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Meanders are the result of erosion-deposition processes tending toward the most stable form in which the variability of certain essential properties is minimized. This minimization involves the adjustment of the planimetric geometry and the hydraulic factors of depth, velocity, and local slope. The planimetric geometry of a meander is that of a random walk whose most frequent form minimizes the sum of the squares of the changes in direction in each successive unit length. The direction angles are then sine functions of channel distance. This yields a meander shape typically present in meandering rivers and has the characteristic that the ratio of meander length to average radius of curvature in the bend is 4.7. Depth, velocity, and slope are shown by field observations to be adjusted so as to decrease the variance of shear and the friction factor in a meander curve over that in an otherwise comparable straight reach of the same river. Since theory and observation indicate meanders achieve the minimum variance postulated, it follows that for channels in which alternating pools and riffles occur, meandering is the most probable form of channel geometry and thus is more stable geometry than a straight or nonmeandering alinement. INTRODUCTION So ubiquitous are curves in rivers and so common are smooth and regular meander forms, they have attracted the interest of investigators from many disciplines. The widespread geographic distribution of such forms and their occurrence in various settings had hardly been appreciated before air travel became common. Moreover, ah* travel has emphasized how commonly the form of valleys not merely the river channels within them assumes a regular sinuosity which is comparable to meanders in river channels. Also, investigations of the physical characteristics of glaciers and oceans as well as landfonns led to the recognition that analogous forms occur in melt-water channels developed on glaciers and even in the currents of the Gulf Stream. The striking similarity in physical form of curves in these various settings is the result of certain geometric proportions apparently common to all, that is, a nearly constant ratio of radius of curvature to meander length and of radius of curvature to channel width (Leopold and Wolman, 1960, p. 774). This leads to visual similarity regardless of scale. When one looks at a stream on a planimetric map without first glancing at map scale, it is not immediately obvious whether it is a large river or a small stream. Explanation of river meanders have been varied. It has been suggested that meanders are caused by such processes as regular erosion and deposition; secondary circulation in the cross-sectional plane; seiche effect analogous to lake seiches. Attempts, however, to utilize these theories to calculate the forms of meanders have failed. Although various phenomena including some of those mentioned above, such as cross circulation are intimately involved in the deviation of rivers from a straight course, the development of meanders is patently related to the superposition of many diverse effects. Although each of these individual effects is in itself completely deterministic, so many of them occur that they cannot be followed in detail. As postulated in this paper, such effects can be treated as if they were stochastic that is, as if they occurred in a random fashion. This paper examines the consequences of this postulate in relation to (1) the planimetric geometry of meanders, and (2) the variations in such hydraulic properties as depth, velocity, and slope in meanders as contrasted with straight reaches. The second problem required new data. These were obtained during the snowmelt season of high discharge in the years 1959-64 by the junior author with the invaluable and untiring assistance of William W. Emmett and Robert M. Myrick. Leon W. Wiard was with us for some of the work. To each of them, the authors are very grateful for their important contribution, measured in part by the internal consistency of the various field data, and the satisfactory closures of surveys made under difficult conditions. HI 112 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

Analytical results for 89 water samples from the Papago Indian Reservation, Arizona