Fluid model of the plasma flow in the magnetic tail of a planet

All planets of the solar system with an active internal dynamo have a their magnetic dipole oriented perpendicularly or nearly perpendicularly to the solar wind during all or part of their orbit  around the Sun. If, in addition, the planetary rotation is slow, or if the angle between dipole and rotation axis is large, planetary field lines crossing the antisolar axis can become stretched to large distances downstream of the planet. Examples where this may occur are Mercury and Uranus at solstice time, respectively. 

Inspired by these examples, we present a tentative one-dimensional magnetohydrodynamic model of the plasma flowing along the antisolar direction. 

Assuming that the radius of curvature R(z) of the planetary field lines is defined locally as R=D/D', where D(z) is a characteristic  transverse scale of the magnetosphere at a distance z downstream of the planet,  we obtain that the plasma velocity u(z) obeys to a Hugoniot type equation  (M²-1) u'/u =  D'/D,  where M=u/v_A is the Alfvén Mach number. 

The solution for a typical profile D(z) will be discussed.