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

Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling

1. 

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

Received  February 2010 Revised  July 2011 Published  August 2011

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.
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," 2nd 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," 2nd 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," 2nd 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," 2nd 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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