• Previous Article
    The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body
  • DCDS Home
  • This Issue
  • Next Article
    Infinitely many radial solutions to elliptic systems involving critical exponents
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 & Continuous Dynamical Systems - A, 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. doi: 10.1073/pnas.0230451100. Google Scholar

[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. doi: 10.5802/aif.233. Google Scholar

[3]

V. I. Arnold, "Mathematical Methods in Classical Mechanics,", Springer, (1974). Google Scholar

[4]

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

[5]

V. I. Arnold and B. A. Khesin, "Topological Methods in Hydrodynamics,", Applied Mathematical Sciences, 125 (1998). Google Scholar

[6]

A. Bossavit, "Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements,", Electromagnetism, (1998). Google Scholar

[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. doi: 10.1007/s10208-008-9030-4. Google Scholar

[8]

M. Desbrun, E. Kanso and Y. Tong, Discrete differential forms for computational modeling,, in, 38 (2008), 287. doi: 10.1007/978-3-7643-8621-4_16. Google Scholar

[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. doi: 10.1016/j.physd.2011.07.011. Google Scholar

[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. doi: 10.1016/0167-2789(96)00104-2. Google Scholar

[11]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations,", Second edition, 31 (2006). Google Scholar

[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. doi: 10.1063/1.1761178. Google Scholar

[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. doi: 10.1006/aima.1998.1721. Google Scholar

[14]

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

[15]

J. R. Holton, "An Introduction to Dynamic Meteorology,", Third edition, (1992). doi: 10.1119/1.1987371. Google Scholar

[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. doi: 10.1002/qj.49711147002. Google Scholar

[17]

H. Lamb, "Hydrodynamics,", Ch. 309, (1932). Google Scholar

[18]

J. Lighthill, "Waves in Fluids,", Ch. 4, (1978). Google Scholar

[19]

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

[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. doi: 10.1017/S0022112007006593. Google Scholar

[21]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,", Second edition, 17 (1999). Google Scholar

[22]

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

[23]

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

[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, 28 (2010), 27. doi: 10.1007/978-94-007-0360-5_3. Google Scholar

[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. doi: 10.1016/j.physd.2010.10.012. Google Scholar

[26]

J. Pedlosky, "Geophysical Fluid Dynamics,", Springer, (1979). Google Scholar

[27]

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

[28]

R. Salmon, "Lectures on Geophysical Fluid Dynamics,", Oxford University Press, (1998). Google Scholar

[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. doi: 10.1017/S0022112003005457. Google Scholar

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. doi: 10.1073/pnas.0230451100. Google Scholar

[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. doi: 10.5802/aif.233. Google Scholar

[3]

V. I. Arnold, "Mathematical Methods in Classical Mechanics,", Springer, (1974). Google Scholar

[4]

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

[5]

V. I. Arnold and B. A. Khesin, "Topological Methods in Hydrodynamics,", Applied Mathematical Sciences, 125 (1998). Google Scholar

[6]

A. Bossavit, "Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements,", Electromagnetism, (1998). Google Scholar

[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. doi: 10.1007/s10208-008-9030-4. Google Scholar

[8]

M. Desbrun, E. Kanso and Y. Tong, Discrete differential forms for computational modeling,, in, 38 (2008), 287. doi: 10.1007/978-3-7643-8621-4_16. Google Scholar

[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. doi: 10.1016/j.physd.2011.07.011. Google Scholar

[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. doi: 10.1016/0167-2789(96)00104-2. Google Scholar

[11]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations,", Second edition, 31 (2006). Google Scholar

[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. doi: 10.1063/1.1761178. Google Scholar

[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. doi: 10.1006/aima.1998.1721. Google Scholar

[14]

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

[15]

J. R. Holton, "An Introduction to Dynamic Meteorology,", Third edition, (1992). doi: 10.1119/1.1987371. Google Scholar

[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. doi: 10.1002/qj.49711147002. Google Scholar

[17]

H. Lamb, "Hydrodynamics,", Ch. 309, (1932). Google Scholar

[18]

J. Lighthill, "Waves in Fluids,", Ch. 4, (1978). Google Scholar

[19]

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

[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. doi: 10.1017/S0022112007006593. Google Scholar

[21]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,", Second edition, 17 (1999). Google Scholar

[22]

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

[23]

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

[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, 28 (2010), 27. doi: 10.1007/978-94-007-0360-5_3. Google Scholar

[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. doi: 10.1016/j.physd.2010.10.012. Google Scholar

[26]

J. Pedlosky, "Geophysical Fluid Dynamics,", Springer, (1979). Google Scholar

[27]

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

[28]

R. Salmon, "Lectures on Geophysical Fluid Dynamics,", Oxford University Press, (1998). Google Scholar

[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. doi: 10.1017/S0022112003005457. Google Scholar

[1]

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

[2]

Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146

[3]

Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093

[4]

Jeffrey K. Lawson, Tanya Schmah, Cristina Stoica. Euler-Poincaré reduction for systems with configuration space isotropy. Journal of Geometric Mechanics, 2011, 3 (2) : 261-275. doi: 10.3934/jgm.2011.3.261

[5]

Emanuel-Ciprian Cismas. Euler-Poincaré-Arnold equations on semi-direct products II. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5993-6022. doi: 10.3934/dcds.2016063

[6]

Young-Sam Kwon, Antonin Novotny. Derivation of geostrophic equations as a rigorous limit of compressible rotating and heat conducting fluids with the general initial data. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 395-421. doi: 10.3934/dcds.2020015

[7]

Haigang Li, Jiguang Bao. Euler-Poisson equations related to general compressible rotating fluids. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1085-1096. doi: 10.3934/dcds.2011.29.1085

[8]

Bruce Hughes. Geometric topology of stratified spaces. Electronic Research Announcements, 1996, 2: 73-81.

[9]

Andrei Cozma, Christoph Reisinger. Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3359-3377. doi: 10.3934/dcdsb.2016101

[10]

Peter Constantin. Transport in rotating fluids. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 165-176. doi: 10.3934/dcds.2004.10.165

[11]

D. Bresch, B. Desjardins, D. Gérard-Varet. Rotating fluids in a cylinder. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 47-82. doi: 10.3934/dcds.2004.11.47

[12]

Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473

[13]

Marin Kobilarov, Jerrold E. Marsden, Gaurav S. Sukhatme. Geometric discretization of nonholonomic systems with symmetries. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 61-84. doi: 10.3934/dcdss.2010.3.61

[14]

Yao Xu, Weisheng Niu. Periodic homogenization of elliptic systems with stratified structure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2295-2323. doi: 10.3934/dcds.2019097

[15]

Vincent Giovangigli, Lionel Matuszewski. Structure of entropies in dissipative multicomponent fluids. Kinetic & Related Models, 2013, 6 (2) : 373-406. doi: 10.3934/krm.2013.6.373

[16]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013

[17]

Chjan C. Lim, Junping Shi. The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 717-740. doi: 10.3934/dcdsb.2009.11.717

[18]

Zoltán Horváth, Yunfei Song, Tamás Terlaky. Steplength thresholds for invariance preserving of discretization methods of dynamical systems on a polyhedron. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2997-3013. doi: 10.3934/dcds.2015.35.2997

[19]

Van-Sang Ngo, Stefano Scrobogna. Dispersive effects of weakly compressible and fast rotating inviscid fluids. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 749-789. doi: 10.3934/dcds.2018033

[20]

Miroslav Bulíček, Eduard Feireisl, Josef Málek, Roman Shvydkoy. On the motion of incompressible inhomogeneous Euler-Korteweg fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 497-515. doi: 10.3934/dcdss.2010.3.497

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (0)

[Back to Top]