Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling

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September 2011, 31(3): 827-846. doi: 10.3934/dcds.2011.31.827

Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling

Marcel Oliver ^1, and Sergiy Vasylkevych ^1,

1.	School of Engineering and Science, Jacobs University, 28759 Bremen, Germany, Germany

Received February 2010 Revised July 2011 Published August 2011

Abstract
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This paper presents a first rigorous study of the so-called large-scale semigeostrophic equations which were first introduced by R. Salmon in 1985 and later generalized by the first author. We show that these models are Hamiltonian on the group of $H^s$ diffeomorphisms for $s>2$. Notably, in the Hamiltonian setting an apparent topological restriction on the Coriolis parameter disappears. We then derive the corresponding Hamiltonian formulation in Eulerian variables via Poisson reduction and give a simple argument for the existence of $H^s$ solutions locally in time.

Keywords: balance models, Semigeostrophic equations, rotating shallow water, Poisson reduction., diffeomorphism group.

Mathematics Subject Classification: Primary: 35Q35, 35A07; Secondary: 76B03, 76U0.

Citation: Marcel Oliver, Sergiy Vasylkevych. Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 827-846. doi: 10.3934/dcds.2011.31.827

References:

[1]	R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2^nd edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
[2]	V. I. Arnold, Sur la géométrie differentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluids parfaits, (French) [On the differential geometry of infinite dimensional Lie groups and its applications], Ann. I. Fourier (Grenoble), 16 (1966), 319-361.
[3]	V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.
[4]	J.-D. Benamou and Y. Brenier, Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère transport problem, SIAM J. Appl. Math., 58 (1998), 1450-1461. doi: 10.1137/S0036139995294111.
[5]	M. Çalik, M. Oliver and S. Vasylkevych, Global well-posedness for models of rotating shallow water in semigeostrophic scaling, submitted for publication, 2010.
[6]	P. R. Chernoff and J. E. Marsden, "Properties of Infinite Dimensional Hamiltonian Systems," Lecture Notes in Mathematics, 425, Springer-Verlag, Berlin-New York, 1974.
[7]	M. J. P. Cullen and W. Gangbo, A variational approach for the 2-dimensional semi-geostrophic shallow water equations, Arch. Rational Mech. Anal., 156 (2001), 241-273. doi: 10.1007/s002050000124.
[8]	D. Ebin, The manifold of Riemannian metrics, in "Global Analysis" (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), AMS, Providence, RI, (1970), 11-40.
[9]	D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92 (1970), 102-163. doi: 10.2307/1970699.
[10]	A. Eliassen, The quasi-static equations of motion with pressure as an independent variable, Geofys. Publ. Norske Vid.-Akad. Oslo, 17 (1949), 1-44.
[11]	A. Eliassen, On the vertical circulation in frontal zones, Geofys. Publ., 24 (1962), 147-160.
[12]	B. J. Hoskins, The geostrophic momentum approximation and the semi-geostrophic equations, J. Atmos. Sci., 32 (1975), 233-242. doi: 10.1175/1520-0469(1975)032<0233:TGMAAT>2.0.CO;2.
[13]	J. Isenberg and J. E. Marsden, A slice theorem for the space of solutions of Einstein's equations, Phys. Rep., 89 (1982), 179-222. doi: 10.1016/0370-1573(82)90066-7.
[14]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems," 2^nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.
[15]	M. Oliver, Classical solutions for a generalized Euler equations in two dimensions, J. Math. Anal. Appl., 215 (1997), 471-484. doi: 10.1006/jmaa.1997.5647.
[16]	M. Oliver, Variational asymptotics for rotating shallow water near geostrophy: A transformational approach, J. Fluid Mech., 551 (2006), 197-234. doi: 10.1017/S0022112005008256.
[17]	M. Oliver and S. Vasylkevych, Generalized LSG models with variable Coriolis parameter, submitted for publication, 2011.
[18]	R. Palais, "Foundations of Global Non-Linear Analysis," W. A. Benjamin, Inc., New York-Amsterdam, 1968.
[19]	I. Roulston and M. J. Sewell, The Mathematical structure of theories of semigeostrophic type, Philos. Trans. Roy. Soc. London Ser. A, 355 (1997), 2489-2517. doi: 10.1098/rsta.1997.0144.
[20]	R. Salmon, New equations for nearly geostrophic flow, J. Fluid Mech., 153 (1985), 461-477. doi: 10.1017/S0022112085001343.
[21]	R. Salmon, Large-scale semi-geostrophic equations for use in ocean circulation models, J. Fluid Mech., 318 (1996), 85-105. doi: 10.1017/S0022112096007045.
[22]	R. Salmon, "Lectures on Geophysical Fluid Dynamics," Oxford University Press, New York, 1998.
[23]	S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics, J. Funct. Anal., 160 (1998), 337-365. doi: 10.1006/jfan.1998.3335.
[24]	R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20 (1975), 32-43. doi: 10.1016/0022-1236(75)90052-X.
[25]	S. Vasylkevych and J. E. Marsden, The Lie-Poisson structure of the Euler equations of an ideal fluid, Dynam. Part. Differ. Eq., 2 (2005), 281-300.

show all references

References:

[1]	R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2^nd edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
[2]	V. I. Arnold, Sur la géométrie differentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluids parfaits, (French) [On the differential geometry of infinite dimensional Lie groups and its applications], Ann. I. Fourier (Grenoble), 16 (1966), 319-361.
[3]	V. I. Arnold and B. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.
[4]	J.-D. Benamou and Y. Brenier, Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère transport problem, SIAM J. Appl. Math., 58 (1998), 1450-1461. doi: 10.1137/S0036139995294111.
[5]	M. Çalik, M. Oliver and S. Vasylkevych, Global well-posedness for models of rotating shallow water in semigeostrophic scaling, submitted for publication, 2010.
[6]	P. R. Chernoff and J. E. Marsden, "Properties of Infinite Dimensional Hamiltonian Systems," Lecture Notes in Mathematics, 425, Springer-Verlag, Berlin-New York, 1974.
[7]	M. J. P. Cullen and W. Gangbo, A variational approach for the 2-dimensional semi-geostrophic shallow water equations, Arch. Rational Mech. Anal., 156 (2001), 241-273. doi: 10.1007/s002050000124.
[8]	D. Ebin, The manifold of Riemannian metrics, in "Global Analysis" (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), AMS, Providence, RI, (1970), 11-40.
[9]	D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92 (1970), 102-163. doi: 10.2307/1970699.
[10]	A. Eliassen, The quasi-static equations of motion with pressure as an independent variable, Geofys. Publ. Norske Vid.-Akad. Oslo, 17 (1949), 1-44.
[11]	A. Eliassen, On the vertical circulation in frontal zones, Geofys. Publ., 24 (1962), 147-160.
[12]	B. J. Hoskins, The geostrophic momentum approximation and the semi-geostrophic equations, J. Atmos. Sci., 32 (1975), 233-242. doi: 10.1175/1520-0469(1975)032<0233:TGMAAT>2.0.CO;2.
[13]	J. Isenberg and J. E. Marsden, A slice theorem for the space of solutions of Einstein's equations, Phys. Rep., 89 (1982), 179-222. doi: 10.1016/0370-1573(82)90066-7.
[14]	J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems," 2^nd edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.
[15]	M. Oliver, Classical solutions for a generalized Euler equations in two dimensions, J. Math. Anal. Appl., 215 (1997), 471-484. doi: 10.1006/jmaa.1997.5647.
[16]	M. Oliver, Variational asymptotics for rotating shallow water near geostrophy: A transformational approach, J. Fluid Mech., 551 (2006), 197-234. doi: 10.1017/S0022112005008256.
[17]	M. Oliver and S. Vasylkevych, Generalized LSG models with variable Coriolis parameter, submitted for publication, 2011.
[18]	R. Palais, "Foundations of Global Non-Linear Analysis," W. A. Benjamin, Inc., New York-Amsterdam, 1968.
[19]	I. Roulston and M. J. Sewell, The Mathematical structure of theories of semigeostrophic type, Philos. Trans. Roy. Soc. London Ser. A, 355 (1997), 2489-2517. doi: 10.1098/rsta.1997.0144.
[20]	R. Salmon, New equations for nearly geostrophic flow, J. Fluid Mech., 153 (1985), 461-477. doi: 10.1017/S0022112085001343.
[21]	R. Salmon, Large-scale semi-geostrophic equations for use in ocean circulation models, J. Fluid Mech., 318 (1996), 85-105. doi: 10.1017/S0022112096007045.
[22]	R. Salmon, "Lectures on Geophysical Fluid Dynamics," Oxford University Press, New York, 1998.
[23]	S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics, J. Funct. Anal., 160 (1998), 337-365. doi: 10.1006/jfan.1998.3335.
[24]	R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20 (1975), 32-43. doi: 10.1016/0022-1236(75)90052-X.
[25]	S. Vasylkevych and J. E. Marsden, The Lie-Poisson structure of the Euler equations of an ideal fluid, Dynam. Part. Differ. Eq., 2 (2005), 281-300.

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