Anisotropic viscous dissipation in transient reconnecting plasmas

Aims. We examine the global energy losses associated with reconnecting coronal plasmas. Methods. Using planar magnetic reconnection simulations we compute resistive and bulk viscous losses in transient coronal plasmas. Resistive scalings are computed for the case of incompressible reconnection powered by large scale vortical flows. These results are contrasted with an example of magnetic merging driven by the coalescence instability. Results. We demonstrate that the large scale advective flows, required to sustain resistive current sheets, may be associated with viscous losses approaching flare-like rates of 10²⁹ erg s⁻¹ . More generally, bulk viscous dissipation appears likely to dominate resistive dissipation for a wide variety of magnetic merging models. We emphasize that these results have potentially important implications for understanding the flare energy budget

Anisotropic viscous dissipation in compressible magnetic X-points

Context: Visco-resistive damping in line-tied magnetic X-points is examined. Aims: The goal is to determine whether fast, Alfvénic energy dissipation is possible for X-point disturbances damped by the plasma resistivity and non-isotropic viscosity. Methods: The response of X-points to planar and axial perturbations is explored numerically by solving the linearized compressible MHD equations in two and a half dimensions. Results: It is demonstrated that fast dissipation is possible in the case of non-reconnective planar disturbances damped by anisotropic viscosity in weakly resistive plasmas. Although perturbations which change the initial X-point topology decay slowly at large times when viscous effects are dominant, there is an initial phase in which a significant fraction of the disturbance energy is removed on an Alfvénic timescale. The decay of incompressive axial field disturbances occurs by a different mechanism, however, that is always formally slow (i.e. dependent on the small viscous and resistive damping coefficients). Conclusions: Computations suggest that fast, visco-resistive energy release in coronal plasmas is possible for compressive X-point disturbances. This result could have important implications for understanding rapid energy release in coronal active regions

An exact solution for steady state magnetic reconnection in three dimensions

An exact three-dimensional solution is derived for the steady state magnetic reconnection of incompressible, resistive plasmas. The analysis provides a natural extension of the analytic, two-dimensional reconnection solution of Craig & Henton. The solution shows how advective motions through the separatrix “spine-curve” lead to global current sheets aligned to the separatrix “fan.

Fast magnetic reconnection and the coalescence instability

The magnetic reconnection that occurs during the nonlinear development of the coalescence instability is considered. The structure of the reconnection region at the time of maximum current as a function of the resistivity η is analyzed in detail using a compressible magnetohydrodynamic fluid code. It is shown that the numerical results concur remarkably well with a simple scaling analysis which predicts the dependence of the reconnection region structure on η. It is argued that the flow topology is crucial in maintaining the ``fast'' reconnection rate. The results indicate a flux pileup solution in which the flux annihilation rate is approximately independent of η, whereas the Ohmic dissipation rate scales as η −1/3. The possibility that these scalings break down at lower values of η is discussed

Resistive and viscous dynamics of finite-amplitude shear waves at a magnetic X-point

The dynamics and dissipation of axial shear waves, superposed on a planar magnetic X-point in a resistive viscous incompressible plasma, are analyzed numerically and analytically. The interplay of viscous and resistive effects is demonstrated by deriving solutions for various values of the scalar coefficients of viscosity and resistivity. These solutions show that viscous-resistive coupling can dramatically affect the global energy dissipation. When either viscosity or resistivity vanishes, the solutions are characterized by oscillatory decaying eigenmodes that maintain equipartition between the magnetic and kinetic energies. This behavior persists when resistivity is the dominant dissipation mechanism. When viscosity is the dominant dissipation mechanism, initial oscillations are followed by exponential decay at sufficiently long times. The applicability of the results to flares in solar active regions, where the viscous Reynolds number can be much smaller than the resistive one, is discussed

Visco-resistive shear wave dissipation in magnetic X-points

We consider the viscous and resistive dissipation of perpendicularly polarized shear waves propagating within a planar magnetic X-point. To highlight the role played by the two-dimensional geometry, the damping of travelling Alfvèn waves that propagate within an unbounded, but non-orthogonal X-point topology is analyzed. It is shown that the separatrix geometry affects both the dissipation time and the visco-resistive scaling of the energy decay. Our main focus, however, is on developing a theoretical description of standing wave dissipation for orthogonal, line-tied X-points. A combination of numerical and analytic treatments confirms that phase mixing provides a very effective mechanism for dissipating the wave energy. We show that wave decay comprises two main phases, an initial rapid decay followed by slower eigenmode evolution, both of which are only weakly dependent on the visco-resistive damping coefficients

Magnetic annihilation and reconnection in two dimensions

The problems of incompressible, planar, magnetic annihilation and reconnection are discussed. We first emphasize that steady-state reconnection solutions can be constructed from annihilation models involving harmonic velocity fields. We show, however, that the only harmonic velocity profile capable of supporting inviscid magnetic annihilation is the traditional stagnation point flow φ= −αxy. The implication is that further steady-state planar reconnection models derived from annihilation solutions are impossible. We go on to show that certain classes of nonharmonic stream functions allow reconnection solutions to be developed, once the constraint of time independence is relaxed. In particular, we construct an exact reconnection model based on cellular inflows into a periodic assemblage of magnetic X-points

Shear wave dissipation in planar magnetic X-Points

The resistive dissipation of nonlinear shear wave disturbances is discussed. We consider an incompressible, “open” X-point geometry, in which mass and energy are free to enter and leave the reconnection region. An analytic treatment is possible which unifies many of the dynamic and steady-state X-point solutions obtained previously. We show that while field disturbances in the plane of the X-point have the potential for rapid energy release when suitably driven, perpendicular shear disturbances dissipate slowly, at a rate η1/2, where η is the plasma resistivity. This behavior can be understood in terms of the absence of flux pileup in nonplanar shear wave disturbances. We conclude that only planar shear waves have the potential for fast magnetic energy release

Magnetic energy release in dynamic fan reconnection models

The problem of dynamic, three-dimensional magnetic reconnection is considered. Analytic “fan current” solutions are derived by superposing plane-wave disturbances on magnetic X-point equilibria. The localization of the wave produces a strong current sheet containing the neutral point. It is shown that the classical rate of resistive dissipation in the sheet, namely Wn~n1/2, represents the slowest possible energy-loss rate for the disturbance. The conditions required for fast coronal reconnection are then discussed. It is pointed out that significant “flare-like” energy release may be possible under physically realizable conditions. Moreover, the small length scales associated with the current sheet widths of order ∆x~n1/2 suggest that conditions are probably collisionless close to the neutral point. It is argued that our results are consistent with magnetic reconnection simulations that display “stalling” of the merging rate at small plasma resistivities

Current singularities at Quasi-separatrix layers and three-dimensional magnetic nulls

The open problem of how singular current structures form in line-tied, three-dimensional magnetic fields is addressed. A Lagrangian magneto-frictional relaxation method is employed to model the field evolution toward the final near-singular state. Our starting point is an exact force-free solution of the governing magnetohydrodynamic equations that is sufficiently general to allow for topological features like magnetic nulls to be inside or outside the computational domain, depending on a simple set of parameters. Quasi-separatrix layers (QSLs) are present in these structures and, together with the magnetic nulls, they significantly influence the accumulation of current. It is shown that perturbations affecting the lateral boundaries of the configuration lead not only to collapse around the magnetic null but also to significant QSL currents. Our results show that once a magnetic null is present, the developing currents are always attracted to that specific location and show a much stronger scaling with resolution than the currents that form along the QSL. In particular, the null-point scalings can be consistent with models of "fast" reconnection. The QSL currents also appear to be unbounded but give rise to weaker singularities, independent of the perturbation amplitude