On residual velocities in sigma coordinates in narrow tidal channels

In shallow coastal regions where water surface fluctuations are non-negligible compared to the mean water depth, the use of sigma coordinates allows the calculation of residual velocity around the mean water surface level. Theoretical analysis and generic numerical experiments were conducted to understand the physical meaning of the residual velocities at sigma layers in breadth-averaged tidal channels. For shallow water waves, the sigma layers coincide with the water wave surfaces within the water column such that the Stokes velocity and its vertical and horizontal components can be expressed in discrete forms using the sigma velocity. The residual velocity at a sigma layer is the sum of the Eulerian velocity and the vertical component of the Stokes velocity at the mean depth of the sigma layer and, therefore, can be referred to as a semi-Lagrangian residual velocity. Because the vertical component of the Stokes velocity is one order of magnitude smaller than the horizontal component, the sigma residual velocity approximates the Eulerian residual velocity. The residual transport velocity at a sigma layer is the sum of the sigma residual velocity and the horizontal component of the Stokes velocity and approximates the Lagrangian residual velocity in magnitude and direction, but the two residual velocities are not conceptually the same.