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\begin{document}

\title{2D parametric model for surface wave development in wind field varying
in space and time}



\author[1]{Vladimir N. Kudryavtsev}%
\author[2]{Maria V. Yurovskaya}%
\author[3]{Bertrand Chapron}%
\affil[1]{Russian State Hydrometeorological University / Marine Hydrophysical Institute}%
\affil[2]{Marine Hydrophysical Institute}%
\affil[3]{IFREMER}%


\vspace{-1em}



  
  \date{May 07, 2024}


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\begin{abstract}
A fully consistent 2D parametric model of waves development under
spatially and temporally varying winds is suggested. The 2D model is
based on first-principle conservation equations, consistently
constrained by self-similar fetch-laws. Derived coupled equations
written in the characteristic form provide practical means to rapidly
assess how the energy, frequency and direction of dominant surface waves
are distributed under varying wind forcing. For young waves, non-linear
interactions are essential to drive the peak frequency downshift, and
the wind energy input and wave breaking dissipation are the governing
sources of the wave energy evolution. With a prescribed wind wave growth
rate, proportional to ustar/c squared, wave breaking dissipation becomes
a power-function of the dominant wave slope. Under uniform wind
conditions, this growth rate imposes solutions for peak frequency and
energy development to follow fetch-laws, with exponents q=-1/4 p=3/4
correspondingly. This set of exponents recovers the Toba's laws, and
imposes the wave breaking exponent equal to 3. A smooth transition from
wind driven seas to swell is obtained. Varying wind direction is the
only source to drive spectral peak direction changes. This can lead to
occurrence of focusing/defocusing wave groups and formation of areas
where wave-rays merge and cross. Solutions predict significant (but
finite) local enhancements of the energy. Further propagating, wave rays
diverge, leading to wave attenuation away from the storm area%
\end{abstract}%



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