Moment Approximations to Magnetic Rotating Shallow Flows

TL;DR

This paper develops higher-order moment approximations for the magnetic rotating shallow water model, capturing vertical profiles of velocities and magnetic fields to improve accuracy while maintaining computational efficiency.

Contribution

It extends the MRSW model to arbitrary order moments, allowing for non-constant vertical profiles in a 2-D framework using polynomial expansions and Galerkin projection.

Findings

Higher-order moments reduce model error

Moment approximations maintain computational efficiency

Third-order moments improve accuracy significantly

Abstract

Originally introduced to describe a transition region in stars, the magnetic rotating shallow water (MRSW) model is now used in many solar physics and geophysical applications. Derived from the 3-D incompressible magnetohydrodynamic system, the shallow nature of these applications motivates depth-averaging of both the velocities and magnetic fields. This is advantageous in terms of computational efficiency -- but at the loss of vertical information, thus limiting the predictive power of the MRSW model. To overcome this problem, we employ higher-order vertical moments, but now in the context of conductive fluids. In doing so, the new approximation maintains non-constant vertical profiles of both the horizontal magnetic fields and horizontal velocities, while still remaining in the simplified 2-D framework corresponding to depth integration. In this work, we extend the derivation of the shallow water moment equations to derive the MRSW moment system of arbitrary order; i.e., we represent the vertical profiles of the velocities -- and now additionally the magnetic fields -- by arbitrary-order polynomial expansions, and close the new expanded 2-D system with evolution equations for these polynomial coefficients, found via Galerkin projection. Through numerical experiments for MRSW moment systems up to third-order, we demonstrate that these moment approximations reduce model error without significantly sacrificing computational efficiency.