Three‐dimensional modeling of Earth's bow shock: Shock shape as a function of Alfvén Mach number

Earth's bow shock changes its three‐dimensional (3‐D) location in response to changes in the solar wind ram pressure P ram , Alfvén Mach number M A , magnetic field orientation, fast mode Mach number M ms , and sonic Mach number M S . Using shock locations from global 3‐D ideal MHD simulations [ Cairns and Lyon , 1995], empirical models are derived for the 3‐D shape and location of Earth's bow shock in the near‐Earth regime as a function of solar wind conditions. Multiple simulations with different M A and P ram but two orientations of the interplanetary magnetic field B IMF are analyzed: θ IMF = 45° and 90° with respect to the solar wind direction v sw . Models for the (paraboloid) flaring parameter b s as a function of M A , azimuthal angle ϕ, and θ IMF = 45° or 90°, show b s decreasing with M A , corresponding to the shock becoming blunter and less swept back (with a larger cross section), as expected. Together with models for the shock's standoff distance (which increases with decreasing M A ) the models for b s ( M A , ϕ) predict the shock's 3‐D location. Variations of b s with ϕ represent eccentricities in the shock's cross section (i.e., a departure from circularity), with the shock extending further perpendicular to v ms (the fast mode speed) than parallel, as M A → 1. An additional effect is observed in which the shock shape is “skewed” for θ IMF = 45° (but not for θ IMF = 90°) in the plane containing B IMF and v sw . These latter two effects are consistent with the fast mode velocity varying with propagation direction relative to B IMF .