# Introduction to Wind Wave Model Formulation Igor V. Lavrenov Prof. Dr., Head of Oceanography Department State Research Center of the Russian Federation.

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Introduction to Wind Wave Model Formulation Igor V. Lavrenov Prof. Dr., Head of Oceanography Department State Research Center of the Russian Federation Arctic and Antarctic Research Institute, Bering 38, 199397, St.Petersburg, Russia, e-mail: lavren@aari.nw.ru.lavren@aari.nw.ru (AARI)

Part 1. Hydro dynamical Problem Formulation of Wind Wave Modelling

1.5 Difficulties of obtaining deterministic problem solution for wind wave modeling A complete system of the equations (1.3), (1.9)–(1.13) for determining the surface evolution at the initial conditions (1.8) presents considerable difficulty for the analysis. Unlike to the usual classical theory of potential waves with the given pressure distribution Pа at the surface, either the surface itself or the pressure are not determined in the wind wave theory. These two unknown functions are not independent, that is why a co-ordinated solution of both equations (1.9)–(1.12) for wave disturbances at z is required for surface determining problem.

Part 2. Wind Wave Energy Balance Equation (Spectral Approach) Wind wave time series

Rayleigh function of wave height distribution

JONSWAP Spectrum approximation

JONSWAP Spectrum Approximation algorithmic scale

JONSWAP Spectrum approximation with cos angular distribution with n=12

JONSWAP Spectrum approximation with cos angular distribution with n=2

JONSWAP Spectrum approximation with JONSWAP angular distribution

Wind wave energy input

Miles model of wind wave energy input. The component Gin of the wind wave energy input is usually determined with a help of the relation based on the model of averaged air flow interaction with waves proposed by J.Miles (1960). In spite of the fact that this model is proposed in 1957, it describes accurately enough the mechanism of wind wave energy input. It is used even nowadays. This mechanism specified by using full-scale observation data (Snyder et al., 1981) can be described as follows: where U10 is the wind speed at a 10 m level; is the angle between the wind speed and the direction of wave spectral component propagation; a1 and a2 are the parameters assumed to be about 1. Nowadays more sofiticated approximations are developed (Chalikov&Belevich, 1995; Makin&Kudriavtcev, 2003)

Non-Linear Energy Transfer in Wind Wave Spectrum The problem of non-linear energy transfer in wind wave spectrum was formulated by K.Hasselmann (1960,1962,1963,1965,1966) and V.Zakharov (1968) in 1960s. As a result of non-linear interaction, the wave spectrum evolution equation can be presented as follows :

Four-wave interaction diagram (Hasselmann, 1963).

Non-linear transfer function for the JONSWAP spectrum with =7: 1 - according to results (Hasselmann S. and Hasselmann K.,1981); 2 – by the Lavrenovs

JONSWAP Spectrum approximation with non-linear energy transfer

Wave Energy Dissipation K.Hasselmann (1974) suggested the wave energy dissipation parameterisation, connected with wave breaking. In his opinion it be considered as random distribution of perturbing forces, making up pressure pulsations with small scales in space and time in comparison with the proper wave length and period.. The wave dissipation used in the WAM model (The WAM model, 1988; Komen et al., 1994) connected with wave breaking is accepted in the form of quasi-linear approximation, as suggested by G.Komen (1984) on the basis of the Hasselmann model: where c, n and m are the model parameters; is the mean frequency of the wave spectrum; PM is the constant of the Pierson-Moskovits spectrum.

Wind wave interaction with non-uniform current and bottom

Freak wave generation in non-uniform current Freak wave collision with the Taganrorsky Zaliv. The Taganrogsky zaliv is a ship of the unrestricted sailing radius. The vessels length is 164.5 m, the largest width is 22 m, the displacement during accident is 15 000 tons, the board height above water is 7 m.

Map of the Southeast Coast (South Africa). 1 – location of abnormal wave accidence (Mallory, 1974); 2 – Location of the "Taganrogsky Zaliv tanker-refrigerator.

Synoptic charts of the southern Indian Ocean on April 27, 1985.

Wave rays arriving to given point in Agulhas current with frequency : 1– 0.20 rad s 1 ; 2-0.38 rad s -1 (at angle 0 = -30°); 3 – 0.76 rad s -1 ; 4 – 0.93 rad s - 1 (at angle 0 =30°)..

Wave breaking in shallow water

Wave Transformation in Water covered with Ice Cover

Ice field distribution in South ocean around Antarctic (February, 1985)

Ice field distribution in South ocean around Antarctic (August, 1973)

3. Wind Wave Model Input and output parameters Input parameters: Wind speed and its direction in every grid point and in 3-6 hours time step; Current speed and its directions in every grid point; Water depth (in shallow water) in every grid point); Ice cover (mainly as a movable boundary)

Output parameters: Wind wave spectrum (two dimensional: frequency –angular) Wave height for swell and wind sea (significant or mean value); Wave period (mean, spectrum maximum); General direction of wave propagation;

Russian Global Wind Wave Model example of global forecast for 09.06.03 www.hydromet.ru

New Book about Wind Wave Modeling by Igor V. Lavrenov, Springer, 2003, 386p.

http//www.aari.nw.ru

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