Collisionless magnetic reconnection with arbitrary guide-field

Collisionless magnetic reconnection with arbitrary guide-field Richard Fitzpatrick Center for Magnetic Reconnection Studies Institute for Fusion Studies Department of Physics University of Texas at Austin Austin, TX 7871 USA Franco Porcelli Burning Plasma Research Group Dipartimento di Energetica Politecnico di Torino 1019 Torino Italy A new set of reduced equations governing two-dimensional, two-fluid, collisionless magnetic reconnection with arbitrary guide-field is derived. These equations represent a significant advance in magnetic reconnection theory, since the existing reduced equations used to investigate collisionless reconnection are only valid in the large guide-field limit. The new equations are used to calculate the linear growth-rate of a strongly unstable, spontaneously reconnecting, plasma instability, as well as the general linear dispersion relation for such an instability. rfitzp@farside.ph.utexas.edu porcelli@polito.it 1

I. INTRODUCTION Magnetic reconnection is a fundamental physical phenomenon which occurs in a wide variety of laboratory and space plasmas: e.g., magnetic fusion experiments, 1 the solar corona, and the Earth s magneto-tail. 3 The reconnection process gives rise to a change in magnetic field-line topology with an accompanying release of magnetic energy. Conventional collisional magnetohydrodynamical (MHD) theory is capable of accounting for magnetic reconnection, but generally predicts reconnection rates which are many orders of magnitude smaller than those observed in high-temperature plasmas. 4 On the other hand, more sophisticated plasma models that neglect collisions (which are irrelevant in high temperature plasmas), and treat electrons and ions as separate fluids, yield much faster reconnection rates which are fairly consistent with observations. 5,6 Magnetic reconnection is generally thought of as a basically two-dimensional (-D) phenomenon in which oppositely directed magnetic field-lines spontaneously merge together. However, in many cases of interest, there is also an essentially uniform magnetic field directed perpendicular to the merging field-lines. This latter field is usually termed the guide-field. Unfortunately, the addition of a guide-field significantly complicates the equations governing -D, two-fluid, collisionless magnetic reconnection. In particular, the guide-field couples the compressional Alfvén (CA) wave into these equations, despite the fact that this wave does not play a particularly significant role in magnetic reconnection. Furthermore, the CA wave becomes increasingly problematic (especially in numerical simulations) as the strength of the guide-field increases. Now, it is common practice in conventional MHD theory to reduce the equations governing MHD phenomena by removing the CA wave. 7 Indeed, this approach has been very successful over the years. In this paper, we present a new set of reduced equations governing -D, two-fluid, collisionless magnetic reconnection which is valid for both large and small guide-fields (measured with respect to the reconnecting field). Our new equations represent a significant advance in magnetic reconnection theory, since the existing reduced equations used to investigate collisionless reconnection are only valid

in the large guide-field limit. 8,9 In addition, we calculate the linear growth-rate of a strongly unstable, spontaneously reconnecting, plasma instability from our new equations, as well as the general linear dispersion relation for such an instability. II. DERIVATION OF REDUCED EQUATIONS A. Basic equations Standard right-handed Cartesian coordinates (x, y, z) are adopted. In the following, we employ a normalization scheme such that all lengths are normalized to a convenient length-scale a, all magnetic fields to a convenient scale field-strength B 0, and all times to τ A = a/v A, where V A = B 0 / µ 0 n m i. Here, n is the electron/ion number density, which is assumed to be constant. We also assume that the ions are singly charged. Furthermore, m i is the ion mass. Finally, the electrons are assumed to be hot, and the ions cold. The normalized electron equation of motion takes the form 10 [ ] Ve ɛ d i t + (V e )V e = d i P (E + V e B), (1) where E is the electric field, B the magnetic field, V e the electron velocity, P the electron pressure (which is assumed to be scalar), d i = (c/ω p i )/a (where ω p i = n e /ɛ 0 m i ) the normalized collisionless ion skin-depth, and ɛ = m e /m i the electron-ion mass ratio. The normalized sum of the electron and ion equations of motion yields (neglecting electron inertia) V t + (V )V = P + J B, () where V = V e +d i J is the ion velocity, and J = B. Also, B = 0, and E = B/ t. Our system of equations is closed by a simple energy equation: where Γ = 5/3 is the ratio of specific heats. ( ) t + V P = Γ V P, (3) 3

B. -D equations Suppose that / z 0. Without loss of generality we can write B = ψ ẑ + B z ẑ, (4) and E z = ψ/ t. The z-component of Eq. (1) yields ( ) ( ) ɛ d i t + V e V e z = t + V e ψ. (5) The z-component of the curl of (1) reduces to ( ) ( ) ɛ d i t + V e + V e ω e z = t + V e B z [V e z, ψ] + V e B z, (6) where and ω e = V e, (7) [A, B] A B ẑ. (8) To lowest order in ɛ, Eqs. (5) and (6) can be rearranged to give ( ) t + V ψ e = d i [ψ, B z ], (9) ( ) t + V B e z = [V z, ψ] + d i [ ψ, ψ] V B e z, (10) respectively. Here, where d e = ɛ d i is the collisionless electron skin-depth. ψ e = ψ d e ψ, (11) B e z = B z d e B z, (1) Equation () can be written ( ) t + V V = P B z B z ψ ψ, (13) ( ) t + V V z = [B z, ψ], (14) where V = (V x, V y, 0). 4

C. Reduction scheme Let P = P (0) + B (0) p 1 + p, (15) B z = B (0) + b z, (16) where P (0) and B (0) are uniform constants, p 1, p, and b z are all O(1), and P (0) B (0) 1. (17) The purpose of the above ordering is to make the compressional Alfvén (CA) wave propagate much faster than any other wave in the system. When this occurs, the CA wave effectively decouples from these other waves, and can therefore be eliminated from our system of equations. We expect the rapidly propagating CA wave to maintain approximate force balance within the plasma at all times: i.e., P + B z B z + ψ ψ 0. (18) Hence, it follows from (17) that p 1 b z. (19) We also expect the CA wave to make the plasma flow almost incompressible, so that V φ ẑ. (0) The curl of Eq. (13) yields where U = ψ. Now, to lowest order, Eqs. (3) and (10) become U t = [φ, U] + [ ψ, ψ], (1) 5

b e z = [φ, b z ] + [V z, ψ] + d i [ ψ, ψ] V B (0), t () p 1 t = [φ, p 1] Γ P (0) B (0) V, (3) where b e z = b z d e b z. We can eliminate V between the above two equations, making use of (19), to give Z e t where Z = b z /c β, Z e = Z c β d e Z, c β = = [φ, Z] + c β [V z, ψ] + d β [ ψ, ψ], (4) β/(1 + β), d β = c β d i, and β = Γ P (0) /[B (0) ]. D. Reduced equations where Our new set of reduced collisionless equations takes the form ψ e = [φ, ψ] + d β [ψ, Z], t (5) Z e = [φ, Z] + c β [V z, ψ] + d β [ ψ, ψ], t (6) U t = [φ, U] + [ ψ, ψ], (7) V z t = [φ, V z] + c β [Z, ψ], (8) φ = U, (9) ψ e = ψ d e ψ, (30) Z e = Z c β d e Z, (31) and d β = c β d i, (3) β c β = 1 + β, (33) β = Γ P (0) [B (0) ]. (34) 6

Note that the parameter β measures the strength of the guide-field. The large guide-field limit corresponds to β 1, whereas β 1 corresponds to the small guide-field limit. Note also that d i only occurs in our reduced equations in the combination d β = [β/(1+β)] 1/ d i. In the small guide-field limit, d β d i. However, in the large guide-field limit d β β d i ρ s. Eqs. (5) (8) are generalized versions of the reduced equations presented in Ref. 11 which take electron inertia into account. E. Zero guide-field limit In the zero guide-field limit, B (0) 0, which corresponds to β, we adopt the modified ordering P = P (0) + p 1, (35) B z = b z, (36) where P (0) 1 is a uniform constant, and p 1, b z are O(1). As before, the purpose of this ordering scheme is to make the compressional Alfvén wave propagate much faster than any other wave in the system. From Eq. (3), the above ordering scheme implies that V = 0. In this limit, the remaining two-fluid equations readily yield Eqs. (5) (8), with c β = 1 and d β = d i. We conclude that Eqs. (5) (8) hold in the zero, small, and large guide-field limits, as long as the compressional Alfvén wave propagates more rapidly than any other wave in the system. III. LARGE GROWTH-RATE A. Introduction Let us calculate the linear growth-rate of a reconnecting instability from Eqs. (5) (8) in the limit in which the tearing stability index 1 is very large. 7

Let where denotes a perturbed quantity. ψ(x, y, t) = 1 x + ψ(x) e i (k y+γ t), (37) Z(x, y, t) = Z(x) e i (k y+γ t), (38) φ(x, y, t) = φ(x) e i (k y+γ t), (39) V z (x, y, t) = Ṽz(x) e i (k y+γ t), (40) Throughout most of the plasma, the perturbation is governed by the equations of ideal- MHD, which yield (d /dx k ) ψ = 0. Ideal-MHD breaks down in a thin layer [of thickness O(d β ) or O(d e )] centered on x = 0. 1 Both d e and d β are assumed to be much less than unity. The layer solution must be asymptotically matched to the ideal-mhd solution via the tearing stability index, = [d ln ψ/dx] 0+ 0. For x 1, Eqs. (5) (8) yield the following linearized layer equations: ( g 1 de d ) ψ = i x dx i d β x Z, (41) ( g 1 cβ d e d dx + c ) β g x Z = i d β x d ψ dx, (4) where g = γ/k. Let g d φ dx = i x d ψ dx, (43) φ(p) = + φ(x) e i p x dx, (44) etc. The above Fourier transform should be understood in the generalized sense, since φ(x) is not square-integrable. The Fourier transformed layer equations take the form g ( g (1 + de p ) ψ = d( φ d β Z), (45) dp 1 + cβ de p c β d ) d(p Z ψ) = d g dp β, (46) dp g p d(p φ ψ) =, (47) dp 8

respectively. B. Inner layer In the limit p c β /g, d 1, Eqs. (45) (47) reduce to β ( ) d r dw Q (1 + c dr 1 + r β dr r ) W = 0, (48) where r = d e p, W = p φ/(1 + c β d e p ), and Q = g/d β. In the limit r 0, the asymptotic solution of the above equation takes the form W = W 0 [ r 1 ν + α r ν + O(r 1 ν ) ], (49) where ν = (1/) ( 1 + 4 Q 1), and α is determined by the condition that the solution to (48) be well-behaved as r. Suppose that Q 1, which implies that 0 < ν 1. For r < 1, Eq. (48) yields ( ) d r dw 0. (50) dr 1 + r dr Integrating, and comparing with (49) (recalling that ν 0), we obtain W = W 0 [ (r 1 r) + α + O(r ) ] (51) for r < 1. For r 1, Eq. (48) reduces to d W dr Q (1 + c β r ) W 0, (5) which is a parabolic cylinder equation. 13 The solution which is well-behaved as r is W U(Q/ c β, Q c β r), (53) where U(a, x) is a standard parabolic cylinder function. 13 For 1 r (Q c β ) 1/, we obtain 13 9

Γ(1/4 + Q/4 c W = W 1 β ) Q c β Γ(3/4 + Q/4 c β ) r + O(r ), (54) where Γ(x) is a standard gamma function. 13 Finally, comparing with Eq. (51), we find α = Q 1 G(Q/c β ), (55) where G(x) = x Γ(1/4 + x/4) Γ(3/4 + x/4). (56) Note that G(x) 1 as x. C. Outer layer Suppose that x d e. In this limit, the real-space layer equations (41) (43) reduce to [ d x d ( g dβ d ) φ (g + x ) d φ ] = 0. (57) dx dx g + cβ x dx dx Now, we expect the solution to the above equation to match to the ideal-mhd solution at (relatively) large x. Thus, φ φ [ 0 1 + ( )] 1 1 x + O x (58) at large x, where is the standard tearing stability index. 1 Let Y = g (d φ/dx)/ φ 0. It follows that at large x. Thus, integrating Eq. (57), we obtain Y g 1 x (59) x d ( g d ) β dy (g + x ) Y = g dx g + cβ x dx. (60) In the large limit (i.e., 1/ 0), the above equation reduces to s d ( ) 1 dy Q (1 + s ) Y = 0, (61) ds 1 + c β s ds 10

where s = x/g. Let V = (1 + c β s ) 1 dy/ds. The above equation can be transformed into ( ) d s dv Q (1 + c ds 1 + s β ds s ) V = 0. (6) Note that this outer layer equation has exactly the same form as the inner layer equation, (48). The small s asymptotic solution of Eq. (6) is V = V 0 [ s 1 ν + α 1 s ν + O(s 1 ν ) ], (63) where α 1 is determined by the condition that V not blow-up as s. However, since Eq. (6) has exactly the same form as Eq. (48), we immediately deduce that α 1 = α = Q 1 G(Q/c β ). (64) Fourier transforming (63), we obtain the asymptotic solution ( ) g 1+ νr W = W [α Γ(1 + ν) cos(ν π/) r 1 ν Γ( ν) sin(ν π/) ν + O(r )] 1 ν de (65) in Fourier space. Comparison with (49) yields α = π in the limit ν 0. It follows from (55) that g d e (66) g 3 = π d e d β G (g/c β d β ). (67) Our derivation of this expression is valid provided d e d β Q 1, (68) and d β d e. These constraints all reduce to the single constraint β m e /m i. D. Discussion We have derived the linear dispersion relation (67) from our reduced equations in the large limit. This dispersion relation yields 14 11

( ) 1/3 g = de 1/3 d /3 β (69) π for m e /m i β (m e /m i ) 1/4, and g = 1 Γ(1/4) d e d i (70) π Γ(3/4) for (m e /m i ) 1/4 β. Note that our derivation of (67) is valid for all β values significantly above m e /m i. The dispersion relation (67) was first obtained by Mirnov, et al. 15 However, the derivation presented in Ref. 15 is only valid for β in the range (m e /m i ) β 1. The improved derivation presented in this paper removes the upper constraint on β, and, thereby, confirms that (67) is valid in both the large and small guide-field limits. IV. SMALL GROWTH-RATE Let us now calculate the linear growth-rate of a reconnecting instability from Eqs. (5) (8) in the limit in which the tearing stability index is relatively small. In the small limit, we expect the tearing layer to be localized on the d e length-scale, and to be governed solely by electron physics. Taking Eqs. (45) and (46), and neglecting the ion terms (i.e., the terms involving φ and d Z/dp ), we obtain ( d r d Z ) Q (1 + c dr 1 + r β dr r ) Z = 0, (71) and ψ = d β d Z 1 + de p dp, (7) where r = d e p and Q = g/d β. The solution to (71) which is well-behaved as r has the small r expansion Z = Z 0 [r 1 + α + O(r)], (73) assuming that Q 1. Here, α is given by Eq. (55), since Eq. (71) has exactly the same form as Eq. (48). Inverse Fourier transforming, we obtain 1

Z = Z [ π 0 d e α + 1 ( )] 1 x + O, (74) x and ψ x Z = Z [ ( )] π x 1 0 d e α + 1 + O. (75) x It follows by matching to the standard ideal-mhd solution, that = π/d e α, or ψ = ψ 0 [ x + 1 + O ( 1 x ) ], (76) = π d e d β g G(g/c β d β ). (77) Hence, we conclude that 14 g = d β π d e (78) for β ( d e ), and g = ( ) Γ(1/4) d i ( d e ) (79) π Γ(3/4) for β ( d e ). The dispersion relation (77) was first obtained by Mirnov, et al. 15 V. GENERAL DISPERSION RELATION Let us now derive a general dispersion relation from Eqs. (5) (8) which holds for large, small, and intermediate values of. For finite, Eqs. (60) and (6) yield s d ( ) 1 dy Q (1 + s ) Y = ˆλ ds 1 + cβ H Q, (80) s ds ( ) d s dv Q (1 + c ds 1 + s β s ) V = ˆλ H Q d ( ) 1, (81) ds ds 1 + s 13

where s = x/g, λ H = π/, and ˆλ H = (λ H /d β )(π/). Note, from Eq. (58), that Y (s) is subject to the constraint 0 Y (s) ds = 1. (8) Our basic expansion parameter is H = G(Q/c β )/Q [see Eq. (56)]. Note that H 1 provided Q 1. It can easily be verified from the following analysis that V O(Q ) and ˆλ H < O(Q/H). It follows that in Eqs. (80) and (81) the inhomogenous term on the right-hand side can be neglected for s H 1, whereas the second term on the left-hand side can be neglected for s H. In the limit s H, Eq. (81) reduces to which can be solved to give ( ) d s dv ˆλ ds 1 + s H Q d ( ) 1, (83) ds ds 1 + s V a 0 ˆλ H Q s + a 1 + a 0 s, (84) where a 0 (Q, c β ) and a 1 (Q, c β ) are constants of integration. Note that the first two terms on the right-hand side of the above expression are negligible for s H 1, and vice versa. In the limit s H 1, the above expression can be asymptotically matched to the inner layer solution described in Sect. III B. In real space, this solution yields (for Q 1) It therefore follows that V V 0 [ π ] d e G(Q/c β ) 1 d β Q s + 1 + O(s). (85) a 0 ˆλ H Q a 1 = π d e G(Q/c β ). (86) d β Q In the limit s H 1, Eq. (81) reduces to d V ds Q (1 + c β s ) V 0. (87) This is a parabolic cylinder equation. 13 The solution which is well-behaved as s has the expansion 14

] Q V V 0 [1 G(Q/c β ) s + O(s ) for s H. Asymptotic matching to Eq. (84) yields Note that for s H 1 we can write where U(a, x) is a parabolic cylinder function. 13 (88) a 0 = Q a 1 G(Q/c β ). (89) U(Q/ c β, Q c β s) V a 1, (90) U(Q/ c β, 0) Dividing Eq. (80) by 1 + s and then integrating from s = 0 to, we obtain 0 s dv 1 + s ds ds = Q ˆλ H Q π, (91) where use has been made of Eq. (8). The dominant contribution to the integral in the above expression comes from the region s 1. Furthermore, ˆλ H < O(Q/H), so the second term on the right-hand side is negligible. Hence, it follows from Eq. (90) that a 1 U(Q/ c β, s= Q c β s) Q, (9) U(Q/ c β, 0) or s=0 Equations (86), (89), and (93) can be combined to give Q G(Q/c β ) ˆλ H Q = π Hence, our general dispersion relation takes the form λ H = π a 1 Q. (93) Q d β G(Q/c β ) d e d e G(Q/c β ), (94) d β Q G(Q/c β ), (95) Q where λ H = π/. This result is valid provided Q 1, which implies that λ H d β, and β m e /m i. The above dispersion relation is consistent with the dispersion relations (67) and (77) which we previously derived in the large and small limits, respectively. Equation (95) was first obtained by Mirnov, et al. 15 using a heuristic argument. We, on the other hand, have derived this result in a rigorous fashion. In the limit β 1, the above expression reduces to a dispersion relation previously obtained by Porcelli. 14 15

VI. SUMMARY We have derived a novel set of reduced equations, (5) (8), governing -D, two-fluid, collisionless magnetic reconnection with (essentially) arbitrary guide-field. These equations represent a major advance in reconnection theory, since the reduced equations previously used to investigate collisionless reconnection 8,9 are only valid in the large guide-field limit. Using our equations, we have calculated the linear growth-rate of a highly unstable, spontaneously reconnecting, plasma instability. Our expression for the growth-rate, (67), is valid in both the large and small guide-field limits. This expression was previously obtained by Mirnov, et al. 15 However, the derivation given in Ref. 15 is only valid in the large guidefield limit. Interestingly enough, expression (67) indicates that the guide-field ceases to have any effect on the linear growth-rate once the plasma β calculated with the guide-field significantly exceeds the rather small critical value (m e /m i ) 1/4 0.16 (for hydrogen). We have also derived the general linear dispersion relation, (95), of a spontaneously reconnecting plasma instability. This result was previously obtained Mirnov, et al. 15 using a heuristic argument. In future work, we intend to investigate the nonlinear properties of our new equations. Acknowledgments This research was funded by the U.S. Department of Energy under contract DE-FG05-96ER-54346 as well as via Cooperative Agreement No. DE-FC0-01ER-5465 under the auspices of the program for Scientific Discovery through Advanced Computing (SIDAC). FP was supported in part by the Euratom Human Capital and Mobility Scheme. 1 F.L. Waelbroeck, Phys. Plasmas 1, 37 (1989). K. Shibata, Adv. Space Res. 17, 9 (1996). 16

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