Spectra of low frequency ocean waves along the Argentine shelf

Deep-Sea Research, 1962, Vol. B, pp. 1515 to 164.

Pergamon Press Ltd.

Printed in Great Britatn

Spectra of low frequency ocean waves along the Argentine shelf DOUGLAS INMAN,* WALTER MUNK* and MARCIANO BALAY'~ (Received 3 January 1961)

Abstract--The low frequency spectra covering the ' quiet ' range between the astronomically induced tides and the wind generated swell was obtained by an analysis of the records from three tide gauge stations along the broad Argentine shelf. The cross-spectral analyses suggest a peaked spectrum (perhaps associated with shelf-resonances)superimposed upon an independently generated spectrum which diminishes monotonically with increasing frequency. The absence of coherence between Quequen and Mar del Plata, separated by a distance of 67 nautical miles, suggests that the seiches at the two locations are not directly related in any simple way. The disturbances are not propagated as simple edge waves between the two areas. INTRODUCTION THE BROAD Argentine continental shelf is noted for the frequent occurrence of open sea seiches up to a metre or more in height, and with periods from a few minutes to several hours. BALAY (1955) has shown that these oscillations frequently occur coincidentally with the passage ~f sharp meteorological pressure fronts coming from central Patagonia. The seiche activity over this broad shelf is indeed unusually high at all times. This suggested to us that it might be worthwhile to carry out crossspectral analysis between tide records from two locations along the shelf. The principle advantage of this technique is that it gives the phase relation as a function of frequency for irregular, complex records and does not depend on the passage of an identifiable pulse. Accordingly, we have analysed the marigrams from three tide stations along the Argentine coast. Two of the stations are situated near Mar del Plata; one on the open sea (Mar Libre), the other three miles to the south inside the protected harbour (Puerto). The third station is inside the harbour of Quequen, about 67 nautical miles west-southwest of Mar del Plata. The continental shelf is approximately 90 nautical miles off Mar del Plata and somewhat wider off Quequen (FIG. 1). The tides in the region are mixed with both semi-diurnal and diurnal components present, and have a maximum diurnal range of almost two metres at Mar del Plata and somewhat greater to the south. The tidal records from each station were analyzed for the ten-day period from 20 to 31 December 1955. All three stations showed rather high levels of background noise throughout the ten-day period. Seiche-like oscillations were especially pronounced during the night of 24-25 December and attained heights of about 80 crn at both stations in Mar del Plata (Fro. 2). In order to obtain spectra with and without these high oscillations, all three records were divided into two continuous series of about five days each and analyzed separately. In each case, the part A records of Figures 3 and 4 contained the high oscillation of 24-25 December 1955. *Scripps Institution of Oceanography of the University of California, La Jolla, California. ~Jefe Division Mareologia, Ministerio de Marina, Republica Argentina. 155

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Fig. 2. Tide record from Mar Libre station at Mar del Plata showing seiche on 25 December 1955. Observations from this record are included in the spectral analysis for this station in Figures 3 (a) and 4 (a).

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DOUGLASINMAN, WALTI~ MUNK and M~mcmNo BALAY

ANALYSIS

PROCEDURE

The records have been analysed according to the method of Turd~Y (BLACKMAN and TUKEY, 1958). For details refer to MUNK, SNODGR~S and TUCKER (1959). The time series were obtained by reading the tide record at five minute intervals to the nearest I mm of sea level, and punching the results on cards. The series were then checked on an I.B.M. 650 for errors (using a first-difference criterion). The tide was reduced by a high-pass filter with 201 weight factors, and the power spectra of the filtered records obtained. The last two operations were performed on the I.B.M. 709 computer at the Western Data Processing Centre, University of California, Los Angeles. At low frequencies the spectra are contaminated by tides owing to an unfortunate choice of the high-pass weight factors. Part A spectra of FIG. 3 and 4 extends from 00h 00" 20 December 1955 to 01 h 05 m 27 December 1955 before high-passing and from 09 h 05 m 22 December to 01 h 05 m 27 December after high-passing.* Part B extends from 00h 00 ~ 27 December 1955 to 23 h 55m 31 December before high-passing, and from 00 h 00m 27 December to 15h 35" 31 December after high-passing. Each of the two parts contain about 1340 values. When analyzed into 100 frequency bands this gives 2 × 1340/100 : 27 degrees of freedom. Standard deviation of the power spectra is roughly (27)-t -- 0.2 times the estimates. In order for the (coherence)2 to be significant it should exceed 4/27 : 0.15. All results are plotted as a function of frequency in FIGS. 3 and 4. The upper plot gives the power spectra for each of the two stations under consideration; the central plot gives the square of the coherence between stations; and the lower plot gives the phase difference between stations. The spectral density of the power spectrum is expressed in centimeters squared per cycle per kilosecond (cm~/c.p.ks.), and can be interpreted as the contribution towards the variance per unit frequency band. The coherency and the phase difference between the records of the two stations can be interpreted as follows : if any two records (x, y) are ' played' through a narrow filter peaked at some frequency f, and the filtered x record lagged relative to the filtered y record by a variable phase $, the plotted phase angle is that value of $ for which the correlation is a maximum, and the plotted (coherence)~ is the square of this maximum correlation. DISCUSSION

Part A spectra for Mar del Plata (both Mar Libre and Puerto), FIG. 3 (a), shows narrow peaks of about 150 cn~2/c.p.ks, at frequencies of 0.20 and 0.34 c.p.ks., and somewhat lower and broader peaks near 0.60 c.p.ks.f The latter is more pronounced for the Puerto spectrum than for Mar Libre. There is good coherence (strong correlation) between Mar Libre and Puerto spectra in the low frequency portions of the spectrum and poor coherence for frequencies higher than about 0.5 c.p.ks. Part B analyses for Mar del Plata (Flo. 3 (b)) show less coherence between Puerto and Mar Libre stations; the peak at 0-2 c.p.ks, is absent from both power spectra, while the broad peak near 0.6 c.p.ks, is only prominent for the Puerto station. *Use of 201 weight factors results in the loss of 100 values from the beginning of Part A and from

the end of Part B records. tThe corresponding periods are 1.44, 0.82 and 0.46 hr respectively.

Fig. 3 (a).

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Spectra of low frequency ocean waves along the Argentine shelf

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Curiously, while both the A and B spectra for Puerto are ' n o i s y ' they show very little power for frequencies of 1.2 c.p.ks, and higher, which more nearly correspond to the fundamental frequencies for the harbour. The good coherence between the spectra from the harbour and the open sea stations at Mar del Plata suggest that the water level disturbances within the harbour follow closely those of the adjacent open sea. Apparently the coupling between sea and harbour is good, and there is little tendency for the harbour to resonate at its fundamental frequency. The spectra for both analyses at Quequen (FIG. 4 (a, b)) show high random background noise throughout all frequencies, with little indication of narrow energy peaks. Further there is little significant coherence between the Quequen and Mar del Plata spectra except at 0.2, 0.34 and possibly at 0.6 c.p.ks., nor is there a convincing relation between the phase of the spectra in the two regions. Taken together the cross-spectral analyses suggest the presence of three separate phenomena: the tides on one extreme, a monotonic spectrum, and shelf seicheswhich form peaks rising above the monotonic spectrum. The shelf seiches are only significant in the Part A spectra, and are undoubtedly associated with the active seiche of 24-25 December 1955 (FIG. 2). The reason for suggesting a distinction between the monotonic spectrum and the shelf seiches is threefold : (I) The troughs between the spectral peaks of the Part A spectra occur at about the level of the monotonic spectra of Part B, as if these shelf peaks were superimposed on the monotonic spectrum. This suggests that without the monotonic spectrum the spectral density between peaks might have been considerably lower. (2) A solid line is drawn in FIG. 3 (a), (b) (bottom) representing a 60 min phase lag of Puerto behind the Mar Libre spectrum, which is due to clock error*. The dashed line in FIG. 3 (a) and (b) represents a 54 min lag of the Puerto station, which is actually a 6 rain lead when the two stations are reduced to the same time datum. A 6 min lead is the published phase difference for tides between these stations. It should be noted that the spectra of FIG. 3 (a) are in agreement with the 6 min phase lead, while that for FiG. 3 (b) indicates no phase difference (other than that due to clock error). From a detailed inspection of the phase differences, one gathers the impression that the 6 min phase difference is associated with the spectral peaks of FiG. 3 (a) at approximately 0.2 and 0.34 c.p.ks., whereas in the troughs the two records are more nearly in phase, as in Fig. 3 (b). (3) The monotonic spectrum, even though low compared with the shelf seiche, is significant as borne out by the meaningful coherence and the systematic phase difference between the spectra of the two stations for frequencies less than about 0.4 c.p.ks. The above three observations suggest that the monotonic spectrum is associated with a different type of wave phenomena than that of the shelf seiche. In particular, Waves associated with the monotonic spectrum are in phase in and out of harbours, whereas the oscillations associated with the peaked spectrum are nott. *This discrepancy of one hr was discovered before we were aware that the clock time at the two stations was in fact different. Upon subsequent inquiries it was found that the Mar Libre clock time was Greenwichminus 4 hr, whereas Puertowas Greenwich minus 3 hr, thus confirming our suspicions. It is not suggested that cross-spectral analysis be routinely employed to obtain clock corrections. tThe reviewer, Dr. W. J. PIERSON,Jr., has kindly pointed out to us that on the basis of some of his unpublished work he finds that truly high coherences are quite drastically reduced if the crossspectra vary rapidly as a function of frequency. Failure to align the records in proper time juxtapositton may cause rapid variation in the cross-spectra at these high frequencies and could perhaps result in the poor coherences that were computed. This casts doubt on the computed coherences for periods short compared to the time displacements between the records, i.e., periods short compared to one hour or frequenciesabove 0.5 c.p.ks.

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DOUGLASINMAN,WALTERMUNKand MARCL~NOBALAY

The principle object of the investigation was to derive information concerning the travel of long waves along the broad Argentinian shelf. In this sense the investigation has proven a disappointment. The three promient low frequency peaks at Mar del Plata are not clearly reproduced at Quequen (FIG. 4 (a)). This result perhaps merely reflects the fact that the width of the shelf changes by a factor of 2 between these two stations (FIG. 1). It is recognized that the problem of shelf seiches is a three-dimensional one, and the application of two dimensional theories is extremely doubtful. Nonetheless, we wish to make some order of magnitude calculations of observed vs. theoretical relations. The calculations in TABLE 1 refer to the three spectral peaks having significant coherence between Mar del Plata and Quequen. The third row in the Table lists our estimate of the phase lag at Mar del Plata relative to Quequen. The lag in degrees is converted to hours in the fourth row, and the travel velocity corresponding to this lag is listed in the fifth row. For comparison we have computed the phase velocity of an edge wave travelling over a shelf with a constant slope of 5 / 10 -4 (LAMa, 1945, p. 447). A comparison between the figures in row 5 and 6 shows that the predicted velocity for edge waves are low by two orders of magnitude.

Table I. Velocity of propagation calculated .from the phase difference between Quequen and Mar del Plata compared with that predicted by edge wave theory. Frequency of spectral peak, c.p.ks.

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*Phase from Fig. 4 (a); Quequen leads Mar del Plata. **Distance Quequen to Mar del Plata 124 km. tPhase velocity for edge wave over shelf with slope 5 × 10-4, LAMa, 1945, p. 447. It is clear from the computations of TABLE 1 that an excitation over the continental shelf at Quequen could not reach Mar del Plata at the observed intervals if the energy were transmitted along the shelf as an edge wave. The evidence, for what it is worth, would rather indicate that a disturbance sets up a wide spread complex oscillation with some of the coupling taking place in deep water beyond the shelf. We have not attempted the prerequisite three-dimensional problem. REFERENCES BALAY, M. A. (1955) La determinacion del nivel medio del Mar Argentino. Ministerio de Marina, Dept. Oceanografia, Republica Argentina, 46 pp. BLACKMAN, R. B. and TUKEY, J. W. 0958) The measurement of power spectra from the point of view of communications engineering. The Bell System Technical Journal, 37, 185-282.

LAMB, H. (1945) Hydrodynamics, 6th ed. Dover, New York, 738 pp. MUNK, W. H., SNODGRASS,F. E. and TUCKER,M. J. 0959) Spectra of low-frequency ocean waves. Bull. Scripps Inst. of Oceanogr., Univ. of Calif., 7, (4), 283-362.