When the water discharge, sediment supply, and base level vary around stable values, an alluvial river evolves toward a mean equilibrium or graded state with small fluctuations around this mean state (i.e., a dynamic or statistical equilibrium state). Here we present analytical relations describing the mean equilibrium geometry of an alluvial river under variable flow by linking channel slope, width, and bed surface texture. The solution holds in river normal flow zones (or outside both the hydrograph boundary layer and the backwater zone) and accounts for grain size selective transport and particle abrasion. We consider the variable flow rate as a series of continuously changing yet steady water discharges (here termed an alternating steady discharge). The analysis also provides a solution to the channel‐forming water discharge, which is here defined as the steady water discharge that, given the mean sediment supply, provides the same equilibrium channel slope as the natural long‐term hydrograph. The channel‐forming water discharge for the gravel load is larger than the one associated with the sand load. The analysis illustrates how the load is distributed over the range of water discharge in the river normal flow zone, which we term the “normal flow load distribution”. The fact that the distribution of the (imposed) sediment supply spatially adapts to this normal flow load distribution is the origin of the hydrograph boundary layer. The results quantify the findings by Wolman and Miller (1960) regarding the relevance of both magnitude and frequency of the flow rate with respect to channel geometry.

An engineered alluvial river (i.e., a fixed‐width channel) has constrained planform but is free to adjust channel slope and bed surface texture. These features are subject to controls: the hydrograph, sediment flux, and downstream base level. If the controls are sustained (or change slowly relative to the timescale of channel response), the channel ultimately achieves an equilibrium (or quasi‐equilibrium) state. For brevity, we use the term “quasi‐equilibrium” as a shorthand for both states. This quasi‐equilibrium state is characterized by quasi‐static and dynamic components, which define the characteristic timescale at which the dynamics of bed level average out. Although analytical models of quasi‐equilibrium channel geometry in quasi‐normal flow segments exist, rapid methods for determining the quasi‐equilibrium geometry in backwater‐dominated segments are still lacking. We show that, irrespective of its dynamics, the bed slope of a backwater or quasi‐normal flow segment can be approximated as quasi‐static (i.e., the static slope approximation). This approximation enables us to derive a rapid numerical space‐marching solution of the quasi‐static component for quasi‐equilibrium channel geometry in both backwater and quasi‐normal flow segments. A space‐marching method means that the solution is found by stepping through space without the necessity of computing the transient phase. An additional numerical time stepping model describes the dynamic component of the quasi‐equilibrium channel geometry. Tests of the two models against a backwater‐Exner model confirm their validity. Our analysis validates previous studies in showing that the flow duration curve determines the quasi‐static equilibrium profile, whereas the flow rate sequence governs the dynamic fluctuations.

[1] Knowledge of hydrological model complexity can aid selection of an optimal prediction model out of a set of available models. Optimal model selection is formalized as selection of the least complex model out of a subset of models that have lower empirical risk. This may be considered equivalent to minimizing an upper bound on prediction error, defined here as the mathematical expectation of empirical risk. In this paper, we derive an upper bound that is free from assumptions on data and underlying process distribution as well as on independence of model predictions over time. We demonstrate that hydrological model complexity, as defined in the presented theoretical framework, plays an important role in determining the upper bound. The model complexity also acts as a stabilizer to a hydrological model selection problem if it is deemed ill-posed. We provide an algorithm for computing complexity of any arbitrary hydrological model. We also demonstrate that hydrological model complexity has a geometric interpretation as the size of model output space. The presented theory is applied to quantify complexities of two hydrological model structures: SAC-SMA and SIXPAR. It detects that SAC-SMA is indeed more complex than SIXPAR. We also develop an algorithm to estimate the upper bound on prediction error, which is applied on five different rainfall-runoff model structures that vary in complexity. We show that a model selection problem is stabilized by regularizing it with model complexity. Complexity regularized model selection yields models that are robust in predicting future but yet unseen data.

The active layer model (Hirano, 1971) is frequently used for modeling mixed‐size sediment river morphodynamic processes. It assumes that all the dynamics of the bed surface are captured by a homogeneous top layer that interacts with the flow. Although successful in reproducing a wide range of phenomena, it has two problems: (1) It may become mathematically ill‐posed, which causes the model to lose its predictive capabilities, and (2) it does not capture dispersion of tracer sediment. We extend the active layer model by accounting for conservation of the sediment in transport and obtain a new model that overcomes the two problems. We analytically assess the model properties and discover an instability mechanism associated with the formation of waves under conditions in which the active layer model is ill‐posed. Numerical simulations using the new model show that it is capable of reproducing two laboratory experiments conducted under conditions in which the active layer model is ill‐posed. The new model captures the formation of waves and mixing due to an increase in active layer thickness. Simulations of tracer dispersion show that the model reproduces reasonably well a laboratory experiment under conditions in which the effect of temporary burial of sediment due to bedforms is negligible. Simulations of a field experiment illustrate that the model does not capture the effect of temporary burial of sediment by bedforms.

When addressing channel response to natural or anthropogenic change of the controls, one needs to distinguish between three channel response phases: the initial, transient, and equilibrium phase (Blom, Arkesteijn, et al., 2017). The initial phase usually only lasts several days or weeks and is here defined as the phase wherein only the flow has time to adjust to the new situation. Adjustment of the river morphodynamic state cannot take place within the initial phase due to its short duration and the typically much longer time scale of morphodynamic adjustment. The equilibrium phase is the stage in which, by definition, the channel has reached the equilibrium or graded state associated with the new specifications of the controls. The transient phase of channel response marks the period in between: it covers the phase of adjustment (with associated downstream or upstream migrating adjustment waves) of channel slope, channel width, and bed surface texture after the initial phase has been completed and before the equilibrium phase has started. The transient phase may last several years, decades, or even centuries. Although not considered in this analysis, the initial phase can be accompanied by sudden morphodynamic change such as a flood-induced avulsion.The time scales of morphodynamic channel response reflect how fast a reach responds in the transient phase regarding adjustment of channel slope, channel width, and bed surface texture to changes in the controls of equilibrium channel geometry (