Seiches on Lake Wakatipu, New Zealand

Transactions and Proceedings of the Royal Society of New Zealand, Volume 83, 1955-56, Page 579

Seiches on Lake Wakatipu, New Zealand By G. A. Bottomley, University of Otago. [Received by Editor, August 1.1955.] Abstract This paper is a report of a co-operative effort by the University of Otago Science Students' Association, and presents selected recordings of the surface oscillations of Lake Wakatipu, showing the principal features which were observed in a series of observations extending over three hundred hours. The fundamental period (52mins) and the first harmonic (27mins) are predominant in most of the results. A portion of the Lake gives rise to the very well developed oscillation of 4.28 minutes recorded at one site. No phenomenon was observed which cannot be explained by serche action in one or other form. Historical Despite a very strong regional belief in the reality of pulsations on Lake Wakatipu, first hand accounts of the phenomena are extremely rare. A Maori legend (Beattie, 1945) and an account by a Government official (Higginson, 1877) provide the sources of the usual description. An appeal through the regional and national papers brought no relevant response. The following article represents an attempt to provide some factual basis for discussion. General By suitable operation of the wind or atmospheric pressure variation the water in any natural basin may be set in oscillation with definite periods which depend upon the depth and cross sectional area. Such oscillations, generally termed seiches, are well known on other large lakes and have been subjected to much experimental and mathematical scrutiny (Landolt-Börnstein, 1952; Murray and Pullar, 1910.) For a long basin the simplest mode of seiche is the fundamental longitudinal one. In this the water at all points moves approximately parallel to the basin bottom so as to produce first a surface elevation and then a depression at each end in turn, with a nodal plane at the centre, and a surface form approximately planar. Various harmonics are possible corresponding to surface forms with two, three and more nodal planes at right angles to the long axis. The actual behaviour of the lake will be some combination of several of these modes of vibration, each with a different amplitude, possibly with the further complication of similar transverse vibrations superimposed, and all these underlying the normal wind-generated surface waves. Experimental Method The apparatus used consisted of a clockwork driven drum, making one revolution per 24 hours, and giving a trace some 10 inches long, on which the rise and fall of the lake surface at a selected point was recorded by a float operated pen. The effect of the surface waves, which are of period less than 5secs, was eliminated by the usual stilling well, in our case a float chamber of cross section 9in by 9in connected to the lake by a pipe about 8 feet long and ¾in diameter, the whole forming a “choke” which has a transmission of about 95% for waves of period more than 3mins, and less than 1% for waves of less than 10secs period. With

the improvised gear some difficulty was experienced in providing adequate protection from the heavy surface swell which arises on the lake in windy weather, and it was generally necessary to build a boulder wave-break at least three feet high around the recorder. Location and Recording Stations Observations were taken at various sites in the preliminary work, but it was soon decided to restrict attention to three stations, Kingston, Bob's Cove, and Kinloch.* The sites may be more exactly described: Kingston. Thirty yards North of the Otago-South-land boundary on the Kingston-Queenstown Road, about two miles from Kingston. Bob's Cove. On the North bank some eight miles West of Queenstown. One hundred yards towards the North end of the lake from the Old Government Wharf on the East side of the Cove. Kinloch. A hundred yards South of the main Kinloch wharf. The first and last of these represent the two extremes of the Lake, South and North respectively, whilst Bob's Cove lies very close to the computed nodal point for the fundamental oscillation. The approximate sites of the stations are indicated on the sketch map of the lake.

Related Theoretical Investigations Various mathematical procedures have been devised for calculating the periods of oscillation for natural lakes. When the shape of the lake approximates to a simple geometrical form, for example uniform width and parabolic longitudinal section, a simple formula due to du Boys (1892) may be used. More rigorous methods are available, though for lakes of less simple form the numerical labour becomes considerable. (Chrystal, 1905; Doodson, 1920; Bergsten, 1926.) Using the bathymetric data for Lake Wakatipu obtained by Lucas (1904), Bottomley (1955) has deduced the periods, the nodal positions, and the surface shapes, for various modes of vibration. This work has been done by arithmetical integration of the fundamental differential equations of motion using as trial solutions values first obtained from a study of a laboratory model hydro-dynamically similar to Lake Wakatipu. The results of this theoretical work which are relevant to the present observational report are given below:— Whole Lake. Fundamental Period 50.9 minutes.* It will be noticed that the harmonics are not exactly one-half, one-third, etc, of the fundamental period. This behaviour arises from the nature of the second order differential equation which describes the liquid motion during oscillation. The extent of the departure from the harmonic series 1, ½, ⅓, etc., depends upon the geometrical shape of the lake basin, special cases are possible in which the series is exactly obeyed (rectangular basins) or even in which the harmonics have the same period as the fundamental, though for real lakes the deviation usually decreases with increasing harmonic number. First Harmonic 27.7 minutes. Second Harmonic 20.2 minutes. Description Of Text-figures Text-Fig. 1. This shows typical traces as obtained at Kingston. The curves are complex in form, and do not retain the same nature for more than a few hours as a rule. Throughout both traces there is a general pattern-main peak, small peak, main peak, small peak, etc. with the main peaks spaced out at intervals of rather less than an hour. (This is shown very well at the left hand end of the lower trace). This pattern is due to the combination of the fundamental oscillation and the first harmonic; analysis of portions of the curves enables the periods to be determined as Fundamental 52, First Harmonic 27mins. The large peaks are thus due to the reinforcement of the fundamental maximum every 50-odd minutes by the every alternate first harmonic swing. The agreement between the observed periods and those computed by Bottomley (1955) is sufficient to make the identification certain; further evidence is presented under Text-Fig. III. The fundamental period is not exactly double that of the first harmonic. The gradual displacement in time of the every second swing of the first harmonic with respect to each swing of the fundamental is shown in the lower trace at the left hand end. A change in the weather occurred in the early hours of 18.5.55, there being strong winds in certain sections of the lake, and the effects on the traces became apparent at about 08.00 hours. Higher harmonics develop and there is a general loss of regularity, but the major oscillation due to the fundamental is still observable.

Text Figure 1

Text Figure 2

The traces in Text-fig. I show an average behaviour; smaller amplitudes have been recorded (some are shown in Text-fig. IV), and much larger ones have been recorded and observed visually. On 28.12.54 the amplitudes at Kingston were larger than the four inches which the machine would record and were estimated to reach a little over five inches. At no time has the hourly fluctuation been less than half an inch as observed at Kingston during occasional observations over the last eight months. Text-Fig. 2. From the position of the nodal planes, and the surface shapes calculated by Bottomley (1955) for the fundamental and the first harmonic motion it is apparent that particular interest attaches to readings taken at or near Bob's Cove. At this site the amplitude of the fundamental should be very small owing to the nearness of the nodal plane, and yet the first harmonic (with a ventral point near Bob's Cove) should be present with some 60% of its end amplitude. The lower curve shows, for the first twelve hours, practically pure first harmonic tracings, and these may be used with advantage to calculate the corresponding period, 27mins. The linear descent of the curve with time represents the fall in the mean lake-level with time due to excess run-off over intake. The slope of the curve and the surface area of the lake indicate a figure of 4,530 cubic feet per second for this particular occasion. By 12.00 hours on 18.5.55 a marked change has come over the traces and there is a very pronounced development of oscillations of very short period. The upper trace shows how these continued to increase on 19.5.55 until towards 12.00 hours on that day the swings were too large to record on the instrument. These oscillations were of period 4.28 minutes, and reached an amplitude which was in excess of eight inches. At the time of these large amplitude oscillations there was virtually no surface wave in the Cove itself and there was not the slightest difficulty in seeing the periodic rise and fall of the surface level on posts and stones in the water. The advance and retreat on the sloping beach was very striking, reaching 10 or 12 feet in some places, yet it was not noticed by a party of twenty casual visitors who spent an hour near the lakeside. There is insufficient evidence to identify positively the particular basin of the Lake which is giving rise to this mode of oscillation. Large amplitude short period waves were recorded at Kinloch during the same day, but no positive correlation can be made between the maxima at the two sites. Regrettably there was no observation station at Queenstown at the material times, but an experienced observer who knew of the phenomenon then occurring at Bob's Cove did not see any similar effect in Queenstown Bay. It is possible that the effect is that of a longitudinal seiche of either (possibly both) the Kinloch arm and the Walter Peak arm. Rough measurements on the model used by Bottomley (1955) show that the fundamental periods for these arms are 23 minutes and 12 minutes respectively, so that it would be necessary to suppose a high harmonic to obtain the required frequency. Direct measurement on a small dam near Dunedin has shown the existence of a fundamental period of about 35 seconds. Bob's Cove is approximately the same area as this dam, and much deeper. As the period decreases with increasing depth, and as open bays have periods roughly half of the corresponding closed bay, it may safely be deduced that the natural period of Bob's Cove is not more than 30 seconds at most, and this disposes of the suggestion that that 4.28 minutes period is that of the Cove itself.

Text Figure 3 The most likely explanation is that the effect is a transverse seiche across the Lake from Bob's Cove to the South bank. Calculation shows that the period of such an action would be 3 ½-4 ½ minutes, depending somewhat on the assumed transverse section and making liberal allowances for errors in the interpretation of Lucas's survey. In the absence of simultaneous measurements on opposite banks with more rapidly revolving drums so that phase relations could be established, further discussion is conjectural. Text-Fig. 3. This shows portions of traces taken simultaneously at the two extremities of the Lake. Both show the general pattern due to a combination of fundamental and first harmonic seiches. At a time when the first harmonic and the fundamental are in phase to give an enhanced upward swing at Kinloch, then it must be that at Kingston the first harmonic and the fundamental are acting in opposite direction, and interfere to produce a reduced swing. The various features of these two curves can be more fully interpreted in this fashion. Text-Fig. 4. This is part of a continuous record extending over five days taken at Kingston in December, 1954. It will be noticed that there is a pronounced long term periodicity with a period of rather more than 12 hours. Similar behaviour is shown (though not so clearly) by the traces taken at Kingston and at Kinloch in May, 1955, but is not detectable at Bob's Cove. It is considered that the effect observed can be attributed to a temperature seiche (for examples and discussions see Honda, 1915 and 1916), that is an oscillation of the general kind so far considered but taking place at the junction

Text-Fig. 4.—The upper and lower traces torm a continuous record taken at Kingston in December, 1954. The twelve-hour periodicity is shown most clearly in the upper trace, but is fairly distinct also in the lower. between the upper hot and the lower cold layers of the lake, the thermocline. Given sufficient data for the relative depths of the two layers and their densities, it is possible to estimate the period of oscillation. Using the data obtained by Miss V. Jolly, on the location of the thermocline, one can show that the period of 12 hours is in the possible range, but much more work, observational and computational, is necessary if the suggestion is to be verified. Correlation Of Observed Seiches With Local Winds At each observation station at three-hourly intervals rough notes were taken of the direction and approximate velocity of the wind, and the state of the surface waves. (Because of the extremely enclosed nature of Bob's Cove very little significance can be attached to weather observations there.) Such evidence as we possess indicated that the development of the 4.28 minute seiche at Bob's Cove depended upon a North wind of considerable intensity blowing down the North Arm. The North wind acting down this arm gave rise to surface waves which increased in size as the Middle Arm was approached, though very little of these waves was transmitted round the corner, and in the shelter of Bob's Cove the 4.28 minute seiches were observed in almost dead calm water. In general it can be said that during a calm spell of weather the lake shows principally its fundamental longitudinal oscillation, whilst the higher harmonics become more pronounced and even predominate during or soon after high winds in any or all arms. Regrettably we have not had the opportunity to watch the seiches undergoing spontaneous damping after a spell of considerable activity. Acknowledgments Assistance has been freely given by so many organisations and individuals to those concerned in this work that it has become impossible to refer to all by name. Part of the work was supported by a grant made to G. A. Bottomley by the Royal Society of New Zealand towards the costs of transport on the lake. The University of Otago Science Students' Association assisted very materially

in providing funds for a party of students who operated the three stations continuously during a week in May, 1955. Apparatus was kindly loaned by the Meteorological Office and very great assistance and kindness was given by residents of the Lake area. To all these persons and organisations we express our indebtedness. Science Students' Association, University of Otago, Dunedin. This paper was written and communicated to the Royal Society by G. A. Bottomley on behalf of the Science Students' Association. It supplements preliminary descriptions given in Science Record, August, 1955. References Beattie, H., 1945 Maori Lore of Lake, Alp and Fiord. p. 36, Dunedin 1945 Bergsten, F, 1926. The Seiches of Lake Vetter. Geogr. Ann. Stockh. 1-73. 1926. Bottomley, G. A., 1955. Free Oscillations of Lake Wakatipu, New Zealand Trans. Amer. geophys. Un., in press. Boys, P. Du, 1892. Essai théorique sin les seiches Arch Sci. Phys. Nat. (Genève). sér 3, XXV 628. 1892. Chrystal, G., 1905 Mathematical Theory of Seiches Trans. roy. Soc. Edinb., 41. pp. 559-649, July, 1905. —— 1905. Calculations of the Seiche Periods and Nodes of Lochs Earn and Treig Trans. roy. Soc. Edinb., 41, pp. 823-850, Nov., 1905. Doodson, A. T., 1920. Theoretical Determination of the Longitudinal Seiches of Lake Geneva Trans. roy. Soc. Edinb., 52, 629-642. 1919-1920. Higginson, H. P., 1877. On Floods in Lake Districts… Trans. N.Z. Institute. Vol. 10, 180. 1877. Honda, K., 1915 Ordinary and Internal Seiches on Lake Tasawa. Sci. Rep., Tohoku Univ. 4, pp. 33-42. 1915. —— 1916. Remarkable Internal Seiches in Lake Inawasiro Sci. Rep., Tohoku Univ. 4.5., 323-331. 1916. Jolly, V., 1954. Private Communications. Landolt-Bornstein, 1952. Zahlenwerte Und Funktonen… Section 32635. Berlin. Springer-Verlag, 1952. Lucas, K., 1904. Bathymetric Survey of the Lakes of New Zealand. Geogr. J., Vol. 23. 1904. Murray, J. and Pullar, L., 1910. Bathymetric Survey of the Scottish Fresh Water Lochs, Vol. 1. Challenger Office, Edinburgh.