Magneto-thermal reconnection processes, related mode momentum and formation of high energy particle populations

In the context of a two-fluid theory of magnetic reconnection, when the longitudinal electron thermal conductivity is relatively large, the perturbed electron temperature tends to become singular in the presence of a reconnected field component and an electron temperature gradient. A finite transverse thermal diffusivity removes this singularity while a finite ‘inductivity’ can remove the singularity of the relevant plasma displacement. Then (i) a new ‘magneto-thermal’ reconnection producing mode, is found with characteristic widths of the reconnection layer remaining significant even when the macroscopic distances involved are very large; (ii) the mode phase velocities can be both in the direction of the electron diamagnetic velocity as well in the opposite (ion) direction. A numerical solution of the complete set of equations has been carried out with a simplified analytical reformulation of the problem. A sequence of processes is analyzed to point out that high-energy particle populations can be produced as a result of reconnection events. These involve mode-particle resonances transferring energy of the reconnecting mode to a superthermal ion population and the excitation of lower hybrid waves that can lead to a significant superthermal electron population. The same modes excited in axisymmetric (e.g. toroidal) confinement configurations can extract angular momentum from the main body of the plasma column and thereby sustain a local ‘spontaneous rotation’ of it.


Introduction
In the context of a two-fluid theory of magnetic reconnection [1,2], when the longitudinal electron thermal conductivity is relatively large, the perturbed electron temperature tends to become singular [3] in the presence of a reconnected field component and an electron temperature gradient. A transverse thermal diffusivity is introduced in order to remove this singularity while a finite 'inductivity' [4] can remove the singularity of the corresponding transverse plasma displacement [1]. Then (i) a new 'magneto-thermal' reconnection producing mode, driven by the electron temperature gradient, and involving a considerable range of scale distances is found [5]; (ii) the characteristic widths of the layers in which magnetic reconnections takes place remain significant even when the macroscopic distances involved in the process are very large; iii) the phase velocities of the modes that are found can be both in the direction of the electron diamagnetic velocity as well as those in the opposite (ion) direction. A numerical solution of the complete set of equations has been carried out with a simplified analytical reformulation of the problem. The mode growth rate is related to the effects of a finite viscous diffusion coefficient or to those of a small electrical resistivity.
The features that can lead to a possible explanation of the fact that high energy particle populations are produced during reconnection events involve mode-particle resonances [6] producing the transfer of energy to super-thermal particle populations and the spatial near-singularity of the electron temperature that can enhance the thermal energy of particles in one region while depleting that of particles in a contiguous region [5] may be an additional factor to be taken into account within this context.
The same low collisionality modes that produce magnetic reconnection can extract momentum from the plasma sheet and when excited in axisymmetric toroidal confinement configurations can sustain a 'spontaneous rotation' [7] of the plasma column by extracting angular momentum from it. This process is to be considered in addition to that of ejection of angular momentum [7] from the edge of the plasma column resulting for instance from the local excitation of electrostatic modes.

Initial equilibrium configuration
For the sake of simplicity, we simulate current carrying wellconfined plasmas with a variety of geometries by a plane configuration with the confining field . The profiles of ( ) B x y related to those of the current density ( ) J x z that have been examined correspond to those produced in experiments. The density and temperature gradients are in the x direction and no equilibrium electric field E is present in the considered frame of reference. Then the equilibrium state is described simply by In order to study magnetic reconnection process in the considered sheared magnetic field configuration, all perturbations from the considered equilibrium state are taken to be of the form ( ) . The surface = x x 0 is chosen to correspond to one on which the electron temperature gradient / T x d d e is significant.

'Outer' asymptotic region and singularities
The layer in which magnetic reconnection takes place is centered around = x x 0 and has a width δ m that is much smaller than all the scale distances associated with the gradients of the plasma pressure and the current density. Outside the layer (the 'outer' asymptotic region), inertia is negligible and the perturbed plasma is described by Then, considering perturbations that only bend the field lines, i.e. ≅ B 0 z , and using ∇ ⋅ = B 0, the ⋅ ∇× e z component of equation (3) gives the (radial) profile of the transverse perturbed magnetic field B x as x y x y y x 2 2 2 (4) Here, x is assumed to be continuous at the regular singular point = x x 0 but its derivative, in general, can be discontinuous. The discontinuity parameter ∆′, . Clearly, the parameter ∆′ is a function of the equilibrium current density gradient, and k y . We choose to include in our analyses equilibrium current density profiles and values of k y for which ∆′ can have a relatively wide range of values. We further add that a positive ∆′ is the driving factor of the drift-tearing types of mode identified in [1].
Next, the 'frozen-in' condition that characterizes the outer region is , the x-component of gives where we have considered ( / )( / ) k k B B 1 z y y z and have intro- Additionally, we find it important to include the 'infinite longitudinal electron thermal conductivity' condition in the outer region. This condition, referring to the electron thermal energy balance equation, implies and it gives x e e (12) where T e is the electron temperature perturbation and . Comparison with equation (10) shows that ξ = − ′ T T x e e in the outer region. It is evident from equations (10) and (12) that, for ≠ B 0 x0 and ≠ ′ T 0 e , both ξ x and T e tend to become singular as → x x 0 as → ⋅ k B 0. These singularities are removed by adopting a two-fluid description [1,3] in dealing with an 'inner' region corresponding to δ − x x~m 0 . A new 'magneto-thermal' reconnection-producing mode, driven by the electron temperature gradient and involving a considerable range of scale distances, is thus found [5].

Basic equations for the 'inner' region
We derive the basic equations for the inner region that are capable of describing the reconnection-producing modes of interest by following the procedure developed in [1,3]. We outline the derivation here.
The quasi-neutrality condition given by involving the gradient of the equilibrium current density, which is the driving factor of the relevant instability and is important in the outer region (see equation (4)), can be neglected in the inner region as it does not contribute to the dispersion relation as indicated by the analysis that follows. In the guiding-center description when polarization drift, finite gyroradius correction and momentum diffusion due to viscosity are added to the ion guiding-center drift velocity is the ion diamagnetic frequency, and µ D is the transverse (to the magnetic field) momentum diffusion coefficient. Electrons are taken to move with the drift velocity u E . We also find from the Ampere's law that 2 in the inner region. Then, the quasi-neutrality condition gives The other equation to be coupled with equation (15) is derived from the longitudinal electron momentum conservation equation Tn nT e e e e , α T T is a numerical coefficient characterizing the thermal force, η is the longitudinal resistivity and iωL J represents the induced electric field, L being the relevant plasma inductivity [4]. Clearly, the inductivity is introduced to break the frozen-in condition, while leading to results consistent with experimental observations [8] concerning modes driven by the current density gradient. In fact, a finite inductivity could represent the coupling of the plasma current channels produced inside the reconnection layer with the effects of other current channels outside it. Next we determine n e from the electron equation of continuity where the term [ ( has been omitted since, as indicated by the analysis that follows, it does not contribute to the dispersion relation due to mode parity x y x e 2 2 (18) We point out that, in reality, a particle transport term repre- , should also be included in the expression of n e , where the particle diffusion coefficient D n can have a classical component (due to electronion collisions) and an anomalous component (due to collective processes). The effect of this term on the reconnection process is considered to be less important than that of the plasma inductivity and has been neglected for simplicity. Then, using equations (9), (14) and (18) in equation (16) we find Here where ⊥ D e and D e are the transverse and longitudinal diffusion coefficients for the electron thermal energy. By using equations (15), (21)and (19) can be written in the alternative form , and the resistive diffusion coefficient η D is considered to be small relative to ωS L . We also point out that the electron inertia term is omitted in equation (16)  The distance that characterizes the 'innermost region', as obtained from equation (21), is where / / ≡ ′ L B B 1 s y , and we note that it remains significant when large macroscopic scale distances are considered. We assume that ρ δ δ δ < ∼ < m i in I , where ρ i is the average ion gyroradius.

Range of phase velocities
The analysis of the modes that can be found involves matching the solution for B x of the inner region equations (15), (21) and (22), with that of the outer region equation by the asymptotic matching condition given by equation (5). The analysis indicates that the nature and the phase velocity of the relevant modes change significantly when the parameter D 0 , defined by is varied. In particular, when the value of D 0 is positive and finite, the phase velocity of the modes that are found is in the direction of the ion diamagnetic velocity as they become of the type reported in [3]. When, instead, D 0 is small such as that for which δ δ m I , as is relevant for applications to plasma configurations for which the involved macroscopic distances are very large (e.g. after space and astrophysics), the direction of the mode phase velocity can be in that of the electron diamagnetic velocity. For our analysis, we choose a set of appropriate dimensionless quantities and rewrite the equations for the inner region (15), (21) and (22) as and  We note that β * p can be considered finite while δ ρ s I 2 2 for the validity of the adopted two-fluid equations. Moreover, with the adopted choice of the variables we may consider that * * B T when x~1, while ε * 1, ε η 1, and ε µ 1. In order to show the possibility that the phase velocity of the relevant modes can change sign, we adopt the constant-B x approximation, that is, , and assume for simplicity that ε ε ε = = = η µ * 0. It is also convenient to use the dimensionless variables  (37) Then, by virtue of equations (33)-(35), the matching condition leading to the dispersion relation can be expressed as ,and d 1 , , modes with frequency / ω ω > * 0 e can be obtained. By examining the order of magnitude of the terms in equations (25)-(27) we conclude that ω ω ≈ * e is a significant limit to investigate. These modes will have phase velocities in the direction of the electron diamagnetic velocity.
In order to facilitate the investigation of ω ω ≈ * e modes, we substitute ω ω δω = + * e , where δω ω < * e in equations (25)-(27) and keeping the leading order terms arrive at the following set of equations to analyze for the inner region and that the width of this region also has to exceed the ion Larmor radius.
When analyzing the 'innermost region', in the limit where ε * can be neglected, it is convenient to use the dimensionless variables (48) x 0 is adopted, ≈ B 1 in equation (48) and the analytic solution of the inhomogeneous equation for U is given by with the asymptotic behaviors: On the basis of this, we may introduce an approximate function representing / * T x d d In particular, the mode growth rate is seen to increase with ε η and with /α 1 T , that is, with the peak value of T e . A (positive) growth rate is also obtained when ε η is neglected and the effects of a finite viscosity represented by ε µ are taken into account.
We note that under conditions where ω ω di , where the analysis reported in [3] is valid, the mode growth rate is associated with the effects of a finite viscosity, that is ε ≠ µ 0, an example being represented by figure 3.
A different case to be considered is the following. In the case where the current density J z is peaked at = x 0 and / ≠ J x d d 0 z at = x x 0 , and the appropriate solution of equation (4) in the outer region is obtained, the solution for the inner region equations will involve an even and an odd component of ( ) * B x unlike the case considered here for simplicity.

Formation of high energy electron and ion populations
We propose that a sequence of mode-particle resonances [6] starting from the excitation of oscillatory modes associated with magnetic reconnection is responsible for the generation of experimentally observed high energy electron populations following magnetic reconnection events. In particular, the mode-particle resonances involving reconnecting modes concern frequencies of the order of ω α ω =    A theoretical scenario that can be analyzed within the limits of quasi linear theory is the formation of a ring, in the velocity space ⊥ v , ∥ v , of a highly superthermal ion population, by a pitch angle scattering process driven by excited reconnecting modes. Indicating their frequency by ω c Re , it is realistic to consider ω Ω c Re ci and the mode-particle resonance producing a pitch angle scattering from the parallel direction to a perpendicular direction to the magnetic field. Here Then lower hybrid (LH) modes excited by the superthermal ion distribution can resonate with the ions and the superthermal electrons simultaneously. The relevant mode-particle resonance conditions involve where > ⊥ k k and > > ⊥ v v v e the i . The transverse ion kinetic energy is thus transferred to the longitudinal electron kinetic energy creating a high energy electron population.
A second envisioned scenario excludes the presence of a pitch angle scattering represented by the mode particle resonance (51) and consider the formation of a superthermal spike in ∥ v out of the tail of the thermal ion distribution due to the mode-particle resonance (52). Then we may argue that a LH mode is excited through the resonance condition where v i LH is located in the region where the slope of the superthermal spike is positive. Following this, the excited lower hybrid modes may undergo non-linear Landau damping [9] interacting with the superthermal tail of the electron distribution and transfer energy to the resonating superthermal electrons. The relevant non-linear Landau damping process can be represented by while E and B are related through the induction equation For electrostatic modes, considered in [11], = B 0, ϕ = −∇ E , where ϕ is the electrostatic potential, and the expression for momentum P reduces to  (64) then we are led to refer to the k-component of the momentum surface density P k , given by