Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields

Physics of Fluids

Volumn 15, Issue 2, 2003, Pages 292-297

  Dellar, Paul J ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

GEOPHYSICS; HAMILTONIANS; MAGNETIC FLUX; MAGNETOHYDRODYNAMICS; VELOCITY;

ENERGY DENSITY;

FLUID DYNAMICS;

GEOPHYSICAL FLOW; INHOMOGENEOUS FLOW; MAGNETIC FIELD; SHALLOW-WATER EQUATION; TEMPERATURE GRADIENT;

EID: 0037331168     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1530576     Document Type: Article

References (25)

1. J. Pedlosky, Geophysical Fluid Dynamics, 2nd ed. (Springer-Verlag, New York, 1987).
2. R. Salmon, Lectures on Geophysical Fluid Dynamics (Oxford University Press, Oxford, 1998).
3. P. Ripa, "On the validity of layered models of ocean dynamics and thermodynamics with reduced vertical resolution," Dyn. Atmos. Oceans 29, 1 (1999).
4. P. Ripa, "Conservation laws for primitive equations models with inhomogeneous layers," Geophys. Astrophys. Fluid Dyn. 70, 85 (1993).
5. R. L. Lavoie, "A mesoscale numerical model of lake-effect storms," J. Atmos. Sci. 29, 1025 (1972).
6. D. L. T. Anderson, "An advective mixed-layer model with applications to the diurnal cycle of the low-level East-African jet," Tellus, Ser. A 36, 278 (1984).
7. P.S. Schopf and M. A. Cane, "On equatorial dynamics, mixed layer physics and sea-surface temperature," J. Phys. Oceanogr. 13, 917 (1983).
8. R. Salmon, "Hamiltonian fluid mechanics," Annu. Rev. Fluid Mech. 20, 225 (1988).
9. P. J. Morrison, "Hamiltonian description of the ideal fluid," Rev. Mod. Phys. 70, 467 (1998).
10. T. G. Shepherd, "Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics," Adv. Geophys. 32, 287 (1990).
11. D. D. Holm, J. E. Marsden, T. Ratiu, and A. Weinstein, "Nonlinear stability of fluid and plasma equilibria," Phys. Rep. 123, 2 (1985).
12. P. J. Dellar, "Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics," Phys. Plasmas 9, 1130 (2002).
13. P. Gilman, "Magnetohydrodynamic 'shallow water' equations for the solar tachocline," Astrophys. J. Lett. 544, L79 (2000).
14. P. J. Morrison and J. M. Greene, "Erratum: Noneanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics," Phys. Rev. Lett. 48, 569 (1982).
15. P. J. Morrison, "Poisson brackets for fluids and plasmas," in Mathematical Methods in Hydrodynamics and Integrability of Dynamical Systems, edited by M. Tabor and Y. M. Treve (American Institute of Physics, New York, 1982), AIP Conf. Proc. 88, 13 (1982).
16. J. E. Marsden, T. Ratiu, and A. Weinstein, "Semidirect products and reduction in mechanics," Trans. Am. Math. Soc. 281, 147 (1984).
17. See National Technical Information Service Document No. ADA296 742/0 [P. Ripa, "Hamiltonian GFD and stability," in Geometrical Methods in Fluid Dynamics, edited by R. Salmon and B. Ewing-Deremer (Woods Hole Oceanographic Institution, Woods Hole, MA, 1994), Report of the 1993 Summer Study Program in Geophysical Fluid Dynamics, WHOI-94-12, pp. 32-36]. Copies may be ordered from the National Technical Information Service, Springfield, VA.
18. H. De Sterck, "Hyperbolic theory of the 'shallow water' magnetohydrodynamics equations," Phys. Plasmas 8, 3293 (2001).
19. P. J. Morrison and J. M. Greene, "Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics," Phys. Rev. Lett. 45, 790 (1980).
20. D. D. Holm and B. A. Kupershmidt, "Hamiltonian theory of relativistic magnetohydrodynamics with anisotropic pressure," Phys. Fluids 29, 3889 (1986).
21. R. Camassa, D. D. Holm, and J. M. Hyman, "A new integrable shallow water equation," Adv. Appl. Mech. 31, 1 (1994).
22. J. S. Allen and D. D. Holm, "Extended-geostrophic Hamiltonian models for rotating shallow water motion," Physica D 98, 229 (1996).
23. P. Ripa, "On improving a one-layer ocean model with thermodynamics," J. Fluid Mech. 303, 169 (1995).
24. D. D. Holm, J. E. Marsden, and T. S. Ratiu, "The Euler-Poincaré equations in geophysical fluid dynamics," in Large-Scale Atmosphere-Ocean Dynamics 2: Geometric Methods and Models, edited by J. Norbury and I. Roulstone (Cambridge University Press, Cambridge, 2002), chao-dyn/ 9903035.
25. R. Salmon, "Practical use of Hamilton's principle," J. Fluid Mech. 132, 431 (1983).