-
Previous Article
Frequency locking of modulated waves
- DCDS Home
- This Issue
-
Next Article
Coherent lists and chaotic sets
Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling
1. | School of Engineering and Science, Jacobs University, 28759 Bremen, Germany, Germany |
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. |
[1] |
Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015 |
[2] |
Werner Bauer, François Gay-Balmaz. Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations. Journal of Computational Dynamics, 2019, 6 (1) : 1-37. doi: 10.3934/jcd.2019001 |
[3] |
Julien Chambarel, Christian Kharif, Olivier Kimmoun. Focusing wave group in shallow water in the presence of wind. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 773-782. doi: 10.3934/dcdsb.2010.13.773 |
[4] |
Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239 |
[5] |
Anna Geyer, Ronald Quirchmayr. Shallow water models for stratified equatorial flows. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4533-4545. doi: 10.3934/dcds.2019186 |
[6] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
[7] |
Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593 |
[8] |
Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 799-818. doi: 10.3934/dcdsb.2010.13.799 |
[9] |
Daniel Guo, John Drake. A global semi-Lagrangian spectral model for the reformulated shallow water equations. Conference Publications, 2003, 2003 (Special) : 375-385. doi: 10.3934/proc.2003.2003.375 |
[10] |
Xiaoping Zhai, Hailong Ye. On global large energy solutions to the viscous shallow water equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4277-4293. doi: 10.3934/dcdsb.2020097 |
[11] |
Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327 |
[12] |
David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629 |
[13] |
Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3765-3818. doi: 10.3934/dcdsb.2018331 |
[14] |
Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103 |
[15] |
Haigang Li, Jiguang Bao. Euler-Poisson equations related to general compressible rotating fluids. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1085-1096. doi: 10.3934/dcds.2011.29.1085 |
[16] |
Bashar Khorbatly. Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022068 |
[17] |
Vikas S. Krishnamurthy. The vorticity equation on a rotating sphere and the shallow fluid approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6261-6276. doi: 10.3934/dcds.2019273 |
[18] |
Daniel Guo, John Drake. A global semi-Lagrangian spectral model of shallow water equations with time-dependent variable resolution. Conference Publications, 2005, 2005 (Special) : 355-364. doi: 10.3934/proc.2005.2005.355 |
[19] |
Andreas Hiltebrand, Siddhartha Mishra. Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography. Networks and Heterogeneous Media, 2016, 11 (1) : 145-162. doi: 10.3934/nhm.2016.11.145 |
[20] |
Aimin Huang, Roger Temam. The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2005-2038. doi: 10.3934/cpaa.2014.13.2005 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]