Plane beds develop under flows in fluvial and marine environments; they are recorded as parallel lamination in sandstone beds, such as those found in turbidites. However, whereas turbidites typically exhibit parallel lamination, they rarely feature dune-scale cross-lamination. Although the reason for the scarcity of dune-scale cross-lamination in turbidites is still debated, the formation of dunes may be dampened by suspended loads. Here, we perform, for the first time, linear-stability analysis to show that flows with suspended loads facilitate the formation of plane beds. For a fine-grained bed, a suspended load can promote the formation of plane beds and dampen the formation of dunes. These results of theoretical analysis were verified with observational data of plane beds under open-channel flows. Our theoretical analysis found that suspended loads promote the formation of plane beds, which suggests that the development of dunes under turbidity currents is suppressed by the presence of suspended loads.

The interactions between fluids and erodible surfaces generate small-scale topographic features called bedforms on both terrestrial surfaces (e.g., riverbeds, deserts, and deep-sea floors) and extra-terrestrial surfaces

Although the reason for the paucity of dune-scale cross-lamination in turbidites is still debated

The relationships between sediment transport modes and the formation of plane beds have received little attention in theoretical works that performed linear-stability analyses. The reason could be that previous studies have succeeded in predicting the wavelength of dunes and antidunes without considering suspended loads

Therefore, in order to investigate the effect of sediment transport mode on the formation of plane beds, we performed a linear-stability analysis of bedforms under open-channel flows carrying suspended loads. The model introduced in

Linear-stability analysis of fluvial bedforms can provide the wavelengths of perturbations (i.e., bed waves) that grow over time

The instability of a system is illustrated as a contour diagram of the perturbation growth rate

Contour map of perturbation growth rate

Stability diagrams described on the

Although the classic

The instability of a system is illustrated as a contour diagram of the perturbation growth rate

Contour maps of the maximum growth rate

Contour maps of the maximum growth rate

The domain

Here we present the formulation of the problem and the method used to solve the differential equations.

The governing equations for flows are the two-dimensional Reynolds-averaged Navier–Stokes equations.
On erodible beds, the flow adjustments occur immediately relative to the bed adjustments

Under the quasi-steady assumption, the dimensionless forms of the Reynolds-averaged Navier–Stokes equations and continuity equation for incompressible flow are described as

We employ a Boussinesq-type assumption to close the flow equations:

In the above equations, the system is non-dimensionalized as follows:

As the flow is continuous, the system can be rewritten using the stream function

Then, Eqs. (

Eliminating

We also assume a quasi-steady state for the advection–diffusion equation for suspended sediment, which is formulated as

Here,

The settling velocity of sediment

The particle Reynolds number

We employ the following transformation of variables to apply the boundary condition at the bed and flow surfaces:

The derivatives with respect to

Additionally, the dimensionless mixing length

Since

The boundary conditions include a vanishing flow component normal to the water surface and vanishing stresses normal and tangential to the water surface as follows:

At the bed, the boundary conditions include the vanishing flow components normal and tangential to the bed.

The boundary conditions for the suspended-sediment flux at the flow surface and bed are as follows:

The basic flow state for linear-stability analysis is a uniform flow over a flat bed. Under this condition, the hydraulic parameters

With Eqs. (

Then, the friction coefficient

Now, we consider the logarithmic law of the open-channel flows as

Under the above uniform-flow condition over a flat bed, Eq. (

Here,

By integrating Eq. (

The development of the bed configuration can be described by the Exner equation considering the suspended load as follows:

Equation (

In this study, dimensionless bed-load discharge per unit width is estimated using the Meyer-Peter and Müller formula modified as described in

In this study, the sediment transport regimes are classified using the threshold conditions of sediment motion in

The coefficients

We impose an infinitesimal perturbation on the basic state. Then, with the use of boundary conditions, we can solve the differential equations to get the growth rate of the perturbation. Please see the Appendix for details of the linear analysis.

The stability diagrams were assessed using an observational dataset pertaining to open-channel flows compiled from the literature, as summarized in Tables

We used the data of plane beds in which the sediment transport mode could be identified, i.e., plane bed with suspension. We identified whether sediment particles were transported as suspended loads or not based on the suspended-sediment concentration. For comparison with the theoretical-analysis results, we used the data of dunes and antidunes with wavenumbers in the range of

To calculate the particle Reynolds number, the kinematic viscosity

The contour maps of

The phase diagrams for the case of the stability analysis with suspension show that a stable region appears at

Comparing the results of theoretical analysis and the observational data, all the plane bed data are within unstable regions in the case without suspension (Fig.

Error rates for the case of fixed

As expected, most dune and antidune data plot in the unstable region, whereas several data points of dunes and antidunes plot in the stable region in both cases with and without suspension (Fig.

The contour maps of the maximum growth rate on the

Error rates for the case of fixed

The role of suspended loads in the formation of plane beds and suppression of dune-scale instabilities is quantitatively illustrated as the broadening of the stable regions (Figs.

We found that dunes are deformed under flows with suspended loads, although further work is needed to investigate the amplitudes of dunes under such conditions. Field surveys have indicated the existence of low-angle dunes in suspended-load-dominated rivers

Ultimately, our linear analyses provide a possible explanation for the absence of dunes in turbidites: suspended loads suppress dune formation and facilitate plane bed formation. Previous research has suggested that the formation of dunes is suppressed due to the insufficient time for dune development

We investigated the influence of suspended loads on the formation of plane beds under open-channel flows. The stability diagrams show that the stable region for finer sediments is wider in the diagram with suspension than that without suspension. Further, the published data of plane beds with suspension coincide well with the stability diagrams where the suspension was considered. Our theoretical analysis found that suspended loads promote the formation of plane beds and suppress the formation of dunes on the fine-grained bed. These results suggest that dune-scale cross-lamination is absent in turbidites because the development of dunes in turbidity currents is restricted by the presence of suspended loads. Additional theoretical work can be improved in future studies by the inclusion of possible mechanisms for the absence of dunes in turbidites.

In Sect.

Conceptual diagram of the flow. The dimensionless parameters

Conceptual diagram of the sediment bed. The origin of the

Contour map of perturbation growth rate

Contour map of perturbation growth rate

Summary of data used for the stability diagram with

Summary of data used for the stability diagram with

Summary of data used for the stability diagram with

Summary of data used for the stability diagram with

Summary of data used for the stability diagram with

We impose an infinitesimal perturbation on the basic state. All the variables are modified using a small amplitude

The subscript 1 denotes a variable at

Here,

Additionally, Eqs. (

We employ a spectral collocation method using Chebyshev polynomials to solve the above differential equations. We expand

The above functions are substituted into Eq. (

By combining the governing equations, boundary conditions, and closure assumptions, we obtain the following system of linear algebraic equations:

Additionally, Eqs. (

Similarly, we solve the eigenvalue problems for the sediment transport equations. By substituting Eq. (

Based on Eqs. (

The boundary conditions give

Here,

We expand

The system is evaluated at the Gauss–Labatte points, and then we obtain

The coefficient

Therefore, the following equation is obtained:

By substituting Eqs. (

Here, using

Thus, we can obtain the growth rate

The datasets and codes used for this study can be found at

KO and NI performed the linear-stability analysis. HN and NI contributed to the interpretation of the results. KO wrote the manuscript and prepared the figures, and then HN and NI provided feedback on the manuscript and figures.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to express our gratitude to Robert Dorrell for his comments. We are thankful to the anonymous referees for their insightful comments on earlier versions of the manuscript.

This research has been supported by the Japan Society for the Promotion of Science (grant no. 18J22211 and 20H01985).

This paper was edited by Daniel Parsons and reviewed by three anonymous referees.