Fournier, Jean-Daniel, 1951- and Galtier, S.. (2001) Meromorphy and topology of localized solutions in the Thomas–MHD model. Journal of Plasma Physics, Vol.65 (No.5). pp. 365-406. ISSN 0022-3778

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Abstract

The one-dimensional MHD system first introduced by J.H. Thomas [Phys. Fluids 11, 1245 (1968)] as a model of the dynamo effect is thoroughly studied in the limit of large magnetic Prandtl number. The focus is on two types of localized solutions involving shocks (antishocks) and hollow (bump) waves. Numerical simulations suggest phenomenological rules concerning their generation, stability and basin of attraction. Their topology, amplitude and thickness are compared favourably with those of the meromorphic travelling waves, which are obtained exactly, and respectively those of asymptotic descriptions involving rational or degenerate elliptic functions. The meromorphy bars the existence of certain configurations, while others are explained by assuming imaginary residues. These explanations are tested using the numerical amplitude and phase of the Fourier transforms as probes of the analyticity properties. Theoretically, the proof of the partial integrability backs up the role ascribed to meromorphy. Practically, predictions are derived for MHD plasmas.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Topology, Functions, Meromorphic, Magnetohydrodynamics, Plasma physics, Wave functions
Journal or Publication Title:	Journal of Plasma Physics
Publisher:	Cambridge University Press
ISSN:	0022-3778
Date:	June 2001
Volume:	Vol.65
Number:	No.5
Page Range:	pp. 365-406
Identification Number:	10.1017/S002237780100887X
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	Ablowitz, M. J., Ramani, A. and Segur, H. 1980 A connection between nonlinear evolution equations and ordinary differential equations of p-type I. J. Math. Phys. 21, 715{721. Arnold, V. I. 1973 Ordinary Differential Equations. Moscow: Mir. Bender, C.M. and Orszag, S.A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw Hill. Bessis, D. and Fournier, J.-D. 1984 Pole condensation and the Riemann surface associated with a shock in Burgers' equation. J. Physique Lett. 45, L833{L841. Bessis, D. and Fournier, J.-D. 1990 Complex singularities and the Riemann surface for the Burgers equation. In: Nonlinear Physics (ed. G. Chaohao, L. Yishen and T. Guizhang). Berlin: Springer-Verlag, pp. 252{257. Bhattacharjee, A., Ng, C. S. and Spangler, S.R. 1998 Weakly compressible magnetohydrodynamic turbulence in the solar wind and the interstellar medium. Astrophys. J. 494, 409{418. Brunelli, J. C. and Das, A. 1997 A Lax description for polytropic gas dynamics. Phys. Lett. A 235A, 597{602. Burgers, J. M. 1939 Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Kon. Ned. Akad. Wet. Verh. 17, 1{53. Burgers, J. M. 1974 The Nonlinear Diffusion Equation. Dordrecht: Reidel. Burlaga, L.F. 1991 Intermittent turbulence in the solar wind. J. Geophys. Res. 96, 5847{5851. Chabat, B. 1990 Introduction �a l'Analyse Complexe. Moscow: Mir. Cole, J. 1951 On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Maths 9, 225{236. Conte, R. and Boccara, N. (eds) 1990 Partially Integrable Evolution Equations in Physics. Dordrecht: Kluwer. Dobrowolny, M., Mangeney, A. and Veltri, P. 1980 Properties of magnetohydrodynamic turbulence in the solar wind. Astron. Astrophys. 83, 26{32. Festou, M. C., Rickman, H. and West, R. M. 1993 Comets. Astron. Astrophys. Rev. Part I 4, 363{447. Flaschka, H., Newell, A. C. and Tabor, M. 1991 Integrability, What is integrability? In: Nonlinear Dynamics (ed. V. E. Zakharov). Berlin: Springer-Verlag, pp. 73{114. Fournier, J.-D. 1986 Propri�et�es locales et singularit�es complexes en dynamique non lin�ea��re. In: M�ethodes Math�ematiques pour l'Astrophysique (ed. M. Auvergue and A. Baglin). Nice: SFSA, pp. 333{383 (available from Dr D. Benest, Observatoire de Nice). Fournier, J.-D. and Bessis, D. 1994 Dealing with the singularities of analytic functions. In: An Introduction to Methods of Complex Analysis and Geometry for Classical Mechanics and Non-Linear Waves (ed. D. Benest and C. Froesuhl�e). Gif sur Yvette: Editions Fronti�eres, pp. 1{45. Fournier, J.-D. and Frisch, U. 1983 L'�equation de Burgers d�eterministe et statistique. J. M�ec. Th�eor. Appl. 2, 699{750. Fournier, J.-D., Levine, G. and Tabor, M. 1988 Singularity clustering in the Duf�ng oscillator. J. Phys. A21, 33{54. Fournier, J.-D., Spiegel, E. A. and Thual, O. 1989 Meromorphic integrals of two nonintegrable systems. In: Non Linear Dynamics (ed. G. Turchetti). Singapore: World Scientic, pp. 366{373. Frisch, U. and Morf, R. 1981 Intermittency in nonlinear dynamics and singularities at complex times. Phys. Rev. A23, 2673{2705. Galsgaard, K. and Nordlund, A. 1996 The heating and activity of the solar corona: boundary shearing of an initially homogeneous magnetic �eld. J. Geophys.Res. 101, 13445{13460. Galtier, S. and Fournier, J.-D. 1998 Shocks and antishocks in the MHD{Thomas model. In: Nonlinear Dynamics in the Heliosphere. Geophysical Research Abstracts (III), EGS Conference, Nice. Galtier, S. and Pouquet, A. 1998 Solar are statistics with a one-dimensional MHD model. Solar Phys. 179, 141{165. Galtier, S., Nazarenko, S.V., Newell, A. C. and Pouquet, A. 2000 A weak turbulence theory for incompressible magnetohydrodynamics. J. Plasma Phys. 63, 447{488. Goldstein, M. L. and Roberts, D.A. 1999 Magnetohydrodynamic turbulence in the solar wind. Phys. Plasmas 6, 4154{4160. Gurbatov, S.N., Simdyankin, S. I., Aurell, E., Frisch, U. and T�oth, G. 1997 On the decay of Burgers turbulence. J. Fluid Mech. 344, 339-374. Heiles, C., Goodman, A. A., McKee, C.F. and Zweibel, E.G. 1993 Magnetic �elds in starforming regions : observations. In: Protostars and Planets III (ed. E. H. Levy, J. I. Lunine, M. Guerrieri and M. S. Matthews). Tucson: University of Arizona Press, pp. 279{326. Hopf, E. 1950 The partial differential equation ut + uux = uxx. Commun. Pure Appl. Mech. 3, 201{230. Kida, S. 1979 Asymptotic properties of Burgers turbulence. J. Fluid Mech. 93, 337{377. Levine, G. and Tabor,M. 1988 Integrating the nonintegrable: analytic structure of the Lorenz system revisited. Physica D33, 189{210. Marsch, E. and Tu, C.Y. 1994 Non-Gaussian probability distributions of solar wind fluctuations. Ann. Geophys. 12, 1127{1138. Newell, A. C. 1985 Solitons in Mathematics and Physics. Philadelphia: SIAM. Ng, C. S. and Bhattacharjee, A. 1997 Scaling of anisotropic spectra due to the weak interaction of shear-Alfv�en wave packets. Phys. Plasmas 4, 605{610. Olver, P. J. and Nutku, Y. 1988 Hamiltonian structures for systems of hyperbolic conservation laws. J. Math. Phys. 29, 1610{1619. Parker, E. N. 1994 Spontaneous Current Sheets in Magnetic Fields. Oxford University Press. Passot, T. 1986 Le test de Painlev�e. In: M�ethodes Math�ematiques pour l'Astrophysique. (ed. M. Auvergne and A. Baglin). Nice: SFSA, pp. 161{170 (available from Dr D. Benest, Observatoire de Nice). Passot, T. 1987 Simulations num�eriques d'�ecoulements compressibles. PhD Thesis, Universit�e de Nice. Passot, T. and Pouquet, A. 1986 The Painlev�e analysis on the Burgers' MHD. Phys. Lett. 118, 121{123. Passot, T. and V�azquez-Semadeni, E. 1998 Density probability distribution in onedimensional polytropic gas dynamics. Phys. Rev. E58, 4501{4510. Pouquet, A., Galtier, S. and Politano, H. 1999 Mechanisms of injection and dissipation of energy and their relation to the dynamics of the interstellar medium. In: New Perspectives on the Interstellar Medium. (ed. A.R. Taylor, T.L. Landecker and G. Joncas). ASP Conference Series, Vol. 168, pp. 417{426. Priest, E. R. 1982 Solar Magnetohydrodynamics. Dordrecht: Reidel. Schwenn, R. and Marsch, E. 1991 Physics of the Inner Heliosphere II: Particles, Waves and Turbulence. Berlin: Springer-Verlag. Shebalin, J.V., Matthaeus, W.H. and Montgomery, D. 1983 Anisotropy in MHD turbulence due to a mean magnetic �eld. J. Plasma Phys. 29, 525{547. Sulem, C., Fournier, J.-D., Frisch, U. and Sulem, P. L. 1979 Remarques sur un mod�ele unidimensionnel pour la turbulence magn�etohydrodynamique. C. R. Acad. Sci. Paris 288, 571{573. Sulem, C., Sulem, P. L. and Frisch, H. 1983 Tracing complex singularities with spectral methods. J. Comput. Phys. 50, 138{161. Taylor, J.B. 1986 Relaxation and magnetic reconnection in plasmas. Rev. Mod. Phys. 58, 741{763. Thomas, J.H. 1968 Numerical experiments on a model system for magnetohydrodynamic turbulence. Phys. Fluids 11, 1245{1250. Thomas, J.H. 1970 Model equations for magnetohydrodynamic turbulence. A gas dynamics analogy. Phys. Fluids 13, 1877{1879. Weiss, J., Tabor, M. and Carnevale, G. 1983 The Painlev�e property for partial differential equations. J. Math. Phys. 24, 522{526. Wild, N., Gekelman, W. and Stenzel, R. L. 1981 Resistivity and energy flow in a plasma undergoing magnetic fi�eld line reconnection. Phys.Rev. Lett. 46, 339{342. Zank, G.P. and Matthaeus, W.H. 1992 The equations of reduced magnetohydrodynamics. J. Plasma Phys. 48, 85{100. Zilbersher, D., Gedalin, M., Newbury, J. A. and Russell C.T. 1998 Direct numerical testing of stationary shock model with low Mach number shock observations. J. Geophys. Res. 103, 26775.
URI:	http://wrap.warwick.ac.uk/id/eprint/822

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