A similarity reduction of the Grad-Shafranov equation

A direct method for finding similarity reductions of partial differential equations is applied to a specific case of the Grad–Shafranov equation. As an illustration of the method, the frequently used Solov’ev equilibrium is derived. The method is employed to obtain a new family of exact analytical solutions, which contain both the classical and nonclassical group-invariant solutions of the Grad–Shafranov equation and thus greatly extends the range of the available analytical solutions. All the group-invariant solutions based on the classical Lie symmetries are shown to be particular cases in the new family of solutions

Modeling sunspot and starspot decay by turbulent erosion

Disintegration of sunspots (and starspots) by fluxtube erosion, originally proposed by Simon and Leighton, is considered. A moving boundary problem is formulated for a nonlinear diffusion equation that describes the sunspot magnetic field profile. Explicit expressions for the sunspot decay rate and lifetime by turbulent erosion are derived analytically and verified numerically. A parabolic decay law for the sunspot area is obtained. For moderate sunspot magnetic field strengths, the predicted decay rate agrees with the results obtained by Petrovay and Moreno-Insertis. The new analytical and numerical solutions significantly improve the quantitative description of sunspot and starspot decay by turbulent erosion

Focused acceleration of cosmic-ray particles in non-uniform magnetic fields

The Fokker–Planck equation for cosmic-ray particles in a spatially varying guide magnetic field in a turbulent plasma is analyzed. An expression is derived for the mean rate of change of particle momentum, caused by the effect of adiabatic focusing in a non-uniform guide field. Results of an earlier diffusion-limit analysis are confirmed, and the physical picture is clarified by working directly with the Fokker–Planck equation. A distributed first-order Fermi acceleration mechanism is identified, which can be termed focused acceleration. If the forward and backward-propagating waves have equal polarizations, focused acceleration operates when the net cross helicity of an Alfvenic slab turbulence is either negative in a diverging guide field or positive in a converging guide field. It is suggested that focused acceleration can contribute to the formation of the anomalous cosmic-ray spectrum at the heliospheric termination shock

Influence of the Hall effect on the reconnection rate at line-tied magnetic X-points

Context. The role of the Hall term in magnetic reconnection at line-tied planar magnetic X-points is explored. Aims. The goal is to determine the reconnection scaling laws and to investigate how the reconnection rate depends on the size of the system in Hall magnetohydrodynamics (MHD). Methods. The evolution of reconnective disturbances is determined numerically by solving the linearized compressible Hall MHD equations. Scaling laws are derived for the decay rate as a function of the dimensionless resistivity and ion inertial length. Results. Although the Hall effect leads to an increase in the decay rate, this increase is shown to be moderated in larger systems. A key finding is that the Hall term contribution to the decay rate, normalized by the resistive decay rate, scales inversely with the system size L, approximately as L-2. Conclusions. The evidence suggests that decay rate enhancements due to Hall effects in line-tied X-points are weakened for large-scale systems. The result may have important implications for modeling energy release in large-scale astrophysical plasma environments, such as solar flares

Particle acceleration scalings based on exact analytic models for magnetic reconnection

Observations suggest that particle acceleration in solar flares occurs in the magnetic reconnection region above the flare loops. Theoretical models for particle acceleration by the reconnection electric field, however, employ heuristic configurations for electric and magnetic fields in model current sheets, which are not solutions to the MHD reconnection problem. In the present study, particle acceleration is discussed within the context of a self-consistent MHD reconnection solution. This has the advantage of allowing poorly constrained local parameters in the current sheet region to be expressed in terms of the boundary conditions and electric resistivity of the global solution. The resulting acceleration model leads to energy gains that are consistent with those for high-energy particles in solar flares. The overall self-consistency of the approach is discussed

Current singularities in planar magnetic X points of finite compressibility

The formation of current singularities in nonresistive, line-tied magnetic X points is addressed. It is pointed out that, although gas pressure suppresses the current singularity development when strictly antiparallel, one-dimensional magnetic fields implode, the pressure is likely to be less effective in the more realistic case of two-dimensional magnetic fields. Detailed nonlinear relaxation computations at various levels of compressibility confirm that singularity is present even in the incompressible limit, but its strength, as determined by the amplitude and morphology of the current density, is considerably reduced. The singularity strength is quantified by computing the scalings of the peak current density with resolution. The scalings show that localized current structures can be expected only for negligible gas pressures. The numerical results imply that the inclusion of gas pressure effectively stalls fast magnetic reconnection in line-tied X-point geometries

Flare energy release by flux pile-up magnetic reconnection in a turbulent current sheet

The power output of flux pile-up magnetic reconnection is known to be determined by the total hydromagnetic pressure outside the current sheet. The maximum energy-release rate is reached for optimized solutions that balance the maximum dynamic and magnetic pressures. An optimized solution is determined in this paper for a current sheet with anomalous, turbulent electric resistivity. The resulting energy dissipation rate Wa is a strong function of the maximum, saturated magnetic field Bs: . Numerically, Wa can exceed the power output based on the classical resistivity by more than 2 orders of magnitude for three-dimensional pile-up, leading to solar flarelike energy-release rates of the order of 1028 ergs s−1. It is also shown that the optimization prescription has its physical basis in relating the flux pile-up solutions to the Sweet-Parker reconnection model

Wave energy dissipation by anisotropic viscosity in magnetic x-points

The viscous dissipation of axial field disturbances in planar magnetic X-points is examined. It is emphasized that an accurate treatment requires a nonisotropic tensor viscosity whose components are governed by the local magnetic field. Numerical solutions are constructed, which compare the buildup of viscous forces using the tensor formulation against a simplified model based on conventional shear viscosity. The scaling of the global energy-loss rate with the viscosity coefficient is shown to follow for both the traditional shear viscosity and the Braginskii bulk viscosity. This suggests that viscous wave dissipation can occur quite rapidly, in a few tens of Alfvén times. The results imply that large-scale disturbances, generated by magnetic reconnection in the solar corona, should dissipate in a time on the order of a few minutes and significantly contribute to coronal heating

Finite-time singularity formation at a magnetic neutral line in Hall magnetohydrodynamics

The formation of a current sheet in a weakly collisional plasma can be modelled as a finite-time singularity solution of magnetohydrodynamic equations. We use an exact self-similar solution to confirm and generalise a previous finding that, in sharp contrast to two-dimensional solutions in standard MHD, a finite-time collapse to a current sheet can occur in Hall MHD. We derive a criterion for the finite-time singularity in terms of initial conditions, and we use an intermediate asymptotic solution for the evolution of an axial magnetic field to obtain a general expression for the singularity formation time. We illustrate the analytical results by numerical solutions

Viscous effects in planar magnetic X-point reconnection

The impact of viscous dissipation is considered on magnetic reconnection in closed line-tied magnetic X-points. It is shown that viscous effects can provide fast energy dissipation for disturbances which do not alter the initial X-point topology. If the X-point topology is altered, then the rate of viscous dissipation depends on both the perturbed topology and the relative magnitudes of viscosity and electric resistivity. New solutions are demonstrated, which derive from the combination of resistive and viscous effects. The solutions are characterized by monotonically decaying modes which are qualitatively different from the previously known oscillatory modes in nonviscous resistive X-point reconnection. These results suggest that viscous heating in magnetic X-points may be an important effect in solar flares