Seiches of Sydney Harbor, N.S. A. K. EASTON

Marine Sciences Branch, Department of the Environment, Ottawa, Ontario

Manuscript received July 16, 1971 Revision accepted for publication April 10, 1972

The periods and responses of the free oscillations are calculated for one-dimensional models of Sydney Harbor. In particular, after Rayleigh's mouth correction is taken into account, it is evident that the prominent wave often recorded at North Sydney is the fundamental seiche. The seiche is induced by the passage of weather systems along the coast of Nova Scotia. Its response could be reduced by building a dam across the North West Arm.

Les pkriodes et les rkponses d'oseillations libres sont calculks pour des modkles unidimen- sionnels du Port de Sydney. En particulier, une fois que Yon tient compte de la correction d'embouchure de Rayleigh, il devient kvident que l'onde probminente, souvent enregistrbe A Sydney Nord, est la seiche fondamentale. La seiche est induite par le passage de systhmes mCtCoroIogiques le long de la NouveIle Ecosse. Sa rhponse pourrait Stre r&uite par l'kdification d'un barrage au travers du Bras Nord-Quest.

Inb-oduction determine the influence of variations in the Waves with periods of about 120 minutes

and ranges of 0.3 m are often recorded by the tide gauge at North Sydney, N.S. (Fig. 1 ) . Honda and Dawson ( 19 1 1 ) observed that they are particularly prominent during stoms and estimated that they are the natural oscillations of Sydney Harbor.

Sydney Harbor is a Y-shaped basin (Fig. 2). The main channel extends from a fairly well defined mouth between Cranberry and Low Points through a constriction at the Southeast Bar. At Point Edward, it divides into the South Arm, which leads to the Sydney River, and the North West Arm.

Honda and Dawson assumed that the free oscillations will favor one of the upper arms and neglected the influence of the other. Then, from calculations based on Merian's formula, they obtained fundamental periods of 120 minutes and 132 minutes, which are identical with those they extracted visually from the records.

The purpose of the present investigation is to determine more accurately the periods and structure of the seiches of Sydney Harbor. Our calculations will be based upon the one-dimen- sional equations of motion. First, we examine an analytical model consisting of three con- nected channels. A numerical model is em- ployed in order to represent the actual varia- tions of width and depth of the harbor and to

harbor depth. After Rayleigh's mouth correc- tion is taken into account, it is clear that the prominent recorded wave, which prompted this investigation, is the fundamental seiche of the harbor. Finally, we show that the seiche may be excited by edge waves which are generated by storms moving along the Nova Scotia shelf and examine how it might be eliminated. Throughout this paper the units of time are minutes, of length kilometers, and of depth and height meters.

Hydrodynamical Equations The linear one-dimensional equations for a

FIG. 1. Prominent seiche shown on the tide record from North Sydney, N.S.-November 25, 1970.

Canadian Journal of Earth Sciences, 9, 857 (1972)

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CANADIAN JOURNAL OF EARTH SCIENCES. VOL. 9, 1972

- - - -

FIG. 2. Bathymetry and principal features of Sydney Harbor. Depth contours are in meters.

channel of gradually varying section are is the breadth. The x-axis is taken along the

aQ 86 principal axis of the channel.

L-11 - + gsa< = 0, If we assume that time dependence is of the at form eiDt, equations [ 11 and [2] become

where Q(x, t ) is the discharge, [(x, t ) is the surface elevation, g is the gravitational accelera- C41 --

dx ob(x)t;(x) = 0.

tion, S ( x ) is the cross-sectional area, and b(x)

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EASTON : SEICHES 859

Boundary conditions are applied at the ends and junctions of the channels. At the head of each arm, Q = 0; at the ocean entrance, [ = 0; at the junction, the surface elevation is con- tinuous and the discharge is conserved.

A three-channel model is suggested by the shape of Sydney Harbor; we call the seaward channel 1, the South Arm channel 2, and the North West Arm channel 3. The dimensions of these channels have been determined from the Canadian Hydrographic Service Chart No. 4315 (Table 1). For this system, equations [3] and [4], subject to the boundary conditions, yield (Defant 196 1 )

4 12 [ 5 ] blel cot o - - b2c2 tan o -- -

el C2

13 b3c3 tan o - = 0,

c 3

where the subscripts identify the channel and cj = (ghj)". The solutions u of [S] are the frequencies of the free oscillations. The corre- sponding periods far the first four modes are 103.1 minutes. 52.0 minutes. 36.6 minutes. and 22.5 minutes ' respectively ' (Table 2) . is much less than the periods calculated by Honda and Dawson (191 1) but lies between the periods of 99.1 minutes and 122.4 minutes, which are obtained by repeating their calcula- tions for the dimensions of the above mentioned chart.

When the derivatives of equations [3] and 141 are replaced by central differences, we ob- tain (Platzman and Rao 1964)

where Di = 2Axbi,

Ax = grid length.

These equations form the basis of the 'leap-frog' method, in which the discharge and surface elevation are calculated at alternate grid steps. The breadth must be known at the [-sections and the cross-sectional area at the Q-sections.

The equally spaced grid chosen for this study of the natural oscillations is shown in Fig. 3. Sections 1 to 19 are in the South Arm, 7l to

TABLE 1. Dimensions and resonant periods of the individual channels of the threechannel model

Channel

Breadth (b km) 3.48 1.10 1.56 Length ( I km) 9.72 9.21 6.14 Depth (h m) 10.9 11.3 9.7 TI (min) 31.3 58.3 41.9

19l in the North West Arm and 20 to 38 be- tween their junction and the sea. The grid spacing Ax = 0.5 12 km.

As initial conditions for equation [6], we take Ql = 0 and c2 -- 1. Then, for a selected CT,

we proceed to calculate Q3, C4, Q5, . . . Q19, 520 using equations [6] and [7] in turn. Next, we set QT1 = 0, assume for the moment that Ss1 = 1, and calculate Qal, . . . , Q191, tZo1. We know from continuity that Czo1 should equal Yzo and thus, since equations [6] and [7] are linear, we obtain the correct values for sections 8l to 20' by multiplying each one by [zo/[201 (Yuen 1969). Then,

and the values of Q and [ may be calculated for sections 21 to 38. If the selected u is the frequency of a normal mode, [38 = 0; other- wise, we choose another a and repeat the calculations. The initial frequencies chosen were those of the three-channel model.

Periods of the first four modes are listed in Table 2. The fundamental period of 107.2 minutes is larger than that of the three-channel model. However, the non-uniformity of the harbor is more marked for & and T3, which are decreased by 10 and 5 minutes, respectively.

Since Sydney Harbor is shallow (the mean depth is 10.8 m), we investigated the influence on the periods of the normal modes of extreme depth variations of 0.9 m above and below mean sea level; such variations are twice as large as the average tide range. There is an increase in the fundamental period of 5.4 minutes when the depth is decreased while there is a decrease of 4.6 minutes when the depth is increased (Table 2) . Possible variations as a result of the choice of the position of the mouth and of the influence of friction were also in- vestigated but found to be negligible.

The 'leapfrog' method was repeated with

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CANADIAN JOURNAL OF EARTH SCIENCES. VOL. 9, 1972

FIG. 3. Grid chosen for 38-section 'leap-frog' numerical model of Sydney Harbor. The continuous lines represent 5. sections and the broken lines Q sections. A and B are proposed positions of barriers which are investigated.

AX = 0.256 km, i.e. half of the original grid spacing. None of the periods of the first four modes differed by more than 0.1 minute from those of the earlier numerical model (Table 2). The only significant changes in the responses were a 20% increase in the amplitude at the head of the North West Arm and a correspond- ing increase in the discharge. Amplitude and discharge responses of this model are taken as being representative of the harbor (Fig. 4). Amplitudes of the fundamental mode decrease slowly from section 2 as far as Southeast Bar. Mode 2 is essentially the oscillation of the upper arms since there is virtually no response be- tween the nodes at their junction and at the sea. Maximum amplitudes for the third mode

are greater at the head of the North West Arm than at the head of the South Arm due to the closeness of T , to the fundamental periods of

TABLE 2. Periods of the free oscillations of Sydney Harbor

-- -- -. - -- --

Period (minutes)

Model TI Tz T3 T4

3 channel 103.1 52.0 36.6 22.5 38 sections -mean water 107.2 39.2 31.5 23.9 - low water 112.6 41.0 32.9 25.1 - high water 102.6 37.7 30.3 22.9

76 sections - mean water 107.2 39.3 31.4 24.0

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EASTON : SEICHES 861

1 4 8 12 16

Dls<ance from the head of the S o ~ t h Arm (kml

1 1 I

4 8 12 16

O8rtance lram tne head of the SJuth Arm !kml

FIG. 4. Responses of the first four modes-76-sec- tion numerical model of Sydney Harbor. The con- tinuous lines represent the structure in South Arm and the main channel and the broken lines the structure in the North West Arm. The numbers indicate the mode.

(a) Amplitude responses (b) Discharge responses

the main channel and the No th West Arm. The constriction at Southeast Bar is very important for the fourth mode since this is a position of a node of the surface elevation and an antinode of the discharge. It is of no consequence for mode 2 for which there is no discharge through this section.

Discussion The 1970 records from the North Sydney

tide gauge showed that prominent waves with periods between 120 minutes and 130 minutes are always recorded during storms and usually continue for at least 12 hours. In November, they continued for almost three days although large waves with amplitudes of up to 0.6 m extended far only one day. On one occasion,

waves with periods of about 20 minutes and amplitudes of 0.01 m were evident. This must be the fourth free mode which would seem from the geometry of Sydney Harbor to be a pre- ferred mode. Its amplitude is small since the North Sydney gauge is situated very near the computed position of the node at the Southeast Bar. If a recorder were placed at the head of one of the arms, it is probable that these waves would be evident at other times.

Rayleigh (Defant 1961 ) showed that, for a basin communicating with the ocean, the Merian period T should be increased to T ( l + E), where

and is Euler's constant. For Sydney Ha~bor , the ratio b/l is approximately 1/6 so that the fundamental period should be increased by 16% to 124 minutes. Thus, it appears that the recorded waves are in fact the fundamental free oscillations of the harbor.

This seiche may be caused by wind and pressure action within the harbor, e.g. the passage of an atmospheric system at a velocity near the free shallow water wave velocity of 10.3 m/s. However, it is more probable that it is initiated externally. It is now well accepted that moving storms generate edge waves on continental shelves (e .g . Munk et al. 1956; Greenspan 1956). If a storm moves with a velocity U along a shelf with slope s, the fre- quency of the induced edge wave is (Groen and \Groves 1962)

In the presence of rotation, the frequency be- comes

where f is the Coriolis parameter. For the Nova Scotia shelf off Cape Breton, typical values are s = 9 x f = 1 x 10-4/s, so that a storm with velocity U = 15 m/s produces edge waves with a period Tf = 152 minutes. This is the fundamental mode edge wave and the one most likely to be excited. For the shelf off Cape Sable, where s = 1.5 x T f = 97 minutes and the nearest tide gauge at Yarmouth shows that waves with a period of 50 minutes are

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862 CANADIAN JOURNAL OF EARTH SCIENCES. VOL. 9, 1972

emphasized there. We see that edge waves in the adjacent ocean do have periods of the right order of magnitude. Thus, it is theorized that the harbor oscillations along the Nova Scotia coast are initiated by storm-induced edge waves. This is supported by the fact that oscillations begin in all harbors during storms and that the relative amplitudes vary from storm to storm. This premise should be investigated further.

We will now examine proposals with the aim of reducing the effect of the fundamental seiche in Sydney Harbor. A partial barrier at the entrance would not be satisfactory. It would allow long waves to pass and would slightly increase the natural period (for an opening with b = 0.5 km and h = 18 m, the computed T 1 = 112.4 min). If the length of the harbor were decreased, the fundamental period would also be decreased. Calculations based on the Yeap-frog' method for the 76-section model show that T 1 is reduced to 103.2 minutes when a barrier A is placed just upstream from Sydney (Fig. 3 ) . This is a slight improvement. If a barrier B were built in the other arm as shown in Fig. 3, T I would be reduced to 93.3 minutes while if both barriers were built, T 1 would be only 88.2 minutes. These represent decreases in T 1 of 13% and 18%, respectively. Both are adequate since then T1 would be appreciably smaller than the periods present outside the harbor. Obviously, many other aspects need to be considered before it is established whether it is desirable to build a barrier.

Conclusions The prominent wave recorded on the tide

chart is the fundamental seiche of Sydney Har- bor. Its calculated period is about 124 minutes which will vary slightly according to the harbor depth.

The influence of irregular topography is evi- dent in the relatively large amplitudes of the fundamental mode upstream of Southeast Bar. Mode 2 is remarkable because of its lack of response in the main channel; it is essentially

the fundamental free oscillation in the upper arms. The principal feature of mode 3 is the occurrence of larger amplitudes in the North West Arm than the South Arm and of mode 4 is its node at the Southeast Bar.

The seichcs are initiated by weather systems and it is suggested that they are probably the response of the harbor to external edge waves. If this is so, the influence of the fundamental seiche could be reduced by building a barrier in the North West Arm. This would also affect the harmonics.

Acknowledgments I wish to thank Dr. G. Godin of the Division

of Oceanographic Research of the Department of the Environment for suggesting this problem and for his interest through all subsequent stages. The computations were carried out by Mr. M. Yuen. The tide records were supplied by Mr. G. Dohler of the Tides and Water Levels Section.

During this research, the author held a Post- doctorate Fellowship of the National Research Council of Canada.

DEFANT, A. 1961. Physical Oceanography, 2. Per- gamon Press, N.Y.

GREENSBAN, H. P. 1956. The generation of edge waves by moving pressure distributions. J . Fluid Mech., 1, pp. 574-592.

GROEN, P. and GROVES, G. W. 1962. Surges. The Sea, M. N. Hill (Ed.), Intersci. Publ. N.Y., 1, pp. 61 1-646.

HONDA, K. and DAWSON, W. B. 1911. On the secon- dary undulations of the Canadian tides. Sci. Rep., Tohoku Univ., Japan, 1, pp. 61-66.

MUNK, W., SNODGRASS, I?., and CARRIER, G. 1956. Edge waves on the continental shelf. Sci., 123, pp. 127-132.

PLATZMAN, G. W. and RAO, D. B. 1964. The free oscillations of Lake Erie. Studies on Oceanog- raphy (Hidaka Volume). K. Yoshida (Ed.), Univ. Washington Press, pp. 359-3 82.

YUEN, K. B. 1969. A numerical study of large-scale motions in a two-layer rectangular basin. MS Rep. No. 14, Mar. Sci. Br., Dep. Energy, Mines and Resources, Ottawa, Ont.

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