THE ELECTRICAL AND MAGNETIC EFFECTS OF TIDAL STREAMS

MONTHLY NOTICES

OF THE

ROYAL ASTRONOMICAL SOCIETY

GEOPHYSICAL SUPPLEMENT

Vol. 5 No. 8 1949 March

T H E ELECTRICAL AND MAGNETIC EFFECTS OF TIDAL STREAMS

M. S. Longuet-Higgins (Communicated by G. E. R. Deacon)

(Received 1948 March 25)

Summary The tidal movement of sea water relative to the Earth’s magnetic field

induces electromotive forces of a few millivolts per kilometre. Recent measurements off Plymouth show that the potential gradient is at right angles to the streams in that part of the English Channel. Observations on cross- channel telephone cables indicate that a considerable flow of electric current takes place, which can be accounted for b assuming the mean conductivity

earth-currents spread into the land on either side of the Channel and have been measured near Lulworth. It may be possible to use them for the measurement of tidal stream velocities.

flow in shallow channels of rectangular or elliptical section are examined. The horizontal gradient in the water is almost independent of vertical variations in water velocity but is affected critically by the depth of the channel and the conductivity of the channel-bed. The induced electric currents can be expected to extend to depths comparable with the width of the channel.

.of the sea-bedto be of the order of 6 X 10- z (ohms-cm.)-l. Tidally generated

In the second part of the paper the potential gradients generated by water

A. EXPERIMENTAL I. Introduction.-Because of their rather local character, tidally induced

earth-currents have not been so widely investigated as those of non-tidal origin, although it seems that in coastal regions they may account for the greater part of the earth-current gradient. Their existence was first predicted by Faraday, soon after his discovery of electromagnetic induction. In his own words: “If a line be imagined passing from Dover to Calais through the sea and returning through the land beneath the water to Dover, it traces out a circuit of conducting matter, one part of which, when the water moves up or down the channel, is cutting the magnetic curves of the earth, while the other is relatively at rest . . . . there is every reason to believe that currents do run in the general direction of the circuit described, either one way or the other as the passage of water is up or down the channel ”.*

* M. Faraday, Phil. Trans. Roy. Soc.. p. 175, 1832.

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286 M . S. Longuet-Higgins

Faraday suspended two copper plates in the tideway at Waterloo Bridge, but could not detect any potential difference between them other than might be due to chemical polarization at his electrodes. It is probable, as pointed out by Young, Gerrard and Jevons *, that the voltage he wished to measure was very much reduced by the conducting river-bed. However, Wollaston t later claimed to have made similar experiments with success near Greenwich.

No evidence of tidally induced earth-currents was published until those of Adams 1 in 1881 but it then appeared that in several previous instances voltages of a tidal character had been observed in telegraph cables earthing in or near the sea. Thus in 1851 Charlton Wollastons measuring the earth-current in a submarine cable between England and France, found it to vary with lunar period, in contrast to the earth-currents previously observed in land cables11 which had indicated a dependence on the solar day. For this reason Wollaston was inclined to discount his own observations, but three years later on communi- cating them to Faraday, was confirmed in his opinion of their lunar period. Similarly in 1858 H. Saundersa, making observations at Valentia in southern Ireland on a fragment of transatlantic cable earthing at about 300 miles from the eastern end, had found, over a period of six and a half days, that times of maximum and minimum potential coincided with times of highand low water. C. Dresing** reported that the current in the Scottish-Norwegian cable reversed sign about four times daily, according to the direction of the tide along the coast.

The measurements of Saunders mentioned above were in contrast to those of J. Gravestt, also at Valentia, who throughout 78 days recorded the voltage in two fragments of cable of length 1820 and 1850 nautical miles. His conclusion was that, apart from non-periodic variations, the voltages were solar-diurnal. Later however, A. J. S. Adams $1, on examining the records, stated that he found them to be lunar-diurnal. A harmonic analysis shows that the solar and lunar semi-diurnal components are both present, in the ratio of about three to one respectively. It seems, therefore, that in the longer cable the general earth- current system, which is solar-diurnal, predominates, but in the shorter length the voltages are governed by the tidal streams, which are probably negligible beyond the edge of the continental shelf.

Adams also found that the earth-currents in the London-Card8 cable, which he recorded for 28 days, were predominantly lunar. This is probably to be accounted fo;by the streams in the Bristol Channel.

In spite of this evidence it was still possible to doubt $5 whether these semi- diurnal variations were different from the solar daily variations normally observed. It was Marc Dechevrens who first demonstrated the distinction between the earth-currents in coastal and continental regions. 11 I[ From 1916 to 1918 Dechevrens measured the earth-currents at Jersey in the Channel Islands by attaching a galvanometer between the water system of St. Louis Observatory,

* F. B. Young, H. Gerrard and W. Jevons, Phil. Mug., @, 149, 1920. t C. Wollaston, J. SOC. Tel. Eng., 10, 51, 1881. 1 A. J. S. Adams, J. Soc. Tel. Eng., 10, 34-43, 1881. 0 C. Wollaston, J. SOC. Tel. Eng., 10, 50, 1881. 11 See W. H. Barlow, Phil. Trans. Roy. Soc., p. 61, 1849. TH. Saunders, J. SOC. Tel. Eng., 10, 46, 1881. ** C. Dresing, J. SOC. Tel. Eng., 10, 71, 1881. tt. J. Graves, J . SOC. Tel. Eng., 2, 102, 1873. #A. J. S. Adams, J. SOC. Tel. Eng., 10,35, 1881. $9 J. SOC. Tel. Eng., 10, Meeting of 1881 February 24. 1111 M. Dechevrens, Rew. Quest. Sci., 83, joz-325, 1923.


The Electrical and Magnetic Eflects of Tidal Streams 287 consisting of about 800 metres of iron piping, and the gas system of St. Helier to the south-west.* The terminals entered the ground about 4 metres apart. This arrangement could not be considered satisfactory because of the large area from which the current was drawn. However, the experiment, when repeated later in a more orthodox fashion with two pairs of electrodes, gave essentially the same results.? The voltages changed sign four times daily, recurring about 50 minutes later each day than the day before. The amplitude of the fluctuation was greatest at new Moon and full Moon and weakest at the quarters, that is to say it obeyed the cycle of spring and neap tides in the sea. The greatest amplitude was recorded at the equinoxes, when the tides also attain their greatest amplitude. In phase the maximum voltage lagged z hours 18minutes behind the times of high water at St. Helier. I t may be noted that the tidal streams between Jersey and. the French coast are strongest to the north-east between two and three hours after high water at St. He1ier.f

An important contribution to the subject was made independently by Young, Gerrard and Jevons of the Admiralty, who in 1918 carried out a number of experiments on electrical disturbances in the sea.§ They observed that the potential difference between a pair of electrodes laid across the entrance to Dartmouth harbour in a line about 30" east of north contained a strong component of tidal period. This component was not in phase with the ebb and flow of water across the electrodes, but depended rather on the tidal streams in the main part of the English Channel. The authors suggested that if the sea-bed were in any degree conducting, as seemed very probable, the streams in the main part of the channel would set up a circulation of electric current which might determine the potential gradient in other parts of the channel where the tidal streams were weaker. In order to demonstrate the existence of these circulating currents, Young, Gerrard and Jevons used drifting electrodes. These being at rest relative to the water measured only the electric currents independently of the water velocity. The measured potential gradients were in fair agreement with their hypothesis.

In the experiments just described the unequal distribution of the tidal streams and the irregularities of the coast-line made a theoretical interpretation relatively uncertain. Recently, however, observations made under much simpler conditions were published by Cherry and Stovold 11 of the Post Office Engineering Depart- ment. During 1945, in the process of restoring cross-channel communications, earth-current potentials were recorded in four different submarine cables. The potentials all varied with tidal period, and times of zero voltage were within half an hour of the predicted times of slack water along each cable. The potential differences between the two shores of the channel, which were in some cases of a volt or more, were from two to six times smaller than might be expected from the mean surface velocities. This indicates a considerable electrical current density which, as will be shown in Section 3, can be accounted for by the actual conductivity of the channel-bed. 7

* M. Dechevrens, Ten. Magn., 23,37-39, and 145-147, 1918; C.R. Acad. Sci., 167,552, 1918. t M. Dechevrens, C.R. Acad. Sci., 169,985, 1919. 1 Admiralty Atlas of Tides and Tidal Streams (Channel Islanak), 1946. 3 F. B. Young, H. Gerrard and W. Jevons, Phil. Mag., 40, 149, 1920. )I D. W. Cherry and A. T. Stovold, Nature, Lond., 157,766, 1946. 7 The following two examples of the observation of tidally induced potentials may also be noted :

M. Bernard, '' Observations du courant tellurique dans un cable sous-marin," onde Electr., 17. 465-468, 1938; and R. W. Guelke and C. A. Schoute-Vanneck, " The measurement of sea water velocities by electromagnetic induction," J. Inst. EZect. Eng~s., 94, 71-4, 1947.

G 23"


288 M. S. Longuet-Higgins

2. Recent Admiralty measurements ofl Plymouth.-The induced electromotive force should theoretically be at right angles to the water velocity and magnetic field and, in northern magnetic latitudes, directed from right to left as one faces downstream. During the recent war several observations made by the Admiralty Mining Establishment in tidal channels, where the streams passed across the electrodes, seemed to verlfy this; in particular (a) off Gilkicker Point in the Solent (streams east-west); (b) in the channel off Southsea (streams north-west to south-east); and (c) in the Clyde estuary near Innellan (streams east-west). In 1946 the Admiralty Research Laboratory, in collaboration with the Underwater Detection Establishment, made records of the potential difference between two pairs of electrodes, each pair about 6000 ft. apart, laid on the sea-bed between Plymouth Sound and Eddystone Lighthouse. The two pairs were approximately at right angles, one north-south and the other east-west, so that the component of potential gradient in each direction could be measured.* The streams in this area are mainly east-west and are almost in phase with those in the main part of the English Channe1.t One would therefore expect tidally induced gradients to be strongest in the north-south record. The southerly component of the streams across the electrodes is small and of uncertain magnitude, but it is known that at a point two miles to the north of the electrodesf near the entrance to Plymouth Sound the stream vector rotates clockwise.

The mean half-hourly values of the potential difference between the north- south pair of electrodes over a period of 15 days are plotted in Fig. IA. I t will be seen that throughout the record there is a steady sinusoidal oscillation of tidal character. The letters E and W denote the predicted times of maximum easterly and westerly stream across the electrodes. These occur close to the times of maximum potential difference, an easterly stream accompanying a northerly gradient and a westerly stream a southerly gradient. There is also an increase in the amplitude of the variation from neap tides on the zznd August to spring tides on the 28th. The mean tidal range at Devonport on these days was 8.8 ft. and 16-8 ft. respectively, agreeing fairly closely with the relative increase in the potential gradient.

Fig. IB shows the potential differences between the east-west pair of electrodes during the same period. Earth currents of pon-tidal origin are more active in this directions and largely conceal the gradients due to the tidal streams. However, the tidal and non-tidal contributions can be partly distinguished by considering the lunar-diurnal and solar-diurnal variations respectively. These are shown in Fig. 2, computed from hourly values during the lunar month August 20 to September 18.11 Since the lunar variation in earth-current records is normally much smaller than the solar variation, the greater part of the voltage in Fig. ZB can be attributed to the tidal streams. This shows that the tidally induced gradient in the east-west direction is only a fraction of that in the north-south direction, as would be expected from the direction of flow of the streams across the electrodes. Also the east-west lunar variation is about 90° out of phase with the north-south variation, and such that the lunar variation vector rotates in a clockwise sense.

* For a diagram of the electrode positions see N. F. Barber, M.N., Geophys. Suppl., 5,258, 1948. t Admiraty Atlas of Tidal S tream (British Isles), 1943. 1 Position ‘‘ A ” on Admiralty Chart 2620. 9 See N. F. Barber, loc. cit. 11 The method used is essentially that described by S. Chapman and J. Bartels, Geomagnetism,

pr. 244-245, Oxford, 1940.


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The Electrical and Magnetic EjJects of Tidal Streams 291

The solar-diurnal variation (Fig. ZA) will also be affected by the tidal streams, but is probably mostly of non-tidal origin. The curve of Fig. ZA is very similar to the solar variation in the east-west earth-current gradient at Greenwich * ; there is a change from westerly to easterly gradient about two hours before noon and also a greater variation during daylight than during the night hours.

3. Comparison of theoretical and observed magnitudes.-The potential distribution in a tidal channel such as the English Channel will depend in a rather complicated way.upon the distribution of the streams, the shape of the coast-line, and the relative conductivity of the water and of the channel-bed. However, the problem may be partly simplified by the following considerations.

First, it is shown in Sections 10, 11 that in a shallow channel the current circulation set up by the horizontal component of magnetic field H , is negligible, so that the potential difference between points on the same horizontal level depends only on the vertical component Hzl. In the neighbourhood of the English Channel HzI may be taken equal to about 0.46 gauss.

Secondly, although the vertical velocity gradient may be quite large it can be shown that the potential due to HzI depends only on the mean velocity in each vertical line (Section 10). The horizontal gradient is therefore the same as if the velocity were locally uniform. The ratio of the mean velocity vm to the surface velocity w8 is likely to depend mainly on the turbulence of the water and the rough- ness of the sea-bed. Van Veen, making measurements in the Straits of Dover +, found that values of wm/w, lay between 079 and 0.87, excluding places where the bottom was exceptionally rough.

without much error. The conductivity of the water is determined chiefly by its temperature and

salinity. In the English Channel, especially in winter, there is a good deal of mixing between upper and lower layers and in any given section the conductivity may be taken as uniform. There is, however, a slight difference in temperature between the eastern and the western ends, and also a rather greater seasonal variation.1 In all, the conductivity varies from about .034 (ohms-cm.)-l at Dover in February to about -043 (ohms-crn.)-l at Plymouth in August.

Much less is known about the conducting properties of the channel-bed. The conductivity of surface rocks and formations may vary from 2 x I O - ~ (ohms- cm.)-l for clays and moist soils to about 10-6 (ohms-cm.)-l for crystalline and igneous rocks.$

That the horizontal gradient depends very critically on the conductivity of the channel-bed may be seen by considering the electrical potential of a uniform stream of water flowing in a long straight channel of elliptical cross-section. The potential of this model is evaluated in Section 11. The horizontal gradient in the water, which is uniform, is given by

We can probably take v, = 0 . 8 3 ~ ~

% = VH, tanhf, a x ( K ~ / K , ) + tanhf, ’

where V is the velocity of the water, tanht, is the ratio of the minor (vertical) to the major (horizontal) axis of the ellipse, and K~ and K~ are the specific conductivities (supposed constant) of the water and of the channel-bed. Now for the

G. B. Airy, Phil. Trans. Roy. Soc., 160, 215-226, 1870. t J. van Veen, 3. Cons. Int. pour 1’Exploration & la M u , 13, 15, 1938.

A t h fiir Temperatur, u.s.w., de* Nor&ee und Ostzee. 5 Report of Imperial Geophysical Survey, p; XI, Cambridge, 1931.

Deutschen Seewarte, Hamburg, 1927.



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( B ) Lunar variation. FIG. 2.-The solar daily and lunar daily variations in the potential dzyerence

betaeen the east-est pair of electrodes at Plymouth. Lunatwn : 1946 August 20-1946 September 18. Ordinates : millivolts (east positive).

English Channel tanhfl is of order IO-~ and, from the data given above, K ~ / ~ C ,

can only be supposed to lie somewhere between 6 x 10-1 and z x I O - ~ . As far as is previously known, therefore, the gradient may take almost any value from zero to a maximum of about 20 millivolts per kilometre per knot.


The Electrical and Magnetic Effects of Tidal Streams 293

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The " shores " of the channel are at P and Q. Q

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For this reason the actual gradients measured are of interest, since they may provide information as to the conductivity in the neighbourhood of the channel. Fig. 3 illustrates the lines of flow of electric current in a cross-section of the channel in the theoretical case when the conductivity of the channel-bed is uniform,



(The circuits are completed by horizontal lines of flow through the water.) The currents extend to depths comparable with the width of the channel and it is therefore to be expected that the conductivity at these depths will affect the potential gradient, unless there are shielding layers of much higher conductivity nearer the Earth’s surface.

The potential difference between the two sides of the channel is given theoretically by

WVHv tanh f1 E = (KOIKl) + tanh 41 ’

where W is the width of the channel. If the observed value of E is taken and the equation solved for K ~ , a certain average value of the sea-bed conductivity will be found which, from an analogy with a similar idea in resistivity surveys *, may be called the “equivalent conductivity’’ and denoted by Z. The details of the calculation of iz from four cross-channel measurements of Cherry and Stovold is set out in Table I. The first three refer to published data t, the fourth

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TABLE I Calculation of Equivalent Conductitit-v of the Sea-bed from

Observations on four Su3mmine Cables

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to an unpublished series of observations, lasting for fourteen hours only, on the Anglo-French cable from St. Margaret’s Bay near Dover to Sangatte near Calais. The latter are quoted by kind permission of the authors. For each cable the width W and the conductivity K~ are taken to be those in the corresponding part of the English Channel ; V is 0.83 times the mean transverse surface velocity V, and el is chosen so that the mean depth of the elliptical section equals that of the actual cross-section of the channel.

4. Potential gradients in the land: experiments at Lu1worth.-From Fig. 3 it will be seen that the theoretical current lines spread out some way into the land on either side of the channel, giving rise to potential gradients perpendicular to the coast-line. In Fig. 4 the potential and potential gradient on the Earth‘s surface are plotted as a function of the distance x from the middle of the channel. Very near to the coast the potential gradient is large, but further away the current lines spread downwards and the potential gradient diminishes. At the coasts also the gradient changes sign, that in the land being the reverse of that in the

* S. Chapman and J. Bartels, Geomagnetism, pp. 417-448, Oxford, 194. t D. W. Cherry and A. T. Stovold, Nature, 157, 766, 1946.


The Electrical and Magnetic Eflects of Tidal Streams 295 water. Thus, whereas an easterly stream should produce a northerly gradient in the water (cf. Fig. I for the Plymouth electrodes) the same stream should produce a southerly gradient in the land.

Experiments were carried out near Lulworth over a short period to verlfy the existence of these tidal potential gradients and to determine their magnitude and sign. Electrodes of similar type were laid in three wells A, B and C (see Fig. 5) on a line running inland from the coast. The distances AB and BC were respectively 2.16 and 1.26 kilometres, and A was about 200 metres from the nearest point on the coast. Wells were used in order to give good contact with the soil.

FIG. 5.-The situation of the electrodes near Lulworth. Inset : Lulworth Cove.

A simultaneous record of the potential differences between A and B and between B and C during six days at spring tides is shown in Fig. 6. Each trace shows a steady oscillation with two maxima and two minima daily. It will be seen that the times of these maxima and minima are closely related to the times, E and W, of maximum easterly and westerly mean velocity in that part of the channel (predicted from tidal data). A northerly gradient accompanies a westerly stream and a southerly gradient accompanies an easterly stream, as expected. There can be little doubt, therefore, that we are measuring on land the effect of the tidal streams in the channel.

The mean amplitude of the tidal variations over the six days shown is estimated as 32.6 mV. for AB and 9.5 mV. for BC. The mean gradient between A and B and between B and C is therefore 15.2 mV./km. and 7-5 mV./km. respectively. This indicates a failing off in potential gradient with distance from the coast in about the expected ratio. However, an exact comparison with the theoretical curve of Fig. 4 would probably not be significant owing to the irregularity of the coast-line and the unknown influence of local differences of conductivity between the electrodes. I t should be noticed that the shorter-period variations in the two


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The Electrical and Magnetic Eflects of Tidal Streams 297 records in Fig. 6, which are due presumably to earth-currents of non-tidal origin, are also reduced in magnitude further inland. This again may be partly due to differences in the conductivity of the ground between the electrodes and partly to the presence of the highly conducting sea water. Non-tidal currents would tend to be concentrated into the water and on passing into the land to spread downwards similarly to the tidal earth-currents.

5. Tidal potential gradients further inland.-The question arises how far inland tidally induced earth-currents are measurable and what influence they have on the diurnal variations at inland stations. The theoretical curve of Fig. 4, which is for a channel of infinite length, indicates that at a distance from the coast equal to the width of the channel the horizontal gradient is 6 per cent of that in the water, and at one and a half times the distance it is about 3 per cent. The gradient in the English Channel as measured by the Cuckmere cable (see Table I ) was about 8.5 mV./km. I t is therefore quite possible that the tides contribute to the lunar earth-current variation at Paris, 19 km. from the coast at Dieppe, where a lunar variation of 0.6 mV./km. has been found.* This is about one-quarter of the total diurnal variation, whereas the lunar magnetic variations in similar latitudes are of the order of one-tenth of the total diurnal variations.

6. Disturbances of the magnetic Jield.-The magnetic field associated with tidal earth-currents is probably quite small at the Earth's surface. Theoretically the currents generated by a uniform stream of water in a long straight channel would flow in closed paths as in a solenoid, and therefore the field at the Earth's surface, being outside the solenoid, would be zero. However, at points below the Earth's surface, which are enclosed by circuits of current, the disturbance may be appreciable. The tidal earth-current density in the English Channel is of the order of 10-* amps/cm.2. At a depth of 50 fathoms this should produce a horizontal field of about ioy. A comparable disturbance might be observed if magnetic measurements were made in a mine-shaft near to the coast.

7. The nieasurentent of tidal streant velocities.-Where the distribution of stream velocities is fairly simple a continuous record of the induced earth-current gradients might be used to give the relative strength of the streams and, for example, times of maximum and minimum velocity. Since, however, the gradients depend critically on the conductivity of the sea-bed it will usually be necessary, in order to obtain absolute magnitudes, to calibrate the measurements by comparison with the known tidal streams at some given time. The chief obstacle to such measurements is the presence of the non-tidal earth-currents: However, the latter are usually quite widespread and could probably be distinguished by a comparison with earth-current records further inland. It might also be possible by a suitable balancing device to eliminate the non-tidal currents altogether. A study of tidal earth-currents measured fairly simply in this way might provide useful information as to the influence of wind strength and atmospheric pressure on tidal stream velocities.

B. THEORETICAL 8. General equations.-The water velocity in any tidal channel will vary

with the time, but if the electrical self-inductance is sufficiently small the time variation may be neglected. Consider a long cylindrical shell of thickness h,

*P. Rougerie, C.R. Acad. Sn'., 205, 1252, 1937.



radius r and conductivity K (in absolute units). The time constant T of a current system flowing in the shell is given by

T = 27r~rh. For comparison with the English Channel we may put h =IOO m., r=50 km. and K =4 x ~o-ll, giving T = 12.6 secs. Since this is negligible compared with a tidal period it is permissible to regard the water motion at any time as being steady.

The electrodynamical equation for a fluid in steady motion in the presence of a steady magnetic field * may be written

where u is the water velocity, l# the magnetic field strength, p the resistivity and i the electric current density.

grad+=u x H - p i , (1)

This equation is equivalent to

l c p i . ds = I; x H . ds, (2)

where C is any closed circuit. In other words the total electromotive force generated in any closed circuit moving with the fluid is equal to the rate at which the circuit is cutting lines of magnetic force.

The function + defined by equation (I) is the potential measured by a pair of stationary electrodes. It is made up of two terms, one depending on the local velocity, the other on the electric current density. I n certain circumstances the second term may vanish; for example, in a channel in which u and H are constant and the boundaries perfect insulators. I n certain circumstances also + may be identically zero (for example if u and Hare constant and the boundaries perfectly conducting). The existence of a potential gradient depends upon the accumulation of volume or surface charges through variations and discontin- uities in u, p or H.

The magnetic field of the induced currents will be small (see Section 6), and therefore in determining 4 the total field H may be taken equal to the applied magnetic field.

curl H = 0. From the condition of continuity :

div i = 0,

it follows, taking the divergence of both sides of equation (I), that

provided p is uniform.

where q is the volume density of electric charge, we see that the accumulation of a volume charge in a fluid of constant resistivity is only possible if the motion is rotational.

If this is of “ external ” origin we have

vZ4 = H. curl u - u . curl H = H . curl u,

vz+ = -4rq,

(3) Comparing this equation with Poisson’s equation

If it is irrotational the potential satisfies Laplace’s equation vz4 = 0.

At a surface of discontinuity of u or p the boundary condition is that the normaf component of i is continuous, or

K(n . grad4 - n . u x H ) (4) * See for example E. J. Williams, “ The induction of electromotive forces in a moving fluid,

and its application to an investigation of the flow of liquid,” J. Phys. SOC., e, 467, 1930.


The Electrical and Magnetic Eflects of Tidal Streams 299

is continuous, where K is the conductivity. At the surface of an insulator, therefore, the above expression must vanish.

9. Method of approach.-Consider a channel of water whose upper and lower surfaces are denoted by S' and S",.and let S"' be the surface of the land on either side of the channel. Let n', n" and n"' be the unit normals to these surfaces. Further, let +o and rjl denote the potential in the channel-bed and in the water respectively, and K~ and K~ the conductivities in the two regions, which will be assumed to be uniform. Then from equations (3) and (4) we have

V2+1 = H . curl v, d.grad+,=n' .vxH, n" . grad +1 = n" . v x H + (KO/K1)n' . grad +o,

n"'. grad v2+0=o* t$o = 0, 1 and

and +o must vanish to a certain order at infinity. Now let x be the function, defined in the same region as +1, satisfying

(6) 1 V2x = H. curl v, n'.gradX=n'.vxH, n" . grad x =n" . v x H,

so that x is the potential that would be generated in the water if K~ were zero. i. e. if the channel-bed were a perfect insulator.

we find that # must satisfy

Writing +1=x+*

(7) I V2* = 0,

n' . grad 4 = 0,

n" . grad I) = (Ko/Kl)R". grad +o.

These equations, together with equations (5) and the further condition

which must hold on s", may be regarded as determining + and +o, that is to say the potential gradients due to the flow of electricity through the conducting bed of the channel. I t will be seen that a,h and do depend only on the value of x on the surface S" and to this extent they are independent of the distribution of water velocity in the channel.

It is convenient therefore to evaluate x first for different distributions of velocity v, supposing that the channel-bed is a perfect insulator. Afterwards we may consider the effect of the finite resistivity of the channel-bed.

10. The effect of unequal distributions of welocity.-Consider a channel whose cross-section is a rectangle of width ZQ and depth b, where b < a. Take Cartesian coordinates x, y, z with the origin at the mid-point of the upper surface of the water, the x-axis horizontally across the channel, the y-axis vertically downwards and the z-axis along the axis of the channel. Let the velocity u be in the z-direction and suppose that the component a, in this direction is given by

If H,, Hy and H, are the (constant) components of magnetic field we have, in Cartesian coordinates,

+ o = x + * ,

Qz = J?af(MY)*

u x H=( - w a y , 0). (8)



Hence the potential x is independent of the component of magnetic field Ha along the axis of the channel. Let

x = x x + xu, where xx and x, are the potentials due to Hx and H, respectively. These two potentials may be evaluated separately ; this follows from the fact that the magnetic field H occurs linearly in the field equations and boundary conditions.

10.1. The evaluation of X,.-From equations (6) and (S), xy must satisfy 1 V2xg = - VqLfYxlg(Y), I I = - VH,f(x)g(y), ax (x = & a)

(9) ax, - aY - -0, J (y = 0, b)

where a prime (’) denotes differentiation. be built up as follows.

and such that

T h e solution of these equations may

Let 8(x,y; ct) he a function defined in the interval ( -a<x<a) , (o<y<b)

1 v2e = 0,

8 is analogous to the Green’s function for the interval ( -a<x<a), and it follows that the solution of equations (9) is given by

where 8, and O2 stand for the values of 0 in ( -a<x<a) ; (a<x<a) respectively. If in equations (ro),g(y) is replaced bycosnny/b,n = I, 2, . . . . it is easilyshown,

by differentiation or otherwise, that the corresponding functions 8, and O2 are given by

[cosh m ( x + n)/b . cosh nn(ct - a)/b - cosh nn(x - a)/b] cos n.rry/b el = 9 (x<a) nn/b . sinh znna/b [coshn.rr(x-a)/b. coshn.rr(u+n)/b-coshn.rr(x-a)/b] cosnnylb e2 = 9 (x>Q.) m / b . sinh anrra/b

and hence

xu = - VHv cos nny/b KqL(x, a)f(ct) du, L where

K,(x, =

cosh n.rr(x - a)/b . sinh nn(u + a)/b

cosh nn(x + a)/b . sinh nn(ol- a)/b

9 (.<X)

3 (%>XI

sinh enna/b

sinh znnalb

When g(y) is replaced by a constant, say unity, it is easily shown that ’ X

xu = - V H , I f ( c t ) du = - VH,F(x) say. J O


The Electrical and Magnetic Eflects of Tidal Streams 301 In the general case suppose g(y) to be given by the cosine series in y :

m

n=l g(y)=iao+ X a,cosn?ry/b, (o<y<b). (12)

Then adding the solutions for each term separately we have formally : oc

xu= -gaoVHvF(x)- VHv C a,cosnny/b n=l

10.2. The evaluation of Xz.-The equations for xz are v2xz = VHxf(xlg'(Y),

a x x ax - =o, (x = k a),

aw aY - =o,

These last equations for w are identical with equations (9) for xu except that H,,f'(x) and G(y) replace Hv, f(x) and g(y). Thus we have immediately

m

w = - &VHzbof(x) - VHz C b, cos n.rry/b c" K,(x, a)f'(a) da, n = l J -a

where b, is the nth coefficient in the series m

n = l G(y)=ib,+ Z b,cosnny/b, (o,<y<b).

Col1ecting:together the solutions so far obtained we have

OD U

n=l -U - Y x cos nry/bj ~ , ( x , a)[aiHvf(a) + b , ~ ~ ( a ) l da.

10.3. Discussion of the solutions.-Since a/b is large (the width of the channel is great compared with its depth) we consider first the form of the solution when a tends to infinity. I n the limit

Since this function diminishes rapidly as I a - X I increases, it follows that the integral

G 24



is almost independent of f(a) for values of 1 a -xi greater than say b/n. Hence the potential gradient at any point (x, y ) is almost independent of values of the velocity beyond a distance b equal to the depth.

When the velocity varies slowly across the stream the terms under the summa- tion sign are small. Writing

we have a,H,f(4 + ~ n W ’ ( a ) = h&),

m lim ~,(x) = J ~ , ( x , a)h,(a) da

alb-t w --m

= &-’b[hn(x -a) - h,(x +a)] da

w(b/n?T)ah;(x) + (b/n?T)*h”’(x) + . . . . bf’(x) < 1,

If it is assumed that

or that the change of velocity in a horizontal distance equal to the depth is small, it follows that I, can be neglected. Under these conditions the potential x is given approximately by the first terms in (13). Now from (12)

b

a, = ( 2 P ) J 0 g(Y)dY=2i7

b0=(2/b)l)G(y)dy=2G, 0

where 2 is the mean value of g ( y ) . Similarly,

where E is the mean value of G(y). Hence x+ VH,f(x)[G(y) -E] - VH$(x)i.

The horizontal component of potential gradient is given by

to the same order of approximation. This is independent of the vertical co- ordinate y and is the gradient that would be generated by a uniform stream moving with the mean velocity on a line above and below the point (x,y). Hence the horizontal potential gradient is the same as though the water moved with its local mean velocity and is constant in any part of the stream. The E.M.F.s are not, of course, constant, since the water velocity varies from top to bottom of the stream. The compensation is brought about by the flow of electric current which, as can be seen from the rapid convergence of the factor Kn(x, a) in (14), circulates locally.

The vertical component of potential gradient is given by

2 = J74f(xlg(y)7 which is identical with the vertical component of E.M.F., and depends only on the velocity at each point. The vertical component of the current density is therefore negligible.

We have discussed so far the case in which the width of the channel is infinite. However, the approximations are valid when the channel is of finite width. In fact, when

( a - x ) > b and (x+a)>b,


The Electrical and Magnetic Eflects of Tidal Streams 303 the functions Kn(x,a) are almost independent of the width a. The effect of the "ends" is therefore negligible beyond a distance about equal to the depth (except for the condition they impose, that the total flow of electricity across any vertical line is zero) and decreases rapidly as the distance increases. This suggests that the effect of a change of depth of the channel is local, and that the potential gradient is not affected by changes in depth at a distance greater than the local depth of the channel. We infer that when a channel is of variable depth and the bottom gradient is small the general conclusions arrived at above for the channel of constant depth still hold good.

Since & is derived from the value of xv on the surface S", it follows that $v depends only on the mean velocity in each vertical line. Thus h, like xv, is independent of the vertical distribution of velocity in the channel. $u may, however, depend on both the velocity and the depth of water in other parts of the channel.

I t might have been expected that the electric current density associated with xs would be negligible, for if H denotes a uniform horizontal field the circulation of u x H round any closed curve C i n the fluid is small compared with the area of the curve. Therefore, by equation (2), so also is the circulation of pi. For a similar reason we may expect that the current density associated with the complete potential +a will also be small. An exception may, however, occur whhe the gradient of the channel-bed is steep, as will be found in the case discussed below.

11. The effect of a conducting channel-bed.-Consider a channel whose section is a shallow ellipse cut along the major (horizontal) axis. Let the lengths of the major and minor axes be 24 and 2b respectively. With the same rectangular coordinates as in Section 10 we make the transformation

where and when y is positive 4 and q take values in the ranges

The surfaces S', S" and S" of Section g are now level surfaces of the coordinates 6, q as follows :

S' is given by 4 = 0 and q =o, r, ((<El) S" is given by f = &,

, o<t<Co; o<q<r.

S"" is given by q = 0, T, (4 243 where

Therefore, supposing x, the potential for a channel-bed of zero conductivity, to be determined, the functions t,b and $o which are to be added (see Section 9) must satisfv

a = k cosh f1, b=ksinh&.


304 M . S. Longuet-Higgzns

Further

0240 = 0, and, since there are no sources of electricity, at infinity 4, must diminish:at least as rapidly as e-€, which is the potential of a dipole at the origin.

Since both 9 and 4, satisfy Laplace's equation, they can be expanded in series of the type

m I

$= C A, coshnt cosnq, (t<tl) n=l

W

4, = C B, rnE. cos nq, (t>tl), n= 1

where A, and B, are constants to be determined from the last two of equations (15). Suppose that, when 5 =el, x is given by the Fourier series

in which the constant C, may be assumed to be zero. equating coefficients of cos nq that

Then it is easily shown by

C,K, cosh m$ cos nq *=- x

K, cosh nfl + tc1 sinh ntl' Cntcl sinh d1 eNe1-f) cos nq

4o =2, tc0 cosh ntl + tcl sinh nt, ' Suppose the water flows with uniform velocity V and consider first the potential

Since the velocity 4, induced by the vertical component of magnetic field, HV. and magnetic field'are uniform

xu = - V H p = - VH,k c0sh-f COST.

Therefore in equation (17) all the coefficients C, are zero except C,, which is given by

Thus

and

T- determine th

VH,FUC, cosh 5 cos q *,= tco+tcl tanhe, ' ( E G )

4u= - tco+tcl tanht, 9 (t>td VH&, sinh El dEl-€) cos q

VH,ktcl tanh t1 cosh ( cos q K, + tcl tanh II

VH#, tanh t1 K, + tcl tanh 5,

(tGt1) 4 =- PI

X. ,= -. -

potential 4, induced by the horizontal magnetic field we have x+ = V H a + constant

= VH,k sinh 4 sin q +constant. (19) Now sin q can be expanded in the cosine series :


The Electrical and Magnetic Eflects of Tidal Streams 305 After substituting in (19) and choosing the constant term suitably we have

Therefore in (18) all the coefficients ofodd suflix vanish (+z is therefore symmetrical about the line x = 0) and we have, writing 2m = n,

4VHzK sinht, ; 4VH& sinh t1

K~ cosh 2n& cos 2nt7 * X = 3r (4m2- 1 ) ( ~ ~ ~ 0 ~ h 2 ~ ~ + ~ ~ s i n h z ~ z ~ ~ ) ' (t Gel)

+ X = - 3r el=l (4m2 - I)(K~ cosh 2mfl + K ~ sinh 2n24,) (t251)

K, sinh zmtl e$m(Ei-E) cos am7

The above series represent the added flow of electric current due to the conductivity of the sea-bed. When the channel is shallow the factor sinhtl multiplying the series is small, and it can be shown that when ,$ is not near to el or 7 not near o or 7~ the contribution of these series to the potential gradient is negligible in comparison with the gradient due to the electromotive force.

At the point (el, o), however, there is a singularity and the horizontal gradient becomes infinite. In fact, when y = o and x tends to a,

There is a similar singularity at the point (tl, T) . These logarithmic singularities at the ends of the elliptic section can be ascribed to the local boundary conditions. The vertical E.M.F. vzHx sets up a distribution of electric charge (see Section 8) on the upper surface S' of the water and a similar charge distribution, of opposite sign, on the lower surface S". At the ends of the channel section, when the water surface terminates abruptly, there is necessarily a singularity if the medium which forms the channel-bed is conducting. With the elliptic section the lower surface S" meets S' at right angles and forms a sharp corner. It can be shown that if this corner were rounded off so that the tangents to S' and S" at their point of meeting were coincident then the singularity would vanish ; for when S' and S" are parallel the charge densities are equal but of opposite sign and the logarithmic terms arising from each cancel one another. The singularities, therefore, result from the particular form of channel section that we have chosen and are of no special significance.

The region in which the logarithmic terms are important is very small and, in the case of the English Channel, would be confined to within a few feet of the shore. Hence we are justified in neglecting the flow of electric current due to H,, the horizontal component of magnetic field.

The total potential is therefore given by

+ =+X ++v.

~ K~ + K ~ tanh 5, The potential in the channel-bed (tat,) can be written

+ = constant x e-6 cos 7. The form of the current lines, which are independent of the depth of the channel, is illustrated in Fig. 3. This shows a vertical cross-section of the channel-bed


306 M. S. Longuet-Higgim in which the shores of the channel (x= +a, y=o) are at the points P and Q. The current lines are, of course, completed through the water, but the channel- bed is so shallow that its interior is not representable with the same vertical and horizontal scales.

The vertical component is equal to the vertical electromotive force, and is independent of the conductivity of the channel-bed, there being no vertical component of current density. The horizontal component a#lx on the other hand is reduced by the flow of electric current and depends critically on the conductivity of the channel-bed. We have

I n the water (5<tl) the potential gradient is uniform.

VHel tanhe, K, + K~ tanh Il (5651). - = -

ax

The effect of the conducting channel-bed is therefore to reduce the horizontal gradient in the ratio S given by

K~ tanh f1 K, + K~ tanh f1 '

Since fl is small, 6 may be written approximately

S=

(22) S+- K l E l

KO + K X l - This result (22) may be interpreted with the aid of a simple physical analogy.

Suppose two points P and Q to be joined by a wire C, of resistance R, in which a total electromotive force E is developed, by batteries or otherwise. If no current flows in C, the potential difference between P and Q will be given simply by

+(P)-+(Q) =E- If, however, P and Q are joined by an external wire Co of resistance R,, it is easily shown that the potential difference between P and Q falls to the new value :

The potential difference is therefore reduced in the ratio

by the introduction of the external circuit. It will be seen that if we write

IIR, =K151,

I I R o = K o ,

A of equation (23) becomes identical with 8 of equation (22). Thus the electrical system consisting of the channel of moving water and the uniformly conducting sea-bed can be compared with the circuit consisting of the wires C, and C, respectively. The constant electromotive force E corresponds to that developed by the moving water between the two sides of the channel. The resistance of Cl must vary inversely both with the specific conductivity of the water and with the depth of the channel, which is proportional to el cosv approximately.

In other words, the water acts as a conducting path whose resistance is inversely proportional to the depth of the channel and the channel-bed acts as an external conducting path of unit length and width. For example, if the external conductivity K~ is large the electromotive force is mostly short-circuited and


The Electrical and Magnetic Eflects of Tidal Streams 307 the measured potential gradient is much smaller than expected. A similar resuk occurs if the channel is very shallow, making the “internal” resistance very high. Thus if 4, tends to zero in equation (21) the horizontal potential gradient vanishes in spite of the fact that a constant electromotive force is still being generated by the water motion.

Acknowledgements The author is indebted to N. F. Barber of the Admiralty Research Laboratory

for assistance in the experimental work and for many valuable suggestions; also to J. T. Crennell of the Admiralty Mining Establishment and to D. W. Cherry and A. T. Stovold of the Post Office Engineering Department for permission to quote unpublished measurements. This paper is published by kind permission of the Admiralty.

Admiralty Research Laboratory, Teddington :

1948 March 20.


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THE ELECTRICAL AND MAGNETIC EFFECTS OF TIDAL STREAMS