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NASA NTRS1975

A B-B-G-K-Y framework for fluid turbulence

A kinetic theory for fluid turbulence is developed from the Liouville equation and the associated BBGKY hierarchy. Real and imaginary parts of Fourier coefficients of fluid variables play the roles of particles. Closure is achieved by the assumption of negligible five-coefficient correlation functions and probability distributions of Fourier coefficients are the basic variables of the theory. An additional approximation leads to a closed-moment description similar to the so-called eddy-damped Markovian approximation. A kinetic equation is derived for which conservation laws and an H-theorem can be rigorously established, the H-theorem implying relaxation of the absolute equilibrium of Kraichnan. The equation can be cast in the Fokker-Planck form, and relaxation times estimated from its friction and diffusion coefficients. An undetermined parameter in the theory is the free decay time for triplet correlations. Some attention is given to the inclusion of viscous damping and external driving forces.

Montgomery, D.↗
NASA NTRS1975

Mirroring in the Fokker-Planck coefficient for cosmic-ray pitch-angle scattering in homogeneous magnetic turbulence

The Fokker-Planck coefficient for pitch-angle scattering, appropriate for cosmic rays in homogeneous stationary magnetic turbulence is computed without making any specific assumptions concerning the statistical symmetries of the random field. The Fokker-Planck coefficient obtained can be used to compute the parallel diffusion coefficient for high-energy cosmic rays propagating in the presence of strong turbulence, or for low-energy cosmic rays in the presence of weak turbulence. Because of the generality of magnetic turbulence allowed for in the analysis, special interplanetary magnetic field features, such as discontinuities or particular wave modes, can be included rigorously.

Goldstein, M. L.↗
NASA NTRS1976

A quasi-linear kinetic equation for cosmic rays in the interplanetary medium

A kinetic equation for interplanetary cosmic rays is set up with the aid of weak-plasma-turbulence theory for an idealized radially symmetric model of the interplanetary magnetic field. As a starting point, this treatment invokes the Vlasov equation instead of the traditional Fokker-Planck equation. Quasi-linear theory is applied to obtain a momentum diffusion equation for the heliocentric frame of reference which describes the interaction of cosmic rays with convecting magnetic irregularities in the solar-wind plasma. Under restricted conditions, the well-known equation of solar modulation can be obtained from this kinetic equation.

Luhmann, J. G.↗
NASA NTRS1982

A note on Compton scattering

Calculations are presented on the energy exchange between free electrons and electromagnetic radiation. The full Klein-Nishina cross section is used in evaluating average scattering coefficients for an electron moving with arbitrary velocity. A number of useful series expansions for the mean energy and mean square energy transfer rates are presented. A Fokker-Planck equation that includes induced scattering is used in deriving a generalized diffusion equation in frequency for multiple scattering of photons of nonrelativistic electrons. The relationship of the Klein-Nishina cross section to that of classical electromagnetic radiation theory is elucidated.

Barbosa, D. D.↗
NASA NTRS1971

Transport of solar flare protons: Comparison of a new analytic model with spacecraft measurements

An analytic solution was obtained to the complete Fokker-Planck equation including the effects of convection, interplanetary deceleration and acceleration, corotation, and anisotropic diffusion. The solution yields a time to maximum for the particle flux of about 10 h and an exponential decay time of about 5 h. Several solar flare particle events were observed with the solar and galactic cosmic ray experiment on OGO-6. Comparisons of the calculated time dependence of the fluxes with these observations of 1 to 70 MeV protons show that the model adequately describes both the rise and decay times.

Lupton, J. E.↗
NASA NTRS1973

The Fokker-Planck coefficient for pitch-angle scattering of cosmic rays

For the case of homogeneous, isotropic magnetic field fluctuations, it is shown that most theories which are based on the quasi-linear and adiabatic approximation yield the same integral for the Fokker-Planck coefficient for the pitch angle scattering of cosmic rays. For example, despite apparent differences, the theories due to Jokipii and to Klimas and Sandri yield the same integral. It is also shown, however, that this integral in most cases has been evaluated incorrectly in the past. For large pitch angles these errors become significant, and for pitch angles of 90 deg the actual Fokker-Planck coefficient contains a delta function. The implications for these corrections relating cosmic ray diffusion coefficients to observed properties of the interplanetary magnetic field are discussed.

Fisk, L. A.↗
NASA NTRS1975

Cosmic ray propagation in interplanetary space

The validity of the test-particle picture, the approximation of static fields, and the spatial-diffusion approximation are discussed in a general way before specific technical assumptions are introduced. It is argued that the spatial-diffusion equation for the intensity per unit energy has a much wider range of applicability than the kinetic (Fokker-Planck) equation it is derived from. This gives strong weight to the phenomenological propagation theory. The general success (and possible failure at small energies) of the phenomenological theory for the modulation of galactic cosmic rays and solar events is described. Apparent effects such as the 'free boundary' are given disproportionate weight since they establish the connection with the detailed plasma physics of the solar wind. Greatest attention is paid to the pitch-angle diffusion theory. A general theory is presented which removes the well-known secularities of the quasi-linear approximation. The possible breakdown of any pitch-angle diffusion theory at very small energies is perhaps connected with the observed 'turn up' of the spectrum at low energies. A first attempt to derive the spatial dependence of the diffusion coefficient in the solar cavity, using such a divergence free scattering theory, is described and compared with recent observations out to 5 AU.

Voelk, H. J.↗
NASA NTRS1972

Solar flare particle propagation: Comparision of a new analytic solution with spacecraft measurements

An analytic solution was obtained to the complete Fokker-Planck equation for solar flare particle propagation including the effects of convection, energy-change, corotation, and diffusion. It is assumed that the particles are injected impulsively at a single point in space, and that a boundary exists beyond which the particles are free to escape. Several solar flare particle events were observed with solar and galactic cosmic ray experiment aboard OGO 6. Detailed comparisons of the predictions of the solution with observations of 1 to 70 MeV protons show that the model adequately describes both the rise and decay times. The solution also yields a time evolution for the vector anisotropy which agrees well with reported observations.

Lupton, J. E.↗
NASA NTRS1979

Comptonization of X-rays by low-temperature electrons

A method is described for calculating the spectrum that results from the Compton scattering of a monochromatic source of X-rays by low-temperature electrons, both for initial-value relaxation problems and for steady-state spatial diffusion problems. The method gives an exact solution of the inital-value problem for evolution of the spectrum in an infinite homogeneous medium if Klein-Nishina corrections to the Thomson cross section are neglected. This, together with approximate solutions for problems in which Klein-Nishina corrections are significant and/or spatial diffusion occurs, shows spectral structure near the original photon wavelength that may be used to infer physical conditions in cosmic X-ray sources. Explicit results, shown for examples of time relaxation in an infinite medium and spatial diffusion through a uniform sphere, are compared with results obtained by Monte Carlo calculations and by solving the appropriate Fokker-Planck equation.

Illarionov, A.↗
NASA NTRS1985

Pitch Angle Distributions of Solar Energetic Particles and the Local Scattering Properties of the Interplanetary Magnetic Field

An approximate solution of the Fokker-Planck equation containing pitch angle scattering and adiabatic focusing is discussed. For modest focusing effects the omnidirectional density obeys an ordinary diffusion equation with a modified diffusion coefficient. The anisotropic part of the distribution function is properly normalized and split into an even and an odd part. The even part is determined by the ratio between the scattering mean free path and the focusing length and by the degree of polarization of the magnetic field fluctuations. The odd part is determined by the deviation of the pitch angle scattering from isotropic scattering. The method supplies a powerful tool to obtain the local characteristics of pitch angle scattering. It is insensitive to long lasting solar injections and to moderate radial variations of the mean free path. The method is applied to solar particle events observed on Helio-1 and -2.

Beeck, J.↗
NASA NTRS1975

Modulation of low-energy cosmic rays

The relation between the diffusion coefficient of cosmic rays in the solar wind and the power spectrum of interplanetary magnetic field fluctuations, established in recent theories, is tested directly for low energy protons (below 80 MeV). In addition, an attempt is made to determine whether the particles are scattered by magnetic field discontinuities or by fluctuations between discontinuities. Predictions of a perturbation solution of the Fokker-Planck equation are compared with observations of the cosmic ray radial gradient. It is found that at energies between 40 and 80 MeV, galactic cosmic ray protons respond to changes in the predicted diffusion coefficients (i.e., the relationship under consideration holds at these low energies). The relation between changes in the proton flux and modulation parameters is best when the contribution of discontinuities is subtracted, which means that scattering is caused by fluctuations between discontinuities. There appears to be no distinct relation between changes in the modulation parameters and changes in the intensity of 20 to 40 MeV protons.

Sari, J. W.↗
NASA NTRS1974

Jupiter's radiation belts

Fluxes of electrons and protons in Jupiter's radiation belts are calculated with the source (radial diffusion inward from the solar wind) and the loss (synchroton radiation). The calculations are tested against the measured radio-wave wavelength distribution, the radio-wave distribution with distance from Jupiter, and the degree of polarization of the radio waves. The Fokker-Planck equation is solved by using the method of Farley and Walt (1971) with the fixed flux at the outer boundary and the zero flux at Jupiter's surface. It is found, in agreement with Brice and McDonough (1973), Jacques and Davis (1973), and Birmingham et al. (1974), that the usual magnetic and electric diffusions, which vary as L to the 10th power and L to the 6th power, respectively, are insufficient to furnish the required electrons and that the diffusion driven by ionospheric winds of Brice and McDonough is strong enough to furnish Jupiter's belts. An additional loss mechanism close to Jupiter is required to remove the electrons, particularly those at low energies.

Stansberry, K. G.↗
NASA NTRS1974

The Fokker-Planck coefficient for pitch-angle scattering of cosmic rays

For the case of homogeneous, isotropic magnetic field fluctuations, it is shown that most theories which are based on the quasi-linear and adiabatic approximations yield the same integral for the Fokker-Planck coefficient for the pitch-angle scattering of cosmic rays. For example, despite apparent differences, the theories due to Jokipii and to Klimas and Sandri yield the same integral. It is also shown, however, that this integral in most cases has been evaluated incorrectly in the past. For small pitch angles, the errors in previous evaluations are fortuitously of minor importance. For large pitch angles, however, these errors become more significant; and for pitch angles of 90 deg, the actual Fokker-Planck coefficient contains a delta function which has been overlooked in the past. The implications of these corrections on the possibility of relating cosmic-ray diffusion coefficients to observed properties of the interplanetary magnetic field are discussed.

Fisk, L. A.↗
NASA NTRS1978

Closure of a kinetic model of plasma in strong turbulence by relaxation

A Fokker-Planck kinetic equation for a turbulent plasma is derived by a repeated cascade decomposition. Calculation of the propagator and the kinetic equation determine the transport coefficients (diffusivity and turbulent viscosity) by means of a closure based on a relaxation procedure governing the approach to equilibrium. The k to the minus third power spectral law is obtained, which governs the coupling between the velocity and the electrostatic field fluctuations.

Tchen, C. M.↗
NASA NTRS1975

Cosmic ray positron and negatron spectra between 20 and 800 MeV measured in 1974

A balloon-borne spark chamber magnetic spectrometer was used to measure separate spectra of positrons and negatrons in two flights during summer, 1974. The total electron flux is about 0.3 m(-2) s(-1) sr(-1) MeV(-1) between 70 and 800 MeV, and increases toward lower energies. The positron spectrum decreases sharply toward lower energies from a value of about 0.08 m(-2) s(-1) sr(-1) MeV(-1) at 650 MeV, and only upper limits are obtained for positrons below 200 MeV. At energies above 180 MeV, the spherically symmetric Fokker-Planck equation provides reasonable fits to both the positron and total electron data. At energies below 180 MeV the data are consistent with a continuation of the same diffusion coefficient and local source of negatrons, or a change in the diffusion coefficient to a constant value.

Hartman, R. C.↗
NASA NTRS1974

Mirroring within the Fokker-Planck formulation of cosmic ray pitch angle scattering in homogeneous magnetic turbulence

The Fokker-Planck coefficient for pitch angle scattering, appropriate for cosmic rays in homogeneous, stationary, magnetic turbulence, is computed from first principles. No assumptions are made concerning any special statistical symmetries the random field may have. This result can be used to compute the parallel diffusion coefficient for high energy cosmic rays moving in strong turbulence, or low energy cosmic rays moving in weak turbulence. Becuase of the generality of the magnetic turbulence which is allowed in this calculation, special interplanetary magnetic field features such as discontinuities, or particular wave modes, can be included rigorously. The reduction of this results to previously available expressions for the pitch angle scattering coefficient in random field models with special symmetries is discussed. The general existance of a Dirac delta function in the pitch angle scattering coefficient is demonstrated. It is proved that this delta function is the Fokker-Planck prediction for pitch angle scattering due to mirroring in the magnetic field.

Goldstein, M. L.↗
NASA NTRS1975

A dynamic model for the time evolution of the modulated cosmic ray spectrum

A recently developed model predicts an energy dependent phase lag in the modulated cosmic ray density U(t) given by U(t) approximately equal to US (t - tau) where US is the solution to the Fokker-Planck equation under time independent conditions and tau is the average time spent by particles inside the modulating region. The delay times tau are functions of modulating parameters R (the radius of the modulating cavity), V (the solar wind velocity), and K (the effective average diffusion-coefficient which is a function of energy). This model is applied to predict the time evolution of the modulated cosmic ray proton spectrum over a simulated solar cycle. A modulation produced mostly by varying R over the solar cycle is less consistent with the observations.

Ogallagher, J. J.↗