Øystein Lie-Svendsen

Kinetic electrons in high‐speed solar wind streams: Formation of high‐energy tails

We study the evolution of the electron velocity distribution function in high-speed solar wind streams from the collision-dominated corona and into the collisionless interplanetary space. The model we employ solves the kinetic transport equation with the Fokker-Planck collision operator to describe Coulomb collisions between electrons. We use a test particle approach, where test electrons are injected into a prescribed solar wind background. The density, temperature, and electric field associated with the background are computed from fluid models. The test electrons are in thermal equilibrium with the background at the base of the corona, and we study the evolution of the velocity distribution of the test electrons as a function of altitude. We find that velocity filtration, due to the energy dependence of the Coulomb cross section, is a small effect and is not capable of producing significant beams in the distribution or a temperature moment that increases with altitude. The distribution function is mainly determined by the electric field and the expanding geometry and consists of a population with an almost isotropic core which is bound in the electrostatic potential and a beam-like high-energy tail which escapes. The trapped electrons contribute significantly to the even moments of the distribution function but almost nothing to the odd moments; the drift speed and energy flux moments are carried solely by the tail. In order to describe the high-speed solar wind observed near 0.3 AU by the Helios spacecraft, we use a multifluid model where ions are heated preferentially. The resulting test electron distribution at 0.3 AU, in this background, is in very good agreement with the velocity distributions observed by the Helios spacecraft.

A Review on Solar Wind Modeling: Kinetic and Fluid Aspects

The paper reviews the main advantages and limitations of the kinetic exospheric and fluid models of the solar wind (SW). The general theoretical background is outlined: the Boltzmann and Fokker–Planck equations, the Liouville and Vlasov equations, the plasma transport equations derived from an “equation of change”. The paper provides a brief history of the solar wind modeling. It discusses the hydrostatic model imagined by Chapman, the first supersonic hydrodynamic models published by Parker and the first generation subsonic kinetic model proposed by Chamberlain. It is shown that a correct estimation of the electric field, as in the second generation kinetic exospheric models developed by Lemaire and Scherer, provides a supersonic expansion of the corona, reconciling the hydrodynamic and the kinetic approach. The modern developments are also reviewed emphasizing the characteristics of several generations of kinetic exospheric and multi-fluid models. The third generation kinetic exospheric models consider kappa velocity distribution function (VDF) instead of a Maxwellian at the exobase and in addition they treat a non-monotonic variation of the electric potential with the radial distance; the fourth generation exospheric models include Coulomb collisions based on the Fokker–Planck collision term. Multi-fluid models of the solar wind provide a coarse grained description of the system and reproduce with success the spatio-temporal variation of SW macroscopic properties (density, bulk velocity). The main categories of multi-fluid SW models are reviewed: the 5-moment, or Euler, models, originally proposed by Parker to describe the supersonic SW expansion; the 8-moment and 16-moment fluid models, the gyrotropic approach with improved collision terms as well as the gyrotropic models based on observed VDFs. The outstanding problem of collisions, including the long range Coulomb encounters, is also discussed, both in the kinetic and multi-fluid context. Although for decades the two approaches have been seen as opposed, in this paper we emphasize their complementarity. The review of the kinetic and fluid models of the solar wind contributes also to a better evaluation of the open questions still existent in SW modeling and suggests possible future developments.

A 16‐moment solar wind model: From the chromosphere to 1 AU

We present a solar wind fluid model extending from the chromosphere to Earth. The model is based on the gyrotropic approximation to the 16-moment set of transport equations, in which we solve for the density, drift speed, temperature parallel and perpendicular to the magnetic field, and transport of parallel and perpendicular thermal energy along the magnetic field (heat flux). The solar wind plasma is created dynamically through (photo) ionization in the chromosphere, and the plasma density in the transition region and corona is computed dynamically, dependent on the type of coronal heating applied, rather than being set arbitrarily. The model improves the description of proton energy transport in the transition region, where classical heat conduction is only retrieved in the collision-dominated limit. This model can serve as a “test bed” for any coronal heating mechanism. We consider heating of protons by a turbulent cascade of Alfven waves in rapidly expanding coronal holes. The resulting high coronal proton temperatures lead to a downward proton energy flux from the corona which is much smaller than what classical transport theory predicts, causing a very low coronal density and an extremely fast solar wind with a small mass flux. Only when some of the wave energy is forcibly deposited in the lower transition region can a realistic solar wind be obtained. Because of the poor proton heat transport, in order to produce a realistic solar wind any viable heating mechanism must deposit some energy in the transition region, either directly or via explicit heating of coronal electrons.

The electron velocity distribution in the high‐speed solar wind: Modeling the effects of protons

The evolution of the electron velocity distribution function (VDF) in high-speed solar wind streams is modeled taking the expanding geometry, the polarization electric field, and Coulomb collisions into account. The VDF we find at the orbit of Mercury is composed of an isotropic, collision-dominated core, a trapped, anisotropic population called “halo” in this study, and a narrow, high-energy “strahl” that escapes along the magnetic field. The distribution function is very similar to the electron VDF observed in the low-density, high-speed solar wind by Pilipp et al. [1987] and Phillips et al. [1989]. The main features of the VDF can be obtained by considering only electron self-collisions; the effect of proton collisions is to make the distribution function more isotropic. At low energies, collisions with protons dominate the angular scattering, but electron self-collisions alone are frequent enough to keep the core of the distribution function quite isotropic. The expanding geometry produces an anisotropic halo and a narrow strahl. The angular scattering by protons reduces the anisotropy of the trapped halo particles and broadens the lower-energy part of the strahl. Along the magnetic field the resulting electron velocity distribution is composed of a relatively cold core and a halo-strahl spectrum that is “flatter” than the coronal spectrum. The two-temperature electron distribution function often observed in the solar wind may therefore be produced by Coulomb collisions and should not be taken as a “proof” of a non-Maxwellian (two-temperature) distribution function in the corona.

Helium Abundance in the Corona and Solar Wind: Gyrotropic Modeling from the Chromosphere to 1 AU

We have developed a solar wind model including helium that extends from the chromosphere to 1 AU. The model is based on the gyrotropic approximation to the 16-moment set of fluid transport equations, which allows it to accommodate temperature anisotropies, as well as nonclassical heat transport. We find that, as in a pure electron-proton solar wind, the flow geometry close to the Sun also has a large impact on helium. In a radially expanding flow, downward proton heat conduction from the corona leads to a high transition region pressure and a large thermal force that pulls helium ions into the corona. In this case α-particles may easily become the dominant species in the corona, resulting in a polar wind type of solar wind in which the light protons are accelerated outward in the electric field set up by the α-particles and electrons. By contrast, applying the same form for the coronal heating in a rapidly expanding geometry intended to simulate a coronal hole, protons become collisionless closer to the Sun, and therefore the downward proton heat flux is smaller, resulting in a lower transition region pressure and a lower thermal force on helium. In this case the helium abundance is low everywhere and helium is unimportant for the acceleration of the solar wind. For the low coronal proton and α-particle densities found in the rapidly expanding flow, where asymptotic flow speeds are typically significantly higher than the gravitational escape speed at the solar surface, the solar wind helium mass flux is determined by the amount of helium available at the top of the chromosphere. In the radially expanding flow, with asymptotic flow speeds lower than the escape speed, the helium mass flux depends on the amount of energy available in the corona to lift helium out of the gravitational potential. In both cases the frictional coupling between helium and hydrogen in the chromosphere, using currently accepted elastic cross sections, is too weak to pull a sufficient number of helium atoms up to the top of the chromosphere and thus obtain a mass flux in agreement with observations. A better understanding of the chromosphere is therefore called for.

The Effect of Transition Region Heating on the Solar Wind from Coronal Holes

Using a 16 moment solar wind model extending from the chromosphere to 1 AU, we study how the solar wind is affected by direct deposition of energy in the transition region, in both radially expanding geometries and rapidly expanding coronal holes. Energy is required in the transition region to lift the plasma up to the corona, where additional coronal heating takes place. The amount of energy deposited determines the transition region pressure and the number of particles reaching the corona and, hence, how the solar wind energy flux is divided between gravitational potential and kinetic energy. We find that when only protons are heated perpendicularly to the magnetic field in a rapidly expanding coronal hole, the protons quickly become collisionless and therefore conduct very little energy into the transition region, leading to a wind much faster than what is observed. Only by additional deposition of energy in the transition region can a reasonable mass flux and flow speed at 1 AU be obtained. Radiative loss in the transition region is negligible in these low-mass flux solutions. In a radially expanding geometry the same form of coronal heating results in a downward heat flux to the transition region substantially larger than what is needed to heat the upwelling plasma, resulting in a higher transition region pressure, a slow, massive solar wind, and radiative loss playing a dominant role in the transition region energy budget. No additional energy input is needed in the transition region in this case. In the coronal hole geometry the solar wind response to transition region heating is highly nonlinear, and even a tiny input of energy can have a very large influence on the asymptotic properties of the wind. By contrast, the radially expanding wind is quite insensitive to additional deposition of energy in the transition region.

MODELING THE ENERGY BUDGET OF SOLAR WIND MINOR IONS: IMPLICATIONS FOR TEMPERATURES AND ABUNDANCES

The outflow of oxygen and silicon ions in the solar wind has been studied using a model that extends from the chromosphere into interplanetary space, with emphasis on understanding the energy budget of the minor ions. The model solves coupled gyrotropic transport equations, which account for temperature anisotropies and heat conduction, for all charge states, and includes ionization and recombination. The minor ions are heated with a constant heating rate per particle in the corona. In the transition region the thermal force causes minor ions to flow faster than protons, with an abundance that can be less than half of the chromospheric abundance. The ions quite suddenly decouple from the proton flow in the corona, and above this point the ion flow is independent of the proton flow. For high heating rates the coronal abundance is comparable to the chromospheric abundance, the ion terminal wind speed is high, and most of the deposited energy is lost into the solar wind. Low heating rates lead to very large coronal abundances and a low terminal flow speed, and the main energy loss is then through collisions with protons and electrons in the corona. The heavy ions become much hotter than protons in the corona, even without preferential heating of the ions. However, preferential heating is necessary to prevent a large abundance enhancement in the corona and to achieve flow speeds close to the speeds observed by the Ultraviolet Coronagraph Spectrometer (UVCS) on the Solar and Heliospheric Observatory (SOHO). The abundance enhancement implies that lowering the heating rate per particle in general leads to an increase in the total energy flux absorbed by the ions.

Elemental Abundances in the Fast Solar Wind Emanating from Chromospheric Funnels

We carry out a model study to determine whether a funnel-type flow geometry in the solar wind source region leads to sufficiently fast hydrogen flow to offset heavy element gravitational settling and can thus explain why solar wind abundances are not much smaller than photospheric abundances. We find that high first ionization potential (FIP) elements are more susceptible to gravitational settling than low-FIP elements, which are pulled up by Coulomb drag from protons, and hence the settling is more sensitive to the charge state of the elements than to their mass. Abundances at the top of the chromosphere, and hence solar wind abundances, can change by many orders of magnitude when the funnel areal expansion factor is changed by a small amount. The observed solar wind neon abundance provides the most severe constraint on the expansion, requiring a total flux tube expansion factor of at least 30-40.

Improved Transport Equations for Fully Ionized Gases

We have developed fluid transport equations for fully ionized gases that improve the description of Coulomb collisions. The aim has been to develop simple and versatile equations that can easily be implemented in numerical models and thus be applied to a large variety of space plasmas, while they still accurately describe thermal forces and energy flows in collision-dominated plasmas. Based on exact solutions to the Boltzmann equation in the collision-dominated limit, the correction term to the velocity distribution function that account for particle flows is assumed to be proportional to the third power of the velocity, leading to a near isotropic core distribution. Applying the fluid equations derived from this new velocity distribution to a collision-dominated electron-proton plasma with a small temperature gradient, the resulting electron heat flux, as well as the thermal force between electrons and protons, deviate less than 25% from the exact results of classical transport theory. The new equations predict a factor of 4 reduction in the thermal force acting on heavy, minor ions caused by an imposed heat flux, compared with fluid equations that are in common use today. The improved description of thermal forces is expected to be important for modeling the composition of stellar atmospheres.

Solar wind originating in funnels : fast or slow?

Aims. We model a hydrogen-helium solar wind originating in funnels, regions of rapid flux tube expansion at the base of the solar corona. Methods. The time-dependent model describes the particle density, flow speed, temperature parallel and perpendicular to the magnetic field, and the heat flow for each ionization state of hydrogen and helium, and for electrons. Results. For a large range of heating parameters, the funnel has two co-existing solutions: both a slow and a fast solar wind solution result from the same heating parameters, depending on the initial state from which the model was started. Though the fast and the slow solar wind can co-exist it is difficult to change from a fast solar wind to a slow solar wind or vice versa. A significant change in the heating parameters is required to “flip” the solution, and it takes a long time, about one month, to reach the other steady state solution. When either the funnel or helium is removed from the model, we no longer have two co-existing states.