Causes of Meander Formation in River Channels and the So-Called Baer’s Law*)

A. Einstein

It is generally known that river channels have a tendency to acquire a winding form instead of following the line of maximum slope of the terrain. It is also well known to geographers that rivers of the northern hemisphere erode chiefly the right bank. Rivers of the southern hemisphere behave in the opposite way (Baer’s law). Many attempts have been made to explain this phenomenon, and I am not sure whether what I shall say below will be new to specialists; some part of my considerations is undoubtedly already known. Nevertheless, having found no one who was fully acquainted with the causes of the effects under discussion, I consider it appropriate to give here a brief qualitative description of them.

First of all it is clear that erosion must be the stronger, the greater the velocity of the current at the place where it touches the bank under consideration; more precisely, erosion must be stronger at that place of the bounding wall where the velocity of the current falls most rapidly to zero. This is true under all circumstances, irrespective of whether erosion is caused by mechanical or physicochemical factors (decomposition of the soil). Therefore we should concentrate our attention on the circumstances that affect the magnitude of the velocity gradient at the wall.

In both cases the asymmetry of the fall in velocity under discussion is indirectly due to the formation of a circulation, on which we shall focus our attention.

I shall begin with a small experiment that anyone can easily repeat. Imagine a cup with a flat bottom, full of tea. On the bottom there are several tea leaves, which remain there because they are heavier than the liquid they displace. If, with the aid of a spoon, the liquid is set rotating, the tea leaves will quickly gather

) Presented at a meeting of the Prussian Academy on January 7, 1926; published in Die Naturwissenschaften, 14* (1926).

in the center of the bottom of the cup. The explanation of this phenomenon is as follows: the rotation of the liquid leads to the appearance of centrifugal forces. These forces by themselves could not bring about a change in the flow of the liquid if the latter rotated like a solid body. But in the vicinity of the walls of the cup the liquid is held back owing to friction, so that the angular velocity with which it rotates proves to be smaller than in other places, closer to the center. In particular, the angular velocity of rotation, and consequently also the centrifugal force, will be smaller near the bottom than farther away from it. The result of this will be a circular motion of the liquid, similar to that illustrated

Fig. 1.              Fig. 2.

in Fig. 1, which increases until, under the influence of friction, it becomes stationary. Tea leaves are carried to the center by the circular motion, thereby proving its existence.

A similar kind of situation occurs for a curved flow (Fig. 2). In each transverse section of its course, where it is curved, the centrifugal force acts in the direction of the outer side of the curve (from \(A\) to \(B\)). This force near the bottom, where the velocity of the current is weakened by friction, proves to be smaller than the corresponding force in the layers situated higher above the bottom. This gives rise to the circular motion indicated in Fig. 2. Even where there are no bends in the river, a circular motion similar to that shown in Fig. 2 will nevertheless occur, although on small scales, as a result of the earth’s rotation. The latter leads to the appearance of the Coriolis force, acting perpendicular to the direction of the current, whose right horizontal component is equal to \(2v\Omega \sin\varphi\) per unit mass of liquid, where \(v\) is the velocity of the current, \(\Omega\) is the angular velocity of the earth’s rotation, and \(\varphi\) is the geographical latitude. Since friction against the ground leads to a decrease of this force on approaching the bottom, this force also leads to the emergence of a circular motion of the type indicated in Fig. 2.

After this preliminary discussion, let us return to the question of the distribution of velocities over the transverse section of the flow, which is the determining factor in erosion. For this purpose we must first of all form a clear idea of how a (turbulent) distribution of velocities develops and is maintained. If water, previously at rest, were suddenly set in motion by the action of a uniformly distributed force, then the distribution of velocities over the transverse section would at first prove uniform.

CAUSE OF THE FORMATION OF MEANDERS IN RIVER CHANNELS AND BAER’S LAW

...ed. A distribution of velocities gradually increasing from the bounding walls toward the center of the cross-section would be established only after some time under the influence of friction against the walls. The disturbance (roughly speaking) of the stationary distribution of velocities over the cross-section would be established only gradually under the influence of the internal friction of the fluid.

Hydrodynamics describes in the following way the process as a result of which this stationary distribution of velocities is established. In a plane (potential) flow all vortex filaments are concentrated at the walls. They separate and slowly move toward the center of the channel cross-section, spreading over a layer of increasing thickness. In connection with this, the velocity gradient at the walls gradually decreases. Under the action of the internal friction of the fluid in the inner part of the cross-section the vortex filaments are gradually absorbed; their place is taken by new ones formed at the walls. Thus a quasi-stationary distribution of velocities is formed. What is important for us is that the attainment of a stationary distribution of velocities is a slow process. This is why relatively insignificant, constantly acting causes are capable of exerting a considerable influence on the distribution of velocities over the cross-section. Let us now consider what influence the circular motion, caused—as was shown in Fig. 2—by a bend of the river or by the Coriolis force, has on the distribution of velocities over the cross-section of a river. The fluid particles moving most rapidly will be farthest from the walls, i.e. in the upper part above the center of the bed. These fastest parts of the water will be carried by the circulation toward the right wall, while water coming from the region near the bottom and having an especially small velocity arrives at the left wall. Consequently, in the case shown in Fig. 2, erosion is inevitably 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 circulatory motion of the water exerts a significant influence on the distribution of velocities, because the regulation of velocities by internal friction, counteracting the influence of the circulatory motion, is likewise a slow process.

We have shown the causes of the formation of river meanders. However, on the basis of what has been said, certain additional details may also be clarified without difficulty. Erosion will prove comparatively stronger not only at the right wall, but also in the right half of the bed, so that a tendency arises toward the formation of the profile shown in Fig. 3.

Fig. 3.

Moreover, water will come to the surface from the left wall, and therefore, especially on the left side, the water will be...

move less rapidly than in the somewhat lower layers. This is in fact what is observed. It should further be noted that circular motion possesses inertia. Therefore the circulation will reach its maximum only after the point of greatest curvature, and the same applies to the asymmetry of erosion. Consequently, in the process of erosion the meandering line of the river must shift in the direction of the current. Finally, in the case of a larger cross-section of the river, the circulatory motion is more slowly destroyed by friction; therefore the undulating line of meander formation will increase with an increase in the river’s cross-section.