1 7 2 D O C . 1 5 4 O N F O R M A T I O N O F M E A N D E R S way. In a regular (potential) flow all the vortex filaments are concentrated at the wall. They detach themselves and slowly move toward the center of the cross- section of the stream, distributing themselves over a layer of increasing thickness. The velocity gradient, at the walls thereby gradually diminishes. Under the action of the internal friction of the liquid the vortex filaments in the interior of the cross- section are gradually absorbed, their place being taken by new ones which form at the wall. A quasi-stationary distribution of velocities is thus produced. The import- ant thing for us is that the attainment of the stationary distribution of velocities is a slow process. That is why relatively insignificant, constantly operative causes are able to exert a considerable influence on the distribution of velocities over the cross-section. Let us now consider what sort of influence the circular motion due to a bend in the river or due to the Coriolis-force, as illustrated in Fig. 2, is bound to exert on the distribution of velocities over the cross-section of the river. The particles of liq- uid in most rapid motion will be located farthest away from the walls, that is to say, in the upper part above the center of the ground. These most rapid parts of the water will be driven by the circulation toward the right-hand wall, while the left-hand wall gets the water which comes from the region near the bottom and has a special- ly low velocity. Hence in the case depicted in Fig. 2, the erosion is necessarily stronger on the right side than on the left. It should be noted that this explanation is essentially based on the fact that the slow circulating movement of the water ex- erts a considerable influence on the distribution of velocities, because the adjust- ment of velocities by internal friction, which counteracts this consequence of the circulating movement, is also a slow process. We have now clarified the cause of the formation of meanders. Certain details can, however, also be deduced without difficulty from these facts. Erosion will be comparatively extensive not merely on the right-hand wall, but also on the right half of the bottom, so that there will be a tendency to assume a profile as illustrated in Fig. 3. Moreover, the water at the surface will come from the left-hand wall, and will therefore, on the left-hand side especially, be moving less rapidly than the water rather lower down. This has, in fact, been observed. It should further be noted that the circular motion possesses iner- tia. The circulation will therefore only achieve its maximum behind the place of the greatest curvature, and the same naturally applies to the asymmetry of the erosion. Hence, in the course of the erosion, an advance of the wave-lines of the meander- formation is bound to take place in the direction of the current. Finally, the larger the cross-section of the river, the more slowly will the circular movement be ab-
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