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NASA NTRS2002

The Statistical Mechanics of Ideal Homogeneous Turbulence

Plasmas, such as those found in the space environment or in plasma confinement devices, are often modeled as electrically conducting fluids. When fluids and plasmas are energetically stirred, regions of highly nonlinear, chaotic behavior known as turbulence arise. Understanding the fundamental nature of turbulence is a long-standing theoretical challenge. The present work describes a statistical theory concerning a certain class of nonlinear, finite dimensional, dynamical models of turbulence. These models arise when the partial differential equations describing incompressible, ideal (i.e., nondissipative) homogeneous fluid and magnetofluid (i.e., plasma) turbulence are Fourier transformed into a very large set of ordinary differential equations. These equations define a divergenceless flow in a high-dimensional phase space, which allows for the existence of a Liouville theorem, guaranteeing a distribution function based on constants of the motion (integral invariants). The novelty of these particular dynamical systems is that there are integral invariants other than the energy, and that some of these invariants behave like pseudoscalars under two of the discrete symmetry transformations of physics, parity, and charge conjugation. In this work the 'rugged invariants' of ideal homogeneous turbulence are shown to be the only significant scalar and pseudoscalar invariants. The discovery that pseudoscalar invariants cause symmetries of the original equations to be dynamically broken and induce a nonergodic structure on the associated phase space is the primary result presented here. Applicability of this result to dissipative turbulence is also discussed.

Shebalin, John V.↗

NASA NTRS1978

Discrepancy in proton flux extrapolation along field lines in the middle Jovian magnetosphere

Fluxes of energetic electrons and protons in Jupiter's outer magnetosphere were observed to be modulated with the 10 hour rotation period of the planet. This modulation was due to the concentration of particles at the magnetic equator: the non-alignment of Jupiter's spin and rotation axes caused Pioneer-10 to oscillate between +20 deg and -19 deg magnetic latitude and hence, between regions of stronger and weaker fluxes. The relationship between electron and proton fluxes observed off the magnetic equator was countered with measurements at the equatorial crossing radii of the same flux tubes by applying Liouville's theorem with the assumption that particles move conserving their magnetic moments. A magnetic model which matches the intensity and direction of the magnetic field along the Pioneer 10 trajectory was used for determining the positions of the equatorial crossings. Energetic electrons compared in this way appear to be consistently described. Protons, on the other hand, show much weaker fluxes at the off-equatorial points than would be predicted by this simple application of Liouville's theorem.

Schardt, A. W.↗

NASA NTRS1994

Comparison between Liouville's theorem and observed latitudinal distributions of trapped ions in the plasmapause region

The presence of anisotropic plasma distributions, trapped at the Earth's magnetic equator, has consequences for the electric field structure which must exist in equilibrium along the magnetic field line. Data from SCATHA and Dynamics Explorer 1 indicate that the core ion distributions at the magnetic equator can be well described as bi-Maxwellian distributions, with a perpendicular temperature an order of magnitude larger than the parallel temperature. A collisionless model is developed for the variation in plasma parameters, following the forms developed by Whipple (1977). If the core electron anisotropy is low, the resulting electric field of approximately 0.1 microV/m is pointed away from the equator. Under these conditions the self-consistent electric field will not overcome the effects of magnetic trapping. The resulting potential distribution results in a local maximum in total plasma density at the equator. Only when the electron distribution is primarily field-aligned can there be a density minimum at the equator. Comparisons are made between this model and the observed variations in DE 1 plasma parameters with latitude.

Olsen, R. C.↗

NASA NTRS1986

Nonlinear upper hybrid drift waves for a longitudinal electric field perpendicular to a uniform magnetic field in the Vlasov-Maxwell approximation

Upper hybrid drift waves are found as a special solution to a Vlasov-Maxwell plasma which has a longitudinal electric field and a perpendicular uniform magnetic field. A single-species plasma with a constant-density mobile neutralizing background supports spatially varying disturbances that oscillate at the upper hybrid frequency. The general functional dependences of the electric field, the plasma number density, and the one-particle distribution function for the special case are found from more general Vlasov-Maxwell equations invariant under a Lie group point transformation. The one-particle distribution function for the plasma is a function of the Liouville invariant, which is the energy in the generalized Bernstein-Greene-Kruskal (BGK) reference frame, and the momentum in the drift direction.

Abraham-Shrauner, B.↗

NASA NTRS1993

Self-consistent electrostatic potential due to trapped plasma in the magnetosphere

A steady state solution for the self-consistent electrostatic potential due to a plasma confined in a magnetic flux tube is considered. A steady state distribution function is constructed for the trapped particles from the constants of the motion, in the absence of waves and collisions. Using Liouville's theorem, the particle density along the geomagnetic field is determined and found to depend on the local magnetic field, self-consistent electric potential, and the equatorial plasma distribution function. A hot anisotropic magnetospheric plasma in steady state is modeled by a bi-Maxwellian at the equator. The self-consistent electric potential along the magnetic field is calculated assuming quasineutrality, and the potential drop is found to be approximately equal to the average kinetic energy of the equatorially trapped plasma. The potential is compared with that obtained by Alfven and Faelthammar (1963).

Miller, Ronald H.↗

NASA NTRS1988

An invariant asymptotic formula for solutions of second-order linear ODE's

An invariant-matrix technique for the approximate solution of second-order ordinary differential equations (ODEs) of form y-double-prime = phi(x)y is developed analytically and demonstrated. A set of linear transformations for the companion matrix differential system is proposed; the diagonalization procedure employed in the final stage of the asymptotic decomposition is explained; and a scalar formulation of solutions for the ODEs is obtained. Several typical ODEs are analyzed, and it is shown that the Liouville-Green or WKB approximation is a special case of the present formula, which provides an approximation which is valid for the entire interval (0, infinity).

Gingold, H.↗

NASA NTRS1994

The streaming-trapped ion interface in the equatorial inner magnetosphere

Spacecraft measurements of core ions on L=4-7 field-lines typically show trapped ion distributions near the magnetic equator, and frequently indicate field-aligned ion streams at higher latitudes. The nature of the transition between them may indicate both the microphysics of hot-cold plasma interactions and overall consequences for core plasma evolution. We have undertaken a statistical analysis and characterization of this interface and its relation to the equatorial region of the inner magnetosphere. In this analysis, we have characterized such features as the equatorial ion flux anisotropy, the penetration of field-aligned ionospheric streams into the equatorial region, the scale of the transition into trapped ion populations, and the transition latitude. We found that most transition latitudes occur within 13 deg of the equator. The typical values of equatorial ion anisotropies are consistent with bi-Maxwellian temperature ratios of T(sub perpendicular)/T(sub parallel) in the range of 3-5. The latitudinal scales for the edges of the trapped ion populations display a rather strong peak in the 2-3 deg range. We also found that there is a trend for the penetration ratio, the anisotropy half width, and the transition scale length to decrease with a higher equatorial ion anisotropy. We may interpret these features in terms of Liouville mapping of equatorially trapped ions and the reflection of the incoming ionospheric ion streams from the equatorial potential peaks associated with such trapped ions.

Lin, J.↗

NASA NTRS1985

Geocoronal structure - The effects of solar radiation pressure and the plasmasphere interaction

The theory of planetary exospheres is extended to incorporate solar radiation pressure in a rigorous manner, and an evaporative geocoronal prototype (classical, motionless exobase) is constructed using Liouville's theorem. Model calculations for density and kinetic temperature at points along the earth-sun axis (solar and antisolar directions) reveal an extensive satellite component, comprising approximately 2/3 of the total hydrogen density near 10 earth radii, and a temperature profile suggestive of an isotropic quasi-Maxwellian velocity distribution for the bound component. A geotail is also evident as an enhancement of the density at local midnight compared to local noon that increases outward (from approximately 25 percent at 10 earth radii to over 60 percent at 20 earth radii). Additional mechanisms acting upon the geocorona alter the basic evaporative case in notable ways. Solar ionization has been included in a simple fashion; the effect is to partially deplete the density without otherwise altering the structure. Interaction with a simple plasmasphere via the Boltzmann equation results in 'heating' the geocorona and enhancing the escape flux at the expense of the density of the bound component, an effect not appreciated in earlier studies; the geotail survives this interaction.

Bishop, J.↗