Barotropic-Baroclinic Splitting for Multilayer Shallow Water Models with Exchanges

TL;DR

The paper develops an exact barotropic-baroclinic operator-splitting framework for multilayer shallow-water models in terrain-following coordinates, enabling a fast barotropic step and a slower baroclinic step that handles vertical exchanges. The scheme preserves total energy in the split sense, satisfies a discrete entropy inequality, and employs well-balancing strategies to maintain geostrophic and lake-at-rest equilibria. A subcycling-based barotropic step and a decoupled treatment of deviations and tracers reduce computational cost, especially at low Froude numbers, while maintaining accuracy. Numerical experiments validate convergence, cost reductions, and robust balance properties across varied test cases, highlighting practical benefits for coastal and ocean-scale multilayer applications.

Abstract

This work presents the numerical analysis of a barotropic-baroclinic splitting in a nonlinear multilayer framework with exchanges between the layers in terrain-following coordinates. The splitting is formulated as an exact operator splitting. The barotropic step handles free surface evolution and depth-averaged velocity via a well-balanced one-layer model, while the baroclinic step manages vertical exchanges between layers and adjusts velocities to their mean values. We show that the barotropic-baroclinic splitting preserves total energy conservation and meets both a discrete maximum principle and a discrete entropy inequality. Several numerical experiments are presented showing the gain in computational cost, particularly in low Froude simulations, with no loss of accuracy. The benefits of using a well-balancing strategy in the barotropic step to preserve the geostrophic equilibrium are inherited in the overall scheme.

Figures (13)

• Figure 1: Notations for the multilayer approach.
• Figure 2: Overview of the scheme. The large baroclinic step with $\Delta t^{n}$ consists of a prediction step where system \ref{['eq:BCnoME']} is solved and a correction step where the mass exchange terms are defined and applied. The computed data of the baroclinic step $(h_{\alpha},u_{\alpha},T_{\alpha})^{n+\frac{1}{2}}$ is used as initial data for the barotropic step. We perform many small barotropic time steps $\delta t^k$ until we reach the final time $\Delta t^{n}$. During this loop we only update the vertically averaged quantities $h$ and $h\bar{u}$. The transported variables $\sigma_{\alpha}$ and $T_{\alpha}$ are updated only once with a large time step $\Delta t^{n}$.
• Figure 3: Upper row: The velocity field for testcase \ref{['sec:analytical_sol_2d_euler']}, along with the highlighted positions for a detailed examination of the vertical velocity profile of $u$. Lower row: The convergence behavior of the vertical velocity profiles of $u$ at various positions $x$.
• Figure 4: $L^2$-norm of the tracer to examine numerical diffusion. The solid line represents the splitting method, while the dashed line indicates the scheme without splitting.
• Figure 5: Comparison of the computational cost and the total error $\| (h_\text{err}, hu_\text{err})^\top \|_2$ for testcase \ref{['sec:analytical_sol_2d_euler']} with different Froude numbers. Top left: $\mathrm{Fr}=1.87$ (using $\alpha = 5$), top right: $\mathrm{Fr}=0.38$ (using $\alpha = 1$), bottom left: $\mathrm{Fr}=0.04$ (using $\alpha = 0.1$).

Theorems & Definitions (40)

• Definition 1.1
• Proposition 1.2
• proof
• Proposition 1.3
• proof
• Remark 1.4
• Remark 1.5
• Proposition 1.6
• proof
• Proposition 1.7