On the integral quantities of a low amplitude solitary wave shoaling on a mild slope

• Integral measures have been investigated for solitary wave propagating over a shoaling region.
• The decrease in horizontal momentum is well predicted by the weight of the wave over the shoaling region.
• The momentum flux parameter shows a minimum over the shoaling region.

A weakly viscous 2D numerical simulation is carried out to investigate the momentum, momentum flux parameter, energy, circulation and angular momentum of a low amplitude solitary wave propagating over uniform depth and then shoaling on a mild slope. The integral measures are studied in detail by analysing separately the inviscid and viscous contributions to their evolution. In the shoaling region, the horizontal momentum is significantly reduced by the horizontal dynamic pressure component caused by the weight of the wave over the shoaling region. The potential/kinetic energy and the circulation are modified by the action of the shear stress on the rigid boundary. For angular momentum, the inviscid contributions of the moment of the pressure and the weight are dominant. It is the resultant torque which acts on the wave which results in the high vertical gradient in the horizontal velocity and in the distinctive overturning shape of a wave close to breaking.