Abstract.
Dissipation can be represented in Hamiltonian mechanics in an extended phase space as a symplectic process. The method uses an auxiliary variable which represents the excitation of unresolved dynamics and a Hamiltonian for the interaction between the resolved dynamics and the auxiliary variable. This method is applied to viscous dissipation (including hyper-viscosity) in a two-dimensional fluid, for which the dynamics is non-canonical. We derive a metriplectic representation and suggest a measure for the entropy of the system.
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V. Arnol'd, A. Givental', in Dynamical Systems IV, edited by V. Arnol'd, S. Novikov (Springer, 2001) pp. 1--138
R. Littlejohn, in Mathematical methods in hydrodynamics and integrability in related dynamical systems, edited by M. Tabor, Y. Treve, AIP Conf. Proc., 88 (AIP, New York, 1982) p. 47
P.J. Morrison, J. Phys. Conf. Ser. 169, 012006 (2009)
G.K. Vallis, G.F. Carnevale, W.R. Young, J. Fluid Mech. 207, 133 (1989)
T.G. Shepherd, J. Fluid Mech. 213, 573 (1990)
F. Gay-Balmaz, D.D. Holmes, Nonlinearity 26, 495 (2013)
N. Padhye, P.J. Morrison, Plasma Phys. Rep. 22, 869 (1996)
P.J. Morrison, Rev. Mod. Phys. 70, 467 (1998)
A.N. Kaufman, Phys. Lett. A 100, 419 (1984)
P.J. Morrison, Phys. Lett. A 100, 423 (1984)
P.J. Morrison, Physica D 18, 410 (1986)
M. Grmela, Physica D 21, 179 (1986)
L.A. Turski, A.N. Kaufman, Phys. Lett. A 120, 331 (1987)
A.N. Beris, B.J. Edwards, J. Rheol. 34, 55 (1990)
D.D. Holm, V. Putkaradze, C. Tronci, J. Phys. A: Math. Theor. 41, 344010 (2008)
A. Bihlo, J. Phys. A: Math. Theor. 41, 292001 (2008)
P. Martin, E. Siggia, H. Rose, Phys. Rev. A 8, 423 (1973)
R. Phythian, J. Phys. A: Math. Gen. 8, 1423 (1975)
R. Phythian, J. Phys. A: Math. Gen. 9, 269 (1976)
R. Phythian, J. Phys. A: Math. Gen. 10, 777 (1977)
O. Cépas, J. Kurchan, Eur. Phys. J. B 2, 221 (1998)
R. Graham, T. Tél, Phys. Rev. Lett. 52, 9 (1984)
R. Graham, T. Tél, J. Stat. Phys. 35, 729 (1984)
R. Graham, T. Tél, Phys. Rev. A 31, 1109 (1985)
G. Carnevale, J. Frederiksen, J. Fluid Mech. 131, 289 (1983)
I. Drummond, J. Fluid Mech. 123, 59 (1982)
G. Carnevale, P. Martin, Geophys. Astrophys. Fluid Dyn. 20, 131 (1982)
A. Navarra, J. Tribbia, G. Conti, PloS ONE 8, e67022 (2013)
T. Lundgren, Lect. Notes Phys. 12, 70 (1972)
P.M. Morse, H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, New York, 1953)
R.W. Atherton, G.M. Homsy, Stud. Appl. Math. 54, 31 (1975)
T.F. Nonnenmacher, in Recent Developments in Nonequilibrium Thermodynamics: Fluids and Related Topics (Springer, 1986) pp. 149--174
T. Shah, R. Chattopadhyay, K. Vaidya, S. Chakraborty, Phys. Rev. E 92, 062927 (2015)
E. Celeghini, M. Rasetti, G. Vitiello, Ann. Phys. 215, 156 (1992)
A. Vanossi, N. Manini, M. Urbakh, S. Zapperi, E. Tosatti, Rev. Mod. Phys. 85, 529 (2013)
M.V. Berry, J.P. Keating, in Supersymmetry and Trace Formulae: Chaos and Disorder, edited by J.P. Keating, D.E. Khmelnitski, I.V. Lerner (Kluwer Academic/Plenum Publishers, New York, 1999)
M.V. Berry, J.P. Keating, SIAM Rev. 41, 236 (1999)
G. Sierra, J. Rodríguez-Laguna, Phys. Rev. Lett. 106, 200201 (2011)
F. Riewe, Phys. Rev. E 53, 1890 (1996)
S. Sieniutycz, Conservation laws in variational thermo-hydrodynamics, Vol. 279 (Springer Science & Business Media, 2012)
P. Névir, M. Sommer, J. Atmos. Sci. 66, 2073 (2009)
R. Salazar, M.V. Kurgansky, J. Phys. A: Math. Theor. 43, 305501 (2010)
R. Blender, G. Badin, J. Phys. A: Math. Theor. 48, 105501 (2015)
B. Leimkuhler, S. Reich, Simulating Hamiltonian Dynamics (Cambridge University Press, 2004)
M. Sommer, P. Névir, Q. J. R. Met. Soc. 135, 485 (2009)
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Blender, R., Badin, G. Viscous dissipation in 2D fluid dynamics as a symplectic process and its metriplectic representation. Eur. Phys. J. Plus 132, 137 (2017). https://doi.org/10.1140/epjp/i2017-11440-x
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DOI: https://doi.org/10.1140/epjp/i2017-11440-x