A conservative, discontinuous Galerkin, tracer transport scheme using compatible finite elements

Abstract

This paper outlines a conservative transport scheme for scalar tracers within a compatible finite element model for geophysical fluid equations. Instead of using the advective transport equation for a mixing ratio, a conservative transport equation is solved for the tracer density of the mixing ratio multiplied by the dry density. This ensures mass conservation in the continuous equations, which can be preserved in the discrete equations with a discontinuous Galerkin transport scheme. Our method is designed to work for two placements of the mixing ratio in a Charney-Phillips vertical staggering: either co-located with the dry density or vertically staggered from it. The new scheme is designed to conserve the tracer density and ensure consistency by maintaining a constant mixing ratio. Additionally, a mass-conserving limiter is developed to ensure non-negativity in the co-located configuration. Tests with terminator toy chemistry and a moist rising bubble show the use of the new transport scheme with physics terms and its ability to accurately model mass conservation of moisture species in a dynamical core setup.

Figures (11)

• Figure 1: The original function spaces for the density and mixing ratio ($\mathbb{V}_{\rho,0},~\mathbb{V}_{\rho,1},~\mathbb{V}_{\theta,0},~\mathbb{V}_{\theta,1}$), and intermediate spaces used in the recovery and embedded DG operations ($\tilde{\mathbb{V}}_1, ~\hat{\mathbb{V}}_{\theta}$). Nodes at a facet enforce continuity between adjacent elements.
• Figure 2: L2 mixing ratio error (left column) and mean tracer density error (right column) in the co-located transport-only tests. Lowest-order elements are shown in the top row and next-to-lowest-order in the bottom row. A characteristic cell length for the cubed-sphere is computed as $\Delta x= \pi R/2 N_e$, where $N_e$ is the number of cells per panel edge. Legend values in the L2 error plots denote the approximate order of convergence from a line of best fit. For both element orders, the conservative tracer scheme shows good mass conservation, as shown by small errors in the tracer density. Larger tracer density errors are present when using $k=0$ spaces, which requires the recovery scheme, compared to $k=1$ spaces, where the fields are transported in their original spaces.
• Figure 3: Co-located transport-only test with the consistency configuration, in which the mixing ratio is held constant while the density is spatially varying. Lowest-order elements are shown on the left and next-to-lowest-order on the right. Both the advective and conservative tracer schemes preserve a constant mixing ratio to a high level of accuracy.
• Figure 4: L2 mixing ratio error (left column) and mean tracer density error (right column) in the vertically staggered transport-only tests. Lowest-order elements are shown in the top row and next-to-lowest-order in the bottom row. Legend values in the L2 error plots denote the approximate order of convergence from a line of best fit. For both element orders, the conservative tracer scheme shows good mass conservation, as shown by very small errors in the tracer density.
• Figure 5: Vertically staggered transport-only test with the consistency configuration, in which the mixing ratio is held constant while the density is spatially varying. Lowest-order elements are shown on the left and next-to-lowest-order on the right. The conservative tracer scheme preserves the constant mixing ratio to a good level of accuracy for both element orders.