Scaling and self‐similarity in one‐dimensional unsteady open channel flow

The conditions under which the Saint Venant equations system for unsteady open channel flow, as an initial–boundary value problem, becomes self‐similar are investigated by utilizing one‐parameter Lie group of point scaling transformations. One of the advantages of this methodology is that the self‐similarity conditions due to the initial and boundary conditions can also be investigated thoroughly in addition to the conditions due to the governing equation. The obtained self‐similarity conditions are compared with the scaling relationships that are derived through the Froude similitude. It is shown that the initial–boundary value problem of a one‐dimensional unsteady open channel flow process in a prototype domain can be self‐similar with that of several different scaled domains. However, the values of all the flow variables (at specified time and space) under different scaled domains can be upscaled to the same values in the prototype domain (at the corresponding time and space), as shown in this study. Distortion in scales of different space dimensions has been implemented extensively in physical hydraulic modelling, mainly because of cost, space and time limitations. Unlike the traditional approach, the distinction is made between the longitudinal–horizontal and transverse–horizontal length scales in this study. The scaled domain obtained by the proposed approach, when scaling ratios of channel width and water depth are equal, is particularly important for the similarity of flow characteristics in a cross‐section because the width‐to‐depth ratio and the inclination angles of the banks are conserved in a cross‐section. It is also shown that the scaling ratio of the roughness coefficient under distorted channel conditions depends on that of hydraulic radius and longitudinal length. The proposed scaling relations obtained by the Lie group scaling approach may provide additional spatial, temporal and economical flexibility in setting up physical hydraulic models. Copyright © 2013 John Wiley & Sons, Ltd.     

Sub‐grid scale parameterization of hillslope runoff and erosion processes for catchment‐scale models of semi‐arid landscapes

The processes of hillslope runoff and erosion are typically represented at coarse spatial resolution in catchment‐scale models due to computational limitations. Such representation typically fails to incorporate the important effects of topographic heterogeneity on runoff generation, overland flow, and soil erosion. These limitations currently undermine the application of distributed catchment models to understand the importance of thresholds and connectivity on hillslope and catchment‐scale runoff and erosion, particularly in semi‐arid environments. This paper presents a method for incorporating high‐resolution topographic data to improve sub‐grid scale parameterization of hillslope overland flow and erosion models. Results derived from simulations conducted using a kinematic wave overland flow model at 0.5 m spatial resolution are used to parameterize the depth–discharge relationship in the overland flow model when applied at 16 m resolution. The high‐resolution simulations are also used to derive a more realistic parameterization of excess flow shear stress for use in the 16 m resolution erosion model. Incorporating the sub‐grid scale parameterization in the coarse‐resolution model (16 m) leads to improved predictions of overland flow and erosion when evaluated using results derived from high‐resolution (0.5 m) model simulations. The improvement in performance is observed for a range of event magnitudes and is most notable for erosion estimates due to the non‐linear dependency between the rates of erosion and overland flow. Copyright © 2013 John Wiley & Sons, Ltd.     

One‐ and two‐dimensional modelling of overland flow in semiarid shrubland,Jornada basin,New Mexico

Two distributed parameter models, a one‐dimensional (1D) model and a two‐dimensional (2D) model, are developed to simulate overland flow in two small semiarid shrubland watersheds in the Jornada basin, southern New Mexico. The models are event‐based and represent each watershed by an array of 1‐m² cells, in which the cell size is approximately equal to the average area of the shrubs. Each model uses only six parameters, for which values are obtained from field surveys and rainfall simulation experiments. In the 1D model, flow volumes through a fixed network are computed by a simple finite‐difference solution to the 1D kinematic wave equation. In the 2D model, flow directions and volumes are computed by a second‐order predictor–corrector finite‐difference solution to the 2D kinematic wave equation, in which flow routing is implicit and may vary in response to flow conditions. The models are compared in terms of the runoff hydrograph and the spatial distribution of runoff. The simulation results suggest that both the 1D and the 2D models have much to offer as tools for the large‐scale study of overland flow. Because it is based on a fixed flow network, the 1D model is better suited to the study of runoff due to individual rainfall events, whereas the 2D model may, with further development, be used to study both runoff and erosion during multiple rainfall events in which the dynamic nature of the terrain becomes an important consideration. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of kinematic wave theory for predicting flash flood hazards on coupled alluvial fan–piedmont plain landforms

A rainstorm that caused a severe flash flood on the piedmont plain at the toe positions of two alluvial fans located to the west of the Organ Mountains in Dona Ana County, New Mexico, USA, is analysed. The space–time distributions of rainfall are evaluated from the Next Generation Weather Radar (NEXRAD) and overland flow is modelled as kinematic wave. The spatial distribution of rainfall shows a topographic control. The greatest rainfall depth, duration, and intensity occurred at the higher elevation mountain slopes and decreased with decreasing elevation from the alluvial fans to the piedmont plain. The alluvial fan–piedmont plain system is modelled by coupling divergent and rectangular overland flow planes. Explicit finite difference approximations, hybridized with the analytical method of characteristics, are made to the kinematic wave equations to account for the spatial and temporal distribution of the rainfall and variable boundary conditions. Simulation results indicate that sheet‐flow floodwater elevations rise (1) in a nonlinear fashion from the apex to toe positions of the alluvial fans, and (2) near linearly from the toe positions of the alluvial fans onto the piedmont plains with the formation of kinematic shocks near the middle to the upstream end of the plane at times between the initiation of the rainstorm and the time of concentration of the plane. Thus, the maximum flooding occurs at the middle or upstream sections of the piedmont plains regardless of the pattern of space–time variability of rainfall. These results are in agreement with observed geomorphologic features suggesting that piedmont plains are naturally flood‐prone areas. This case study demonstrates that flood hazards on piedmont plains can exceed those on alluvial fans. The models presented in this study suggest that the flood hazard zones on coupled alluvial fan–piedmont plain landforms should be delineated transverse to the flow directions, as opposed to the flood hazard zones with boundaries in the longitudinal direction of the axis of an alluvial fan. Copyright © 2003 John Wiley & Sons, Ltd.     

Interactions and connectivity between runoff generation processes of different spatial scales

Monitoring runoff generation processes in the field is a prerequisite for developing conceptual hydrological models and theories. At the same time, our perception of hydrological processes strongly depends on the spatial and temporal scale of observation. Therefore, the aim of this study is to investigate interactions between runoff generation processes of different spatial scales (plot scale, hillslope scale, and headwater scale). Different runoff generation processes of three hillslopes with similar topography, geology and soil properties, but differences in vegetation cover (grassland, coniferous forest, and mixed forest) within a small v‐shaped headwater were measured: water table dynamics in wells with high spatial and temporal resolution, subsurface flow (SSF) of three 10 m wide trenches at the bottom of the hillslopes subdivided into two trench sections each, overland flow at the plot scale, and catchment runoff. Bachmair et al. ( 2012 ) found a high spatial variability of water table dynamics at the plot scale. In this study, we investigate the representativity of SSF observations at the plot scale versus the hillslope scale and vice versa, and the linkage between hillslope dynamics (SSF and overland flow) and streamflow. Distinct differences in total SSF within each 10 m wide trench confirm the high spatial variability of the water table dynamics. The representativity of plot scale observations for hillslope scale SSF strongly depends on whether or not wells capture spatially variable flowpaths. At the grassland hillslope, subsurface flowpaths are not captured by our relatively densely spaced wells (3 m), despite a similar trench flow response to the coniferous forest hillslope. Regarding the linkage between hillslope dynamics and catchment runoff, we found an intermediate to high correlation between streamflow and hillslope hydrological dynamics (trench flow and overland flow), which highlights the importance of hillslope processes in this small watershed. Although the total contribution of SSF to total event catchment runoff is rather small, the contribution during peak flow is moderate to substantial. Additionally, there is process synchronicity between spatially discontiguous measurement points across scales, potentially indicating subsurface flowpath connectivity. Our findings stress the need for (i) a combination of observations at different spatial scales, and (ii) a consideration of the high spatial variability of SSF at the plot and hillslope scale when designing monitoring networks and assessing hydrological connectivity. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Stochastic overland flows

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.     

On the relationship between hydrographs and chemographs

The spatial representativeness of gauging stations was investigated in two low‐mountainous river basins near the city of Trier, southwest Germany. Longitudinal profiles during low and high flow conditions were sampled in order to identify sources of solutes and to characterize the alteration of flood wave properties during its travel downstream. Numerous hydrographs and chemographs of natural flood events were analysed in detail. Additionally, artificial flood events were investigated to study in‐channel transport processes. During dry weather conditions the gauging station was only representative for a short river segment upstream, owing to discharge and solute concentrations of sources contiguous to the measurement site. During artificial flood events the kinematic wave velocity was considerably faster than the movement of water body and solutes, refuting the idea of a simple mixing process of individual runoff components. Depending on hydrological boundary conditions, the wave at a specific gauge could be entirely composed of old in‐channel water, which notably reduces the spatial representativeness of a sampling site. Natural flood events were characterized by a superimposition of local overland flow, riparian water and the kinematic wave process comprising the downstream conveyance of solutes. Summer floods in particular were marked by a chronological occurrence of distinct individual runoff components originating only from a few contributing areas adjacent to the stream and gauge. Thus, the representativeness of a gauge for processes in the whole basin depends on the distance of the nearest significant source to the station. The consequence of our study is that the assumptions of mixing models are not satisfied in river basins larger than 3 km². Copyright © 2006 John Wiley & Sons, Ltd.     

A kinematic‐wave‐based distributed watershed model using FEM,GIS and remotely sensed data

For the appropriate management of water resources in a watershed, it is essential to calculate the time distribution of runoff for the given rainfall event. In this paper, a kinematic‐wave‐based distributed watershed model using finite element method (FEM), geographical information systems (GIS) and remote‐sensing‐based approach is presented for the runoff simulation of small watersheds. The kinematic wave equations are solved using FEM for overland and channel flow to generate runoff at the outlet of the watershed concerned. The interception loss is calculated by an empirical model based on leaf area index (LAI). The Green‐Ampt Mein Larson (GAML) model is used for the estimation of infiltration. Remotely sensed data has been used to extract land use (LU)/land cover (LC). GIS have been used to prepare finite element grid and input files such as Manning's roughness and slope. The developed overland flow model has been checked with an analytical solution for a hypothetical watershed. The model has been applied to a gauged watershed and an ungauged watershed. From the results, it is seen that the model is able to simulate the hydrographs reasonably well. A sensitivity analysis of the model is carried out with the calibrated infiltration parameters, overland flow Manning's roughness, channel flow Manning's roughness, time step and grid size. The present model is useful in predicting the hydrograph in small, ungauged watersheds. Copyright © 2007 John Wiley & Sons, Ltd.     

Simulation of the direct runoff hydrograph at basin outlet

A simple model describing the transformation of effective rainfall to direct runoff through the overland flow mechanism is presented. The model is based on the classical representation of a watershed by a combination of planes and channels. The dynamics of overland flow in each plane is simulated by the non-linear kinematic wave, but the outflow from a given plane is concentrated in the middle of the corresponding drainage channel. The water routing in the channels is carried out by a piece-wise linearized formulation in space of the kinematic wave approximation. Using synthetic events on 10 watersheds, the model was tested by comparing it with results obtained by applying the non-linear kinematic wave to all the elements of the watershed. The model was found to be adequate, even in a form that simplifies the geometric features of the planes through an averaging procedure based on the Horton–Strahler ordering scheme of the watershed. © 1998 John Wiley & Sons, Ltd.     

Hortonian overland flow from Japanese forest plantations—an aberration,the real thing,or something in between?

There is a growing opinion that poorly managed plantation forests in Japan are contributing to increased storm runoff and erosion. Here we present evidence to the contrary from runoff plots at two scales (hillslope and 0·5 × 2 m plots) for several forest conditions in the Mie and Nariki catchments. Runoff coefficients from small plots in untended hinoki forests were variable but typically higher than from better managed or deciduous forests during small storms at Nariki; at Mie, runoff during small events was highly variable from all small plots but runoff coefficients were similar for hinoki plots with and without understory vegetation, while the deciduous plot had lower runoff coefficients. Storm runoff was less at the hillslope scale than the plot scale in Mie; these results were more evident at sites with better ground cover. During the largest storms at both sites, differences in runoff due to forest condition were not evident regardless of scale. Dynamic soil moisture tension measurements at Nariki indicated that during a large storm, flow in the upper organic‐rich and root‐permeated soil horizons was 3·2 times higher than measured overland runoff from a small hinoki plot with poor ground cover and 8·3 times higher than runoff from a deciduous forest plot. On the basis of field observations during storms, at least a portion of the monitored ‘Hortonian overland flow’ was actually occurring in this near‐surface ‘biomat’. Therefore our field measurements in both small and large plots potentially included biomat flow in addition to short‐lived Hortonian runoff. Because overland flow decreased with increasing scale, rill erosion did not occur on hillslopes. Additionally, runoff coefficients were not significantly different among cover conditions during large storms; thus, the ‘degraded’ forest conditions appear not to greatly enhance peak flows or erosion potential at larger scales, especially when biomat flow is significant. Copyright © 2007 John Wiley & Sons, Ltd.     

Diverging overland flow

This study investigates divering overland flow utilizing kinematic wave theory, which does not appear to have been dealt with previously. Explicit analytical solutions are derived in dimensionless form for space-time invariant rainfall. Analytical solutions do not seem to be tractable for time-varying rainfall. Depending upon the duration of rainfall, equilibrium and partial equilibrium cases are distinguished explicitly. The effect of divergence parameter on the hydrograph shape is shown. The adequacy of kinematic approximation for characterization of diverging overland flow is tested against laboratory watershed results. The diverging overland flow model is found to yield results which compare well with observations and with those of a plane model.     

Development and application of a spatially‐distributed Arctic hydrological and thermal process model (ARHYTHM)

A process‐based, spatially distributed hydrological model was developed to quantitatively simulate the energy and mass transfer processes and their interactions within arctic regions (arctic hydrological and thermal model, ARHYTHM). The model first determines the flow direction in each element, the channel drainage network and the drainage area based upon the digital elevation data. Then it simulates various physical processes: including snow ablation, subsurface flow, overland flow and channel flow routing, soil thawing and evapotranspiration. The kinematic wave method is used for conducting overland flow and channel flow routing. The subsurface flow is simulated using the Darcian approach. The energy balance scheme was the primary approach used in energy‐related process simulations (snowmelt and evapotranspiration), although there are options to model snowmelt by the degree‐day method and evapotranspiration by the Priestley–Taylor equation. This hydrological model simulates the dynamic interactions of each of these processes and can predict spatially distributed snowmelt, soil moisture and evapotranspiration over a watershed at each time step as well as discharge in any specified channel(s). The model was applied to Imnavait watershed (about 2·2 km²) and the Upper Kuparuk River basin (about 146 km²) in northern Alaska. Simulated results of spatially distributed soil moisture content, discharge at gauging stations, snowpack ablations curves and other results yield reasonable agreement, both spatially and temporally, with available data sets such as SAR imagery‐generated soil moisture data and field measurements of snowpack ablation, and discharge data at selected points. The initial timing of simulated discharge does not compare well with the measured data during snowmelt periods mainly because the effect of snow damming on runoff was not considered in the model. Results from the application of this model demonstrate that spatially distributed models have the potential for improving our understanding of hydrology for certain settings. Finally, a critical component that led to the performance of this modelling is the coupling of the mass and energy processes. Copyright © 2000 John Wiley & Sons, Ltd.     

Improvement of surface run‐off in the hydrological model ParFlow by a scale‐consistent river parameterization

We propose an improvement of the overland‐flow parameterization in a distributed hydrological model, which uses a constant horizontal grid resolution and employs the kinematic wave approximation for both hillslope and river channel flow. The standard parameterization lacks any channel flow characteristics for rivers, which results in reduced river flow velocities for streams narrower than the horizontal grid resolution. Moreover, the surface areas, through which these wider model rivers may exchange water with the subsurface, are larger than the real river channels potentially leading to unrealistic vertical flows. We propose an approximation of the subscale channel flow by scaling Manning's roughness in the kinematic wave formulation via a relationship between river width and grid cell size, following a simplified version of the Barré de Saint‐Venant equations (Manning–Strickler equations). The too large exchange areas between model rivers and the subsurface are compensated by a grid resolution‐dependent scaling of the infiltration/exfiltration rate across river beds. We test both scaling approaches in the integrated hydrological model ParFlow. An empirical relation is used for estimating the true river width from the mean annual discharge. Our simulations show that the scaling of the roughness coefficient and the hydraulic conductivity effectively corrects overland flow velocities calculated on the coarse grid leading to a better representation of flood waves in the river channels.     

Stability and accuracy of two-dimensional kinematic wave overland flow modeling

A two-dimensional finite element based overland flow model was developed and used to study the accuracy and stability of three numerical schemes and watershed parameter aggregation error. The conventional consistent finite element scheme results in oscillations for certain time step ranges. The lumped and the upwind finite element schemes are tested as alternatives to the consistent scheme. The upwind scheme did not improve on the stability or the accuracy of the solution, while the lumped scheme provided stable and accurate solutions for time steps twice the size of time steps needed for the consistent scheme. A new accuracy based dynamic time step estimate for the two-dimensional overland flow kinematic wave solution is developed for the lumped scheme. The newly developed dynamic time step estimates are functions of the mesh size, and time of concentration of the watershed hydrograph. Due to lack of analytical solutions, the time step was developed by comparing numerical solutions of various levels of discretization to a reference solution using a very fine mesh and a very small time step. The time step criteria were tested on a different set of problems and proved to be adequate for accurate and stable solutions. A sensitivity analysis for the watershed slope, Manning’s roughness coefficient and excess rainfall rate was conducted in order to test the effect of parameter aggregation on the stability and accuracy of the solution. The results of this analysis show that aggregation of the slope data resulted in the highest error. The roughness coefficient had a smaller effect on the solution while the rainfall intensity did not show any significant effect on the flow rate solution for the range of rainfall intensity used. This work pioneers the challenge of providing guidelines for accurate and stable numerical solutions of the two-dimensional kinematic wave equations for overland flow.