Rotating Shallow Water Equations (RSWE)

Updated 23 January 2026

RSWE are a fundamental set of equations in geophysical fluid dynamics that describe shallow rotating flows with exact conservation of mass, energy, and potential vorticity.
They capture critical phenomena such as Rossby and inertia–gravity waves, providing insights into geostrophic balance and large-scale atmospheric and oceanic motions.
Advanced numerical methods, including mimetic, variational, and Hamiltonian–Poisson schemes, preserve RSWE invariants, enabling stable and accurate simulations.

The rotating shallow water equations (RSWE) constitute a fundamental model in geophysical fluid dynamics, describing the evolution of a shallow, rotating fluid layer under the influence of gravity and planetary rotation. The RSWE capture the essential dynamical balances governing large-scale atmospheric and oceanic motions, such as Rossby waves, inertia–gravity waves, and geostrophic balance. They serve as the prototypical nonlinear dispersive PDE system admitting exact conservation laws, geophysical wave phenomena, and energetic cascades, and are central both as a physical model and as a testbed for numerical algorithms and structure-preserving discretizations.

1. Mathematical Formulation and Fundamental Properties

The RSWE on a rotating sphere or in the plane are typically written in vector-invariant form: $\begin{aligned} \partial_t h + \nabla\cdot(h\,\mathbf{u}) &= 0, \ \partial_t \mathbf{u} + (\mathbf{u}\cdot\nabla)\mathbf{u} + f\mathbf{k}\times\mathbf{u} + g\nabla h &= 0, \end{aligned}$ where $h$ is the fluid depth, $\mathbf{u}$ is the horizontal velocity, $f$ is the Coriolis parameter (which may be spatially varying), and $g$ is gravity (Thuburn et al., 2014, Duru et al., 5 Jan 2026, Bauer et al., 2023). In this form, the system is hyperbolic for $h > 0$ and admits both inertia–gravity and geostrophic modes.

A key feature is the exact conservation of mass, energy, and a noncanonical Hamiltonian Poisson structure. The relative and absolute vorticity, as well as potential vorticity (PV) $q = (\nabla\times\mathbf{u} + f)/h$ , are central diagnostic fields, with total PV and enstrophy serving as Casimir invariants (Bauer et al., 2023). These invariant structures underlie the long-time stability and the correct representation of balanced and unbalanced motions in numerical discretizations.

2. Variational and Hamiltonian Structure

The RSWE derive from an Euler–Poincaré variational principle on the diffeomorphism group, with the Lagrangian

$\ell(\mathbf{u},h) = \int \left( \tfrac{1}{2} h |\mathbf{u}|^2 + h\mathbf{R}\cdot\mathbf{u} - \tfrac{1}{2}g h^2 \right) dS,$

where $\mathbf{R}$ encodes rotational effects (Brecht et al., 2018, Bauer et al., 2017). The resulting equations conserve total energy and mass, and possess a noncanonical Lie–Poisson structure. In the presence of stochastic transport or Casimir-selective dissipation, this structure can be preserved pathwise or modified to dissipate only enstrophy, leaving the total energy invariant (Bauer et al., 2023, Brecht et al., 2021).

Potential vorticity is materially conserved by the RSWE flow, playing a key role in nonlinear wave interactions, slow manifold dynamics, and secondary balance models (Dritschel et al., 2017, Owen et al., 2018). The framework conveniently generalizes to 2-layer systems (Owen et al., 2018), location-uncertainty or stochastic extensions (Brecht et al., 2021), and variationally derived balance models (Dritschel et al., 2017).

3. Structure-Preserving Numerical Discretizations

High-fidelity simulation of RSWE requires discrete schemes that respect the system's key conservation and balance properties. Leading approaches include:

Mimetic and Finite Element Exterior Calculus Methods: FEEC organizes discrete variables into compatible spaces forming an exact de Rham complex, enabling algebraic preservation of mass, vorticity, PV, energy, and enstrophy (Cotter et al., 2012, Thuburn et al., 2014). Primal-dual mimetic methods employ compound elements and Hodge star operators on polygonal meshes, with PV advection solved on the dual mesh for improved accuracy and consistency (Thuburn et al., 2014, Lee et al., 2017). The mixed mimetic spectral element framework uses edge-based/histopolant bases and incidence matrices to encode topological operators, guaranteeing exact conservation up to quadrature and time-stepping errors (Lee et al., 2017, Lee et al., 2018).
Variational and Geometric Integrators: Discrete Euler–Poincaré methods are constructed from a mesh-based variational principle, yielding schemes that exactly preserve discrete mass, energy, and circulation, and display long-term stability even on highly irregular simplicial meshes (Brecht et al., 2018, Bauer et al., 2017).
Hamiltonian–Poisson Based Compatible Schemes: Fully discrete Hamiltonian FE schemes incorporating upwinding in velocity and/or depth achieve energy conservation and controlled enstrophy dissipation, leveraging bracket antisymmetry and Poisson time-integrators for high robustness (Wimmer et al., 2019).
Casimir-Dissipative and Stochastic Schemes: Selective Casimir-(enstrophy)-dissipative modifications stabilize explicit high-resolution runs by removing only grid-scale enstrophy without damping total kinetic energy, outperforming standard biharmonic/Laplacian diffusions in preserving ensemble spread and large-scale jet dynamics (Bauer et al., 2023). Stochastic modelings using location-uncertainty principles extend RSWE to physically consistent random ensembles with energy-conserving spatial cores, showing superior probabilistic skill in benchmark tests (Brecht et al., 2021).

4. Boundary Conditions, Well-Posedness, and Analytical Results

Recent work gives a rigorous characterization of well-posed, energy-stable, and entropy-stable initial boundary value problems (IBVP) for RSWE in vector-invariant form (Duru et al., 5 Jan 2026). Subcritical BCs based on Riemann invariants, mass flux, and Bernoulli potential are shown to guarantee maximal energy dissipation at inflow/outflow, yielding existence and uniqueness for both linear and nonlinear IBVPs. High-order stable discretizations employing summation-by-parts (SBP) operators and simultaneous approximation terms (SAT) weakly enforce BCs, with third-order convergence and discrete preservation of energy/entropy stability on complex meshes.

For the 1D RSWE, symmetrization and appropriate parabolic regularization allow local well-posedness and global regularized solutions in energy spaces for data near rest (Bedjaoui et al., 2021). In the rapidly rotating limit, analytical studies establish uniform-in-ε bounds for classical solutions and rigorous convergence to the geostrophic slow manifold, with explicit projection operators onto the space of zonal geostrophic flows (Cheng et al., 2019).

5. Wave Solutions, Balance Models, and Nonlinear Interactions

The RSWE admit a rich variety of exact and approximate solutions:

Wave dynamics: Linearization reveals three bands—geostrophic and Poincaré (inertia–gravity)—with quantum geometric tensor analysis identifying Chern numbers and related topological invariants of RSWE wave bands (Ganeshan et al., 15 Jan 2026). In two-layer models, barotropic and baroclinic modes couple to produce both fast–fast–fast and fast–slow–fast resonant triads, subject to strong enstrophy-based selection rules (Owen et al., 2018).
Nonlinear and conditionally invariant solutions: The conditional symmetry method yields large families of implicit exact solutions depending on arbitrary functions of Riemann invariants (Huard, 2010). Lie symmetry and similarity reductions produce closed-form, traveling, oscillatory, and scaling solutions for polytropic RSWE (Paliathanasis, 2019).
Variational balance models: Asymptotic analysis of RSWE produces a continuum of variational balance models, of which the Salmon L₁-member is uniquely regular and accurate on long (O(1/Ro)) timescales, while others develop unphysical high-wavenumber artifacts (Dritschel et al., 2017).

6. Numerical Accuracy, Conservation, and Practical Benchmarks

All structure-preserving RSWE schemes are evaluated on canonical test problems—e.g., Williamson tests, Galewsky barotropic jet, and Rossby–Haurwitz waves—using both planar and spherical geometries (Thuburn et al., 2014, Shipton et al., 2017, Bauer et al., 2023, Lee et al., 2018). Mimetic and variational methods achieve O(Δx^{2)–O(Δx⁴⁾} convergence in L₂ and L∞ errors, with exact or near-exact conservation of mass, total energy, and global vorticity. Selective enstrophy dissipation removes grid-scale noise with minimal impact on large-scale dynamics, while stochastic and Casimir-dissipative schemes outperform standard Laplacian methods for uncertainty quantification and high-resolution ensemble spread (Brecht et al., 2021, Bauer et al., 2023).

7. Extensions and Current Directions

Ongoing research explores:

Extension to higher-order temporal and spatial discretizations, parallel scalability, and adaptation to three-dimensional baroclinic or compressible systems (Thuburn et al., 2014, Lee et al., 2018).
Implementation of quantum geometric and topological diagnostics, including physical probing via parametric driving (Ganeshan et al., 15 Jan 2026).
Rigorous mathematical analysis of multi-layer extensions, resonance, and averaging in the limit of strong rotation and stratification (Cheng et al., 2019, Owen et al., 2018).
Structure-preserving schemes on irregular and adaptive meshes, and practical deployment in operational numerical weather and climate models (Bauer et al., 2017, Lee et al., 2017).