers, we verify that both models well approximate the vertically averaged 3-D flow. Overall, the GN model is found to be substantially more accurate than the SW model. However, this accuracy is only achieved using an explicit reformulation of the equations in which the vertically integrated non-hydrostatic pressure is found from a novel linear elliptic equation. This reformulation is shown to extend straightforwardly to multi-layer flows having uniform density layers.

JFM classification

Geophysical and Geological Flows: Shallow water flows Mathematical Foundations: General fluid mechanics Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A33
Copyright
© The Author(s), 2020. Published by 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

REFERENCES

Beltrami, E. 1871 Sui principii fondamentali dell'idrodinamica razionali. Memor. Accad. Sci. Inst. Bologna, Ser. 3 1, 431–476.Google Scholar
Bolin, B. 1955 Numerical forecasting with the barotropic model. Tellus 7, 27–49.CrossRefGoogle Scholar
Charney, J. 1955 The use of the primitive equations of motion in numerical prediction. Tellus 7, 22–26.CrossRefGoogle Scholar
Dellar, P. J. 2003 Dispersive shallow water magnetohydrodynamics. Phys. Plasmas 10, 581–590.CrossRefGoogle Scholar
Dellar, P. J. & Salmon, R. 2005 Shallow water equations with a complete Coriolis force and topography. Phys. Fluids 17 (10), 106601.CrossRefGoogle Scholar
Demirbilek, Z. & Webster, W. C. 1999 The Green–Naghdi theory of fluid sheets for shallow–water waves. In Developments in Offshore Engineering: Wave Phenomena and Offshore Topics (ed. Herbich, J. B.), pp. 1–54. Gulf Publishing Company.Google Scholar
Dritschel, D. G. 1989 On the stabilization of a two-dimensional vortex strip by adverse shear. J. Fluid Mech. 206, 193–221.CrossRefGoogle Scholar
Dritschel, D. G. & Ambaum, M. H. P. 1997 A contour-advective semi-lagrangian algorithm for the simulation of fine-scale conservative fields. Q. J. R. Meteorol. Soc. 123, 1097–1130.CrossRefGoogle Scholar
Dritschel, D. G., de la Torre Juárez, M. & Ambaum, M. H. P. 1999 On the three-dimensional vortical nature of atmospheric and oceanic flows. Phys. Fluids 11 (6), 1512–1520.CrossRefGoogle Scholar
Dritschel, D. G., Gottwald, G. A. & Oliver, M. 2017 Comparison of variational balance models for the rotating shallow water equations. J. Fluid Mech. 822, 689–716.CrossRefGoogle Scholar
Dritschel, D. G. & Jalali, M. R. 2019 On the regularity of the Green–Naghdi equations for a rotating shallow fluid layer. J. Fluid Mech. 865, 100–136.CrossRefGoogle Scholar
Dutykh, D., Clamond, D., Milewski, P. & Mitsotakis, D. 2013 Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations. Eur. J. Appl. Maths 24 (5), 761–787.CrossRefGoogle Scholar
Ertekin, R. C. 1984 Soliton generation by moving disturbances in shallow water. PhD thesis, University of California, Berkeley, CA, USA.Google Scholar
Ford, R., McIntyre, M. E. & Norton, W. A. 2000 Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57, 1236–1254.2.0.CO;2>CrossRefGoogle Scholar
Fox, D. G. & Orszag, S. A. 1973 Pseudospectral approximation to two-dimensional turbulence. J. Comput. Phys. 11, 612–619.CrossRefGoogle Scholar
Ghader, S., Ghasemi, A., Banazadeh, M. R. & Mansoury, D. 2012 High-order compact scheme for Boussinesq equations: implementation and numerical boundary condition issue. Intl J. Numer. Meth. Fluids 69, 590–605.CrossRefGoogle Scholar
Gill, A. E. 1982 Atmosphere-Ocean Dynamics. Academic Press.Google Scholar
Green, A. E., Laws, N. & Naghdi, P. M. 1974 On the theory of water waves. Proc. R. Soc. Lond. A 338 (1612), 43–55.Google Scholar
Green, A. E. & Naghdi, P. M. 1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78 (2), 237–246.CrossRefGoogle Scholar
Holm, D. D. 1988 Hamiltonian structure for two-dimensional hydrodynamics with nonlinear dispersion. Phys. Fluids 31 (8), 2371–2373.CrossRefGoogle Scholar
Le Métayer, O., Gavrilyuk, S. & Hank, S. 2010 A numerical scheme for the Green–Naghdi model. J. Comput. Phys. 229 (6), 2034–2045.CrossRefGoogle Scholar
Miles, J. W. & Salmon, R. 1985 Weakly dispersive, nonlinear gravity waves. J. Fluid Mech. 157, 519–531.CrossRefGoogle Scholar
Mohebalhojeh, A. R. & Dritschel, D. G. 2001 Hierarchies of balance conditions for the $f$-plane shallow water equations. J. Atmos. Sci. 58 (16), 2411–2426.2.0.CO;2>CrossRefGoogle Scholar
Mohebalhojeh, A. R. & Dritschel, D. G. 2004 Contour-advective semi-Lagrangian algorithms for many-layer primitive equation models. Q. J. R. Meteorol. Soc. 130, 347–364.CrossRefGoogle Scholar
Mohebalhojeh, A. R. & Dritschel, D. G. 2007 Assessing the numerical accuracy of complex spherical shallow water flows. Mon. Weath. Rev. 135 (11), 3876–3894.CrossRefGoogle Scholar
Norbury, J. & Roulstone, I. (Eds) 2002 a Large-Scale Atmosphere–Ocean Dynamics, Analytical Methods and Numerical Models, vol. 1. Cambridge University Press.CrossRefGoogle Scholar
Norbury, J. & Roulstone, I. (Eds) 2002 b Large-Scale Atmosphere–Ocean Dynamics, Geometric Methods and Models, vol. 2. Cambridge University Press.CrossRefGoogle Scholar
Pearce, J. D. & Esler, J. G. 2010 A pseudo-spectral algorithm and test cases for the numerical solution of the two-dimensional rotating Green–Naghdi shallow water equations. J. Comput. Phys. 229 (20), 7594–7608.CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Ritchie, H. 1988 Application of the semi-Lagrangian method to a spectral model of the shallow water equations. Mon. Weath. Rev. 116, 1687–1698.2.0.CO;2>CrossRefGoogle Scholar
Saint-Venant, A. J. C. 1871 Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et a l'introduction de marées dans leurs lits. C. R. Acad. Sci. 73, 147–154, 237–240.Google Scholar
Serre, F. 1953 Contibution à l’étude des écoulements permanents et variables dans les canaux. La Houille Blanche 8 (12), 830–887.CrossRefGoogle Scholar
Shields, J. J. & Webster, W. C. 1988 On direct methods in water-wave theory. J. Fluid Mech. 197, 171–199.CrossRefGoogle Scholar
Smith, R. K. & Dritschel, D. G. 2006 Revisiting the Rossby–Haurwitz wave test case with Contour Advection. J. Comput. Phys. 217 (2), 473–484.CrossRefGoogle Scholar
Su, C. H. & Gardner, C. S. 1969 Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation. J. Math. Phys. 10, 536–539.CrossRefGoogle Scholar
Vallis, G. K. 2008 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.Google Scholar
Viúdez, Á. 2001 The relation between Beltrami's material vorticity and Rossby–Ertel's potential vorticity. J. Atmos. Sci. 58, 2509–2517.2.0.CO;2>CrossRefGoogle Scholar
Waugh, D. W. & Dritschel, D. G. 1991 The stability of filamentary vorticity in two-dimensional geophysical vortex-dynamics models. J. Fluid Mech. 231, 575–598.CrossRefGoogle Scholar