## 1 Introduction

### 1.1 Background

[2] Turbidity currents are sediment-laden, gravity-driven underflows that occur in lakes and oceans. They are responsible for delivering sediment into these subaqueous environments, where they can shape the largest sedimentary features on the modern earth [*Middleton*, 1993]. Complex interactions between turbidity currents and their surrounding environment take place as they travel and evolve. First of all, the difference in density between the turbidity current and the ambient fluid causes interfacial instability, which acts to entrain ambient fluid into the current body. As a result, the current thickens and becomes dilute. Second, these currents actively exchange sediment with the bed surface material by depositing sediment through downward settling and entraining sediment through shear, creating depositional or erosional features in the process. These interactions are keys to explaining the formation of important subaqueous morphologies such as submarine channels, levees, and canyons.

[3] One of the most important factors associated with these interactions is the internal structure of the current. For example, the near-bed velocity gradient and the turbulent energy determine the erosive power of a current; the near-bed sediment concentration determines settling flux onto the bed. On the other hand, as the current evolves, settling and entrainment processes will in turn adjust its internal structure. Such a system can be further complicated when the dispersive component, sediment in our case, self-stratifies. Due to the tendency for sediment to settle, the concentration tends to decrease upward within the current body. This stratification in density requires additional consumption of energy from the mean flow in order to mix the concentration against the density gradient. As turbulence dampens, the flow becomes less effective at mixing momentum, resulting in a velocity profile that varies more strongly in the vertical [*Vanoni*, 1946; *Einstein and Chien*, 1955; *Coleman*, 1981, 1986; *Lyn* 1988].

[4] Various efforts have been devoted to understanding the hydrodynamics as well as the morphodynamic response associated with turbidity currents through numerical modeling. These studies are often composed of a depth-averaged, three-equation model [*Ellison and Turner*, 1959; *Pantin*, 1979] for flow dynamics and an equation for bed movement. For example, *Imran et al*. [1998] examined the condition for self-channelization of turbidity currents on a submarine fan using a two-dimensional three-equation model; *Kostic and Parker* [2003] implemented the model as a submodel to study the depositional pattern of fine material beyond a prograding delta foreset; *Bradford and Katopodes* [1999] extended the model to include multiple-grain sizes; *Parker et al*. [1986] included an additional equation for the balance of turbulent kinetic energy, resulting in a four-equation model, to restrain unrealistic self-acceleration of turbidity currents that appeared in the three-equation model. While these depth-averaged models are more computationally efficient and are often applied to large-scale simulations, various empirical relations and/or assumptions are required to close the problem, thus compromising the accuracy of the models. An example is the use of a specified, constant near-bed concentration ratio, *r*_{0} = *c*_{b}/*C*, where *c _{b}* and

*C*are the near-bed and depth-averaged volumetric sediment concentration, respectively. Such an assumption is intuitively inaccurate since the value of

*r*

_{0}should vary at least with the grain size in the flow. More specifically, for the same layer-averaged concentration, an increase in sediment size should result in a higher near-bed concentration as the particles become increasingly difficult to keep in suspension in the flow. Similar arguments can be made in regard to the use of a constant resistant coefficient

*C*when computing the bed shear stress.

_{f}[5] Finding suitable relations for these parameters, however, has been challenging. Field observations in the deep water environment have recently become feasible [*Xu et al*., 2004; *Xu*, 2010] but are nevertheless difficult to obtain due to the unpredictable nature of turbidity currents and the depth at which they often occur. Often, the evidence of an event is only retrieved from the resulting deposit, i.e., turbidites, as observed in surface expression cores or seismics [e.g., *Babonneau et al*., 2002; *Fildani et al*., 2006], or in outcrop [e.g., *Ito and Saito*, 2006]. On the other hand, turbidity currents in the laboratory can be produced and recorded under more controlled conditions. However, due to scale effects, viscous effects tend to be overemphasized in laboratory turbidity currents and hence limitations arise in regard to application of the results at field scale.

[6] Several numerical efforts have attempted to address this issue by using depth-resolving Reynolds-averaged Navier-Stokes (RANS) models to capture certain level of details of turbulence, the vertical structure of the flow, and hence reduce empiricism [e.g., *Huang et al*., 2004; *Choi and Garcia*, 2002]. These models, when coupled with an equation that accounts for mass conservation of the bed, serve as a good tool for understanding and predicting the morphodynamic responses in the deep water environment. For example, *Khan and Imran* [2008] used a one-equation turbulence closure with the Exner equation to investigate the flow characteristics and the filling process of minibasins by turbidity currents on the continental slope; *Abd El-Gawad et al*. [2012a, 2012b] implemented the Mellor-Yamada model at a field scale to examine the flow and depositional pattern in a submarine-meandering channel.

[7] The stratification effects in these depth-resolving RANS models are typically incorporated using a buoyancy production term which, under stably stratified conditions, acts to damp turbulent kinetic energy and reduce mixing. While these models have achieved good results compared to laboratory data and been widely applied to field-scale simulations, the limitations to such an approach in capturing the stratification effects have not been examined in detail. As a result, fine-tuning of some parameters cannot be avoided in general to ensure the success of such models. Moreover, the results obtained from these models are usually problem-specific, varying according to boundary and initial conditions. Hence, they are not readily used to generate physically based rules that enhance our basic knowledge on the nature of the flow or improve modeling efficiency in large-scale simulations.

### 1.2 Turbidity Current With a Roof

[8] The concept of a turbidity current with a roof (TCR) proposed by *Cantero et al*. [2009], hereinafter noted as C09, is designed to provide this basic insight. A recapitulation of their work is given as follows. The configuration for TCR is shown in Figure 1. The tank is filled with fresh water and, in the absence of sediment, the ambient fluid in the channel is still and the local pressure is hydrostatic. Upon the release of the sediment at the upstream end of the channel, gravity drives the sediment to form a turbidity current which travels down the slope. The sediment and water are free to mix, but only within the channel. The roof of the channel prevents further entrainment of the ambient water into the channel, and hence sets a maximum thickness of the flow (i.e., roof height). Under appropriate conditions, the near-bed downward (depositional) sediment flux associated with sediment fall velocity can be fully compensated for by resuspension into the flow by turbulence, yielding zero net sediment deposition on the channel bed.

[9] The flow in the channel is treated as a single-phase fluid where the suspended particles are assumed to follow closely the turbulence movement up to a fall velocity. In reality, this is true only when the size of the particles is small compared to the Kolmogorov scale of turbulence. For larger particles, separation between the particles and the fluid may lead to vortex shedding and enhance turbulence [*Gore and Crowe*, 1991; *Nino and Garcia*, 1998]. In C09 and the present work, however, such effects are neglected regardless of the flow conditions in order to investigate only the role of density stratification on turbulence attenuation.

[10] In C09, the details of a rough boundary are not modeled. The details of the region at the interface between the bed and the flow are replaced with a thin molecular sublayer, in which sediment can be sequestered. This is achieved by introducing an artificial “molecular” diffusivity of sediment concentration. In this configuration, sediment is deposited into, and entrained by turbulence from, this thin “molecular” layer rather than the bed itself.

[11] In such a configuration, the turbidity current in the channel can eventually reach normal (steady and streamwise uniform when averaged over turbulence) conditions either sufficiently far downstream or after a sufficient period of time has passed. By focusing on this normal flow condition, which is made possible by the presence of a roof, we can understand the behavior, and in particular, that associated with sediment stratification effects, of turbidity currents in a way that allows extraction of useful generalities and information/rules. Due to the presence of the roof, TCR does not give a precise model of what may be found in natural turbidity currents. The results of TCR are, however not problem-specific, but rather dependent only on physically based dimensionless parameters.

[12] In this study, we investigate the performance and limitations of RANS models in characterizing turbidity currents under the setting of a TCR. Three of most widely implemented turbulence closures for stratified flows are used here: standard k-ϵ [*Rodi*, 1993], Mellor-Yamada [*Mellor and Yamada*, 1974, 1982] (referred to as M-Y hereinafter), and quasi-equilibrium k-ϵ [*Burchard et al*., 1998] (referred to as QE k-ϵ hereinafter). We examine the capability of these models to reproduce the mean flow and turbulence fields given by the direct numerical simulation (DNS) results from C09. It is found that all three models can capture the stratification effects, but only up to a certain threshold. Two failings common to the RANS models used in this study are identified. First of all, the “fish trap” effect manifested in the sharp density gradient near the velocity maximum separates the sediment concentration into two distinct regions. Such a phenomenon is associated with the overemphasized reduction of the eddy diffusivity caused by the structure of the closures. The artificial sequestration of the sediment in the near-bed region also implies errors when using these models for large-scale flow and morphodynamic predictions.

[13] If the flow is stratified, a portion of the energy is also transferred to and dissipated in the form of the internal waves. Such a process is typically described by wave-wave theory [*Gregg*, 1989; *Müller et al*., 1986], which is fully independent from the classical turbulence energy cascade theory. For weakly stratified shear flows like most natural turbidity currents, turbulence dominates and the presence of internal waves has rarely been considered in the literature as an additional energy sink. Under strongly stratified conditions, however, a treatment of the wave energy has to be incorporated to account for the additional energy loss. This leads to the second failing of the RANS models. It is found that beyond a criterion where flow relaminarization occurs, significant errors arise in all RANS model predictions, due to both inappropriate specification of boundary conditions and the inability of RANS closures to capture the level of turbulence damping. By examining the various length scales associated with turbulence and buoyancy, we relate such failure to the inability of the RANS models to describe energy dissipation due to internal waves.

[14] The three models examined in this study represent a small but representative (due to their popularity) subset of a variety of second-moment turbulence closures. More recently, the standard k-ϵ model has been criticized for a degree of lack of physical soundness, in that it uses small-scale turbulence to determine the macroscale of turbulence [*Mellor and Yamada*, 1982; *Kantha*, 2004]. The M-Y model rectified this by introducing an equation for a macroscale length. In addition to the differences in the length scale equations, the role of stability functions is also crucial. We show that the models which incorporate stability functions in the calculation of the eddy viscosity and eddy diffusivity, such as the M-Y and QE k-ϵ models, outperform the standard k-ϵ model in terms of agreement with the DNS results.

[15] The rest of the paper is laid out as follows. In section 2, we introduce the governing equations and turbulence closures. We also derive and examine the boundary conditions in our models in detail. In section 3, we review some of the flow settings and important findings of C09 which set the foundation for our model comparison. In section 4, numerical methods and model validation using two special cases are presented. Results for Regime I (below the threshold for the failure of RANS models) and Regime II flow conditions (above this threshold, as defined in section 3) are presented and discussed in sections 5 and 6, respectively. In section 7, we apply the model to examine the sensitivity in flow characteristics to variation in two important dimensionless parameters, the dimensionless settling velocity and shear Richardson number. The effect of multiple-grain sizes on the mean flow and turbulent field is also examined. A summary and the conclusions of this study are given in section 8.