Global MHD simulations are used by several research groups to gain a global view of the dynamic processes in the coupled solar wind-magnetosphere-ionosphere system (Lyon et al., 2004; Raeder et al., 1995; Janhunen, 1996; Gombosi et al., 2000). While the codes differ in many details, basically they all solve the (ideal) MHD equations in a large box extending out to at least 30 RE in the Sunward direction, about 60 RE in the perpendiculardirections, and several hundred RE in the downtail direction. This way, the box completely encompasses the magnetosphere in all but the tailward direction, and as the flow is supersonic at the tailward boundary, there is no feedback from that boundary to the other parts of the simulation. With the modern computing capabilities, the variable-size grids can resolve minimum cell sizes of only a fraction of RE, which allows resolving structures to the thermal ion gyroradius scale.
Near the inner boundary, the MHD magnetosphere is coupled to an electrostatic model of the ionosphere. The details of how the ionosphere is treated in the simulations varies quite significantly, and both the coupling to the ionosphere as well as the details of the ionospheric solution are still being refined many of the modeling groups. The inner boundary of the MHD simulation box is located between 2 and 4 RE to avoid very short time steps associated with the large wave speeds in the high-field region near the Earth. The field-aligned currents obtained from the MHD simulation at the inner boundary are mapped to the ionospheric altitude along dipolar magnetic field lines. The ionospheric module solves the potential equation using the field-aligned currents, ionospheric conductances, and Ohm’s law in the ionosphere to get the horizontal ionospheric currents and potential pattern. The electric field computed from the ionospheric potential is then mapped to the inner boundary of the magnetosphere, where it is used as a boundary condition for the MHD simulation part. The ionospheric conductances are determined by the solar illumination and in many models by parametrized electron precipitation from the magnetosphere. The space between the ionosphere and the inner boundary of the MHD simulation part is a passive medium, which only transmits electromagnetic effects along dipolar magnetic field lines. Thus there is no plasma or currents perpendicular to the magnetic field in that region. Figure 8 shows a diagram of the simulation scheme.
The simulations are driven by the solar wind and IMF values at the Sunward boundary of the simulation box and the F10.7 flux that determines the level of ionospheric ionization and hence the conductivity. The models can either use idealized solar wind and IMF conditions to gain understanding of the generic dynamic properties of the magnetosphere or use real spacecraft measurements to model the evolution of individual events.
In cases where the global MHD simulations solve the ideal MHD equations with zero resistivity, magnetic reconnection occurs in the simulations only through numerical diffusion. In case the resistivity is explicitly accounted for, it is most often parametrized to scale with the current density; the magnetospheric plasmas are fully collisionless and hence the classical collisional resistivity is zero. This and the lack of microphysical processes below the MHD scale naturally limits the applicability of the models to describe the details of the reconnection process either at the magnetopause or in the magnetotail. However, it has been shown that in the large scale, the simulation results are consistent with the conceptual understanding of when and where reconnection in the magnetosphere should occur (Hones Jr, 1979) and with case studies where in-situ satellite observations of reconnection events are available (Pulkkinen and Wiltberger, 2000).
Figure 9 shows MHD simulation results compared with observations during the storm on April 2000. The top left panel shows the 𝜖 parameter as a reference for the level of energy input to the magnetosphere. The next two panels show geosynchronous orbit magnetic field measurements from two GOES spacecraft. As the geostationary field is under normal conditions dominated by the strong dipole field pointing northward, the fact that Bz changes sign and becomes strongly negative is a clear indication that the satellites have crossed the magnetopause boundary and entered in the magnetosheath. This time period is highlighted with the green shading. The fourth panel in Figure 9 shows the empirical Shue et al. (1998) model for the subsolar magnetopause position shaded in dark blue (Equation 2). The model correctly predicts that the magnetopause was near or inside the geostationary orbit at 6.6 RE for a large part of the storm main phase. The black curves show two measures of the magnetopause position in the MHD simulation: the upper curve is determined from the maximum current density flowing at the boundary, while the lower curve is defined from the open-closed field-line boundary separating field lines tying to the Earth from the disconnected solar wind field lines. The bottom panel shows Geotail measurements from just upstream of the magnetopause (blue). The measurements trace the IMF Bz variations but are much enhanced, indicating that Geotail was located in the shocked magnetosheath flow between the magnetopause and the shock. The shaded region shows the time period when the simulation (shown black) predicts Geotail position in the solar wind upstream of the shock, and thus overpredicts the amount of inward motion of the shock. However, as the two snapshots of plasma density in the equatorial plane on the right show, Geotail was very close to the shock, and thus even a small error in the shock location may lead to erroneous prediction of the region where the spacecraft resides.
In addition to reconnection modeling, the inner magnetosphere also poses significant challenges to the MHD simulations. As the simulations include only one ion population, it describes the entire distribution with a single temperature. This is not realistic in the inner magnetosphere, where the ring current is carried by ions with significantly higher energies than the average plasma sheet population. Furthermore, the ion sources from the ionosphere (plasmasphere, ion outflows associated with geomagnetic activity) are not included in most simulations. As a consequence, the MHD simulations predict a much more dipolar inner magnetosphere than is observed, and do not reproduce the strong inner magnetosphere field depression associated with stormtime ring current. A variety of efforts to couple the inner magnetosphere simulation part with other, more detailed models of the inner magnetosphere are currently underway to address this issue.
Even given their limitations in describing details of reconnection, multicomponent plasma systems, or the physics associated with non-adiabatic ion motion or anisotropic pressures, the global MHD simulations are presently the best tool to obtain a large-scale view of the magnetospheric activity. As such, they are widely developed for use in space weather applications (Raeder et al., 2001; Manchester IV et al., 2004).
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