Wave- and Wind-Driven Flow in Water of Finite Depth

Thickness-Weighted Mean Theory for the Effect of Surface Gravity Waves on Mean Flows in the Upper Ocean

“…Their solution nevertheless differs from that of some previous authors, notably Madsen 51 (1978) and Xu and Bowen (1994), who include an additional surface stress that they associate with the 52 so-called virtual wave stress (VWS) of Longuet-Higgins (1953, 1960. So far, the most rigorous framework 53…”

“…Interestingly, although the surface 540 boundary condition used by Xu and Bowen (1994) corresponds to our case of no variation of the tangential 541 stress, corresponding to step (iv) in our derivation (see the previous subsection), these authors appear to 542 have arrived at (iv) without apparently using steps (i)-(iii) (they do not take explicit account of the thin 543 viscous boundary layer). The large difference between the different solutions is apparent from Figure 3 544 where the left panel shows the solution of Huang (1979) and Polton et al (2005) for two different wave 545 amplitudes, while the right panel shows to the corresponding solution of Xu and Bowen (1994) for the 546 same two wave amplitudes. Clearly, the different surface boundary conditions applied to the waves can 547 have a big effect on the resulting EM velocity, not only at the surface but throughout the water column.…”

“…this to (73a)-(73e) yields stress tensor in Cartesian coordinates to P ss and P sn acting on the s-component of velocity, and P ns and 724 P nn acting on the n-component of velocity, The formula of LH69 has been adopted in Xu and Bowen (1994), Piedra-Cueva (1995), Ng (2004), and the present study (see our Eq. 76).…”

“…The boundary condition of Xu and Bowen (1994) is (u c 2z , v c 2z )| z=−δ = 538 (τ 2 /(ρν)+2σk 2 a 2 , 0), which corresponds to our boundary condition for the TWM velocity, ( u 2z , v 2z )| z=−δ = 539 (τ 2 /(ρν)+ 4σk 2 a 2 , 0) in the case of no variation in the tangential stress. Interestingly, although the surface 540 boundary condition used by Xu and Bowen (1994) corresponds to our case of no variation of the tangential 541 stress, corresponding to step (iv) in our derivation (see the previous subsection), these authors appear to 542 have arrived at (iv) without apparently using steps (i)-(iii) (they do not take explicit account of the thin 543 viscous boundary layer).…”

“…applying a boundary condition of no pressure disturbance at the free surface to the waves (implying no 590 form stress) whereas the solutions of Madsen (1978) and Xu and Bowen (1994) correspond to applying 591 a condition of no variations in the tangential component of surface stress to the waves. In this second 592…”