Three-Dimensional Wave-Induced Current Model Equations and Radiation Stresses

After the approach by Mellor(2003, 2008), the present paper reports on a repeated effort to derive the equations for three-dimensional wave-induced current. Via the vertical momentum equation and a proper coordinate transformation, the phase-averaged wave dynamic pressure is well treated, and a continuous and depth-dependent radiation stress tensor, rather than the controversial delta Dirac function at the surface shown in Mellor(2008), is provided. Besides, a phase-averaged vertical momentum flux over a sloping bottom is introduced. All the inconsistencies in Mellor(2003, 2008), pointed out by Ardhuin et al.(2008) and Bennis and Ardhuin(2011), are overcome in the presently revised equations. In a test case with a sloping sea bed, as shown in Ardhuin et al.(2008),the wave-driving forces derived in the present equations are in good balance, and no spurious vertical circulation occurs outside the surf zone, indicating that Airy's wave theory and the approach of Mellor(2003, 2008) are applicable for the derivation of the wave-induced current model.