Moment Approximations to Magnetic Rotating Shallow Flows

TL;DR

The paper tackles the loss of vertical information in depth-averaged magnetic rotating shallow water (MRSW) models by developing a hierarchical moment closure (MRSWME) that represents vertical profiles of horizontal velocity and magnetic fields through arbitrary-order polynomials. Using Legendre-based bases and Galerkin projection, it derives evolution equations for both the mean fields and the vertical-coefficient moments, yielding a closed 2-D system augmented by a Godunov-Powell nonconservative term to handle the divergence-free constraint. The authors implement path-conservative central-upwind schemes for both the moment system and a vertically resolved reference model, demonstrating that higher-order moments substantially reduce model error with modest computational cost in 1-D tests and magneto-geostrophic adjustment scenarios. A key finding is that while MRSWME improves accuracy, it incurs loss of global hyperbolicity for M≥2, indicating a need for future hyperbolicity-preserving strategies. Overall, the work advances efficient, higher-fidelity modeling of conductive shallow flows with significant vertical structure, with potential applications in solar and geophysical contexts.

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.

Figures (16)

• Figure 3.1: Hyperbolic region of the system \ref{['eq:firstorder']} for values $b_m\in[-5,5]$ and $\widetilde{\beta},\ \widetilde{\eta}\in[-10,10]$ (left column) and for the zoomed region $b_m\in[-2,2]$ and $\widetilde{\beta},\ \widetilde{\eta}\in[-4,4]$ (right column), where $\widetilde{\beta},\ \widetilde{\eta}$ are defined in \ref{['eq:q_scalings']}. Note the slice $b_m = 0$, which is always hyperbolic, is omitted for visibility of the rest of the region. Also note that the colors are purely for the purpose of providing depth.
• Figure 5.1: Example 1: Reference system solutions $h$ (left) and $v_m$ (right) for the four different polynomial vertical profiles.
• Figure 5.2: Example 1: Solutions for $h$ (top row) and $v_m$ (bottom row) computed using the 2-D reference system, the 1-D standard shallow water (SW), and the SWME for the different vertical profile cases of $M = 0$ (first column), $M = 1$ (second column), $M = 2$ (third column), and $M = 3$ (last column).
• Figure 5.3: Example 2: Reference system solutions $h$ (left) and $v_m$ (right) for the four different polynomial vertical profiles.
• Figure 5.4: Example 2 (Linear case): The solutions for $h$ (left) and $v_m$ (right) of the $Mth$-order MRSWME, $M = 0,\dots,3,$ against the reference solution.

Theorems & Definitions (1)

• Remark 2.1