## 1 Introduction

[2] The planform of a delta and its distributary network are set by the hydrodynamics and depositional patterns at the mouths of distributary channels. In the absence of significant waves and tides, the hydrodynamics usually consists of a bounded plane turbulent jet (Figure 1). The jet determines the sediment particle flow paths, and the evolving sediment bed influences the jet. This morphodynamic interaction then sets the number of distributaries, the rugosity of the shoreline, and ultimately the delta planform [*Edmonds and Slingerland*, 2007, 2008; *Jerolmack and Swenson*, 2007; *Edmonds et al*., 2009; *Jerolmack*, 2009; *Hoyal and Sheets*, 2009; *Martin et al*., 2009; *Chatanantavet et al*., 2012]. More recent studies have investigated the effect of wind waves [*Nardin and Fagherazzi*, 2012; *Nardin et al*., 2013] and tides [*Leonardi et al*., 2013] on jet hydrodynamics, but here we restrict our discussion to exclude them.

[3] It is unfortunate that most laboratory investigations of plane jets debouching into quiescent bodies are of the unbounded type, i.e., the experiments are performed in a way to minimize the effect of the bounding walls. Measurements are usually taken in a zone spanning from the orifice to *x*/*h* = 1, where *x* is distance along the jet axis and *h* is the distance between the bounding surfaces [e.g., *Bradbury* 1965; *Goldschmidt and Young*, 1975; *Everitt and Robins*, 1978; *Ramaprian and Chandrasekhara*, 1983, among others]. *Dracos et al*. [1992] pointed out at the time of his work, that few studies were performed on bounded plane jets especially for *x*/*h* » 1. *Foss and Jones* [1968] and *Holdemann and Foss* [1975], in fact, studied bounded rectangular jets, but only for *x*/*h* < 10. Also, these earlier studies only consider orifice geometries with aspect ratios *B*/*h* smaller than 2 where *B* is channel top width. These ratios are smaller than observed in typical delta distributaries. Only recently have experiments been conducted on bounded planar jets with *B*/*h* » 1 [*Rowland et al*., 2009, 2010].

[4] The turbulence characteristics of these narrow bounded jets differ from those of unbounded ones [*Dracos et al*., 1992]. Using the depth as a scaling parameter, *Draco et al*. [1992] defined three fields for a bounded jet: near (*x*/*h* < 2), middle (2 < *x*/h < 10), and far (*x*/*h* > 10). In the near field the flow is essentially the same as a classical unbounded jet, i.e., the effect of the bounding walls is not yet felt and the *x*-directed velocity *u* is uniform along the vertical axis *z*. In the middle field, secondary flows are present that affect the entire depth (*u* is nonuniform along *z*). For *x*/*h* > 10, the intensity of the turbulence is roughly constant, the jet begins to meander, and it is flanked by two set of counter-rotating vortices. The influence of the boundary is clearly seen in the turbulence spectrum of a bounded jet. In the near field the spectrum at small wave numbers is typical of a three-dimensional cascading turbulent flow with a −5/3 wave number dependence [*Goldschmidt and Young*, 1975]. This is also the case of the near field in a bounded jet, but in the far field the energy transfer at small wave numbers follows a −3 wave number dependence [*Dracos et al*., 1992; *Landel et al*., 2012] typical of quasi-two-dimensional turbulence characterized by an enstrophy cascade [*Batchelor*, 1969]. This means that part of the turbulent energy is transferred from the inertial subrange to smaller wave number, i.e., to larger scale eddies. This turbulent energy transfer leads to unstable jets due to the formation of large-scale counter-rotating eddies around which the mean flow meanders.

[5] The transition from a stable, planar jet to an unstable meandering jet should be an important threshold in delta growth, because it would change the depositional patterns at the river mouth [e.g., *Rowland et al*., 2010]. Linear stability analysis is a typical theoretical approach to defining the threshold and general stability behavior of shallow jets, wakes, and mixing layers [*Jirka*, 1994; *Socolofsky and Jirka*, 2004; *Chen and Jirka*, 1997; *Jirka*, 2001; *van Prooijen and Uijttewaal*, 2002]. In this approach a parallel flow is considered, i.e., spreading of the jet is neglected. The shallow water equations are linearized, which leads to a modified Orr-Sommerfeld equation, including turbulence viscosity (*ν*^{T}) and bottom friction (*c _{f}*) as dissipative terms. Analyses of this type indicate jet stability has a strong dependence on friction and aspect ratio [

*Jirka*, 1994]. This is embedded in the stability parameter

*S*for shallow jets:

where *h* is the water depth, *L* is a jet length scale which for expanding jets we take to be equal to half mouth river width *B* [*Jirka*, 1994], and *c _{f}* is the friction factor in the formulation , with

*τ*the bottom shear stress and

*u*the local one-directional velocity (note that

*c*can also be written as a function of the Chezy coefficient

_{f}*C*usually adopted in shallow water simulations, i.e., ). Below a critical jet stability parameter,

*S*, jet instabilities grow unimpeded; above this threshold instabilities are dampened by bottom friction. Theory and experiment suggest

_{c}*S*ranges from 0.06 to 0.6 [

_{c}*Uijttewaal and Booij*, 2000;

*Jirka*, 2001;

*vanProoijen and Uijttewaal*, 2002;

*Socolofsky and Jirka*, 2004].

[6] The influence of jet instability on sedimentation patterns at a river mouth has not been fully explored, a notable exception though is the recent work by *Rowland et al*. [2010]. In this work, we are motivated by the following considerations: first, to our knowledge, no detailed numerical investigations of the transition between stable and unstable shallow jets for field-scale river mouths have been carried out. Second, most experimental investigations of shallow jets [e.g., *Rowland et al*., 2009, 2010] are limited to a single geometrical configuration. More experiments are needed with different river mouth aspect ratios, friction values, and incoming discharges to cover the entire stability parameter space. Third, the available analytical solutions for shallow jet dynamics [*Borichansky and Mikhailov*, 1966; *Wright and Coleman*, 1974; *Ozsoy*, 1977; *Ozsoy and Unluata*, 1982; *Ozsoy*, 1986; *Wang*, 1984; *Izumi et al*., 1999; *Ortega-Sánchez et al*., 2008] cannot take into account the coupled evolution of the flow field and the bottom, and can only predict the initial depositional pattern [*Ozsoy*, 1986; *Syvitski et al*., 1998]. Moreover, they cannot capture jet meandering but only predict time-averaged quantities.

[7] The aim of this paper is to define a threshold for instability (jet meandering) of a turbulent, bounded, high aspect ratio jet and to explore how frictional effects and jet instability coevolve with the patterns of sedimentation at the distributary mouth. We employ a three-dimensional model, Delft3D, that is presented in section 'Numerical Model'. Section 'Results' describes the hydrodynamic results of a jet debouching into a quiescent body of water, from which a stability diagram is deduced. Section 'Morphodynamics of Stable and Unstable Jets' shows the results of morphodynamic simulations and effect of instability on the depositional patterns. Discussions and conclusions are drawn in sections 'Discussion' and 'Conclusions', respectively.