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The Intermediate Water Depth Limit of the Zakharov Equation and Consequences for Wave Prediction

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  • 1 European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom
  • | 2 Dipartimento di Fisica Generale, Università di Torino, Turin, Italy
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Abstract

Finite-amplitude deep-water waves are subject to modulational instability, which eventually can lead to the formation of extreme waves. In shallow water, finite-amplitude surface gravity waves generate a current and deviations from the mean surface elevation. This stabilizes the modulational instability, and as a consequence the process of nonlinear focusing ceases to exist when kh < 1.363. This is a well-known property of surface gravity waves. Here it is shown for the first time that the usual starting point, namely the Zakharov equation, for deriving the nonlinear source term in the energy balance equation in wave forecasting models, shares this property as well. Consequences for wave prediction are pointed out.

Corresponding author address: Peter A. E. M. Janssen, European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading, RG2 9AX, United Kingdom. Email: p.janssen@ecmwf.int

Abstract

Finite-amplitude deep-water waves are subject to modulational instability, which eventually can lead to the formation of extreme waves. In shallow water, finite-amplitude surface gravity waves generate a current and deviations from the mean surface elevation. This stabilizes the modulational instability, and as a consequence the process of nonlinear focusing ceases to exist when kh < 1.363. This is a well-known property of surface gravity waves. Here it is shown for the first time that the usual starting point, namely the Zakharov equation, for deriving the nonlinear source term in the energy balance equation in wave forecasting models, shares this property as well. Consequences for wave prediction are pointed out.

Corresponding author address: Peter A. E. M. Janssen, European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading, RG2 9AX, United Kingdom. Email: p.janssen@ecmwf.int

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