A new method for solving the MHD equations in the magnetosheath

Abstract

We present a new analytical method to derive steady-state magnetohydrodynamic (MHD) solutions of the magnetosheath in different levels of approximation. With this method, we calculate the magnetosheath's density, velocity, and magnetic field distribution as well as its geometry. Thereby, the solution depends on the geomagnetic dipole moment and solar wind conditions only. To simplify the representation, we restrict our model to northward IMF with the solar wind flow along the stagnation streamline. The sheath's geometry, with its boundaries, bow shock and magnetopause, is determined self-consistently. Our model is stationary and time relaxation has not to be considered as in global MHD simulations. Our method uses series expansion to transfer the MHD equations into a new set of ordinary differential equations. The number of equations is related to the level of approximation considered including different physical processes. These equations can be solved numerically; however, an analytical approach for the lowest-order approximation is also presented. This yields explicit expressions, not only for the flow and field variations but also for the magnetosheath thickness, depending on the solar wind parameters. Results are compared to THEMIS data and offer a detailed explanation of, e.g., the pile-up process and the corresponding plasma depletion layer, the bow shock and magnetopause geometry, the magnetosheath thickness, and the flow deceleration

Statistical survey of day-side magnetospheric current flow using Cluster observations: Bow shock

“…The geometric parameters c BS,t represent the bow shock curvature in t = y and t = z direction. Nabert et al (2013) deduce values of…”

Solar wind reconstruction from magnetosheath data using an adjoint approach

“…Analogously to the previous studies, the reconstruction can be used to provide solar wind information if in situ solar wind observations are not available. This determines the solar wind parameters, which are the drivers of the magnetospheric current system and are usually required in magnetospheric and ionospheric models (e.g., Nagata et al, 2008;Korth et al, 2010;Zhang et al, 2013;Baker et al, 2013) We reconstruct the solar wind conditions during a spacecraft magnetosheath passage using the magnetosheath model introduced by Nabert et al (2013). The solar wind parameters are boundary conditions of the magnetosheath model and they are varied until the model predictions match the magnetosheath data.…”

“…For the solar wind reconstruction, we chose the zeroth order MHD (magnetohydrodynamics) model presented in Nabert et al (2013). This model uses series expansions of density, velocity, gas pressure, and magnetic field up to the first order away from the stagnation streamline to simplify the stationary MHD equations.…”

“…Note that the y and z directions are called tangential hereafter. The series expansion procedure provides the following set of ordinary differential equations (Nabert et al, 2013):…”

“…The vacuum permeability is µ 0 = 4π × 10 −7 N/A 2 . The tangential velocity derivatives u y10 and u z01 , as well as the magnetic field derivative B x01 , are given by u y10 = 0.8 u/x BS , u z01 = u/x BS , and B x01 ≈ 0 (Nabert et al, 2013). Here, u denotes the x component of the velocity difference across the bow shock, which is calculated using Rankine-Hugoniot relations.…”