Three-dimensional solutions of the magnetohydrostatic equations: rigidly rotating magnetized coronae in cylindrical geometry

School of Mathematics and Statistics, University of St. Andrews, St. Andrews KY16 9SS, UK e-mail: na76@st-andrews.ac.uk;nalsalti@squ.edu.om

Received: 23 November 2009
Accepted: 1 February 2010

Abstract

Context. Solutions of the magnetohydrostatic (MHS) equations are very important for modelling astrophysical plasmas, such as the coronae of magnetized stars. Realistic models should be three-dimensional, i.e., should not have any spatial symmetries, but finding three-dimensional solutions of the MHS equations is a formidable task.

Aims. We present a general theoretical framework for calculating three-dimensional MHS solutions outside massive rigidly rotating central bodies, together with example solutions. A possible future application is to model the closed field region of the coronae of fast-rotating stars.

Methods. As a first step, we present in this paper the theory and solutions for the case of a massive rigidly rotating magnetized cylinder, but the theory can easily be extended to other geometries, We assume that the solutions are stationary in the co-rotating frame of reference. To simplify the MHS equations, we use a special form for the current density, which leads to a single linear partial differential equation for a pseudo-potential U. The magnetic field can be derived from U by differentiation. The plasma density, pressure, and temperature are also part of the solution.

Results. We derive the fundamental equation for the pseudo-potential both in coordinate independent form and in cylindrical coordinates. We present numerical example solutions for the case of cylindrical coordinates.