Application of a hindcast model to waves in Bass Strait, Australia

Application of a hindcast model to waves in Bass Strait, Australia D. R. Blackman and A. D. McGowan Department of Mechanical Engineering, Monash University, Australia (Received June 1985; revised May 1986)

This paper describes the application of a spectral hindcast model in computing the wave climate in Bass Strait, Australia, from historical wind fields. The model is based on the numerical solution of the spectral energy balance equation. The conclusion reached is that the physical data available are inadequate to calibrate the model, and that less elaborate models may suffice. Keywords:

hindcast

The results presented here are part of a project which involves collecting wave data, particularly swell, and numerically modelling the generation, propagation and decay of waves in Bass Strait, Australia. Bass Strait washes the most densely populated part of the continent and is therefore of economic significance, but little systematic work has been carried out until recently and wave data are sparse and largely subjective. The Southern Ocean is a source of strong swell to which the whole southern half of the continent, and Bass Strait in particular, is exposed. The depth of the Strait means that waves with periods of 10s or more will be affected by interaction with the bottom. This limits the development of local wind-sea and causes it to steepen, a quite notorious feature of the area. The impinging swell feels this effect strongly and suffers, in consequence, rather severe dissipation and refraction. When this project was being planned, the treatment of this swell was thought to be of particular importance. Indeed, the Norswam model’ was set up and applied to Bass Strait and adjacent parts of the Southern Ocean, but the discrete part of the model appeared too coarse to adequately describe swell in the Strait, and the version available could not be easily modified to include shallow water effects such as shoaling, bed friction and refraction. It was, therefore, decided to develop a discrete spectral model for application to Bass Strait, using existing empirical growth terms. In this way, swell could be more accurately resolved, shallow water effects could be easily included, and the source terms could be later modified to take advantage of more accurate approximations to the wave-wave interaction terms, e.g. Resio,2 as they were developed.

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model,

Bass Strait, spectral

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Since calibration data were almost non-existent, work also commenced on a programme of field trials to measure incident swell. Data from the three successful deployments are still being evaluated, but the portion relevant to the wave modelling is presented in this paper.

Model development Early numerical wave hindcast modelss6 were generally of the discrete spectral type. These models are based on the numerical solution of the spectral energy balance equation, carried out over a range of discrete frequency and direction bands, and include empirically-derived source terms for describing the physical mechanisms associated with wave growth and decay. Hasselmann et al.’ concluded from the results of the JONSWAP study that nonlinear wave-wave interactions were the dominant processes in determining the non-equilibrium wind-sea spectrum. Early discrete spectral models4,5 attempted approximations to these processes but their approximations could lead to significant errors for some spectral shapes.2.8 Problems formulating realistic approximations to the wave-wave interaction terms have, until recently, remained the main limitations to further development of this type of model. As the inclusion of rigorous wave-wave interaction terms is very expensive with current computer speeds a simplified parametric type of model was proposed.9,‘0 With this model, the spectral energy balance equation is projected onto a set of prognostic equations in the main parameters of the JONSWAP spectrum; in this way the effects of the nonlinear wave-wave interaction terms are included implicitly. Owing to the way in which they are formulated, however, parametric models do 0307-904X/87/01002-09/$03.00 0 1987 Butterworth & Co. (Publishers) Ltd

Application

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to waves:

D. R. Blackman

and A. D. McGowan 7

1

Exact solution 3rd order Hermitian

0.9

3rd order upstream 2nd order centred

0.8

1 st order wstream

0.6

-0.3

t -0.4

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Distance Ax

Figure 7 Comparison

of performance

of different

transport

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not model situations with rapid variations in wind speed or direction particularly well, and do not recognize swell.

Spectral energy balance equation The shallow water form of the spectral equation can be given as:

energy

balance

$(ECCJ +c, cos 0: (ECC,)

these three separate aspects of equation (1) is dealt with in more detail below. The formulation of equation (1) in terms of wave action density, instead of the more normal wave energy density, E, has the advantage that shoaling effects are included implicitly and do not need to be included separately. For modelling Bass Strait itself, shallow water effects such as shoaling, refraction and bed friction need to be included. For the approaches to Bass Strait, only the deep water version of the spectral energy balance equation need be considered. Transport

+ C, sin 0;

sin i32 i

(ECC,)

+ 2

- cos 05

i(ECC,)

= S

1

where E is the spectral wave energy density for frequencyf, direction 0, position X, y and time t; C denotes the phase celerity for frequencyf; C, is the group celerity for frequency f; and S represents source / sink terms. The first line of the left-hand side of equation (1) is simply a two-dimensional transport equation describing the transport of wave action density, ECC,, from one point to another. The remainder of the left-hand side of equation (1) describes the effects of refraction and is not required in deep water. The right-hand side comprises various source terms used to describe wave generation and decay mechanisms. The treatment of

model&

In the approaches to the Bass Strait region, swell propagating from the Southern Ocean makes a significant contribution to the local wave action. It is therefore essential that the left-hand side of equation (l), which is responsible for describing the propagation of the swell, be modelled accurately to ensure that swell predictions will not be compromised by numerical errors. High accuracy transport schemes are generally more complex and require more computer time than simpler less accurate methods. Several possible finite difference methods were evaluated, and it was concluded that the third order upstream method of Leonard” provided a satisfactory compromise between the conflicting requirements of accuracy and computational effort (see Figure 0 In one-dimensional form, the third order upstream difference scheme describing the transport of one component E(f,e) of the spectrum can be given as:

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0

6’ 110-E

100”E Figure 2

of a hindcasting

li

I

120”E

Area map and computational

D. R. Blackman

I

130”E

(2)

a = l/3 - O/2 + Cr?/6 b = l/2 + Cr - Cr*/2 c = -1 - Cr/2 + Cr2/2 d=1/6-Cr*/6 At upwind boundaries, the sea state is assumed to be duration-limited at ocean boundaries, i.e. no net transfer of wave energy density across the boundary, and fetch-limited at land boundaries, i.e. no transfer of wave energy density across the boundary. For both land and ocean downstream boundaries, a radiation condition is applied using simple first order differences. Source terms Following Hasselman et al.’ the source term in equation (1) can be expressed as a sum of terms, each of which represents a class of physical processes. This is given schematically as:

s,,

(3)

where Si, is the energy input from the atmosphere, S,, the nonlinear transfer of energy into this frequency by conservative wave-wave interactions, S,, denotes dissipative processes, and S,, represents additional shallow water effects. It is now generally accepted that S,, may

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140°E

where j denotes grid point jh; y1denotes time step nAt; and Cr = C,At/A.x is the Courant number. Also:

S”, + S& +

18

150”E

160”E

170”E

grid for Southern Ocean

E,,” = E;’ - C~(UE,+, + bE, + cE,_, + dEj_,)”

s = Si” +

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1987, Vol. 11, February

be of the same order as Si”. Some of the main phenomena attributed to S,, include the sharply-peaked spectra of developing seas, the rapid growth rates on the steep forward face of the spectrum, and the tendency of growing spectral components to overshoot their equilibrium values. In the present form of the model, relatively simple empirical growth terms are used in which the effects of S,, are included implicitly. Wave growth The growth term takes a form similar to that of Pierson et a1.3: Si, = (A + BE) x

(4)

where A is a linear growth term representing the turbulent resonance mechanism of Phillips’*; B is an exponential growth term representing the shear flow mechanism of Miles13; E, is an upper limit to wave growth. The actual A, B and E, values used in the determination of Si” follow from those used by Karlsson,6 and as developed by Sand et al.‘“. The linear growth term A which is responsible for initiating wave growth was set by Barnett4 proportional to the sixth power of the wind speed, U6. Dexter’j modified Barnett’s original term to become proportional to the fourth power of the wind speed, U4. Following Sand

Application

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et al. ,I4 Barnett’s original formulation, as shown in equation (5), is used unchanged for wind speeds less than 15ms-I. For higher wind speeds, CT6 is replaced by 225 U4. Barnett’s4 original formulation is expressed as:

(1.357 x lO-%J~(~/ [0.332(w/

U)2.231/6)

U)?.j6 + (kX - w/ U)2][0.522(w/

angular frequency klcos0l kIsin81 wave number angle relative to wind direction

The exponential growth term, BE, is included to represent the rapid growth rates observed in nature. In the present model the exponential BE term of Inoue16 is used for this purpose: (6) where: by u* = k,U/g

The Pierson-Moskowitz spectral value E, for a fully developed sea is imposed as the upper limit to wave growth. This takes into account the fact that wave breaking ultimately limits any further growth of waves. Both the exponential growth term, BE and the limiting spectral energy density, E,, are assumed to have a cosinesquared distribution for directions within 90” of the wind direction. E, is given by:

E,(w,U)

=

8.1 x lo-3g2 . e-o.74(g/ E w’

uo$

(7)

The validity of the chosen growth terms was checked by comparing numerically determined growth rates against those obtained from the SMB nomograms in reference 17, and the JONSWAP results.18 Overall, it was found that the numerical results compared quite favourably.‘9

Wave dissipation The use of the limiting spectrum in equation (4) empirically includes dissipation mechanisms, such as white capping, for growing and fully-developed seas. It does not, however, consider dissipation effects once the energy source is removed. To take this into account, Karlsson,6 following the work of Gelci and Devillaz,*” proposed a turbulent dissipation term of the form: s,, = &H?(W)4

D. R. Blackman

and A. D. McCowan

low frequency wave energy remains relatively unaffected. This is, qualitatively at least, what is observed in nature. For the present application it was found that best results were obtained with a damping coefficient, CY= 10 x 10-6.

Implementation

where:

shear velocity, approximated E,* von Karman’s constant, 0.4 C wave phase speed

to waves:

U)‘.y + k_;] (5)

k”, k, k 0

model

(8)

where CYis a dissipation coefficient and H, is the significant wave height. This term is highly frequency-dependent, and leads to rapid dissipation of high frequency wave energy while

Spectral models are computationally expensive. In the current environment, one time step of the deep water model on a grid of 400 points using 12 directions and 13 frequency bins (Figure 2) takes about 60s of cpu time and 9min elapsed time on a Perkin Elmer 3220. The results described later therefore represent a very extensive season of processing.

Wind fields The weather in the region is characterized by prevailing westerly winds, modulated by depressions recurring with a period of four or five days. These depressions operating in the Southern Ocean give rise to a considerable swell of quite long period (often in excess of 20 s). Unlike the European situation, where there is a high density of land observing stations, permanent weather ships and many casual ships reporting at sea, there is a great paucity of observational data for the region. The permanent meteorological stations on the Australian mainland lie wholly on the northern boundary of the Southern Ocean model area and are frequently very widely spaced. Shipping is sparse and predominantly coastal; almost no vessels now ply the open spaces of the Southern Ocean. Conventional reporting is, therefore, largely absent. The standard weather satellites do not provide quantitative data on wind fields, less still of sea state. One must resort, therefore, to very indirect estimates of the wind field. The traditional method of estimating sea level wind speed and direction is to compute the geostrophic wind from isobaric charts and apply some factor to correct for height and stability. Even when data are available, the method is very tedious and requires considerable skill if it is to give reliable estimates.?’ There are no stability data available in the present case, which limits the accuracy of this method at the outset. The geostrophic wind derived from the pressure field has been corrected to sea level by applying a factor of 0.7 to the magnitude and 15” to the direction. For the Southern Ocean the 1OOOmbar winds taken from the analysis model run by the Australian Bureau of Meteorology’” have been adopted. This model has, as input, both quantitative measurements and qualitative satellite data which are linked using a variational form of the equations of motion for the atmosphere. The wind field from this model was made available in machine readable form at 12 h intervals on a 254 km grid. This is rather coarse for the present purpose, not only in space but also in time; it does not give wind at the surface, and the data from which this can be recovered cannot be obtained from the wind fields. Nevertheless, the wind fields have the virtue of selfconsistency and comparisons with observations suggest that the errors may be no worse than those from individual weather stations. These last errors are, in fact, considerable. The wind speeds of the regional analysis wind

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model are corrected by a factor of 0.8 to take boundary layer effects into account.

Field measurements

of swell

Three periods of field measurements have been completed, the results from two of which are used here. The instrument used was a bottom-resident pressuresensing recorder. The output from the pressure element is almost linear with pressure. The recorder is a microcomputer-controlled digital cassette drive capable of conditional sampling of the input. The intelligent part of the instrument was turned on at an interval of 4 h. At each awakening, the instrument first determines a ‘sea-state’ parameter by noting the maximum and minimum values of pressure encountered during a 10min sensing. At the end of the sensing period, the difference between these extrema is compared with a firmware value selected with the kind of records sought and the site of the instrument in mind. If the difference between extrema does not exceed this threshold, then the instrument logs the water depth, perceived sea state, water temperature and battery voltage. If it decides there is sufficient sea running it logs conditions as just described and then enters an extended recording mode. In the present experiments, this involved samples at 4s intervals, for a period of 30min in the first trial, and 40 in later trials. The 4s sampling period gives a Nyquist period of 8s. In the depth of water (55 m), the effect of waves of this period at the bottom is l/300 of that at the surface, so that aliasing is not a serious problem. This cut-off period also means that the instrument is insensitive to local wind-sea. The sea-state parameter is convenient to measure with the limited resources of the microprocessor. It is, however, statistically difficult, since measured extrema are notoriously hard to interpret. It is therefore not so convenient for subsequent comparison with model prediction, since some assumptions about the probability distribution of the wave heights are needed to arrive at the root mean square (rms) from this parameter. The sensing period corresponds only to about 17 swell periods. The authors have confirmed peaks approximating to a Rayleigh distribution, so the observed sea-state might be expected, on average, to correspond to about 6u, where (T is the rms value of surface elevation. The extended records, based on a much greater number of samples, provide a far better estimate of u and confirm the statistics (see Figure 3). For convenience, Figures 3,5 and 6 are plotted in terms of the sea-state parameter. The traditional significant wave height, 4a, is 2/3 of this value. The station for the measurements was located at the edge of the continental shelf and with a good aspect to the Southern Ocean, 2.5 km northwest of the Black Pyramid Island (144” 20’E, 40” 28’S). The sea bed was not as smooth as desired, being a low rocky ridge rising 15 m above the general surroundings to a depth of 55 m below MWL. In the first trial, the instrument ran for 40 days (November/ December 1981) giving 232 logged records, and, in this time, six events reached the threshold value and resulted in wave records. In the second trial (July 1982) the threshold was set lower and the battery was exhausted after 31 wave records had been taken in 21 days. In a third trial (March 1983) an array of three linked recorders operated for 13 days.

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July 1982 Figure 3 Measured and predicted sea state parameters at Black Pyramid, 25 July 1982. Spectra derived from 40 min records: (---I computed; (---_) measured; (0) from spectra

60 7-

t

Frequency (Hz) Figure 4 sured

Spectra on 25 July 1982: (---I computed;

(-1

mea-

Model results The sensitivity of the model to input parameters is discussed in McGowan and Blackman’” and performance is satisfactory. For comparisons of model results against the field measurements from Black Pyramid, the deep water model applied to the Southern Ocean, as shown in Figure 2, was used. The model used a 127 km square grid, half that of the regional analysis wind model, and a time step of 1 h. The direction-frequency spectrum was divided into 16 equally spaced direction bands and 12 frequency bands covering a range of wave periods from 4-26 s, with a uniform spacing of 2 s. Wind speeds for each time step were obtained by linear interpolation between the 12-hourly frames provided. Although results from the two field measurement periods at Black Pyramid showed that, for much of the time, the sea-state parameter (the difference between the extrema detected in a 5min period) was generally in the range of 0.5-1.5 m with values of 2 m rarely being exceeded, much higher sea-states (up to 4.1 m) had been

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November/December 1981 figure 5

Sea state parameters

at Black Pyramid for period 5 November-15

December 1981:

(e) measured; (-1

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July 1982 Figure 6 Sea state parameters at Black Pyramid for period 5-25 July 1982. Predictions at end of period differ slightly from Figure 3 because slightly different model parameters were used: (@) measured; (---) computed

recorded from 0600-1800 hours on 25 July 1982. Examination of pressure charts showed that during the period 20-26 July 1982, a series of quite strong depressions, as low as 956 mbar, crossed the Southern Ocean at latitudes of 5&60” south. These depressions gave rise to persistent strong westerly winds of 20ms-’ (and in places, 25 ms-I or more) blowing over the central and southern parts of the model, but for much of the period 20-26 July 1982 winds over Bass Strait were less than 12 ms-I. Thus, the high waves measured were due predominantly to swell arriving from the Southern Ocean. The model was run for the five-day period up to 26 July and Figure 3 compares measured and computed sea-states. Measured and computed surface elevation spectra are compared in Figure 4. With high energy densities at low frequencies, TP - 18 s, these show that the wave energy was indeed predominantly swell. Thus, the model has performed well for this event. The encouraging agreement shown by this result is not, however, sustained in a more extended comparison. Figure 5 shows the comparison of observations with the model predictions for the whole period of the first field trial, and Figure 6 that for the second. This agreement is poor.

The cause of these poor results is inadequate or inaccurate wind fields. The sensitivity of wave models to their input wind field is notorious. Although the wind fields used are internally consistent, the physical data needed to keep the values in touch with reality over the open ocean are lacking. This can be tested for a period in 1978 when wind data in the Southern Ocean were secured by the Seasat satellite. To make this comparison, the very large number of Seasat observations has been reduced by averaging in the following way. The model grid was subdivided at intervals of 0.1 mesh lengths and a square drawn around each grid point. Any Seasat observation falling in that square will contribute to an hourly average wind value at the grid point. A comparison of 1114 such averages over three months is made in Figure 7 from which it may be seen that, in spite of the averaging, the agreement with the value interpolated from the windfields supplied is unconvincing. This comparison does not really test one feature of the wind field of importance for the wave model, which is the way the wind vectors are organized; the comparison deals only with wind magnitude, and the result of

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rn

r

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Wind speed from wind model (m s-’

Figure 7

figure

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Comparison

Predicted

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contours

Seasat estimates

of windspeed

and model windfields

of swell in Bass Strait. Incident field is westerly

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) during

swell of height 3 m

1978

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and A. D. McGowan

a direct path to this point is achieved only for swell originating from quite selective directions. When diffraction and bottom dissipation are added to this it becomes clearer why westerly swell is of little significance in the eastern part of the Strait.

Conclusions N

1 Bass Strait

The disagreeable conclusion to be made from the foregoing is that a method as complicated and expensive as the spectral model is not required to predict the wave field in Bass Strait, at least for weather from the south and west. Within the confines of the Strait, the wind-sea relationship can probably be treated adequately by traditional empirical, and certainly by parametric, methods. The swell component, although important on the western boundary, appears to lose some significance owing to refraction and bottom dissipdtion. This is fortunate since the physical data do not permit predictions of adequate accuracy to be made of the incident swell. This is not, as we have tried to show, because the model is defective but because the wind fields available for the open ocean do not agree well enough with the physical reality. The inability to bring waves from the open ocean to the boundaries of the Strait remains a serious problem since infrequently occurring weather situations, such as arose in the last week of December 1984, when strong easterly winds blew for several days, can lead to heavy easterly swells with economic consequences.

Acknowledgements Fjgure 9 Ray paths for 18 s swell to Barwon heads. Diffraction effects ignored. Paths formed by tracing backwards from target point; figures on rays give swell direction of ray at target point

this less stringent test is not good. It seems unlikely then that wind directions in the model wind fields are any more reliable. But, even if they were, the unreliability of the speed means that the wave model will not generally predict wave conditions well.

Waves in Bass Strait There are no long runs of wave observations in Bass Strait. Data are now being collected on an oil rig in the far east of the Strait and at Bernie in Tasmania, a point very well sheltered from westerly swell. It became clear, however, with the accumulation of subjective observations, that westerly swell is rapidly dissipated. Application of the full equations (1) adds substantially (50%) to the computational effort, and when finally implemented the model results confirm these qualitative observations. Contours of equal wave height for a westerly swell propagating into the Strait are shown in Figure 8. 24 Some insight into the reasons for this somewhat unintuitive result may be obtained by examining the effect of the bottom topography independently of the other mechanisms included in Figure 8. A ray-tracing technique based on that of Abernethy and Gilbert25 was used to determine the ray windows open to a point near Barwon Heads. In the present application, the rays are tracked backwards from the target point to the open ocean. A typical result for a wave period of 18s, shown in Figure 9, indicates that

The numerical model was developed by A. D. McGowan whose research fellowship has been supported by AMSTAC Grant 80/0040. Mr McGowan wishes to acknowledge the assistance of Dr 0. BrinkKjaer of the Danish Hydraulic Institute in the selection of the source terms used. The cooperation of the Australian Bureau of Meteorology in supplying analysis model wind fields and Seasat data is gratefully acknowledged.

References Weare, J. and Worthington, B. A. ‘A numerical model hindcast of severe wave conditions for the North Sea’, Proc. NATO Symp. Turbulent Fluxes Through Sea Surface-Wave Dyn. Prediction (eds A. Favre and K. Hasselmann), Plenum, NY, 1978.617-628 Resio, D. T. ‘The estimation of wind-wave generation in a discrete spectral model’.l. Phys. Oceanogr., 1981,11(J), 51C-25 Pierson, W. J., Tick, L. J. and Baer, L. ‘Computer based procedures for preparing global wave forecasts and wind field analyses capable of using wave data obtained by a spacecraft’, 6th Nav. Hydrodyn. Symp., Ojjice Nav. Res., Washington, DC, 1966, 499-532 Barnett, T. P. ‘On the generation, dissipation and prediction of ocean wind waves’, J. Geophys. Res., 1968,73(2), 513-29 Ewing, J. A. ‘A numerical wave-prediction method for the North Atlantic Ocean’, Dtsch. Hydrogr. Z., 1971,24(6). 241-61 Karlsson, T. ‘An ocean wave forecasting scheme for Iceland’, Rep. Science Inst., Div. Earth Sci., Univ. Iceland, 1972 Hasselmann. K. ef al. ‘Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP)‘, Dtsch. Hydrogr. Z., 1973, Supplement AS, 12 Young, I.,R. and Sobey, R. J. ‘The numerical prediction of tropical cyclone wind-waves’. Res. Bull. No. CS20. 1981, DUD. Civ. and Syst. Eng.. James Cook Univ., h’orlh Queenslund, Australia Cardone, V. J. and Ross, D. B. ‘State-of-the-art wave prediction methods and data requirements’. in ‘Ocean wave climate’ (eds M. D. Earle and A. Malahoff). Plenum, NY. 1978. pp. 61-92

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Hasselmann, K., Ross, D. B., Miiller, P. and Sell, W. ‘A parametric wave prediction model’, J. Phys. Oceanogr., 1976, 6(2), 200-28 Leonard, B. P. ‘A stable and accurate convective modelling procedure based on quadratic upstream interpolation’, Comput. Methods Appl. Mech. and Eng., 1979,19,59-98 Phillips, 0. M. ‘On the generation of waves by turbulent wind’, .I. Fluid Mech., 1957,2(5), 417-45 Miles, J. W. ‘On the generation of surface waves by shear flows’, J. Fluid Mech., 1957,3(2). 185-204 Sand, S. E., Brink-Kjaer, 0. and Nielsen, .I. B. ‘Directional numerical models as a design basis’, Proc. Conf Wind Wave Directionality, Paris, 1981 Dexter, P. E. ‘Tests on some programmed numerical wave forecast models’, _I. Phys. Oceanogr., 1974,4(4), 635-44 Inoue, T. ‘On the growth of the spectrum of a wind-generated sea according to a modified Miles-Phillips mechanism and its Ph.D. thesis, Sch. Arts Sci., application to wave forecasting’, NY Univ., 1967 ‘Shore protection manual’, CERC (Costa1 Eng. Res. Cent.), 1917 Carter, D. S. T. ‘Prediction of wave height and period for a

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constant wind velocity using the JONSWAP results’, Ocean Eng., 1982,9(l), 17-33 McGowan. A. D. and Blackman, D. R. ‘A wave hindcast model for the Southern Ocean’, Proc. 6th Aust. Conf. Coastal and Ocean Eng., 1983,145-150 Geici, R. and Devillaz, E. ‘Le calcul numtrique de l’etat de la mer’, Houille Blanche, 1970,25,117-136 Harding, J. and Binding, A. A. ‘The specification of wind and pressure fields in the North Sea and some areas of the North Atlantic during 42 gales from the period 1966 to 1976’, Inst. Oceanogr. Sci. Rep. 55, 1978 Seaman, R. S., Falconer, R. L. and Brown, J. ‘Application of a variational blending technique to numerical analysis.in the Australian region’. Aust. Meteorol. Map.. 1977.25(l). 3-23 Blackman, D. R. and Evans, R. J.“An intelligent tide and wave recorder’, Proc. 6th Aust. Conf. Coastal and Ocean Eng., 1983, 306307 Blackman, D. R. and McGowan, A. D. ‘Wave hindcast modelling for Bass Strait’, in ‘Computational techniques and applications: CTAC-83, North Holland, 1984, pp. 438-448 Abernethy, C. L. and Gilbert, C. ‘Refraction of wave spectra’, Hydraul. Res. Station Rep. INTI17, 1975