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A Hyperbolic Moment Based Shallow Water Model for Coupled Bedload Suspended Load Morphodynamics with Variable Density

This paper introduces the HSWEMED model, a hyperbolic shallow water framework that couples variable-density momentum, suspended-load, and bedload transport equations with erosion and deposition to provide a mathematically well-posed and accurate tool for morphodynamic simulations.

Original authors: Afroja Parvin, Giovanni Samaey, Julian Koellermeier

Published 2026-04-21

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CC BY 4.0

Original authors: Afroja Parvin, Giovanni Samaey, Julian Koellermeier

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 a river not just as a flat sheet of water flowing over a flat floor, but as a complex, living system where the water, the sand, and the riverbed itself are all dancing together.

This paper introduces a new, smarter way to predict how that dance happens. The authors, Afroja Parvin and her team, have built a mathematical model called HSWEMED. Let's break down what that means using some everyday analogies.

The Problem: The "Flat" View vs. Reality

For a long time, scientists used "Shallow Water Models" to predict floods and river changes. Think of these old models like looking at a river from a drone flying high above. They see the water as a single, flat layer. They assume the water moves at the same speed from the surface all the way down to the muddy bottom.

The Flaw: In reality, water near the bottom is slow and sticky because of friction with the riverbed, while water at the surface is fast and free.

Why it matters: If you want to know how much sand gets washed away (erosion) or where it settles (deposition), you need to know exactly how fast the water is moving right at the bottom. If your model thinks the bottom water is moving as fast as the surface water, it will guess that way too much sand is being washed away.

The Solution: The "Moment" Approach

The authors decided to stop treating the water as a flat sheet. Instead, they used a technique called the "Moment Method."

The Analogy: Imagine the river's vertical speed profile as a musical chord.

Old Model (SWEED): Only hears the "root note" (the average speed).
New Model (HSWEMED): Hears the root note plus the harmonics. It uses math (Legendre polynomials) to reconstruct the shape of the speed curve from top to bottom. It knows the water is slow at the bottom and fast at the top.

This allows the model to calculate the bottom velocity much more accurately, which is the key to knowing exactly how much sand gets kicked up.

The New Ingredients: Mixing Sand and Water

The model doesn't just look at water speed; it treats the river as a mixture of water and sand.

Variable Density: When you stir sand into water, the mixture gets heavier. The model accounts for this. A river full of sand is "thicker" and heavier than clear water, which changes how it flows.
The Two Types of Sand:
- Bedload: Heavy rocks and pebbles that roll or hop along the bottom (like bowling balls on a lane).
- Suspended Load: Fine sand and silt that float in the water column (like dust in a sunbeam).
- The model tracks both. It knows when sand is being eroded (kicked up) and when it's being deposited (settling down).

The "Hyperbolic" Safety Net

Mathematical models can sometimes go crazy and produce impossible results (like negative water depth or infinite speeds) if the equations get too complicated. This is called losing "hyperbolicity."

The Analogy: Imagine driving a car. If you turn the steering wheel too fast, the car might spin out of control.

The authors added a "safety mechanism" (hyperbolic regularization) to their model. It's like a stability control system in a car. It ensures that even when the river is chaotic (like a dam breaking), the math stays stable, the numbers stay real, and the simulation doesn't crash.

The Test: The Dam Break

To prove their model works, they simulated a dam break.

The Scenario: A dam holding back water suddenly bursts. A massive wave rushes out, hitting a riverbed made of either light plastic pellets (PVC) or heavy sand.
The Result:
- The Old Models either washed away too much sand (because they thought the bottom water was too fast) or didn't wash away enough (because they ignored the sand settling).
- The New Model (HSWEMED) matched real-life experiments much better. It correctly predicted how far the wave went, how deep the water got, and exactly how the riverbed changed shape.

Why This Matters

This isn't just about math; it's about safety and engineering.

Flood Control: Better predictions mean better flood defenses.
River Management: We can better predict how dams affect riverbeds downstream.
Coastal Protection: Understanding how sand moves helps us protect coastlines from erosion.

In a nutshell: The authors built a "smart river simulator" that doesn't just see the water as a flat sheet. It understands the vertical layers, the weight of the sand, and the friction at the bottom, all while keeping the math stable enough to run on a computer. It's a significant upgrade from the "flat map" view to a "3D movie" view of how rivers really work.

1. Problem Statement

Sediment transport in shallow water environments (rivers, estuaries, coastal zones) involves complex interactions between fluid hydrodynamics and bed morphology. Existing models face significant limitations:

Classical Shallow Water Exner (SWE) Models: These rely on depth-averaged velocities and empirical friction laws (e.g., Manning, Chézy). They fail to resolve the vertical velocity structure, which is critical for accurately estimating near-bed shear stress, erosion rates, and bedload transport.
Standard Moment Models: While Shallow Water Moment (SWM) models use polynomial expansions (Legendre polynomials) to resolve vertical velocity profiles, they often lack hyperbolicity at higher orders ( $N \geq 2$ ), leading to numerical instabilities. Furthermore, previous hyperbolic extensions (HSWEM) neglected suspended load, erosion/deposition processes, and variable fluid density caused by sediment concentration.
Coupling Gap: There is no existing hyperbolic framework that simultaneously simulates bedload and suspended load transport while accounting for erosion/deposition and the resulting variable mixture density.

2. Methodology

The authors derive a new coupled model, the Hyperbolic Shallow Water Exner Moment model with Erosion and Deposition (HSWEMED), starting from the incompressible Navier-Stokes equations for a water-sediment mixture.

A. Governing Equations Derivation

Hydrodynamics: The model employs a moment-based approach. The horizontal velocity $u(t,x,z)$ $u (t, x, z)$ is expanded using Legendre polynomials in the vertical coordinate $\zeta \in [0,1]$ $ζ \in [0, 1]$ .
- $u(t,x,\zeta) = u_m(t,x) + \sum_{j=1}^N \alpha_j(t,x)\phi_j(\zeta)$ , where $u_m$ is the depth-averaged velocity and $\alpha_j$ are the moments.
- The system is derived via Galerkin projection of the Navier-Stokes equations, resulting in evolution equations for water depth ( $h$ ), discharge ( $hu_m$ ), and moments ( $h\alpha_j$ ).
Variable Density: The mixture density $\rho$ is treated as a function of the depth-averaged sediment concentration $c_m$ : $\rho = \rho_w + c_m(\rho_s - \rho_w)$ . This density variation introduces baroclinic forcing terms in the momentum equations.
Suspended Load: A depth-averaged transport equation for sediment concentration ( $c_m$ ) is introduced. To close the erosion-deposition flux, the authors reconstruct a quadratic vertical concentration profile consistent with the velocity moments. This allows the calculation of the near-bed concentration ( $c_b$ ) required for erosion/deposition laws without solving a full 3D concentration equation.
Bedload and Morphology: The Exner equation models bed evolution ( $h_b$ ). The sediment discharge ( $Q_b$ ) is calculated using the Meyer-Peter & Müller formula, driven by the bottom velocity ( $u_b = u_m + \sum \alpha_j$ ) rather than the depth-averaged velocity.
Source Terms: The system includes source terms for:
- Erosion ( $E$ ) and Deposition ( $D$ ): Coupling the water column and the bed.
- Friction: Non-linear Manning friction at the bed and Newtonian friction within the fluid.

B. Hyperbolic Regularization

A major theoretical hurdle is that high-order moment systems often lose hyperbolicity (complex eigenvalues). The authors apply a hyperbolic regularization technique (based on [36]):

They modify the transport matrix by forcing higher-order moments ( $\alpha_2, \dots, \alpha_N$ ) to zero in the sub-matrix governing the fluid transport, while retaining the full non-linear coupling in the source terms.
This ensures the system remains strictly hyperbolic (real eigenvalues) for any order $N$ , guaranteeing numerical stability.

C. Numerical Scheme

The model is solved using a path-conservative finite-volume scheme based on a Lax–Friedrichs flux.
Simulations are performed on fine meshes ( $N_x \approx 1000-1200$ ) to control numerical diffusion, as the focus is on model structure rather than high-order numerical reconstruction.

3. Key Contributions

Novel Coupled Framework (HSWEMED): The first hyperbolic shallow water moment model that simultaneously couples bedload and suspended load transport with erosion/deposition dynamics and variable mixture density.
Mathematical Rigor:
- Proof of hyperbolicity for the regularized system via characteristic polynomial analysis.
- Derivation of dissipative energy balance laws for lower-order models (SWEMED0 and SWEMED1), proving the system satisfies physical energy dissipation.
Physical Fidelity:
- Explicit calculation of near-bed velocity and shear stress using velocity moments, eliminating the need for empirical friction profiles to estimate bed shear.
- Dynamic feedback of suspended sediment concentration on hydrodynamics via variable density (buoyancy effects).
- Algebraic reconstruction of the vertical concentration profile to determine near-bed concentration for erosion/deposition closures.

4. Results

The model was validated against laboratory experiments (from [50]) involving dam-break flows over erodible beds with two materials: PVC pellets (low density) and uniform coarse sand (high density).

Accuracy vs. SWEED (Standard Model): HSWEMED significantly outperformed the standard SWEED model. The SWEED overestimated erosion because it used depth-averaged velocity (which is higher than the actual near-bed velocity). HSWEMED, using the reconstructed bottom velocity, predicted more realistic erosion rates and bed evolution.
Accuracy vs. HSWEM (No Erosion/Deposition): HSWEMED captured suspended sediment concentration and bed changes that HSWEM could not, as HSWEM neglects the mass exchange between the bed and the flow.
Material Dependence:
- PVC Pellets: Showed significant differences between models due to lower density and slower settling velocity, leading to higher suspended concentrations. HSWEMED accurately captured the broader concentration profiles.
- Sand: Differences between models were smaller due to higher density and faster settling, but HSWEMED still provided superior accuracy in bed evolution, particularly when friction coefficients were increased.
Wave Propagation: The model correctly captured wavefront speeds and hydraulic jumps, with characteristic speeds varying appropriately based on sediment density and concentration.

5. Significance

Theoretical Robustness: By ensuring hyperbolicity, the model provides a mathematically well-posed framework for complex morphodynamic simulations, avoiding the numerical instabilities common in high-order moment models.
Computational Efficiency: It offers a middle ground between computationally expensive 3D simulations and inaccurate depth-averaged models. It captures vertical velocity structures and density effects at a 1D cost.
Practical Application: The model is particularly valuable for scenarios involving non-equilibrium sediment transport, wet/dry fronts, and variable density flows (e.g., turbidity currents, flash floods over erodible beds), where classical models fail to predict accurate erosion and deposition patterns.
Future Direction: The work establishes a foundation for future development of well-balanced, positivity-preserving schemes and higher-order reconstructions for sediment concentration.