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High order tracer variance stable transport with low order energy conserving dynamics for the thermal shallow water equations

This paper presents a hybrid numerical scheme for the thermal shallow water equations that couples a low-order, energy-conserving mixed finite element dynamics solver with a high-order, tracer-variance-conserving discontinuous Galerkin transport method via a nested mesh hierarchy, thereby preserving rich turbulent solutions without compromising model stability.

Original authors: David Lee, Kieran Ricardo, Tamara Tambyah

Published 2026-05-13

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: David Lee, Kieran Ricardo, Tamara Tambyah

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to simulate the weather on a computer. To do this, you need to solve two very different types of problems at the same time:

The Big Picture (Dynamics): How the air and water move in large, slow swirls (like a giant ocean current or a storm system). This needs to be calculated very carefully to ensure the total energy of the system doesn't magically appear or disappear.
The Details (Transport): How specific "tracers" (like heat, humidity, or pollution) get carried along by that moving air. This needs to be calculated with high precision so the tracers don't get smeared out or lose their shape.

The problem is that the best tools for the "Big Picture" are often too slow and clunky for the "Details," while the best tools for the "Details" are often too unstable or energy-wasting for the "Big Picture."

This paper presents a clever hybrid solution that combines the best of both worlds. Here is how it works, using some everyday analogies:

The "Low-Order" Engine: The Reliable Truck

Think of the low-order dynamics solver as a sturdy, reliable delivery truck. It's not the fastest vehicle, and it doesn't have a super-suspension, but it's incredibly stable. It's designed to carry heavy loads (energy) without spilling a drop.

Why use it? It's mathematically proven to keep the system's total energy balanced. If you drive it around a track, the fuel (energy) you put in equals the fuel you get out.
The downside: Because it's built for stability, it's a bit "blurry." If you try to paint a sharp, detailed picture with it, the edges get fuzzy.

The "High-Order" Paintbrush: The Precision Artist

Now, think of the high-order transport scheme as a master artist with a fine-tipped paintbrush. This artist can draw incredibly sharp, detailed lines and preserve the exact shape of a painting (the "tracer variance") even when moving it around.

Why use it? It prevents the "paint" (heat or moisture) from getting smeared or losing its definition. It keeps the details crisp.
The downside: If you let this artist drive the truck alone, they might drive too fast and crash, or they might accidentally create energy out of thin air, breaking the laws of physics.

The Hybrid Solution: The Artist Riding in the Truck

The authors of this paper figured out how to let the Artist (High-Order) ride inside the Truck (Low-Order) without causing a crash.

The Map (Mesh Hierarchy): They created a special map where the "Truck" drives on a coarse grid (big blocks), but the "Artist" works on a much finer grid (tiny blocks) that fits perfectly inside the Truck's path. It's like having a high-resolution photo printed on a low-resolution grid; the details align perfectly with the big picture.
The Handoff: The Truck tells the Artist where the wind is blowing (the velocity and depth). The Artist then uses this information to move the "paint" (buoyancy/heat) with extreme precision.
The Safety Net (Upwinding): To stop the Artist from getting too excited and creating chaos (numerical noise), they added a "damping" mechanism called upwinding. Imagine this as a gentle hand on the Artist's shoulder, smoothing out tiny, unrealistic wiggles in the paint without ruining the overall picture.

The Results: Why This Matters

The paper tested this new hybrid vehicle in two main ways:

The "Spinning Top" Test: They spun a tracer around in a circle. The high-order method kept the tracer's shape perfectly sharp, whereas the old low-order method would have turned it into a blurry smear.
The "Storm" Test: They simulated a chaotic, turbulent storm.
- The old low-order method with upwinding was too safe; it smoothed out the storm so much that all the interesting details vanished (like watching a storm through a foggy window).
- The old low-order method without upwinding was too wild; it kept the details but eventually became unstable and noisy, like a car shaking apart at high speed.
- The New Hybrid Method was the Goldilocks solution. It kept the storm's details sharp and realistic (thanks to the Artist) but remained stable and didn't lose energy (thanks to the Truck).

The Bottom Line

This paper doesn't claim to solve global warming or predict tomorrow's weather perfectly. Instead, it introduces a new mathematical engine for weather models.

It proves that you can have your cake and eat it too: you can use a slow, stable, energy-conserving engine for the big movements, while using a fast, precise, detail-preserving engine for the heat and moisture moving inside it. This results in a computer model that is stable enough to run for a long time but detailed enough to show rich, realistic weather patterns that older models would either blur out or crash.

Technical Summary: High Order Tracer Variance Stable Transport with Low Order Energy Conserving Dynamics

Problem Statement
Numerical modeling of geophysical fluid dynamics often requires a compromise between computational efficiency and physical fidelity. Low-order numerical schemes (finite difference, finite volume, and finite element) are frequently employed for implicit atmospheric solvers because their lower condition numbers facilitate faster convergence and matrix assembly. However, these schemes typically rely on upwinding to suppress high-frequency artifacts, leading to excessive numerical dissipation that degrades the resolution of turbulent features. Conversely, high-order methods (such as Discontinuous Galerkin) are effective for resolving inertial motions and constructing accurate numerical fluxes but are computationally expensive and often lack the robust stability properties of low-order implicit solvers. Furthermore, standard high-order transport schemes often struggle to conserve tracer variance (a numerical entropy of the system) without introducing instability, while low-order schemes often fail to preserve the energy-conserving structure required for long-term stability in non-canonical Hamiltonian systems.

Methodology
The authors propose a hybrid numerical framework that couples a low-order mixed finite element solver for dynamics with a high-order Discontinuous Galerkin (DG) method for the material transport of thermodynamic tracers, specifically within the context of the thermal shallow water equations.

Low-Order Dynamics: The dynamics solver utilizes a mixed finite element formulation (lowest polynomial order) that preserves the energy-conserving structure of the equations. This solver operates on a coarse mesh and computes prognostic variables for velocity ( $u$ ), depth ( $h$ ), and depth-weighted buoyancy ( $S$ ).
High-Order Transport: The buoyancy transport is handled by a high-order DG scheme operating on a nested, refined mesh. The buoyancy is treated as a material variable ( $s$ ) rather than a flux variable ($S=sh$). The scheme employs a skew-symmetric expansion of the advection operator at the continuous level to ensure that the discrete formulation can provably conserve tracer variance.
Coupling Strategy: The two solvers are coupled via a nested hierarchy of meshes. The low-order mesh is embedded within the high-order transport mesh such that the cell centers of the low-order elements coincide with the Gauss-Legendre quadrature points of the high-order elements. This creates a 1:1 mapping between degrees of freedom.
- The high-order buoyancy field is diagnosed from the low-order prognostic variables at the start of each time step.
- The high-order transport equation is advanced using the low-order mass flux and depth fields (averaged over the time step).
- The resulting high-order buoyancy is projected back to the low-order space to update the dynamics solver.
Conservation Properties:
- Energy: The coupling preserves the anti-symmetry of the flux-form buoyancy transport and the pressure gradient, ensuring discrete energy conservation for the low-order dynamics.
- Tracer Variance: The high-order transport scheme is designed to conserve tracer variance in space. To ensure stability and prevent grid-scale noise from nonlinear aliasing, an upwinding term is included, which allows for controlled dissipation of tracer variance (damping) rather than unbounded growth.
Time Integration: The system uses a stiffly stable third-order Runge-Kutta (SSP-RK3) integrator for the transport step and a quasi-Newton approach (solved via GMRES) for the coupled dynamics.

Key Contributions

Hybrid Formulation: The paper introduces a consistent formulation that harmonizes low-order energy-conserving dynamics with high-order tracer-variance-conserving transport.
Material Form Transport: Unlike previous tracer-variance-conserving formulations that rely on flux forms, this work expresses the discrete advection operator in material form to couple effectively with low-order flux-form dynamics.
Mesh Hierarchy: The implementation of a specific mesh hierarchy where low-order cell centers align with high-order Gauss-Legendre points facilitates a direct, efficient coupling without complex interpolation errors.
Stability Mechanism: The method demonstrates that using high-order transport with upwinding provides a stable solution that preserves richer turbulent structures compared to purely low-order methods, without the excessive dissipation typical of low-order upwinding.

Results
The method was validated through three standard test cases:

Solid Body Rotation: The high-order DG scheme demonstrated 4th-order convergence for upwinded fluxes and 3rd-order for centered fluxes. Mass conservation was exact, and tracer variance conservation errors decayed with resolution, confirming the stability of the scheme.
Thermogeostrophic Balance: In a steady-state test, the full coupled system showed second-order convergence for all variables, limited by the low-order dynamics. The scheme maintained machine-precision conservation of mass, depth-weighted buoyancy, and energy, with no long-term drift in relative vorticity.
Thermal Instability (Nonlinear Turbulence): In a highly nonlinear regime, the high-order transport scheme with upwinding produced stable solutions that preserved fine-scale features.
- Comparison: Purely low-order centered fluxes became unstable due to uncontrolled grid-scale aliasing. Purely low-order upwinded fluxes were excessively diffusive, smearing out turbulent structures.
- Hybrid Performance: The coupled high-order/low-order scheme maintained stability (tracer variance decayed moderately) while preserving the richness of the turbulent solution, outperforming both purely low-order alternatives.

Significance and Claims
The authors claim that this approach successfully bridges the gap between the computational efficiency of low-order implicit solvers and the physical fidelity of high-order transport schemes. While the overall accuracy of the system is formally limited to second order by the low-order dynamics, the use of high-order, tracer-variance-conserving transport significantly improves model stability and the preservation of turbulent solutions.

The paper modestly notes that the current implementation uses a GMRES solver which may not scale efficiently for very large parallel domains compared to geometric multigrid methods, suggesting this as a potential area for future extension. The authors assert that the coupling strategy is generalizable to other non-canonical Hamiltonian systems involving thermodynamic tracers, such as compressible Euler equations for dry and moist atmospheres. The primary achievement is demonstrating that energy conservation and tracer variance stability can be maintained simultaneously in a coupled system, preventing the excessive dissipation or instability often found in traditional low-order atmospheric models.

The "Low-Order" Engine: The Reliable Truck

The "High-Order" Paintbrush: The Precision Artist

The Hybrid Solution: The Artist Riding in the Truck

The Results: Why This Matters

The Bottom Line

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