## 1. Introduction

[2] The oscillatory motion of the tide in a lagoon produces water fluxes inside the basin and at the inlets that are of fundamental importance for several nearshore processes. For example, from a geomorphic point of view, the tidal flow is the main force responsible for the creation and evolution of dendritic channels that often dissect tidal flats and salt marshes. Clearly the tidal flow is determinant for the sediment budget and the dispersion of contaminants in shallow lagoons, and its correct modeling becomes crucial in assessing an eventual environmental risk.

[3] A powerful and widely used technique for the determination of tidal fluxes in a basin is based on the numerical resolution of the shallow water equations, given the basin shape and tidal characteristics. The numerical resolution of this problem, accomplished with several refined methods [see *Velyan*, 1992], yields to results of remarkable precision, when compared to measured tidal velocities [e.g., *Wang et al.*, 1998]. However, in determinated situations, a correct analysis of the equations and the application of suitable simplifications that enhance the leading terms can be of great scientific interest and provide simplified but physically based tools for different important applications. In the context of tidal basins it is natural to wonder what is the influence of the basin shape on the tidal flow, and whether it is possible to separate this influence from the effect of the basin bottom topography.

[4] This paper shows that, under precise hypotheses, it is possible to split the tidal flow in a component dependent on the basin shape and a component dependent on the basin bottom topography. This mathematical development is not only a mere exercise, but opens the door to physical applications of great interest. In cases where the bathymetry of tidal basins is not available, say for example when remote sensing data are used, a simplified method that estimates the tidal discharge within the basin, neglecting the bathymetry, becomes helpful. Clearly we are also interested in knowing what correction we need to add to this solution in order to obtain realistic discharges with different bottom configurations. Furthermore, the basin boundaries and the basin bottom are features of the coastal landscape that evolve at different timescales. The boundaries, often determined by the shape of river paleo-valleys, usually vary with sea level oscillations, i.e., in thousands of years. On the other hand, the basin bottom is modified by several agents, including for example a change in the sediment input from rivers and the dredging of channels for navigation, that typically act at a smaller timescale, from decades to centuries. Thus a method able to separate the component of the flow field due to the basin shape from the component determined by the bottom topography results straightforward in studying the short term evolution of tidal basins and their channel networks.

[5] When the tide enters a basin of limited dimensions, the signal is reflected at the mainland boundaries, and the tide assumes characteristics of a standing wave, with strong currents around mid-tide and minimum velocity at high and low water (slack water) [*Wright et al.*, 2000].

[6] The fact that in small basins the tidal wave is similar to a standing wave was pointed out by *Schuttelaars and De Swart* [2000], as compared with the progressive character of the tide in a large estuary [*Lanzone and Seminara*, 1998; *Friedrichs and Aubrey*, 1994]. In a small basin an intuitive approach is to consider water elevation to be flat and oscillating synchronously with the tide at the inlet. This assumption is common in classical studies of tidal inlets, where the tidal basin is treated as a reservoir with oscillating elevation and the flow along the inlet is calculated by means of open channel hydraulics [*Bruun*, 1978].

[7] *Schuttelaars and De Swart* [1996] showed that, for small tidal basins, the tidal wavelength is long with respect to the basin dimensions, so that the water surface can be considered flat as a first approximation (i.e., the phase difference in different locations of the basin is negligible and the water surface is oscillating synchronously everywhere). This hypothesis has been previously utilized to calculate the discharge [*Boon*, 1975] and to explain the velocity asymmetry [*Pethick*, 1980] in salt marsh creeks. *Healey et al.* [1981] questioned the applicability of this simplification in salt marshes, and *Rinaldo et al.* [1999b] showed that discrepancies arise from the complex nonlinear phenomenon of wetting and drying of the marsh surface. The same hypothesis of a flat surface was adopted to study the equilibrium bottom configuration in a rectangular tidal basin [*Schuttelaars and de Swart*, 1999], to single out the drainage area of salt marsh creeks [*Rinaldo et al.*, 1999a], and to model the cross-sectional evolution of marsh channels [*Fagherazzi and Furbish*, 2001].

[8] Even though the hypothesis of basin dimensions small with respect to the tidal wavelength (hereinafter referred to as “small embayment hypothesis”) and the consequent flat water surface has already been utilized in different studies, there is an evident need to develop in detail the simplifications that this hypothesis yields to the tidal flow. We will first present the simplified unidimensional formulation that allows us to understand the basic concepts; the rest of the paper is then the extensions of the same concepts to the full bidimensional shallow water equations.

[9] In shallow water the unidimensional propagation of a wave of small amplitude can be formulated with two equations, continuity and conservation of linear momentum [*Stoker*, 1957, p. 24],

with η the water surface elevation with respect to Mean Sea Level (M.S.L), *h* the depth of the bottom with respect to M.S.L., *u* the water velocity averaged over the vertical, *g* the gravity acceleration, *t* time and *x* the space coordinate. Equations (1a) and (1b) were derived considering an incompressible fluid, eliminating inertial and diffusion terms, neglecting the friction at the bottom and averaging the unidimensional equations over the vertical direction. A sinusoidal tidal wave traveling in one direction in an unconfined domain is then governed by the two following equations, derived from equation (1):

where *a* is the wave amplitude, ω is the angular frequency, and the wave celerity in shallow water. Both surface elevation and velocity vary in space and time, but they are in phase, in the sense that *u* and η simultaneously reach their maximum at a given time. Moreover the velocity *u* scales with *c* = ωλ, where λ is the tidal wavelength. On the contrary, if the tidal wave reaches an obstacle, say a vertical wall, it gets reflected, forming a standing wave. The equations for the standing wave, obtained by summing two waves given by equation (2) traveling in opposite directions, are

where now *x* is the distance from the wall. In this case the surface elevation and the velocity are out of phase, with the maximum of velocity occurring for zero elevation and vice versa. At a distance *L* from the wall small with respect to the wavelength then ω*L*/*c* = *L*/λ is small and the approximations hold. Equation (3) then becomes

with the velocity proportional to *L*ω. The same equations are obviously valid for each point closer to the wall. Consequently, equation (4) show that for *x* < *L* the water elevation is flat and oscillates synchronously with the tide, whereas the velocity decreases proportionally to the distance from the wall, going to zero for *L* = 0.