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February  2014, 34(2): 477-509. doi: 10.3934/dcds.2014.34.477

Variational discretization for rotating stratified fluids

1. 

Applied Geometry Lab, Computing+Mathematical Sciences, Caltech, 1200 E. California Blvd, Pasadena, CA 91125,, United States

2. 

Computational and Mathematical Engineering, Stanford University, 450 Serra Mall, Stanford, CA 94305-2004, United States

3. 

CNRS/LMD, École Normale Supérieure, Paris,, France

4. 

LMD, École Normale Supérieure, UPMC, Paris, France

Received  March 2013 Revised  April 2013 Published  August 2013

In this paper we develop and test a structure-preserving discretization scheme for rotating and/or stratified fluid dynamics. The numerical scheme is based on a finite dimensional approximation of the group of volume preserving diffeomorphisms recently proposed in [25,9] and is derived via a discrete version of the Euler-Poincaré variational formulation of rotating stratified fluids. The resulting variational integrator allows for a discrete version of Kelvin circulation theorem, is applicable to irregular meshes and, being symplectic, exhibits excellent long term energy behavior. We then report a series of preliminary tests for rotating stratified flows in configurations that are symmetric with respect to translation along one of the spatial directions. In the benchmark processes of hydrostatic and/or geostrophic adjustments, these tests show that the slow and fast component of the flow are correctly reproduced. The harder test of inertial instability is in full agreement with the common knowledge of the process of development and saturation of this instability, while preserving energy nearly perfectly and respecting conservation laws.
Citation: Mathieu Desbrun, Evan S. Gawlik, François Gay-Balmaz, Vladimir Zeitlin. Variational discretization for rotating stratified fluids. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 477-509. doi: 10.3934/dcds.2014.34.477
References:
[1]

R. V. Abramov and A. J. Majda, Statistically relevant conserved quantities for truncated quasigeostrophic flow, Proc. Natl. Acad. Sci. USA., 100 (2003), 3841-3846. doi: 10.1073/pnas.0230451100.

[2]

V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.

[3]

V. I. Arnold, "Mathematical Methods in Classical Mechanics," Springer, New York, 1974.

[4]

D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica, 15 (2006), 1-155. doi: 10.1017/S0962492906210018.

[5]

V. I. Arnold and B. A. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.

[6]

A. Bossavit, "Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements," Electromagnetism, Academic Press, Inc., San Diego, CA, 1998.

[7]

N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin Integrators on Lie Groups. Part I: Introduction and Structure-Preserving Properties, Foundations of Computational Mathematics, 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4.

[8]

M. Desbrun, E. Kanso and Y. Tong, Discrete differential forms for computational modeling, in "Discrete Differential Geometry," Oberwolfach Semin., 38, Birkhäuser, Basel, (2008), 287-324. doi: 10.1007/978-3-7643-8621-4_16.

[9]

E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D, 240 (2011), 1724-1760. doi: 10.1016/j.physd.2011.07.011.

[10]

I. Gjaja and D. D. Holm, Self-consistent Hamiltonian dynamics of wave mean-flow interaction for a rotating stratified incompressible fluid, Physica D, 98 (1996), 343-378. doi: 10.1016/0167-2789(96)00104-2.

[11]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations," Second edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006.

[12]

F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Physics of Fluids, 8 (1965), 2182-2189. doi: 10.1063/1.1761178.

[13]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.

[14]

D. D. Holm and V. Zeitlin, Hamilton's principle for quasigeostrophic motion, Phys. Fluids, 10 (1998), 800-806. doi: 10.1063/1.869623.

[15]

J. R. Holton, "An Introduction to Dynamic Meteorology," Third edition, Academic Press, 1992. doi: 10.1119/1.1987371.

[16]

B. J. Hoskins, M. E. McIntyre and A. W. Robertson, On the use and significance of isentropic potential vorticity maps, Q. J. R. Met. Soc., 111 (1985), 877-946. doi: 10.1002/qj.49711147002.

[17]

H. Lamb, "Hydrodynamics," Ch. 309, 310, Dover, 1932.

[18]

J. Lighthill, "Waves in Fluids," Ch. 4, Cambridge University Press, Cambridge-New York, 1978.

[19]

B. Kadar, I. Szunyogh and Q. J. Devenyi, On the origin of model errors, J. Hung. Meteor. Soc., 101 (1998), 71-107.

[20]

R. C. Kloosterziel, P. Orlandi and G. F. Carnevale, Saturation of inertial instability in rotating planar shear flows, J. Fluid Mech., 583 (2007), 413-422. doi: 10.1017/S0022112007006593.

[21]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems," Second edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.

[22]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[23]

J. Marshall and F. Molteni, Toward a dynamical understanding of planetary-scale flow regimes, J. Atmos. Sci., 50 (1993), 1792-1818.

[24]

S. Medvedev and V. Zeitlin, Parallels between stratification and rotation in hydrodynamics, and between both of them and external magnetic field in magnetohydrodynamics, with applications to nonlinear waves, in "IUTAM Symposium on Turbulence in the Atmosphere and Oceans," IUTAM Bookseries, 28, Springer, Netherlands, (2010), 27-37. doi: 10.1007/978-94-007-0360-5_3.

[25]

D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden and M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D, 240 (2011), 443-458. doi: 10.1016/j.physd.2010.10.012.

[26]

J. Pedlosky, "Geophysical Fluid Dynamics," Springer, NY, 1979.

[27]

R. Plougonven and V. Zeiltin, Nonlinear development of inertial instability in a barotropic shear, Physics of Fluids, 21 (2009), 106601. doi: 10.1063/1.3242283.

[28]

R. Salmon, "Lectures on Geophysical Fluid Dynamics," Oxford University Press, New York, 1998.

[29]

V. Zeitlin, G. M. Reznik and M. Ben Jelloul, Nonlinear theory of geostrophic adjustment. Part 2. Two-layer and continuously stratified primitive equations, J. Fluid Mech., 491 (2003), 207-228. doi: 10.1017/S0022112003005457.

show all references

References:
[1]

R. V. Abramov and A. J. Majda, Statistically relevant conserved quantities for truncated quasigeostrophic flow, Proc. Natl. Acad. Sci. USA., 100 (2003), 3841-3846. doi: 10.1073/pnas.0230451100.

[2]

V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.

[3]

V. I. Arnold, "Mathematical Methods in Classical Mechanics," Springer, New York, 1974.

[4]

D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica, 15 (2006), 1-155. doi: 10.1017/S0962492906210018.

[5]

V. I. Arnold and B. A. Khesin, "Topological Methods in Hydrodynamics," Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998.

[6]

A. Bossavit, "Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements," Electromagnetism, Academic Press, Inc., San Diego, CA, 1998.

[7]

N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin Integrators on Lie Groups. Part I: Introduction and Structure-Preserving Properties, Foundations of Computational Mathematics, 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4.

[8]

M. Desbrun, E. Kanso and Y. Tong, Discrete differential forms for computational modeling, in "Discrete Differential Geometry," Oberwolfach Semin., 38, Birkhäuser, Basel, (2008), 287-324. doi: 10.1007/978-3-7643-8621-4_16.

[9]

E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D, 240 (2011), 1724-1760. doi: 10.1016/j.physd.2011.07.011.

[10]

I. Gjaja and D. D. Holm, Self-consistent Hamiltonian dynamics of wave mean-flow interaction for a rotating stratified incompressible fluid, Physica D, 98 (1996), 343-378. doi: 10.1016/0167-2789(96)00104-2.

[11]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations," Second edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006.

[12]

F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Physics of Fluids, 8 (1965), 2182-2189. doi: 10.1063/1.1761178.

[13]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.

[14]

D. D. Holm and V. Zeitlin, Hamilton's principle for quasigeostrophic motion, Phys. Fluids, 10 (1998), 800-806. doi: 10.1063/1.869623.

[15]

J. R. Holton, "An Introduction to Dynamic Meteorology," Third edition, Academic Press, 1992. doi: 10.1119/1.1987371.

[16]

B. J. Hoskins, M. E. McIntyre and A. W. Robertson, On the use and significance of isentropic potential vorticity maps, Q. J. R. Met. Soc., 111 (1985), 877-946. doi: 10.1002/qj.49711147002.

[17]

H. Lamb, "Hydrodynamics," Ch. 309, 310, Dover, 1932.

[18]

J. Lighthill, "Waves in Fluids," Ch. 4, Cambridge University Press, Cambridge-New York, 1978.

[19]

B. Kadar, I. Szunyogh and Q. J. Devenyi, On the origin of model errors, J. Hung. Meteor. Soc., 101 (1998), 71-107.

[20]

R. C. Kloosterziel, P. Orlandi and G. F. Carnevale, Saturation of inertial instability in rotating planar shear flows, J. Fluid Mech., 583 (2007), 413-422. doi: 10.1017/S0022112007006593.

[21]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems," Second edition, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.

[22]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[23]

J. Marshall and F. Molteni, Toward a dynamical understanding of planetary-scale flow regimes, J. Atmos. Sci., 50 (1993), 1792-1818.

[24]

S. Medvedev and V. Zeitlin, Parallels between stratification and rotation in hydrodynamics, and between both of them and external magnetic field in magnetohydrodynamics, with applications to nonlinear waves, in "IUTAM Symposium on Turbulence in the Atmosphere and Oceans," IUTAM Bookseries, 28, Springer, Netherlands, (2010), 27-37. doi: 10.1007/978-94-007-0360-5_3.

[25]

D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden and M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D, 240 (2011), 443-458. doi: 10.1016/j.physd.2010.10.012.

[26]

J. Pedlosky, "Geophysical Fluid Dynamics," Springer, NY, 1979.

[27]

R. Plougonven and V. Zeiltin, Nonlinear development of inertial instability in a barotropic shear, Physics of Fluids, 21 (2009), 106601. doi: 10.1063/1.3242283.

[28]

R. Salmon, "Lectures on Geophysical Fluid Dynamics," Oxford University Press, New York, 1998.

[29]

V. Zeitlin, G. M. Reznik and M. Ben Jelloul, Nonlinear theory of geostrophic adjustment. Part 2. Two-layer and continuously stratified primitive equations, J. Fluid Mech., 491 (2003), 207-228. doi: 10.1017/S0022112003005457.

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