Volume 10, 1877
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### Art. VI.—On the Influence of the Earth's Rotation on Rivers.

[Read before the Philosophical Institute of Canterbury, 4th October, 1877.]

In his address to this Institute, delivered on April 5th, 1877, Dr. von Haast devotes some space to the explanation of the important fact that rivers, whose banks are composed of loose materials, wear away their right banks in the northern hemisphere and their left in the southern, at the same time building up the opposite banks.

Dr. von Haast gives an account of the theory of Von Baer, who first showed that the observed changes in the courses of rivers might be explained as a consequence of the earth's rotation. Von Baer's explanation depends upon a well known mechanical theorem, by which the westward motion of the trade winds had previously been accounted for, and which may be thus stated—“A body moving on a meridian tends to be deflected towards the right in the northern hemisphere, and towards the left in the southern, in consequence of the change in its eastward velocity as it approaches or recedes from the earth's axis.” The change in the eastward velocity, it is necessary to observe, accounts for only a part of the deflecting force. The direction of motion in space is also changing as the earth revolves. A railway truck moving on a meridian in the southern hemisphere has its line of motion turned round in the same direction as that of the hands of a watch. There must consequently be a pressure against the truck towards the right, and an equal pressure against the rails towards the left, which must be added to that caused by the change of eastward velocity.

The explanation given by Von Baer does not account for the fact that where the course of a river is east and west the banks are worn away in the same manner as where the course is on a meridian. To explain this it must be shown that a body moving at right angles to the meridian tends to be deflected.

A body resting on the earth's surface, and free to move in any direction upon it, is maintained in equilibrium by attraction directed towards the earth's centre, and centrifugal force directed away from the axis. If the

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centrifugal force ceased, the body would evidently move towards the nearest pole, as down a hill. From the poles to the equator may therefore be regarded as up-hill—bodies free to move being prevented from going down towards the poles by centrifugal force. Suppose, now, a body to move from west to east, that is, in the same direction as the earth revolves; the centrifugal force of the body is increased and there is a tendency to move up-hill, towards the equator. If the motion be from east to west, the centrifugal force is diminished, and the body tends towards the pole. In each case the tendency is towards the right in the northern hemisphere and towards the left in the southern.

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The deflecting force arising from the earth's rotation being a horizontal force acting always at right angles to the direction of motion, its effect on a stream in the southern hemisphere must be to raise the water-level at the left bank and lower it at the right, and this difference of level by increasing the depth would increase the velocity, and consequently the erosive power at the left bank. It might appear that the wearing away of the left banks of rivers in the southern hemisphere is thus accounted for, but examination will make it evident that this explanation is insufficient. It will be shown that the difference of level at the opposite sides of a stream in lat. 45°, whose mean velocity is three miles an hour, is 1/71023 of the width, which in a stream a mile wide is only 9/10 inch. The effect of the small difference in the erosive power, due to that difference of level—in causing unequal wearing away of the banks—would be neutralized, if the left bank were composed of slightly harder material than the right; or if the left bank were a little higher, so that the quantity of material to be removed for each foot cut away horizontally would be greater than at the right. The small difference of erosive power could not explain how it is that, as a general rule, the bank which is being worn away is much the highest, the opposite bank being, in many rivers, below flood level, especially when it is taken into consideration that the material below the water-line at the high bank has been consolidated by superincumbent pressure, and made more compact and difficult to break up than the loose recent deposit of the river, of which the low bank consists. Take for instance the lower course of the river Rangitata, where there is a high terrace on the left, and a low plain on the right. The erosive power at the left bank would have to be many times, instead of a small fraction, greater than at the right, to account for the high terrace being cut away instead of the low plain.

I shall try to show that the changes in river-courses are due to the unequal velocities of the surface and bottom layers of running water.

In ordinary streams the velocity increases nearly uniformly from the bottom to the surface, the deflecting force being proportional to the

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velocity of the moving water. The water near the surface is urged towards the left bank with greater force than that near the bottom. To ascertain the kind of motion caused by these unequal forces:—Suppose the stream to be straight, of uniform cross-section, to be everywhere of the same depth, and imagine it to be divided into a great number of layers parallel to the surface, each moving with different velocity. Then, the increase of pressure against the left bank, due to the earth's rotation, equals the sum of the deflecting forces, which sum is the same as if the mean deflecting force acted on every layer. Therefore, the water-level at the left bank is raised to the same height, and the surface-line of the cross-section is inclined to the horizon at the same angle as if the mean deflecting force acted on every layer, and the tangent of the inclination is the latter force divided by the force of gravitation.

The accompanying cut shows the cross-section of a stream, the angle A being very much exaggerated.

Let a be a small cube of water in any part of the cross-section whose volume is dx dy dz.

h, h1 the depths below the surface of the centres of the left and right sides of the cube respectively.

f, f1 the deflecting forces acting on the cube a and on a similar cube in the middle layer respectively.

w, the weight of an unit of volume of water.

A, the angle of inclination of the surface of the stream from left to right.

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F, the resultant force acting on the cube a. Then F = w (h—h1) dy dz—f = tan A. w dx dy dz—f And tan A =f/w dx dy dz Therefore F = f1f

That is, the resultant force acting on a particle in any part of the cross-section is the difference of the deflecting forces acting on that particle and a particle in the middle layer. This quantity is of different sign for particles situated above and below the middle line, showing that the resultant force acts in opposite directions, above and below that line. These forces must evidently cause a circulation, as shown by the arrows in the figure, which motion will be combined with that down the stream, so that the actual motion of any particle is inclined at a very small angle to the direction of the channel. It is clear that a very slow motion of the bottom layer from left to right must cause a transfer to the right side of the

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river-bed of shingle and sand rolled along by the current. The effect of this is plainly to make the right shore more shelving and the left steeper, and to place the deepest part of the stream nearer the left bank. This will cause the velocity and consequently the erosion to be greater at the left bank. The right shore being more shelving will be more favourable to the deposition of sediment during the subsidence of floods. The continual building-up of the right bank explains how a river cuts away on its left bank harder material than that of which the right is composed; while the latter, instead of being cut away, is being added to. The wearing away of a high terrace instead of a low flat on the opposite side of the river is similarly explained.

The principal facts in connection with the changes of river courses are thus accounted for, which a minute difference in the erosive power at the opposite banks of rivers, caused directly by a difference of level, appears inadequate to explain.

To find the deflecting force acting on a body moving on the earth's surface in any direction:—

Let p be the position of the body on the great circle A A′, the inclination of whose plane to the plane of the equator is a, the motion of the body being towards A′. Let O A′ be the axis of x, O C the axis of z, that of y being perpendicular to the plane of the paper.

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Let θ be the angular velocity of the earth round the polar axis P P′; θ1, θ3, the angular velocities round the axes of x and z respectively. Then θ1 = θ sin a, θ3 = θ cos a Let x = r cos ø, y = r sin ø

r being the radius of the earth, and ø being measured from the axis of x towards that of y.

O p D being a spherical triangle, and the angle D a right angle, sin p D = sin a cos ø = sin Lat.

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The equations of motion of a particle, referred to axes moving in any manner about a fixed origin, are—the mass being unity,— X = du/dt - vθ2 + wθ2     v = dx/dt - yθ3 + zθ2 Y = dv/dt - wθ1 + uθ3    v = dy/dt - zθ1 + xθ3 Z = dw/dt - vθ2 + vθ1    w = dz/dt - xθ2 + yθ1

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Therefore X = —2 dy/dtθ3xθ23 = — 2 θ V cos a cos ø — r θ2 cos2 a cos ø Y = —2 dx/dtθ3yθ21yθ23 = — 2 θV cos a sin ø — r θ2 sin ø Z = 2 dy/dtθ1 + xθ1θ3 = 2θV sin a cos ø + r θ2 sin a cos a cos ø = 2 θ V sin Lat. + r θ2 sin a cos a cos ø

V being the velocity of the body on the earth's surface. The terms in X, Y, Z, not containing V, are the forces parallel to the axes, acting on a body at rest on the earth's surface. Their resultant is therefore balanced by centrifugal force, and reaction of the earth's surface. The terms containing V in X and Y, being resolved along the tangent at p to the great circle A A′, cancel, showing that the earth's rotation has no effect in accelerating or retarding a moving body. The term in Z, 2 θ V sin Lat. represents a force acting in a tangent to the earth's surface, and at right angles to the line of motion of the body, the positive sign showing that the constraining force is directed towards the left, the body having an equal tendency towards the right which is the deflecting force.

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Let the mean velocity of a stream be three miles an hour (the stream being assumed to be everywhere of the same depth, and the velocity to be equal at equal depths below the surface), then the deflecting force being proportional to the velocity, the mean deflecting force acting on the stream is that due to a velocity of three miles an hour, and it has been shown that the angle of inclination of the surface line of the cross-section to the horizon, is that whose tangent is the mean deflecting force divided by the force of gravitation = f1/g suppose— And f1/g = 2 θ V sin Lat./g where θ is the angular motion of the earth per second which is 0.0000729 V = 3 miles an hour or 4.4 feet per second Sin Lat. = sin 45°= 0.707 Then f1/ g = 2 × 0.0000729 × 0.707 × 4.4/32.2=0.00001408 =1/71023 which multiplied by 63360, the number of inches in a mile, gives 9/10 inch nearly, which is the difference of level at the opposite sides of a stream a mile wide.