Shallow-Water Equations

Updated 12 January 2026

Shallow-water equations are hyperbolic PDEs that govern the depth-averaged flow of inviscid fluids, capturing gravity-driven dynamics and geophysical phenomena.
They incorporate effects like variable bathymetry, Coriolis forces, and additional source terms, using conservation laws and Riemann invariants to ensure precise modeling.
Numerical techniques such as finite-volume and Galerkin methods are employed to achieve well-balanced, shock-capturing simulations for practical applications like tsunami forecasting.

The shallow-water equations (SWEs) are a fundamental family of hyperbolic partial differential equations governing the depth-averaged horizontal flow of a thin layer of inviscid, incompressible fluid under gravity, typically with variable bathymetry and possibly subject to rotational and additional physical effects. SWEs provide the core dynamical framework for geophysical fluid dynamics, modeling ocean, atmospheric, and hydraulic flows where vertical accelerations are negligible compared to horizontal scales.

1. Mathematical Formulation and Core Properties

The classical (1D and 2D) non-dispersive shallow-water equations in conservation form are given by: $\begin{aligned} & \partial_t h + \nabla\cdot(h\,\mathbf{u}) = 0,\ & \partial_t(h\,\mathbf{u}) + \nabla\cdot(h\,\mathbf{u}\otimes\mathbf{u} + \tfrac{1}{2} g h^2\,\mathbf{I}) = -g h \nabla z, \end{aligned}$ where $h(x,t)$ is the local fluid depth, $\mathbf{u}(x,t)$ is the depth-averaged horizontal velocity, $z(x)$ is the bottom topography, and $g$ is gravitational acceleration. This balance can be supplemented by Coriolis force, wind stress, friction, and other source terms depending on application (Alexeenko et al., 2016).

Key mathematical properties:

The system is strictly hyperbolic whenever $h > 0$ .
Riemann invariants and characteristic velocities in 1D: $z_\pm = u \pm 2\sqrt{h}$ with speeds $c_\pm = u \pm \sqrt{h}$ (Alexeenko et al., 2016).
For generic smooth initial data, shock formation is generic in finite time, but there exist sufficient conditions (monotonic, sign-definite Riemann invariants; e.g., ‘unidirectional’ flows) guaranteeing global smooth solutions (Alexeenko et al., 2016).

2. Generalizations: Variable Bottom, Geometry, and Boundaries

Variable Bathymetry and Covariant Extensions

Shallow-water equations accommodate variable bottom by:

Adding $-g h \nabla z$ as the source in the momentum equation.
Interpreting the conservation laws on curved surfaces via differential geometry: turning $(x,y)$ into local coordinates $(\xi^1, \xi^2)$ on a manifold, introducing metric tensors $g_{ij}$ , covariant derivatives, and source terms that account for both geometric curvature and gravity (Fent et al., 2017, Abreu et al., 2022).

Key developments include:

Depth-integration along normals to complex terrains (as opposed to the gravity direction), yielding ‘normally averaged’ shallow-water models (Fent et al., 2017).
Fully intrinsic Lagrangian–Eulerian schemes for flows on manifolds, employing covariant divergence and locally-adapted mesh motion (Abreu et al., 2022).

Boundary Conditions and Riemann Invariants

For bounded domains, characteristic boundary conditions based on the Riemann invariants and local flow regime (supercritical/subcritical) are used to ensure transparent, reflection-free propagation of outgoing waves (Antonopoulos et al., 2015, Kounadis et al., 2019).

Flow Regime	Boundary Data Prescribed	Reference
Supercritical	Both $\eta$ , $u$ at inflow	(Antonopoulos et al., 2015)
Subcritical	Incoming invariants at ends	(Antonopoulos et al., 2015, Kounadis et al., 2019)

3. Physical Extensions: Dissipation, Moisture, Capillarity

Viscous and Moment-Enhanced Models

The classical SWE are inviscid and employ depth-averaged horizontal velocity; however, practical scenarios often require viscous or higher-moment corrections:

Moment-enhanced models expand the vertical structure of velocity using depth-moment expansions, systematically capturing vertical shears (Zhou et al., 26 May 2025).
Robust source treatment is necessary for non-slip boundary conditions. A modified source-term approach addresses stiffness and ensures correct hydrostatic and dynamic balances compared with full Navier-Stokes reference solutions (Zhou et al., 26 May 2025).

Moist Shallow Water Models

Moist shallow-water models introduce additional prognostic fields (moisture species, buoyancy) and source terms that capture condensation, evaporation, and latent heat effects. Multiple compatible model formulations exist, categorized by how physics-dynamics feedback couples to mass and buoyancy (Hartney et al., 2024). A three-state (vapor-cloud-rain) finite-element discretization provides a flexible framework for exploring physics-dynamics coupling.

Capillary and Dispersive Modifications

For high-gradient or vibrational settings, dispersive and surface-tension extensions are relevant:

Augmented Euler–Korteweg or skew-symmetric formulations permit general capillary energies and enable stable, energy-consistent numerics when $\nabla h$ is large (Vila et al., 2019).
Strong vertical vibration induces weak dispersion akin to a surface-tension effect, leading to depression-type solitary and periodic waves (Ilin, 2017).

4. Numerical Methods and Well-Balanced Schemes

Robust numerical integration of the SWEs for geophysical applications requires properties such as positivity, well-balancing (exact preservation of the ‘lake-at-rest’ state), invariant domain preservation, and shock-capturing.

Finite-volume/Godunov methods: Employ approximate or exact Riemann solvers, often with hydrostatic reconstruction for exact well-balancing (Hu, 2018, Guermond et al., 2024).
Staggered schemes: Decoupled or MAC-type layouts enable simple, explicit, and entropy-consistent algorithms with formal first- or quasi-second-order accuracy (Herbin et al., 2019, Herbin et al., 2021).
Galerkin finite element methods: Convergent schemes of high order in both space and time, with rigorous $L^2$ error control and well-balanced properties under suitable quadrature and data projection (Kounadis et al., 2019, Antonopoulos et al., 2015).
Lagrange-projection / all-regime methods: Decompose the equations to handle acoustic and transport scales separately; provide asymptotic correctness in the low-Froude number regime and positivity (Chalons et al., 2019).
Kinetic frameworks: Modern discrete-velocity kinetic approaches allow independent bulk/shear viscosity control and robust hydrodynamic limit recovery (Hosseini et al., 10 May 2025).

5. Gauge-Theoretic and Topological Reformulations

Recent developments connect the SWEs to gauge theory and topological field theory:

The 2D shallow-water system can be recast as a 2+1D Abelian gauge theory with Chern–Simons and mixed BF couplings, with fluid height, velocity, and vorticity mapped onto gauge field strengths (Tong, 2022).
The gauge-theoretic formalism reveals the deep connection between chiral edge modes of Maxwell–Chern–Simons theory and classical coastal Kelvin waves. Bulk gapped excitations correspond to Poincaré waves with dispersion $\omega = \pm\sqrt{c^2 k^2 + f^2}$ , while the topological edge state decays offshore on the scale $c/f$ and manifests as a unidirectional (coastal) Kelvin wave (Tong, 2022).

6. Analytical Results and Global Well-posedness

The SWE are genuinely nonlinear and strictly hyperbolic (except at dry states), with generic formation of shocks in finite time as predicted by the Lax theorem. However, rigorous nonlinear analysis can identify special classes of initial data (e.g., monotonic, sign-definite Riemann invariants, or strong unidirectionality) guaranteeing global classical solutions with no gradient blow-up (Alexeenko et al., 2016).

Condition on Initial Data	Global Classical Solution?	Reference
Riemann invariants $\leq 0$ , monotonic inc.	Yes, for all $t$	(Alexeenko et al., 2016)
Generic smooth initial data	No (finite-time shocks)	(Alexeenko et al., 2016)

7. Applications and Physical Regimes

The SWEs are the principal PDE system for tsunami modeling, flood prediction, atmospheric large-scale flow, and industrial hydrodynamics, underpinning operational models and high-resolution simulation codes. Their generalizations, including curvature effects, stratification, and multi-layer coupling, enable realistic modeling across scales from laboratory flumes to global ocean–atmosphere dynamics.

Advanced developments, such as the connection between the SWEs and gauge theory, highlight the potential for the emergence of topologically protected modes (e.g., Kelvin waves) and a new perspective on wave phenomena in geophysical fluids (Tong, 2022).

References:

(Alexeenko et al., 2016) Alexeenko, Dontsova, Pelinovsky, "Global solutions to the shallow-water system".
(Antonopoulos et al., 2015) Antonopoulos, Dougalis, "Notes on Galerkin-finite element methods for the Shallow Water equations with characteristic boundary conditions".
(Kounadis et al., 2019) Kounadis, Dougalis, "Galerkin finite element methods for the Shallow Water equations over variable bottom".
(Hu, 2018) Hu, "A simple numerical scheme for the 2D shallow-water system".
(Guermond et al., 2024) Guermond et al, "A high-order explicit Runge-Kutta approximation technique for the Shallow Water Equations".
(Herbin et al., 2019) Herbin et al, "A decoupled staggered scheme for the shallow water equations".
(Herbin et al., 2021) Boutin et al, "A Consistent Quasi-Second Order Staggered Scheme for the Two-Dimensional Shallow Water Equations".
(Zhou et al., 26 May 2025) Guo, Bauerle, "Moment-enhanced shallow water equations for non-slip boundary conditions".
(Ilin, 2017) Ilin, "Shallow-water models for a vibrating fluid".
(Abreu et al., 2022) Bachini et al, "A geometrically intrinsic Lagrangian-Eulerian scheme for 2D Shallow Water Equations with variable topography and discontinuous data".
(Fent et al., 2017) Fent et al, "Modeling Shallow Water Flows on General Terrains".
(Hartney et al., 2024) Hartney et al, "Exploring forms of the moist shallow water equations using a new compatible finite element discretisation".
(Vila et al., 2019) Bouchut et al, "Augmented Skew-Symetric System for Shallow-Water System with Surface Tension Allowing Large Gradient of Density".
(Tong, 2022) Tong, "A Gauge Theory for Shallow Water".
(Hosseini et al., 10 May 2025) Coretti, "Kinetic framework with consistent hydrodynamics for shallow water equations".