Chaotic magnetic fields in Vlasov-Maxwell equilibria Abhijit Ghosh, M. S. Janaki, Brahmananda Dasgupta, and Alak Bandyopadhyay Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 013117 (2014); doi: 10.1063/1.4865253 View online: http://dx.doi.org/10.1063/1.4865253 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Force-free Jacobian equilibria for Vlasov-Maxwell plasmas Phys. Plasmas 20, 102117 (2013); 10.1063/1.4826502 Vlasov-Maxwell equilibria: Examples from higher-curl Beltrami magnetic fields Phys. Plasmas 19, 032113 (2012); 10.1063/1.3694751 Instability of nonsymmetric nonmonotone equilibria of the Vlasov-Maxwell system J. Math. Phys. 52, 123703 (2011); 10.1063/1.3670874 Vlasov-Maxwell plasma equilibria with temperature and density gradients: Weak inhomogeneity limit Phys. Plasmas 14, 042103 (2007); 10.1063/1.2718911 Vlasov–Maxwell equilibrium solutions for Harris sheet magnetic field with Kappa velocity distribution Phys. Plasmas 12, 070701 (2005); 10.1063/1.1941047

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CHAOS 24, 013117 (2014)

Chaotic magnetic fields in Vlasov-Maxwell equilibria Abhijit Ghosh, M. S. Janaki, Brahmananda Dasgupta,a) and Alak Bandyopadhyayb) Saha Institute of Nuclear Physics, I/AF Bidhannagar, Calcutta 700 064, India

(Received 6 September 2013; accepted 29 January 2014; published online 12 February 2014) Stationary solutions of Vlasov-Maxwell equations are obtained by exploiting the invariants of single particle motion leading to linear or nonlinear functional relations between current and vector potential. For a specific combination of invariants, it is shown that Vlasov-Maxwell equilibria have an associated C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4865253] Hamiltonian that exhibits chaos. V Plasma equilibria are suitable starting points for investigating many phenomena in space and laboratory plasmas. For collisionless plasmas, the most relevant equilibria are self-consistent solutions of the Vlasov-Maxwell equations. Such equilibria lead to an understanding of the topological properties of the magnetic fields in a plasma. Since chaotic magnetic fields are responsible for several interesting phenomena in plasmas, in the present paper, we attempt to connect the plasma distribution function and selfconsistent magnetic fields that exhibit chaotic behavior. Exact stationary solutions of Vlasov-Maxwell equations for a magnetized plasma are obtained by constructing the distribution function as some chosen function of the invariants of single particle motion in a magnetic field (e.g., energy, canonical momentum). Jeans’ theorem1 forms the fundamental starting point for obtaining the stationary equilibria of Vlasov-Maxwell equations. Such formulation leads to linear or nonlinear functional relations between current and vector potential. The nonlinear relations support various special types of magnetized plasma configurations2–7 including multiple current sheets and magnetic field discontinuities leading to singular current layers. Further, it can be shown that the equilibrium equations for the magnetic vector potential are equivalent to the motion of a pseudo-particle in a two-dimensional potential well. In this work, we demonstrate that for some specific combination of the invariants employed in the construction of the distribution function, the nonlinear functional relations between the current and the vector potential can be reduced to a set of coupled nonlinear equations for the two components of the vector potential. The first integral of this set of coupled equations for the vector potentials (which can be called the “Hamiltonian” of the two pseudo-particle system) corresponds to a pseudo-particle motion in a bounded potential with quadratic terms and quartic coupling term. This first integral (i.e., the Hamiltonian) is entirely similar to the Hamiltonian for a classical SU(2) Yang Mills Higgs system,8,9 which exhibits chaotic behavior. Magnetic field line equations constitute a Hamiltonian dynamical system10–12 and thus all the features of the a)

Present address: Center for Space Plasma and Aeronomic Research, University of Alabama in Huntsville, Huntsville, Alabama 35805, USA. b) Present address: Department of Computer Science Alabama, A & M University Normal, Alabama 35762, USA. 1054-1500/2014/24(1)/013117/4/$30.00

Hamiltonian dynamics can be attributed to magnetic fields. An important question is the integrability of the magnetic field line equations: being a Hamiltonian system nonintegrability is to be expected13 when there is no apparent symmetry and many studies12–15 corroborate this. Thus, the prevalence of chaotic magnetic fields in nature also is to be expected. The study16 of the structure of magnetic fields including formation of magnetic surfaces as well as field line chaos is of much help in understanding the problems of plasma confinement and instabilities in the context of fusion devices. Chaotic magnetic fields in astrophysical environments and fusion physics have been directly and indirectly postulated by many earlier workers. Parker17 was the first to point out the effect of irregular and chaotic magnetic field on charged particle motion in cosmic plasma. These works were subsequently elaborated in a series of papers by Jokipii18 to study the cosmic ray propagation in a random magnetic field. Lee and Parks19 studied the evolution of nonlinear magnetic field in MHD plasmas by casting these equations in the form of a forced Duffing’s equation which showed chaotic behavior. In fusion physics, existence of chaotic magnetic fields has been conjectured by several authors20,21 due its relevance to enhanced heat transport. Recently, the ubiquity of the chaotic magnetic fields in an asymmetric combination of current carrying wire loop system have been demonstrated.22,23 In this work we show that the magnetic field obtained from a Vlasov-Maxwell equilibria has a chaotic nature. We consider a stationary plasma configuration, consisting of electrons and ions, with the magnetic field lying in the x–y plane and all variations are with respect to the z-coordinate. The magnetic field is obtained from a vector potential A ¼ (Ax(z), Ay(z), 0). For the sake of simplicity, we assume that the electric field is zero and the plasma is fully neutral. Thus we have Bx ¼ 

dAy ; dz

By ¼

dAx : dz

(1)

The current J is obtained from the Ampere’s law l 0 Jx ¼ 

d 2 Ax ; dz2

l0 Jy ¼ 

d2 Ay : dz2

(2)

For the stationary case, the Hamiltonian of the particle motion, HE given by

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 1  HE ¼ ms v2x þ v2y þ v2z 2

(3)

C 2014 AIP Publishing LLC V

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is an integral of motion. Further, in this case, since x and y coordinates are cyclic, the corresponding canonical momenta

The differential equations governing Ax and Ay are of the form

Pxs ¼ ms vx þ es Ax ;

(4)

Pys ¼ ms vy þ es Ay ;

(5)

d2 Ax;y þ bAx;y þ cAx;y A2y;x ¼ 0; (12) dz2 P P where b ¼ 2l0 s b1s m3s e2s hv2xs ihv2ys i; c ¼ 2l0 s b1s ms e4s hv2xs i. Exact analytical solutions to the above equation are usually impossible to obtain, however a constant of motion governing the above equation can be obtained as

are conserved. In the above equations, the suffix s ¼ i,e is for ions and electrons respectively. Any distribution function which is a function fs of the above three integrals of motion, i.e., fs ¼ fs(HE, Pxs, Pys) is a solution of the stationary Vlasov equation v

@ es @ fs ðHE ; Pxs ; Pys Þ þ ðv  BÞ  fs ðHE ; Pxs ; Pys Þ ¼ 0: c @r @v (6)

The functional form of fs is assumed to be fs(HE) gs(Pxs, Pys) with fs ðHE Þ ¼ ðms bs =2pÞ3=2 expðbs HE Þ; bs ¼ 1=kB Ts , where the functional form of gs needs to be appropriately chosen. Following Kocharovsky et al.,24 we construct a general distribution function which is a combination of polynomials given by the form X r X fs ðHE Þ bns ðms vx þ es Ax Þ2n f ðHE ; Px ; Py Þ ¼ s

n¼0

  ðms vy þ es Ay Þ2n ;

(7)

where bns are constants and r is a appropriately chosen number. It is evident that the above form of the distribution function (7) will not generate any constant current and also remains positive definite for 1 < vx;y < 1. We first note that the density of ions and electrons are given as ð (8) Ni ¼ Ni ðAx ; Ay Þ ¼ fi ðHE ; Pxi ; Pyi Þd3 v; ð

Ne ¼ Ne ðAx ; Ay Þ ¼ fe ðHE ; Pxe ; Pye Þd 3 v;

(9)

and the charge-neutrality is ensured by imposing the following condition Ni ðAx ; Ay Þ ¼ Ne ðAx ; Ay Þ and assuming that the two species composing the plasma (electrons and ions) are of opposite charge. The charge neutrality condition implies that the equilibria do not support any electric fields. For the choice r ¼ 1, we obtain For the choice r ¼ 1, we obtain h fs ðHE ; Px ; Py Þ ¼ fs ðHE Þ b0s þ b1s ðms vx þ es Ax Þ2 i  ðms vy þ es Ay Þ2 :

 2  2 1 dAx 1 dAy 1 þ þ ðA2x þ A2y Þ þ lA2x A2y ¼ C; H¼ 2 dz 2 dz 2 (13) pffiffiffi where the normalized variables z ¼ bz, and l ¼ c/b have been used. The Hamiltonian H in Eq. (13) corresponds to that of a system of nonlinearly coupled oscillators with Ax and Az playing the role of coordinates. The quantity l  1=ðqL kÞ2 where qL is the finite Larmor radius and k is the inverse of the characteristic size L of the spatial gradients in the system. In the framework of Vlasov-Maxwell equilibria, finite Larmor radius effects give rise to a pressure tensor that is non isotropic and non gyrotropic.25 The system is integrable for l ¼ 0. (a)

Assuming Ay ¼ 0 (when the magnetic field is unidirectional), we get a simple harmonic oscillator equation for Ax d2 Ax þ Ax ¼ 0: dz2

(b)

(14)

The general solution is obtained as Ax ¼ A1 cos z þ A2 sin z with A21 þ A22 ¼ 2C. Another set of solutions can be obtained by considering in and out of phase oscillations for the two components of the field components with Ay ðzÞ ¼ 6Ax ðzÞ, so that Ax satisfies the equation d 2 Ax þ Ax þ lA3x ¼ 0; dz2

(15)

for which the integral of motion is  2 1 dAx 1 l þ ðA2x Þ þ A4x ¼ C: 2 dz 2 4

(16)

A particular solution of Eq. (15) is Ax ðzÞ ¼ A0 cn½az þ b; k

(10)

(17)

with The complete expressions for Jx,y in this case are obtained as  X 2b1s m3s e2s hv2xs ihv2ys iAx þ 2b1s ms e4s hv2xs iAx A2y ; l0 Jx ¼ l0 s   X l0 Jy ¼ l0 2b1s m3s e2s hv2xs ihv2y iAy þ 2b1s ms e4s hv2xs iA2x Ay : s

(11)

a2 ¼ 1 þ A20 l; k2 ¼

A20 l ; 2ð1 þ A20 lÞ

(18)

where cn(z, k) is the Jacobi elliptic cosine of argument z and modulus k, with A0, a, b are arbitrary constants of which one is independent. The period of cn function if 4 K(k) where K(k) is the complete elliptic integral of

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the first kind. In the limit l ! 0, the cn functions reduce to cosine functions. Order-Chaos transition for Vlasov-Maxwell Equilibria: The Hamiltonian obtained in Eq. (13) is similar to that obtained in the context of a charged particle moving in a hyperbolic magnetic field with an “X”-type magnetic neutral line. In this context (  2 ) 1 eA p2 þ p2y þ pz  H¼ 2m x c

VðKG ¼ 0; Ax Þ ¼

16l2 A4x þ 2lA2x þ 1 : 4lð6lA2x  1Þ

(21)

The minimal energy on the zero curvature line is given by

A ¼ xyk; with k being a unit vector along the z-axis. This Hamiltonian arises by restricting the motion to the x–y plane when pz ¼ 0. The gauge used corresponds to a magnetic induction field B B ¼ xi  yj: This, however, is a Hamiltonian for particle motion, where as Eq. (13) is a Hamiltonian for magnetic fields. Dynamics of charged particle motion in x2y2 and other similar quartic potentials exhibiting large scale chaos have been extensively studied.26–28 A chaos-order transition for a Hamiltonian system can be studied using the Chirikov criterion,8 Lyapunov exponents29 or the theory of quantal overlapping resonances.30 The overlap condition is particularly useful to describe the destruction of a particular kind of invariant set, but possess limitations.31 In the present section, we study following Manfredi and Salasnich,32 the chaotic behaviour of the Vlasov-Maxwell equilibria for a magnetized plasma using the Toda33 Gaussian curvature criterion for potential energy. A more useful technique that can predict global properties of the system is provided by plotting Poincare sections.34 The potential energy for the Hamiltonian given in Eq. (13) is given by 1 VðAx ; Ay Þ ¼ ðA2x þ A2y Þ þ lA2x A2y : 2

(19)

The Gaussian-curvature of the potential energy surface is given by the following expression 2  2 @2V @2V @ V  @A2x @A2y @Ax @Ay (20) KG ðAx ; Ay Þ ¼   2  2  2  2 2 : @ V @ V þ 1þ @A2x @A2y The potential has a minimum at Ax ¼ Ay ¼ 0 with V ¼ 0. At low energy, the dynamics near the minimum of the potential where the Gaussian curvature is positive is periodic or quasiperiodic. When the energy is increased, for certain initial conditions, the system will be in a region of negative curvature, where the dynamics is chaotic. The energy for the chaos-order transition is equal to the minimum value of the line of zero-Gaussian curvature KG(Ax, Ay) on the potential energy surface. The potential energy on the zero-curvature line is given by

FIG. 1. Surface of section plots in the Ax–Ay plane at By ¼ 0 with l ¼ 0.05. (a) E ¼ 10, (b) E ¼ 30, (c) E ¼ 50.

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pffiffiffiffiffiffi 3 Ec ¼ Vmin ðKG ¼ 0; Ax ¼ 1= 2lÞ ¼ : 4l

Chaos 24, 013117 (2014) 1

(22)

Thus, there is a order-chaos transition when the energy E of the system is increased. It is important to realize that this and several other criteria mentioned in this section are only indicative of a signature of chaos, but do not constitute a proof of nonintegrability of the system. Such a proof might be possible by using the Melnikov35 criterion, but this is beyond the scope of the present work. For l ¼ 0, the system is integrable and corresponds to a two dimensional harmonic oscillator. For finite values of l the system corresponds to coupled oscillators. All cases with l 6¼ 0 are likely to be nonintegrable, but the homoclinic tangle being small, the plots for very small values of l show mostly invariant curves. According to equation (22), for a choice of l ¼ .05, the system should display36 chaotic orbits for most initial conditions for E > 15. Fig. 1 shows the surface of section plot where certain regular orbits are shown for E ¼ 10 while the system shows mostly chaotic orbits for E ¼ 50 for a coupling parameter l ¼ 0.05. However, Fig. 1(c) still has some invariant sets near the elliptic points, which is a characteristic nonergodic behaviour displayed by such systems. A more interesting study in this context would be investigation of destruction of particular families of invariant sets, such as the last magnetic surface surrounding the plasma. The distribution function defined in Eq. (10) for the values of the parameters as discussed here corresponds to the distribution function for chaotic magnetic fields. An analysis of the equilibrium magnetic field is important in understanding the plasma behaviour since the microscopic parameters defining the distribution functions depend on the macroscopic parameters describing the magnetic field. The authors are sincerely thankful to the referee for his helpful and constructive suggestions which improved the quality of this work. B.D. acknowledges supports from NSF grant AGS1062050 and Individual Investigator Distinguished Research (IIDR) award from the University of Alabama in Huntsville.

J. H. Jeans, Mon. Not. R. Astron. Soc. 76, 71 (1915). E. G. Harris, Nuovo Cimento 23, 115 (1962). D. Correa-Restrepo and D. Pfirsch, Phys. Rev. E 55, 7449 (1997). 4 V. Krishan, T. D. Sreedharan, and S. M. Mahajan, Mon. Not. R. Astron. Soc. 249, 596 (1991). 5 T. Neukirch, F. Wilson, and M. G. Harrison, Phys. Plasmas 16, 122102 (2009). 6 M. G. Harrison and T. Neukirch, Phys. Rev. Lett. 102, 135003 (2009). 7 G. Bertin and M. Stiavelli, Rep. Prog. Phys. 56, 493 (1993). 8 G. K. Savvidy, Phys. Lett. B 159, 325 (1985). 9 L. Salasnich, Mod. Phys. Lett. A 12, 1473 (1997). 10 M. S. Janaki and G. Ghosh, J. Phys. A: Math. Gen. 20, 3679 (1987). 11 J. R. Cary and R. G. Littejohn, Ann. Phys. 151, 1 (1983). 12 P. J. Morrison, Phys. Plasmas 7, 2279 (2000). 13 J. Moser, Stable and Random Motions in Dynamical Systems (Prince-ton University Press, 1973). 14 M. A. Zugasti, J. A. G omez, D. Garcıa-Pablos, F. R. Llorente, and A. F. Ra~ nada, Chaos, Solitons Fractals 4, 1943 (1994). 15 K. Bajer and H. K. Moffatt, J. Fluid Mech. 212, 337 (1990). 16 Review of Plasma Physics, edited by M. A. Leontivich (Consultants Bureau, 1966); Structure of Magnetic Fields, edited by A. I. Morozov and L. S. Solovev (Consultants Bureau, 1966). 17 E. N. Parker, J. Geophys. Res. 69, 1755, doi:10.1029/JZ069i009p01755 (1964). 18 J. R. Jokipii, Astrophys. J. 146, 480 (1966). 19 N. C. Lee and G. K. Parks, Theory, Geophys. Res. Lett. 19, 637 (1992); Geophys. Res. Lett. 19, 641, doi: 10.1029/92GL00635 (1992). 20 A. B. Rechester and M. N. Rosenbluth, Phys. Rev. Lett. 40, 38 (1978). 21 E. C. da Silva, I. L. Caldas, and R. L. Viana, IEEE Trans. Plasma Sci. 29, 617 (2001). 22 M. T. Hosoda, K. Imagawa, and K. Nakamura, Phys. Rev. E 80, 067202 (2009). 23 A. K. Ram and B. Dasgupta, Phys. Plasmas 17, 122104 (2010). 24 V. V. Kocharovsky, V. V. Kocharovsky, and V. Ju. Martyanov, Phys. Rev. Lett. 104, 215002 (2010). 25 F. Mottez, Ann. Geophys. 22, 3033 (2004). 26 P. Dahlqvist and G. Russberg, Phys. Rev. Lett. 65, 2837 (1990). 27 R. F. Martin, Jr. and H. Matsuoka, Phys. Rev. E 47, 721 (1993). 28 R. F. Martin, Jr., J. Geophys. Res. 91, 11985, doi:10.1029/JA091iA11p11985 (1986). 29 T. Kawabe and S. Ohta, Phys. Rev. D 44, 1274 (1991). 30 L. Salasnich, Phys. Rev. D 52, 6189 (1995). 31 D. del-Castillo-Negrete and P. J. Morrison, Phys. Fluids A 5, 948 (1993). 32 V. R. Manfredi and L. Salasnich, Mod. Phys. Lett. A 12, 1951 (1997). 33 M. Toda, Phys. Lett. A 48, 335 (1974). 34 H. Poincare, New Methods of Celestial Mechanics (NASA Washington DC, 1967), Vol. 3, Chap. 27. 35 P. J. Holmes and J. E. Marsden, Commun. Math. Phys. 82, 523 (1982). 36 R. A. Pullen and A. R. Edmonds, J. Phys. A 14, L477 (1981). 2 3

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Chaotic magnetic fields in Vlasov-Maxwell equilibria.

Stationary solutions of Vlasov-Maxwell equations are obtained by exploiting the invariants of single particle motion leading to linear or nonlinear fu...
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