European Journal of Mechanics B/Fluids 30 (2011) 137–146 Contents lists available at ScienceDirect European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu The localized Zakharov equation: Derivation and validation O. Gramstad a,∗ , Y. Agnon b , M. Stiassnie b a Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway b Faculty of Civil and Environmental Engineering, Technion IIT, Haifa 32000, Israel article info Article history: Received 12 January 2010 Received in revised form 7 October 2010 Accepted 13 October 2010 Available online 18 November 2010 Keywords: Water-waves Nonlinear interaction Phase-resolving model abstract In Zakharov’s equation, the spectral function represents the entire horizontal plane. In practical applications, one often has to use a finite number of Fourier modes that are determined for limited regions of the horizontal plane, but vary from region to region. To overcome this shortcoming, we utilize a discrete windowed Fourier transform to obtain a new localized Zakharov equation (LZE), which can handle spatial variations in a more transparent way. The LZE is successfully validated by comparing different aspects of its performance with results from the Zakharov equation and a modified nonlinear Schrödinger equation. © 2010 Elsevier Masson SAS. All rights reserved. 1. Introduction Zakharov’s equation [1] is the main existing model to study the weakly nonlinear evolution of sea states with a broad band of wavelengths. Zakharov’s equation is deterministic, i.e. no stochas- tic assumptions were made in the course of its derivation. How- ever, it is also a very convenient starting point for the derivation of Hasselmann’s stochastic model [2]. The Zakharov equation is formulated in terms of a generalized amplitude spectrum ˆ b(k, t ), which is determined from the Fourier transform of the surface elevation η(x, t ) and the Fourier transform of the velocity potential at the free surface ψ(x, t ) by ˆ b(k, t ) = 1 2π ∫   g 2ω(k)  1/2 η(x, t ) + i  ω(k) 2g  1/2 ψ(x, t )  e −ik·x dx. (1) Here, · denotes a scalar product; k = (k x , k y ) is the wavenumber vector; x = (x, y) are the horizontal space coordinates; t is time; g is gravity; and ω is the angular frequency in water of constant depth h given by the dispersion relation ω 2 = g |k| tanh(|k|h). Zakharov has shown that the temporal evolution of ˆ b(k, t ) is ∗ Corresponding author. E-mail addresses: oding@math.uio.no (O. Gramstad), agnon@tx.technion.ac.il (Y. Agnon), miky@tx.technion.ac.il (M. Stiassnie). governed by i ∂ ˆ b ∂ t − ω(k) ˆ b = ∫ T 0,1,2,3 ˆ b ∗ 1 ˆ b 2 ˆ b 3 δ 0+1−2−3 dk 1,2,3 , (2) which is now called the Zakharov equation. Here, we have used the compact notation, e.g. dk 1,2,3 = dk 1 dk 2 dk 3 and δ 0+1−2−3 = δ(k + k 1 − k 2 − k 3 ). The kernel T 0,1,2,3 = T (k, k 1 , k 2 , k 3 ) can be found, for example, in [3], or in an alternative form with modifications necessary for the Zakharov equation to be Hamiltonian [4]. Early numerical simulations of (2) can be found in [5], and more recent references are given by [6]. Note that, strictly speaking, (1) also includes bound modes in addition to the leading-order free modes, whereas (2) is for the free modes only. Details about obtaining the bound modes once the free modes are known can be found in [3]. The inverse of (1) is η = 1 2π ∫ ω 2g  1/2  ˆ be ik·x + c.c.  dk, (3a) ψ = −i 2π ∫  g 2ω  1/2  ˆ be ik·x − c.c.  dk. (3b) The main goal of the current paper is to tackle the following challenge. In the derivation of Zakharov’s equation (2), the Fourier transform (1) is applied over the entire horizontal plane, resulting in ˆ b(k, t )—a ‘global’ amplitude spectrum. In practice, however, i.e. in field or laboratory applications, the Fourier transform is applied only to a limited region of the horizontal plane, resulting in a ‘local’ amplitude spectrum which varies from region to region. In Section 2 of this paper, we develop a localized discrete Za- kharov equation (LZE), tailored to overcome the above-mentioned 0997-7546/$ – see front matter © 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2010.10.004