Cyclic steps: A phenomenon of supercritical shallow flow from the high mountains to the bottom of the ocean

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Abstract

Cyclic steps constitute a characteristic bedform of Froude-supercritical shallow flow over an erodible bed. They are long-wave features that are bounded by hydraulic jumps and migrate upstream. They can be seen in alluvial streams, stream in cohesive sediment, bedrock streams, and on the seafloor in response to turbidity currents. Recent progress in the modeling of cyclic steps is summarized.

Introduction

Shallow Froude-supercritical flow over an erodible bed is generally unstable. In the case of freelyerodible, noncohesive alluvium, this instability results in the formation of bedforms that interact with the flow. Perhaps the most common kind of such bedforms is the antidune. Antidunes are short-wave bedforms that interact with the flow such that undulations on the water surface are nearly in phase with those on the bed. Although antidunes can migrate downstream (e.g. Carling and Shvidchenko, 2002), they are most commonly observed to migrate upstream.

Upstream-migrating antidunes tend to be ephemeral in that they form within localized trains, grow in amplitude, and finally collapse like a row of dominos, releasing a bore in the process. The process of growth and collapse is then repeated elsewhere.

There is, however, a long-wave manifestation of the instability, which Parker (1996) has called “cyclic steps.” These steps consist of a train of upstream-migrating bed undulations bounded by hydraulic jumps. The presence of the hydraulic jumps stabilizes the morphodynamics of the flow, so that a pattern of permanent or quasi-permanent form can migrate upstream. An example of such a train of steps in loose alluvium is shown in Fig. 1.

The structure of cyclic steps is schematized in Fig. 2. The flow on the upstream side of a step is Froude-subcritical, and on the downstream side of the step it is Froude-supercritical. The flow is bounded at either end of the step by hydraulic jumps. In the relatively slow-moving zone of subcritical flow, sediment deposition is favored, whereas sediment erosion is favored in the relatively swift zone of supercritical flow.

Here it is shown that cyclic steps can develop under a wide variety of settings, including openchannel flow over alluvium, cohesive soil and bedrock, and the flow of submarine turbidity currents festations is the presence of shallow flow that would be Froude-supercritical in the absence of the undulations. In the case of deep-sea turbidity currents, the relevant Froude number is the densimetric Froude number.

Section snippets

Common framework for subaerial and subaqueous (submarine) cyclic steps

In the work presented here, it is necessary to distinguish between shallow flow in the subaerial and submarine setting. Here “subaerial” refers to water flowing over air, as shown in Fig. 1. The relevant Froude number Fr isFr=UgHwhere h denotes flow depth, U denotes a depthaveraged flow velocity and g denotes gravitational acceleration. Subaerial flows are thus openchannel flows, or more specifically river flows. Such flows are driven by the action of gravity pulling the water downslope. The

Cyclic steps in alluvium

Perhaps the most familiar case for which cyclic steps are manifested consists of subaerial flow over a noncohesive, alluvial bed. Steps of this form are shown in Fig. 1.

The history of their identification as a distinct bedform is somewhat obscure, but they seem to be identifiable with the poorly-documented “chute-and-pool” morphology described in Simons et al. (1965). Fukuoka et al. (1982), on the other, hand, describe a wide range of bedforms in supercritical flow, one of them identifiable as

Submarine manifestations of cyclic steps

Turbidity currents constitute a major mechanism for delivering sediment from the shoreline to the deep sea. The seafloor exhibits many examples of rhythmic structures which are likely to be due to a morphodymamic interaction between flowing turbidity currents and the bed over which they flow.

Turbidity currents on submarine fans often form well-defined leveed channels. The levees that bound a channel are typically 10's to 100's of meters high, and can often have more relief than the channel

Conclusions

Cyclic steps represent a characteristic bedform that arises in response to Froude-supercritical flow. The steps are bounded by hydraulic jumps, and migrate upstream in cohort. They can occur in alluvial rivers, channels in cohesive sediment, bedrock, channels and on the seafloor in response to the supercritical flow of turbidity currents. They thus can be observed over a range from the slopes of mountains to the bottom of the sea. They can be successfully modeled in the laboratory. In addition,

Acknowledgements

This research has been supported by ExxonMobil. It is dedicated to the memory of William H. Normark and Bruno Savoye.

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