Mixing in an Internal Wave Bottom Boundary Layer: A Simple Model Predicts Entrainment above a Sloping Bed during the Stabilizing Phase of Strain-Induced Periodic Stratification

Abstract Mixed layer models have been applied to diverse environments, yet a complete parameterization does not exist for cases with significant boundary-parallel density gradients, where strain-induced periodic stratification (SIPS) influences mixed layer thickness. We developed a parameterization for entrainment into an internal wave bottom boundary layer above a sloping bed. This parameterization accounts for two turbulent buoyancy fluxes. The first flux ( F e ) is proportional to entrainment, whereas the second ( F s ) redistributes buoyancy as required to maintain a mixed layer in the presence of stabilizing SIPS. The turbulent energy loss associated with F s reduces entrainment, with reduction proportional to the ratio h / L MO , where h is the mixed layer thickness and L MO is a Monin–Obukhov length defined using F s . This ratio resembles the Simpson number often used to analyze SIPS in estuaries. Entrainment is almost unaffected by F s when h / L MO ≪ 1, whereas entrainment halts when h / L MO ∼ 1. The entrainment parameterization was used to close boundary layer–averaged equations of momentum and buoyancy. The resulting model accurately described the thickening, velocity, and buoyancy of mixed layers observed above a sloping lakebed. Accounting for F s improved predictions. Therefore, the simple parameterization improves model skill and provides a scaling quantifying the importance of SIPS to near-bed mixing. Significance Statement Mixing near sloping topography is important for the transport of heat, chemicals, and organisms in oceans and lakes. For example, near-bed mixing may influence mean vertical circulation, even very far from ocean or lake boundaries. Through the process of “strain-induced periodic stratification” (SIPS), oscillating upslope and downslope flows associated with internal waves and tides cause major fluctuations in near-bed stratification and mixing. This study derives a very simple model that conserves momentum and buoyancy, while accounting for SIPS. The model accurately described the thickening, velocity, and buoyancy of a mixed layer observed above a sloping lakebed. Parameterization of SIPS improves model skill and provides a scaling determining the importance of SIPS to near-bed mixing.