Linear shoaling in Boussinesq-type wave propagation models Gonzalo Simarro a, ⁎, Alejandro Orfila b , Alvaro Galan c a ICM (CSIC), Passeig Maritim de la Barceloneta 37-49, 08003 Barcelona, Spain b IMEDEA (CSIC-UIB), Carrer Miquel Marques 21, 07190 Esporles, Spain c ETSICCP (UCLM), Avenida Camilo Jose Cela 2, 13071 Ciudad Real, Spain abstract article info Article history: Received 18 October 2012 Received in revised form 23 May 2013 Accepted 27 May 2013 Available online 29 June 2013 Keywords: Linear shoaling Linear dispersion Boussinesq-type equations This work focuses on linear shoaling performance of low order Boussinesq-type equations. It is shown that the linear shoaling errors can be important in well known equations in the literature. New sets of coefficients are presented for three well known sets of equations. The sets are found so as to minimize a global linear error that includes celerity and shoaling errors. Finally, a new set of enhanced bilayer low order equations is presented, with much improved linear behavior (errors in wave celerity and wave amplitude below 1% up to kh = 20). For completeness, the equations are written in their fully nonlinear version, and the nonlinear coefficients are also given. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Boussinesq-Type Equations (“BTEs” hereafter) can be understood as an extension of Nonlinear Shallow Water Equations (NSWEs) that includes dispersive effects. Dispersive effects are present in mid to deep waters (i.e., for κ ≡ ω 2 h/g ≳ 0.1 with ω the wave angular fre- quency, h the water depth and g the gravitational acceleration), where waves with different angular frequencies travel with different celerities. For example, wave celerity in deep waters (i.e., κ ≳ 3.0) is c ≈ g/ω (Dean and Dalrymple, 1984). The behavior in shallow waters (ω 2 h/g ≲ 0.1) is very different: the linear wave celerity is c≈ ffiffiffiffiffi gh p , i.e., independent of the wave angular frequency. Therefore, waves with different angular frequencies travel at the same celerity and there is no dispersion. NSWEs are valid for nondispersive conditions. To increase the limited range of application of NSWEs (κ ≲ 0.1) to deeper waters, BTEs include weakly dispersive terms in a perturbative fashion. The first approach considered weak nonlinearities for weakly dispersive conditions over flat beds (Boussinesq, 1872), and it was ex- tended to uneven bottoms by Peregrine (1967). Today there are several well known sets of fully nonlinear and weakly dispersive equations valid over uneven bathymetries (Lynett and Liu, 2004b; Madsen and Schaffer, 1998, 1999; Wei et al., 1995). The above mentioned BTEs include all the weakly dispersive terms up to order O kh ð Þ 2   , k being the wave number. From the dispersion relationship for Airy theory, κ = kh tanh(kh), so that κ ≈ kh in deep waters. Because BTEs include terms O kh ð Þ 2   , they have improved dispersive performance compared to NSWEs: roughly speaking, they give good results up to kh ≈ 1. In order to further increase the applicability of BTEs to deeper waters, the literature offers a range of approaches. For example, the natural extension to include higher order dispersive terms has been explored (Gobbi et al., 2000; Kennedy et al., 2002; Madsen et al., 2002, 2003) lead- ing to a set of equations with improved dispersive behavior but also with higher order derivatives (up to five). Other approaches (Lynett and Liu, 2004a,b) consider splitting the vertical domain into several layers, using within each layer low order approximations. Now the number of unknowns and equations is increased: in the most general 2DH case, the number of equations is increased from three to five. Besides high-order and multi-layer approaches, there are at least two ways that have shown to allow improvements of the linear dispersion of the Boussinesq equations without increasing the order of derivatives nor the number of unknowns: the enhancement technique by Madsen and Sorensen (1992) and the use of the velocity at an arbitrary level z α as proposed by Nwogu (1993)—extended to the fully nonlinear case by Wei et al. (1995) and Kennedy et al. (2001). These two ideas have been combined (Madsen and Schaffer, 1998). Of note, multi-layer equa- tions by (Lynett and Liu, 2004a,b), besides splitting the vertical domain, use arbitrary-level velocities (and interfaces). While high-order and multi-layer procedures allow a “full” improve- ment of the equations' behavior by getting closer to the real solution, both the enhancement and arbitrary-level velocity techniques intro- duce coefficients that are chosen to mimic some well known linear and weakly nonlinear properties in mid to deep waters (Galan et al., 2012; Madsen and Schaffer, 1998; Nwogu, 1993; Schaffer, 1996). The linear properties usually used for comparison are wave celerity and group celerity (Lynett and Liu, 2004b; Madsen and Schaffer, 1998; Coastal Engineering 80 (2013) 100–106 ⁎ Corresponding author. Tel.: +34 932309500. E-mail address: simarro@icm.csic.es (G. Simarro). 0378-3839/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.coastaleng.2013.05.009 Contents lists available at SciVerse ScienceDirect Coastal Engineering journal homepage: www.elsevier.com/locate/coastaleng