Solutions to bi-Maxwellian transport equations for SAR-arc conditions

We have compared solutions obtained from the bi-Maxwellian based 16-moment transport equations with those obtained from the Maxwellian based 13-moment transport equations for conditions leading to the steady state, subsonic flow of a fully-ionized electron-proton plasma along geomagnetic field lines in the vicinity of the plasmapause. The bi-Maxwellian based equations can account for large temperature anisotropies and the flow of both parallel and perpendicular thermal energy, while the Maxwellian based equations account for small temperature anisotropies and only the total heat flow. Our comparison indicates that for Stable Auroral Red arc (SAR-arc) conditions leading to strong field-aligned heat flows (temperatures of 8000 K and temperature gradients of4K. km⁻¹ at 1500 km), the bi-Maxwellian based equations predict a different thermal structure in the topside ionosphere than the less rigorous Maxwellian based equations. In particular, the bi-Maxwellian based equations predict proton and electron temperature anisotropies with T > T, while the Maxwellian based equations predict the opposite behavior for the same boundary conditions. This difference is related to the way in which the temperature anisotropies and heat flows are treated in the two formulations. For the bi-Maxwellian based equations, the inclusion of separate heat flows for parallel and perpendicular thermal energy allows for the development of a pronounced tail in both the electron and proton distribution functions, which leads to temperature anisotropies with T > T. For the Maxwellian based equations, on the other hand, the tail development is restricted because only the total heat flow is considered. Consequently, as the heat flows down, the presence of an increasing magnetic field acts to produce an anisotropy with T > T, and this process dominates tail formation for the Maxwellian based equations.     

Equations of anisotropic hydrodynamics for electron component of the ionospheric plasma

In this paper we have derived a set of transport equations for thermal electron component of the ionospheric plasma in the presence of an anisotropy of the electron energy distribution. Expressions are calculated in a 16-moment approximation for the moments of integrals of elastic and inelastic collisions of thermal electrons with basic neutral ionospheric components. The obtained moments determine variations of the hydrodynamical parameters, such as macroscopic velocity, pressure tensor, viscosity tensor, heat fluxes in respective equations due to collisions. The results have been obtained for an arbitrary degree of electron temperature anisotropy.     

Thermal response properties of the Earth's ionospheric plasma

The thermal response of the Earth's ionospheric plasma is calculated for various suddenly applied electron and ion heat sources. The time-dependent coupled electron and ion energy equations are solved by a semi-automatic computational scheme that employs Newton's method for coupled vector systems of non-linear parabolic (second order) partial differential equations in one spatial dimension. First, the electron and composite ion energy equations along a geomagnetic field line are solved with respect to a variety of ionospheric heat sources that include: thermal conduction in the daytime ionosphere; heating by electric fields acting perpendicular to the geomagnetic field line; and heating within a stable auroral red are (SAR-arc). The energy equations are then extended to resolve differential temperature profiles, first for two separate ion species (H⁺, O⁺) and then for four separate ion species (H⁺, He⁺, N⁺, O⁺) in addition to the electron temperature. The electron and individual ion temperatures are calculated for conditions within a night-time SAR-arc excited by heat flowing from the magnetosphere into the ionosphere, and also for typical midlatitude daytime ionospheric conditions. It is shown that in the lower ionosphere all ion species have the same temperature; however, in the topside ionosphere above about 400 km, ion species can display differential temperatures depending upon the balance between thermal conduction, heating by collision with electrons, cooling by collisions with the neutrals, and energy transfer by inter-ion collisions. Both the time evolution and steady-state distribution of such ion temperature differentials are discussed.The results show that below 300km both the electrons and ions respond rapidly (<30s) to variations in direct thermal forcing. Above 600 km the electrons and ions display quite different times to reach steady state, depending on the electron density: when the electron density is low the electrons reach steady state temperatures in 30 s, but typically require 700 s when the density is high; the ions, on the other hand, reach steady state in 700 s when the density is high, and 1500–2500 s when the density is low. Between 300 and 600 km, a variety of thermal structures can exist, depending upon the electron density and the type of thermal forcing; however steady state is generally reached in 200–1000 s.     

Effect of diffusion-thermal processes on the high-latitude topside ionosphere

We have studied the extent to which diffusion-thermal heat flow affects H⁺ temperatures in the high-latitude topside ionosphere. Such a heat flow occurs whenever there are H⁺?O⁺ relative drifts. From our study we have found that at high-latitudes, where H⁺ flows up and out of the topside ionosphere, diffusion-thermal heat flow acts to reduce H⁺ temperatures by 500–600 K at altitudes above about 900 km.     

Ion transition heights from topside electron density profiles

Theoretical electron density profiles are calculated for the topside ionosphere to determine the major factors controlling the profile shape. Only the mean temperature, the vertical temperature gradient and the $\text{O}{}^{\mathrm{+}}\text{H}{}^{\mathrm{+}}$ ion transition height are important. Vertical proton fluxes alter the ion transition height but have no other effect on the profile shape. Diffusive equilibrium profiles including only these three effects fit observed profiles, at all latitudes, to within experimental accuracy.Values of plasma temperature, temperature gradient and ion transition height h_tT were determined by fitting theoretical models to 60,000 experimental profiles obtained from Alouette l ionograms, at latitudes of 75°S–85°N near solar minimum. Inside the plasmasphere h_T varies from about 500 km on winter nights to 850 km on summer days. Diurnal variations are caused primarily by the production and loss of O⁺ in the ionosphere. The approximately constant winter night value of h_T is close to the level for chemical equilibrium. In summer h_T is always above the equilibrium level, giving a continual production of protons which travel along lines of force to aid in maintaining the conjugate winter night ionosphere. Outside the plasmasphere h_T is 300–600 km above the equilibrium level at all times. This implies a continual near-limiting upwards flux of protons which persists down to latitudes of about 60° at night and 50° during the day.     

Electron distribution function and ion concentrations in the earth's lower ionosphere from Boltzmann-Fokker-Planck theory

Using the Boltzmann-Fokker-Planck method and the local approximation we derive coupled non-linear equations for the electron and ion concentrations and the energy-dependent electron distribution function in the Earth's lower ionosphere. These equations are new and give the appropriate generalization of the standard electron-ion continuity equations in the local approximation when electron-neutral particle impact ionization is treated rigorously. We report stable, numerical solutions to these equations and compare our calculated electron concentration to the experimentally determined result for a rocket experiment where the electron concentration and solar EUV spectral flux were measured simultaneously.     

Incoherent scatter spectrum of ionospheric plasma with an anisotropic temperature ion distribution

The expression of anisotropic temperature ion distribution function under the 13-moment approximation is obtained by solving a set of moment equations based on the Boltzmann equation for a relaxation collision model and with consideration of the anisotropic temperature ion distribution. And the incoherent scatter spectrum with an anisotropic temperature ion distribution is simulated in different directions based on the electromagnetic radiation theory of Sheffield. The effects of different electrical field strengths, ratios of electron temperature to ion temperature, and ion-neutral collision frequencies on the incoherent scatter spectrum are all discussed. Finally, the value of theoretical simulation is compared with the measured value of incoherent scattering spectrum. The result show that the incoherent scatter spectrum of ions seriously deviates from the form of the Maxwellian distribution in the equilibrium state. This phenomenon can be attributed to the effects of anisotropic temperature ion distribution, the larger convection electric field, and other factors in high latitude ionosphere.     

An equation of state at subnuclear densities

An equation of state for cold matter above white dwarf densities is evaluated. The gas is considered to be a mixture of degenerate neutrons, protons and electrons combined with nuclei of one type (that is only oneA andZ value). We derive the equilibrium equations for the mixture and calculate the number densities as well as theA andZ of the nucleus. Finally we calculate an equation of state, which smoothly goes over to that of a neutron, proton electron gas mixture at a density of 5×10¹³ g/cm³.     

Solutions of the two-fluid solar wind equations: Adiabatic and conduction dominated solutions

The equations governing the two-fluid spherically symmetric models of the solar wind have been solved numerically for a wide range of base conditions. As predicted from an asymptotic analysis we find a whole domain of solutions which are asymptotically adiabatic with the proton and electron temperatures tending to equality and varying like r ^{- 4/3}. In these 4/3 solutions the electron and proton heat conduction is asymptotically negligible and if it is neglected the resulting equations can be integrated analytically and shown to have the 4/3, 4/3 behaviour.Proceedings of the 14th ESLAB Symposium on Physics of Solar Variations, 16–19 September 1980, Scheveningen, The Netherlands.     

Effects of Wave-Particle Interactions on Double-Hump Distributions of the H + Polar Wind

A Monte Carlo simulation is used in order to study the effects of wave-particle interactions (WPI) on H⁺ distributions in the polar wind outflow. The simulation also considers effects of the gravity, the polarization electric field, the divergence of geomagnetic field lines and H⁺−O⁺ Coulomb collisions. The proton velocity distribution function (VDF) and the profiles of its moments (density, bulk velocity, parallel and perpendicular temperatures, heat flux…) are found for different levels of WPI, i.e., for different values of the normalized diffusion rate in the velocity space (D _⊥). We find that the wave-particle interactions accelerate the polar wind and can have important effects on the double-hump H⁺ distribution obtained in the transition region between the collision-dominated low altitudes and the collisionless high altitude regions.     

The penetration of soft electrons into the ionosphere

Calculations are presented of energy spectra and angular and spatial distributions of electron fluxes in the ionosphere resulting from precipitation ofmonoenergetic (E = 25, 50 and 100 eV) electrons. The incident electrons are assumed to be isotropic over the downward direction. It is found that the resulting steady-state electron fluxes above ca. 300 km are highly anisotropic, and that the pitch angle distribution is energy dependent. About 15 per cent of the incident electrons are backscattered elastically to the protonosphere. A much larger number of electrons escape after they have deposited a part of their energy in the atmosphere. The mean energy of the escaping electrons is about half that of the incident electrons. About 50% of the incident energy is absorbed in the atmosphere, the remainder being returned to the protonosphere. The rate of absorption of energy is a maximum at heights between 300 and 400 km. Most of the energy is absorbed in ionization and excitation of atomic oxygen. An appreciable amount of energy is, however, absorbed as heat by the ambient electron gas. Altitude profiles are presented of the rates of ionization, excitation, and electron heating caused by soft electron precipitation.     

Thermal proton flow in the plasmasphere: The morning sector

Vertical profiles of electron density obtained in the vicinity of the plasmapause using the Alouette-II topside sounder have been analyzed to assess the presence of H⁺ flow in the topside ionosphere. The observations in the midnight sector show clearly the presence of the plasmapause; i.e. there is a sharp boundary separating the poleward regions of polar wind H⁺ flow and the more gentle conditions of the plasmasphere where light ions are present in abundance. In contrast, in the sunlit morning sector upwards H⁺ flow is deduced to be present to invariant latitudes as low as 48° (L = 2·2) in the regions normally known to be well inside the plasmasphere. The upwards H⁺ flux is sufficiently large (3 × 10⁸ ions cm^?2 sec^?1) that the plasmapause cannot be seen in the latitudinal electron density contours of the topside ionosphere. The cause for this flow remains unknown but it may be a result of a diurnal refilling process.     

Standing slow magnetosonic waves in a dipole-like plasmasphere

A problem of the structure and spectrum of standing slow magnetosonic waves in a dipole plasmasphere is solved. Both an analytical (in WKB approximation) and numerical solutions are found to the problem, for a distribution of the plasma parameters typical of the Earth's plasmasphere. The solutions allow us to treat the total electronic content oscillations registered above Japan as oscillations of one of the first harmonics of standing slow magnetosonic waves. Near the ionosphere the main components of the field of registered standing SMS waves are the plasma oscillations along magnetic field lines, plasma concentration oscillation and the related oscillations of the gas-kinetic pressure. The velocity of the plasma oscillations increases dramatically near the ionospheric conductive layer, which should result in precipitation of the background plasma particles. This may be accompanied by ionospheric F2 region airglows modulated with the periods of standing slow magnetosonic waves.     

The effect of the lower ionosphere on hydromagnetic modes in the plasmasphere

Dimensionless resonant frequencies of hydromagnetic modes have been calculated for a simple model plasmasphere including a lower ionosphere. Results for the Alfvén mode are broadly consistent with those obtained by Hughes and Southwood [1976]. It is further concluded that the lower ionosphere, despite its strong damping effect for part of the day, does not provide much dissipative coupling between adjacent magnetic field shells in the Alfvén mode. The fast mode is found to be only slightly damped for horizontal wavelengths of global extent.     

Simulation of ionosphere-plasmasphere coupling taking into account ion inertia and temperature anisotropy

In this paper we offer a model for the Earth's ionosphere and plasmasphere, allowing for the inertia and anisotropic energy distribution of thermal plasma. A procedure for simultaneous solution of equations of continuity and motion for the O⁺ and H⁺ ions, subject to inertia terms, is described. The model also includes transfer equations for longitudinal and transversal thermal energies. The system of simulating equations and the kinetic equation for superthermal electron spectra are concordantly solved along geomagnetic field lines. Within the framework of the model we developed a study is made of the dynamics of filling of the evacuated plasmaspheric reservoir after a magnetospheric disturbance. It is shown that the filling of the tubes offorce with L ? 3.5 proceeds with supersonic speeds during the first several days and the character of filling differs very much from a diffusion-equilibrium one. The spatio-temporal behavior of electron and ion temperature anisotropy that is formed in the process of filling, is considered. It is found that the value of electron anisotropy can be large. A brief analysis is made of the causes of electron and ion temperature anisotropy.     

Propagation of Hydrogentic waves in the upper F2-region>

The theory of hydromagnetic-wave in the upper F2-region, in which electrons are in a transitional regime from collisional to collisionless conditions and ions are in a collisionless state, is examined. Derivation of the governing equations is based on the fact that the isotropic electrons are fluid-like, and the anisotropic ions follow kinetic equations modified by ion-electron collisions. Magneto-acoustic waves of a period of 0.2–10 sec are dissipated by ion Landau damping and electron thermal conduction and viscosity. Numerical solutions under ionospheric conditions show that the dissipation of hydromagnetic waves is insufficient to modify the large scale heating of the ionosphere.     

A comparison of the temperature and density structure in high and low speed thermal proton flows

The continuity, momentum and energy hydrodynamic equations for an H⁺-O⁺ topside ionosphere have been solved self-consistently for steady state conditions similar to those found outside the plasmasphere. Results are given for undisturbed and trough conditions with a range of H⁺ outflow velocities yielding subsonic and supersonic flow. In the formulation of the equations, account was taken of the velocity dependence of ion-neutral, ion-ion and ion-electron collision frequencies. In addition, parallel stress and the nonlinear acceleration term were retained in the H⁺ momentum equation. Results computed from this model show that, as a result of Joule (frictional) heating, the H⁺ temperature rises with increasing outflow velocity in the subsonic flow regime, reaching a maximum value of about 4000 K. For supersonic flow other terms in the H⁺ momentum equation become important and alter the H⁺ velocity profile such that convection becomes a heat sink in the 1000–1500 km altitude range. This, together with the reduced Joule heating resulting from the high-speed velocity dependence of the H⁺ collision frequencies, results in a decrease in the H⁺ temperature as the outflow velocity increases. However, for all outward flows the H⁺ temperature remains substantially greater than the O⁺ temperature. With identical upper boundary velocities, the H⁺ flow velocity is higher at low altitudes for trough conditions compared with non-trough conditions, but the H⁺ temperature in the trough is lower. The form of the H⁺ density profiles for supersonic flow does not in general differ greatly from those obtained with wholly subsonic flow conditions.     

Jupiter's ionosphere and magnetosphere

The distribution of ionization and the temperature variation in the Jovian ionosphere is determined by simultaneously solving the momentum and chemical equations for electrons, ions and neutrals together with their respective heat transport equation. The boundary conditions at the bottom of the ionosphere are chosen in accordance with recent infra-red and occultation measurements. The ionosphere is hotter than previously thought. The electron temperature may be as high as 1500 K. A considerable flux of particles can escape from the ionosphere. These particles are trapped in the Jovian magnetosphere by a two-stream instability. A Gledhill disk will form. The variation of plasma density along Io's orbit is calculated.     

The formation and lifetime of whistler ducts

It is assumed that whistler ducts are formed by electric fields interchanging magnetospheric flux tubes of ionization. It is found that such ducts end several hundred kilometres above the transition level, that is usually in the altitude range of 1000–1500 km. Further, the enhancement factor is found to increase towards the equator if the background density has little latitudinal variation. Both of the above properties make such ducts ideal for trapping whistlers.The half-life of whistler ducts is estimated to be of the order of one day. During quiet times ducts decay through enhanced plasma flow into the underlying ionosphere, whereas during storm times, when the plasmasphere is depleted of ionization, large upward plasma flows reduce the enhancement factors of ducts.