Predicting sediment transport by interrill overland flow on rough surfaces

Modelling soil erosion requires an equation for predicting the sediment transport capacity by interrill overland flow on rough surfaces. The conventional practice of partitioning total shear stress into grain and form shear stress and predicting transport capacity using grain shear stress lacks rigour and is prone to underestimation. This study therefore explores the possibility that inasmuch as surface roughness affects flow hydraulic variables which, in turn, determine transport capacity, there may be one or more hydraulic variables which capture the effect of surface roughness on transport capacity suffciently well for good predictions of transport capacity to be achieved from data on these variables alone. To investigate this possibility, regression analyses were performed on data from 1506 flume experiments in which discharge, slope, water temperature, rainfall intensity, and roughness size, shape and concentration were varied. The analyses reveal that 89·8 per cent of the variance in transport capacity can be accounted for by excess flow power and flow depth. Including roughness size and concentration in the regression improves that explained variance by only 3·5 per cent. Evidently, flow depth, when used in combination with excess flow power, largely captures the effect of surface roughness on transport capacity. This finding promises to simplify greatly the task of developing a general sediment equation for interrill overland flow on rough surfaces. Copyright © 1998 John Wiley & Sons, Ltd.     

Evolution of overland flow connectivity in bare agricultural plots

Soil surface roughness not only delays overland flow generation but also strongly affects the spatial distribution and concentration of overland flow. Previous studies generally aimed at predicting the delay in overland flow generation by means of a single parameter characterizing soil roughness. However, little work has been done to find a link between soil roughness and overland flow dynamics. This is made difficult because soil roughness and hence overland flow characteristics evolve differently depending on whether diffuse or concentrated erosion dominates. The present study examined whether the concept of connectivity can be used to link roughness characteristics to overland flow dynamics. For this purpose, soil roughness of three 30‐m² tilled plots exposed to natural rainfall was monitored for two years. Soil micro‐topography was characterized by means of photogrammetry on a monthly basis. Soil roughness was characterized by the variogram, the surface stream network was characterized by network‐based indices and overland flow connectivity was characterized by Relative Surface Connection function (RSCf) functional connectivity indicator. Overland flow hydrographs were generated by means of a physically‐based overland flow model based on 1‐cm resolution digital elevation models. The development of eroded flow paths at the soil surface not only reduced the delay in overland flow generation but also resulted in a higher continuity of high flow velocity paths, an increase in erosive energy and a higher rate of increase of the overland flow hydrograph. Overland flow dynamics were found to be highly correlated to the RSCf characteristic points. By providing information regarding overland flow dynamics, the RSCf may thus serve as a quantitative link between soil roughness and overland flow generation in order to improve the overland flow hydrograph prediction. Copyright © 2016 John Wiley & Sons, Ltd.     

Effects of rainfall,overland flow and their interactions on peatland interrill erosion processes

Interrill erosion processes on gentle slopes are affected by mechanisms of raindrop impact, overland flow and their interaction. However, limited experimental work has been conducted to understand how important each of the mechanisms are and how they interact, in particular for peat soil. Laboratory simulation experiments were conducted on peat blocks under two slopes (2.5° and 7.5°) and three treatments: Rainfall, where rainfall with an intensity of 12 mm h^?1 was simulated; Inflow, where upslope overland flow at a rate of 12 mm h^?1 was applied; and Rainfall + Inflow which combined both Rainfall and Inflow. Overland flow, sediment loss and overland flow velocity data were collected and splash cups were used to measure the mass of sediment detached by raindrops. Raindrop impact was found to reduce overland flow by 10 to 13%, due to increased infiltration, and reduce erosion by 47% on average for both slope gradients. Raindrop impact also reduced flow velocity (80–92%) and increased roughness (72–78%). The interaction between rainfall and flow was found to significantly reduce sediment concentrations (73–85%). Slope gradient had only a minor effect on overland flow and sediment yield. Significantly higher flow velocities and sediment yields were observed under the Rainfall + Inflow treatment compared to the Rainfall treatment. On average, upslope inflow was found to increase erosion by 36%. These results indicate that overland flow and erosion processes on peat hillslopes are affected by upslope inflow. There was no significant relationship between interrill erosion and overland flow, whereas stream power had a strong relationship with erosion. These findings help improve our understanding of the importance of interrill erosion processes on peat. Copyright © 2017 John Wiley & Sons, Ltd.     

Interstorm surface preparation and sediment detachment by vehicle traffic on unpaved mountain roads

Road survey and field rainfall simulation experiments have shown that the erodibility of a road surface is dynamic. In the absence of extreme runoff events, dynamic erodibility results from the generation and removal of easily entrained surface material by human road surface maintenance activities, vehicular detachment and overland flow events. Maintenance activities introduce easily transportable material to the road surface where it can be entrained by overland flow. Traffic in dry conditions detaches material that is quickly removed during subsequent overland flow events. The pre‐storm erodibility of a road is therefore largely a function of maintenance and vehicle activity since the last overland flow event. During rainstorms, vehicle passes increase sediment production by detaching/redistributing surface material and creating efficient overland flow pathways for sediment transport. However, if incision of tracks by overland flow does not occur, post‐pass sediment transport quickly returns to pre‐pass rates. Field rainfall simulation data suggest that sediment transport resulting from during‐storm vehicle passes is greatly influenced by the presence of existing loose material, which again is a function of prior road usage and maintenance activities. Incorporation of vehicular effects into physically based road erosion models may be possible by parameterizing both during‐storm and inter‐storm changes in the supply of loose surface material as changes in surface erodibility. Copyright © 2001 John Wiley & Sons, Ltd.     

Terrestrial laser scanning soil surfaces: a field methodology to examine soil surface roughness and overland flow hydraulics

Conventional roughness–resistance relationships developed for pipe and open‐channel flows cannot accurately describe shallow overland flows over natural rough surfaces. This paper develops a new field methodology combining terrestrial laser scanning (TLS) and overland flow simulation to provide a high‐resolution dataset of surface roughness and overland flow hydraulics as simulated on natural bare soil surfaces. This method permits a close examination of the factors controlling flow velocity and a re‐evaluation of the relationship between surface roughness and flow resistance. The aggregate effect of flow dynamics, infiltration and depression storage on retarding the passage of water over a surface is important where runoff‐generating areas are distant from well‐defined channels. Experiments to separate these effects show that this ‘effective resistance’ is dominated by surface roughness. Eight measurements of surface roughness are found to be related to flow resistance: standard deviation of elevations, inundation ratio, pit density (measured both perpendicular and parallel to the flow direction), slope, median depth, skewness of the depth distribution and frontal area. Hillslope position is found to affect the significant roughness measures. In contrast, infiltration rate has little effect on the velocity of water fronts advancing over the soil surfaces examined here and the effect of depression storage is limited. Overland flow resistance is depth dependent where complex microtopographic structures are progressively inundated. Copyright © 2010 John Wiley & Sons, Ltd.     

Effect of saltating sediment on flow resistance and bed roughness in overland flow

The acceleration of saltating grains by overland flow causes momentum to be transferred from the flow to the grains, thereby increasing flow resistance and bed roughness. To assess the impact of saltating sediment on overland flow hydraulics, velocity profiles in transitional and turbulent flows on a fixed sand-covered bed were measured using hot-film anemometry. Five discharges were studied. At each discharge, three flows were measured: one free of sediment, one with a relatively low sediment load, and one with a relatively high sediment load. In these flows from 83 to 90 per cent of the sediment was travelling by saltation. As a result, in the sediment-laden flows the near-bed velocities were smaller and the velocity profiles steeper than those in the equivalent sediment-free flows. Sediment loads ranged up to 87·0 per cent of transport capacity and accounted for as much as 20·8 per cent of flow resistance (measured by the friction factor) and 89·7 per cent of bed roughness (measured by the ratio of the roughness length to median grain diameter). It is concluded that saltating sediment has a considerable impact on overland flow hydraulics, at least on fixed granular beds. Saltation is likely to have a relatively smaller effect on overland flow on natural hillslopes and agricultural fields where form and wave resistance dominate. Still, saltation is generally of greater significance in overland flow than in river flow, and for this reason its effect on overland flow hydraulics is deserving of further study. © 1998 John Wiley & Sons, Ltd.     

Influences of spatially heterogeneous roughness on flow hydrographs

Spatially uniform roughness, which although may not be adequate in all cases, is often conveniently adopted for rainfall-runoff simulations in conventional watershed hydrology. In this study, effects of spatially heterogeneous roughness on hydrological response were investigated systematically. Three types of surface roughness scenarios, including: roughness decreasing in a downstream direction (Scenario 1), roughness increasing in a downstream direction (Scenario 2) and roughness distributed at random (Scenario 3), were assigned to an overland plane. A non-inertia wave model was developed to generate hydrographs at the end of the overland plane for certain rainstorms. The hydrographs in the three scenarios showed that when rainfall duration was less than the time of concentration of the overland plane, peak discharges generated from the spatially heterogeneous roughness scenarios significantly exceeded those from the spatially uniform roughness condition. The temporal equilibrium discharge which was usually observed on an overland plane under the spatially uniform roughness condition was not found in the present study. The flow peak in Scenario 1 occurred before that in Scenario 2. Nevertheless, the runoff peak could occur at the preceding, the middle or the later part of the hydrograph in Scenario 3, and this depended on the spatial distribution of the roughness. Reasons for the variation of peak discharge and the time to peak discharge were analyzed in detail based on water depth and flow velocity. Spatially heterogeneous roughness had an obviously influence on runoff generation, which ought to be handled with care in hydrological simulations.     

Hydraulics of overland flow influenced by litter incorporation under extreme rainfall

Water erosion on hillslopes is a worldwide environmental problem, which is a rainfall‐induced process, especially extreme rainfall. The great intensity of extreme rainfall strongly enhances the power of overland flow to detach soil and transport sediment. Plant litter is one of the most important constituents of ecosystems that often covers the soil surface and can be incorporated into topsoil. However, little attention has been paid to its effect on flow hydraulics owing to the veiled nature. This study aimed to examine the effects of incorporated litter on the hydraulic properties under extreme rainfall condition. To reach this goal, six litter rates of 0, 0.05, 0.10, 0.20, 0.35, and 0.50 kg m^?2 and four litter types collected from deciduous trees, coniferous trees, shrubs, and herbs were incorporated into topsoil. Then, simulated rainfall experiments were performed on five slope gradients (5°, 10°, 15°, 20°, and 25°) with an extreme rainfall intensity of 80 mm h^?1. The results showed that Froude number and flow velocity of the overland flow decreased, whereas flow resistance increased exponentially with litter incorporation rate. Litter type had an influence on flow hydraulics, which can mainly be attributed to the variations in surface coverage of the exposed litter and the litter morphology. Flow velocity and Darcy–Weisbach coefficient increased markedly with slope gradient. However, the variation of slope gradient did not modify the relationships between flow hydraulics and incorporated litter rate. The random roughness, resulting from heterogeneous erosion due to the uneven protection of surface exposed litter, increased linearly with litter incorporated rate. As rainfall proceeded, flow hydraulics varied with incorporated litter rate and slope gradient complicatedly due to the increases in flow rate and coverage of the exposed litter and the modification of soil surface roughness.     

An experimental study on the evolution law of roll wave parameters in overland flow

Roll waves commonly occur in overland flow and have an important influence on the progress of soil erosion on slopes. This study aimed to explore the evolution and mechanism of roll waves on steep slopes. The potential effects of flow rate, rainfall intensity and bed roughness on the laws controlling roll wave parameters were investigated. The flow rates, rainfall intensities and bed roughness varied from 5 to 30 L/min, 0 to 150 mm/h, and 0.061 to 1.700 mm, respectively. The results indicate that roll waves polymerize significantly along the propagation path, and bed roughness and rainfall affect the generation and evolution of roll waves. The wave velocity, length and height decreased with bed roughness, whereas the wave frequency increased with increasing bed roughness under fixed flow rate and rainfall intensity conditions. Rainfall increased the wave velocity and wavelength and decreased the wave frequency. The wave velocity, height and wavelength tended to increase with an increasing flow rate. Rainfall promoted the generation of roll waves, whereas bed roughness had the opposite effect. The generation of roll waves is closely related to the Froude number (Fr) and flow resistance. In this experiment, the range of the Reynolds number for the roll waves generated in the laminar region was 142–416, and the range of the flow resistance coefficient was 0.64–4.85. The critical value of the Fr for flow instability in the laminar region was approximately 0.57. Exploring the generation and evolution law of roll waves is necessary for understanding the processes and dynamic mechanisms of slope soil erosion.     

Study on effect of surface roughness on overland flow from different geometric surfaces through numerical simulation

Effect of variability in surface roughness on overland flow from different geometric surfaces is investigated using numerical solution of diffusion wave equation. Three geometric surfaces rectangular plane, converging and diverging plane at slopes 1 to 3% are used. Overland flow is generated by applying rainfall at constant intensity of 10 mm/h for period 30 min and 100 min. Three scenarios of spatial roughness conditions viz. roughness increasing in downstream direction, roughness decreasing in downstream direction and roughness distributed at random are considered. Effect of variability of roughness on overland flow in terms of depth, velocity of flow and discharge along the distance from upstream to downstream for different geometric surfaces are discussed in detail. Results from the study indicate that roughness distribution has significant effect on peak, time to peak and overall shape of the overland flow hydrograph. The peak occurs earlier for the scenario when roughness increases in downstream direction as compared to scenario when roughness is decreasing in downstream for all three geometric surfaces due to very low friction factor and more velocity at the top of the domain. The converging plane attains equilibrium state early as compared to rectangular and diverging plane. Different set of random values result in different time to peak and shape of hydrograph for rectangular and diverging plane. However, in case of converging plane, the shape of computed hydrographs remains almost similar for different sets of random roughness values indicating stronger influence of converging geometry than effect due to variation of roughness sequence on computed runoff hydrograph. Hierarchically, the influence of geometry on overland flow is stronger than the influence of slope and the influence of slope is stronger than the influence of roughness. Copyright © 2013 John Wiley & Sons, Ltd.     

Hillslope runoff and erosion as affected by rolled erosion control systems: a field study

A replicated field study using rainfall simulation and overland flow application was conducted in central Oahu, Hawaii, on a clay‐dominated Oxisol with a 9% slope. Three main treatment groups were examined: a bare treatment, a group of four rolled erosion control systems (RECSs) with open weave designs, and a group of five randomly oriented fibre RECSs. A total of 1122 measurements of runoff and erosion were made to examine treatment differences and to explore temporal patterns in runoff and sediment flux. All erosion control systems significantly delayed the time required to generate plot runoff under both simulated rainfall (35 mm h^?1) and the more intense trickle flow application (114 mm h^?1). Once runoff was generated during the rainfall application phase, the bare treatment runoff coefficients were significantly lower than those from the two groups of RECSs, as surface seal disruption by rilling is inferred to have enhanced infiltration in the bare treatments. During the more intense phase of overland flow application, the reverse pattern was observed. Interrill contributing‐area roughness was reduced on the bare treatment, facilitating increased runoff to well‐developed rill networks. Meanwhile, the form roughness associated with the RECSs delayed interrill flow to the poorly organized rills that formed under some of the RECSs. Regardless of runoff variations between treatments, sediment output was significantly lower from all surfaces covered by RECSs. The median cumulative sediment output from the bare surfaces was 6·9 kg, compared with 1·2 kg from the open‐weave RECSs and 0·2 kg from the random‐fibre RECSs. The random‐fibre systems were particularly effective under the more stressful overland flow application phase, with 63 times less sediment eroded than the bare treatments and 12 times less than that from the open‐weave systems. Architectural design differences between the two groups of RECSs are discussed in light of their relation to erosion process dynamics and shear stress partitioning. Copyright © 2006 John Wiley & Sons, Ltd.     

Macroscale surface roughness and frictional resistance in overland flow

The hydraulics of overland flow on rough granular surfaces can be modelled and evaluated using the inundation ratio rather than the flow Reynolds number, as the primary dimensionless group determining the flow behaviour. The inundation ratio describes the average degree of submergence of the surface roughness and is used to distinguish three flow regimes representing partially inundated, marginally inundated and well-inundated surfaces. A heuristic physical model for the flow hydraulics in each regime demonstrates that the three states of flow are characterized by very different functional dependencies of frictional resistance on the scaled depth of flow. At partial inundation, flow resistance is associated with the drag force derived from individual roughness and therefore increases with depth and percentage cover. At marginal inundation, the size of the roughness elements relative to the depth of flow controls the degree of vertical mixing in the flow so that frictional resistance tends to decrease very rapidly with increasing depth of flow. Well-inundated flows are described using rough turbulent flow hydraulics previously developed for open channel flows. These flows exhibit a much more gradual decrease in frictional resistance with increasing depth than that observed during marginal inundation. A data set compiled from previously published studies of overland flow hydraulics is used to assess the functional dependence of frictional resistance on inundation ratio over a wide range of flow conditions. The data confirm the non-monotonic dependence predicted by the model and support the differentiation of three flow regimes based on the inundation ratio. Although the percentage cover and the surface slope may be of importance in addition to the inundation ratio in the partially and marginally inundated regimes, the Reynolds number appears to be of significance only in describing well-inundated flows at low to moderate Reynolds numbers. As these latter conditions are quite rare in natural environments, the inundation ratio rather than the Reynolds number should be used as the primary dimensionless group when evaluating the hydraulics of overland flow on rough surfaces. © 1997 by John Wiley & Sons, Ltd.     

Evaluation of an interrill soil erosion model using laboratory catchment data

Physically based soil erosion simulation models require input parameters of soil detachment and sediment transport owing to the action and interactions of both raindrops and overland flow. A simple interrill soil water transport model is applied to a laboratory catchment to investigate the application of raindrop detachment and transport in interrill areas explicitly. A controlled laboratory rainfall simulation study with slope length simulation by flow addition was used to assess the raindrop detachment and transport of detached soil by overland flow in interrill areas. Artificial rainfall of moderate to high intensity was used to simulate intense rain storms. However, experiments were restricted to conditions where rilling and channelling did not occur and where overland flow covered most of the surface. A simple equation with a rainfall intensity term for raindrop detachment, and a simple sediment transport equation with unit discharge and a slope term were found to be applicable to the situation where clear water is added at the upper end of a small plot to simulate increased slope length. The proposed generic relationships can be used to predict raindrop detachment and the sediment transport capacity of interrill flow and can therefore contribute to the development of physically‐based erosion models. Copyright © 1999 John Wiley & Sons, Ltd.     

An analysis of the characteristics of rough bed turbulent shear stresses in an open channel

Entrainment of sediment particles from channel beds into the channel flow is influenced by the characteristics of the flow turbulence which produces stochastic shear stress fluctuations at the bed. Recent studies of the structure of turbulent flow has recognized the importance of bursting processes as important mechanisms for the transfer of momentum into the laminar boundary layer. Of these processes, the sweep event has been recognized as the most important bursting event for entrainment of sediment particles as it imposes forces in the direction of the flow resulting in movement of particles by rolling, sliding and occasionally saltating. Similarly, the ejection event has been recognized as important for sediment transport since these events maintain the sediment particles in suspension. In this study, the characteristics of bursting processes and, in particular, the sweep event were investigated in a flume with a rough bed. The instantaneous velocity fluctuations of the flow were measured in two-dimensions using a small electromagnetic velocity meter and the turbulent shear stresses were determined from these velocity fluctuations. It was found that the shear stress applied to the sediment particles on the bed resulting from sweep events depends on the magnitude of the turbulent shear stress and its probability distribution. A statistical analysis of the experimental data was undertaken and it was found necessary to apply a Box-Cox transformation to transform the data into a normally distributed sample. This enabled determination of the mean shear stress, angle of action and standard error of estimate for sweep and ejection events. These instantaneous shear stresses were found to be greater than the mean flow shear stress and for the sweep event to be approximately 40 percent greater near the channel bed. Results from this analysis suggest that the critical shear stress determined from Shield's diagram is not sufficient to predict the initiation of motion due to its use of the temporal mean shear stress. It is suggested that initiation of particle motion, but not continuous motion, can occur earlier than suggested by Shield's diagram due to the higher shear stresses imposed on the particles by the stochastic shear stresses resulting from turbulence within the flow.     

Adaptive criterion curves describing incipient motion of sediment under wave and current conditions

Sediment incipient motion is a fundamental issue in sediment transport theory and engineering practice. Whilst Shields curve often is used to determine the threshold of sediment movement under unidirectional current conditions, it is unclear whether it can be directly applied for the wave or combined wave-current conditions. The study developed adaptive criterion curves describing incipient motion of sediment under wave and current conditions based on the flow pattern around the sediment particles. Firstly, the flow pattern law for fixed particles was recognized based on the friction law under various dynamic conditions (wave, current, and their combinations), and the flow pattern demarcations for incipient sediment motion were obtained with the threshold conditions for sediment movement under various dynamic conditions combined. Secondly, the exact shape of the Shields curve in each flow regime was derived under the current condition. By combining the flow pattern demarcations for incipient sediment motion under the wave condition, the criterion curve under the wave condition was derived. By combining the flow pattern demarcations for incipient sediment motion under the combined current-wave condition, the criterion curve for sediment incipient motion under the combined current-wave condition was derived. The results indicated that the flow pattern around incipient particles includes laminar, laminar-rough turbulent transition, and rough turbulent regimes. The criterion curves for sediment incipient motion under various dynamic conditions stayed the same in the laminar and rough turbulent regimes, but different in the transition regime. Depending on the relative strengths of the currents and waves, the shape of the criterion curve under the combined current-wave condition transitions adaptively between the criterion curve under the current condition and the criterion curve under the wave conditions.     

RUNOFF GENERATION AND SOIL EROSION IN SMALL AGRICULTURAL CATCHMENTS WITH LOESS-DERIVED SOILS

Over a two-year period, rainfall, runoff and sediment output were measured in six small agricultural catchments (3–10 ha) in south Limburg (The Netherlands). These measurements were needed for validation of an erosion model for South Limburg (LISEM). In this paper, results of the measurements are presented and processes that determine surface runoff and sediment yield during winter and summer rainfall are identified. Before the start of the measurement programme, surface slaking and crust formation on the erodible loess soils were thought to be the main cause of overland flow and soil erosion in South Limburg. This was the starting point for soil conservation measures in the area. The measurement results discussed in this paper show that in some catchments much runoff occurred in winter and that soil moisture storage capacity may be just as important for runoff generation as infiltration capacity. Therefore, when modelling soil erosion and optimizing erosion control measures for South Limburg, runoff generation through Hortonian as well as through saturation overland flow must be considered.     

The errors in surface runoff prediction by neglecting the relationship between infiltration rate and overland flow depth

Many simplifications are used in modeling surface runoff over a uniform slope. A very common simplification is to determine the infiltration rate independent of the overland flow depth and to combine it afterward with the kinematic-wave equation to determine the overland flow depth. Another simplication is to replace the spatially variable infiltration rates along the slope i(x, t) due to the water depth variations h(x,t) with an infiltration rate that is determined at a certain location along the slope. The aim of this study is to evaluate the errors induced by these simplications on predicted infiltration rates, overland flow depths, and total runoff volume. The error analysis is accomplished by comparing a simplified model with a model where the interaction between the overland flow depth and infiltration rate is counted. In this model, the infiltration rate is assumed to vary along the slope with the overland flow depth, even for homogeneous soil profiles. The kinematic-wave equation with interactive infiltration rate, calculated along the slopy by Richard's equation, are then solved by a finite difference scheme for a 100-m-long uniform slope. In the first error analysis, we study the effect of combining an ‘exact’ and ‘approximate’ one-dimensional infiltration rate with the kinematic-wave equation for three different soil surface roughness coefficients. The terms ‘exact’ and ‘approximate’ stand for the solution of Richard's equation with and without using the overland flow depth in the boundary condition, respectively. The simulations showed that higher infiltration rates and lower overland flow depths are obtained during the rising stage of the hydrograph when overland flow depth is used in the upper boundary condition of the one-dimensional Richard's equation. During the recession period, the simplified model predicts lower infiltration rates and higher overland flow depths. The absolute relative errors between the ‘exact’ and ‘approximate’ solutions are positively correlated to the overland flow depths which increase with the soil surface roughness coefficient. For this error analysis, the relative errors in surface runoff volume per unit slope width throughout the storm are much smaller than the relative errors in momentary overland flow depths and discharges due to the alternate signs of the deviations along the rising and falling stages. In the second error analysis, when the spatially variable infiltration rate along the slope i(x, t) is replaced in the kinematic-wave equation by i(t), calculated at the slope outlet, the overland flow depth is underestimated during the rising stage of the hydrograph and overestimated during the falling stage. The deviations during the rising stage are much smaller than the deviations during the falling stage, but they are of a longer duration. This occurs because the solution with i(x, t) recognizes that part of the slope becomes dry after rainfall stops, while overland flow still exists with i(t) determined at the slope outlet. As obtained for the first error analysis, the relative errors in surface runoff volume per unit slope width are also much smaller than the relative errors in momentary overland flow depths and discharges. The relation between the errors in overland flow depth and discharge to different mathematical simplifications enables to evaluate whether certain simplifications are justified or more computational efforts should be used.     

Stochastic overland flows Part 1: Physics-based evolutionary probability distributions

A theoretical solution framework to the nonlinear stochastic partial differential equations (SPDE) of the kinematic wave and diffusion wave models of overland flows under stochastic inflows/outflows, stochastic surface roughness field and stochastic state of flows was obtained. This development was realized by means of an eigenfunction representation of the time-space overland flow depths, and by transforming the problem into the phase space. By using Van Kampen's lemma and the cumulant expansion theory of Kubo-Van Kampen-Fox, the deterministic partial differential equation (PDE) for the evolutionary probability density function (pdf) of overland flow depths was finally obtained. Once this deterministic PDE is solved for the time-varying pdf of overland flow depths, then the time-space varying pdf of overland flow depths can be obtained by a transformation given in the text. In this solution framework it is possible to incorporate the stochastic dynamic behavior of the parameters and of the forcing functions of the overland flow process. For example, not only the individual rainfall duration and fluctuating rain intensity characteristics but also the sequential behavior of rainfall patterns is incorporated into the evolutionary probability density function of overland flow depths.     

Incremental distributed modelling investigation in a small agricultural catchment: 2. Erosion and phosphorus transport

Distributed physically based erosion and phosphorus (P) transport models, run by the overland flow model described in Taskinen and Bruen (2006. Hydrological Processes 20 : this issue), are described. In the erosion model, the additional components to the basic model were the outflow of the particles by infiltration and a new model component, i.e. deposition when rainfall stops. Two ways of calculating the shielding factor due to the flow depth were compared. The P transport model had both dissolved P (DP) and particulate P (PP) components. The processes included in the DP model were desorption from the soil surface, advection, storage in the overland flow and infiltration. The PP model accounted for advection, storage in the flow, infiltration, detachment from the soil surface by flow and rainfall and deposition both when transport capacity of suspended solids (SS) is exceeded and when rainfall ceases. When the models were developed and validated in small agricultural fields of cohesive soil types in southern Finland, comparisons were made between corresponding processes and the significance of added components were estimated in order to find out whether increased model complexity improves the model performance. The sedigraphs were found to follow the dynamics of rainfall, emphasizing the importance of the rainfall splash component. The basic model was too slow to react to changes in rainfall and flow rates, but infiltration and deposition that acts during the cessation in rainfall improved the model significantly by enabling the modelled SS to fall sharply enough. The shielding effect of flow depth from the splash detachment was found to play a significant role. Transport capacity should also be included in erosion models when they are applied to cohesive soils. In this study, the Yalin method worked well. A strong correlation was obvious between the measured SS and total P concentrations, indicating that the main form of P in runoff is PP. This emphasizes the importance of a good sediment transport model in P transport modelling. The submodel used for DP desorption from the soil surface produced plausible results without any calibration. Copyright © 2006 John Wiley & Sons, Ltd.