BED-SEDIMENT ENTRAINMENT BY DEBRIS FLOWS: JUMP CONDITIONS THAT CONSTRAIN DEPTH-INTEGRATED MODELS

Growing use of depth-integrated equations to model the dynamics of debris flows that entrain bed material motivates a review and reformulation of the underlying theory. Conservation of mass and momentum in a two-layer continuum consisting of a moving upper layer and static lower layer requires that erosion or deposition rates at the interface between layers must in general satisfy three jump conditions. These conditions impose constraints on valid erosion formulas, and they help determine the correct forms of conservation equations used to simulate flow dynamics. The jump conditions can be expressed compactly in a normalized form that uses the bed-parallel velocity of entrained material as the velocity scale: -∂z/∂t = τ-(σ/τ), w = σ/τ, and ρ = (τ²/σ)-1. Here -∂z/∂t represents the nondimensional bed-normal erosion rate, w represents the nondimensional bed-normal velocity of material undergoing entrainment, ρ represents the nondimensional contrast in bulk density between static and entrained material, and τ and σ represent jumps in nondimensional shear traction and normal traction, respectively, across the eroding interface. Additionally, τ can be interpreted as the nondimensional excess boundary shear traction, and w can be interpreted as the dilatancy of bed sediment undergoing entrainment. Many models assume that w = 0, implying that bed material moves exclusively in the depth-averaged flow direction as it is entrained. In this case the only applicable jump condition is -∂z/∂t = τ. Even for this simple case, however, precise formulation of depth-integrated momentum-conservation equations requires a clear distinction between boundary shear tractions that exist in the presence or absence of bed erosion. Neglect of this distinction leads to spurious inferences about the role of momentum exchange between moving flows and static bed material. Moreover, because valid erosion-rate formulas used in depth-integrated models must be consistent with the momentum jump equation, -∂z/∂t = τ-(σ/τ), some widely used empirical formulas are untenable.