B. Dasgupta

Local and nonlocal advected invariants and helicities in magnetohydrodynamics and gas dynamics I: Lie dragging approach

In this paper advected invariants and conservation laws in ideal magnetohydrodynamics (MHD) and gas dynamics are obtained using Lie dragging techniques. There are different classes of invariants that are advected or Lie dragged with the flow. Simple examples are the advection of the entropy S (a 0-form), and the conservation of magnetic flux (an invariant 2-form advected with the flow). The magnetic flux conservation law is equivalent to Faradays equation. The gauge condition for the magnetic helicity to be advected with the flow is determined. Different variants of the helicity in ideal fluid dynamics and MHD including: fluid helicity, cross helicity and magnetic helicity are investigated. The fluid helicity conservation law and the cross-helicity conservation law in MHD are derived for the case of a barotropic gas. If the magnetic field lies in the constant entropy surface, then the gas pressure can depend on both the entropy and the density. In these cases the conservation laws are local conservation laws. For non-barotropic gases, we obtain nonlocal conservation laws for fluid helicity and cross helicity by using Clebsch variables. These nonlocal conservation laws are the main new results of the paper. Ertels theorem and potential vorticity, the Hollman invariant, and the Godbillon?Vey invariant for special flows for which the magnetic helicity is zero are also discussed.

Vlasov-Maxwell equilibria: Examples from higher-curl Beltrami magnetic fields

Stationary solutions of Vlasov-Maxwell equations are obtained by exploiting the invariants of single particle motion and lead to linear or nonlinear functional relations between current and vector potential. The nonlinear relations support various special types of magnetic configurations including multiple current sheets and magnetic field discontinuities leading to singular current layers. It is demonstrated through the examples that in one dimension, the description of the equilibrium magnetic fields obeys double or higher-curl Beltrami equation. For the linear case, such representation gives the advantage of obtaining exact analytic solutions that are expressed as a superposition of the single-curl Beltrami fields.

Exclusion of Tiny Interstellar Dust Grains From the Heliosphere

The distribution of interstellar dust grains (ISDG) observed in the Solar System depends on the nature of the interstellar medium‐solar wind interaction. The charge of the grains couples them to the interstellar magnetic field (ISMF) resulting in some fraction of grains being excluded from the heliosphere while grains on the larger end of the size distribution, with gyroradii comparable to the size of the heliosphere, penetrate the termination shock. This results in a skewing the size distribution detected in the Solar System. We present new calculations of grain trajectories and the resultant grain density distribution for small ISDGs propagating through the heliosphere. We make use of detailed heliosphere model results, using three‐dimensional (3‐D) magnetohydrodynamic/kinetic models designed to match data on the shape of the termination shock and the relative deflection of interstellar H° and He° flowing into the heliosphere. We find that the necessary inclination of the ISMF relative to the inflow direction results in an asymmetry in the distribution of the larger grains (0.1 μm) that penetrate the heliopause. Smaller grains (0.01 μm) are completely excluded from the Solar System at the heliopause.

Dynamics of charged particles in spatially chaotic magnetic fields

The spatial topology of magnetic field lines can be chaotic for fields generated by simple current configurations. This is illustrated for a system consisting of a circular current loop and a straight current wire. An asymmetric configuration of the current system leads to three-dimensional spatially chaotic magnetic fields. The motion of charged particles in these fields is not necessarily chaotic and exhibits intriguing dynamical properties. Particles having initial velocities closely aligned with the direction of the local magnetic field are likely to follow chaotic orbits in phase space. Other particles follow coherent and periodic orbits; these orbits being the same as in the symmetric current configuration for which the field lines are not chaotic. An important feature of particles with chaotic motion is that they undergo spatial transport across magnetic field lines. The cross-field diffusion is of interest in a variety of magnetized plasmas including laboratory and astrophysical plasmas.

Kinetic model of Janaki et al.'s bifurcated current sheet

Space satellite observations show that current sheets in the space plasma environment are often characterized by bifurcated structures. In a recent paper a two-fluid model of the bifurcated current sheet was constructed on the basis of the pseudo potential method. A straightforward generalization of Janaki et al.s two-fluid model to kinetic formalism, however, leads to an unphysical situation of negative particle velocity distribution function occurring over a certain range of velocity space. If one assumes an isotropic background plasma population, however, one can show that a rigorous, physically valid kinetic solution of the bifurcated current sheet can be obtained.

Particle transport and acceleration in a chaotic magnetic field: Implications for seed population to solar flare and CME

In large Solar Energetic Particle (SEP) events, ions and electrons are accelerated to GeV/nucleon and keV in energy. These very high energetic particles are likely accelerated at fast coronal shocks. Observations have shown that the seed population (the particles that participate in the shock acceleration process) is not the bulk solar wind, but the suprathermal population. In this work, we propose to investigate a novel pre-acceleration mechanism that may provide the needed seed population for the subsequent shock acceleration in large SEP events. We examine the transport and acceleration of charged particles by chaotic electric and magnetic fields during the pre-eruptive period. It is demonstrated that a realistic chaotic magnetic field can be produced by any asymmetric current configurations - one such configuration is an asymmetric current wire loop system (CWLS). Observational studies have established the existence of current loops and current filaments at the solar surface and simple configurations as CWLSs inevitably exist in solar active regions. This suggests that the magnetic field at an active region is very much chaotic and time variation of these current configurations induces time-varying electric fields. Therefore, charged particles can be naturally accelerated. We outline an approximate model to study the pre-acceleration process of seed particles in a solar active region prior to eruptions by considering the transport and acceleration of charged particles in a time-dependent chaotic magnetic field.

Particle Motion and Energization in a Chaotic Magnetic Field

In nature there are many systems where macroscopic and time varying currents exist. For example, in solar flares, currents in the form of filaments and/or loops have been observed. In a simple asymmetric configuration where steady state currents flow through a straight wire and a loop, a somewhat surprising feature is that the resulting magnetic field can become chaotic depending on the relative size of the currents. This implies that studying particle motion in a time‐dependent chaotic magnetic field is a fundamental problem in space plasma physics and it has profound implications to astrophysical phenomena as the solar flares. Because these currents are time dependent, they induce electric fields. Consequently, electrons and ions moving in the field can experience alternating acceleration and deceleration, leading to a second order Fermi acceleration. We discuss here charged particle motion in a chaotic magnetic fields and the energization process. Particle trajectories are obtained by following single ...

Particle energization through time-periodic helical magnetic fields.

We solve for the motion of charged particles in a helical time-periodic ABC (Arnold-Beltrami-Childress) magnetic field. The magnetic field lines of a stationary ABC field with coefficients A=B=C=1 are chaotic, and we show that the motion of a charged particle in such a field is also chaotic at late times with positive Lyapunov exponent. We further show that in time-periodic ABC fields, the kinetic energy of a charged particle can increase indefinitely with time. At late times the mean kinetic energy grows as a power law in time with an exponent that approaches unity. For an initial distribution of particles, whose kinetic energy is uniformly distributed within some interval, the probability density function of kinetic energy is, at late times, close to a Gaussian but with steeper tails.

Hamiltonians and variational principles for Alfvén simple waves

The evolution equations for the magnetic field induction B with the wave phase for Alfven simple waves are expressed as variational principles and in the Hamiltonian form. The evolution of B with the phase (which is a function of the space and time variables) depends on the generalized Frenet–Serret equations, in which the wave normal n (which is a function of the phase) is taken to be tangent to a curve X, in a 3D Cartesian geometry vector space. The physical variables (the gas density, fluid velocity, gas pressure and magnetic field induction) in the wave depend only on the phase. Three approaches are developed. One approach exploits the fact that the Frenet equations may be written as a 3D Hamiltonian system, which can be described using the Nambu bracket. It is shown that B as a function of the phase satisfies a modified version of the Frenet equations, and hence the magnetic field evolution equations can be expressed in the Hamiltonian form. A second approach develops an Euler–Poincare variational formulation. A third approach uses the Frenet frame formulation, in which the hodograph of B moves on a sphere of constant radius and uses a stereographic projection transformation due to Darboux. The equations for the projected field components reduce to a complex Riccati equation. By using a Cole–Hopf transformation, the Riccati equation reduces to a linear second order differential equation for the new variable. A Hamiltonian formulation of the second order differential equation then allows the system to be written in the Hamiltonian form. Alignment dynamics equations for Alfven simple waves give rise to a complex Riccati equation or, equivalently, to a quaternionic Riccati equation, which can be mapped onto the Riccati equation obtained by stereographic projection.

Energization of charged particles in regular and chaotic magnetic fields

The dynamics of charged particles is studied in stationary magnetic fields that are obtained as solutions of nonlinear coupled equations varying in one dimension. The chosen equation can give both regular and chaotic solutions depending on the chosen coupling parameter. For chaotic numerical solution of the equation, rms values of fluctuation are increased by changing the coupling parameter, whereas for regular analytical solution, rms values of fluctuation are increased by changing the amplitude of fluctuation. Energy gain of an ensemble of particles is studied in both cases in the presence of a uniform electric field. Ensemble averaged energy gain is shown to decrease with the increase in rms values of fluctuation in the first case and increase with the increase in rms values of fluctuation in the second case.