Diffusion Character in Four-Dimensional Volume-Preserving Map

Due to the existence of invariant tori, chaotic sea and hyperbolic structures in higher dimensional phase space of a volume-preserving map, the diffusion route of chaotic orbits will be complicated. The velocity of diffusion will be very slow if the orbits are near an invariant torus. In order to realize this complicated diffusion phenomenon, in this paper we study the diffusion characters in the different regions, i.e., chaotic, hyperbolic and invariant tori's regions. We find that for the three different regions, the diffusion velocities are different. The diffusion velocity in the vicinity of an invariant torus is the slowest one. This revised version was published online in July 2006 with corrections to the Cover Date.     

Directional diffusion coefficients of solar protons inside and outside the bow shock

The directional diffusion coefficients of low-energy (? 0.3 MeV) solar protons inside and outside the bow shock are examined during the solar flare event of 24 January 1969. The data are derived from simultaneous observations obtained by Explorer 33 inside the magnetosheath and by Explorer 35 in the interplanetary medium. Although the gross properties of the spin-averaged intensities on a diffusion-type plot appear to be the same in both media, the directional intensities show significant variations. It is shown that directional intensities of low-energy protons can be described reasonably well by anisotropic diffusion with an associated diffusion coefficient. Directional diffusion coefficients are found to differ by a factor as much as three among different directions in space, and from the spin-averaged diffusion coefficient. This suggests that anisotropic diffusion does indeed take place and that so called ‘isotropic’ diffusion coefficients derived in the past from spin-averaged intensities may actually be directional diffusion coefficients in cases where substantial anisotropies (> 50 per cent) exist. The typical postulated ratio of field aligned to cross-field diffusion coefficients is $\text{κ⊥}\text{κ∥}\text{< 0.1}$. The present data would indicate a ratio of ?0.3. This value of the anisotropy is to be taken only as an upper limit of the ratio because of the limitations introduced by the wide field of view of the detectors (~90°) and the lack of directional measurements over the entire sphere. Comparison between directional diffusion coefficients in the interplanetary medium and magnetosheath derived from identical directions in space implies changes in the parameters of the interplanetary magnetic field as it interacts with the bow shock.     

Diffusion of trapped particles due to the bounce-drift resonance interaction in the magnetosphere

Within the framework of the quasi-linear approximation, the hybrid diffusion process due to the bounce-drift resonance interaction between trapped particles and low-frequency field fluctuations is examined. The diffusion coefficients obtained, which are valid for particles with large pitch angles, cover the previous results in a few limiting cases. In general, the diffusion coefficients depend strongly on the spatial structure of the power spectrum along field lines, as well as the frequency dependence. The relative importance of the radial diffusion and field-aligned acceleration for ringcurrent particles is discussed. It is shown that the field-aligned acceleration exceeds the inward penetration of the particles near the plasmapause.     

Saturated magnetic field amplification at supernova shocks

Cosmic ray streaming instabilities at supernova shocks are discussed in the quasi-linear diffusion formalism which takes into account the feedback effect of wave growth on the cosmic ray streaming motion. In particular, the non-resonant instability that leads to magnetic field amplification in the short wavelength regime is considered. The linear growth rate is calculated using kinetic theory for a streaming distribution. We show that the non-resonant instability is actually driven by a compensating current in the background plasma. The non-resonant instability can develop into a non-linear regime generating turbulence. The saturation of the amplified magnetic fields due to particle diffusion in the turbulence is derived analytically. It is shown that the evolution of parallel and perpendicular cosmic ray pressures is predominantly determined by non-resonant diffusion. However, the saturation is determined by resonant diffusion which tends to reduce the streaming motion through pitch angle scattering. The saturated level can exceed the mean background magnetic field.     

Resonant interaction of hydromagnetic waves with charged particles

The effect of the resonant interaction of a distribution of hydromagnetic waves on a distribution of particles is described by a diffusion equation in momentum space. The diffusion coefficients and other coefficients describing systematic acceleration and diffusion in energy space are derived in general and for a number of particular cases. It is shown that the resonant acceleration of slow ions by hydromagnetic waves is ineffective. The time evolution of the energy spectrum for ultrarelativistic particles due to interaction with hydromagnetic waves is found and applied to the case of the Crab Nebula in an accompanying article.     

Modeling Eri and Asteroseismic Tests of Element Diffusion

Taking into account the helium and metal diffusion,we explore the possible evolutionary status with a seismic analysis,of the MOST (Microvariability and Oscillations of STars) target:the star e Eft.We adopt different input parameters to construct models to fit the available observational constraints in,e.g.,Teff,L,R and [Fe/H].From the computation we obtain the average large spacings of ∈ Eri to be about 194±1 μHz.The age of the diffused models was found to be about 1 Gyr,which is younger than the age determined previously by models without diffusion.We found that the effect of pure helium diffusion on the internal structure of the young low-mass star is slight,but that of metal diffusion is obvious.The metal diffusion leads the models to have much higher temperature in the radiative interior,and,correspondingly a higher sound speed there,hence a larger frequency and spacings.     

Venus' superrotation,mixing length theory and eddy diffusion: A parametric study

The Hadley mechanism is adopted to describe the axisymmetric four day superrotation in the Venus atmosphere, with solar driven meridional winds redistributing energy and momentum, and eddy diffusion describing the actions of three dimensional transient eddies. We address the question how the eddy diffusion coefficients are related to the properties of the circulation. For the atmosphere of a slowly rotating planet such as Venus, we show that a form of the non-linear closure is suggested by the mixing length hypothesis, which constrains the magnitude of the eddy diffusion coefficients. Combining this constraint with the concept of the Rossby radius of deformation yields zonal velocities on the order of 100 m sec^–1. A steady state, non-linear, one-layer spectral model is used for a parametric study to find a relationship between heat source, meridional circulation and eddy diffusion coefficients, which yields the large zonal velocities observed. This analysis leads to the following conclusions: (1) Proportional changes in the heat source and eddy diffusion coefficients do not significantly change the zonal velocities. (2) The meridional velocity is virtually constant for large eddy diffusion coefficients. (3) Below a threshold in the diffusion rate, the meridional velocity decreases, commensurate with the mixing length hypothesis. Eddy heat conduction becomes important and shares with the Hadley cell advection in balancing the solar heating. The zonal velocities then reach large values near 100 m sec^–1. (4) For large eddy diffusion and small heating rates, the zonal velocities decrease with decreasing planetary rotation rates. However, under condition (3), the zonal velocities are independent of the planetary rotation rate. Ramifications are discussed for related parameterizations in GCMs, emphasizing that eddy diffusion coefficients are governed by solar forcing and cannot be chosen independently.     

Formulation of molecular diffusion in planetary atmospheres

We report on a formulation of molecular diffusion for ionized multi-component atmospheres that is valid in the diffusion and small electron mass limits. The formulation is based on the construction of successive approximations of the diffusion matrix by means of the projective iterative algorithm of Ern and Giovangigli [Projected iterative algorithms with application to multicomponent transport. Lin. Alg. and its Appl. 250, 289–315], and allows naturally for different temperatures for the neutral, ion and electron constituents of the gas. The reported expressions incorporate the effect of electric forces preventing charge separation, are explicit in the driving forces and mass conservative. Yet approximate, their accuracy can be easily tested and improved upon by going to a higher approximation of the diffusion matrix. We have illustrated the formulation with a model that solves the composition of Mars’ atmosphere. The continuity equations of the model are linearized and marched in time with an implicit numerical scheme, allowing thus for large time steps. It is found that the first and second approximations of the diffusion matrix are probably optimal trade-offs between computational cost and accuracy. Finally, the formulation is tested against more conventional approximations of the molecular diffusion velocities of neutral and ion species, showing the importance of the various assumptions that may restrict their applicability.     

Radial diffusion revisited

The radial diffusion of equatorially mirroring particles (J = 0) is considered for Jupiter. A steady-state phase-space density distribution is obtained for (i) source-loss-free diffusion; (ii) diffusion with synchrotron radiation losses only and (iii) diffusion with synchrotron radiation plus the resonant wave-particle interaction losses. The resonant wave-particle interaction is assumed to occur when particles are in phase with a wave propagating across the magnetic field. The interaction of particles which go through a B drift with electrostatic plasma waves is shown to alter the phase-space density which is observed byPioneer 10 and 11 flybys.     

Chirikov diffusion in the asteroidal three-body resonance (5, −2, −2)

The theory of diffusion in many-dimensional Hamiltonian system is applied to asteroidal dynamics. The general formulation developed by Chirikov is applied to the Nesvorny-Morbidelli analytic model of three-body (three-orbit) mean-motion resonances (Jupiter-Saturn-asteroid). In particular, we investigate the diffusion along and across the separatrices of the (5, −2, −2) resonance of the (490) Veritas asteroidal family and their relationship to diffusion in semi-major axis and eccentricity. The estimations of diffusion were obtained using the Melnikov integral, a Hadjidemetriou-type sympletic map and numerical integrations for times up to 10⁸ years.     

Star Formation and the Hall Effect

The breakdown of flux-freezing in molecular clouds and protostellar discs is usually approximated by ambipolar diffusion at low densities or by resistive diffusion at high densities. Here an intermediate regime is discussed in which the Hall term in the conductivity tensor is significant, and the vector evolution of the magnetic field, and therefore the evolution of the system under consideration is dramatically altered. Calculations of charged particle abundances in dense gas in molecular clouds and protostellar discs demonstrate that Hall diffusion is important over a surprisingly broad range of conditions.     

Scaling for the intensity of radiation in spherical and aspherical planetary nebulae

Planetary nebulae are imaged using three different physical processes. The first process is the expansion of the shell, which can be modelled by the canonical laws of motion in the spherical case and by momentum conservation when gradients of density are present in the interstellar medium. The second process is the diffusion of particles that radiate from the advancing layer. Three-dimensional diffusion from a sphere and one-dimensional diffusion with drift are analysed. The third process is the composition of an image through an integral operation along the line of sight. The framework developed is applied to A39, the Ring nebula and the etched hourglass nebula MyCn 18.     

Vertical diffusion of O and N2 in the thermosphere

The vertical transport velocities of O and N₂ in the height range 200–500 km are derived for simple models of the thermosphere. Also, we discuss the Brunt-Väisälä, diffusion and collisional time constants. If the O/N₂ ratio is perturbed, recovery to steady state is controlled by diffusion; the diffusion time constant exceeds 1 day below about 150 km.     

Pitch-angle diffusion of ring current protons caused by electrostatic ion loss-cone waves

Pitch-angle diffusion of ring current protons is investigated, assuming that the scattering is caused by resonant interaction with electrostatic ion loss-cone waves. Bounce averaged diffusion coefficients are presented for different wave energy distributions. It is found that wave amplitudes about 1 mV/m are required to give strong diffusion of protons with average energy.     

Electron precipitation in a non-uniform glow aurora

Electron intensities at 5 keV >18 keV and >45 keV, were measured on a Petrel rocket flown from Kiruna, Sweden, into a non-uniform glow aurora during the recovery phase of a magnetic bay. The intensities depend on pitch-angle in a way that is consistent with the precipitation being caused by pitch-angle diffusion from reservoirs of geomagnetically-trapped electrons. The scattering process that causes pitch-angle diffusion, and leads to three regions of relatively high intensity, appears to have properties different from the scattering process that leads to two intervening regions of low intensity. A spatial structure in electron reservoir intensity, is attributed mainly to variations in the rate of erosion by pitch-angle diffusion of an initially nearly-uniform reservoir intensity. An expression is derived for the minimum lifetime of trapped electrons undergoing strong pitch-angle diffusion.     

Regimes of contradiction in the study of strong diffusion

Various aspects of Etchetoet al.'s theory are studied in detail. It is shown that the inequalitya <b <c fails at lowL shells. A comparative study of wave spectral densities generated during weak/strong diffusion events suggests that this theory is strictly not applicable at lowL shells. It is found that different criterea for measuring the diffusion of energetic electrons by ELF waves (100–3000 Hz) contradict each other. If one region is strong diffusion region for one criteria, it may be a weak diffusion region according to another criteria. It is found that whereas critical wave intensities (1–2 pT²/Hz) can cause strong diffusion of energetic electrons, if one applies Fokker—Planck's diffusion approach, very large wave intensities are needed for strong diffusion if we apply Etchetoet al.'s approach.     

A study of the force-field equation for the propagation of galactic cosmic rays

A new development is given of the solution of the equation of the force-field approximation for the propagation of galactic cosmic rays in the interplanetary region. It leads to simpler methods for determining the force-field parameters. A method is given for determining the separable diffusion coefficient from observations of galactic electron spectrum and near-Earth electron spectra; it is shown that this diffusion coefficient is not unique but may have a periodic-like dependence upon rigidity; and the method is used to obtain diffusion coefficients for 1965 and 1968. Approximate formulae relating small changes in intensity and diffusion coefficient are developed and some applications of these noted; in one it is shown that the form of, and changes in, diffusion coefficient deduced previously for a neutron monitor event during June–September 1969 are unnecessarily constrained and therefore probably not correct.     

The Role of Hyperbolic Invariant Sets in Stickiness Effects

With the standard map model, we study the stickiness effect of invariant tori, particularly the role of hyperbolic sets in this effect. The diffusion of orbits originated from the neighborhoods of hyperbolic points, periodic islands and torus is studied. We find that they possess similar diffusion rules, but the diffusion of orbits originated from the neighborhood of a torus is faster than that originated near a hyperbolic set. The numerical results show that an orbit in the neighborhood of a torus spends most of time around hyperbolic invariant sets. We also calculate the areas of islands with different periods. The decay of areas with the periods obeys a power law, and the absolute values of the exponents increase monotonously with the perturbation parameter. According to the results obtained, we conclude that the stickiness effect of tori is caused mainly by the hyperbolic invariant sets near the tori, and the diffusion speed becomes larger when orbits diffuse away from the torus.     

Arnold diffusion: an overview through dynamical astronomy

The aim of this work is to review the fundamental ideas behind the stability problem, emphasizing the differences between two well-known mechanisms that could lead to chaos, namely overlap of resonances and Arnold diffusion. Here we restrict the discussion to multidimensional autonomous Hamiltonian systems which are of major relevance in Dynamical Astronomy. Arnold diffusion is reviewed in a standard mathematical language, by means of different tools such as heuristic reasoning, graphic and geometrical considerations and numerical experiments. In this direction the pioneer work due to Chirikov [PhR 52 (1979) 263] is followed, but including additional notes, further examples and useful discussions that may well illuminate the understanding of Arnold diffusion. We also summarize the main difficulties when coping with this instability, from both the analytical and numerical sides of the problem. The discussion whether Arnold diffusion could play any role in the dynamical evolution of, for instance elliptical galaxies, is also included.     

Replenishment of electrons lost in the south atlantic anomaly by gyroresonant pitch-angle diffusion

Experimental data describing the effect of the South Atlantic anomaly on E? 280 keV electron flux at L = 2 and high B values, are compared to the numerical solution of a pitch-angle diffusion equation with a varying loss cone. The diffusion coefficient needed to explain replenishment of the electrons lost over the anomaly is found to be 3.2 × 10^?2 sec^?1 Calculation of the diffusion coefficient due to cyclotron resonant interaction with VLF electro-magnetic waves leads to the conclusion that the observed wave spectral density can yield the needed diffusion coefficient.