## 1. Introduction

[2] The term “hydraulic geometry” connotes the relationships between the mean stream channel form and discharge both at-a-station and downstream along a stream network in a hydrologically homogeneous basin. The channel form includes the mean cross-section geometry (width, depth, etc.), and the hydraulic variables include the mean slope, mean friction, and mean velocity for a given influx of water and sediment to the channel and the specified channel boundary conditions. *Leopold and Maddock* [1953] expressed the hydraulic geometry relationships for a channel in the form of power functions of discharge as

where B is the channel width, d is the flow depth, V is the flow velocity, Q is the flow discharge, and a, b, c, f, k, and m are parameters. Also added to equation (1a) are:

where n is Manning's roughness factor, S is slope, and N, p, s, and y are parameters. Exponents b, f, m, p and y represent, respectively, the rate of change of the hydraulic variables B, d, V, n, and S as Q changes; and coefficients a, c, k, N, and s are scale factors that define the values of B, d, V, n, and S when Q = 1. The hydraulic geometry relations (1a) and (1b) are of great practical value in prediction of channel deformation; layout of river training works; design of stable canals and intakes, river flow control works, irrigation schemes, and river improvement works. These relations through their exponents can also be employed to discriminate between different types of river sections [*Richards*, 1976] as well as in planning for resource and impact assessment [*Allen et al.*, 1994].

[3] The hydraulic variables, width, depth, and velocity, satisfy the continuity equation:

Therefore the coefficients and exponents in equation (1a) satisfy

[4] The at-a-site hydraulic geometry entails mean values over a certain period, such as a week, a month, a season, or a year. The concept of downstream hydraulic geometry involves spatial variation in channel form and process at a constant frequency of flow. *Richards* [1982] has noted that the downstream hydraulic geometry involving channel process and form embodies two types of analyses, both of which are expressed as power functions of the form [*Rhoads*, 1991] given by equations (1a) and (1b). The first type of analysis is typified by the works of *Leopold and Maddock* [1953] and *Wolman* [1955], who formalized a set of relations, such as equations (1a) and (1b), to relate the downstream changes in flow properties (width, mean depth, mean velocity, slope, and friction) to the mean discharge. This type of analysis describes regulation of flow adjustments by channel form in response to increases in discharge downstream, and has been applied at particular cross sections as well as in the downstream direction.

[5] The second type of analysis is a modification of the original hydraulic geometry concept and entails variation of channel geometry for a particular reference discharge downstream with a given frequency. Implied in this analysis is an assumption of an appropriate discharge that is the dominant flow controlling channel dimensions [*Knighton*, 1987; *Rhoads*, 1991]. For example, for perennial rivers in humid regions, the mean discharge or a discharge that approximates bank-full flow (Q_{b}), such as Q_{2} and Q_{2.33}, with a return period of 2 and 2.33 years, respectively, is often used in equations (1a) and (1b). This concept is similar to that embodied in the regime theory [*Blench*, 1952, 1969]. It should, however, be noted that the coefficients and exponents are not constrained by the continuity equation when the selected discharge substantially differs from the bank-full flow. On the other hand, *Stall and Yang* [1970] related hydraulic geometry to flow frequency and drainage area.

[6] The mean values of the hydraulic variables of equations (1a) and (1b) are known to follow, according to *Langbein* [1964] and *Yang et al.* [1981], necessary hydraulic laws and the principle of the minimum energy dissipation rate (or stream power). As a consequence, these mean values are functionally related and correspond to the equilibrium state of the channel. This state is regarded as the one corresponding to the maximum sediment transporting capacity. The implication is that an alluvial channel adjusts its width, depth, slope, velocity, and friction to achieve a stable condition in which it is capable of transporting a certain amount of water and sediment. In other words, the average river system tends to develop in such a way as to produce an approximate equilibrium between the channel and the water and sediment it must transport [*Leopold and Maddock*, 1953]. *Knighton* [1977] observed that at cross sections undergoing a systematic change, the potential for adjustment toward some form of quasi-equilibrium in the short term is related to the flow regime and channel boundary conditions; and that the approach to quasi-equilibrium or establishment of a new equilibrium position is relatively rapid.

[7] The relations of equations (1a) and (1b) have been calibrated for a range of environments, using both field observations and laboratory simulations. *Dury* [1976] confirmed the validity of power function relations for hydraulic geometry using extended sets of data at the 1.58-year mean annual discharge. *Chong* [1970] stated, without a firm basis, that hydraulic geometry relations of equations (1a) and (1b) were similar over varying environments. *Parker* [1978] analyzed the cause of this systematic behavior for gravel rivers. Thus it seems that the regional generalizations proposed in the literature are acceptable for rivers that have achieved “graded-time” equilibrium [*Phillips and Harlin*, 1984]. *Parker* [1979] has stated that the scale factors, a, c, and k, vary from locality to locality but the exponents, b, f, and m, exhibit a remarkable degree of consistency, and seem independent of location and only weakly dependent on channel type. From an analysis of a subalpine stream in a relatively homogeneous environment, *Phillips and Harlin* [1984] found that hydraulic exponents were not stable over space. *Knighton* [1974] emphasized variations in exponents as opposed to mean values. *Rhodes* [1978] noted that the exponent values for high-flow conditions can be vastly different than those for low-flow conditions.

[8] Using data from 318 alluvial channels in the midwestern United States and 50 Piedmont sites, *Kolberg and Howard* [1995] showed that the discharge-width exponents were distinguishable, depending on the variations in materials forming the bed and banks of alluvial channels. Both midwestern and piedmont data indicated that the width-discharge exponents ranged from 0.35 to 0.46 for groups of streams with width to depth ratios less than 45. For groups of streams with width to depth ratios greater than 45, the width-discharge exponents decreased to values below 0.15, suggesting a systematic variation in the exponents and a diminished influence of channel shape. These results are in agreement with the findings of *Osterkamp and Hedman* [1982]. *Howard* [1980] asserted that the variations among channel types are not discrete but can be viewed as continuous. This assertion was supported by *Rhoads* [1991].

[9] *Rhoads* [1991] examined the factors that produce variations in hydraulic geometry parameters. He hypothesized that the parameters are functions of channel sediment characteristics and flood magnitude, and that the parameters vary continuously rather than discretely. Analyzing the variation of channel width with downstream discharge, *Klein* [1981] found that the value b = 0.5 was a good average. The low b values normally occur for small basins (in lower flows) and for very big basins (in very high flows). Thus the b = 0.5 value, being a good average, tends to smooth out deviations from the average. The value of b ranged from 0.2 to 0.89. Klein argued that the simple power function for hydraulic geometry was valid for small basins and that did not hold over a wide range of discharges.

[10] The above discussion shows that the exponents and coefficients of hydraulic geometry relations of equations (1a) and (1b) vary from location to location on the same river and from river to river, as well as from high-flow range to low-flow range. This is because the influx of water and sediment and the constraints (boundary conditions) that the river channel is subjected to vary from location to location as well as from river to river. This means that for a fixed influx of water and sediment a channel will exhibit a family of hydraulic geometry relations in response to the constraints imposed on the channel. It is these constraints that force the channel to adjust its allowable hydraulic variables. For example, if a river is leveed on both sides, then it cannot adjust its width and is therefore left to adjust other variables, such as depth, friction, slope, and velocity. Likewise, if a canal is lined, then it cannot adjust its friction. This aspect does not seem to have been fully explored in the literature.

[11] Various approaches have been employed for deriving functional relationships among the aforementioned hydraulic variables for downstream hydraulic geometry or equations (1a) and (1b). These approaches are based on the following theories: (1) empirical theory (e.g., regression theory [*Leopold and Maddock*, 1953], regime theory [*Blench*, 1952]), (2) tractive force theory [*Lane*, 1955] and its variants-threshold channel theory [*Li*, 1974] and stability theory [*Stebbings*, 1963], (3) hydrodynamic theory [*Smith*, 1974], (4) thermodynamic entropy theory [*Yalin and Da Silva*, 1997, 1999], (5) minimum extremal theories (e.g., minimum channel mobility theory [*Dou*, 1964], minimum energy dissipation rate theory or its simplified versions of minimum unit stream power theory [*Yang and Song*, 1986] and minimum stream power theory [*Chang*, 1980, 1988; *Yang et al.*, 1981], minimum energy dissipation theory [*Rodriguez-Iturbe et al.*, 1992], minimum energy degradation theory [*Brebner and Wilson*, 1967], minimum entropy production theory [*Leoplod and Langbein*, 1962], principle of least action [*Huang and Nanson*, 2000], and minimum variance theory [*Langbein*, 1964]), and (6) maximum extremal theories (maximum friction theory [*Davies and Sutherland*, 1983], maximum sediment discharge theory [*White et al.*, 1982], maximum sediment discharge and Froude number theory [*Ramette*, 1980], and maximum entropy theory [*Deng and Zhang*, 1994]). Each hypothesis leads to unique relations between channel form parameters and discharge, and the relations corresponding to one hypothesis are not necessarily identical to those corresponding to another hypothesis.

[12] The objective of this first part of the two-part paper is to apply the principles of minimum energy dissipation rate and maximum entropy to derive downstream hydraulic geometry relations. Inherent in the derivation is an explanation for self-adjustment of channel morphology. It is shown that by combining the hypotheses based on the principles of maximum entropy and minimum energy dissipation rate a family of hydraulic geometry relations is obtained. This family may encompass many of the hydraulic geometry relations corresponding to other hypotheses. The paper is organized as follows. Introducing hydraulic geometry relations in section 1, derivation of hydraulic geometry relations using the principles of maximum entropy and minimum energy dissipation rate is presented in section 2. The discussion of the derived equations is given in section 3. The paper is concluded in section 4, followed by an appendix and the cited literature.