  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.
