Time Series Water Wave Modeling using a Modified Euler Momentum Conservation Equation and a Momentum Equilibrium Equation

From an energy perspective, the water-particle velocity equations derived from the Euler or Navier Stokes momentum conservation formulations show an imbalance: kinetic energy changes occur through variations in velocity, but no corresponding source of potential energy is included. To supply this missing potential energy, a Momentum Equilibrium Equation is introduced, and its horizontal velocity is superimposed on the velocity from the momentum conservation equation. Three modifications are applied to the Euler momentum equation: reducing the driving force, applying a weighted Taylor series to the total acceleration, and treating the convective acceleration as a hydrodynamic force. Model results on a sloping bottom show that the approach can simulate shoaling and breaking effectively, pass through the breaking phase, and continue into shallow water.