Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T10:25:15.464Z Has data issue: false hasContentIssue false
  • English
  • Français

Variations on a beta-plane: derivation of non-traditional beta-plane equations from Hamilton's principle on a sphere

Published online by Cambridge University Press:  14 March 2011

PAUL J. DELLAR
PAUL J. DELLAR*
Affiliation:
OCIAM, Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK
*
†Email address for correspondence: dellar@maths.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Starting from Hamilton's principle on a rotating sphere, we derive a series of successively more accurate β-plane approximations. These are Cartesian approximations to motion in spherical geometry that capture the change with latitude of the angle between the rotation vector and the local vertical. Being derived using Hamilton's principle, the different β-plane approximations each conserve energy, angular momentum and potential vorticity. They differ in their treatments of the locally horizontal component of the rotation vector, the component that is usually neglected under the traditional approximation. In particular, we derive an extended set of β-plane equations in which the locally vertical and locally horizontal components of the rotation vector both vary linearly with latitude. This was previously thought to violate conservation of angular momentum and potential vorticity. We show that the difficulty in maintaining these conservation laws arises from the need to express the rotation vector as the curl of a vector potential while approximating the true spherical metric by a flat Cartesian metric. Finally, we derive depth-averaged equations on our extended β-plane with topography, and show that they coincide with the extended non-traditional shallow-water equations previously derived in Cartesian geometry.

JFM classification

Geophysical and Geological Flows: Ocean circulation Geophysical and Geological Flows: Rotating flows Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 674 , 10 May 2011 , pp. 174 - 195
Copyright
Copyright © Cambridge University Press 2011

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

Abarbanel, H. D. I. & Holm, D. D. 1987 Nonlinear stability analysis of inviscid flows in three dimensions: incompressible fluids and barotropic fluids. Phys. Fluids 30, 3369–3382.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Cullen, M. J. P. 1993 The unified forecast/climate model. Met. Mag. 122, 81–94.Google Scholar
Cushman-Roisin, B. 1982 Motion of a free particle on a β-plane. Geophys. Astrophys. Fluid Dynam. 22, 85–102.CrossRefGoogle Scholar
Dellar, P. J. & Salmon, R. 2005 Shallow water equations with a complete Coriolis force and topography. Phys. Fluids 17, 106601.Google Scholar
Eckart, C. 1960 a Hydrodynamics of Oceans and Atmospheres. Pergamon.Google Scholar
Eckart, C. 1960 b Variation principles of hydrodynamics. Phys. Fluids 3, 421–427.Google Scholar
Fu, L.-L. 1981 Observations and models of inertial waves in the deep ocean. Rev. Geophys. 19, 141–170.CrossRefGoogle Scholar
Gates, W. L. 2004 Derivation of the equations of atmospheric motion in oblate spheroidal coordinates. J. Atmos. Sci. 61, 2478–2487.Google Scholar
Gerkema, T. & Shrira, V. I. 2005 Near-inertial waves on the non-traditional β-plane. J. Geophys. Res. 110, C10003.Google Scholar
Gerkema, T., Zimmerman, J. T. F., Maas, L. R. M. & van Haren, H. 2008 Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Rev. Geophys. 46, RG2004–33.Google Scholar
Gill, A. E. 1982 Atmosphere Ocean Dynamics. Academic Press.Google Scholar
Goldstein, H. 1980 Classical Mechanics, 2nd edn. Addison-Wesley.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, 237–246.CrossRefGoogle Scholar
Grimshaw, R. H. J. 1975 A note on the β-plane approximation. Tellus A 27, 351–357.CrossRefGoogle Scholar
van Haren, H. & Millot, C. 2005 Gyroscopic waves in the Mediterranean Sea. Geophys. Res. Lett. 32, L24614.Google Scholar
Herivel, J. W. 1955 The derivation of the equations of motion of an ideal fluid by Hamilton's principle. Proc. Camb. Phil. Soc. 51, 344–349.Google Scholar
Hinkelmann, K. H. 1969 Primitive equations. In Lectures in Numerical Short Range Weather Prediction, pp. 306–375. World Meteorological Organization, Regional Training Seminar, Moscow, WMO No. 297.Google Scholar
Holton, J. R. 1992 An Introduction to Dynamic Meteorology, 3rd edn. Academic Press.Google Scholar
Hua, B. L., Moore, D. W. & Le Gentil, S. 1997 Inertial nonlinear equilibration of equatorial flows. J. Fluid Mech. 331, 345–371.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1976 Mechanics, 3rd edn. Pergamon.Google Scholar
LeBlond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.Google Scholar
Miles, J. & Salmon, R. 1985 Weakly dispersive nonlinear gravity waves. J. Fluid Mech. 157, 519–531.Google Scholar
Müller, R. 1989 A note on the relation between the ‘traditional approximation’ and the metric of the primitive equations. Tellus A 41, 175–178.CrossRefGoogle Scholar
Munk, W. & Phillips, N. 1968 Coherence and band structure of inertial motion in the sea. Rev. Geophys. 6, 447–472.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.Google Scholar
Phillips, N. 1973 Principles of large-scale numerical weather prediction. In Dynamic Meteorology: Lectures Delivered at the Summer School of Space Physics of the Centre National d'Études Spatiales (ed. Morel, P.), pp. 3–96. D. Reidel.Google Scholar
Phillips, N. A. 1966 The equations of motion for a shallow rotating atmosphere and the traditional approximation. J. Atmos. Sci. 23, 626–628.2.0.CO;2>CrossRefGoogle Scholar
Phillips, N. A. 1968 Reply to comment by Veronis. J. Atmos. Sci. 25, 1155–1157.Google Scholar
Queney, P. 1950 Adiabatic perturbation equations for a zonal atmospheric current. Tellus 2, 35–51.Google Scholar
Ripa, P. 1981 Symmetries and conservation laws for internal gravity waves. In Nonlinear Properties of Internal Waves (ed. West, B. J.), AIP Conference Proceedings, vol. 76, pp. 281–306. AIP.Google Scholar
Ripa, P. 1982 Nonlinear wave–wave interactions in a one-layer reduced-gravity model on the equatorial beta plane. J. Phys. Oceanogr. 12, 97–111.2.0.CO;2>CrossRefGoogle Scholar
Ripa, P. 1993 Conservation laws for primitive equations models with inhomogeneous layers. Geophys. Astrophys. Fluid Dyn. 70, 85–111.Google Scholar
Ripa, P. 1997 ‘Inertial’ oscillations and the β-plane approximation(s). J. Phys. Oceanogr. 27, 633–647.Google Scholar
Rossby, C. G. & Collaborators, 1939 Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. J. Mar. Res. 2, 38–55.Google Scholar
Salmon, R. 1982 a Hamilton's principle and Ertel's theorem. In Mathematical Methods in Hydrodynamics and Integrability of Dynamical Systems (ed. Tabor, M. & Treve, Y. M.), AIP Conference Proceedings, vol. 88, pp. 127–135. AIP.Google Scholar
Salmon, R. 1982 b The shape of the main thermocline. J. Phys. Oceanogr. 12, 1458–1479.Google Scholar
Salmon, R. 1983 Practical use of Hamilton's principle. J. Fluid Mech. 132, 431–444.Google Scholar
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20, 225–256.CrossRefGoogle Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.Google Scholar
Serrin, J. 1959 Mathematical principles of classical fluid mechanics. In Handbuch der Physik, vol. 8/1, pp. 125–263. Springer.Google Scholar
Shutts, G. 1989 Planetary semi-geostrophic equations derived from Hamilton's principle. J. Fluid Mech. 208, 545–573.CrossRefGoogle Scholar
Stewart, A. L. & Dellar, P. J. 2010 Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. J. Fluid Mech. 651, 387–413.Google Scholar
Thuburn, J., Wood, N. & Staniforth, A. 2002 Normal modes of deep atmospheres. I: Spherical geometry. Q. J. R. Meteorol. Soc. 128, 1771–1792.Google Scholar
van der Toorn, R. & Zimmerman, J. T. F. 2008 On the spherical approximation of the geopotential in geophysical fluid dynamics and the use of a spherical coordinate system. Geophys. Astrophys. Fluid Dyn. 102, 349–371.CrossRefGoogle Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.Google Scholar
Verkley, W. T. M. 1990 On the beta plane approximation. J. Atmos. Sci. 47, 2453–2459.Google Scholar
Veronis, G. 1963 On the approximations involved in transforming the equations of motion from a spherical surface to the β-plane. I. Barotropic systems. J. Mar. Res. 21, 110–124.Google Scholar
Veronis, G. 1968 Comment on Phillips' proposed simplification of the equations of motion for a shallow rotating atmosphere. J. Atmos. Sci. 25, 1154–1155.2.0.CO;2>CrossRefGoogle Scholar
Veronis, G. 1973 Large-scale ocean circulation. Adv. Appl. Mech. 13, 1–92.Google Scholar
Veronis, G. 1981 Dynamics of large-scale ocean circulation. In Evolution of Physical Oceanography (ed. Warren, B. A. & Wunsch, C.), pp. 140–183. Massachusetts Institute of Technology Press.Google Scholar
White, A. A. 2002 A view of the equations of meteorological dynamics and various approximations. In Large-Scale Atmosphere–Ocean Dynamics 1: Analytical Methods and Numerical Models (ed. Norbury, J. & Roulstone, I.), pp. 1–100. Cambridge University Press.Google Scholar
White, A. A. & Bromley, R. A. 1995 Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force. Q. J. R. Meteorol. Soc. 121, 399–418.Google Scholar
White, A. A., Hoskins, B. J., Roulstone, I. & Staniforth, A. 2005 Consistent approximate models of the global atmosphere: shallow, deep, hydrostatic, quasi-hydrostatic and non-hydrostatic. Q. J. R. Meteorol. Soc. 131, 2081–2107.Google Scholar
Zdunkowski, W. & Bott, A. 2003 Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press.CrossRefGoogle Scholar