Is it appropriate to model turbidity currents with the three-equation model?

TL;DR

This paper analytically and numerically examines the validity of the three-equation model (TEM) for turbidity currents, demonstrating its consistency under certain conditions and comparing it favorably to the four-equation model (FEM).

Contribution

The study provides an analytical proof that the steady TEM does not violate energy balance if the bed drag coefficient is realistic, and introduces a novel method for setting boundary conditions.

Findings

Self-accelerating TCs in TEM do not violate energy balance with realistic bed drag.

Stronger bed erosion increases turbulent kinetic energy due to potential energy conversion.

TEM can produce self-accelerating flows under supercritical conditions with proper boundary values.

Abstract

The three-equation model (TEM) was developed in the 1980s to model turbidity currents (TCs) and has been widely used ever since. However, its physical justification was questioned because self-accelerating TCs simulated with the steady TEM seemed to violate the turbulent kinetic energy balance. This violation was considered as a result of very strong sediment erosion that consumes more turbulent kinetic energy than is produced. To confine bed erosion and thus remedy this issue, the four-equation model (FEM) was introduced by assuming a proportionality between the bed shear stress and the turbulent kinetic energy. Here we analytically proof that self-accelerating TCs simulated with the original steady TEM actually never violate the turbulent kinetic energy balance, provided that the bed drag coefficient is not unrealistically low. We find that stronger bed erosion, surprisingly, leads to more production of turbulent kinetic energy due to conversion of potential energy of eroded material into kinetic energy of the current. Furthermore, we analytically show that, for asymptotically supercritical flow conditions, the original steady TEM always produces self-accelerating TCs if the upstream boundary conditions ("ignition" values) are chosen appropriately, while it never does so for asymptotically subcritical flow conditions. We numerically show that our novel method to obtain the ignition values even works for Richardson numbers very near to unity. Our study also includes a comparison of the TEM and FEM closures for the bed shear stress to simulation data of a coupled Large Eddy and Discrete Element Model of sediment transport in water, which suggests that the TEM closure might be more realistic than the FEM closure.