Maximum Entropy Theory by Using the Meandering Morphological Investigation LEVENT YlLMAZ Technical University of Istanbul, Hydraulic Division, Civil Engineering Department, Maslak, 80626, Istanbul, Turkey.; E-mail: lyilmaz@itu.edu.tr Received: November 02, 2006 Accepted: November 14, 2006 Abstract: Based on the principle of maximum entropy the primary morphologic equation is derived, and then the equations for hydraulic geometry of longitudinal profile and cross-section are established. For V-shaped cross-sections the relevant morphologic equations which are derived are compared with the existing empirical and semi-empirical formulae. They show good agreement with the prototypes. Key words: Principle of maximum entropy, hydraulic geometry, morphologic equation Introduction Einstein (1950) once pointed out that entropy theory is the first theory for overall science. The principle of maximum entropy has been extensively applied in many domains of natural science. In view of this fact, to apply the principle of maximum entropy is adopted to deal with characteristics of alluvial channels capable of carrying given water and sediment load in meandering boundary layers without causing excessive aggradation and degradation. Width adjustment may take place over a wide range of scales in time and space at meandering channels. In the past engineering analyses of channel width have tended to concentrate on prediction of the equilibrium width for stable channels. Most commonly the regime; extremal hypothesis, and rational (mechanistic) approaches are used. By meandering channels, more recently, attention has switched to channels that are adjusting their morphology either due to natural instability or in response to changes in meandering watershed land use, river regulation, or channel engineering. Characterizing and explaining the time-dependent behavior of width in such channels requires and understanding of the fluvial hydraulics of unstable channels, especially in the near-bank regions. Useful engineering tools are presented, and gaps requiring further field and laboratory research are identified. Finally, this research will consider the mechanics of bank retreat due to flow erosion and deposition at meandering bends, mass failure under gravity, and bank advance due to sedimentation and berm building. It will be demonstrated that, while rapid progress is being made, most existing analyses of bank mechanics are still at the stage of being research tools that are not yet suitable for design applications. Most mathematical models, however, neglect time- dependent channel width adjustments and do not simulate processes of bank erosion or deposition at meandering channels. Although changes in channel depth caused by aggradation or degradation of the river bed can be simulated, changes in width cannot. Meandering channel morphology usually changes with time, and adjustment of both width and depth, in addition to changes in planform, roughness, and other attributes are the rule rather than the exception (Leopold et al., 1964; simon and Thorne, 1996). As a result, the ability to model and predict changes in river morphology and their engineering impacts is limited. The meandering river width adjustments can seriously impact floodplain dwellers, riparian ecosystems, bridge crossings, bank protection works, and other riverside structures, through bank erosion, bank accretion, or bankline abandonment by the active river channel, which are very important for sustainable development of European Mediterranean countries. Considerable research effort has recently been directed towards improving this situation. The objectives of the river width adjustment research were as follows: • Review the current understanding of the fluvial processes and bank mechanics involved in river width adjustment • Evaluate methods (including regime analysis, extremal hypotheses and rational, mechanistic approaches) for predicting equilibrium river width • Assess the present capability to quantify and model width adjustment • Identify current needs to advance both state-of-the-art research and the solution of real world problems faced by practicing engineers To achieve these objectives, river width adjustments may occur due to a wide range of morphological changes and channel responses. Widening can occur by erosion of one or both banks without substantial incision (Everitt, 1968; Burkham, 1972; Hereford, 1984; pizzuto, 1992). Widening in sinuous channels may occur when outer bank retreat, due to toe scouring, exceeds the rate of advance of the opposite bank, due to alternate or point bar growth (Nanson and Hickin, 1983: Pizzuto, 1994) while, in braided rivers, bank erosion by flows deflected around growing braid bars is a primary cause of widening (Leopold and Wolman, 1957; Best and Bristow, 1993; Thorne et al., 1993). In degrading streams, widening often follows incision of the channel when the increased height and steepness of the banks causes them to become unstable. Bank failures can cause very rapid widening under these circumstances (Thorne et al., 1981; Little et al., 1982; Harvey and Watson, 1986; simon, 1989). Widening in coarse-grained, aggrading channels can occur when flow acceleration due to a decreasing cross-sectional area, coupled with current deflection around growing bars, generates bank erosion (simon and Thorne, 1996). Morphological adjustments involving channel narrowing are equally diverse. Rivers may narrow through the formation of in-channel berms, or benches at the margins. Berm/bench growth often occurs when bed levels stage following a period of degradation and can eventually create a new, low-elevation floodplain and establish a narrower, quasi-equilibrium channel (Woodyer, 1968; Harvey and Watson, 1986; simon, 1989; Pizzuto, 1994). Narrowing in sinuous channels occurs when the rate of alternate or point bar growth exceeds the rate of retreat of the cut bank opposite (Nanson and Hickin, 1983; Pizzuto, 1994). Croachment of riparian vegetation into the channel is often satisfied as contributing to the growth, stability, and initiation of berm or bench features (Hadley, 1961; Schumm and Lichty, 1963; Harvey and Watson, 1986; Simon, 1989). In braided channels, narrowing may result when a marginal anabranch in the braided system is abandoned (Schumm and Lichty, 1963). Sediment is deposited in the abandoned channel until it merges into the floodplain. Also, braid bars or islands may become attached to the floodplain, especially following a reduction in the formative discharge. Island tops are already at about floodplain elevation and attached bars are built up to floodplain elevation by sediment deposition on the surface of the bar, often in association with establishment of vegetation. Attached islands and bars may, in time, become part of the floodplain bordering a much narrower, sometimes single-threaded channel (Williams, 1978; Nadler and Schumm, 1981). If the flow regime and sediment supply are quasi-steady over periods of decades, the morphology of the river adjusts to create a metastable, equilibrium form (Schumm and Lichty, 1965). Such rivers are described as being graded or in regime (Mackin, 1948; Leopold and Maddock, 1953; Wolman, 1955; Leopold et al. 1964; Ackers and charlton, 1970a). Although the width of an equilibrium stream may change due to the impact of a large flood or some other extreme event, the stable width is eventually recovered following such perturbations (costa, 1974; Gupta and Fox, 1974; Wolman and Gerson, 1978). Unfortunately, predicting the time-averaged morphology of equilibrium channels remains, despite years of effort, a difficult problem (Ackers, 1992; Ferguson, 1986; Bettess and White, 1987). Many rivers, however, cannot be considered to have equilibrium channels even as an engineering approximation. These rivers display significant morphological changes. Under the assumption that the only information available on a drainage basin is its mean elevation, the connection between entropy and potential energy is explored to analyze drainage basins morphological characteristics. Nearly, 30 years ago, Leopold and Langbein (1962) applied for the first time the concepts of physical entropy to study the behavior of streams. Their application was based on the analogy between heat energy and temperature in a thermodynamic system and potential energy and elevation, respectively, in a stream system. Two ther-modynamic principles were applied. The first principle is that the most probable state of a system is the one of maximum entropy. The second is the principle of minimum entropy production rate. Using these principles, Yang (1971) derived for a stream system the law of average stream fall, and the law of the least rate of energy expenditure . Yang (1971) and others have since applied the latter law to a range of problems in hydraulics. The connection between entropy and potential energy, which these workers so successfully exploited to investigate river engineering, sediment transport, and other problems, was not exploited in hydrology. In this work we pursue this connection to derive relations between entropy and mean elevation for a drainage basin network and to derive relations for the river profiles. Much of the work employing the entropy concepts in hydrology has been with the application of informational entropy. The beginnings of such a work can be traced to Lienhard (1964), who used a statistical mechanical approach to derive a dimensionless unit hydrograph of a drainage basin. It may be visualized that the study of the landscape is the study of constraints imposed by geologic structure, lithology, and history. The way in which some constraints affect the river profile can be evaluated if one considers the profile to approximate its maximum probable condition under a given set of constraints. The most important observations are summarized as below: a) The absence of all constraints leads to no solution. b) The longitudinal profile of a stream system subject only to the constraint of base level is exponential with respect to elevations above base level. c) The profile of a stream subject only to the constraint of length is exponential with respect to stream length which is a logarithmic function with respect to elevation. d) Introduction of the constraint of a partial base level above that of the sea adds a measure of convexity in the profile. Principle of maximum discrete ENTROPY For a discrete variable X, Shannon (1948) defines quantitatively the entropy in terms of probability as: H(X) = -fjP(Xi)LnP(Xi) ,=i (1) where P(Xi) is the probability of a system being in state X. which is a member of i » {X., i = 1, 2, ...}, and ZjP(X.) = L It has i ¡=1 been proved that H(X) defined by Equation 1 is the only function to satisfy the following three conditions: a) H(X) is the continuous function of P(Xi). b) If and only if all P(X^ are equal, H (X) attains its maximum value. This conclusion is known as the principle of maximum discrete entropy. c) If the states X and Y are mutually inde -pendent, then H(XY) = H(X)+H(Y). Jaynes (1957) have proved that an equilibrium system under steady constraints tends to maximize its entropy. Based on this statement, the entropy of a river system, having reached its dynamic equilibrium, should approach its maximum value, also the principle of maximum entropy should be valid too for the case of regime rivers. Stream Power Although many formulas for sediment transport have been devised, most can be expressed in terms of stream power as suggested by Bagnold (1960). Power is an important factor in the formulation of the hydraulic geometry of river channels. As explained by Bagnold, the stream power at flows sufficiently great to be effective in shaping the river channel is directly related to the transport of sediment, whose movement is responsible for the channel morphology. Laursen (1958) gives several typical equations for the transport of sediment, based on flume experiments and the average relation shows sediment transport in excess of the point of incipient motion to vary about as (vDS)15 where vDS is the stream power per unit area. In terms of sediment per unit discharge, that is the concentration, C, the several equations average out as C x n (vD)0 5 S15, a result that is consistent with the conclusion reached by Bagnold (1960). There is in addition to be considered the effect of sediment size. Examination of several equations indicates that sediment transport varies inversely as about the 0.8 power of the particle size. There have been several attempts to relate particle size to the friction factor n and by using the Strickler relation that the value of n varies as the 1/6 power of the particle size. It is realized full well that both the sediment transport and the friction factor are influenced by many other factors such as bed form and the cohesive-ness, sorting, and texture of the material. These are the kinds of influences, themselves effects of the river, that prevent a straightforward solution of river morphology. In order to limit the number of variables only the effect of particle size on transport will be considered, as this factor varies systematically along a river from headwater to mouth. Thus, sediment transport concentration is given as C x (vD)05 S15 / n4. The sediment transport per unit discharge in the river system will be recognized as a hydrologic factor that is independent of the hydraulic geometry of a river in dynamic equilibrium. Consequently sediment concentration may be considered constant. Thus, there are three equations: continuity, hydraulic friction, and sediment transport. There are five unknowns. The two remaining equations will be derived from a consideration of the most probable distribution of energy and total energy in the river system. The probability of a given distribution of energy is the product of the exponential functions of the ratio of the given units to the total as pcoe E e E e E ...etc. (1a) The ratios of the units of energy E1, E2, etc., representing the energy in successive reaches along the river sufficiently long to be statistically independent, to the total energy E in the whole length, are E1/E ; E2/E;......En/E. The product of the exponentials of these is the probability of the particular distribution of energy. As previously, the most probable condition is when this joint probability, p, is a maximum and this exists when E1 = E2 = E3 ... = En. Thus energy tends to be equal in each unit length of channel (Leopold and Langbein, 1962). Equable distribution of energy corresponds to a tendency toward uniformity of the hydraulic properties along a river system. Considering the internal energy distribution, uniform distribution of internal energy per unit mass is reached as the velocity and depth tend toward uniformity in the river system. Since the energy is largely expended at the bed equable distribution of energy also requires that stream power per unit of bed area tend toward uniformity. An opposite condition is indicated by Prigogine's (1955) rule of minimization of entropy production which leads to the tendency that the total rate of work, ZQSAQ in the system as a whole be a minimum. Because S x Qz, then ZQ1+z AQ^a minimum. For a given drainage basin this condition is met as z takes on increasingly large negative values. However, there is a physical limit on the value of z, because for any drainage basin the average slope ZSAQ/^AQ must remain finite. This condition is met only for values of z greater than -1, and therefore z must approach -1 or 1 + z approaches zero. The condition of minimum total work tends to make the profile concave; whereas the condition of uniform distribution of internal energy tends to straighten the profile. Hence, we seek the most probable state. The most probable combination is the one in which the product of the probabilities of deviations from expected values is a maximum. It is unnecessary to evaluate the probability function, provided one can assume normality, as we can then state directly that the product of the separate probabilities is a maximum when their variances are equal. \2 F \ y V ( G F \ y \2 G F V 3 — etc. where F2, F3 represent the several functions. The standard deviations c , c _ c , and m f z' cy represent the variability of the several factors as may occur along a river system. Since these values are not known initially, the problem must be solved by iteration (Leopold and Langbein, 1962). Fortunately, the solution is not sensitive to the values of the several standard deviations, so the solution converges rapidly. Therefore, Fx Fi —-—l--— Fx G F = 0 2 y (3) for which there are two possible solutions: =0 f \ Fx F2 —-—l--- g f G F V 's (4) or / =0 (5) To summarize, we have introduced three statements on the energy distribution: (2) 1 -z-y->0 2 m + / + z-> 0 (l + z)—» 0 (6), (7),(8) The absolute values of the standard deviations need not be known, as we can infer their relative values. For example, letting Fj = (1/2)z - y , the standard deviation of Fj is ^(CT z/2)2+CT (9) and F2 = m + f+ z ; oFl = ^ F2=m+f+z (10) Leopold and Maddock (1953) describe and evaluate from field data the hydraulic geometry of river channels by a set of relations as follows: v