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General function bases moment models for suspended sediment with vertical profile of concentration
Celia Caballero-Cárdenas, José Garres-Díaz, Irene Gómez-Bueno, Tomás Morales de Luna
TL;DR
Standard shallow-water models lose vertical variation, although suspended sediment can have sharply stratified concentration profiles. The paper derives a hydrostatic shallow-moment framework using adaptable, orthogonalized bases and tests continuous piecewise-linear bases. The reported profiles improve over Legendre polynomial reconstructions, particularly for sedimentation with sharp concentration gradients.
Problem
Standard shallow-water models assume vertically uniform velocity, while suspended sediment can exhibit sharply varying vertical concentration profiles.
Source: §1 · p. 1Method
The paper derives a hydrostatic shallow-moment model with numerically orthogonalized arbitrary bases, allowing velocity and concentration to use different spaces, and explores continuous piecewise-linear bases.
Source: §1 · p. 2Results
P1 bases give accurate boundary-layer velocity approximations with fewer modes and qualitatively accurate sedimentation concentration profiles, unlike oscillatory Legendre reconstructions.
Source: §1 · p. 2, §6 · p. 22Takeaways & Limitations
Non-uniform P1 bases and separate approximation spaces let the vertical discretization reflect the differing profile structures of velocity and concentration.
Source: §1 · p. 2, §6 · p. 22The shallow moment model is not expected to be hyperbolic in general for N > 1.
Source: §4 · p. 10
Abstract
from arXiv · showhide
We introduce a general hydrostatic shallow moment framework for suspended sediment transport with vertically varying horizontal velocity and concentration profiles. The formulation is constructed over arbitrary vertical functional bases that are numerically orthogonalised via the Gram-Schmidt algorithm. This approach allows the approximation spaces for the vertical profile to be adapted to the specific structure of each physical variable, where also the velocity and sediment concentration can be projected onto different functional spaces. To overcome the limitations due to the inherent oscillatory nature of global Legendre polynomials, we explore a specific continuous piecewise linear $\mathbb{P}_1$ finite-element basis. Furthermore, non-uniform vertical partitions are considered, enabling an enhanced representation of the vertical profiles in specific regions. Numerical experiments involving boundary layers and suspended sediment transport demonstrate that the proposed framework is accurate to reproduce the expected vertical profiles. The improvement is particularly significant for sediment concentration profiles with sharp vertical variations, where polynomial reconstructions exhibit pronounced oscillations leading to physically meaningless profiles, whereas the $\mathbb{P}_1$ bases produce a qualitatively accurate representation of the vertical profiles observed in the sedimentation processes.
1 Introduction
Standard shallow-water models lose vertical structure, while suspended sediment can form sharply varying concentration profiles. The paper derives a flexible shallow-moment framework and tests piecewise-linear bases that improve the representation of these profiles.
1 Introduction
- Standard shallow-water models assume vertically uniform horizontal velocity, and classical sediment models often assume uniform concentration despite strong vertical stratification.The paper identifies bottom-concentrated sediment as a setting where the velocity profile also departs substantially from uniformity.Source: §1 · p. 1
- Moment models are more computationally efficient than multilayer methods, while multilayer methods offer greater flexibility for arbitrary profiles.Global polynomials can also oscillate when representing nearly constant regions, such as boundary-layer velocity profiles.Source: §1 · p. 2
- Prior work leaves the derivation of a shallow-moment model with vertically stratified suspended-sediment concentration unaddressed.The paper distinguishes this gap from existing multilayer sediment models and moment models for bedload transport.Source: §1 · p. 2
- The framework derives a hydrostatic shallow-moment model that expands velocity and sediment concentration in arbitrary orthogonalized bases, which may differ by variable.Gram-Schmidt orthogonalization is applied numerically after selecting the basis.Source: §1 · p. 2
- P1 bases and non-uniform vertical partitions improve the represented velocity and concentration profiles over polynomial bases, especially for sedimentation.The paper reports qualitatively accurate sedimentation profiles using P1 where polynomial reconstructions oscillate.Source: §1 · p. 2
2 Derivation of a hydrostatic moment model for a general functional basis
The derivation begins with a hydrostatic fluid–sediment mixture model and transforms it to a fixed vertical coordinate. It then projects the governing equations onto a numerically orthogonalized functional basis to obtain evolution equations for depth, velocity modes, and concentration modes.
2.1 Governing equations
- The derivation treats a single sediment species in one horizontal dimension and adopts the Boussinesq approximation and hydrostatic pressure.It assumes no horizontal slip between phases and represents their vertical slip with an empirical settling law whose effective settling speed decreases with concentration.Source: §2 · p. 3, §2.1 · p. 3, Closure relation for the vertical slip · p. 4
- The starting system comprises mixture incompressibility, phase continuity, and momentum equations, with concentration evolving under vertical slip and settling.The pressure is hydrostatic, and the settling law accounts for particle interactions through a concentration-dependent effective settling velocity.Source: §2.1 · p. 3, Closure relation for the vertical slip · p. 4
2.2 Moment approach in orthogonal bases
- The moment approach multiplies the transformed governing equations by basis functions and integrates vertically to evolve the velocity and concentration moments.The basis is chosen to include the constant function and is orthogonalized with Gram-Schmidt; orthogonality avoids coefficient-matrix inversion.Source: §2.2 · p. 4, §2.2.1 · p. 5, §2.2.2 · pp. 5–6
2.2.1 σ-coordinates
- The σ-coordinate change maps the moving fluid column to the fixed interval ξ ∈ [0, 1].The mapped vertical coordinate is defined through the depth and bottom topography.Source: §2.2.1 · p. 4
2.2.2 Integration procedure
- For an arbitrary basis, the model derives moment equations by vertical integration, with the constant basis mode recovering the depth average.The free-surface pressure gradient affects only the mean velocity, while baroclinic interactions couple sediment concentration to the vertical velocity distribution.Source: §2.2.1 · p. 5, §2.2.2 · pp. 5–6
3 Final hydrostatic shallow moment model
The final model accommodates arbitrary orthogonal bases and can use different approximation spaces for velocity and concentration. Its P1 option uses a potentially non-uniform vertical partition to better represent sharp profiles, while the asymmetric formulation allows more modes for concentration than for velocity.
3 Final hydrostatic shallow moment model
- The final hydrostatic shallow-moment system is formulated for an arbitrary orthogonal basis whose first function is constant.The derivation accommodates evolution equations for the moment modes of the model variables.Source: §3 · p. 7
3.1 Asymmetric Formulation (N̸ = M)
- The asymmetric formulation permits concentration to use more modes than velocity, matching the expectation that concentration profiles can require a richer approximation space.A shared basis lets the required coefficients be computed once over the maximum index range.Source: §3.1 · pp. 7–9
3.2 A particular basis: P1 functions
- Global polynomial bases can oscillate when representing sharp gradients or nearly constant zones, motivating the P1 option for sediment concentration profiles.The paper contrasts this limitation with polynomial accuracy for smooth configurations.Source: §3.2 · p. 9
- The P1 basis uses continuous piecewise-linear functions on a possibly non-uniform vertical partition, with nodes clusterable near critical regions such as the bottom boundary layer.Numerical Gram-Schmidt orthogonalization preserves the orthogonal-basis formulation while avoiding the complexity of a non-orthogonal system.Source: §3.2 · p. 9
4 Numerical approximation
The numerical method splits horizontal moment evolution from vertical sedimentation, using approximate wave speeds for the first step and a fine-grid reconstruction–evolution–projection procedure for the second. The full moment model is not generally hyperbolic for N > 1, but the authors report no related numerical issues in their tests.
4 Numerical approximation
- The two-step finite-volume method first solves the system without sedimentation, then updates sedimentation through a vertical reconstruction–evolution–projection algorithm.The first step uses a hyperbolic-system framework with non-conservative products; the second resolves sedimentation dynamics.Source: §4 · p. 10
4.1 Step 1: Hyperbolic system with non-conservative products
- For N > 1, shallow moment systems are not generally hyperbolic, so the model’s eigenvalues cannot be computed exactly and are approximated for the HLL flux.The authors use an approximation based on the Linearized Moment system and incorporate suspended-sediment effects.Source: §4 · p. 10, §4.1 · p. 12
- The eigenvalue approximation recovers exact limits for vertically uniform velocity and concentration, and for linear velocity with constant concentration when A111 = 0.The A111 condition holds for shifted Legendre bases and P1 bases with N = 1; for the stated P1 family, A111 = 0 at N = 1 and tends to zero as N increases.Source: §4.1 · p. 12
4.2 Step 2: Vertical sedimentation
- Sedimentation is updated by reconstructing concentration on a fine vertical grid, transporting it with an upwind scheme, and projecting the result back onto the moment basis.The upwind flux accounts for saturation in the lower receiving cell; zero boundary fluxes keep sediment within the water column, and the decoupled update can be parallelized.Source: §4.2 · pp. 12–13
- The concentration reconstruction is filtered to keep values in [0, cmax], since moment-based profiles may contain inadmissible concentrations.The method uses cmax as a numerical admissibility threshold; the scheme is stated to be stable under the usual CFL restriction, with vertical transport potentially requiring a check for large settling velocities.Source: §4.2 · pp. 12–13
5 Numerical experiments
The numerical experiments assess velocity and sediment-concentration bases in boundary-layer, sedimentation, and coupled-transport tests. They compare how the bases represent vertical structure and report concentration profiles and CPU-cost measurements for P1 and polynomial reconstructions.
5.1 P1-bases and local refinement for boundary-layer profiles
- In this test case, projection error falls from 4.09 × 10^-2 with Legendre and 5.17 × 10^-2 with uniform P1 to 7.14 × 10^-3 with non-uniform P1, enabled by local refinement at the same mode count.Source: §5.1 · p. 16
5.2 Evolution of a boundary-layer velocity profile
- During boundary-layer evolution, non-uniform P1 with N = 3 resolves the near-bottom profile more sharply, comparably to Legendre with N = 6, using four velocity coefficients.Uniform P1 and Legendre with N = 3 behave similarly and have limited resolution near the bottom, although all four approximations produce comparable large-scale dynamics.Source: §5.2 · p. 16, Figure 3 · p. 17, Figure 4 · p. 18
5.3 Vertical sedimentation of an initially uniform concentration
- In the sedimentation test, Legendre reconstructions oscillate at sharp concentration gradients, while non-uniform P1 better localizes the saturated and depleted regions.The reference solution shows sediment settling into a concentrated lower layer and diminishing upper concentration; uniform P1 is less oscillatory than Legendre but diffuses the transition.Source: §5.3 · p. 18, Figure 5 · p. 19
5.4 Coupled velocity–sediment transport over a smooth bump
- Over the smooth bump, P1 velocity with N = 2 captures dominant low-order shear but is restricted, whereas P1 with N = 5 gives richer structure and Legendre with N = 5 captures shear.With M = 5 concentration modes in each simulation, the P1-based concentration solutions remain comparatively localized and regular as sediment is transported and settles.Source: §5.4 · p. 20, Figure 7 · p. 21
- P1-based models produce qualitatively physically correct sediment concentration profiles, while polynomial bases show pronounced oscillations and physically meaningless profiles, especially at intermediate and final times.The comparison is for coupled velocity–sediment transport over a smooth bump.Source: §5.4 · p. 22
- Total CPU time was approximately 2.70 s for P1-2 and 7.05 s for P1-5 (factor about 2.61); sedimentation-update time was approximately 1.88 s and 2.84 s (factor about 1.51).The sedimentation update's share of total CPU time fell from about 69.7% for P1-2 to about 40.3% for P1-5; measurements compared uniform P1-2 and P1-5 bases.Source: §5.4 · p. 22
- At higher moment counts, CPU cost is mainly associated with the hyperbolic coupled moment system, while a coarser grid could reduce sedimentation-update cost; all cases used Nz = 100.The cited cost drivers include evaluation of moment fluxes and non-conservative terms.Source: §5.4 · p. 22
6 Conclusions
The framework combines arbitrary, Gram–Schmidt-orthogonalised vertical bases with continuous piecewise-linear P1 elements and non-uniform partitions. It reproduces boundary-layer velocity profiles and sharply varying sediment concentrations, while allowing velocity and concentration to use different approximation spaces.
6 Conclusions
- The framework expands velocity and concentration in arbitrary functional bases that are numerically orthogonalised with Gram–Schmidt.It also permits non-uniform vertical partitions to add resolution in specific regions.Source: §6 · p. 22
- Piecewise-linear reconstructions give qualitatively accurate sedimentation concentration profiles, whereas global Legendre bases can produce severe oscillations and physically meaningless results.P1 approximations also provide accurate boundary-layer velocity profiles with fewer modes.Source: §6 · p. 22
- Velocity and concentration can be projected onto different subspaces, which is useful because concentration often requires a richer approximation space than velocity.The authors identify application to other physical contexts with complex vertical profiles as future work.Source: §6 · p. 22