## 1. Introduction

[2] Long-term tidal morphologies are largely controlled by the net exchange of sediments between the enclosed tidal basins and the adjacent seas. The evaluation of such exchanges has traditionally been carried out by focusing on control sections, typically tidal inlets, where cross-sectional forms adjust to prevailing hydrodynamic and sediment transport conditions. The basic relationship employed for coupling hydrodynamic and morphodynamic processes is an originally empirical linkage of cross-sectional area of tidal inlets, say Ω, with spring (i.e., maximum astronomical) tidal prism, *P*

where the scaling coefficient *α* typically lies in the range 0.85–1.10 [e.g., *O'Brien*, 1931, 1969; *Jarrett*, 1976; *Hughes*, 2002]. Such a relationship embodies the complex and site-specific feedbacks between tidal channel morphology and tidal flow properties occurring both in inlet and sheltered channel sections [e.g., *O'Brien*, 1931, 1969; *Jarrett*, 1976; *Bruun*, 1978; *Friedrichs*, 1995; *de Swart and Zimmerman*, 2009].

[3] Various attempts [e.g., *Escoffier*, 1940; *Krishnamurthy*, 1977; *Marchi*, 1990; *Hughes*, 2002] have been carried out to substantiate, from a theoretical point of view, the existence of a relationship of the form (1) for tidal inlets. All of these approaches assume a sinusoidal tide, and that at equilibrium the maximum bottom shear stress is represented by the critical threshold for incipient motion of bed sediment. The analyses proposed by *Krishnamurthy* [1977] and *Hughes* [2002] rely on the assumption of a given velocity profile (either logarithmic or described by a power law relationship with exponent equal to 1/8) which is then integrated across the inlet cross section, in order to obtain the flowing discharge and, eventually, the tidal prism. On the other hand, the theoretical derivation pursued by *Marchi* [1990] defines a general framework to rationally include all of the various theoretical treatments of the subject. As summarized in Appendix A, the problem was tackled by considering the one-dimensional propagation of the tide along a straight rectangular inlet channel connecting the open sea with a schematic tidal basin, short enough to be treated as oscillating statically (i.e., with a nearly horizontal water surface). The exact value *α* = 6/7 was derived in that context. For these reasons *D'Alpaos et al.* [2009] have proposed, in their review of the origins of equation (1), to term it the O'Brien-Jarrett-Marchi (OBJM) “law”.

[4] The extension of equation (1) to sheltered sections, the subject of this paper, bears notable practical implications on the predictability of long-term morphodynamics. Indeed, tidal networks exert a fundamental control on hydrodynamic, sediment and nutrient exchanges within tidal environments, which are characterized by highly heterogeneous landscapes and physical and biological properties [e.g., *Adam*, 1990; *Perillo*, 1995; *Rinaldo et al.*, 1999a, 1999b; *Allen*, 2000; *Friedrichs and Perry*, 2001; *D'Alpaos et al.*, 2005, 2007a; *Savenije*, 2006; *Kirwan and Murray*, 2007; *Marani et al.*, 2007].

[5] Migration of watershed divides induced by elaboration of tidal channelization has a strong feedback on local prisms and on the regime of inner cross sections. Should a synthesis like (1) apply, owing to the different time scales characterizing tidal network contractions/expansions (with fast adaptation of the channel cross section) and the dynamics of unchanneled tidal landforms (salt marshes and tidal flats), long-term predictions of tidal morphologies would be within reach. For example, when a bifurcation appears, prompting a new tidal channel to cut through a salt marsh in a network expansion phase [e.g., *D'Alpaos et al.*, 2007a], the progressively rearranged prisms would define the evolutionary trends for local cross sections. Thus, the validation of equation (1) as a general tool within tidal landscapes would constitute a major predictive tool for long-term tidal geomorphology.

[6] Here we are interested in substantiating the applicability of the equation (1) to sheltered tidal channel sections on the basis of field observations and hydrodynamic and morphodynamic models, assessing its validity and limitations. The interest toward simplified relationships of this type, in fact, is central to the development of models describing the long-term ecomorphodynamic evolution of tidal systems, and, hence to address key conservation issues related to the effects of climate changes and increasing human pressure. This task can be pursued with a reasonable computational effort only through the formulation of reliable, though suitably simplified, model components, including the most relevant morphological processes responsible for shaping the tidal landscape [e.g., *Rinaldo et al.*, 1999a, 1999b; *D'Alpaos et al.*, 2005, 2006, 2007a; *Kirwan and Murray*, 2007; *Marani et al.*, 2007]. For example, the complex tidal network structures generated through the model proposed by *D'Alpaos et al.* [2005], are obtained under the assumption, implied by relations of the type (1), that channel cross sections are in dynamic equilibrium with the local tidal prism. The validity of such an assumption, however, was assessed only indirectly [*D'Alpaos et al.*, 2005, 2007a], by observing that the synthetically generated networks meet distinctive real network statistics, reproducing several observed characteristics of geomorphic relevance, such as, e.g., the probability distribution of unchanneled lengths [*Marani et al.*, 2003]. In one rare instance [*D'Alpaos et al.*, 2007b] a constructed salt marsh allowed monitoring of the developing tidal network which the model reproduced reasonably.

[7] Although the validity of equation (1) for sheltered channels not exposed to littoral transport or open sea has been to some extent addressed by *Friedrichs* [1995], *Rinaldo et al.* [1999b], *Schuttelaars and de Swart* [1996, 2000], *Lanzoni and Seminara* [2002], and *van der Wegen et al.* [2008] among others, a synthesis is still missing. In fact, in principle morphodynamic relationships for channelized tidal embayments associated with the interior of tidal marshes or lagoons could work somewhat differently from inlets on open coasts. This stems, among other factors, from the relative lack of direct, intense wave attack and littoral drift in sheltered sites. Moreover, spatial gradients in tidal amplitude and phase are usually less pronounced within inner regions of a tidal setting than in the inlet zone, and highly nonlinear hydrodynamic feedbacks may enter the picture whenever adjacent tidal flats or salt marshes are subject to wetting and drying processes (an occurrence not accounted for in *Marchi*'s [1990] analysis).

[8] A further matter of concern is the fact that the relation (1) holds only for tidal systems believed to have achieved an asymptotic stable equilibrium. One thus wonders about the chances to face consistently such a condition anywhere and at any time within a tidal landscape. Within a different landscape evolution context, it has been argued that the main features of an erosional system are reached relatively soon in the evolutionary history of the relevant geomorphology, thus making the steady state regime assumption not so unreasonable an approximation in most real life cases [*Rinaldo et al.*, 1993]. The empirical and theoretical findings of *Friedrichs* [1995] and *Rinaldo et al.* [1999b] seen in that context thus make sense, as they have explored the relationship between cross-sectional size, Ω, and the coevolving local peak discharge, *Q*. They found, in agreement with previous field observations [e.g., *Myrick and Leopold*, 1963; *Nichols et al.*, 1991], that a significant proportionality between Ω and *Q* (which is directly related to the tidal prism) exists for sheltered estuarine or tidal network cross sections. Other well-defined power law relationships between channel width, cross-sectional area, watershed area and peak discharges were also documented in other contexts [*van Dongeren and de Vriend*, 1994] using the morphometric parameters proposed by *Myrick and Leopold* [1963].

[9] The validity of OBJM-like relations has also been confirmed by the theoretical and numerical analyses carried out by *Schuttelaars and de Swart* [1996, 2000], and by *Lanzoni and Seminara* [2002] concerning tidal channels with negligible intertidal storage of water over tidal flats and salt marshes. Some information on the effects of intertidal areas, though on the relationship between cross-sectional size, Ω, and the coevolving local peak discharge, *Q*, has been provided by *D'Alpaos et al.* [2006] through the numerical modeling of the morphodynamic evolution of a generic cross section composed by a tidal channel and an adjacent marsh platform, which drains a given tidal watershed. Their results indicate that when the marsh platform evolves in the vertical direction, the ratio between cross-sectional area and peak discharge remains nearly constant, thus suggesting that the cross section adapts quite rapidly to changes in water discharge and, therefore, in the related tidal prism. The results obtained by *Lanzoni and Seminara* [2002], on the other hand, suggest a departure from equation (1) for smaller cross sections, likely associated to wetting and drying. A similar departure from a power law relationship, though in the opposite direction, has also been observed by *Rinaldo et al.* [1999b] who, however, interpreted such deviation as an artifact of the poor morphologic resolution of the small channels (i.e., characterized by widths smaller than 20 m) in their data.

[10] In the present contribution we explore systematically the broad applicability of equation (1) to sheltered tidal channel sections. To this purpose we will use extensive observations of morphological data of the highest available resolution and the results of a number of one- and two-dimensional models, set up on the basis of progressively more realistic and detailed formulations, to analyze the relationships between given or evolving cross-sectional areas, Ω, and the related tidal prisms, *P*.

[11] The paper is organized as follows. Section 2 describes the field data and, jointly with suitable auxiliary material (Text S1), recalls the mathematical models used to substantiate the validity of equation (1) for arbitrary tidal channel cross sections. Section 3 summarizes the results of empirical and theoretical validations. The range of validity of the investigated law, as well as its implications, are discussed in section 4. Finally, section 5 reports our conclusions. Appendix A emphasizes how the theoretical analysis proposed by *Marchi* [1990] provides a comprehensive framework capable of generalizing previous theoretical treatments of the problem.