Sustainable river management often requires long-term morphological simulations. As the future is unknown, uncertainty needs to be accounted for, which may require probabilistic simulations covering a large parameter domain. Even for one-dimensional models, the simulation times can be long. One of the strategies to speed up simulations is simplification of models by neglecting terms in the governing hydrodynamic equations. Examples are the quasi-steady model and the diffusive wave model, both widely used by scientists and practitioners. Here, we establish under which conditions these simplified models are accurate. Based on the results of linear stability analyses of the St. Venant-Exner equations, we assess migration celerities and damping of infinitesimal, but long riverbed perturbations. We did this for the full dynamic model, i.e. no terms neglected, as well as for the simplified models. The accuracy of the simplified models was obtained from comparison between the characteristics of the riverbed perturbations for simplified models and the full dynamic model. We executed a spatial-mode and a temporal-mode linear analysis and compared the results with numerical modelling results for the full dynamic and simplified models. The numerical results match best with the temporal-mode linear stability analysis. The analysis shows that the quasi-steady model is highly accurate for Froude numbers up to 0.7, probably even for long river reaches with large flood wave damping. Although the diffusive wave model accurately predicts flood wave migration and damping, key morphological metrics deviate more than 5% (10%) from the full dynamic model when Froude numbers exceed 0.2 (0.3).
Sustainable river management can be supported by models predicting long-term morphological developments. Even for one-dimensional morphological models, run times can be up to several days for simulations over multiple decades. Alternatively, analytical tools yield metrics that allow estimation of migration celerity and damping of bed waves, which have potential for being used as rapid assessment tools to explore future morphological developments. We evaluate the use of analytical relations based on linear stability analyses of the St. Venant-Exner equations, which apply to bed waves with spatial scales much larger than the water depth. With a one-dimensional numerical morphological model, we assess the validity range of the analytical approach. The comparison shows that the propagation of small bed perturbations is well-described by the analytical approach. For Froude numbers over 0.3, diffusion becomes important and bed perturbation celerities reduce in time. A spatial-mode linear stability analysis predicts an upper limit for the bed perturbation celerity. For longer and higher bed perturbations, the dimensions relative to the water depth and the backwater curve length determine whether the analytical approach yields realistic results. For higher bed wave amplitudes, non-linearity becomes important. For Froude numbers ≤0.3, the celerity of bed waves is increasingly underestimated by the analytical approach. The degree of underestimation is proportional to the ratio of bed wave amplitude to water depth and the Froude number. For Froude numbers exceeding 0.3, the net impact on the celerity depends on the balance between the decrease due to damping and the increase due to non-linear interaction.