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Adjustment of vorticity fields with specified values of Casimir invariants as initial condition for simulated annealing of an incompressible, ideal neutral fluid and its MHD in two dimensions

Published online by Cambridge University Press:  15 June 2015

Y. Chikasue  and
M. Furukawa
Y. Chikasue
Affiliation:
Graduate School of Frontier Sciences, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8561, Japan
M. Furukawa*
Affiliation:
Graduate School of Engineering, Tottori University, Minami 4-101, Koyama-cho, Tottori-shi, Tottori 680-8552, Japan
*
†Email address for correspondence: furukawa@damp.tottori-u.ac.jp
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Abstract

A method is developed to adjust a vorticity field to satisfy specified values for a finite number of Casimir invariants. The developed method is tested numerically for a neutral fluid in two dimensions. The adjusted vorticity field is adopted as an initial condition for simulated annealing (SA) of an incompressible, ideal neutral fluid and its magnetohydrodynamics (MHD), where SA enables us to obtain a stationary state of the fluid. Since the Casimir invariants are kept unchanged during the annealing process, the obtained stationary state has the required values of the Casimir invariants specified by our method.

JFM classification

Mathematical Foundations: Computational methods Mathematical Foundations: Hamiltonian theory Mathematical Foundations: Mathematical Foundations
Type
Papers
Information
Journal of Fluid Mechanics , Volume 774 , 10 July 2015 , pp. 443 - 459
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Copyright
© 2015 Cambridge University Press 

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References

Arnol’d, V. I. 1965 Conditions for nonlinear stability of stationary plane curvilinear flows of ideal fluid. Dokl. Akad. Nauk SSSR 162, 975–978 (English translation in Sov. Maths 6, 773–777).Google Scholar
Bretherton, F. P. & Haidvogel, D. B. 1976 Two-dimensional turbulence above topography. J. Fluid Mech. 78, 129–154.CrossRefGoogle Scholar
Carnevale, G. F. & Vallis, G. K. 1990 Pseudo-advective relaxation to stable states of inviscid two-dimensional fluids. J. Fluid Mech. 213, 549–571.CrossRefGoogle Scholar
Chikasue, Y. & Furukawa, M. 2015 Simulated annealing applied to two-dimensional low-beta reduced magnetohydrodynamics. Phys. Plasmas 22, 022511.Google Scholar
Flierl, G. R. & Morrison, P. J. 2012 Hamiltonian–Dirac simulated annealing: application to the calculation of vortex states. Physica D 240, 212–232.Google Scholar
Grad, H. & Rubin, H. 1958 In Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy, vol. 31, pp. 190–197. United Nations.Google Scholar
Hazeltine, R. D. & Meiss, J. D. 2003 Plasma Confinement. Dover.Google Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123 (1–2), 1–116.Google Scholar
Ishioka, K. 2013 A proof for the equivalence of two upper bounds for the growth of disturbances from barotropic instability. J. Met. Soc. Japan 91, 843–850.Google Scholar
Miller, J. 1990 Statistical mechanics of Euler equations in two dimensions. Phys. Rev. Lett. 65, 2137–2140.CrossRefGoogle ScholarPubMed
Morrison, P. J. 1986 ${\it\delta}^{2}F$ : a generalized energy principle for determining linear and nonlinear stability. Bull. Am. Phys. Soc. 31, 1609.Google Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70 (2), 467–521.CrossRefGoogle Scholar
Morrison, P. J. & Greene, J. M. 1980 Non-canonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics. Phys. Rev. Lett. 45 (10), 790–794.Google Scholar
Morrison, P. J. & Hazeltine, R. D. 1984 Hamiltonian formulation of reduced magnetohydrodynamics. Phys. Fluids 27, 886–897.Google Scholar
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310.CrossRefGoogle Scholar
Shafranov, V. D. 1958 On magnetohydrodynamical equilibrium configurations. Sov. Phys. JETP 6, 545.Google Scholar
Shepherd, T. G. 1988 Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows. J. Fluid Mech. 196, 291–322.Google Scholar
Shepherd, T. G. 1990 A general method for finding extremal states of Hamiltonian dynamic systems, with applications to perfect fluids. J. Fluid Mech. 213, 573–587.Google Scholar
Strauss, H. R. 1976 Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks. Phys. Fluids 19, 134–140.Google Scholar
Vallis, G. K., Carnevale, G. F. & Young, W. R. 1989 Extremal energy properties and construction of stable solutions of the Euler equations. J. Fluid Mech. 207, 133–152.Google Scholar