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Ocean Engng, Vol. 24, No. 9, pp. 879-885, 1997 © 1997 Elsevier Science Ltd. All rights reserved

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T E C H N I C A L N O T E

I N T E R - C O M P A R I S O N O F M O D E L - P R E D I C T E D W A V E H E I G H T S W I T H S A T E L L I T E A L T I M E T E R M E A S U R E M E N T S

IN T H E N O R T H I N D I A N O C E A N

Abhijit Sarkar, M. Mohan and Raj Kumar Meteorology and Oceanography Division, Space Applications Centre (ISRO), Ahmedabad 380053, India

(Received 4 May 1996; accepted in final form 4 July 1996)

Abstraet--A numerical model based on a wind-wave energy transport formulation of Toba is developed to generate hindcast wave height data for the equatorial and the north Indian Ocean, which is otherwise a data-sparse region. The intercomparison between model-predicted wave heights for three years (1987-1989) obtained utilising analysed surface wind fields' data, and model grid averaged GEOSAT Altimeter significant wave height data showed moderate match, particularly for Hs greater than 1 m. © 1997 Elsevier Science Ltd.

1. I N T R O D U C T I O N

In many oceanic regions (e.g., in the Indian Oceans) where wave measurements are sparse, numerical wave forecast models can be extremely helpful in bridging the data gaps and creating chronologically sequential wave fields, an important requirement for most offshore industrial and research activities. While a global third-generation ocean-wave-prediction model (WAMDI Group, 1988) is undergoing refinements, the National Meteorological Services of many countries continue to use the second generation wave models (World Meteorological Organisation, 1988), similar to the one used in the present work. An inter- comparison exercise conducted by the Sea Wave Modeling Project (SWAMP Group, 1985) brought out the strengths and weaknesses of the wave models of all types. Eventually, progress in research with different models will lead to a situation where different models will be operated in tandem or the stronger elements of different models will be absorbed in the future-generation wave model which can closely describe the realistic wave fields in space and time.

In the present work, a numerical wave forecast model developed upon the transport equation for average wave energy was used with three years' analysed surface wind field data over the equatorial and the north Indian Ocean region as input. For evaluation of the model-forecast wave heights, satellite-altimeter-measured significant wave height data were averaged over the model grids. The results of intercomparison between model and GEOSAT-bome altimeter wave heights are presented.

2. THE WAVE MODEL

The wave forecast (or hindcast) model in our work is based on the single parameter wind-wave transport equation, originally formulated by Toba (Toba, 1978). The source

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880 A. Sarkar et aL

function used in this formulation parametrically combines the effects of wind forcing, dissipative losses and non-linear wave-wave interactions, and physically represents the fraction of the momentum flux from wind fields to the waves of the total momentum transferred from the winds to the sea surface. Following Toba (1978) and Toba et al. (1985), one can write the transport equation as:

OE~3 ~ - + A ( u . ) E ~ a V g V ( E 2/3) = S (1)

where:

E = spectrally integrated energy of wind waves; A(u.) = a function of u. (friction velocity); g = acceleration due to gravity; Vg = unit vector in the group velocity direction; and S = the source function.

Toba (1978) suggested a stochastic form in terms of an error function for S:

S = P(u.){ 1 - erf[Q(u.)E 1/3] } (2)

where P and Q are functions of friction velocity. The source function S used in this work is the one developed in our earlier work (Mohan

et al., 1994):

S = Ga{ 1 - tanhS[(/3bE~/3) r] } (3)

where:

and

G = 2.4 x 10 - 4uSj3g - 1/3

b = 0.12u.- 4/3g2/3

The quantities or,/3, F and ~ are four adjustable coefficients that depend only on the wind speed. The use of the above source function [Equation (3)] increases the computational efficiency of the model and yields higher accuracy (Mohan et al., 1994). The wave model with this source function was run with analysed surface wind vectors in the area encompassing the Arabian Sea, the Bay of Bengal and a part of the Indian Ocean as shown in Fig. 1. This area ( - 5 ° to 25°N latitude; 45 ° to 100°E longitude) is represented by a rectangular grid of 250 x 250 km spacing with appropriately defined coastal and deep ocean boundaries. The time step size of numerical integration is chosen to be 30 minutes. Analysed surface wind fields at each integration time step were created by linear interp- olation of the 12 hourly analyses generated by the European Centre for Meteorological Weather Forecast (ECMWF). Beyond the southern boundary of the model area (i.e,, south of - 5 ° latitude) wind fields are assumed to be equal to those in the adjacent grids north of them. At the sea-land boundaries, waves were not allowed to grow for seaward compo- nents of winds since there is no fetch in the down-wind direction, while for landward components, no such restriction was applied. Thus a series of conditions on wind direction

Technical Note 881

for wave growth decided by the shape of land-sea boundary contour were introduced for each grid adjacent to land.

The mechanism of swell generation by wind waves and swell propagation are incorpor- ated into the model specified by the following criteria (Joseph et al., 1981).

1. If wind waves move into a region where the wind direction differs by an angle (1 01) less than 30 ° or if the wind has reduced from the earlier value, the wave energy in excess of the "Pierson-Moskowitz fully developed wave energy (EpM)" is transferred into swells (Es = Ew - EpM) and both wind waves and swells are made to propagate in the new wind direction.

2. If 30 ° -< 1 01(60 o, the smaller of the two energies, [Ew cos2(lsol), EpM], is made to propagate as wind waves in the new wind direction while the rest continues to propagate as swells in the old wave direction.

3. If I~01 - 60 °, the whole of Ew is transformed into swells propagating in the old wave direction and new wind waves are generated by the local winds propagating in the wind direction.

3. GEOSAT ALTIMETER

Basically, a radar altimeter works by transmitting a narrow pulse, and getting it back- scattered from the ocean surface. The shape of the transmitted pulse is altered by scattering at the rough ocean surface and provides information on the surface conditions. The leading edge or the rising portion of the return pulse can be processed to determine the significant wave height (SWH) and the skewness of the sea surface height distribution.

In October 1986, through a series of manoeuvres, the orbits of the GEOSAT satellite were altered to produce sea surface traces within a few kilometers of previously realised Seasat satellite tracks. This new orbit has a repeat period of about 17 days, and the mission is referred to as the GEOSAT Exact Repeat Mission (ERM). Altimeter data of GEOSAT ERM for three years, viz., 1987, 1988 and 1989 over the study area (Fig. 1) were used in this work. A satellite altimeter observes averaged values of significant wave heights over footprint circles of about 5 km radius along the satellite pass. The GEOSAT altimeter with its onboard altimeter collected such data during the period 1985-1989 with a repeat cycle of 17.05 days and provided data at a resolution of approximately 150 kms at the equator (Cheney et al., 1987).

4. ANALYSIS

For the convenience of numerical computation, Equation (1) is recast in terms of M, a quantity proportional to momentum into:

~M ~M 8~ + Vg -~s = S (4)

where

M = E 2/3

Vg = 1.3g2/3u. - 1/3M1/2

s = space coordinate in the direction of group velocity.

The numerical integration of Equation (4) follows a grid interpolation scheme which is

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a modified version of the one used by Joseph et al. (1981). In the original scheme, the computation of wave growth at a grid point involves a spatial interpolation of the wave growths in the neighbouring grid points which very often leads to wave-growth errors near land-sea boundaries, particularly under inhomogeneous and time-varying wind con- ditions. So, this scheme was modified to take care of all types of wind field situations. In our version, first, the wave growth at a grid point, say P, during the nth time step is computed using a fourth order Runge-Kutta scheme by ignoring the spatial derivative term in the left hand side of Equation (4) as:

M . + l = M . + ~t.S (5)

Then a point P' is identified near P in the upwind direction at a distance of

d = [Vg(M.) + Vg(M. + ,)]6t/2 (6)

The M. values at the four comers of the box around P' are interpolated to P' to obtain M'., and then substituted back into Equations (5) and (6) to recompute M. + ~ and d. This procedure is repeated until M. + 1 converges to a stable value. It should also be mentioned that after trials with second and third orders, the fourth order Runge-Kutta scheme was found adequate for the numerical from the point of view of solution accuracy.

At every time step, the model checks the conditions stated for the creation of swells at each grid point. The swell propagation is realised through a wave packet following scheme (Joseph et al., 1981). They have been treated as free waves traveling with the same group velocities they had at the time of generation, undergoing a small amount of damping (Kawai et al., (1979); Joseph et al., (1981). Both wind waves and swells propagating in the landward direction are absorbed at the land-sea boundaries without any reflection. The

Technical Note 883

total wave height at any desired time step is computed by shifting the swell packets to the nearest grid points and by combining swell energy with wind-wave energy.

The model was run, as stated earlier, in hindcast mode with the ECMWF-analysed 12 hourly ocean surface winds over the study area (Fig. 1) for every month of the period 1987-1989. The model-predicted wave heights were then correlated with the coincident GEOSAT altimeter SWH measurements. For this, a box of 2.5 ° x 2.5 ° size was defined around each grid point and the mean of all the altimeter measurements within the box was taken as the wave height at that grid point. The time of pass of the sub-satellite point through a box is approximated to the nearest time step of the model run. The correlations between the model predictions and altimeter measurements for the three years 1987, 1988 and 1989, as well as for the entire three years' period 1987-1989, were conducted. Corre- lations for all the wave heights and for the subset consisting of those wave heights which are greater than 1 m were computed separately and the results discussed below.

5. RESULTS AND DISCUSSIONS

The scatter plot of model-predicted wave heights with those measured by GEOSAT altimeter is shown in Fig. 2. Year-wise regression equations for the best-fit lines and the correlation coefficients for the model-forecast and GEOSAT-measured SWH are provided in Table 1.

As can be seen in the table, the correlation coefficients for the entire data set (i.e., for the 1987-1989 period) are 0.58 and 0.68 for all SWH and for SWH > 1 m, respectively. It is also seen that the scatter plots are broader at the base of the diagrams (Fig. 2) where the wave heights are small. One possible reason for the low correlations for the low wave heights is that the low sea state conditions are not detected significantly above the measure- ment inaccuracy of the altimeter which, for GEOSAT was about 0.5 m. This same effect can be seen in the general increase of the correlation coefficients when only wave heights greater than 1 m are considered for comparison. Further, the ECMWF surface wind speeds are not exactly at the sea surface but at 1000 mb, which may also contribute to a general degradation of the correlations.

However, considering the fact, that the short term fluctuations in the wind fields also contribute to the altimeter-measured wave heights but are not reflected in the input-ana- lysed wind data (viz., ECMWF objectively analysed wind fields), the correlation coefficient of around 0.7 for waves greater than 1 m, can be considered to be fairly good. The negative biases in Fig. 2 show that the model computed Hs are found to be slightly underestimating with respect to those measured by the GEOSAT altimeter for all the three years and care should be taken when the satellite data are considered for assimilation in the wave models.

Table 1. Regression equations for the best-fit lines and correlation coefficents for the model-forecast vs GEOS- TAT-measured SWH

Correlation coefficient (number of data)

Year Best-fit line For all SWH For SWH > 1 m

1987 y = 0.559 × - 0.198 0.58 (6854) 0.68 (1508) 1988 y = 0.524× - 0.148 0.56 (6054) 0.69 (1359) 1989 y = 0.604X - 0.191 0.61 (2879) 0.68 (955)

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Acknowledgements The authors are grateful to the Department of Ocean Development, Government of India, who provided the financial support for this work. The authors thank Dr Pranav Desai for suggesting several improvements in the paper. The authors are also grateful to Dr P. C. Pandey, Dr M. M. Ali, Ms Rashmi Sharma, Mr H. I. Andharia and Ms S. B. Karthikeyan for scientific discussions and help in data handling.

R E F E R E N C E S

Cheney, R. E., Douglas, B. C., Agreen, R. W., Miller, L., Porter, D. L. and Doyle, N. (1987) GEOSATAltimeter Geophysical Data Record: User Handbook. NOAA, Rockville.

Joseph, P. S., Kawai, S. and Toba, Y. (1981) Prediction of ocean waves based on the single parameter growth equation of wind waves. (II) Introduction of grid method. Journal of the Oceanographic Society of Japan 37, 9-20.

Kawai, S., Joseph, P. S. and Toba, Y. (1979) Prediction of ocean waves based on the single parameter growth equation of wind waves. Journal of the Oceanographic Society of Japan 35, 151-167.

Mohan, M., Sarkar, A. and Kumar, R. (1994) A source function for wave forecast model: Duration limited case. Indian Journal of Marine Science 2,3, 105-107.

SWAMP Group (1985) Ocean Wave Modeling. Plenum Press, New York. Toba, Y. (1978) Stochastic form of the growth of wind waves in a single parameter representation with physical

implications. Journal of Physical Oceanography 8, 469-507. Toba, Y., Kawai, S. and Joseph, P. S. (1985) The TOHOKU Wave Model in Ocean Wave Modelling. Plenum

Press, New York, pp. 201-210. WAMDI Group (1988) The WAM Model--A third generation ocean wave prediction model, Journal of Physical

Oceanography 18, 1775-1810. World Meteorological Organisation (1988) Guide to Wave Analysis and Forecasting, WMO no. 702. WMO,

Switzerland.