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\leftline{HG1 : Computer simulations of multiple scale processes in
space plasmas\qquad\qquad\quad P}

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{\bf {\large
Hierarchical cubic grid and locally varying timestep in
global ionosphere--magnetosphere
coupling simulations}}

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{\obeylines
  P.~Janhunen,
  Finnish Meteorological Institute,
  Department of Geophysics,
  P.O.B. 503, FIN-00101, Helsinki, Finland
  Phone: +358-0-1929535, Fax: +358-0-1929539, email: Pekka.Janhunen@fmi.fi
}
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The ionosphere and the magnetosphere form a coupled system which is
affected by the solar wind. The presence of multiple spatial and
temporal scales presents a major problem for any numerical simulation
of such a system. In this paper we describe a new simulation model
(GUMICS2), which is especially suited for studying dynamical events
such as geomagnetic substorms which occur relatively close to the Earth.

The magnetosphere is modelled using MHD equations, possibly also
containing the Hall term. We use a hierarchically (recursively)
refined cubic grid, which is dense ($\approx$ 0.1 Earth radii) near the
Earth and sparse (several Earth radii) in the far tail. The temporal
discretization uses a novel technique: the time step is not the same
for all grid points, but it varies from point to point. This saves a
lot of computing time, because small time steps ($< 0.1 $s) are only needed
near the Earth, where the Alfv\'en speed is high. In most of the
simulation box the time step can be more than one second without
violating the Courant condition. In our implementation the possible
time steps are inverse powers of two. The largest possible
time step (the {\em time leap}) is currently four seconds.

The magnetosphere is coupled to an electrostatic ionosphere in the
following standard way: The field-aligned current (FAC) density is computed from the
MHD magnetic field near the Earth. The FAC is mapped to the
ionospheric plane, where it is used as a source term to an elliptic
equation representing current continuity. The solution of this
elliptic equation gives the ionospheric potential. The potential is
differentiated and mapped back to the magnetosphere to yield the
electric field ${\bf E}$. The associated ${\bf E}{\rm\times}{\bf B}$
velocity is used as a boundary condition to the MHD equations. This
procedure is executed at every time leap.

In addition, the ionospheric Pedersen and Hall conductances are
computed from solar EUV flux and magnetospheric electron
precipitation. The latter is computed dynamically during the run. The
interplay between current/electric field coupling on one hand and
conductivity coupling on the other hand is the main source of
complicated dynamical features observed in the system.

The simulation model is very flexible. By redefining the grid point
density function one can shift emphasis to various magnetospheric
regions. In this paper we have tuned the grid for studying near-Earth
nightside phenomena.

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\centerline{\bf References}
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[1] Janhunen P., T.I. Pulkkinen and K. Kauristie, Auroral fading in
ionosphere-magnetosphere coupling model: Implications for possible
mechanisms, Geophys. Res. Lett., 22, 2049-2052, 1995.


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