## A model hierarchy for ionospheric plasma modeling.(English)Zbl 1177.76456

This paper deals with the modeling of the ionospheric plasma. Starting from the two-fluid isothermal Euler-Maxwell equations, first a Hall-MHD model is introduced which includes ion-neutral, electron-neutral and ion-electron collisions as well as the electron pressure. Then, from the Hall-model two hierarchies of models are developed, the MHD hierarchy and the dynamo hierarchy. The MHD hierarchy consists of the successive limits $$\kappa\rightarrow0$$, $$\tau\rightarrow0$$, $$\beta\rightarrow0$$, where $$\kappa$$ is the ratio of the collision frequency against neutrals to the cyclotron frequency in the earth’s magnetic field, $$\tau$$ describes the time scale of the processes of interest (here the mean time between the ion-neutral collisions) and $$\beta$$ measures the strength of the self-consistent magnetic field perturbation induced by the dynamics of the plasma relative to the earth’s magnetic field. The dynamo hierarchy is obtained through $$\tau\rightarrow0$$, $$\beta\rightarrow0$$, $$\kappa\rightarrow0$$. The dynamo hierarchy is suitable for the standard ionospheric situation with rather low density, while the MHD hierarchy is better suited to an ionosphere with abnormal high density (e.g. of about $$10^{15}$$ m$$^{-3}$$ at 300 km altitude which might occur at thermonuclear explosions). It is shown that the limit $$\beta\rightarrow0$$ leads to simpler models (e.g. the dynamo model), if it is preceded by the limit $$\tau\rightarrow0$$. Most of the models encompassed by the dynamo hierarchy are classical ones, but in the paper a unified presentation of them is given, which brings a new insight into their interrelations. By contrast, the MHD hierarchy involves a relatively unknown massless-MHD model. The massless-MHD model is found after letting $$\kappa\rightarrow0$$ and $$\tau\rightarrow0$$. It is a diffusion system for the density and magnetic field, which could be of great practical interest. The both considered hierarchies of models terminate with the classical Striation model, which is investigated in detail. It is shown that the Striation model is a reduction of the dynamo model in which the electric field is forced to be orthogonal to the magnetic field, thus giving rise to a two-dimensional elliptic equation. This equation is still coupled with the 3D density and (massless) momentum equations.

### MSC:

 76W05 Magnetohydrodynamics and electrohydrodynamics 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 76N99 Compressible fluids and gas dynamics 82D10 Statistical mechanics of plasmas
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