Entropy and Energy Dissipation in Water Resources

A state-of-the-art description of the theory and applications of the various entropy optimization principles is given. These principles include Jaynes’ maximum entropy principle (MaxEnt), Kullback’s minimum cross-entropy principle (MinxEnt), generalised maximum entropy and minimum cross-entropy principles, inverse entropy optimization principles, minimum interdependence principle, minimax entropy principle and finally, the dual entropy optimization principles. The relation between information-theoretic entropy and thermodynamic entropy is specially recalled in the context of the more general relationship that exist between what are designated as primary and secondary entropies.

Entropy and the principle of maximum entropy are being increasingly applied to a wide range of problems in hydrology and water resources. This paper reviews some of the hydrologic applications of entropy, and comments on the entropy-based approaches. Entropy is a powerful model-building tool, and its potential in explaining the great many hydrologic processes remains largely untapped.

The science of mechanics has been developed along two parallel lines, that is, the vectorial and the variational approaches. The vectorial approach is based on force and momentum while the variational approach is based on entropy, energy, or energy dissipation rate. This paper provides a review and comparison of the basic characteristics, strengths and weaknesses, and interrelationships between the two approaches. Some of the basic difficulties of solving hydraulic problems from the vectorial approach alone are cited to indicate the need for the variational approach. Examples of applications of principles and theories based on variational approach are given to demonstrate the flexibility and applicability of the approach to solve or explain complicated and diversified phenomena from a simple and unified point of view.

The concept of entropy, which originated in classical thermodynamics, has found versatile uses in hydrology and water resources. The investigations in one group of applications basically rely on the concept of “thermodynamic entropy”, where problems associated with river morphology and river hydraulics are handled by a rather non-probabilistic approach. The second group of studies use the concept of “informational entropy” within a probabilistic context to define uncertainties in hydrologic variables, hydrologic systems and their models, and parameters of probability distribution functions. Although it has a very short history in hydrology and water resources, informational entropy has found a wider range of applications in this field, as compared to the thermodynamic entropy. The presented paper discusses the versatile uses of informational entropy in water resources, summarizing the progress obtained so far in developing the concept into a widely accepted technique. Besides the already covered areas of application, new fields where entropy can be used effectively are proposed to cover basically problems in environmental engineering. In view of current research results, the merits and limitations using entropy in water resources engineering problems are discussed, followed by the conclusion that there is a definite need for further investigations so that entropy becomes a principal technique in hydrology and water resources.

This communication deals with the needs for improving the general knowledge in hydrological sciences. In particular a special effort is devoted to define the concept of “entropy” as function of the knowledge, and to derive some simple reference ideas in order to analyze the general problem of the hydrological knowledge.First, the general problem of space and time resolution in hydrological data is introduced, starting from known problems and their related budget of data, then introducing new technological improvements and the consequent change in terms of frequency and volume of data.By means of the known images of time and space variance for hydro-meteorological phenomena, a few examples of the possible optimum sampling is described, but the basic goals of economy and efficiency are declared as fundamental tools for the abatement of the knowledge entropy.

Commonly, order and disorder are scientific concepts strictly related to entropy. However, the definition of entropy is not unique, as it concerns either a macroscopic or microscopic level of description.

In this paper a measure of the entropy associated with a channel network is defined, according to the Shannon informational definition, as the expectation of (-log P_j), where p_j is the ratio of the existing paths at the bifurcation level j to the total number of paths. Here j and p_j are respectively proportional to the arrival time of water to the network outlet and to the number of water parcels arriving from the distance j. Then, the expression for the channel network entropy proposed in this paper is well suited for hydrologic purposes. By analyzing a river network (and related sub-networks) of an Italian basin with surface area of 123 km2, it is shown that the network entropy is strictly related to basin characteristics such as average elevation, Horton order, and magnitude.

The 116-year rainfall record of San Jose, California, includes series of unusually dry periods, two to eight years long, and excessive wet periods, one to three years long. These consecutive dry and wet sequences appear random and unrelated. The entropy concept was used to define: (a) the relationship of drought and flood sequences to the average rainfall; (b) the relationship among selected drought sequences; (c) the relationship among selected flood sequences; (d) the drought and flood patterns; and (c) the worth of the information transfer in the rainfall record. This analysis is important in understanding the ongoing 5-year California drought.

Inadequacy of streamflow records, in many situations, is well recognized by water resources managers. Among other causes responsible for inadequacy of the streamflow records, is the existence of intermittent missing data gaps. This paper addresses this issue and proposes data infilling procedures based on pattern recognition techniques. The characteristics and relationships of distinct groups of data, rather than the entire time series as a whole, forms the basis of model development. Two types of models are proposed, including the models for infilling missing values based on the characteristics and relationships of only the streamflow time series with missing data values; and the models which also incorporate relevant information on the characteristics and relationships of the other time series of nearby rivers. As expected, the latter type of models are found to perform better. Further investigations into the relative efficacy of the proposed models with those existing in literature are continuing.

This paper aims to improve the statistical performances of the regional flood frequency estimator based on a TCEV distribution and on a hierarchical approach. Traditional techniques, using the ML method, may produce unacceptable solutions; other techniques, such as the POME, do not have these problems if we use a simplified method, but there are some numerical problems. If one uses the complete iterative scheme some failures still occur. A theoretical consideration of the feasibility space of the parameters and a large number of numerical experiments, based on Monte Carlo techniques, leads to two different improvement proposals. Their statistical properties are shown and suggestions for further investigation are given.

This study describes a methodology for functional and quantitative analysis of the possible ways of malfunctioning of a Hydrological Forecasting System (HFS) and its unavailability. Historical data on failures of operational HFSs provided information on the expected failure rates and the out-of-service time contributors of the most common HFS equipment. A dynamic event tree methodology was employed for a probabilistic analysis of the HFS. A computer code generated the whole universe of events of interest and produced a selective reading of the event histories available. The probabilities of occurrence of specific failures, of partial and total success of the HFS in performing its functions were analysed and discussed in this study. The costs of different repair and maintenance practices were compared with the HFS reliability to perform its hydrological forecasts. Finally, the entropy approach was employed and its results were compared with the probabilistic model of the HFS. The analysis provided information about the most appropriate preventive and emergency maintenance schedules.

A qualitative approach to assessing redundancy of water distribution networks using entropy theory is proposed. The redundancy measures derived from this approach are able to assess redundancy of supply for individual nodes and for the network as a whole. The measures themselves are based on the fundamental entropy function of Shannon modified to include such network features relevant to redundancy as the number of paths available to supply flow to each demand node, the capacities of these alternate paths, the degree of dependence among the paths, the possibility of flow reversal, and the desirability of having links of equal capacity incident on demand node. The measures are demonstrated by application to eight network layouts subject to the same demand conditions.

Two optimisation formulations for design of water distribution networks under reliability considerations are presented. The formulations incorporate entropy derived statements of redundancy to impose redundancy, and therefore reliability of supply, in the network. One formulation maximises the network wide redundancy subject to a maximum allowable network cost. This model is demonstrated by application to a simple example network. The variation in the layout and redundancy values of this network arising from changes in the levels of maximum allowable cost are analysed. The second formulation minimises network capital cost subject to minimum acceptable levels of nodal redundancy. This model is demonstrated by application to a well known network from the literature and the resulting design compared to the design generated by an established technique. A close similarity was found in the designs produced by the two techniques.

Both the water quantity and water quality processes constitute an integral part of the natural hydrologic environment. These processes are in continuous dynamic interaction so that proper assessment, development and management of water resources require a full understanding of these processes. More specifically, water quality is particularly needed for pollution control and is one of the basic factors to determine the amount of available water that can be used to meet a specific water demand.The general trend in water quality management has been to gather and use information on water quality variables for purposes of planning, design and operation of water resources systems and wastewater treatment facilities. However, growing concern for environmental quality has given rise to a new trend in respect of the impact of water quality variables on human health and life conditions. Thus, there is the need for a better understanding of how water quality processes evolve both in time and space under natural and man-made conditions. This accentuates the need for better methods of extracting information from collected water quality data.Water quality monitoring is a complex, difficult, and costly process. Despite all the efforts and investment made on monitoring, the current status of existing networks shows that the accruing benefits are low. That is, the results of most practices do not fulfill what is expected of monitoring.Summarizing the role of water quality and the need for water quality in water resources assessment, development and management, an attempt is made to examine water quality networks existing in both the developed and developing countries. The existing water quality networks suffer from a lack of compatibility between collected data and water quality management objectives, resulting in “data-rich but information-poor” monitoring practices. Other problems with the networks pertain to selection of variables to be observed, selection of sampling frequencies, selection of sampling sites, duration of monitoring of certain variables at certain sites, and reliability of collected data.Finally, a methodology is proposed for designing an efficient and cost-effective water quality monitoring network. The methodology is based on the entropy concept which permits alleviation of shortcomings of existing networks. It presents some perspectives on design of networks in the future.

The design of water quality monitoring networks is still a controversial issue despite the variety of methodologies proposed. Existing networks are marked with unsolved problems, and the data collected are often of a “messy” character. The basic difficulty lies in the lack of a precise definition for “information” expected from and produced by a network so that it is fairly difficult to assess the efficiency of monitoring practices. The same problem prevails in the evaluation of cost- effectiveness of a network where costs are easy to estimate, but where benefits are often described indirectly in terms of other parameters. In essence, benefits of monitoring can only be measured by means of the information conveyed by collected data. Since no design methodology up-to-date has provided a quantitative measure of information, benefits-of monitoring networks still remain as unquantifiable parameters in the decision making process.The presented study proposes an information-based perspective for the technical design of a network with two basic objectives: maximization of information produced by the network and minimization of accruing costs. Both are evaluated by the entropy principle which provides an information based statistical measure to evaluate the efficiency and cost-effectiveness of a monitoring network, considered here as an “information system”.The entropy concept serves four main objectives of network design: temporal design, spatial design, combined temporal/spatial design, selection of variables and determination of sampling duration. The application of the method for the first three objectives is demonstrated in case of the suspended sediment data of Dicle river basin in Turkey. The strengths and shortcomings of the proposed methodology are evaluated, with recommendations presented for future research on the application of the entropy principle in network design.

In this paper we present some applications of maximum entropy techniques in atmospheric physics and in environmental problems. In the first case we analyze the possibility of using entropy as a regularization operator in the solution of Fredholm integral equations. We discuss in detail the inversion of aerosol size distributions from ground measurements. The results of regularization by entropy are compared with those obtained using the derivatives of the unknown function, as usually suggested in the literature [1]. In the second case we use maximum entropy techniques in statistics in order to study the behaviour of an area perturbed by pollution problems (thermoelectric plant) using the biological composition species as an indicator of environmental conditions. We show that statistics derived from entropy principles are very suitable to delineate homogeneous areas and transition areas which require careful monitoring.

Methods to apply the probability and entropy concepts in open channel hydraulics are presented. Recent applications of an approach based on these concepts produced results that had been impossible to obtain by using only the traditional, deterministic principles. The approach has great potentials in future efforts to obtain knowledge still missing in open channel hydraulics today.

A new method for bridge pier scour prediction is derived using an energy approach. The method permits determination of the maximum scour corresponding to a flood flow of a given recurrence interval. The scour method uses the concept of an effective depth, which is the point in the flow above which the flow essentially has no downward velocity component and therefore does not contribute to scour. A formula for the maximum scour that is related to the velocity profile below the effective depth is given. The method was used to analyze the scour at a published field installation and gave very good results (R2 =.924).

When a regional electrical energy demand is far from stationary and features daily peaks, convenient ways to operate the energy system are by peak turbogas plants or by energy recovery from storage systems. The present tendency is to provide for large storage capacities, since peak plants are rather expensive; the only feasible store proven so far is the hydro storage, which plays the role of an important water resource for energy uses. In this paper the thermodynamic and environmental benefits of operating hydro stores versus peak plants as turbogas will be demonstrated under certain circumstances, adding extra convenience to the choice of large storage capacities, for which lower first and operating costs are already ascertained. The method that is followed is a purely thermodynamic one, consisting in a first and second law analysis of dissipations.

By referring to the simplified schematization of monochromatic wave motion and of a permeable rectangular breakwater of constant porosity, an analysis is presented of the law which characterises energy dissipation inside a porous media. The problem was tackled both by adopting the theoretical approach of classical hydrodynamics, expressed in the definition and solution of differential equations of continuity, momentum and energy, and by using the maximum entropy principle. The latter approach, the results of which are very close to those of the former, has the advantage of introducing fewer parameters to calculate in the analytical formulas.

The paper studies the relation between the structure of river networks and the features of their geomorphologic hydrologic response. The hydrologic response of a channel network is defined by decomposing the process of runoff formation into two distinct contributions, one accounting for the mechanisms of travel time within individual reaches (hydrodynamic dispersion), and the other accounting for the morphology of the network structure (geomorphological dispersion). The variance of the resulting travel time distributions is shown to be made up by two additive contributions corresponding to the two dispersion mechanisms considered. The geomorphologic dispersion coefficient is shown to depend on the ratios of bifurcation, length and area of the network suggesting that, at the scale of organized network, heterogeneities other than those related to the convection field shape the dispersive character of transport. In particular, a significant application of the general solution to Hortonian channel networks suggests that models based on accurate specification of the geometry and the topology of the network and a simplified dynamics capture the foremost features of the travel time distributions in a broad range of dispersivities within individual reaches. We then conclude that the form of a network explains most features of the hydrologic response of its drainage basin.

Spatial variability of morphological characteristics and flow in river networks, and its relation to power distribution are analytically and empirically investigated. It is assumed, and positively tested, that Horton-type laws describe the downstream change in link morphological and topological characteristics. Accordingly, surrogates to the traditional stream length and area ratios are provided by the link number ratio, and the total link number ratio, respectively. The opposite statistical behaviour of stream and link length, as being dependent and independent variables, respectively, is found to be reversed in the case of link and stream heights. This property leads to an identical trend in the spatial variability of slope in both cases. On the other hand, assessment the of the self-similar model of link altitudinal geometry [Gupta & Waymire, 1989] reveals that previous testing, upon which the model has been refuted by Tarboton et al. [1989] was inadequately performed. However, our results show an increasing structured departure from simple-scaling for the n-th order moments. Finally, using Horton-type laws for height and flow yields the distribution of power to be characterized by a state of maximum spatial uniformity for a given flow quantile, for which the scaling exponent of mean link slope equals the one that describes mean flow pattern. This result is found to be implicitly explained by using the informational entropy principles as introduced by Kapoor [1990] for river networks.

In this paper an effort is made to develop a new physically-based overland flow model which accounts for raindrop mixing and for the complex overland flow geometry. Theoretical curves relating overland flow parameters are found and compared to field data. The curves are deduced analytically by means of a kinematic model of thin sheets of flow under rainfall and by means of a geometric model simulating a natural rill network by rectangular channels. The micro-channels may merge together at larger flow depths, thus reducing the relative roughness. This way, the water dissipates (relatively speaking) less energy in flowing downhill and the energy in excess is available for the erosion process which stabilizes the rill network.

The characteristics of drought phenomena at mid-latitudes are analyzed by means of a simplified general circulation model. The geophysical system is excited by a temperature wave to simulate the persistence of an anticyclonic high pressure center over a geographical region. A positive-feedback mechanism, by which an anomalous thermal equilibrium is induced in the system, is shown to cause a water storage depletion corresponding to a drought. The peak severity of the drought occurs one year after the priming and it takes several years to recover from such a moisture depletion. Priming intensity and drought characteristics are shown to be non-linearly related. Therefore, in principle, drought severities and durations could be reduced by acting adequately upon the geophysical system during the priming.

Based on the theory of irreversible thermodynamics and a number of numerical and experimental examples of fluid mechanics and water resources, energy dissipation has been shown to be the primary stabilizing force that determines the direction of change towards an equilibrium condition. Possible future research directions on the use of energy methods in dealing with complex water-resources related problems are suggested.

Hydrodynamic phenomenon, occurring at the junction where a dividing flow is present, is extremely complex and greatly varies according to modifications in discharge conditions near the junction. This is a theoretical and experimental study of free surface dividing flow phenomena in rectangular, horizontal channels with particular attention to energy losses. Relations to quantify energy losses depending on some specific parameters of the phenomenon are studied.

This paper investigates the type of hydraulic jump occurring in the event of a drop known as W_max-jump, and especially investigates the evaluation of piezoraetric head at the step and energy losses. Some aspects characteristic of wave profiles have also been studied.

The well known inadequacy of the momentum equation, written in the usual form, to reconstruct fast transients with satisfactory results, has induced the writers to examine the simplifying hypotheses on which it is based in greater detail. Attention has been focused on two points: criteria for evaluating the friction term and the variability of the Coriolis momentum factor. A local approach has been followed and experimental velocity distributions in unsteady flow have been examined. Even though no conclusive result has been attained because of the little experimental data available, the peculiarities of unsteady flow velocity profiles as well as the notable importance of more precise knowledge of the local characteristics of the flow field have been distinctly shown.

Pore pressures in soils decrease the effective stress level and consequently the shear strength on the slip surface, so reducing the slope safety factor against sliding. Then, drainage systems are involved to limit this dangerous condition. The numerical solution of the equations governing the seepage phenomenon is not straightforward because of the non-linearities involved and possible moving boundaries in the computational domain. Then, the choice of appropriate mathematical models makes the phenomenon to be simulated and numerically evaluated with different orders of accuracy depending on both the exemplification introduced into the model and the computational method chosen.In this paper the two-dimensional seepage problem is considered, for which an efficient and simple second-order accurate in space (and first order in time) implicit finite difference scheme has been set up, capable both to rapidly compute the steady-state solution and to accurately represent the unsteady seepage.

A lemma, useful to prove the equivalence of different variational formulations of related problems, is proposed. Extremal hypotheses about regime alluvial channels are then examined and an equivalence class between them is pointed out, including the minimum stream power and the maximum transport principles. Evidence is produced of contradictions between statements derived from this class of hypotheses and well-established opinions about regime channel. The internal consistency of the theory of minimum dissipation rate is examined, and it is shown that it can’t be derived from the principles of mechanics. The physical basis of extremum hypotheses about alluvial channels is analyzed, and a functional is proposed whose minimum should be equivalent to transversal equilibrium of the channel section.

The present paper discusses the possibility of applying either variational or rational methods in order to evaluate the stable configuration of alluvial channels and the effects of man-made narrowings along degraded and aggraded rivers. A variational approach based on the Minimum Stream Power principle is here proposed, results of which agree fairly well with laboratory and field data. Nevertheless, due to the lack of a consolidated theoretical framework, it cannot certainly be concluded that a variational approach is the best method to apply, though, until now, no better results have been obtained by using existing rational methods.

In the first part of this paper I will deal with distributions of statistical quantities in the normal sections of different streams. The statistical quantities I will deal with will be the local mean velocity, the three components of the turbulent velocity and Reynolds’ stresses like uv. Furthermore, the streams which I will deal with will be mainly the uniform flow (pipe flow or open channel flow) and the boundary layer flow (with different pressure gradients).At the end of this review, in a second part of the paper I will analyse the distributions of only the local mean velocity in the case of solid transport comparing them with the previous ones. Finally, in a third part, I will deal with the local mean velocity distribution as it can be found on the ground of the principle of the maximum entropy. This analysis will be made as an exemple of comparison between quantities that result from entropy principles and analog quantities which are experimentally known and result from the classical frame of the turbulent flows.

Some kinds of turbulent phenomena in fluid motion, like intermittency, can be explained in terms of vortex structures evolution. Simple interactions between a vortex structure and a bluff body can exhibit complex underlying characteristics, and an unpredictable behavior. In the present work the interaction between a vortex ring and a moving rigid sphere is analyzed, in an ideal flow. We start by assuming the motion perfectly axisymmetric to define the main features of the flow; then an analysis of a full three-dimensional evolution is taken into account. The motion can be written in Hamiltonian form, and exhibits definite singular points. When the sphere oscillates there can be a ‘stochastization’ of the hyperbolic point and the behavior becomes unpredictable. The Melnikov technique is used to observe if this is the case, and proves that the vortex ring trajectory can be unpredictable and chaotic. An axisymmetric numerical simulation is performed in correspondence of unpredictable evolution, and chaotic ring trajectories are examined. A fully three-dimensional simulation is also performed in order to highlight the limitations of three-dimensional computational vortex methods.