98 research outputs found
Topics in Energy Release and Particle Acceleration in the Heliosphere
No full textThis thesis investigates both the release of energy in solar flares, and the acceleration and transport of particles in various astrophysical situations. While numerical simulations are central to this thesis, these are always motivated by analytical arguments.
A review of flare energy release is given in Chapter 2, with results presented in Chapters 3 and 4. The main goal of the flare work is to investigate the effect of viscosity on energy release rates. Scaling arguments and exact solutions of the magnetohydrodynamic equations are used to interpret the results of two-dimensional numerical simulations of magnetic reconnection. The results support viscous energy dissipation accounting for a significant fraction of flare energy release.
Chapter 5 contains an introduction to astrophysical particle acceleration, using the Fokker-Planck formulation. The theory introduced in this chapter is used to study electron transport in solar flare loops (Section 5.5). A key aspect of the analysis is the expression of the Fokker-Planck equation as a system of stochastic differential equations. A generalisation to the flare loop hard X-ray emission prediction of Conway et al. (1998) is obtained, giving a stronger dependence on density for dispersed initial distributions.
Chapter 6 uses the methods of the previous chapter to study the acceleration of cosmic-rays at the heliospheric termination shock. The applicability of the focused acceleration mechanism of Schlickeiser and Shalchi (2008) is examined using numerical simulations, which are interpreted using analytical arguments based on averaging the stochastic equations. The results show significant limitations in assuming a near-isotropic distribution, a requirement for the focused acceleration mechanism. In addition, momentum diffusion provides a significant effect that cannot be neglected. The theory is extended to include focused deceleration and pure momentum diffusion
Focused acceleration of cosmic-ray particles in non-uniform magnetic fields
No full textThe 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
Wave energy dissipation by anisotropic viscosity in magnetic x-points
No full textThe 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
Particle acceleration scalings based on exact analytic models for magnetic reconnection
No full textObservations 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
Modeling sunspot and starspot decay by turbulent erosion
No full textDisintegration 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
Proton acceleration in analytic reconnecting current sheets
No full textParticle acceleration provides an important signature for the magnetic collapse that accompanies a solar flare. Most particle acceleration studies, however, invoke magnetic and electric field models that are analytically convenient rather than solutions of the governing magnetohydrodynamic equations. In this paper a self-consistent magnetic reconnection solution is employed to investigate proton orbits, energy gains, and acceleration timescales for proton acceleration in solar flares. The magnetic field configuration is derived from the analytic reconnection solution of Craig and Henton. For the physically realistic case in which magnetic pressure of the current sheet is limited at small resistivities, the model contains a single free parameter that specifies the shear of the velocity field. It is shown that in the absence of losses, the field produces particle acceleration spectra characteristic of magnetic X-points. Specifically, the energy distribution approximates a power law ~ξ-3/2 nonrelativistically, but steepens slightly at the higher energies. Using realistic values of the “effective” resistivity, we obtain energies and acceleration times that fall within the range of observational data for proton acceleration in the solar corona
Finite-time singularity formation at a magnetic neutral line in Hall magnetohydrodynamics
No full textThe 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
Steady and unsteady visco-resistive reconnection in the presence of the Hall effect
No full textIn this thesis we investigate the effects of viscosity and the Hall effect on magnetic reconnection. Magnetic reconnection is a process of releasing large amounts of magnetic energy as observed in solar flares. In the first two chapters, we describe the basic mathematics and early models of reconnection.
In Chapter 3, we search for a visco-resistive length scale in reconnection solutions. This is demonstrated in reconnective annihilation and a quasi-one-dimensional series expansion. We find that the visco-resistive length scale appears organically unless a specific geometry is chosen. Upon adding small scale perturbations, the visco-resistive length scale always appears.
In Chapter 4, we build on Litvinenko’s (2007) self-similar solution that showed singularities appear with a Hall MHD X-point geometry for a certain set of initial conditions. These singularities signal current sheet formation. We consider a general set of initial conditions and find that the singularities will form in this self-similar solution unless the axial field is many orders of magnitude larger than the planar field.
In Chapter 5, we review the Craig and McClymont (1991) linear, oscillatory model of reconnection. In Chapter 6, we attempt to quantify a general model that includes viscosity, pressure and axial effects, the Hall effect and electron inertia. We perform a dimensional analysis to find order-of magnitude estimates for how the aforementioned effects perturb the Craig and McClymont (1991) solution. We verify these estimates with numerical simulations.
In Chapter 7, we give an overview of the thesis and make suggestions for future work
A similarity reduction of the Grad-Shafranov equation
No full textA 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
Estimating the size of the cosmic-ray halo using particle distribution moments
No full textContext: Particle transport in many astrophysical problems can be described either by the Fokker–Planck equation or by an equivalent system of stochastic differential equations. Aims: It is shown that the latter method can be applied to the problem of defining the size of the cosmic-ray galactic halo. Methods: Analytical expressions for the leading moments of the pitch-angle distribution of relativistic particles are determined. Particle scattering and escape are analyzed in terms of the moments. Results: In the case of an anisotropic distribution, the first moment leads to an expression for the halo size, identified with the particle escape from the region of strong scattering. Previous studies are generalized by analyzing the case of a strictly isotropic initial distribution. A new expression for the variance of the distribution is derived, which illustrates the anisotropization of the distribution. Conclusions: Stochastic calculus tools allow one to analyze physically motivated forms for the scattering rate, so that a detailed realistic model can be developed
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