Auroral ion velocity distribution function: The Boltzmann model revisited

The velocity distribution of ion populations is calculated for auroral conditions where strong convection electric fields exist. The Boltzmann equation has been solved for the E and F regions of the ionosphere where plasma is weakly dense, weakly ionized and where the ion-neutral collision frequency is small in regard to the ion cyclotron frequency. The ion distribution function has been expanded in a generalized orthogonal polynomial series about a bi-Maxwellian “temperature” varying weight function. This generalized Grad solution expansion enables us to obtain good approximations for electric field strengths as large as 75 mV m^?1 and 115 mV m^?1 respectively, for both the resonant charge exchange and the polarization collision models. The instability threshold of these distribution functions appears to be higher than the two respective electric field strengths considered above.     

Auroral ion velocity distributions using a relaxation model

For application to the auroral ionosphere we have calculated ion velocity distributions for a weakly-ionized plasma subjected to crossed electric and magnetic fields. By replacing the Boltzmann collision integral with a simple relaxation model, we have been able to obtain an exact solution to Boltzmann's equation. This solution has the advantage over a series expansion in that all the higher order velocity moments are inherent in it. The exact solution is particularly advantageous when studying large departures of the distribution from its Maxwellian form because these departures are caused by the higher velocity moments. In general, however, a simple relaxation model can only be used to obtain qualitative information on the distribution function. Consequently, we can determine when the higher order velocity moments affect the ion velocity distribution and the nature of their effect, but we cannot obtain accurate quantitative results. The higher velocity moments have their greatest effect on the distribution function above about 120 km, where the ion-neutral collision frequency is less than the ion cyclotron frequency. As the magnitude of the electric field increases, these higher moments act to decrease the number of ions at the peak of the distribution function. Peak densities are reduced by a few per cent for perpendicular electric fields of about 20 mV m^?1.     

Behaviour of ion velocity distributions for a simple collision model

For application to studies of the high latitude ionosphere, we have calculated ion velocity distributions for a weekly-ionized plasma subjected to crossed electric and magnetic fields. An exact solution to Boltzmann's equation has been obtained by replacing the Boltzmann collision integral with a simple relaxation model. At altitudes above about 150 km, where the ion collision frequency is much less than the ion cyclotron frequency, the ion distribution takes the shape of a torus in velocity space for electric fields greater than 40 mV m^?1. This shape persists for 1–2 hr after application of the electric field. At altitudes where the ion collision and cyclotron frequencies are approximately equal (about 120 km), the ion velocity distribution is shaped like a bean for large electric field strengths. This bean-shaped distribution persists throughout the lifetime of ionospheric electric fileds. These highly non-Maxwellian ion velocity distributions may have an appreciable affect on the interpretation of ion temperature measurements.     

Ion velocity distributions in the auroral ionosphere

For application to studies of the auroral ionosphere we have calculated the velocity distribution of the ions in a weakly-ionized plasma subjected to crossed electric and magnetic fields. We have retained enough terms in the series expansion of the distribution to enable us to determine under what conditions departures from the Maxwellian form become significant and what the nature of these departures is, but we cannot calculate precise values of the distribution function when the departures are large. Departures are negligibly small under conditions appropriate to the auroral ionosphere at low altitudes, where the ion-neutral collision frequency is much larger than the ion cyclotron frequency. At altitudes above about 120 km, however, the magnitude of the departures varies little with altitude. Electric fields greater than 25 mV m⁻¹ cause departures from the Maxwellian distribution that are greater than 20 per cent at random velocities equal to or greater than twice the mean thermal speed of the ions. Under almost all conditions we find that the distribution is depleted in ions moving parallel to the magnetic field relative to those moving perpendicular, an effect that might be detectable in ionospheric measurements of ion temperature.     

Ion cyclotron instability triggered by drifting minor ion species: Cascade effect and exact results

On the basis of bi-Maxwellian velocity distribution functions it has been recently shown that the combined effect of heavy ion thermal anisotropy and drift velocity can trigger ion-cyclotron instabilities beyond the corresponding heavy ion-cyclotron frequency. (Proton-cyclotron instability induced by the thermal anisotrophy of minor ions. J. Geophys. Res. 107 (2002) 1494; Ion-cyclotron instability due to the thermal anisotrophy of drifting ion species. J. Geophys. Res. 108 (2003) 1050.) Here we show that the cascade-type mechanism proposed by Gomberoff and Valdivia (2002, 2003) can take place in the region where main heating of the fast solar wind seems to occur (i.e. within 10 solar radii). We also compare some of the results obtained by using the semi-cold approximation with the exact kinetic dispersion relation.     

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.     

Effect of Cold Plasma Injection on Whistler Mode Instability Triggered by Perpendicular Ac Electric Field at Uranus

The effect of cold plasma injection on whistler mode instability has been studied separately for a bi-Maxwellian and a loss-cone hackground plasma with perpendicular AC electric field. The cold plasma is described by a simple Maxwellian distribution, whereas a generalized distribution function with index j that reduces to a bi-Maxwellian for j = 0 and to a loss-cone for j = 1 has been derived for a plasma in the presence of a perpendicular AC electric field, to form a hot/warm background. The dispersion relation is obtained using the method of characteristic solutions and kinetic approach. An expression for the growth rate of a system with added cold plasma injection has been calculated. Results of sample theoretical calculations for representative values of parameters suited to the magnetosphere of Uranus has been obtained. The salient features of the analysis and the results obtained in both cases have been compared and discussed. It is inferred that it is not the magnitude but the frequency of the AC field which influences the growth rate and a loss-cone background plasma has a triggering effect on the growth rate, increasing the value of the real frequency and maximum growth rate by an order of magnitude. These results may go a long way to enable one to get a better understanding of whistlers and diagnostics of plasma parameters in the Uranian magnetosphere.     

Auroral ion velocity distribution function: Generalized polynomial solution of Boltzmann's equation

The ion distribution function is calculated for the E and the F regions of the auroral latitudes. In these regions the plasma is weakly ionized and there exist convective electric fields which may attain very significant intensities. Boltzmann's equation is solved in the limit where the ion-neutral collision frequency is much lower than the ion gyrofrequency. This solution is obtained in the form of a generalized polynomial series expansion starting from a good zeroth-order approximation. With this weight function, and considering a development to the order of the 32 moments, good approximations are obtained for high electric fields. The resonant charge exchange model and the polarization model are successively considered. The Post-Rosenbluth instability threshold is discussed for the above two models.     

Whistler Mode Instability in a Lorentzian (κ) Magnetoplasma in the Presence of Perpendicular A.C. Electric Field and Cold Plasma Injection

The dielectric tensor, modified plasma dispersion function and dispersion relation for Whistler mode instability in an infinite magnetoplasma are obtained in the case of cold plasma injection to background hot anisotropic generalized bi-Lorentzian (κ) plasma in the presence of external perpendicular a.c. electric field. The method of characteristics solutions using perturbed and unperturbed particle trajectories have been used to determine the perturbed distribution function. Integrals and modified plasma dispersion function Z_κ ^*(ξ ) are reduced in power series expansion form. Numerical methods using computer technique have been used to obtained temporal growth rate for magnetospheric plasma at geostationary height. The bi-Lorentzian (κ) plasma is reducible to various forms of distribution function by changing the spectral index κ. The results of bi-Lorentzian (κ) plasma are compared with those of bi-Maxwellian plasma. It has been found that the addition of cold plasma injection gives different frequency spectra. The a.c. frequency of moderate amplitude increases the growth rate and instability in K space to lower range. Growth rate maximum is not affected by a.c. frequencies. However, it shifts the maximum to lower K space in both cases, rather than on the variation of the magnitude. Thus a physical situation like this may explain emission of various high frequency whistler emissions by cold plasma injection. The potential application of controlled plasma experiments in the laboratory and for planetary atmosphere are indicated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of the H ring velocity distribution function near the Martian and Venusian bow shocks by an analytical model

We have studied the H⁺ velocity distribution function at Mars and Venus near the bow shock both in the solar wind and in the magnetosheath by a simple analytical one-dimensional model. We found that over half of the ions in the ring velocity distribution which moved towards the magnetosheath were scattered back into the bow shock. The original ring distribution is destroyed in less than an ion gyro period. Ions contained in the magnetosphere which hit the bow shock bounce back into the solar wind with a maximum energy exceeding twice the energy of solar wind protons. The ions finite gyroradius causes an asymmetric flow in the magnetosheath with respect to the direction of the convective electric field, which can be observed already a few ion gyroradius downstream of the bow shock.     

The magnetic field near Mars: A comparison between a hybrid model, Mars Global Surveyor and Mars Express observations

Mars Express (MEX) does not carry its own magnetometer which complicates interpretation of ASPERA-3/MEX ion measurements. The direction of the interplanetary magnetic field (IMF) is especially important because it, among other things, determines the direction of the convective electric field and orientation of the cross tail current sheet and tail lobes. In this paper we present a case study to show the properties of the magnetic field near Mars in a quasi-neutral hybrid (QNH) model at the orbits where the Mars Global Surveyor (MGS) has made measurements, present a method to derive the IMF clock angle by comparing fields in a hybrid model and the direction of the magnetic field measured by MGS by deriving the IMF clock angle. We also use H⁺ ring velocity distribution observations upstream of the bow shock measured by the IMA/ASPERA-3 instrument on board MEX spacecraft. These observations are used to indirectly provide the orientation of the IMF. We use a QNH model (HYB-Mars) where ions are modeled as particles while electrons form a mass-less charge neutralizing fluid. We found that the direct MGS and non-direct IMA observations of the orientation magnetic field vectors in non-crustal magnetic field regions are consistent with the global magnetic field draping pattern predicted by the global model.     

Some ideas concerning the problem of anomalous polar E region heating due to Farley-Buneman turbulence

The energetics of the excitation of the Farley-Buneman instability is considered, which is recently observed in the auroral and equatorial E regions of the Earth's ionosphere at altitudes between 100 km and 120 km. In the magnetic field of the Earth the Farley-Buneman instability is excited under the condition of a strong enough external electric field in the case of ion-neutral collisions with frequencies much larger than the ion gyrofrequency and electron-neutral collisions with frequencies much below the electron gyrofrequency. It is shown that the linear increase of the wave amplitudes is caused by a small disbalance between the processes of nonlinear energy pumping into the wave from an external electric field and the energy loss because of the collisions of the electrons and ions with the neutral particles. During the nonlinear energy pumping energy of the external electric field is transferred into a nonlinear current of second order, which is connected with the oscillating motion of the electrons in the wave. The oscillating electron motion takes place perpendicular to wave propagation. From the estimations follows that the energy pumped into a Farley-Buneman wave during one period of pulsation is much larger than the wave energy itself. A new and simply to understand derivation of the anomalous diffusion coefficient is presented, related to the study of the behaviour of a test wave with frequency much above the frequencies of the Farley-Buneman turbulence in developed stage can cause an additional macroscopic nonlinear Pedersen current directed along the external electric field. It is found that the nonlinear Pedersen current can reach the order of the usual Pedersen current and should contribute to the effective heating of the ionospheric plasma.     

Solar wind energy transfer regions inside the dayside magnetopause—I. Evidence for magnetosheath plasma penetration

PROGNOZ-7 high temporal resolution measurements of the ion composition and hot plasma distribution in the dayside high latitude boundary layer near noon have revealed that magnetosheath plasma may penetrate the dayside magnetopause and form high density, high β, magnetosheath-like regions inside the magnetopause. We will from these measurements demonstrate that the magnetosheath injection regions most probably play an important role in transferring solar wind energy into the magnetosphere. The transfer regions are characterized by a strong perpendicular flow towards dawn or dusk (depending on local time) but are also observed to expand rapidly along the boundary layer field lines. This increased flow component transverse to the local magnetic field corresponds to a predominantly radial electric field of up to several mV m^?1, which indicates that the injected magnetosheath plasma causes an enhanced polarization of the boundary layer. Polarization of the boundary layer can therefore be considered a result of a local MHD-process where magnetosheath plasma excess momentum is converted into electromagnetic energy (electric field), i.e. we have primarily an MHD-generator there. We state primarily because we also observe acceleration of “cold” ions inside the magnetopause as a result of this radial electric field. A few cases of polarity reversals suggest that the polarization is sometimes quite localized.The perhaps most significant finding is that the boundary layer is observed to be charged up to tens of kilovolts, a potential which may be highly variable depending on e.g. the presence of a momentum exchange by the energy transfer regions.     

Non Gaussian and Non Local Transport in the Earth's Distant Magnetotail

We study the ion dynamics in a magnetic field reversal with a constant electric field and with a model of three dimensional magnetic turbulence. By computing the mean square displacements in the plane of the current sheet we find superdiffusive and superballistic transport regimes. Since velocity increases with the length of the free path, we have accelerated Lévyflights. The possibility to generate power law velocity distribution functions is pointed out, as well as the long memory effects and non local properties of ion transport. This revised version was published online in July 2006 with corrections to the Cover Date.     

A New Inner Heliosphere Proton Parameter Dataset from the <Emphasis Type="Italic">Helios</Emphasis> Mission

In the near future, Parker Solar Probe and Solar Orbiter will provide the first comprehensive in-situ measurements of the solar wind in the inner heliosphere since the Helios mission in the 1970s. We describe a reprocessing of the original Helios ion distribution functions to provide reliable and reproducible data to characterise the proton core population of the solar wind in the inner heliosphere. A systematic fitting of bi-Maxwellian distribution functions was performed to the raw Helios ion distribution function data to extract the proton core number density, velocity, and temperatures parallel and perpendicular to the magnetic field. We present radial trends of these derived proton parameters, forming a benchmark to which new measurements in the inner heliosphere will be compared. The new dataset has been made openly available for other researchers to use, along with the source code used to generate it.     

Thermal protons in the morning magnetosphere: Filling and heating near the equatorial plasmapause

Thermal H⁺ distributions have been measured as the European Space Agency GEOS-1 satellite passed through the late morning equatorial magnetosphere, plasmapause and plasmasphere. The unique capabilities of the on-board Supralhermal Plasma Analysers (SPA) have been used to overcome the retarding floating potential of the satellite and measure the velocity distribution of the cold protons. In the magnetosphere an enhanced source cone of such ions with a temperature of ~ 0.5 eV is a signature of the filling process occurring outside the plasmapause where flux tubes are relatively empty. In the plasmasphere the thermal H⁺ is essentially isotropic with a temperature less than 0.5 eV but the motion of the satellite introduces apparent drift.These measurements of cold proton velocity distribution now permit a reappraisal of the definition of the “plasmapause”. It becomes inappropriate to use an arbitrary empirical density, e.g. the conventional 10 cm ^?3, in order to establish a boundary. It is now possible to identify a plasmapause interaction region where the two cold proton populations co-exist. This region generally lies Earthward of the 10 cm ^?3 density level, has a width which is strongly dependent on magnetic activity and the temperature is typically between 0.5 and 1.5 eV. The change from “filled” to “unfilled” flux tubes relates to the physical processes which are occurring and the controlling electric field configuration; in particular, the last closed equipotential. Throughout this region, in going from the plasmasphere to the magnetosphere, the plasma drift motion is expected to change from corotation to a convection which is controlled by E ×B, and is predominantly Sunward due to the dawn-dusk electric field. Crossing the plasmapause on the morning side, little change in drift direction should occur but subtle variations in the ionic velocity distribution do reflect the change in the degree of flux tube density equilibrium.Our first direct measurement of the magnetospheric E × B drift has been reported previously but here measurements from a selected six day period show how the plasma in the plasmapause region responds to changing magnetospheric activity. The drift velocities cannot he derived with high accuracy but the analysis shows that the technique can provide a valid mapping of the magnelospheric electric field. In addition, since the magnetospheric cold plasma distribution is observed after it has come from the ionosphere, a distance of many Earth radii, the scattering and accelerating mechanisms along the flux tube can be studied. For this particular data-set taken in the late morning, the maximum potential drops along the flux tubes were less than a volt. The ionospheric proton source cone is observed to be broad, pitch angle scattering persists up to 40 or even 70°.Although these results throw new light on the plasmaspheric filling process one must recognise that, however the plasmapause is defined, it is not a simple matter to map this boundary from the equatorial plane down to low altitudes and the mid-latitude trough.     

Temporal Evolution of Whistler Instability Due to Cold Plasma Injection in the Presence of Perpendicular AC Electric Field in the Magnetosphere of Uranus

Temporal evolution of whistler instability has been studied due to cold plasma injectionin the presence of a perpendicular AC electric field in the magnetosphere of Uranus. Ageneralized distribution function with index j, which is a reducible to a bi-Maxwellianfor j = 0 and to a loss-cone for j = 1, for a plasma in the presence of a perpendicularAC electric field, has been derived from a hot/warm background plasma and atime-dependent plasma described by a simple Maxwellian distribution has been considered to represent the injected cold plasma. An expression for the growth rate of a system with added time-dependent cold plasma injection has been calculated using the method of characteristics and kinetic approach The results obtained for representative value of the parameters suited to the Uranian magnetosphere in both cases have been compared and discussed. It is inferred that the temperature anisotropy remains the major source of free energy whereas a loss-cone background acts as an additional source of free energy for the instability. It is not the magnitude but the frequency of the AC field which Influences the growth rate. In comparison to the Uranian magnetosphere this effect is more significant in Earth's magnetosphere. As the ionisation time of the time-dependent injected cold plasma increases, the growth rate goes on increasing, this effect being much greater in a loss-cone background in comparison to a bi-Maxwellian background plasma time-dependence of thecold plasma has been considered since it represents a more realistic situation in injection experiments.     

A Monte Carlo Simulation of the H+ Polar Wind: Effect of Velocity Distributions with Kappa Suprathermal Tails

A Monte Carlo simulation is used to study the effects of Kappa H⁺distributions in the polar wind. We consider the gravity, the polarization electric field, the divergence of geomagnetic field lines and Coulomb collisions of H⁺ in a background of O⁺ ions. The aim is to study the consequences of a velocity distribution function with an enhanced high energy tail instead of a Maxwellian distribution as assumed in earlier Monte Carlo simulations. The transformation of the velocity distribution function of H⁺ ions as a function of the altitude is presented. Effects resulting from the acceleration of the particles by the polarization electric field and from Coulomb collisions depend on the energy of the particles. Coulomb collisions mainly affect low energy particles while high energy particles are more efficiently accelerated by the upward directed ambipolar electric field. The combination of both effects results in double-hump velocity distribution functions developing in the transition region. We study consequences of suprathermal tails distributions on the shape of the double-hump and on the moments of the velocity distribution function. This revised version was published online in July 2006 with corrections to the Cover Date.     

Helium ion outflow from the terrestrial ionosphere

Extensive calculations have been made of the behaviour of He⁺ for situations where ion outflow occurs from the topside ionosphere. For these circumstances, steady state solutions for the He⁺ continuity, momentum and energy equations have been obtained self-consistently, yielding density, velocity and temperature profiles of He⁺ from 200 to 2000 km altitude. To model the high latitude topside ionosphere, a range of background H⁺O⁺ ionospheres was considered with variations in the H⁺ outflow velocity, the presence of a perpendicular electric field and different peak O⁺ densities. In addition, the atmospheric density of neutral helium was chosen to model typical observed winter and summer densities. From our studies we have found that: (a) The outflowing He⁺ has density profiles of similar shape to those of H⁺, for basically different reasons; (b) The effect of the perpendicular electric field differs considerably for H⁺ and He⁺. This difference stems from the fact that an electric field acts to alter significantly the O⁺ density at high altitudes and this, in turn, changes the H⁺ escape flux through the O⁺+H charge exchange reaction. A similar situation does not occur for He⁺ and therefore the He⁺ escape flux exhibits a negligibly small change with electric field; (c) The fractional heating of He⁺ due to the He⁺O⁺ relative flow is not as effective in heating He⁺ as the H⁺O⁺ relative flow is in heating H⁺; (d) During magnetospheric disturbances when the N₂ density at the altitude of the He⁺ peak (600 km) can increase by a factor as large as 50, the He⁺ peak density decreases only by approximately a factor of 2; and (e) The He⁺ escape flux over the winter pole is approximately a factor of 20 greater than the He⁺ escape flux over the summer pole. Consequently, on high latitude closed field lines there could be an interhemispheric He⁺ flux from winter to summer.     

Directional currents in nocturnal E-region layers

In the mid-latitude E-region, the wind-shear mechanism produces thin ionized layers at levels where the vertical ion velocity is zero. We show that such layers conduct electric current only towards the magnetic equator, and not in the zonal direction. We surmise that this property may influence the electric field distribution in the nocturnal ionosphere, and possibly also the coupling between ion drifts and neutral air winds in the F-region. Detailed case studies of nocturnal layers located near the peak of ion Pedersen conductivity (around 130km) are needed to test this idea.