The turbulent stress components of the WOS and the WS cases are further examined in Figs. 5(a) and 5(b), respectively. The stresses are presented normalized by their corresponding turbulent kinetic energy. As it can be seen in the figure, the components of the stress tensor in the WS case are closer to each other than the WOS case, which implies that the presence of swell increases the turbulence isotropy slightly. To examine the turbulence isotropy of the two cases, the anisotropy invariant map of the kinematic shear stresses is plotted in the turbulence triangle [27] in Fig. 5(c). The stress components of the WOS case are mainly located to the right limit of the triangle which indicates that the turbulence in the WOS case has an axisymmetric (prolate) shape with one large eigenvalue. In the WS case, the turbulence near the surface has a spherical isotropic shape, where the eigenvalues of the normalized stress tensor are almost zero. Above the surface, the turbulence has an axisymmetric (prolate) shape, similar to the WOS case, with one large eigenvalue. The first invariant (η) of the normalized stress tensor decreases, while the second invariant (ς) increases with the height up to about the height of the velocity maximum (25 m above the surface), which means that the turbulence goes more toward the two-component axisymmetric (the base of the triangle) near the velocity maximum. Further above the velocity maximum, the turbulence returns gradually to the spherical isotropic shape following the axisymmetric (prolate) line. There are regions of alternating increasing/decreasing stress magnitudes in the WS case in Fig. 5(b), while they generally decrease with the height in the WOS case in Fig. 5(a) (except for the *vv* component which increases due to the spiral profile of the Ekman layer). More interesting is the presence of a region of positive streamwise shear stress (*uw*) in the WS case. This positive shear stress region is believed to be caused by the swell-induced stress τ_{swell}, which acts against the turbulent shear stress [2,3,12]. To test this hypothesis, the shear stress (*uw*) is further decomposed into viscous, turbulence (filtered and SGS), and swell components. The viscous term is neglected here due to the high Reynolds number. The other components are displayed for the lower part of the MABL of the WS case in Fig. 6. The partition of the total stress is noticeably different from the one of the MABL over short sea waves (see Fig. 2(b)). The total shear stress and its components τ_{gs} and τ_{sgs} of the WS case are much lower than the counterpart stresses of the WOS case. Both the filtered turbulent shear stress τ_{gs} and the swell-induced stress τ_{swell} of the WS case have their maximum at the surface, and they are almost equal in magnitude. Therefore, the total shear stress at the surface is very small, and it is mainly due to the subgrid scale stress. The low shear stress at the surface results in a low velocity gradient above the surface. The reduced velocity gradient produces less turbulence, and therefore the τ_{gs} also reduces. The τ_{swell} decays exponentially above the surface, and it acts against the result of τ_{gs} and τ_{sgs}. At some height above the surface, the τ_{swell} balances the turbulent stress, and the total stress becomes zero. Above this height, the turbulence stress becomes even lower than the swell-induced stress, and the total stress changes its sign (i.e., becomes positive, upward momentum transfer). This upward momentum accelerates the wind to a velocity higher than its free stream velocity and results in so-called supergeostrophic wind jet that can be seen in the velocity profile.