The purpose of this paper is to demonstrate how the MHD equilibrium theory can be used to describe the global magnetic field configuration of Earth's magnetosphere and its time evolution under the influence of magnetospheric convection. The MHD equilibrium theory represents magneto-hydrodynamics in the slow-flow approximation. In this approximation time scales are long compared to typical Alfvén wave travel times, and plasma flow velocities are small compared to the Alfvén speed. Under those conditions, the inertial term ρ(dv/dt) in the MHD equation of motion is a small second order term which can be neglected. The MHD equilibrium theory is not a static theory, though, because time derivatives and flow velocities remain first order quantities in the continuity equation, in the thermodynamic equation of state, and in the induction equation. Therefore one can compute slowly time-dependent processes, such as magnetospheric convection, in terms of series of static equilibrium states. However, those series are not arbitrary; they are constrained by thermodynamic conditions according to which the magnetosphere evolves in time.
It is an interesting question, whether or not the magnetosphere, driven by slow, lossless, adiabatic, earthward convection of magnetotail flux tubes, can reach a steady state. There exist magnetospheric equilibria in which magnetotail flux tubes satisfy the steady-state condition d/dt (Pρ−γ) = 0. Those configurations exhibit a deep magnetic field minimum in the equatorial plane, near the inner edge of the tail plasma sheet. The magnetosphere becomes tearing-mode unstable in the neighborhood of such a minimum, thus leading to periodic onsets of substorms in the inner plasma sheet. This explains why distinct magnetic field minima have not been observed in this region. Magnetic substorms seem to be an inevitable element of the global convection cycle which inhibit the establishment of an ultimate steady state.
MHD equilibria discussed in this paper result from linear and non-linear solutions to the two-dimensional Grad-Shafranov equation for isotropic thermal plasma pressure.