This work aims to develop a new framework for the interaction of a subsurface flow and surface gravity water waves, based on a perturbation and multiple-scales expansion. Surface waves are assumed of a narrow band δ (δ ), indicating they can be expressed as a carrier wave whose amplitude varies slowly in space and time relative to its phase. Using the Direct Integration Method proposed in Li & Ellingsen (2019), the effects of the vertical gradient of a subsurface flow are taken into account on the linear wave properties in an implicit fashion. At the second order in wave steepness ϵ, the forcing of the sub-harmonic bound waves is considered that plays a role in the primary equations for a subsurface flow.
The novel framework derives the continuity and momentum equations for a subsurface flow in two different formats, including both the depth integrated as well as the depth resolved version. The former compares with Smith (2006) to examine the roles of the rotationality of wave motions in the subsurface flow equations. The latter employs the sigma coordinate system proposed in Mellor (2003, 2008, 2015) and extends the framework therein to allow for quasi-monochromatic surface waves and the effects of the shear of a current on linear surface waves. Compared to Mellor (2003, 2008, 2015), the vertical flux/vertical radiation stress term in the proposed framework is approximated to one order of magnitude higher, i.e. O(ϵ2δ2).
Li, Y., Ellingsen, S. Å. A framework for modeling linear surface waves on shear currents in slowly varying waters. J. Geophys. Res. C: Oceans, (2019) 124(4), 2527-2545.
Mellor, G. L. The three-dimensional current and surface wave equations. J. Phys. Oceanogr., (2003) 33, 1978–1989.
Mellor, G. L. The depth-dependent current and wave interaction equations: a revision. J. Phys. Oceanogr., (2008) 38(11), 2587-2596.
Smith, J. A. Wave–current interactions in finite depth. Journal of Physical Oceanography, (2006) 36(7), 1403-1419.