## 1. Introduction

[2] Tidal sandwaves form a prominent bed feature in tide-dominated offshore areas of shallow shelf seas. They occur in rhythmic patches, with crests spaced several hundreds of meters apart (Figure 1), several meters high and oriented almost perpendicularly to the principal direction of the tidal flow [*Terwindt*, 1971].

[3] Sandwaves are highly dynamic, with typical timescales of formation of the order of years and migration speeds up to 10 m yr^{−1}. Insight in sandwave dynamics, particularly their migration, has several practical applications, as it may help optimizing dredging strategies, and ascertaining pipeline safety [*Németh et al.*, 2003].

[4] Using a process-based morphodynamic model, *Hulscher* [1996] showed that the formation of tidal sandwaves can be explained as a linear instability of a flat seabed, driven by the vertical flow circulations induced by tide-topography interactions. This theory was later extended and refined with respect to the description of hydrodynamics and solution procedure [*Gerkema*, 2000; *Komarova and Hulscher*, 2000; *Besio et al.*, 2003a]. Of practical interest is the modeling of sandwave migration by including a residual current [*Németh et al.*, 2002] and tidal asymmetry [*Besio et al.*, 2003b; *Besio et al.*, 2004]. In the studies cited here the main focus is on the so-called fastest growing mode (fgm), that is, on the wavelength for which the growth rate attains its maximum. The models therefore essentially describe rhythmic sandwave patterns of infinite spatial extent.

[5] In the present study, however, we focus on the evolution of a local, or isolated topographic disturbance on an otherwise flat seabed. How will such a disturbance evolve in time, according to the hydrodynamic and morphodynamic mechanisms of sandwave formation?

[6] We follow an approach which fully relies on the linear theories cited above. We particularly use the dispersion relationship for sandwaves, that is, the relation between growth rate (and migration rate) of sandwaves and the topographic wave numbers. The study of a spatially constrained topography, however, involves the dynamics of the full spectrum rather than that of the fastest growing mode only. Essential is that we use the general properties of the dispersion relationship, so we do not need to revisit the details of the underlying hydrodynamics and sediment transport. The solution to the problem, that is, the bed topography as a function of both space and time, follows from a Fourier integral which can generally not be evaluated in closed form. We explore and compare two solution methods: numerical integration and an analytical approximation of the dispersion relation. The former method is direct and accurate, but not transparent and computationally expensive, especially in three dimensions. The approximate method, based on an asymptotic Gaussian model used previously in a hydrodynamic context [*Benjamin*, 1961; *Gaster*, 1968; *Gaster and Davey*, 2003; *Gaster*, 1981, 1982], combines a quadratic approximation of the dispersion relation (around the fgm) with suitable initial conditions. This Gaussian model is also known as Method of Steepest Descent or Saddle-Point Method [*Morse and Feshbach*, 1953]. The analytical solution, while giving insight in the underlying physics, turns out to be quick and to perform well. To obtain the input parameters of this model, we use the three-dimensional extension, neglecting Coriolis effects, of the code of *Besio et al.* [2003a] (hereinafter referred to as BBF) (for a description of this model, see Appendix B).

[7] Finally, we apply the theory to the practical case of a dredged channel or a sandpit, focusing on length scales of sandwaves (several hundreds of meters). What is the influence of such a human intervention on the short and intermediate term morphodynamics of the seabed? We compare our results with those from a different model [*van de Kreeke et al.*, 2002], in which a depth-averaged flow approach has been adopted, neglecting the details of the vertical flow structure and, hence, neglecting the dynamics of sandwaves.

[8] This paper is organized as follows. Section 2 provides a general background on sandwave dynamics. In section 3 we present our analysis of the sandwave packets and a description of the numerical method. In section 4, we compare the analytical approximation with the numerical method and we apply the analysis to study the evolution of a dredged trench. Finally, sections 5 and 6 contain the discussion and conclusions, respectively.