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 an inhomogeneous background that is general enough to represent the Earth's magnetosphere using a multiscale analysis and the Kreiss theorem. The most important effect beyond homogeneous RMHD is the propensity for background Alfvén speed gradients to drive nonlinear wave effects. This work has immediate application to the magnetosphere-ionosphere coupling problem in the Earth's magnetosphere, where the Alfvén speed varies by an order of magnitude over a short distance. We estimate the turbulent energy transfer driven by this. In Earth's magneotsphere, a substantial portion of energy (>20%) could be transferred in this way along auroral field lines. A revised field aligned current density mapping technique to account for this energy loss mechanism is proposed to replace using j∥/B0 being constant along the field line. We also predict the existence of a ``saturation current density,'' where there is a maximum j∥ that exist before driving a nonlinear cascade. At Earth, this value is ∼ 5 μA/m2 at the ionosphere, which corresponds to the maximum observed value by ground magnetometers. Furthermore, this study makes use of a novel methodology, a computer algebra system, in performing 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.
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BibTeX
@misc{desjardin2026equations,
title = {{The Equations of Reduced Magnetohydrodynamics in Dipole Coordinates}},
author = {Ian M. DesJardin and Christopher Bard and Natalia Y. Buzulukova and John C. Dorelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.06617},
eprint = {2608.06617},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.06617},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Equations of Reduced Magnetohydrodynamics in Dipole Coordinates
AU - Ian M. DesJardin
AU - Christopher Bard
AU - Natalia Y. Buzulukova
AU - John C. Dorelli
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2608.06617
UR - https://arxiv.org/abs/2608.06617
AB - 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 an inhomogeneous background that is general enough to represent the Earth's magnetosphere using a multiscale analysis and the Kreiss theorem. The most important effect beyond homogeneous RMHD is the propensity for background Alfvén speed gradients to drive nonlinear wave effects. This work has immediate application to the magnetosphere-ionosphere coupling problem in the Earth's magnetosphere, where the Alfvén speed varies by an order of magnitude over a short distance. We estimate the turbulent energy transfer driven by this. In Earth's magneotsphere, a substantial portion of energy (>20%) could be transferred in this way along auroral field lines. A revised field aligned current density mapping technique to account for this energy loss mechanism is proposed to replace using j_∥/B_0 being constant along the field line. We also predict the existence of a ``saturation current density,'' where there is a maximum j_∥ that exist before driving a nonlinear cascade. At Earth, this value is ∼ 5 μA/m^2 at the ionosphere, which corresponds to the maximum observed value by ground magnetometers. Furthermore, this study makes use of a novel methodology, a computer algebra system, in performing 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.
ER -