Conformal mapping for the magnetosheath modeling problems

Determining the plasma flow and magnetic field in the terrestrial and planetary magnetosheath region is a challenge both in theoretical and observational studies in space plasma physics. We are motivated by the discovery of the Kobel-Flückiger exact solution for the Laplace equation to the scalar shielding potential for the parabolic geometry of the magnetosheath, and develop a novel algorithm of the conformal mapping to exactly transform the Kobel-Flückiger scalar potential onto a user-specified, arbitrary geometry of the magnetosheath. The algorithm starts with the outer and innter boundary models (e.g., bow shock and magnetopause locations in the case of planetary magnetospheres). The shell variable v is constructed by smoothly interpolating between the two boundaries, and the connector variable u (connecting between the two boundaries in an orthogonal fashion to the shell variable) is determined by evaluating the gradient of the shell variable along the shell segment under the Cauchy-Riemann relations. The conformal mapping method is computationally by far inexpensive, and retains the exactness character of the steady-state magnetosheath solution. The method has a wide range of applications such as validating the numerical simulations, planning the space (planetary and heliospheric) missions, and even estimating the solar wind condition from the magnetosheath data.