Patricio Bohorquez

Stochastic interpretation of the advection‐diffusion equation and its relevance to bed load transport

The advection diffusion equation is one of the most widespread equations in physics. It arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Phenomenological laws are usually sufficient to derive this equation and interpret its terms. Stochastic models can also be used to derive it with the significant advantage that they provide information on the statistical properties of particle activity. These models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. Among these stochastic models, the most common approach consists of random walk models. For instance, they have been used to model the random displacement of tracers in river streams. Here we explore an alternative approach, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. Birth-death Markov processes are well-suited to this objective. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received no attention. We therefore look into the possibility of deriving the advection diffusion equation (with a source term) within the framework of birth-death Markov processes. We show that in the continuum limit (when the cell size becomes vanishingly small), we can derive an advection diffusion equation for particle activity. Yet, while this derivation is formally valid in the continuum limit, it runs into difficulty in the practical applications involving cells or meshes of finite length. Indeed, within our stochastic framework, particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particles velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due to velocity fluctuations), with the important consequence that local measurements depend on both the intrinsic properties of particles displacement and the dimensions of the measurement system.

Dynamic simulation of catastrophic late Pleistocene glacial-lake drainage, Altai Mountains, central Asia

ABSTRACT Numerical simulations of the catastrophic draining of Pleistocene glacial-lake Kuray–Chuja quantify the discharge history of the draining event in detail. The plan-view basin flows are modelled as water emptied due to the instantaneous failure of the impounding ice-dam when the lake was at maximum capacity. The Chuja Basin water exited as a jet-flow into the Kuray Basin via a narrow conjoining valley. The peak discharge from the Chuja Basin is determined to be 1.20 × 107 m3 s−1, and the peak discharge (3.19 × 107 m3 s−1 > Q ≤ 2.0 × 107 m3 s−1) that flowed from the Kuray Basin at the failed impoundment is also calculated for two limiting conditions. The variations in lake volume and depth indicate complete drainage within 50 h. In both basins, fields of relict gravel bedforms reflect sediment transport due to entrained lake-bed sediments. Thus, in addition to the general overview of drainage, the detailed temporal and spatial evolutions of drainage parameters are reported, including for the locations of the bedform fields. Local flow above the bedforms is considered in relation to thresholds for sediment motion, bedform development, and orientations. Within the simple bathymetry of the Chuja Basin, the flow field was fairly uniform with flow conducive to bedform evolution only occurring close to the exit from the basin, which accords with field evidence. In contrast, within the Kuray Basin, the flow responded sensitively to the complex bathymetry, which included rapid changes in flow direction due to interaction of the Kuray water with the jet-flow from Chuja, and as submerged ridges shoaled. Thus the Kuray flow field was complex but with time-dependent flow conditions in accordance with bedform development. It is concluded that the location of the bedforms can be explained in terms of the flow modelling and suggestions are made as to how future drainage models might be improved.

Competition between kinematic and dynamic waves in floods on steep slopes

We present a theoretical stability analysis of the flow after the sudden release of a fixed mass of fluid on an inclined plane formally restricted to relatively long time scales, for which the kinematic regime is valid. Shallow-water equations for steep slopes with bed stress are employed to study the threshold for the onset of roll waves. An asymptotic solution for long-wave perturbations of small amplitude is found on background flows with a Froude number value of 2. Small disturbances are stable under this condition, with a linear decay rate independent of the wavelength and with a wavelength that increases linearly with time. For larger values or the Froude number it is shown that the basic flow moves at a different scale than the perturbations, and hence the wavelength of the unstable modes is characterized as a function of the plane-parallel Froude number Fr-p and a measure of the local slope of the free-surface height phi by means of a multiple-scale analysis in space and time. The linear stability results obtained in the presence of small non-uniformities in the flow, phi > 0, introduce substantial differences with respect to the plane-parallel flow with phi = 0. In particular, we find that instabilities do not occur at Froude numbers Fr-cr much larger than the critical value 2 of the parallel case for some wavelength ranges. These results differ from that previously reported by Lighthill & Whitham (Proc. R. Soc. A, vol. 229, 1955, pp. 281-345), because of the fundamental role that the non-parallel, time-dependent characteristics of the kinematic-wave play in the behaviour of small disturbances, which was neglected in their stability analyses. The present work concludes with supporting numerical simulations of the evolution of small disturbances, within the framework of the frictional shallow-water equations, that are superimposed on a base state which is essentially a kinematic wave, complementing the asymptotic theory relevant near the onset. The numerical simulations corroborate the cutoff in wavelength for the spectrum that stabilizes the tall of the darn-break flood.

Entrainment, motion, and deposition of coarse particles transported by water over a sloping mobile bed

In gravel bed rivers, bed load transport exhibits considerable variability in time and space. Recently, stochastic bed load transport theories have been developed to address the mechanisms and effects of bed load transport fluctuations. Stochastic models involve parameters such as particle diffusivity, entrainment, and deposition rates. The lack of hard information on how these parameters vary with flow conditions is a clear impediment to their application to real-world scenarios. In this paper, we determined the closure equations for the above parameters from laboratory experiments. We focused on shallow supercritical flow on a sloping mobile bed in straight channels, a setting that was representative of flow conditions in mountain rivers. Experiments were run at low sediment transport rates under steady nonuniform flow conditions (i.e., the water discharge was kept constant, but bed forms developed and migrated upstream, making flow nonuniform). Using image processing, we reconstructed particle paths to deduce the particle velocity and its probability distribution, particle diffusivity, and rates of deposition and entrainment. We found that on average, particle acceleration, velocity, and deposition rate were responsive to local flow conditions, whereas entrainment rate depended strongly on local bed activity. Particle diffusivity varied linearly with the depth-averaged flow velocity. The empirical probability distribution of particle velocity was well approximated by a Gaussian distribution when all particle positions were considered together. In contrast, the particles located in close vicinity to the bed had exponentially distributed velocities. Our experimental results provide closure equations for stochastic or deterministic bed load transport models.

Hydrodynamic Instabilities in Well-Balanced Finite Volume Schemes for Frictional Shallow Water Equations. The Kinematic Wave Case

We report the developments of hydrodynamic instabilities in several well-balanced finite volume schemes that are observed during the computation of the temporal evolution of an out-balance flow which is essentially a kinematic wave. The numerical simulations are based on the one-dimensional shallow-water equations for a uniformly sloping bed with hydraulic resistance. Subsequently, we highlight the need of low dissipative high-order well-balanced filter schemes for non-equilibrium flows with variable cut-off wavenumber to compute the out-balance flow under consideration, i.e. the kinematic wave.

Analysis of Sediment Transport from Recorded Signals of Sediments in a Gravel-Bed River: Role of Sediment Availability

Geophones were used as a proxy for determining the behavior of sediment transport in boulder mountain river. Field observations aggregated by a statistical analysis (PCA) allowed to identifying periods in which the behaviour between the water discharge (Qw), the sediment flux (Qs) and the meteorologic conditions was correlated. Within each period a relationship between Qs and Qw could be found and summarized by two parameters: the critical drag discharge for motion initiation (Qd) and the mean transport rate (Tmean). The coefficients of determination were close to 0.8, which showed the consistency of the linear relationship. The variability in the relationship Qs ~ Qw could be related to the availability to motion of sediments. In contrast with laboratory experiments, in which sediment storage is indefinitely large, natural systems can suffer from sediment shortage under certain conditions. This may explain why bedload transport equations determined from laboratory experiments significantly overestimate the bedload transport rate.

100 Years of Competition between Reduction in Channel Capacity and Streamflow during Floods in the Guadalquivir River (Southern Spain)

Reduction in channel capacity can trigger an increase in flood hazard over time. It represents a geomorphic driver that competes against its hydrologic counterpart where streamflow decreases. We show that this situation arose in the Guadalquivir River (Southern Spain) after impoundment. We identify the physical parameters that raised flood hazard in the period 1997–2013 with respect to past years 1910–1996 and quantify their effects by accounting for temporal trends in both streamflow and channel capacity. First, we collect historical hydrological data to lengthen records of extreme flooding events since 1910. Next, inundated areas and grade lines across a 70 km stretch of up to 2 km wide floodplain are delimited from Landsat and TerraSAR-X satellite images of the most recent floods (2009–2013). Flooded areas are also computed using standard two-dimensional Saint-Venant equations. Simulated stages are verified locally and across the whole domain with collected hydrological data and satellite images, respectively. The thoughtful analysis of flooding and geomorphic dynamics over multi-decadal timescales illustrates that non-stationary channel adaptation to river impoundment decreased channel capacity and increased flood hazard. Previous to channel squeezing and pre-vegetation encroachment, river discharges as high as 1450 m3·s−1 (the year 1924) were required to inundate the same areas as the 790 m3·s−1 streamflow for recent floods (the year 2010). We conclude that future projections of one-in-a-century river floods need to include geomorphic drivers as they compete with the reduction of peak discharges under the current climate change scenario.

Discussion of "Computing Nonhydrostatic Shallow-Water Flow over Steep Terrain"

The authors are to be commended for their valuable and unique comparison of experimental data for clear-water dam-break floods on steep slopes and for their nonhydrostatic shallow-water model. The dam-break experiments down a steep slope at the USGS outdoor laboratory Logan and Iverson 2007 mimic the behavior of natural flows and shed light on different mechanisms inertia, viscous dissipation, pressure gradient that control the flow dynamics. The discusser has studied roll-wave development in dam breaks and similar problems on steep slopes restricted to relatively long timescales, for which the kinematic regime is reached, and the authors’ fine results present a great opportunity to consider some interesting features of the dam-break wave. It is understood that there is a lack of validation of asymptotic solutions for dam-break waves on steep slopes based on the kinematic wave approximation Lighthill and Whitham 1955 with both experimental data and numerical simulations. Even the earlier solutions by Hunt 1982, 1984 and Weir 1983 for shallow slopes have not yet been validated. As discussed by Hunt 1984: “A comparison with experiment shows good qualitative agreement, but more comparisons with experiment should be made in order to assess the qualitative accuracy of the solution.” This question remains still unresolved e.g., Singh 2002, and references therein. To address this point, the value of enhanced gravity g must