Collisionless magnetic reconnection in the presence of a sheared velocity field

Collisionless magnetic reconnection in the presence of a sheared

velocity field

M. Faganello,1F. Pegoraro,2F. Califano,2and L. Marradi2,3

1

École Polytechnique, LPP, Palaiseau, 91128 France 2

Department of Physics, University of Pisa and CNISM, Pisa, 56127 Italy 3

Université de Nice Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, 06304 Nice, France

共Received 18 March 2010; accepted 26 April 2010; published online 4 June 2010兲

The linear theory of magnetic field lines reconnection in a two-dimensional configuration in the presence of a共Kelvin–Helmholtz stable兲 sheared velocity field is investigated within a single fluid model, where the onset of magnetic field line reconnection is made possible by the effect of electron inertia in the so called large⌬

⬘

regime. © 2010 American Institute of Physics.

关doi:10.1063/1.3430640兴

I. INTRODUCTION

Inhomogeneous current distributions and sheared veloc-ity fields are ubiquitous features of laboratory and space plasmas. In fact, together with pressure gradients in the pres-ence of curved magnetic field lines or gravity forces, they represent two of the most important sources of instability in a magnetized plasma. Velocity field shear may lead to the onset of the Kelvin–Helmholtz instability1,2 that causes the formation of velocity vortices that, in the nonlinear phase of the instability, roll up and pair.3–5Current layers on the con-trary tend to tear apart, causing magnetic field lines to reconnect.6–8 The Kelvin–Helmholtz instability is mainly a hydrodynamic, ideal instability while magnetic reconnection is related to the evolution of the magnetic field in the plasma. Nevertheless, the fact that at low frequencies and large spa-tial scales in a high conductivity plasma, the dynamics of the magnetic field and of the plasma are tightly linked by the so called “frozen-in” condition brings the evolution of the ve-locity and of the magnetic field together.

In principle, magnetic reconnection is a local instability that should be expected to come into play once faster, large scale magnetic instabilities occurring on the Alfvèn time scale have developed in the plasma. However, even if the breaking of the field lines occurs only locally, the resulting restructuring of the magnetic connection between the plasma fluid elements is global and, in addition, if externally driven, magnetic reconnection can occur on time scales that are not very different from the ideal Alfvèn times and from the hy-drodynamical time scales.9–13

For these reasons in the literature, the interplay between the Kelvin–Helmholtz instability and magnetic field line re-connection has long since been studied with two main aims: investigate the stabilizing effect of a velocity shear field on the development of magnetic reconnection in configurations that are Kelvin–Helmholtz stable14–18 on the one hand and, on the other, investigate how the evolution of the Kelvin– Helmholtz instability can stretch the magnetic field lines and lead to field line reconnection, provided the resulting field tension be weak enough not to inhibit the onset and the evo-lution of the Kelvin–Helmholtz instability.1,2 The Kelvin– Helmholtz instability can drive magnetic reconnection in two

different ways. If the magnetic field has an inversion line关we refer, for simplicity, to a two-dimensional 共2-D兲 configura-tion, see below兴, the fluid velocity field, modified by the linear evolution of Kelvin–Helmholtz, forces the magnetic lines to reconnect and to follow the evolution imposed by the Kelvin–Helmholtz instability.18–22 On the contrary, if the magnetic field initially has no inversion line, magnetic recon-nection cannot occur during the linear phase of the Kelvin– Helmholtz instability. However, the nonlinear evolution of the Kelvin–Helmholtz vortices is able to roll up and stretch the magnetic lines, producing inversion regions and leading to the formation of reconnection unstable current layers.12,13,21,23–26

The modification of the reconnection process that occurs in the presence of a Kelvin–Helmholtz stable, sheared velocity field has been mainly investigated in 2-D configurations,14–18 where the sheared velocity field and the inhomogeneous magnetic field have coincident null lines in the x-y plane, while z is an ignorable coordinate. In these configurations, the magnetic field component along z, if present, is taken to be homogeneous over the spatial scales of interest. Furthermore, fluid regimes were generally consid-ered where the local violation of the perfect conductivity condition was caused by the effect of a small but finite resistivity.14–18This is not an accurate model when we aim at studying almost dissipationless plasma regimes where a ki-netic plasma description should be introduced, at least in the regions were the current layers form and where, even if a fluid description is still adopted, resistivity is not the main effect that violates the perfect conductivity condition.

In the present paper, we revisit analytically the linear theory of magnetic field lines reconnection in a 2-D configu-ration共i.e., we consider perturbations that do not depend on the z coordinate兲 within a single fluid model where, however, the onset of magnetic field line reconnection is made pos-sible by the effect of electron inertia27,28instead of resistivity. This magnetohydrodynamic共MHD兲 model with electron in-ertia has a number of interesting differences with respect to the resistive model. First it maintains the Hamiltonian, i.e., dissipation free, character of high temperature dilute plasma regimes,29as would be the case for a full kinetic description.

Second, it strongly discriminates between the “reconnecting” 共large ⌬

⬘

兲 regime from the “tearing” 共small ⌬

⬘

, also called constant-␺兲 regime. In fact, in the MHD plus electron inertia description, the growth rate of the tearing instability turns out to be too small to be of relevance as it scales,30,31in the absence of a velocity field, proportionally to共de/L兲3. Here, L is a macroscopic scale length and deⰆL is the collisionless inertial skin depth. On the contrary, in the large⌬

⬘

regime, the instability growth rate scales27,31 as de/L. Thus in this article we will consider the large⌬

⬘

regime only.

We will show that in a magnetic inverted equilibrium that is Kelvin–Helmholtz stable, the growth rate of the col-lisionless magnetic reconnection instability is modified by the presence of a sheared velocity field. In particular, al-though the large⌬

⬘

growth rate still scales as de/L 共recon-necting regime兲, reconnection is partially stabilized by the sheared velocity field. Furthermore, we will show that even in the large⌬

⬘

regime, the reconnection instability is fully stabilized by the velocity field when the slope of the equilib-rium velocity profile equals the slope of the Alfvén velocity profile at the magnetic null line. As shown analytically in Refs. 14, 15, and 17 and numerically in Refs. 16 and 18, when the fluid velocity profile is steeper than the Alfvén velocity profile a transition to the ideal Kelvin–Helmholtz instability occurs. In this case, reconnection is not an insta-bility but is simply driven by the far faster ideal instability.18–22

II. PLASMA MODEL

A. Governing equations and equilibrium configuration

We start from the dissipationless MHD set of equations in the incompressible limit with electron inertia included in Ohm’s law. We write this set in the form

冋

⳵U ⳵t +共U · ⵜ兲U

册

= −ⵜP + 共ⵜ ⫻ B兲 ⫻ B, 共1兲 E +U⫻ B c = de 2

冋

⳵J ⳵t +共U · ⵜ兲J

册

, 共2兲 ⵜ ⫻ E = −⳵B ⳵t, ⵜ ⫻ B = J, 共3兲 ⵜ · U = 0, ⵜ · B = 0, 共4兲

where dimensionless units are used normalized on the mag-netic equilibrium spatial scale length LB, on the Alfvén ve-locity UA= B0/

冑

4␲␳0with␳0the constant mass density, and on the corresponding Alfvén time␶A= LB/UA. Here, deis the normalized electron collisionless skin depth.

We consider a one-dimensional equilibrium configura-tion where the three-dimensional magnetic and velocity fields depend on the inhomogeneity coordinate x only

B_eq= By共x兲yˆ + Bz共x兲zˆ, Ueq= Uy共x兲yˆ + Uz共x兲zˆ. 共5兲 Since we are interested here in the investigation of how an equilibrium velocity field can modify the development in the reconnection instability, we consider a velocity jump⌬Uy/z

that is lower than the Alfvén velocity UA,y/z. In this case, the magnetic tension is strong enough to prevent the develop-ment of the Kelvin–Helmholtz instability.1,2

B. Linearized equations

Now we consider small perturbations of the equilibrium fields which we Fourier transform along the homogeneous directions y and z. Thus, the linearized equations, obtained by neglecting the second order terms, are

共␥+ i␣G兲共W

⬙

−␣2W兲 − i␣G

⬙

W = i␣F共␺

⬙

−␣2␺兲

− i␣F

⬙

␺, 共6兲

共␥+ i␣G兲␺− i␣FW = de2兵共␥+ i␣G兲共␺

⬙

−␣2␺兲 − i␣F

⬙

W其, 共7兲 where␥is the growth rate,␣= kLBis the normalized module of the wave vector, k2= ky2+ kz2, and

F共x兲 =k · Beq共x兲 kB0

, G共x兲 =k · U共x兲eq共x兲 kUA

. 共8兲

Note that the incompressibility limit allows us to describe the linear evolution using only the perturbed magnetic flux

␺= Bpert,x/B0and the x component W = Upert,x/UAof the per-turbed velocity, and to take into account equilibrium varia-tion only along the wave vector direcvaria-tion. In fact, F共x兲 and G共x兲 measure the magnetic and velocity equilibrium varia-tions along the wave vector direction, respectively. The in-compressibility condition 共4兲 is well satisfied if there is a strong guide magnetic field perpendicular to the plane where the instability develops 共i.e., the plane defined by the x-direction and the wave vector k兲.

III. BOUNDARY LAYER APPROACH

For the sake of simplicity, we assume that the magnetic field has only one inversion point at x = 0, i.e., F共0兲=0. Fur-thermore, we assume that the velocity profile is monotonic and that G共0兲=0. This last condition can always be satisfied by a change in the reference frame.

Thus, if we neglect all the terms proportional to the small parameter de

2

in Eqs.共6兲 and共7兲, for ␥Ⰶ1, the corre-sponding ideal MHD equations are singular at the inversion point x = 0. As usual, we consider a boundary layer approach and divide the whole space in an external region, where the ideal MHD equations hold, and in an internal layer across the singular point x = 0, where the small d_e2 terms are able to remove the singularity.

A. External region

In the external region, we can neglect all terms propor-tional to the small parameters de

2

and␥. Thus Eqs.共6兲and共7兲 correspond to stationary ideal MHD equations

i␣G␺− i␣FW = 0. 共10兲 These equations correspond to the usual zeroth order expan-sion of the of the Hydromagnetic equations analyzed in Refs. 14 and 17. The internal solution must match the external solution at all orders in the small expansion parameters␥and de

2

, but it is sufficient to consider the zeroth order of the external solution in order to obtain a physical solution.

Following the standard approach,7we estimate the value of the internal solutions ␺, W considering the limit of Eq. 共10兲 for x→0. This “use of the frozen-in law for internal solutions” leads to

␺⬃ i␣F

⬘

共0兲⑀

␥+ i␣G

⬘

共0兲⑀W, 共11兲 where⑀Ⰶ1 is the width of the internal region. This scaling is valid in both the external and internal region.14,15,17

B. Internal region

Let us consider an internal layer of width ⑀Ⰶ1 across the singular point x = 0. In this “very thin” region, the deriva-tives of␺and W became so large that the d_e2terms in Eqs.共6兲 and共7兲 can no longer be neglected.

We consider a “stretched” variable␰= x/⑀and the Taylor series in x = 0 of the equilibrium fields G共x兲=G

⬘

共0兲x + 1/2G

⬙

共0兲x2_{+ O共x}3_{兲 and F共x兲=F}

_⬘

_{共0兲x+1/2F}

_⬙

_共0兲x2_{+ O共x}3_兲. Equations共6兲 and共7兲 now read

冋

␥ ␣F

⬘

共0兲⑀+ i G

⬘

共0兲 F

⬘

共0兲␰+ 1 2i G

⬙

共0兲 F

⬘

共0兲⑀␰ 2

册

⳵ 2_W ⳵␰2 − i G

⬙

共0兲 F

⬘

共0兲⑀W =

冋

i␰+1 2i F

⬙

共0兲 F

⬘

共0兲⑀␰ 2

册

⳵ 2_␺ ⳵␰2− i F

⬙

共0兲 F

⬘

共0兲⑀␺+ O共⑀ 2_{兲, 共12兲}

冋

␥ ␣F

⬘

共0兲⑀+ i G

⬘

共0兲 F

⬘

共0兲␰+ 1 2i G

⬙

共0兲 F

⬘

共0兲⑀␰ 2

册

_{␺− i␰W} − 1 2i F

⬙

共0兲 F

⬘

共0兲⑀␰ 2_W = de 2 ⑀2

再

冋

␥ ␣F

⬘

共0兲⑀+ i G

⬘

共0兲 F

⬘

共0兲␰+ 1 2i G

⬙

共0兲 F

⬘

共0兲⑀␰ 2

册

⳵ 2_␺ ⳵␰2 − iF

⬙

共0兲 F

⬘

共0兲⑀W

冎

+ O共⑀ 2_{兲. 共13兲}

As usual, we suppose that inside the internal layer the mode amplitude variation along x is far more important than those along y and z, i.e.,⳵/⳵␰Ⰷ␣.

C. Matching condition

The solutions obtained in the external and internal region must overlap in the limit兩x兩→0 and 兩␰兩→⬁. We define the standard matching parameter7 ⌬

⬘

as

⌬

⬘

=

冏

1 ␺ext共x兲 ⳵␺ext共x兲 ⳵x

冏

0₋ 0+ =

冏

1 ⑀␺int共␰兲 ⳵␺int共␰兲 ⳵␰

冏

−⬁ +⬁ . 共14兲 The value of the parameter ⌬

⬘

is set by the external ideal solution and thus it can be strongly modified by the

occur-rence of an equilibrium velocity field.14,17 However, if the large scale magnetic and velocity field are such that G共x兲 ⬀F共x兲, the value of ⌬

⬘

is unaffected with respect to its value ⌬U=0

⬘

in the case of a stationary plasma. The fact that⌬

⬘

is strongly affected by the velocity field should not appear sur-prising, since the magnetic field in the external region is frozen-in the plasma motion.

IV. KINK MODE

In the case of a stationary plasma equilibrium U_eq= 0, the collisionless kink mode27,31has a growth rate proportional to de, while the collisionless tearing mode, or so called constant-␺, scales as de3.

30,31

For this reason, the kink mode has attracted more attention than the tearing mode. These two regimes are usually characterized by the value of the matching parameter⌬

⬘

Ⰷ1 for the kink mode and ⌬

⬘

⬃1 for the tearing mode and are known as the fast growth regime

␥⬃兩␣F

⬘

共0兲⑀兩 and the slow growth regime ␥Ⰶ兩␣F

⬘

共0兲⑀兩, respectively.17

In general, the collisionless tearing mode is too slow to be competitive with other plasma processes that influence the plasma dynamics. Therefore, we focus our attention on the kink mode evolution, or fast growth regime, where

冏

␥

␣F

⬘

共0兲⑀

冏

⬃ 1. 共15兲 Modifications resulting from the presence of a large scale flow Ueq⫽0 in Eqs.共6兲 and共7兲 are controlled by the parameter G

⬘

共0兲/F

⬘

共0兲 which represents the ratio of the ve-locity and magnetic shear at the magnetic null point.

As shown analytically in Refs. 14, 15, and 17 and numerically in Refs. 16 and 18, there can be a transition to the ideal Kelvin–Helmholtz instability共much faster than the reconnection instability兲 when this parameter is larger than 1. In the following we will consider the regime 兩G

⬘

共0兲/F

⬘

共0兲兩ⱗ1.

A. Scaling laws

In the internal layer and using the frozen-in condition given by Eq.共11兲, in the fast growth regime Eq.共15兲allows us to state that

␺⬃ W. 共16兲

At zeroth order in the small parameters ⑀and de2, Eqs. 共12兲 and共13兲read

冋

␥ ␣F

⬘

共0兲⑀+ i G

⬘

共0兲 F

⬘

共0兲␰

册

⳵2_W ⳵␰2 = i␰ ⳵2_␺ ⳵␰2, 共17兲

冋

␥ ␣F

⬘

共0兲⑀+ i G

⬘

共0兲 F

⬘

共0兲␰

册

␺− i␰W =de 2 ⑀2

冋

␥ ␣F

⬘

共0兲⑀+ i G

⬘

共0兲 F

⬘

共0兲␰

册

⳵2_␺ ⳵␰2. 共18兲 The balance between the various terms of Eqs.共17兲and共18兲 gives

␥ ␣F

⬘

共0兲⑀⬃ 1, ␰⬃ 1 ⇒ de 2 ⑀2 ␥ ␣F

⬘

共0兲⑀⬃ 1, 共19兲 and thus we have

• Internal layer width⑀⬃deand • Growth rate␥⬃␣F

⬘

共0兲de⬀de.

These scaling laws do not depend on the value of 兩G

⬘

共0兲/F

⬘

共0兲兩, provided it is comparable or smaller than 1. For this reason, these scaling laws are the same as those obtained for the collisionless kink mode in a stationary plasma.27,31

B. Small velocity shear regime

When the velocity shear at the magnetic null point is small compared to the magnetic shear, i.e., 兩G

⬘

共0兲/F

⬘

共0兲兩 Ⰶ1, it is possible to obtain the value of the growth rate in the kink regime, i.e., for ⌬

⬘

Ⰷ1, explicitly with an expansion procedure. Equations共17兲and共18兲can be rewritten as

⳵ ⳵␰

冉

A2 ⳵␾ ⳵␰

冊

= −␰ ⳵2_␺ ⳵␰2, 共20兲 A共␺−␰␾兲 =de 2 ⑀2 ⳵2_␺ ⳵␰2A, 共21兲 where A = ␥ ␣F

⬘

共0兲⑀+ i G

⬘

共0兲 F

⬘

共0兲␰, ␾= iW A . 共22兲

By integrating Eq.共20兲we can define the new variable X

␰⳵␺

⳵␰ −␺= − A2 ⳵␾

⳵␰ + C0⬅ X, 共23兲 and from Eq.共21兲, we have

de2 ⑀2

冉

⳵2_X ⳵␰2 − 2 ␰ ⳵X ⳵␰

冊

= X + ␰2 A2共X − C0兲. 共24兲

Furthermore we can state that

冏

⳵␺ ⳵x

冏

₀− 0+ =

冕

0− 0+⳵2_␺ ⳵x2dx = 1 ⑀

冕

−⬁ +⬁_⳵2_␺ ⳵␰2d␰= 1 ⑀

冕

−⬁ +⬁₁ ␰ ⳵X ⳵␰d␰. 共25兲 Since, as we will see,␺→C0for兩␰兩→ ⫾⬁, the usual match-ing condition can be rewritten as

⌬

⬘

= 1 ⑀C0

冕

−⬁ +⬁₁ ␰ ⳵X ⳵␰d␰. 共26兲

We can thus consider the power expansions in the small pa-rameter兩G

⬘

共0兲/F

⬘

共0兲兩 X =

兺

n=0 ⬁ Xn

冋

i G

⬘

共0兲 F

⬘

共0兲

册

n =

兺

n=0 ⬁ Xn⌫ n , 共27兲 ␥=

兺

n=0 ⬁ ␥n

冋

i G

⬘

共0兲 F

⬘

共0兲

册

n =

兺

n=0 ⬁ ␥n⌫n. 共28兲 At zero order in the small parameter ⌫, Eqs. 共22兲–共24兲 become A = ␥0 ␣F

⬘

共0兲⑀+ O共⌫兲, 共29兲 de 2 ⑀

冉

⳵2_X 0 ⳵␰2 − 2 ␰ ⳵X0 ⳵␰

冊

= X0+ ␣2_F

_⬘

_共0兲2_⑀2 ␥02 ␰2_共X 0− C0兲. 共30兲 The asymptotic behavior of the zeroth order solution is thus given by X0→ C0− C0␥0 2 ␣2_F

_⬘

_共0兲2_⑀2 1 ␰2 for 兩␰兩 → + ⬁. 共31兲 From the matching condition共26兲 we have, in the Kink re-gime⌬

⬘

⑀Ⰷ1, C0= 0, X0= e−␰ 2_/2 , de 2 ⑀2= ␣2_F

_⬘

_共0兲2_⑀2 ␥02 = 1, 共32兲 and thus • ␥0=␣F

⬘

共0兲de=␥U_eq=0 • ⑀= de.

Note that this is the solution of the singular Sturm–Liouville problem given by Eq.共30兲with C0= 0. The solution共32兲has no inversion point and thus represents the fundamental state of this problem, corresponding to the smallest eigenvalue

␣2_F

_⬘

_共0兲2_⑀2_/␥ 0

2_{, and thus to the largest growth rate ␥} 0. The singularity is related to the existence of the continuous spec-trum. Its contribution to the perturbation evolution is propor-tional to t−兩a兩and is thus negligible compared to the exponen-tial instability.32–34

At the first order in⌫ we have

A = ␥0 ␣F

⬘

共0兲⑀+

冋

␰+ ␥1 ␣F

⬘

共0兲⑀

册

⌫ + O共⌫ 2_兲 = 1 +

冉

␰+␥1 ␥0

冊

⌫ + O共⌫2_{兲, 共33兲} 1 A2= 1 − 2

冉

␰+ ␥1 ␥0

冊

⌫ + O共⌫2_{兲, 共34兲} ⳵2_X 1 ⳵␰2 − 2 ␰ ⳵X1 ⳵␰ =共1 +␰2兲X1− 2␰2

冉

␥1 ␥0 +␰

冊

X₀. 共35兲 Once that the zeroth order solution X0 is fixed by Eq.共32兲, we can determine the first order solution X1that satisfies the proper boundary condition given by X1→0 for 兩␰兩→+⬁

X1= 1 3␰

3_e−␰2/2_{, ␥}

1= 0. 共36兲

In order to obtain a correction to the stationary configuration growth rate␥0=␥U_eq=0, we are forced to consider the second order of the expansion series in ⌫. At this order, Eqs. 共22兲–共24兲read

A = 1 +⌫␰+␥2 ␥0 ⌫2_{+ O共⌫}3_{兲, 共37兲} 1 A2= 1 − 2⌫␰+

冉

3␰ 2_{− 2}␥2 ␥0

冊

⌫2_{+ O共⌫}3_{兲, 共38兲} ⳵2_X 2 ⳵␰2 − 2 ␰ ⳵X2 ⳵␰ =共1 +␰2兲X2− 2␰3X1+␰2

冉

3␰− 2 ␥2 ␥0

冊

X0. 共39兲 Given the lower order solutions X0 and X1, we obtain the second order solution

X2= 1 18␰ 6_e−␰2/2₋1 4␰ 4_e−␰2/2_, ␥2 ␥0 =1 2. 共40兲

Thus we obtain for the growth rate␥U_eq⫽0

␥U_eq⫽0⬃␣F

⬘

共0兲de兵1 − 关G

⬘

共0兲/2F

⬘

共0兲兴2其

=␥U_eq=0兵1 − 关G

⬘

共0兲/2F

⬘

共0兲兴2其. 共41兲 In the small velocity shear regime, the effect of the equilib-rium velocity field is to provide a small reduction in the growth rate of the kink mode.

Note that for consistency, ⌫2=关G

⬘

共0兲/F

⬘

共0兲兴2 must be greater than the small parameter⑀⬃deas we have neglected all the terms O共⑀兲.

C. Comparable shear regime

As stated before, the scaling laws obtained in Sec. IV A do not depend on the value of G

⬘

共0兲/F

⬘

共0兲. Thus, the kink growth rate is proportional to de also when G

⬘

共0兲/F

⬘

共0兲 ⱗ1.

It is important to notice that there can be a transition to the ideal Kelvin–Helmholtz instability when 兩G

⬘

共0兲/F

⬘

共0兲兩 →1,14–18

and thus a suppression of the reconnection instabil-ity.

The Kelvin–Helmholtz ideal instability is far faster than the reconnection instability. Furthermore, it has been shown that the velocity field, generated by Kelvin–Helmholtz insta-bility, is able to force the magnetic field lines to reconnect.18–22This reconnection is not an instability but is induced by the ideal Kelvin–Helmholtz instability. In fact, for a resistive plasma, the rate at which it happens is inde-pendent from the value of the resistivity and is related only to the Kelvin–Helmholtz growth rate.18,20

It is also possible to show that the collisionless kink mode becomes stable for 兩G

⬘

共0兲/F

⬘

共0兲兩→1. In fact when 兩G

⬘

共0兲/F

⬘

共0兲兩 approaches 1, the constant-␺approximation is valid also for⌬

⬘

Ⰷ1, as shown in the resistive case in Refs. 14and17. There is thus a transition from the kink mode to the slower tearing mode, which is stable for 兩G

⬘

共0兲/F

⬘

共0兲兩 →1. This not only leads to a slower growth rate, but also to a total stabilization of the collisionless kink mode 共see Ap-pendix for analytical results兲. In order to confirm the scaling law ␥⬀de and the stabilization of the Kink mode for 兩G

⬘

共0兲/F

⬘

共0兲兩→1, Eqs.共6兲and共7兲have been integrated nu-merically in the parameter ranges 10⬍⌬

⬘

⬍20, 0.1⬍de ⬍0.5, and 5000⬍S_␩⬍10 000, where S_␩ is the resistive

Lundquist number. Although a finite but small resistivity, re-quired for numerical stability, is included in the linearized mode equations, the numerical and the analytical results are in good qualitative agreement.

V. CONCLUSION

With the help of a dissipationless MHD model that in-cludes the effect of electron inertia, it has been shown that a sheared velocity field in a 2-D equilibrium configuration with a magnetic inversion line modifies the onset conditions and the growth rate of the collisionless kink mode. In par-ticular, the sheared velocity field is able to stabilize partially the kink mode and to lower the value of the growth rate at which reconnection proceeds. This stabilization does not de-pend on whether the velocity field is parallel or antiparallel to the magnetic field, but only on the absolute value of the gradient of the equilibrium velocity compared to the gradient of the inverted magnetic field component inside the recon-necting layer.

When the absolute value of ratio between the equilib-rium velocity gradient and the Alfvén velocity gradient 共as-sociated with the inverted magnetic field component兲, inside the reconnecting layer, is equal to 1, the collisionless kink mode is totally stabilized. In particular, when this ratio ap-proaches 1, the use of the constant-␺approximation is valid also in the kink regime. This allows us to show the total kink stabilization explicitly共see Appendix兲.

When the ratio between the equilibrium velocity gradi-ent and the Alfvén velocity gradigradi-ent is greater than 1, the system usually becomes Kelvin–Helmholtz unstable. In that case, the reconnection occurring during the development of the Kelvin–Helmholtz instability is thus driven by the faster ideal instability.18–22

APPENDIX: SLOW GROWTH REGIME

Let us consider the slow growth regime, where

␥/␣F

⬘

共0兲⑀Ⰶ1, in the “comparable shear” regime, where 兩G

⬘

共0兲兩/F

⬘

共0兲⬃1. For the sake of simplicity, we take

F

⬙

共0兲 = G

⬙

共0兲 = 0, F

⬘

共0兲 ⬎ 0, G

⬘

共0兲 ⬎ 0. 共A1兲 From Eqs.共12兲and共13兲that describe the plasma dynamics inside the internal layer, we see that the only way to balance the different terms is to consider, once again, d_e2/⑀2_⬃1. From the use of the frozen-in law inside the internal region 共11兲and remembering that 兩G

⬘

共0兲/F

⬘

共0兲兩=O共1兲, we can es-timate the relative importance of␺and W

␺⬃ i␣F

⬘

共0兲⑀

␥+ i␣G

⬘

共0兲⑀W⇒␺⬃ W. 共A2兲 This estimate is different from the standard stationary plasma estimate.7 This is not surprising since, in our case, the con-vective term G

⬘

共0兲/F

⬘

共0兲 is far more important that the time derivative term␥/␣F

⬘

共0兲⑀.

In order to obtain the growth rate, we can consider the power expansions of the solution in the small parameter

W =

兺

n=0 ⬁ Wn关␥/共␣F

⬘

共0兲⑀兲兴n, ␺=

兺

n=0 ⬁ ␺n关␥/共␣F

⬘

共0兲⑀兲兴n. 共A3兲 The zeroth order of Eqs.共12兲and共13兲in the small parameter

␥/␣F

⬘

共0兲⑀gives us iG

⬘

共0兲 F

⬘

共0兲␰ ⳵2_W 0 ⳵␰2 = i␰ ⳵2_␺ 0 ⳵␰2 , 共A4兲 iG

⬘

共0兲 F

⬘

共0兲␰␺0− i␰W0= i G

⬘

共0兲 F

⬘

共0兲␰ ⳵2_␺ 0 ⳵␰2 , 共A5兲 where we have taken⑀= de. The constant-␺solution

G

⬘

共0兲

F

⬘

共0兲␺0= W0= G

⬘

共0兲

F

⬘

共0兲C0 共A6兲

is the only one able to match the external solution. As in the case of resistive reconnection in the presence of an equilib-rium velocity field,14,15,17 the collisionless “small growth” 共comparable shear兲 regime corresponds to the constant-␺ re-gime. At first order in the small parameter ␥/␣F

⬘

共0兲⑀, we obtain iG

⬘

共0兲 F

⬘

共0兲␰ ⳵2_W 1 ⳵␰2 = i␰ ⳵2_␺ 1 ⳵␰2 , 共A7兲 ␺0+ i G

⬘

共0兲 F

⬘

共0兲␰␺1− i␰W1= i G

⬘

共0兲 F

⬘

共0兲␰ ⳵2_␺ 1 ⳵␰2 . 共A8兲 It is noteworthy that Eq. 共A8兲is singular in ␰= 0. This sin-gularity is related, once again, to the existence of the con-tinuous spectrum.32–34 As stated before, its contribution to the perturbation evolution is proportional to t−兩a兩 and is thus negligible compared to the exponential instability. Further-more, we will see that the logarithmic singularity in zero is of the form q ln q, and thus will not pose any problem in founding the discrete spectrum. The matching with the exter-nal solution imposes␺1→0 for 兩␰兩Ⰷ1, and thus, from Eqs. 共A7兲and共A8兲, we obtain

W1 G

⬘

共0兲 F

⬘

共0兲 =␺1, 共A9兲 G

⬘

共0兲 F

⬘

共0兲␺0− i␰

再

1 −

冋

G

⬘

共0兲 F

⬘

共0兲

册

2

冎

␺1= i

冋

G

⬘

共0兲 F

⬘

共0兲

册

2 ␰⳵2␺1 ⳵␰2 . 共A10兲 The last equation can be rewritten as

␰⳵2␺1 ⳵␰2 + K 2_␰␺ 1+ I = 0, 共A11兲 with K2=兵1−关G

⬘

共0兲/F

⬘

共0兲兴2其/关G

⬘

共0兲/F

⬘

共0兲兴2 and I = i␺0 ⫻关G

⬘

共0兲/F

⬘

共0兲兴.

If we assume that兵1−关G

⬘

共0兲/F

⬘

共0兲2_{兲⬍0其 and} K = − i

再

兵关G

⬘

共0兲/F

⬘

共0兲兴

2_{− 1其} 关G

⬘

共0兲/F

⬘

共0兲兴2

冎

1/2

, 共A12兲

the solution of Eq.共A11兲, compatible with the boundary con-ditions, is given by

␺1共␰兲 = − I

K关ci共K␰兲sin共K␰兲 − si共K␰兲cos共K␰兲兴, 共A13兲 where si共K␰兲 = −

冕

K␰ +⬁_{sin q} q dq, ci共K␰兲 = −

冕

K␰ +⬁_{cos q} q dq. 共A14兲 The functions si, ci can be written using the functions inte-gral sine Si and inteinte-gral cosine Cin, analytic over the whole complex plane

si共q兲 = Si共q兲 −␲/2, 共A15兲

ci共q兲 = C + ln q − Cin共q兲, arg q 苸 关−␲,␲兴, 共A16兲

Cin共− q兲 = Cin共q兲, 共A17兲

Si共− q兲 = − Si共q兲. 共A18兲

The asymptotic behavior of Cin and Si is given by Si共x兲 ⯝␲ 2 − cos x x

冉

1 − 2! x2+ . . .

冊

− sin x x

冉

1 x− 3! x3+ . . .

冊

, 共A19兲 Cin共x兲 ⯝ C + ln x −sin x x

冉

1 − 2! x2+ . . .

冊

+cos x x

冉

1 x− 3! x3+ . . .

冊

, 共A20兲

for Re x⬎0. Thus we have, for 兩␰兩Ⰷ1, Im␰⬎0,

␺1共␰兲 ⯝ − i ␺0 1 −关G

⬘

共0兲/F

⬘

共0兲兴2

冋

G

⬘

共0兲 F

⬘

共0兲

册

1 ␰. 共A21兲

It is thus necessary to consider the second order terms in the small parameter ␥/␣F

⬘

共0兲⑀ in order to match the internal and external solutions. At this order Eqs.共12兲 and共13兲 be-come ⳵2_W 1 ⳵␰2 + i G

⬘

共0兲 F

⬘

共0兲␰ ⳵2_W 2 ⳵␰2 = i␰ ⳵2_␺ 2 ⳵␰2 , 共A22兲 ␺1+ i G

⬘

共0兲 F

⬘

共0兲␰␺2− i␰W2= i G

⬘

共0兲 F

⬘

共0兲␰ ⳵2_␺ 2 ⳵␰2 . 共A23兲

From Eq.共A23兲we obtain

冕

−⬁ +⬁_⳵2_W 2 ⳵␰2 d␰= G

⬘

共0兲 F

⬘

共0兲

冕

_−⬁ +⬁_⳵2_␺ 2 ⳵␰2d␰. 共A24兲

If we insert Eq.共A24兲into Eq.共A22兲and integrate, it yields

冋

G

⬘

共0兲 F

⬘

共0兲

册

2

冕

−⬁ +⬁_⳵2_␺ 2 ⳵␰2d␰=

冕

−⬁ +⬁_⳵2_␺ 2 ⳵␰2d␰+ i

冕

−⬁ +⬁₁ ␰ ⳵2_W 1 ⳵␰2 d␰. 共A25兲 From the definition of⌬

⬘

冕

−⬁ +⬁_⳵2_␺ 2 ⳵␰2d␰= ⑀⌬

⬘

␺0 关␥/共␣F

⬘

共0兲⑀兲兴2, 共A26兲 we have

再

冋

G

⬘

共0兲 F

⬘

共0兲

册

2 − 1

冎

⑀⌬

⬘

␺0 兵␥关␣F

⬘

共0兲⑀兴其2 =

冕

−⬁ +⬁ G

⬘

共0兲 F

⬘

共0兲␺0+ i␰

冋

冉

G

⬘

共0兲 F

⬘

共0兲

冊

2 − 1

册

␺1 关G

⬘

共0兲/F

⬘

共0兲兴3_␰2 . 共A27兲 Thanks to the asymptotic behavior of␺₁and to its behavior at␰= 0, we can close the integral path in the upper half of the complex plane and consider the indentation around␰= 0. As stated before, the logarithmic singularity at ␰= 0 does not give any contribution to our mode. In fact we obtain

⑀⌬

⬘

␺0

冉

␥ ␣F

⬘

共0兲⑀

冊

2= i␲ i␺1兩␰=0

冉

G

⬘

共0兲 F

⬘

共0兲

冊

3 = −␲ 2 2 ␺0

冏

G

⬘

共0兲 F

⬘

共0兲

冏

−3

再

冋

G

⬘

共0兲 F

⬘

共0兲

册

2 − 1

冎

−1/2 , 共A28兲 which yields the following expression for the growth rate:

␥2_{= −} 2 ␲2␣ 2_关F

_⬘

_共0兲兴2_d e 3_⌬

_⬘

冏

G

⬘

共0兲 F

⬘

共0兲

冏

3

再

冋

G

⬘

共0兲 F

⬘

共0兲

册

2 − 1

冎

1/2 . 共A29兲 We note that the scaling of the growth rate is different from that of the tearing mode in a stationary plasma. Furthermore, the small growth mode is stable for关G

⬘

共0兲/F

⬘

共0兲兴2_{⬎1 and} ⌬

⬘

⬎0 共while ⌬

⬘

⬎0 is the usual tearing instability condi-tion兲. It is important to note that usually the system is Kelvin–Helmholtz unstable for关G

⬘

共0兲/F

⬘

共0兲兴2_{⬎1, thus the} tearing instability for ⌬

⬘

⬍0 has a growth rate negligible with respect to that of the faster ideal Kelvin–Helmholtz instability.14–18 From the assumption ␥/␣F

⬘

共0兲⑀Ⰶ1 and

from Eq. 共A29兲, we obtain ⌬

⬘

⑀兵关G

⬘

共0兲/F

⬘

共0兲兴2_{− 1其}1/2_Ⰶ1, which justifies the constant-␺approximation for⌬

⬘

⬃1.

However, it is important to observe that the constant-␺ approximation is also justified for ⌬

⬘

Ⰷ1 when 关G

⬘

共0兲/F

⬘

共0兲兴2_{→1. We can thus state that the ⌬}

_⬘

_{Ⰷ1 Kink} mode is totally stabilized by the equilibrium velocity field when关G

⬘

共0兲/F

⬘

共0兲兴2→1.

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