Supplementary material to “Solar-Terrestrial Interactions Workshop: STIMM-2”
Joseph Lemaire, Belgian Institute for Space Aeronomy, Brussels, Belgium; Marius Echim, Institute for Space Sciences, Bucharest, Romania.
Citation:
Lemaire, J. F., and M. Echim (2008), Solar-Terrestrial Interactions Workshop: STIMM-2, Eos Trans. AGU, 89(9), 86.
[Full Article (pdf)]
Solar-Terrestrial Interactions from Microscale to Global Models (STIMM2) June 12-16, 2007, Sinaia, Romania
Solar and Polar Winds models : argument & summary
STIMM2 Modeling Focus Group
Introduction : In 1958, Parker found that the hydrostatic models of the solar corona convey too large kinetic pressure compared to what was expected in the distant heliosphere and interstellar medium. He proposed the first hydrodynamic model for the solar corona expansion with a supersonic radial bulk velocity and zero kinetic pressure at infinity. This supersonic expansion was then compared with the flow of gases in a classical de Laval nozzle.
A similar stationary hydrodynamic model was proposed a decade later by Banks and Holzer for the supersonic escape of ionospheric plasma over the polar caps along “open” magnetotail field lines. Like in the de Laval nozzle analogy, the ionospheric plasmas are considered to be driven supersonic by the steep plasma pressure gradients. In 1969 Lemaire and Scherer show that the thermal protons are accelerated to supersonic velocities via an outward directed polarization electrostatic field. In 1970, Jockers arrives independently to the similar conclusions for the solar wind. The electrostatic field is induced by the trend of the free electrons to evaporate much faster than the heavier and slower ions out of the gravitational potential well. This discovery gave some new momentum to kinetic models of the solar wind (SW) and polar wind (PW). It is interesting to realize that in the hydrodynamic model of Banks and Holzer the electric field is “hidden” in the pressure gradient term of the fluid momentum equation, a property not generally appreciated in the 70's.
Since that time, challenging model improvements (kinetic and hydrodynamic) have been proposed and confronted. At Sinaia, Romania, a dozen of scientists from different orientations discussed the complementary kinetic and hydrodynamic approaches. The aims of this Modeling Focus Group were primo: to review the various models; secundo: to outline advantages and limitations of the two approaches.
By kinetic models, we mean those obtained by calculating the velocity distribution function (VDF) of particles species. The spatial distribution and temporal evolution of these VDFs are solutions of kinetic equations: the Boltzmann equation (BE), or the Fokker-Planck equation (FPE). The density, bulk velocity and any higher moments of the VDFs are derived by analytical or numerical integration of the VDF over the appropriate velocity domains. At Sinaia, these kinds of models have been reviewed by Lamy, Brussels, and Pierrard, Brussels & Louvain-La-Neuve.
On the other hand, by hydrodynamic or fluid models, we mean those where the spatial distributions and temporal evolution of these moments are obtained by numerically integrating differential equations corresponding to various approximations of the (macroscopic) transport equations. Note that the magnetohydrodynamic equations, as well as the ideal MHD ones are critical approximations of these transport equations where the electrons and ion species are merged and form a single (quasi-neutral) fluid, with the consequence that the electric charge separation or the relative diffusion velocities of the ions and electrons are neglected. The displacement currents are null in fluid approaches, however, diamagnetic currents are different from zero.
At Sinaia, hybrid SW and PW models and time-dependent simulations of multi-moment and multi-species transport equations have been presented by Sunny Tam, Tainan, Taiwan and by Blelly, Orléans. A new non-classical expansion of the VDF to improve the fluid equations was proposed by Lie-Svendsen and co-workers from Oslo.
Advantages and limitations
Fluid models. The main attraction for fluid models in space plasma physics resulted initially from the long tradition in using the macroscopic transport equations in classical hydrodynamics of collision dominated neutral gases, where the Knudsen number is very small.
Non-stationary boundary conditions are rather convenient to implement in MHD or hydrodynamic models. Standard integration methods for coupled system of partial differential equations are available to solve the moments and transport equations. Full time-dependent, 3-D models can be constructed more easily than corresponding kinetic models. Reactive processes, such as ionization, recombination, charge exchange, wave-particle interactions can also be more easily included by considering amenable source terms.
The spatial integration domain can span in hydrodynamic transport equations a vast range of altitudes, from the ionosphere or chromosphere up to the exobase, where the Knudsen number becomes larger than unity. Provided appropriate boundary conditions are chosen, hydrodynamic models based on suitable moments equations can even be extended into the collisionless exosphere. But there is always the difficult contingency that the values of the associated VDF should not become negative anywhere in the velocity space.
At Sinaia, the various approximations of moment equations and their applications to the SW and PW have been illustrated and compared. Lie-Svendsen pointed out that higher-order fluid equations are not necessarily “better” than low-order approximations for the moment equations. The gain obtained from a high-order closure assumption may be small compared to the cost (in the form of increased complexity). Unless the Ansatz for the VDF, fo(v), is a priori chosen in such a way that its functional form fits as closely as possible the (actual) full VDF, f0 [1 + ϕ(v)], the results can be rather misleading. Indeed, the collision terms, diffusion coefficient, heat conduction coefficient, viscosity coefficient (which are part of the transport equations) can only be adequately evaluated/approximated. The shape assumed for f0(v) affects the value of collision terms, especially for plasmas where the Coulomb collision cross-section is strongly energy dependent. Except in the usual situations of collision dominated gases, where the VDF is generally close to drifting Maxwellian, these “generally admitted” constrains jeopardize, however, the application of Chapman-Enskog theory, as well as Grad's approximations in collisionless plasmas.
Kinetic models. Despite the rather dissenting truncation of the VDF the exospheric models contributed also to better understand the origin of the electrostatic acceleration of the SW and PW ions. They also emphasize the consequences of coronal temperature enhancements, of super-thermal electron populations (kappa VDF), of non-radial magnetic field configurations (spiral B-field lines), on the acceleration and radial distribution of plasmas in the SW and PW.
Furthermore, collisional kinetic models which are solutions of Fokker-Planck equations, have demonstrated how the polar wind develops gradually across the transition region separating the collision dominated regime and the collisionless exosphere. The main feature is a double-hump VDF with a secondary peak at supersonic velocity, in addition to the main isotropic Maxwellian peak centered at null radial velocity. The later peak corresponds to the velocity distribution of particles which are collisionally bound to the background ions , while the secondary peak corresponds to those particles which have enough energy to be practically collisionless. Within the Knudsen transition region, the primary Maxwellian peak of the VDF fades away at the expense of the secondary supra-thermal peak, with increasing altitudes. This special behavior of the VDF was unanticipated, since it is very different from that postulated (admitted) in hydrodynamic models.
Direct Monte Carlo Simulations (DMCS) also belonging to the class of kinetic approaches have been applied to the PW by Barakat et al. in 1995. The results of DMCS, reviewed at Sinaia by Barakat, confirm the existence of this quite fundamental transformation, showing the metamorphose of the VDF from a single peaked function centered around v =0 (at low altitudes), to a doubly peaked function of v, in the Knudsen transition region. The secondary peak carries most of the escape flux of the particles, as well as the energy flux leaking out of the polar ionosphere. This indicates that the applications of DMCS to the SW and PW escape have been most useful to crosscheck the results derived by the other kinetic approaches based on polynomial expansions or on finite difference representations of the VDF.
What are the main limitations or shortcomings of kinetic models? The arbitrary truncation of the VDF in exospheric models is an often put forward as a shortcoming, and has been recalled in Lamy's presentation. Note that this shortcoming disappears for the kinetic solutions of the FPE which are smooth functions in the velocity space. Of course, like for any other kind of approaches, kinetic solutions depend on boundary conditions at the bottom and top of the domain of integration. Usually, stationary boundary conditions are assumed for the sake of simplicity. For convenience, the simplest and straightforward boundary conditions (e.g. half range Maxwellian VDFs) have usually been taken so far to determine the solutions of the FPE reported by Viviane Pierrard at Sinaia.
So far no coupled FPEs for the SW and PW electrons and ions; furthermore, no time-dependent boundary conditions have ever been applied to kinetic solutions. This would however be a promising way to go in future, using either (i) the finite element method developed by Lie-Svendsen and Rees in 1996, (ii) the polynomial expansion method developed by Pierrard and Lemaire in 1998, and Pierrard et al. in 1999, or (iii) the DMCS introduced by Barakat et al. in 1995.
Fluid and kinetic approaches were combined in a hybrid model developed by Tam in 1995. The complementariness of the two approaches in SW and PW applications, as demonstrated by the respective versions of the hybrid model, was discussed at Sinaia.
Besides these presentations focused on SW and PW modeling, Echim presented challenging kinetic simulations of diamagnetic plasma blobs drifting across a background magnetic field. His results are supplementing those obtained in the framework of MHD.
All abstracts and power point files are available on the STIMM-2 web site : http://iss30.nipne.ro/gpsm/ws_ro/stimm2/pres.php
A.R. Barakat, P.-L. Blelly, M.M. Echim (co-convener), H. Lamy, J.F. Lemaire (convener), O. Lie-Svensden, V. Pierrard, S.W.Y Tam
9 December 2007