Waves and Tides

### Effect of Friction on Tides and Tidal Currents

In shallow water the tide and the tidal currents will be modified by the friction to which the waters are subjected when moving over the bottom. This bottom friction influences the currents to a considerable distance from the boundary surface, owing to the turbulent character of the flow (p. 480).

The effect of friction on the tide can be illustrated by considering a co-oscillating tide in a bay of constant depth and width. In the absence of friction the tide will have the character of a standing wave that can be considered composed of two waves traveling in opposite direction, the incoming wave and the reflected wave. In the presence of friction the tide can still be considered as composed of two such waves, but the combination no longer results in a single standing oscillation because the amplitudes of both waves must decrease in their directions of progress. In general, it can be assumed that the amount of energy that is dissipated is always proportional to the total energy of the wave. If this is true, the friction leads to a logarithmic decrease of the amplitude, provided the depth is constant. Assume that the waves progress in the x direction, that the influence of friction begins at x = 0 and that reflection takes place at x = l. On these assumptions the amplitude of the incoming wave will be (Fjeldstad, 1929)

and of the reflected wave Here μ represents a coefficient of damping. The amplitude of the tide is found by adding η1 and η2, and the result can be written in the form

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From this equation it follows that the oscillation can be considered as brought about by two standing waves of phase difference π/2 or one quarter of a period (p. 552).

Let us consider a bay the length of which is ⅜ L, where L is the length of the wave in the bay. This means introducing l = ⅜ L, kl = ¾ π, and kx = 2πx/L. Let us furthermore assume that at the opening of the bay the tide can be represented by the equation η0 = Z cos σt, which means that at x = 0 the amplitude is Z and high water occurs at t = 0. In the absence of friction the standing wave in the bay will show a node at a distance of one quarter wave length from the opening, and inside of the node high water will occur at t = 6h if the period of the wave is 12h.

#### Effect of friction on amplitude and phase of the cooscillating tide in a bay, the length of which is 3/8 of the length of the tide wave.

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The variations along the length of the bay of amplitude and phase are shown in fig. 146, by the curves marked 0. The effect of friction will depend upon the value of μ and, in order to illustrate the effect, we introduce three numerical values μ = 8/(15L), μ = 4/(3L), and μ = 4/L, corresponding to a decrease of the amplitude of the tide wave to one half of its value on a distance equal to 1.17 L, 0.52 L, and 0.17 L, respectively. The corresponding variations along the length of the bay of amplitude and phase of the tide are shown in fig. 146 by the curves marked 1, 2, and 3. The dashed line in the upper part of the figure shows the change in phase on three eighths of a wave length of a progressive wave.

By means of fig. 146 three effects of friction are brought out: (1) the node at which the amplitude of the tide is zero disappears and, instead, a region with minimal range is found; (2) the abrupt change of phase disappears and is replaced by a gradual change; (3) the phase difference between the opening and the end of the bay is decreased and approaches

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the value found by considering a progressive wave only. It is evident that if the effect of friction is very great, the wave takes the character of a progressive wave the amplitude of which decreases exponentially because the reflected wave becomes insignificant when the amplitude of the incoming wave has decreased at the end of the bay to a small fraction of its original amount.

The most striking example of the influence of friction on tides is found on the wide shelf along the Arctic coast of eastern Siberia. There the tide wave reaches the shelf from the north after having entered the Polar Sea through the wide opening between Spitsbergen and Greenland and having crossed the deep portions of the Polar Sea. Between longitudes 150°E and 180°E the width of the North Siberian Shelf exceeds 300 miles and in the greater part of that area the depth of the water is between 20 and 40 meters. The sea is ice-covered nearly throughout the year and, owing to the resistance which the ice offers, the tidal currents are subjected to frictional influences from the ice on top as well as from the bottom. The total effect of friction is therefore so great that on the coast the tide nearly vanishes (Sverdrup, 1927, Fjeldstad, 1929 and 1936). The decrease of the amplitude when approaching the coast is brought out by the data in table 72, which shows the amplitude and phase of the term M2 near the border of the shelf and at two localities on the coast. Of these two localities, Ayon Island lies a little south of Four Pillar Island, but the tide wave reaches Four Pillar Island later because the direction of progress of the wave is altered near the coast owing to the configuration of the bottom (Sverdrup, 1927).

AMPLITUDE AND PHASE OF THE SEMIDIURNAL TIDE, M2, ON THE NORTH SIBERIAN SHELF
Locality Latitude N Longitude E M2
Amplitude (cm) Phase (degrees) Difference in phase
Near border of shelf 74°33′ 167°10′ 13.75 158 0
Ayon Island 69 52 167 43 1.78 347 189
Four Pillar Island 70 43 162 35 0.98 60 262

It is seen that the later the tide the smaller the amplitude is, and it can be readily verified that the logarithm of the amplitude is nearly a linear function of the phase difference, as should be expected if the wave length remained constant, because in that case μx = μLα/2π where α = kx represents the phase difference.

The fact that the tide practically vanishes on the coast shows that when crossing the wide shelf the energy of the incoming tide wave is

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nearly dissipated by friction against the bottom and the ice. This feature is of importance to the tides of the Atlantic Ocean, from which the principal semidiurnal components enter the Polar Sea (p. 581).

In several adjacent seas the effect of friction has been studied by H. Jeffreys, who used a method developed by G. I. Taylor and first applied to conditions in the Irish Sea. The principle is simply that under stationary conditions the net amount of tidal energy which is brought into an area must equal the amount which is lost in the same area by dissipation due to friction. Therefore a determination of the net amount of tidal energy which is brought into an area represents also a determination of the dissipation.

#### Combined influence of friction and the rotation of the earth on tidal currents in shallow water in the Northern Hemisphere (according to Sverdrup). For explanation, see text.

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These studies have found an interesting application. It appears to be established by astronomers that the speed of rotation of the earth is very slowly decreasing, so that during a century the length of the day increases on an average by about one thousandth of a second. This slowing up may be caused by the dissipation of tidal energy, because estimates of the dissipation give values which correspond to the energy needed for bringing about the observed change in the earth's period of rotation.

So far, we have considered the effect of friction on the tides. In order to study theoretically the effect of friction on tidal currents, it is necessary to add the frictional terms (p. 475) in the equations of motion applicable to long gravitational waves (p. 555), and to integrate the equations. Such integration was performed by Sverdrup (1927) on the assumption that only the vertical turbulence need be considered and that the coefficient of eddy viscosity was constant. The boundary conditions were that at the free surface the shearing stresses should be zero and at the bottom the velocity should be zero. The results give some idea about the effect of friction, although the assumption of a constant eddy viscosity is not in agreement with more recent results according to which the eddy viscosity near the bottom increases rapidly with increasing distance from the bottom.

The more important conclusions can be summarized as follows. Near the bottom there exists a “layer of frictional influence” the thickness of which depends upon the ratio s = (2T sin ϕ)/T0 and upon the value of the eddy viscosity, and above which the tidal currents have the

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same character as in the absence of friction. Within a progressive wave in the Northern Hemisphere, with which the investigation deals, the major axis of the current ellipse is in the direction of progress and maximum current occurs at high water (fig. 147B). Close to the bottom, in the layer of frictional influence, the current ellipse is more narrow, it is turned to the right, and maximum current occurs earlier (fig. 147C). As a consequence the current near the bottom will flow against the direction of progress. The velocities in the direction of progress at different time intervals and as function of depth are shown in fig. 147A. A longitudinal section of the wave (fig. 147D) shows that the lines along which the velocity is zero are no longer vertical lines as in the case of no friction (p. 517), but are curved forward. The angle which the current ellipse forms with the direction of progress does not increase throughout the layer of frictional resistance but reaches a maximum at some distance above the bottom. If the depth to the bottom is small, the effect of friction may reach to the surface, in which case the maximum surface current will no longer coincide with the direction of progress but will deviate to the right and, in very shallow water, the deviation may decrease towards the bottom.

#### Tidal currents in the North Sea, lat. 58°17′N, long. 2°27′E, depth 80 m, demonstrating the effect of friction when approaching the bottom. Measurements by Helland-Hansen on August 7 and 8, 1906.

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Figure 148 shows an example of current measurements in the North Sea which appear to confirm the above conclusions. Other examples are found in Sverdrup's discussion (1927) of current measurements on the North Siberian Shelf, but in several of these cases it was necessary to take into account that the ice offered a resistance to the tidal motion and also that occasionally a nearly discontinuous increase in density at some depth brought complications. In the latter case an approximation could be obtained by introducing two layers of constant eddy viscosity separated by a layer of no eddy viscosity, the latter being the layer of very great stability.

The theoretical treatment of the subject has been expanded by Fjeldstad (1929, 1936) who has found integrals of the equations of wave motion in cases in which the eddy viscosity can be represented as a simple function

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of depth, and who has developed methods of numerical integration which are applicable to other cases. His conclusions are that the essential features which were found by assuming a constant eddy viscosity remain unaltered. In sufficiently deep water the current ellipse will at some distance from the bottom show a maximum deflection to the right (to the left in the Southern Hemisphere) and when approaching the bottom the time of maximum current will be more and more advanced. It is shown that the value of the eddy viscosity near the bottom is of greatest importance, for which reason the character of the currents depends mainly upon the value of the eddy viscosity at the bottom and upon the rate of increase directly above the bottom.

#### (A) Observed variations with depth of tidal currents at different lunar hours, according to measurements by Sverdrup on August 1, 1925, in lat. 76°36′N, long. 138°30′E. (B) Computed variation with depth of tidal currents, assuming an eddy viscosity which increases linearly from the bottom to the surface (according to Fjeldstad).

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At the bottom one should expect, from analogies with experimental work in laboratories (p. 479), that the eddy viscosity will be small, having a value which depends upon the roughness of the bottom and the “friction velocity.” Near the bottom the eddy viscosity should increase linearly with increasing distance, the increment being proportional to the friction velocity. At some greater distance from the bottom, stability of the stratification may influence the eddy viscosity, and in very shallow water the eddy viscosity must reach a maximum below the free surface and decrease to a small value at the very surface. In homogeneous shallow water it may be expected, however, that the introduction of an eddy viscosity which increases linearly from the bottom to the surface will give a good approximation because conditions close to the bottom exercise the greatest influence upon the character of the motion and because at some distance from the bottom the value of the eddy viscosity is of minor importance. This is illustrated by the example in fig. 149. To the left are represented the components of the tidal current in the direction of progress of the tide wave, at the time of maximum current at the surface (marked I) and at the five following tidal hours. The curves are based on observations at three depths—0, 12, and 20 m—on

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August 1, 1925, in lat. 76°36′N and long. 138°30′E, where the depth to the bottom was 22 m. The current was at all depths rotating clockwise, and at 0, 12, and 20 m the ratios between the axes of the current ellipses were 0.62, 0.53, and 0.61 respectively. To the right in the same figure are shown the currents which have been computed by Fjeldstad assuming A = 0.202 (z + 68) g/cm sec, where z is the distance from the bottom in centimeters. The computed values have been adjusted to give the observed average current, therefore emphasis must be put on the fact that the computed variation of the current with increasing distance from the bottom agrees with the observed. The computed ratios between the axes of the current ellipses are 0.60, 0.56, and 0.56 in 0, 12, and 20 m, respectively. Both observations and computation show that in this case the ratio decreases very slowly when approaching the bottom.

Observations of tidal currents at different distances from the bottom and within the layer of frictional resistance are not available from many localities and the factual information as to the effect of friction on tidal currents is therefore meager. Measurements from the North Sea off the coast of Germany have been discussed by Thorade (1928), who has studied the influence of friction by a different method of attack. In the North Sea the gravitational forces can be directly determined because the slope of the surface due to the tide wave can at any time be derived from tidal observations at coastal stations. Furthermore, Corioli's force and the accelerations can be derived from the current measurements and the frictional forces can therefore be found by means of the equations of wave motion because all other terms in the equations are known. Thorade's results are, in general, in agreement with the conclusions which have been presented, but many details need further examination. It is of particular interest, however, to observe that on an average during one tidal period Thorade finds that the eddy viscosity is very small at the bottom, increases rapidly with increasing distance from the bottom, but decreases again when approaching the surface. The general character of this variation is in agreement with the above considerations as to the variation of the eddy viscosity.

The influence of friction on tidal currents is also evident from studies of the tidal currents in the Dover Straits by J. van Veen (1939). He finds there that the velocity distribution between the surface and the bottom can be represented by means of a function of the form v = az1/n where n equals about 5.2. This implies that the eddy viscosity is approximately proportional to z4.2/5.2, meaning that the increase is somewhat less than that corresponding to a linear law, but no conclusions can be drawn as to the numerical values of the coefficient.

The effect of lateral mixing on tidal currents has so far not been examined, but it is possible that friction arising from lateral turbulence is of importance close to coasts.

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Waves and Tides