Stability Diagrams of Bed Evolution for Vertically Averaged and Moment (VAM) Models

Mohamed Hassan Elgamal () and Mohd Aamir Mumtaz
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Mohamed Hassan Elgamal: Civil Engineering Department, College of Engineering, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
Mohd Aamir Mumtaz: Civil Engineering Department, College of Engineering, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia

Mathematics, 2025, vol. 13, issue 12, 1-33

Abstract: This study presents, for the first time, a detailed linear stability analysis (LSA) of bedform evolution under low-flow conditions using a one-dimensional vertically averaged and moment (1D-VAM) approach. The analysis focuses exclusively on bedload transport. The classical Saint-Venant shallow water equations are extended to incorporate non-hydrostatic pressure terms and a modified moment-based Chézy resistance formulation is adopted that links bed shear stress to both the depth-averaged velocity and its first moment (near-bed velocity). Applying a small-amplitude perturbation analysis to an initially flat bed, while neglecting suspended load and bed slope effects, reveals two distinct modes of morphological instability under low-Froude-number conditions. The first mode, associated with ripple formation, features short wavelengths independent of flow depth, following the relation F 2 = 1 / (kh ), and varies systematically with both the Froude and Shields numbers. The second mode corresponds to dune formation, emerging within a dimensionless wavenumber range of 0.17 to 0.9 as roughness increases and the dimensionless Chézy coefficient C ∗ decreases from 20 to 10. The resulting predictions of the dominant wavenumbers agree well with recent experimental observations. Critically, the model naturally produces a phase lag between sediment transport and bedform geometry without empirical lag terms. The 1D-VAM framework with Exner equation offers a physically consistent and computationally efficient tool for predicting bedform instabilities in erodible channels. This study advances the capability of conventional depth-averaged models to simulate complex bedform evolution processes.

Keywords: stability analysis diagram; evolution of bedforms; low-regime bedforms; vertically averaged and moment models; velocity over varied topography; moment-based Chezy formula (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:12:p:1997-:d:1680837

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