A physical model for the magnetosphere of Uranus at solstice time

Uranus is the only planet in the Solar System whose rotation axis and orbital plane are nearly parallel to each other. Uranus is also the planet with the largest angle between the rotation axis and the direction of its magnetic dipole (roughly $59^\circ$). Consequently, the shape and structure of its magnetospheric tail is very different to those of all other planets in whichever season one may consider. We propose a magnetohydrodynamic model for the magnetic tail of Uranus at solstice time. One of the main conclusions of the model is that all magnetic field lines forming the extended magnetic tail follow the same qualitative evolution from the time of their emergence through the planet's surface and the time of their late evolution after having been stretched and twisted several times downstream of the planet. In the planetary frame, these field lines move on magnetic surfaces that wind up to form a tornado-shaped vortex with two foot points on the planet (one in each magnetic hemisphere). The centre of the vortex (the eye of the tornado) is a simple double helix with a helical pitch (along the symmetry axis $z$) $\lambda=\tau[v_z+B_z/(\mu_0\rho)^{1/2}],$ where $\tau$ is the rotation period of the planet, $\mu_0$ the permeability of vacuum, $\rho$ the mass density, $v_z$ the fluid velocity, and $B_z$ the magnetic field where all quantities have to be evaluated locally at the centre of the vortex. In summary, in the planetary frame, the motion of a typical magnetic field of the extended Uranian magnetic tail is a vortical motion, which asymptotically converges towards the single double helix, regardless of the line's emergence point on the planetary surface.