Thomas J. Birmingham

Partially averaged field approach to cosmic ray diffusion

The kinetic equation for particles interacting with turbulent fluctuations is derived by a new nonlinear technique which successfully corrects the difficulties associated with quasi‐linear theory. In this new method the effects of the fluctuations are evaluated along particle orbits which themselves include the effects of a statistically averaged subset of the possible configurations of the turbulence. The new method is illustrated by calculating the pitch angle diffusion coefficient Dμμ for particles interacting with ’’slab model’’ magnetic turbulence, i.e., magnetic fluctuations linearly polarized transverse to a mean magnetic field 〈B〉. Results are compared with those of quasi‐linear theory and also with those of Monte Carlo calculations reported in a companion paper. The major effect of the nonlinear treatment in this illustration is the determination of Dμμ in the vicinity of 90° pitch angles where quasi‐linear theory breaks down. The spatial diffusion coefficient κ∥ parallel to 〈B〉 is evaluated usin...

Charged Particle Diffusion by Violation of the Third Adiabatic Invariant

An equation which describes statistically the motion of charged particles in response to fluctuating electric and magnetic fields is derived. The particles are assumed to be moving in a mirror‐type magnetic geometry. In addition to a static magnetic field there are small superposed fields fluctuating randomly on such a time scale that the first and second adiabatic invariants, M and J, are conserved, but the third or flux invariant φ is violated. By using second adiabatic theory a two dimensional diffusion equation is obtained valid on a much longer time scale than that of the fluctuations. Elements of the diffusion tensor are time—space correlations of fluctuation‐induced perturbations in the guiding center drifts. These drift perturbations are systematically derived and shown to reduce simply in various special cases.

Computer simulation of the velocity diffusion of cosmic rays

Monte Carlo simulation experiments have been performed in order to study the velocity diffusion of charged particles in a static turbulent magnetic field. By following orbits of particles moving in a large ensemble of random magnetic field realizations with suitably chosen statistical properties, a pitch‐angle diffusion coefficient is derived. Results are presented for a variety of particle rigidities and rms random field strengths and compared with the predictions of standard quasi‐linear theory and the nonlinear partially averaged field theory.

Birkeland currents in an anisotropic, magnetostatic plasma

An expression for the parallel current density is derived for a plasma characterized by negligible bulk flow (magnetostatic) velocity and a two-component (anisotropic) pressure tensor by expanding the equilibrium Vlasov equation for each species in the adiabatic parameter until such point as a nonvanishing moment j∥ = ∫ d³ υυ∥ ƒ is identified. The result is a nonlocal one: it relates j∥ at one point s along a field line to j∥ at another (reference) point s0 plus an integral function of the pressure and magnetic field between them. It is a generalization and elaboration of results obtained by Bostrom (1975), Heinemann (1990), and Heinemann and Pontius (1991). The expression could have been obtained by integrating the current continuity equation with −▽ · j⊥ as a source term and j⊥ given by perpendicular momentum balance. We explicitly show the equivalency. The widely used Vasyliunas (1970) equation follows when P⊥ is set equal to P∥ and s and s0 are taken to be at the ionosphere and the equator. An extended discussion of the relationship of results derived here to others in the literature is carried out in an effort to bring unity and perspective to this problem area.

Resonant diffusion in the presence of strong plasma turbulence

The diffusion equation which describes the evolution of the average one‐particle distribution function for an ensemble of strongly turbulent plasmas is derived. The diffusion tensor is a time integral of the autocorrelation tensor of the fluctuations as observed by particles moving along statistically distributed orbits. These orbits contain the effects of fluctuations and thus differ from those encountered in weak turbulence theory. Two statistical orbit effects quadratic in the strength of the fluctuations affect the magnitude of the diffusion: (a) modification of the ensemble average orbits by the fluctuations, and (b) statistical dispersion in particle orbits about the average. The plasma trajectory equations are used to relate each to the diffusion tensor itself when the turbulence is electrostatic. The diffusion tensor is explicitly evaluated for a strongly turbulent unmagnetized plasma.

WAVE GROWTH IN A STRONGLY TURBULENT PLASMA.

An equation for the average nonlinear growth (damping) rate 〈γk〉 of an electrostatic mode with wave vector k for an ensemble of electrostatically turbulent, unmagnetized plasmas is derived. Strong turbulence alters particle orbits during the course of wave growth and hence brings particles in and out of resonance with the wave. The wave‐growth rate in a strongly turbulent plasma, therefore, depends on the initial velocity distribution of particles not only at u = k·v = αk/k , the wave speed, but in the vicinity of this speed as well. Two orbital effects are considered: turbulence modification of ensemble average orbits and turbulence‐produced orbital dispersion about the average orbits. The first of these leads to a shift in the central u of the resonance. Both contribute to a resonance broadening.

Cosmic rays in a random magnetic field: breakdown of the quasilinear derivation of the kinetic equation

The problem of deriving a kinetic equation for the cosmic ray distribution function in a random magnetic field is considered. A model is adopted which is mathematically simple but which contains the essential physics. The perturbation expansion upon which the quasi-linear treatment is based is investigated. The existence of resonant particles causes the breakdown of the adiabatic approximation frequently used in this theory. Resonant particles cause a general secular growth of higher order terms in the expansion which invalidates the entire perturbative approach.

Propagators in strong plasma turbulence.

Straightforward relationships among Weinstocks propagator, UA, the Vlasov propagator, U, and the ensemble average Vlasov propagator, 〈U〉, are derived. U and 〈U〉 U are related to the characteristic trajectories of the Vlasov equation.

The effects of pressure anisotropy on birkeland currents in dipole and stretched magnetospheres

The expression for the Birkeland current density in an anisotropic P⊥, P∥ plasma derived in a companion paper (Birmingham, this issue) is simplified by linearization: for purposes of computing j∥ the magnetic configuration is assumed to be axisymmetric, either a dipole or a dipole stretched by the field arising from an azimuthal current sheet. The anisotropy is that of a bi-Maxwellian and characterized by equatorial values P⊥ (s0) and r, the equatorial temperature (pressure) ratio. Parameter j∥ is proportional to the azimuthal derivatives of P⊥ (s0) and r, with coefficients that are integral functions along the background B of the pressure and field strength from the equator s0, where there is (by assumption) no Birkeland current, to the point of reference. Values of j∥ at the ionosphere are compared between the two magnetic models for 0.1 1) than for one favoring P∥(r < 1). The growth away from the equator is particularly steep in the stretched model because of the strong curvature of the field lines, especially at large r0: plasma is trapped in the weak equatorial field, confined on either side by the strong B which arises from the azimuthal current sheet.

Collisional Absorption and Emission of Longitudinal Waves in a One‐Dimensional Plasma

Collisional absorption and emission of longitudinal waves in a one‐species, one‐dimensional plasma are investigated by numerical experiments. The theory of Birmingham, Dawson, and Kulsrud for longitudinal bremsstrahlung is extended to the one‐dimensional case. The results of the numerical experiments are found to be in agreement with the theory. Low‐frequency electric field fluctuations for long‐wavelength modes are also found and are associated with the random motion of the particles.