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The flow of water in a tide, flood or tsunami is often turbulent. We model large scale, but shallow, turbulent flow using the well established $k$-$\omega$ model of turbulence. Vertical turbulent mixing underlies the existence of a slow manifold model. As a first step, in this article the flow is assumed laterally homogeneous. Then constructing the slow manifold discovers the evolution of the average lateral velocity and average turbulent energy. We focus upon the influence on the mathematical analysis of key physical factors affecting the dynamics: turbulent mixing, energy production due to shear, volume energy dissipation and gravitational forcing on sloping ground. Further research will incorporate large scale lateral variations in the flow in order to predict tides, floods and tsunamis.

The dynamics of the vertical structure of turbulence in flood flows