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A one-dimensional long-wave model of an unsteady three-layer flow of astratified fluid under a lid is proposed, taking into account turbulent mixingin the intermediate layer. In the Boussinesq approximation, the equations ofmotion are reduced to an evolutionary system of balance laws, which ishyperbolic for a small difference in velocities in the layers. Classes ofstationary solutions are studied and the concept of subcritical (supercritical)three-layer stratified flow is introduced. Oscillating solutions areconstructed that describe the spatial evolution of the mixing layer. Theproblem of transcritical flow over an obstacle is considered. Solutions areobtained that describe qualitatively different flow regimes on the leeward sideof the obstacle. The proposed model is validated using experimental data andfield observations on the entrainment of ambient fluid.

Long nonlinear internal waves and mixing in a three-layer stratified flow in the Boussinesq approximation