ALBERT

Description: We hypothesize that total hillslope water loss for a rainfall–runoff event is inversely related to a function of a lognormal random variable, based on basin- and point-scale observations taken from the 21 km2 Goodwin Creek Experimental Watershed (GCEW) in Mississippi, USA. A top-down approach is used to develop a new runoff generation model both to test our physical-statistical hypothesis and to provide a method of generating ensembles of runoff from a large number of hillslopes in a basin. The model is based on the assumption that the probability distributions of a runoff/loss ratio have a space–time rescaling property. We test this assumption using streamflow and rainfall data from GCEW. For over 100 rainfall–runoff events, we find that the spatial probability distributions of a runoff/loss ratio can be rescaled to a new distribution that is common to all events. We interpret random within-event differences in runoff/loss ratios in the model to arise from soil moisture spatial variability. Observations of water loss during events in GCEW support this interpretation. Our model preserves water balance in a mean statistical sense and supports our hypothesis. As an example, we use the model to generate ensembles of runoff at a large number of hillslopes for a rainfall–runoff event in GCEW.

Description: A methodology is presented to understand the role of the statistical self-similar topology of real river networks on scaling, or power law, in peak flows for rainfall-runoff events. We created Monte Carlo generated sets of ensembles of 1000 random self-similar networks (RSNs) with geometrically distributed interior and exterior generators having parameters pi and pe, respectively. The parameter values were chosen to replicate the observed topology of real river networks. We calculated flow hydrographs in each of these networks by numerically solving the link-based mass and momentum conservation equation under the assumption of constant flow velocity. From these simulated RSNs and hydrographs, the scaling exponents β and φ characterizing power laws with respect to drainage area, and corresponding to the width functions and flow hydrographs respectively, were estimated. We found that, in general, φ 〉 β, which supports a similar finding first reported for simulations in the river network of the Walnut Gulch basin, Arizona. Theoretical estimation of β and φ in RSNs is a complex open problem. Therefore, using results for a simpler problem associated with the expected width function and expected hydrograph for an ensemble of RSNs, we give heuristic arguments for theoretical derivations of the scaling exponents β(E) and φ(E) that depend on the Horton ratios for stream lengths and areas. These ratios in turn have a known dependence on the parameters of the geometric distributions of RSN generators. Good agreement was found between the analytically conjectured values of β(E) and φ(E) and the values estimated by the simulated ensembles of RSNs and hydrographs. The independence of the scaling exponents φ(E) and φ with respect to the value of flow velocity and runoff intensity implies an interesting connection between unit hydrograph theory and flow dynamics. Our results provide a reference framework to study scaling exponents under more complex scenarios of flow dynamics and runoff generation processes using ensembles of RSNs.

Description: An analytical theory is developed that obtains Horton laws for six hydraulic–geometric (H–G) variables (stream discharge Q, width W, depth D, velocity U, slope S, and friction n') in self-similar Tokunaga networks in the limit of a large network order. The theory uses several disjoint theoretical concepts like Horton laws of stream numbers and areas as asymptotic relations in Tokunaga networks, dimensional analysis, the Buckingham Pi theorem, asymptotic self-similarity of the first kind, or SS-1, and asymptotic self-similarity of the second kind, or SS-2. A self-contained review of these concepts, with examples, is given as "methods". The H–G data sets in channel networks from three published studies and one unpublished study are summarized to test theoretical predictions. The theory builds on six independent dimensionless river-basin numbers. A mass conservation equation in terms of Horton bifurcation and discharge ratios in Tokunaga networks is derived. Assuming that the H–G variables are homogeneous and self-similar functions of stream discharge, it is shown that the functions are of a power law form. SS-1 is applied to predict the Horton laws for width, depth and velocity as asymptotic relationships. Exponents of width and the Reynolds number are predicted and tested against three field data sets. One basin shows deviations from theoretical predictions. Tentatively assuming that SS-1 is valid for slope, depth and velocity, corresponding Horton laws and the H–G exponents are derived. Our predictions of the exponents are the same as those previously predicted for the optimal channel network (OCN) model. In direct contrast to our work, the OCN model does not consider Horton laws for the H–G variables, and uses optimality assumptions. The predicted exponents deviate substantially from the values obtained from three field studies, which suggests that H–G in networks does not obey SS-1. It fails because slope, a dimensionless river-basin number, goes to 0 as network order increases, but, it cannot be eliminated from the asymptotic limit. Therefore, a generalization of SS-1, based on SS-2, is considered. It introduces two anomalous scaling exponents as free parameters, which enables us to show the existence of Horton laws for channel depth, velocity, slope and Manning friction. These two exponents are not predicted here. Instead, we used the observed exponents of depth and slope to predict the Manning friction exponent and to test it against field exponents from three studies. The same basin mentioned above shows some deviation from the theoretical prediction. A physical reason for this deviation is given, which identifies an important topic for research. Finally, we briefly sketch how the two anomalous scaling exponents could be estimated from the transport of suspended sediment load and the bed load. Statistical variability in the Horton laws for the H–G variables is also discussed. Both are important open problems for future research.

Description: An analytical theory is presented to predict Horton laws for five Hydraulic-Geometric (H-G) variables (stream discharge Q, width W, depth D, velocity U, slope S, and friction n'). The theory builds on the concept of dimensional analysis, and identifies six independent dimensionless River-Basin numbers. We consider self-similar Tokunaga networks and derive a mass conservation equation in the limit of large network order in terms of Horton bifurcation and discharge ratios. It is applied to obtain self-similar solutions of type-1 (SS-1), and predict Horton laws for width, depth and velocity as asymptotic relationships. Exponents of width and the Reynold's number are predicted. Assuming that SS-1 is valid for slope, depth and velocity, corresponding Horton laws and the H-G exponents are derived. The exponent values agree with that for the Optimal Channel Network (OCN) model, but do not agree with values from three field experiments. The deviations are substantial, suggesting that H-G in network does not obey optimality or SS-1. It fails because slope, a dimensionless River-Basin number, goes to 0 as network order increases, but, it cannot be eliminated from the asymptotic limit. Therefore, a generalization of SS-1, based in self-similar solutions of Type-2 (SS-2) is considered. It introduces two anomalous scaling exponents as free parameters, which enables us to show the existence of Horton laws for channel depth, velocity, slope and Manning's friction. The Manning's friction exponent, y, is predicted and tested against observed exponents from three field studies. We briefly sketch how the two anomalous scaling exponents could be estimated from the transport of suspended sediment load and the bed load. Statistical variability in the Horton laws for the H-G variables is also discussed. Both are important open problems for future research.

Description: We present the construction of a dynamic area fraction model (DAFM), representing a new class of models for an earth-like planet. The model presented here has no spatial dimensions, but contains coupled parameterizations for all the major components of the hydrological cycle involving liquid, solid and vapor phases. We investigate the nature of feedback processes with this model in regulating Earth's climate as a highly nonlinear coupled system. The model includes solar radiation, evapotranspiration from dynamically competing trees and grasses, an ocean, an ice cap, precipitation, dynamic clouds, and a static carbon greenhouse effect. This model therefore shares some of the characteristics of an Earth System Model of Intermediate complexity. We perform two experiments with this model to determine the potential effects of positive and negative feedbacks due to a dynamic hydrological cycle, and due to the relative distribution of trees and grasses, in regulating global mean temperature. In the first experiment, we vary the intensity of insolation on the model's surface both with and without an active (fully coupled) water cycle. In the second, we test the strength of feedbacks with biota in a fully coupled model by varying the optimal growing temperature for our two plant species (trees and grasses). We find that the negative feedbacks associated with the water cycle are far more powerful than those associated with the biota, but that the biota still play a significant role in shaping the model climate. third experiment, we vary the heat and moisture transport coefficient in an attempt to represent changing atmospheric circulations.

Description: Over the last two decades, concepts of scale invariance have come to the fore in both modeling and data analysis in hydrological precipitation research. With the advent of the use of the multiplicative random cascade model, these concepts have become increasingly more important. However, unifying this statistical view of the phenomenon with the physics of rainfall has proven to be a rather nontrivial task. In this paper, we present a simple model, developed entirely from qualitative physical arguments, without invoking any statistical assumptions, to represent tropical atmospheric convection over the ocean. The model is analyzed numerically. It shows that the data from the model rainfall look very spiky, as if generated from a random field model. They look qualitatively similar to real rainfall data sets from Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE). A critical point is found in a model parameter corresponding to the Convective Inhibition (CIN), at which rainfall changes abruptly from non-zero to a uniform zero value over the entire domain. Near the critical value of this parameter, the model rainfall field exhibits multifractal scaling determined from a fractional wetted area analysis and a moment scaling analysis. It therefore must exhibit long-range spatial correlations at this point, a situation qualitatively similar to that shown by multiplicative random cascade models and GATE rainfall data sets analyzed previously (Over and Gupta, 1994; Over, 1995). However, the scaling exponents associated with the model data are different from those estimated with real data. This comparison identifies a new theoretical framework for testing diverse physical hypotheses governing rainfall based in empirically observed scaling statistics.