The Equations of Reduced Magnetohydrodynamics in Dipole Coordinates

Abstract

The equations of reduced magnetohydrodynamics (RMHD) isolate the Alfvénic energy transfer and turbulence dynamics from MHD in a computationally tractable way. In this study, we derive the equations of RMHD for a low $β$ plasma in a dipole coordinate system aligned with the background magnetic field using a multiscale analysis. The Kreiss theorem is used to derive the equilibrium conditions that the background conditions must satisfy on the time scale of MHD turbulence. From these, we find a connection between plasma flow along flux tubes and changes in the Alfvén speed that may drive nonlinear wave effects. The final equations demonstrate an intimate coupling between field aligned currents and plasma vorticity including the effects of nonuniform flux tube area and realistic plasma density profiles. This work has immediate application to the magnetosphere-ionosphere coupling problem in the Earth's magnetosphere as it provides a way to link dynamically evolving field aligned currents from the magnetosphere, especially important during geomagnetic storms and substorms, with the development of magnetohydrodynamic turbulence at low altitudes in a self-consistent manner. This also clarifies the role of Alfvén waves as a transfer mechanism of energy under these inhomogeneous circumstances while remaining simple enough to make predictions. Furthermore, this study makes use of a novel methodology, a computer algebra system, in performing the brunt of algebraic work under complicated coordinate systems. We hope this approach serves as a template for performing reproducible and verifiable multiscale perturbation analysis under arbitrarily complex geometries.