The waves which have been dealt with so far are characterized by maximum vertical displacements at the surface. For short waves the vertical displacement of the water particles decreases exponentially downwards, and for long waves the vertical displacement decreases linearly with depth, being zero at the bottom (p. 521). These waves will now be called ordinary waves. They are the only ones possible in homogeneous
The theory of the internal waves was first developed by Stokes (Lamb, 1932, p. 370) in the simple case of two layers of different density, and the general theory of progressive internal waves in heterogeneous water was developed by Fjeldstad (1933). Both theories have found application to oceanographic phenomena.
In a fluid consisting of two layers of infinite thickness, one lower layer of density p′ and one upper layer of density p′, waves at the boundary surface between the two layers will have a velocity of progress as given by
When dealing with internal waves in water which has a free surface but consists of two homogeneous layers of different density, the kinematic and dynamic boundary conditions must be fulfilled both at the free surface and at the internal boundary surface, and the equation of continuity must be satisfied. This leads to a quadratic equation for c2 which, for short waves, has the approximate rootsequations (XIV, 47) are reduced to
Applied to the ocean, this means that wherever there exists a thin top layer of water of small density, two types of waves are possible: the ordinary surface waves that progress with velocity c1; and the internal waves at the boundary between the light top layer and the heavier water underneath, that progress with velocity c2. Ekman (1904) has availed himself of this conclusion in order to explain the phenomenon known as “dead water.” In the time of the sailing vessels many captains reported that with a light breeze their vessels occasionally appeared to “stick” in the water, behaving sluggishly and making little headway. The experience was particularly common in Arctic waters in the presence of a thin top layer of nearly fresh water produced by melting of ice, and off rivers from which fresh water spread out. Slowly moving steamers have had similar experiences, but when their speed was increased to a few knots the unusual resistance disappeared. According to Ekman's theoretical studies and the results of his numerous experiments, this dead water is due to the fact that a slowly moving vessel may create internal waves at the lower boundary of a thin fresh-water layer the thickness of which is not much less nor much greater than the draft of the vessel. The energy otherwise applied towards overcoming the ordinary resistance of the water will now be used also for generating and maintaining internal waves, for which reason the vessel appears to “stick” in the water. The velocity of progress of internal waves as given by equation (XIV, 48) is, however, small. If the velocity of the vessel is greater than this small value, no internal waves are created and the vessel can proceed normally. With p − p′ = 0.025, nearly corresponding to a layer of fresh water on top of sea water of temperature 10°C and salinity 30 ‰, and with h′ = 400 cm, one obtains c2 = 100 cm/sec = 1.9 knots. These numerical values indicate that at a speed of a few knots no internal waves are created, which is in agreement with the general experience that dead water is not encountered at speeds above a few knots.
The short internal waves that have been dealt with so far may be present anywhere in the ocean, but escape observation on the high seas where the variation of density with depth is less conspicuous. In the open ocean long internal waves exist, however, and these have in recent years received much attention. When dealing with two layers and neglecting the effect of the earth's rotation the velocities of progress of the ordinary long wave and of the internal wave are obtained from the equations
The internal wave is characterized by having its maximum amplitude at the boundary surface. At the free surface the amplitude of the internal wave does not entirely disappear but is reduced tofig. 152B). The change in amplitude with depth is therefore equal to Z/h′ in the upper layer, and to −Z/h in the lower layer. The amplitude of the horizontal particle velocities can be derived from the vertical amplitudes because the relation exists
This numerical example shows that internal waves are characterized by large horizontal particle velocities. The corresponding velocity of progress of the internal wave is 78 cm/sec, whereas the ordinary long wave proceeds at a velocity of 4430 cm/sec.
(A) Schematic representation of an internal wave at the boundary between two liquids of densities p and p′. (B) Schematic representation of the variation with depth of the amplitudes of the vertical displacements p and p′, and of the maximum horizontal velocities U′ and U.
The character of the internal wave at the boundary between two liquids of different density is illustrated in fig. 152A, which shows the deformation of the boundary surface and the directions of the horizontal velocities within the two layers. The wave is supposed to progress from left to right. At the line marked a the horizontal currents in the upper layer are divergent, for which reason the lower boundary surface must rise, and the horizontal velocities in the lower layer are convergent, for which reason also the boundary surface must rise. At the line marked b the boundary surface must sink for similar reasons, and the wave must therefore progress from left to right, as stated. In the figure it is also indicated that the vertical displacement of the free surface is opposite in phase to that of the boundary surface, but the displacement of the free surface has been greatly exaggerated. The pressure at the bottom remains constant and equal to the hydrostatic pressure because the vertical accelerations are negligible. The amplitude of the deformation of the free surface can be computed from the hydrostatic equation and the result is as before, η0 = −Z(p − p′)/p.
If several boundary surfaces are present, several internal waves can occur simultaneously, and the greater the number of boundary surfaces, the greater the number of possible internal waves. On the basis of this reasoning, when the density varies continuously with depth one should expect an unlimited number of possible internal waves. That such is the case has been shown by Fjeldstad (1933), who has developed the theory of internal waves in water in which the density is a continuous function of depth. He deals with progressive waves only, and presents a complete
(A) Variation with depth of the vertical displacements corresponding to internal waves of first, second, third, and fourth order at Michael Sars Station 115 (according to Fjeldstad). The density distribution is shown by the curve marked σt. (B) Variation with depth of the amplitudes of the horizontal velocities corresponding to an internal wave of first, second, third, and fourth orders (according to Fjeldstad). Vertical displacements and amplitudes are plotted on an arbitrary scale.
Figure 153A shows Fjeldstad's computed vertical displacements and horizontal velocities as functions of depth for the internal waves of first, second, third, and fourth orders, corresponding to the distribution of density as shown in the same figure, which was observed at Michael Sars station 115 (Helland-Hansen, 1930), where the depth to the bottom was 580 meters. The amplitudes of the accompanying horizontal velocities
Fjeldstad's method also leads to determination of the velocity of progress of waves of different orders, provided that the depth is constant and that the distribution of density remains unaltered in the direction of progress; but the periods of the waves cannot be determined theoretically and must be derived from observation. At Michael Sars station 115 the velocities of progress were c1 = 70 cm/sec, c2 = 39 cm/sec, c3 = 26 cm/sec, and c4 = 19.5 cm/sec; and for a wave of period 24 lunar hours the corresponding wave lengths are 62.5 km, 34.8 km, 23.2 km and 17.4 km, respectively. Evidently, the internal waves are short compared to tide waves. It should be observed that the velocity of progress increases when the difference in density between the upper and lower layers decreases, and also increases with increasing depth to the bottom. In low and middle latitudes the velocity of progress of the first-order wave will, however, rarely exceed 300 cm/sec. For diurnal or semidiurnal waves the corresponding wave lengths are 268 km or 134 km, respectively, and the waves of higher order are correspondingly shorter.
Observations indicating vertical displacements of water masses which may be related to internal waves have been made on numerous occasions when oceanographic observations have been repeated in the same locality at short time intervals. If observations of temperature at different depths are made at, say, hourly intervals, from an anchored vessel or from a vessel which maneuvers in such a manner that its position changes only one or two miles, it is often found that the temperature varies more or less periodically at all depths. Assuming that these variations are due to vertical displacements, one can find the vertical displacements at the different depths if the average temperature distribution is known. If, for instance, the average temperature at 200 m is 12.40° and at 220 m is 12.17°, it may be concluded when a temperature of 12.17° is observed at 200 m that water which under undisturbed conditions should be found at 220 m has been displaced 20 m upwards. If the temperature oscillation at a given depth d is periodic and has an amplitude of A°, the amplitude of the corresponding vertical oscillation is found by dividing the amplitude A° by the average temperature gradient at that depth, (dθ/dz)d. Similar conclusions may be based on observations of salinity and oxygen, and when all these elements have been observed good
If the observations have been carried out during a sufficiently long time, it is possible to find the period length of the oscillations. In a number of cases period lengths have corresponded to tidal period, and it has therefore been concluded that internal waves of tidal periods commonly occur in the ocean. It is not probable that such internal waves are caused directly by the tide-producing forces but it is more nearly probable, as suggested by Defant, that they are caused by the periodic variations of the actual tidal currents which may lead to periodic changes in the inclination of isosteric surfaces in the sea. Besides these internal waves of tidal periods, waves of other periods also exist.
The first observations of short-period variations which indicated the existence of internal waves were discussed by Helland-Hansen and Nansen (1909). On the Michael Sars Expedition to the North Atlantic in 1910, repeated serial observations were made at several stations, and on one occasion simultaneous observations in the Faeroe-Shetland Channel were conducted from the Michael Sars and the Scottish research vessel, the Goldseeker, the two vessels being about 106 km (57 mi) apart. The possible vertical displacements derived from the temperature observations can be well represented by two periodic oscillations of period length 12 and 24 lunar hours. The results of Helland-Hansen's harmonic analysis (1930) of these data are given in table 73. It appears that the oscillations at the two stations were different in respect to the vertical variations of amplitude and phase of the two waves, and in respect to the relative magnitude of the semidiurnal and diurnal oscillations. This might be expected if the oscillations were associated with progressive internal waves. If waves of different order are present, the combined result may be a complicated variation with depth of amplitudes and phase angles (fig. 156, p. 598), and the velocity of progress of such waves is so small, 0.7 to 2.5 km per hour, that different phases must be found at stations which are 106 km apart. Furthermore, the amplitudes may vary along a line at right angles to the direction of progress, owing to the earth's rotation, as in a Kelvin wave (p. 555).
In his discussion Helland-Hansen draws special attention to the fact that the observed variations of temperature may be caused by variations of horizontal currents and not by internal waves. In the Faeroe-Shetland Channel all isothermal surfaces slope considerably, and lateral displacement might therefore give rise to such variations as were recorded. The same reservation must always be made when interpreting oscillations of temperature in a region where a lateral temperature gradient exists.
Recent measurements by Seiwell (1937) have substantiated the view, however, that observed oscillations are due to vertical displacements and not to horizontal movement, because he selected a locality for repeated serial measurements in the region NNW of Bermuda within which, on several of the Atlantis cruises, very small horizontal gradients had been found. On July 12 and 13, 1936, Seiwell (1937) observed very large vertical displacements, reaching total ranges during 24 hours up to 80 m at depths of 500 to 600 m. As an example the observed temperatures at 500 m are shown in fig. 154 where the corresponding vertical displacements are also entered. The latter were computed by dividing the temperature
|Depth (m)||Semidiurnal vertical displacements||Diurnal vertical displacements|
|Michael Sars 115||Sc||Michael Sars 115||Sc|
|Amplitude (m)||Phase (lunar hours)||Amplitude (m)||Phase (lunar hours)||Amplitude (m)||Phase (lunar hours)||Amplitude (m)||Phase (lunar hours)|
Harmonic analysis showed that at all levels the major part of the observed oscillations could be represented as the sum of three oscillations of periods 24, 12, and 8 lunar hours. The amplitudes of the harmonic terms varied with depth, but the 24-hour term dominated at all levels and the 8-hour term was smallest at most levels. The facts that in this case horizontal motion cannot account for the observed variations of temperature and that the oscillations were periodic strongly suggest the presence of some kind of wave motion.
|Depth (m)||Semidiurnal waves||Diurnal waves|
|Current||Vertical displacement||Current||Vertical displacement|
|Maximum current towards||Velocity (cm/sec)||Phase (lunar hours)||Amplitude (m)||Phase (lunar hours)||Maximum current towards||Velocity (cm/sec)||Phase (lunar hours)||Amplitude (cm)||Phase (lunar hours)|
|50||N 30 W||9.4||3.0||N 12°W||10.3||11.8|
|100||N 72 E||5.4||1.9||N 33 W||4.8||15.9|
|150||N 67 E||11.6||3.3||7||3.7||N 84 W||9.9||6.1||4||7.4|
|200||N 61 W||11.1||6.8||10||4.0||N 63 W||6.7||3.8||8||16.9|
|300||N 42 W||9.4||6.3||7||3.6||N 49 W||4.7||5.4||6||23.4|
|500||N 51 W||5.7||3.2||7||3.8||N 14 W||11.9||17.2||6||6.4|
The existence of internal waves is also confirmed by the results of numerous current measurements from vessels anchored in deep water (Ekman and Helland-Hansen, 1931, Defant, 1932, Lek, 1938). Observations from different depths show that currents of tidal periods dominate; but, instead of being uniform from surface to bottom as would be expected if the currents were ordinary tidal currents, the amplitude and the time of maximum current (the phase) vary in a complicated manner with depth, and at some levels the semidiurnal currents are strongest, and at others, the diurnal. This is illustrated by the results of current measurements and repeated serial observations at Meteor anchor station 176 on the Mid-Atlantic Ridge in the South Atlantic Ocean, lat. 21°29.8′S, long. 11°41.5′W
It is evident that extremely complicated currents may be found if several internal waves of different orders, different phases, and different tidal periods are present, and if the currents associated with these waves are superimposed upon the ordinary tidal currents. At first glance it may appear hopeless to separate the latter from the currents of the internal waves of tidal periods, but fortunately these “internal tidal currents” can be eliminated if observations are available from a sufficient number of depths. Because (p. 588)
It follows from equation (XIV, 53) that currents associated with internal waves are eliminated by computing the average currents between the surface and the bottom, provided that observations from a sufficient number of depths are available. Such elimination was attempted by Defant when he derived the tidal currents from observations at anchor stations in the Atlantic Ocean (see p. 584), but the available data were mostly from the upper layers only, and this accounts perhaps for the fact that he found much greater velocities than those corresponding to the range of the tide.
So far, the effect of the earth's rotation and of friction have been neglected. It cannot be expected that Corioli's force alters the variation with depth of amplitude of the internal wave, and Fjeldstad's method should therefore in all cases give correct results as to the relative amplitudes. On the rotating earth the accompanying currents, however, will also be rotating if the period of the internal wave approaches the period of a pendulum day. Transverse currents will accordingly be present, but these must also satisfy equation (XIV, 53) and can therefore be eliminated if observations from many depths are available. The velocity
An internal wave in water consisting of two layers of different density can be considered as an oscillation of the boundary surface. Defant (1940) has shown that on the rotating earth the period of a free oscillation of a boundary surface in the sea approaches the period of the inertia oscillation when the dimensions of the oscillating system are great. In a basin of length l the longest period of the free oscillations is, with the previous notations and neglecting the earth's rotation,
The period of the inertia oscillation is Te = 2π/2ω sin ϕ. Defant obtains the result that on the rotating earth the period of the free oscillation is
As a numerical example values from the Baltic may be introduced, ρ − ρ′ = 2 × 10−3, h′ = 25 m, and h = 35 m. With these values one obtains Tr = 3.75 l (l in meters). If it is required that Te/Tr = 0.1, one obtains in latitude 57°49′, l = 136 km. In a basin of these dimensions, the difference between the periods of the free oscillations and the inertia oscillation would be only 1 per cent, and it seems therefore probable that a disturbance which would develop motion in the inertia circle would set up free oscillations of the boundary surface. A deeper basin or a basin in lower latitudes would have to be of greater dimensions; thus, with ρ − ρ′ = 2 × 10−3, h′ = 500 m, and h = 1500 m, one obtains Tr = 0.736 l, and with Te/Tr = 0.1, one finds that in the latitude 57°49′, l = 692 km, and in latitude 30°,l = 1170 km.
The friction will lead to a dissipation of the energy of the internal wave, and unless the wave is maintained by a periodic disturbance it will gradually die off. The current observations from the Baltic by Gustafson and Kullenberg, which were discussed on page 438 (fig. 104), can be interpreted as inertia oscillations that may be associated with an internal wave and as showing the gradual dissipation of the energy of the wave.
It is evident from this discussion that the internal waves greatly complicate the actual movement of the water masses and lead to the
Variation in depth of stated σt values on June 23 and 24, 1930, according to observations of the Snellius Expedition at Station 2534 in lat. 1°47.5′S, long. 126°59.4′E.
An illustration of the application of Fjeldstad's theory is given by Lek (1938) in his discussion of results of current and serial measurements of the Snellius Expedition in the eastern part of the Netherlands East Indies, 1929–1930. On this expedition current measurements were made at a number of anchor stations and at one of these, station 253A, lat. 1°47.5′S, long. 126°59.4′E, very complete observations comprised hourly measurements during 26 hours of currents at 0, 50, 100, 200, 350, and 500 m, and hourly observations of temperature, salinity, and oxygen at seven depths between the surface and 800 m. The depth to the bottom was 1740 m. From the temperature and salinity data the density was computed. In fig. 155 is shown the variation in depth of different σt curves during the period of observation. From this graph one obtains immediately the impression that large vertical oscillations took place, some of which appear to have been of a diurnal, others of a semidiurnal, period. The phases of the oscillations appear to have varied with depth, the vertical oscillation at a depth of about 50 m being opposite in phase to the oscillation at a depth of about 230 m. Hence,
In order to examine the character of these waves, Lek, in cooperation with Fjeldstad, computed the relative amplitudes of the internal waves of first, second, third, and fourth orders by means of the average distribution of density between the surface and the bottom. Consider first the semidiurnal oscillations. It is evident that any observed variation with depth of amplitude and phase can be represented by a sufficiently large number of the theoretical internal waves of different orders, because the absolute values of the theoretical amplitudes and the phase angles of the theoretical displacements can be adjusted to fit the observed data. A certain check on the theory is, however, obtained if the number of the theoretical internal waves is smaller than the number of depths of observations.
(A) Variation with depth of the amplitudes of internal waves of order 1 to 4 at Snellius Station 253A. The phases of the different waves are shown in the figure. (B) Curves show the variation with depth of amplitude η and phase α, as derived from the curves in (A). Crosses and dots indicate observed values (according to Lek and Fjeldstad).
In the present case observations from five levels were used for determining the amplitudes and phases to be assigned to the internal waves or orders one to four, meaning that an adjustment was made, and that the validity of the theory could be checked to a certain extent by examining how closely the observed values would agree with the theoretical after making such adjustments. Figure 156A shows the adjusted amplitudes and phase angles of the internal waves of order one to four between the surface and 500 m. From this figure it is evident that the wave of third order is dominant. By combining these four waves one obtains
|Depth (m)||Amplitude (cm/sec)||Phase|
These results cannot be considered as conclusive evidence as to the character of the observed displacements, because the numerical values of the theoretical terms have been adjusted to fit the data, but an entirely independent check can be obtained by computing the currents corresponding to the theoretical internal waves and comparing these computed values with the observed ones. When doing this, it should be borne in mind that the theory presupposes the existence of progressive waves, and agreement in phase of computed and observed currents would indicate that the internal waves actually were of the progressive type. It should also be borne in mind that the observed currents include ordinary tidal currents besides those associated with internal waves of tidal periods, and that for this reason certain discrepancies must be expected. The semidiurnal currents were not used as a check on the theory because there were reasons for believing that these were not of the simple progressive type, but the computed and observed diurnal currents which are shown in table 75 are in surprisingly good agreement, particularly when considering the reservations which were made. Lek points out that the greatest discrepancies between computed and observed values occur at a depth of 350 m, where according to the theory the current due to internal waves should nearly vanish, and that therefore the discrepancy may be due to the presence of actual tidal currents. It thus appears that in this case the complicated variations of the currents
Lek and Fjeldstad find the following velocities of progress: c1 = 234 cm/sec, c2 = 116 cm/sec, c3 = 77 cm/sec, c4 = 58 cm/sec. The corresponding lengths of the waves of period 24 lunar hours are 210 km, 104 km, 69 km, and 52 km, respectively.
This example illustrates the numerous complications which may be encountered anywhere in the ocean, and serves to emphasize the fact that many observations of currents over long periods of time are needed in order to obtain information as to the many types of motion present in the sea.
Standing internal waves may be present in bays or basins. The probability of such standing waves is great, because in heterogeneous water in a bay or a basin of a given form a large number of internal waves of different wave lengths are possible, corresponding to waves of different order and corresponding to different period lengths. An intermittent disturbance or a disturbance of tidal period may therefore bring about an oscillation which corresponds to one of the possible free oscillations of the system, particularly because a small amount of energy is needed for creating an internal wave.
In a bay of constant depth and width the periods of oscillation of free standing waves in the presence of two layers are
A computation of this nature may give approximately correct results if a distinct boundary surface is present and if the geometrical shape
In conclusion, two effects of internal waves should be emphasized because they have bearing on general oceanographic problems. In the first place the internal waves probably are of importance to the process of mixing. Where internal waves are present, large velocity gradients are often met with which lead to great values of the eddy viscosity. Furthermore, owing to the dissipation of energy by friction, a given water mass will never return exactly to the locality from which it started out, even in the absence of general currents, and consequently an exchange of water in horizontal direction must take place. The intensity of the mixing processes which are maintained by internal waves has not yet been examined, but it is probable that these processes are not negligible.
In the second place, it has been pointed out, particularly by Seiwell (1937), that owing to the existence of internal waves the distribution of mass along any vertical will be subject to periodic variations and, as a consequence, the geopotential height of the free surface relative to a given isobaric surface will vary periodically. This agrees with the statement (p. 588) that when dealing with internal waves on a boundary between two liquids the free surface will also show a wave motion of a small amplitude. The variations in height of the free surface are so small that when dealing with the internal wave they can be disregarded, but when examining results of dynamic computations they cannot be left out of account because these variations are of the same order of magnitude as the horizontal differences in geopotential height which may occur on distances up to 100 km or more. As an illustration, fig. 157 shows the variation of geopotential height of the free surface above the 800-decibar surface as derived from the serial observations at Snellius station 253A. It is seen that during 26 hours the height of the surface varies no less than 14.5 dyn cm. Seiwell has computed that, due to internal waves at Atlantis station 2639 (p. 453) the variations of the
Variation with time of the dynamic height of the surface over the 800-decibar surface at Snellius Station 253A.