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Statistical mechanics and numerical modelling of geophysical fluid dynamics
Dubinkina, S.B.
Publication date
2010
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Citation for published version (APA):
Dubinkina, S. B. (2010). Statistical mechanics and numerical modelling of geophysical fluid
dynamics.
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Bibliography
[1℄ Abramov,R. andMajda, A.J.,Statistically relevant conserved quantities
for truncated quasi-geostrophic flow.Pro .Natl.A ad.S i.U.S.A.100(7) (2003),38413846.
[2℄ Arakawa,A.,Computational design for long-term numerical integration of
the equations of fluid motion: two-dimensional incompressible flow. Part I.J.Comput.Phys.1(1966),119143.
[3℄ Arnold, V.I., Mathematical methods of classical mechanics. Springer-Verlag,NewYork,se onded., 1989.
[4℄ Benettin,G.andGiorgilli,A.,On the Hamiltonian interpolation of near to
the identity symplectic mappings with application to symplectic integration algorithms.J.Stat.Phys.74(1994),11171143.
[5℄ Bokhove, O. andOliver,M., Parcel Eulerian-Lagrangian fluid dynamics
of rotating geophysical flows.Pro .R.So .Lond.Ser.AMath.Phys.Eng. S i.462(2073) (2006),25752592.
[6℄ Bond,S.D.,Leimkuhler,B.J.andLaird,B.B.,The Nos´e-Poincar´e method
for constant temperature molecular dynamics.J. Comput. Phys. 151 (1) (1999),114134.
[7℄ Bridges, Th.J. and Rei h, S., Multi-symplectic integrators: numerical
schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A284(2001),184193.
[8℄ Bulga , A. and Kusnezov, D., Canonical ensemble averages from
pseu-domicrocanonical dynamics.Phys.Rev.A 42(1990),50455048.
[9℄ Bühler, O., Statistical mechanics of strong and weak point vortices in a
cylinder.Physi sofFluids14(2002),21392149.
[10℄ Bühler, O., A Brief Introduction to Classical, Statistical, and Quantum
Mechanics. Courant Le ture Notes, vol. 13, AMS Bookstore, Rhode Is-land,2006.
[11℄ Carnevale,G.F.andFrederiksen,J.S., Nonlinear Stability and Statistical
Mechanics of Flow Over Topography.J.FluidMe h.175(1987),157181. [12℄ Chartier, P. and Faou, E., Geometric integrators for piecewise smooth
98 Bibliography
[13℄ Chavanis,P.H.,Statistical mechanics of 2D turbulence with a prior
vortic-ity distribution.Phys.DNonlinearPhenomena237(14-17)(2008),1998 2002.
[14℄ Cohen, D., Hairer, E. and Lubi h, C. Conservation of energy,
momen-tum and actions in numerical discretizations of nonlinear wave equations.
Numer.Math. 110(2008),113143.
[15℄ Cottet,G.H.andKoumoutsakos,P.D.,Vortex Methods: Theory and
Prac-tice.CambridgeUniversityPress,Cambridge,2000.
[16℄ Cotter,C.,Frank,J.andRei h,S.,Hamiltonian particle-mesh method for
two-layer shallow-water equations subject to the rigid-lid approximation.
SIAMJ.Appl.Dyn.Syst.3(1)(2004),6983(ele troni ).
[17℄ Cotter,C.andRei h,S.,Geometric Integration of a Wave-Vortex Model. Appl.Numer.Math. 48(2004),293305.
[18℄ Dubinkina,S.andFrank,J.,Statistical mechanics of Arakawa’s
discretiza-tions.J.Comput.Phys.227(2007),12861305.
[19℄ Dubinkina,S.andFrank,J.,Statistical relevance of vorticity conservation
with the Hamiltonian Particle-Mesh method. J. Comput. Phys. (2010), publishedonline: doi:10.1016/j.j p.2009.12.012.
[20℄ Dubinkina, S., Frank, J. and Leimkuhler, B., A thermostat closure for
point vortices.submitted(2009).
[21℄ Durran,D.R.,Numerical methods for wave equations in geophysical fluid
dynamics. Texts in Applied Mathemati s, vol. 32, Springer-Verlag, New York, 1999.
[22℄ Ellis,R.S.,Haven,K.and Turkington,B.,Nonequivalent statistical
equi-librium ensembles and refined stability theorems for most probable flows.
Nonlinearity15(2)(2002),239255.
[23℄ Feynman, R.P., Statistical mechanics. Perseus Books, Advan ed Book Program,Reading, MA,1998.
[24℄ Frank, J., Geometric space-time integration of ferromagnetic materials. Appl.Numer.Math. 48(3-4)(2004),307322.
[25℄ Frank, J., Gottwald, G. and Rei h, S., A Hamiltonian particle-mesh
method for the rotating shallow-water equations.InMeshfree methods for
partial differential equations,Le t.NotesComput.S i.Eng..26,Springer, Berlin,2003,131142.
[26℄ Frank,J.,Huang,W.andLeimkuhler,B.,Geometric integrators for
[27℄ Frank,J.,Moore,B.E.andRei h,S.Linear PDEs and numerical methods
that preserve a multisymplectic conservation law.SIAMJ. S i. Comput. 28(1) (2006),260277(ele troni ).
[28℄ Frank,J.andRei h,S.,Conservation properties of smoothed particle
hy-drodynamics applied to the shallow water equation. BIT 43 (1) (2003), 4155.
[29℄ Frank, J.and Rei h, S., The Hamiltonian Particle-Mesh Method for the
Spherical Shallow Water Equations.Atmospheri S ien eLetters5(2004), 8995.
[30℄ Frenkel, D.and Smit,B.,Understanding Molecular Simulation: from
al-gorithms to applications.A ademi Press,SanDiego,2002.
[31℄ Frutos,J.,Ortega,T.andSanz-Serna,J.M.,A Hamiltonian, explicit
algo-rithm with spectral accuracy for the ’good’ Boussinesq equation.Comput. MethodsAppl.Me h.Egrg.80(1990),417423.
[32℄ Frutos, J. and Sanz-Serna, J.M., An easily implementable fourth-order
method for the time integration of wave problems.J.Comput. Phys.103 (1992),160168.
[33℄ Ge,Z. and Marsden J.E., Poisson Hamilton-Jacobi theory and
Lie-Poisson integrators.Phys.Lett.A133(1988),134139.
[34℄ Hairer, E., Backward analysis of numerical integrators and symplectic
methods.Annals ofNumer.Math. 1(1994),107132.
[35℄ Hairer, E., Lubi h, C. and Wanner, G., Geometric Numerical
Integra-tion: Structure-Preserving Algorithms for Ordinary Differential Equa-tions. Springer Series in Computational Mathemati s, se ond ed., vol. 31,Springer-Verlag,Berlin,2006.
[36℄ Hairer, E., Nørsett, S.P. and Wanner, G., Solving Ordinary Differential
Equations I: Nonstiff Problems.SpringerSeries inComputational Math-emati s,vol.8,se onded.,Springer-Verlag,Berlin,1993.
[37℄ Hoover, W.G., Canonical dynamics: Equilibrium phase-space
distribu-tions.Phys.Rev.A 31(1985),16951697.
[38℄ Hundsdorfer, W. and Verwer, J., Numerical solution of time-dependent
advection-diffusion-reaction equations. SpringerSeries in Computational Mathemati s,vol.33,Springer-Verlag,Berlin,2003.
[39℄ Khin hin, A.I.,Mathematical Foundations of Statistical Mechanics.New York: Dover,1960.
100 Bibliography
[41℄ Kullba k, S. and Leibler, R.A., On Information and Sufficiency. Ann. Math.Statist.22(1)(1951),7986.
[42℄ Lan zos,C., The Variational Principles of Mechanics. Toronto: Univer-sityofTorontoPress,1970.
[43℄ Leimkuhler, B., Generalized Bulgac-Kusnezov Methods for Sampling of
the Gibbs-Boltzmann Measure.Inpreparation,2009.
[44℄ Leimkuhler,B.,Noorizadeh,E. andTheil,F.,A Gentle Stochastic
Ther-mostat for Molecular Dynamics.J.Stat.Phys.135 (2)(2009),261277. [45℄ Leimkuhler,B. and Rei h, S., Simulating Hamiltonian Dynamics,
Cam-bridgeMonographsonAppliedandComputationalMathemati s,vol.14, CambridgeUniversityPress,Cambridge,2004.
[46℄ Lorenz,E.N.,Energy and numerical weather prediction.Tellus12(1960), 364-373.
[47℄ Lynden-Bell,D.,Statistical mechanics of violent relaxation in stellar
sys-tems.Mon.Not.R. astr.So .136(1967),101121.
[48℄ Majda, A.J. andWang, X., Non-linear dynamics and statistical theories
for basic geophysical flows.CambridgeUniversityPress,Cambridge,2006. [49℄ Marsden,J.E.andRatiu,T.,Mechanics and Symmetry.Springer-Verlag,
NewYork,se ond ed.,1998.
[50℄ M La hlan,R.I.,Explicit Lie-Poisson integration and the Euler equations. Phys.Rev.Lett.71(19)(1993),30433046.
[51℄ M La hlan, R.I., Symplectic integration of Hamiltonian wave equations. Numer.Math. 66(4)(1994),465492.
[52℄ M Williams, J.C., Fundamentals of geophysical fluid dynamics. Cam-bridgeUniversityPress,NewYork, 2006.
[53℄ Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E., Equation of State Calculations by Fast Computing Machines. J.Chem. Phys.21(1953),10871092.
[54℄ Miller, J., Statistical mechanics of Euler equations in two dimensions. Phys.Rev.Lett.65(17)(1991),21372140.
[55℄ Miller,J.,Wei hman,P.B.andCross,M.C.,Statistical mechanics, Euler’s
equation, and Jupiter’s Red Spot.Phys.Rev.A45(4)(1992),23282359. [56℄ Mol hanov, V., Particle-Mesh and Meshless Methods for a Class of
[57℄ Morrison, P.J., Hamiltonian description of the ideal fluid. Rev. Modern Phys.70(2)(1998),467521.
[58℄ Morrison,P.J.andGreene,J.M.,Noncanonical Hamiltonian Density
For-mulation of Hydrodynamics and Ideal Magnetohydrodynamics.Phys.Rev. Lett.45(1980),790794.
[59℄ Nambu, Y., Generalized Hamiltonian dynamics.Phys. Rev. D 7 (1973), 24052412.
[60℄ Névir, P. and Blender, R., A Nambu representation of incompressible
hydrodynamics using helicity and enstrophy.J. Phys. A 26 (22) (1993), L1189L1193.
[61℄ Nosé, S.,A molecular dynamics method for simulations in the canonical
ensemble.Mol. Phys.52(1984),255268.
[62℄ Nosé, S., A unified formulation of the constant temperature molecular
dynamics methods.J.Chem. Phys.81(1984),511519.
[63℄ Oliver,M.andBühler,O.,Transparent boundary conditions as dissipative
subgrid closures for the spectral representation of scalar advection by shear flows.J.Math. Phys.48(6)(2007),065502-065502-26.
[64℄ Olver, P.J., Applications of Lie Groups to Differential Equations. Springer-Verlag,NewYork,1986.
[65℄ Onsager, L., Statistical hydrodynamics. Nuovo Cimento, Suppl. 6 (2) (1949),279287.
[66℄ Pavliotis, G.A. and Stuart, A.M., Multiscale Methods: Averaging and
Homogenization.Springer,NewYork,2008.
[67℄ Pedlosky, J., Geophysical Fluid Dynamics. Springer, New York, se ond ed.,1987.
[68℄ Petersen, K.E.,Ergodic theory.CambridgeUniversity Press,Cambridge, 1989.
[69℄ Rei h, S.,Numerical integration of the generalized Euler equation. Te h-ni alReportTR93-20,UniversityoftheBritishColumbia,1993.
[70℄ Rei h, S., Backward error analysis for numerical integrators. SIAM J. Numer.Anal.36(1999),475491.
[71℄ Robert, R., A maximum-entropy principle for two-dimensional perfect
fluid dynamics.J.Statist.Phys.65(3-4)(1991),531553.
[72℄ Robert, R. and Sommeria, J., Statistical equilibrium states for
102 Bibliography
[73℄ Salmon, R., Hamiltonian fluid mechanics. Ann. Rev. Fluid Me h. 20 (1988),225256.
[74℄ Salmon, R., Lectures on geophysical fluid dynamics. Oxford University Press,NewYork,1998.
[75℄ Salmon,R.,A general method for conserving quantities related to potential
vorticity in numerical models.Nonlinearity18(5)(2005),R1R16. [76℄ Salmon,R.,HollowayG.andHendershottM.C.,The Equilibrium
Statis-tical Mechanics of Simple Quasi-Geostrophic Models. J. Fluid Me h. 75 (1976),691703.
[77℄ Sanz-Serna, J.M., Runge-Kutta schemes for Hamiltonian systems. BIT 28,877883.
[78℄ Sanz-Serna, J.M., Symplectic integrators for Hamiltonian problems: an
overview.A tanumeri a(1992),243286.
[79℄ Sanz-Serna,J.M.andCalvo,M.P.,Numerical Hamiltonian problems. Ap-pliedMathemati s and Mathemati alComputation, vol. 7,Chapman & Hall,London,1994.
[80℄ S hli k, T., Molecular Modeling and Simulation: An Interdisciplinary
Guide.Springer,NewYork, vol.21,2002.
[81℄ Shannon, C.E., A Mathematical Theory of Communication. Bell Syst. Te h.J.27(1948),379423and623656.
[82℄ Shepherd,T.G.,Symmetries, conservation laws, and Hamiltonian
struc-ture in geophysical fluid dynamics.Adv.Geophys.32(1990),287-338. [83℄ ShinS. and Rei h S., Hamiltonian particle-mesh simulations for a
non-hydrostatic vertical slice model.Atmos.S i.Lett.(2009),publishedonline. [84℄ Stuart, A.M. and Humphries, A.R., Dynamical Systems and Numerical
Analysis.CambridgeMonographs onAppliedandComputational Math-emati s,vol.2,CambridgeUniversityPress,Cambridge,1996.
[85℄ Walters, P., An Introduction to Ergodic Theory. Springer-Verlag, New York, 1982.
[86℄ Weiss,J.B.and M Williams,J.C.,Nonergodicity of point vortices.Phys. FluidsA3 (5part1) (1991),835844.
[87℄ Wilkinson, J.H., Error analysis of floating-point computation. Numer. Math.2(1960),319340.
[88℄ Zeitlin,V., Finite-mode analogues of 2D ideal hydrodynamics: Coadjoint