Relativistic magnetic reconnection at X-type neutral points
aa r X i v : . [ a s t r o - ph . H E ] A p r Astronomy & Astrophysics manuscript no. kojima c (cid:13)
ESO 2018November 7, 2018
Relativistic magnetic reconnection at X-type neutral points
Y. Kojima, J. Oogi, and Y. E. Kato
Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526, Japane-mail: [email protected]
Preprint online version: November 7, 2018
ABSTRACT
Context.
Relativistic effects in the oscillatory damping of magnetic disturbances near two-dimensional X-points areinvestigated.
Aims.
By taking into account displacement current, we study new features of extremely magnetized systems, in whichthe Alfv´en velocity is almost the speed of light.
Methods.
The frequencies of the least-damped mode are calculated using linearized relativistic MHD equations for wideranges of the Lundquist number S and the magnetization parameter σ . Results.
The oscillation and decay times depend logarithmically on S in the low resistive limit. This logarithmic scalingis the same as that for nonrelativistic dynamics, but the coefficient becomes small as ∼ σ − / with increasing σ . Thesetimescales approach constant values in the large resistive limit: the oscillation time becomes a few times the lightcrossing time, irrespective of σ , and the decay time is proportional to σ and therefore is longer for a highly magnetizedsystem. Key words.
Magnetohydrodynamics (MHD) – Magnetic reconnection – Relativistic processes
1. Introduction
The importance of magnetic reconnection manifests itself invarious energetic astrophysical phenomena, including rela-tivistic objects such as pulsars, magnetars, active galacticnuclei and gamma ray bursts. The characteristic propaga-tion velocity for magnetic disturbances, the Alfv´en velocity,depends on the magnetization parameter σ : 2 σ representsthe ratio of the magnetic to the rest mass energy densityof the plasma. When the magnetization parameter σ ≫ σ ≪
1. Inthis paper, we consider some inherently relativistic featuresthat may appear in the magnetic reconnection when σ islarge.In a simple analysis of the Sweet-Parker type recon-nection, the structure of the reconnection layer dependson two large dimensionless numbers: σ and the Lundquist(or magnetic Reynolds) number S , an inverse of resistiv-ity (Lyutikov & Uzdensky (2003)). For σ ≪ S , the in-flow velocity is nonrelativistic, and the reconnection is verysimilar to the classical Sweet-Parker model. However, for σ ≫ S ≫
1, the inflow velocity becomes relativistic.Lyubarsky (2005) incorporated the compressibility of mat-ter and found that the inflow velocity is always sub-Alfv´enicand remains much less than the speed of light, contradict-ing Lyutikov & Uzdensky (2003). The reconnection rate isstill estimated by substituting c for the Alfv´en velocity inthe nonrelativistic formula, even in the relativistic regime.Small differences may also originate from the as-sumption of a steady state. Numerical simulationsof an anti-parallel magnetic configuration in two di-mensions have been performed without assuming asteady state using relativistic resistive MHD code(Watanabe & Yokoyama (2006)), a relativistic two-fluid model(Zenitani et al. (2009)), and PIC simulations on ki-netic scale (Zenitani & Hoshino (2005, 2007, 2008)). Seealso Komissarov (2007) for the numerical schemes of theresistive relativistic MHD. These approaches have clearlydemonstrated the relativistic dynamics, but simulation ina wide range of parameters would be time-consuming.Moreover, the resolution becomes poor for small resistiv-ity. The dynamics at an X-type null point, where a cur-rent sheet forms and the magnetic energy is dissipated,have been studied previously. In context of nonrelativis-tic dynamics, Craig & McClymont (1991) considered thebehavior of MHD waves near the X-point in the coldplasma approximation using linear perturbation theory.They showed the remarkable result that the dissipationtime behaves as ∼ (ln S ) . This logarithmic dependence,in contrast to the normal power behavior ∼ S α , indi-cates fast decay. Subsequently, the problem was studiedanalytically(Hassam (1992)) and by considering the prop-agation of linearized waves(McLaughlin & Hood (2004)).Some physical properties of a more realistic system havealso been included, such as non-linear waves with thermalpressure(McClymont & Craig (1996); McLaughlin et al.(2009)), electron inertial effects(McClements et al. (2004)),and viscosity(Craig et al. (2005); Craig (2008)). See a re-cent review of this topic given by McLaughlin et al. (2010)and references therein.The main concern of this paper is to explore relativis-tic effects on the dynamical reconnection at an X-point. Inparticular we consider whether the reconnection is qualita-tively modified for a highly magnetized system with σ ≫ Y. Kojima et al.: Relativistic magnetic reconnection at X-type neutral points we will calculate complex normal frequencies, which deter-mine the oscillatory damping of the magnetic disturbanceswith small amplitudes, neglecting thermal pressure, viscos-ity and so on. The problem may be solved as an initial valueproblem, but the initial data inevitably contain electromag-netic waves besides MHD waves, and subsequent evolutionmay be complex. In section 2, we discuss our numericalmethods and boundary conditions. Our results are shownin section 3. Section 4 contains our conclusions.
2. Model
We consider a two-dimensional problem, assuming ∂/∂z =0. In our model, the magnetic field B is located on a planeand the electric field is perpendicular to it, E = E e z . Theelectric current j e z is also perpendicular to the plane, andthe charge density consistently vanishes, since ∇ · E = 0.These electromagnetic fields can be expressed in terms ofonly the z -component of a vector potential A = A e z as B = ∇ A × e z , E = − c ∂A∂t . (1)The flux function A satisfies with a wave equation with asource term: (cid:18) − c ∂ ∂t + ∇ (cid:19) A = − πjc , (2)where the displacement current is included in contrast tothe usual nonrelativistic treatment.The dynamics of the plasma flow is determined by thecontinuity equation ∂ρ∂t + ∇ · ( ρ v ) = 0 , (3)and the momentum equation with the Lorentz force ρ (cid:18) ∂∂t + v · ∇ (cid:19) γ v = jc e z × B = jc ∇ A, (4)where γ = (1 − ( v/c ) ) − / , ρ is the mass number den-sity in the laboratory frame, and the proper one is ρ/γ .In eq. (4), the Coulomb force vanishes and thermal effectsin the pressure and internal energy are neglected in thecold limit. This cold plasma approximation simplifies theproblem:The slow magnetoacoustic wave is absent. In non-relativistic dynamics, it is found that propagation of thefast one causes the current density to accumulate at the Xpoint, where the energy is dissipated (McLaughlin & Hood,2004; McLaughlin et al., 2009). Thermal pressure is ne-glected, since our concern is the propagation in linearizedsystem. The finite pressure is meaningful in fully non-lineardynamics, where coupling and mode conversion betweenMHD waves are important in the neighborhood of the dis-sipation zone.Ohm’s law with resistivity η can be written as E + 1 c ( v × B ) z = 4 πηγc j, (5)which, in terms of A , is (cid:18) ∂∂t + v · ∇ (cid:19) A = − πηγc j. (6) The relativistic motion reduces the resistivity by theLorentz factor γ . (See, e.g, Blackman & Field (1993);Lyutikov & Uzdensky (2003).) However, this factor may beset to γ = 1 for a linear perturbation from a static back-ground. We consider the dynamics of small perturbation in thevicinity of an X-point, which is governed by current-free( j = 0), static ( v = 0) background fields with uniformdensity ( ρ = ρ ).The magnetic potential A of the background field canbe written in the Cartesian ( x, y ) or polar coordinates ( r, θ )as A = B L ( − x + y ) = − B L r cos(2 θ ) , (7)where L is a normalization constant for the length and B is a constant representing the magnetic field at r = L .The linear perturbation approximation for eqs. (2)-(6)reduces to a single equation for δA : (cid:18) η ∂∂t + ( ∇ A ) πρ (cid:19) (cid:18) − c ∂ ∂t + ∇ (cid:19) δA − ∂ ∂t δA = 0 . (8)By using normalized length ¯ r = r/L and time ¯ t = v t/L ,where v = B / (4 πρ ) / , eq. (8) becomes (cid:18) s ∗ ∂∂ ¯ t + ¯ r (cid:19) (cid:18) − σ ∂ ∂ ¯ t + ¯ ∇ (cid:19) δA − ∂ ∂ ¯ t δA = 0 , (9)where s ∗ and σ are non-dimensional parameters given by s ∗ = v Lη , (10) σ = B πρ c = v c . (11)The magnetization parameter σ has been introducedthrough the displacement current, and hence eq. (8) be-comes eq. (2.4) of Craig & McClymont (1991) when theD’Alembertian − σ ∂ ∂ ¯ t + ¯ ∇ is replaced by the Laplacian ¯ ∇ in the limit of σ = 0. It should be noted that v representsthe Alfv´en velocity at radius L only in the nonrelativisticcase. The Alfv´en velocity at L is in general given by V A ≡ cσ / / ( σ + 1) / = v / ( σ + 1) / . For highly magnetizedcases where σ ≫
1, we have V A ≈ c , whereas V A ≈ v for σ ≪
1. Although eq. (9) is used for mathematical calcula-tion, the physical results are presented after normalizationby V A . The Lundquist number S characterizing the systemis defined in terms of the Alfv´en velocity V A , the radius L and resistivity η as S = V A Lη . (12)The related parameter s ∗ is s ∗ = ( σ + 1) / S .Equation (9) exhibits two different behaviors near toand far from the origin. For large ¯ r , the dissipating termwith s ∗ can be neglected, so that we have (cid:20) − σ ¯ r + 1¯ r ∂ ∂ ¯ t + ¯ ∇ (cid:21) δA = 0 . (13) . Kojima et al.: Relativistic magnetic reconnection at X-type neutral points 3 This is exactly the equation in the cold plasma limit for thepropagation of a fast magnetoacoustic wave, whose velocityat ¯ r is given by the Alfv´en velocity v A (¯ r ) ≡ v ¯ r ( σ ¯ r + 1) / = cσ / ¯ r ( σ ¯ r + 1) / . (14)On the other hand, close to the origin, the term with ¯ r can be neglected in eq. (9). After integrating by ¯ t once, wehave (cid:20) − σ ∂ ∂ ¯ t − s ∗ ∂∂ ¯ t + ¯ ∇ (cid:21) δA = 0 . (15)This is the so-called telegraphist’s equation, in which the ef-fect of the finiteness of the velocity c on the resistive losses,or the effect of resistivity on the wave equation, is takeninto account. (See, e.g.,Morse & Feshbach (1953).) In thelimit of σ = 0, the equation becomes the diffusion equa-tion. Thus, eq. (9) leads to an advection-dominated outerregion described by eq. (13) and a diffusion dominated in-ner one described by eq. (15). The diffusion region may behighly modified in nature for large σ , as electromagneticwave propagation becomes important even in the diffusionzone for a highly magnetized system. The critical radius ¯ r c ,which separates the two regions, will be determined by thefollowing normal mode analysis.We solve eq. (9) as an eigenvalue problem in the form δA = f (¯ r ) exp( imθ ) exp( − i ¯ ω ¯ t )= f (¯ r ) exp( imθ ) exp( − i ¯ ωV A ( σ + 1) / t/L ) , (16)where ¯ ω is a complex number. We only consider the axiallysymmetric m = 0 mode, which is relevant to reconnectionat the origin, as discussed in Craig & McClymont (1991).Another type of reconnection for m = 0 is discussed byOfman et al. (1993) and Vekstein & Bian (2005), but thatnot is considered here. Equation (9) becomes1¯ r dd ¯ r ¯ r dd ¯ r f + ¯ ω (cid:18) σ + 1¯ r − i ¯ ωs − ∗ (cid:19) f = 0 . (17)From this, a natural choice of the core radius ¯ r c is of or-der ( | ¯ ω | /s ∗ ) / ∼ S − / and ¯ r c corresponds to the usualskin depth (Craig & McClymont (1991)). The dissipativeterm is dominant for ¯ r < ¯ r c , whereas outside the criticalradius eq. (17) represents wave propagation, since the termwith | ¯ ωs − ∗ | = ¯ r c can be neglected. The current density isconcentrated around the null point.A series solution inside the radius ¯ r c may be expressedas f = 1 −
14 (¯ ω σ + is ∗ ¯ ω )¯ r + · · · , (18)where we have normalized to f = 1 at the origin. We solveeq. (17) with boundary condition (18), from ¯ r = ¯ r c to 1,assuming a complex number ¯ ω . The boundary conditionimposed on the circle ¯ r = 1 is f = 0. This means that themagnetic flux is frozen and δE = δj = δv = 0 there. Thus,we have a one-dimensional eigenvalue problem for ¯ ω .Our main concern is not whole eigenfrequency spec-trum, but rather the lowest frequency mode, which persistsfor a long time in the magnetic reconnection. In particu-lar, we will study the effect of the magnetization parameteron it. For this purpose, we first calculate ¯ ω for the case σ = 0, and then repeat the calculation, gradually changingthe parameter S or σ .
3. Results
The oscillation time t osc is defined in terms of the real partof the eigenfrequency ¯ ω by t osc = 2 πL/ (( σ +1) / Re(¯ ω ) V A ).(A factor ( σ +1) / comes from our normalization of ¯ ω . (Seeeq. (16).) Figure 1 shows the normalized time V A t osc /L asa function of S for several values of σ . Craig & McClymont(1991) showed that the relation V A t osc /L ≈ S ≈ . S holds for a wide range of S with σ = 0. The ori-gin of this relation can be understood by considering thetraveling time of an MHD wave from the outer boundaryto the resistive region, t osc ∼ Z r ∗ Lv A (¯ r ) d ¯ r. (19)The velocity in the limit of σ = 0 is scaled by v A ∝ ¯ r , andthe dominant contribution in eq. (19) comes from a smallcore region. By choosing the lower boundary ¯ r ∗ as ¯ r c , wehave t osc ∝ − ln ¯ r c ∝ ln S .When σ is included, the oscillation time deviates fromthe relation V A t osc /L ≈ S . The normalized time, ingeneral, becomes smaller than that at σ = 0, as shownin Fig. 1. The logarithmic dependence with S can be seenonly in the larger regime, and the coefficient in front ofln S becomes smaller as σ increases. The Alfv´en veloc-ity becomes relativistic for σ > r N ≈ σ − / for σ ≫
1, andthe velocity is almost equal to c outside this radius. Thewave traveling time in eq. (19) is almost determined by theslow region inside ¯ r N , and the system size may be regardedas being effectively reduced to σ − / L . We therefore have V A t osc / ( σ − / L ) ≈ S , i.e, V A t osc /L ≈ σ − / ln S forthe large S regime. This property can be seen from thecurves around log S ≈
50 in Fig. 1, except for σ = 10 . Afactor of ( σ + 1) / instead of σ / may provide a betterextension to σ = 0, but a simple correction is used here.Figure 1 also shows that V A t osc /L approaches a con-stant in the small S regime, for sufficiently large σ .Asymptotically the value of this constant as S → V A t osc /L ≈ ct osc /L ≈ .
5, which is independent of σ , as far as σ ≥ . In our model, the core size increasesas ¯ r c ∝ S − / , and hence the traveling time (19) becomessmaller with decreasing S , but the lower bound is a fewtimes the light crossing time for a region of size L .The critical value S c , which discriminates between con-stant V A t osc /L for smaller S and V A t osc /L ∝ ln S for larger S , is given approximately by ln S c ∼ σ / , or log S c ∼ . σ / . The transition is not very sharp but the relationdoes give the approximate boundary between two distinctbehaviors. Because log S c ∼ σ = 10 and log S c ∼ σ = 10 , which are located at the edges of Fig. 1, the twodifferent behaviors are not clearly shown for these param-eters. This critical value S c also characterizes a transitionin the decay time as will be discussed below.The decay time is related to the imaginary part of ¯ ω , t decay = L/ (( σ + 1) / | Im(¯ ω ) | V A ). Figure 2 shows the nor-malized decay time V A t decay /L as a function of S for sev-eral values of σ . The time for σ = 0 scales as V A t decay /L = 2(ln S ) /π (Craig & McClymont (1991)). This scalingrelation is also broken by the inclusion of σ . The smalland large S regimes are different, as they are for the os-cillation time. A typical example is given by the curve for Y. Kojima et al.: Relativistic magnetic reconnection at X-type neutral points æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ôç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç Σ = Σ = Σ = Σ = Σ = Σ = Τ o s c Fig. 1.
Normalized oscillation time τ osc ≡ V A t osc /L as afunction of Lundquist number S , for magnetization param-eter values σ = 0 , , , , . and 10 . æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ôç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç Σ = Σ = Σ = Σ = Σ = Σ = Τ d eca y Fig. 2.
Normalized decay time τ decay ≡ V A t decay /L as afunction of Lundquist number S , for magnetization param-eter values σ = 0 , , , , . and 10 . σ = 10 : the critical value is log S c ∼ . σ / ∼
13 for thiscase. Logarithmic dependence can be seen for log
S >
S <
7. Therelation V A t decay /L ∝ (ln S ) can be seen in the large S regime, S ≫ S c , except for σ = 10 , but the timescale isreduced to approximately V A t decay /L ≈ σ − / (ln S ) /π for σ ≫
1. The factor σ − / can be interpreted as beingdue to an effective reduction of the system’s size, as con-sidered for the oscillation time. The normalized decay timebecomes the minimum around S c .In the small S regime, S ≪ S c , normalized decay timeapproaches a constant value V A t decay /L ≈ . σ . The nor-malized decay time for fixed S increases with the magneti-zation parameter σ . The limit of σ → ∞ corresponds to thevacuum, in which there is no matter ( ρ = 0) and the dis-sipation time becomes infinite. This σ -dependence comesfrom taking account of the finiteness of c in the resistivelosses. (See eq. (15).) This effect can be neglected in thelarge S regime, where the approximation of instantaneousdissipation is good. However, the effect becomes evident inthe small S regime.The energy E of perturbation decreases due to theOhmic dissipation dEdt = − η Z j dV. (20) The linearized form with Fourier component provides anexpression of the decay time as V A t decay L = 2 S R ( δ ¯ ε B + δ ¯ ε E + δ ¯ ε M )2 π ¯ rd ¯ r R | δ ¯ j | π ¯ rd ¯ r , (21)where δ ¯ ε is dimensionless energy density of magnetic field,electric field, kinetic energy of the fluid, and δ ¯ j is dimen-sionless current density. Their explicit forms are given by δ ¯ ε B = 18 π | δ ¯ B | = 12 | dfd ¯ r | , (22) δ ¯ ε E = 18 π | δ ¯ E | = σ | ¯ ωf | , (23) δ ¯ ε M = 12 ρ | δ ¯ v | = ¯ r | ¯ ω | | ( 1¯ r dd ¯ r ¯ r dd ¯ r + ¯ ω σ ) f | , (24)and | δ ¯ j | = | ( 1¯ r dd ¯ r ¯ r dd ¯ r + ¯ ω σ ) f | . (25)Spatial distributions of these energy densities are displayedin Fig. 3 for S = 10 , σ = 10 and in Fig. 4 for S = 10 , σ = 10 . These functions are calculated by numerical so-lution outside ¯ r c , and by the analytic asymptotic formeq. (18) inside it. Note that a sharp peak in δ ¯ ε M and δ ¯ ε B is located within ¯ r c . Both kinetic energy of matterand magnetic energy are accumulated from outer part tothe core( ∼ ¯ r c ), and are dissipated in the central region.However, distribution of electric energy is flat. These overallfeatures are not so much different in Figs. 3 and 4, althoughthe sharp peak shifts by ¯ r c = (¯ ω/ (( σ + 1) / S )) / .The magnitude of δ ¯ ε E is much smaller than that of δ ¯ ε B in Fig. 3 ( σ = 10 ), whereas δ ¯ ε E becomes comparable to δ ¯ ε B in Fig. 4 ( σ = 10 ). The electric energy is approxi-mately proportional to σ , as shown in eq.(23), and signif-icantly contributes to the sum of energy. Hence, the de-cay time becomes longer with the increase of σ for fixed S , since the total energy increases. (See eq.(21).) In thelarge S regime, however, the functions δ ¯ ε B and δ ¯ ε M aremuch larger than δ ¯ ε E , so that the electric energy can beneglected. The decay time does not increase with σ in thisregime.
4. Discussion and conclusions
Relativistic MHD differs, in general, from the nonrelativis-tic case in at least three ways: (i) the Lorentz factor γ ,(ii) the Coulomb force ρ e E , and (iii) the displacement cur-rent c − ∂E/∂t in Maxwell’s equation. The Lorentz factorappears in the flow velocity and also in the resistivity ofOhm’s law as a Lorentz contraction. The difference is oforder ( v/c ) in magnitude. Since we considered a linearperturbation from the static state, the inflow velocity isnot very large and the Lorentz factor may approximate to γ = 1. The magnitude of ρ e E is of order ( v/c ) times theLorentz force j × B , and is hence neglected in nonrelativis-tic MHD. Moreover, the charge density is always zero dueto the 2D X-point geometry considered here, so that theCoulomb force ρ e E vanishes exactly. This leaves the dis-placement current as a possible factor for the difference be-tween relativistic and nonrelativistic MHD. We have stud-ied its effects, especially on the dynamics of the magnetic . Kojima et al.: Relativistic magnetic reconnection at X-type neutral points 5 δε E / δε Bmax -25 δε M / δε Bmax δε B / δε Bmax
Fig. 3.
Normalized energy density δ ¯ ε as a function of x =ln ¯ r for S = 10 and σ = 10 . The function δ ¯ ε M has a sharppeak, and is shown with a reduction factor 4 × − , while δ ¯ ε E is magnified by 2 × . δε E / δε Bmax -14 δε M / δε Bmax δε B / δε Bmax
Fig. 4.
Normalized energy density δ ¯ ε as a function of x =ln ¯ r for S = 10 and σ = 10 . The function δ ¯ ε M is shownwith a factor 6 × − , while δ ¯ ε E is shown with a factor 6.reconnection using a simplified system based on linearizedequations in the cold plasma limit. The magnetization pa-rameter σ is incorporated in the basic equation through thedisplacement current and the oscillation and decay timesfor the least-damped mode were calculated numerically forparameters S =10-10 and σ = 0-10 .In the system with σ = 0, for which the displacementcurrent can be neglected, the oscillation and decay times areproportional to ln S and (ln S ) , respectively. By including σ , these timescales are modified in different ways, in tworegimes, which are characterized by S ≫ S c or S ≪ S c for S c ≈ exp( σ / ). For low resistivity, S ≫ S c , a logarithmicdependence with S can seen, but the timescales normalizedby the boundary radius L and the Alfv´en velocity V A be-come smaller with increasing σ . The smaller timescales canbe explained as being due to an effective reduction in thesize of the system, or the enlargement of the outer regionwhere MHD waves propagate at almost the speed of lightand the traveling time is negligible. On the other hand, forhigh resistivity, S ≪ S c , a new feature appears in boththe oscillation and decay times, which do not depend on S . The oscillation time is a few times the light crossing timeand does not depend on σ . The dissipation time becomeslonger in proportion to σ and goes to infinity in the limit of σ → ∞ , that is, no dissipation in the vacuum. Reconnectionat the X point is thought to be “fast”, since the dissipationtime is scaled with (ln S ) . Actual time is of the order of10-10 times crossing time with Alfv´en velocity. The dis-placement current significantly spoils the good property,and the timescale increases with σ in high resistive region.The increase of the decay time is related with deficiency ofmatter, which is involved in the Ohmic dissipation.Magnetic reconnection is expected to be an importantprocess of abrupt energy release in the solar and magne-tar flares. For example, the explosive tearing-mode recon-nection in the magnetar like the solar flares is discussed(Lyutikov (2006); Masada et al. (2010)). Dimensionless pa-rameters are however quite different in them: σ ∼ − and S ∼ in solar corona, whereas it is likely that σ ≫ S ≫ t ∼ . σL/V A ∼ − σ ( L/ cm) s under highly magne-tized environment. The spiky rise time ( < . < σL . The energy of the flare ∆ E ( ∼ erg)should be a part of magnetic energy within the volume L : B L ∼ ρ σL > ∆ E . These two conditions provide anupper limit of σ as σ < . ( ρ / (g / cm )) / . In such highenergy events, radiation and possibly pair creation may beimportant in the energy transfer. Further study is neededfor these effects. However, the results in this paper demon-strate that the dynamics significantly depends on the mag-netization parameter through the displacement current. Acknowledgements
This work was supported in part by a Grant-in-Aidfor Scientific Research (No.21540271) from the JapaneseMinistry of Education, Culture, Sports, Science andTechnology.
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