a. LES setup

We use an LES methodology of turbulent airflow over surface waves that is identical to previous studies (Sullivan et al. 2014; Hara and Sullivan 2015; Sullivan et al. 2018; Husain et al. 2019; Part I), all of which use pressure-driven channel flow over a wavy surface propagating through a domain with doubly periodic horizontal boundary conditions and a free-slip flat-top boundary. Here, we define time t, the along-wave x coordinate, the cross-wave y coordinate, and the vertical coordinate z pointing upward in the positive direction with z = 0 at the mean water surface. Velocities (u, υ, w) are in the (x, y, z) directions.

We consider a monochromatic wave train propagating in the x direction ($\partial /\partial y=0$) with h(x, t) = a cos(kx − ωt), where a is the wave amplitude, k is the wavenumber, ω is the angular frequency, and $c=\omega /k=\sqrt{g/k}$, identical to Part I. In LES, wind is driven by imposed horizontal pressure gradient $\nabla p=\left(\partial p/\partial x,\partial p/\partial y\right)$, which is balanced by the surface wind stress $\left|{\tau }_s\right|=u_{*s}^2=\left|\nabla p\right|l_{\zeta }$, where l_ζ is the domain height. The total wind stress vector θ is always pointed in the direction of −∇p, the angle of the external negative pressure gradient. Since we apply the free slip condition at the top boundary, the total wind stress linearly decreases from the wavy surface to the top.

For the actual LES, the waves always propagate in the positive x direction and the pressure gradient force (i.e., negative pressure gradient) is applied in the directions of 0°, 22.5°, 45°, 67.5°, 180°, 202.5°, 225°, and 247.5° (measured from the x axis in the counterclockwise direction; see Fig. 1, left panel). However, for the data analysis and discussion, the wave direction for the last two cases is reversed. Therefore, for the eight wind-wave conditions examined in this study, the wave directions are (0°, 0°, 0°, 0°, 180°, 180°, 180°, 180°) and the wind stress directions θ are (0°, 22.5°, 45°, 67.5°, 0°, 22.5°, 45°, 67.5°), respectively (see Fig. 1, right panel).

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Schematic of wind stress, wave phase speed, and surface wave elevation in the ξ–y coordinate for (left) the LES simulation and (right) data analysis. For simplicity, only the cases of θ = 0° (solid) and 45° (dashed) are shown. In LES, the direction of wave phase speed (c) is from left to right both for waves following wind (orange) and opposing wind (blue) with opposing wind stress directions determined by an external pressure gradient. In our data analysis, we present waves opposing wind (blue) with c directed from right to left and the wind stress in the same direction as waves following wind (orange).

Citation: Journal of Physical Oceanography 52, 1; 10.1175/JPO-D-21-0044.1

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The subgrid and surface stress parameterizations are identical to those in Part I. The background surface roughness z_ob along the resolved wavy surface is set at a constant kz_ob = 2.7 × 10⁻³, which accounts for the form drag of unresolved waves and the surface viscous stress. As in Part I we focus on a steep wave train of ak = 0.27. The wind forcing is set at $\left|c\right|/u_{*}=1.4$ because we are mainly interested in strongly forced waves that support the bulk of the air–sea momentum flux (Donelan et al. 2012; Reichl et al. 2014).

Each simulation is run for approximately 100 000 time steps and averaged over the last 20 000 time steps after the wind field has reached a statistically steady state. Sullivan et al. (2014) and Sullivan et al. (2018) provide a full description of the LES algorithm and numerical methods used to solve the governing equations.

b. Data analysis

Using the triple decomposition (separating all relevant variables ψ into phase average $\overline{\psi }$ and turbulent ψ′ components, and separating $\overline{\psi }$ into a horizontal mean 〈ψ〉 and wave coherent component $\tilde{\psi }$; see Hara and Sullivan 2015; Buckley and Veron 2016; Husain et al. 2019; Part I), we can analyze the partition of the wind stress (momentum budget) in terms of turbulent and wave-coherent stresses as well as pressure stress in the along-wave direction. The addition of oblique wind adds a cross-wave (y) momentum budget to the along-wave (x) budget. The momentum budget is discussed in more detail in section 3d.

a. 2D phase-averaged airflow

In this section, all flow fields presented are normalized by $u_{*s}$ and k. In Figs. 2 (except the rightmost column), 3, 4, and 5, we display the two-dimensional phase-averaged flow fields (phase-average denoted by an overbar). In the top (bottom) two rows waves propagate from left to right (from right to left) in the positive (negative) x direction. In the first and third rows wind blows from left to right in the positive x direction. In the second and fourth rows the pressure gradient force and the resulting wind stress are in the direction rotated by 45° from the positive x direction. Consequently, the wind direction is also positively rotated from the x direction (by more than 45°, as discussed later). All the phase averaged flow fields are presented in the coordinate moving with the wave so that the flow fields are independent of t.

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Normalized phase-averaged flow fields in the ξ–z coordinate (top two rows) for waves following wind (oblique wind in second row) and (bottom two rows) for waves opposing wind (oblique wind in fourth row). From left to right: along-wave velocity [$\left(\overline{u}-c\right)/u_{*s}$], cross-wave velocity ($\overline{\upsilon }/u_{*s}$), vertical velocity ($\overline{w}/u_{*s}$), and pressure ($\overline{p}/u_{*s}^2$). Rightmost plots show the surface stress distribution for the normal stress $\overline{{\tau }_n}/u_{*s}^2$ (panels labeled A and C; solid line is total normal stress and dotted line is pressure only) and for the tangential stress $\overline{{\tau }_t}/u_{*s}^2$ in the x direction (panels labeled B and D) for waves (top two panels) following wind (orange lines) and (bottom two panels) opposing wind in aligned (thick lines) and oblique (thin lines) wind conditions.

Citation: Journal of Physical Oceanography 52, 1; 10.1175/JPO-D-21-0044.1

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Figure 2 includes the streamwise velocity [$\left(\overline{u}-c\right)/u_{*s}$], spanwise velocity [$\left(\overline{\upsilon }\right)/u_{*s}$], vertical velocity ($\overline{w}/u_{*s}$), and pressure ($\overline{p}/u_{*s}^2$) plotted in rectangular (ξ–z) coordinates. In the rightmost column, the surface stress distribution is plotted for the total normal stress (pressure plus the turbulent normal stress), pressure only, and the along-wave turbulent tangential stress. Figure 3 includes the TKE magnitude ($\overline{e}/u_{*s}^2$), the dissipation rate [$\epsilon /\left(k{u}_{*s}^3\right)$], and the horizontal vorticity magnitude $[\overline{{\omega }_h}/\left(k{u}_{*s}\right)=\sqrt{{\overline{{\omega }_x}}^2+{\overline{{\omega }_y}}^2}/\left(k{u}_{*s}\right)]$. Here, $\overline{{\omega }_x}=-\left(\partial \overline{\upsilon }/\partial z\right)+\left(\partial \overline{w}/\partial y\right)$ and $\overline{{\omega }_y}=\left(\partial \overline{u}/\partial z\right)-\left(\partial \overline{w}/\partial x\right)$ are dominated by the vertical shear of $\overline{\upsilon }$ and $\overline{u}$, respectively. Therefore, $\overline{{\omega }_h}$ is dominated by the vertical shear of oblique horizontal wind. Figure 4 shows the horizontal vorticity magnitude (same as Fig. 3) and the two components of the horizontal vorticity, $\overline{{\omega }_y}/\left(k{u}_{*s}\right)$ and $-\overline{{\omega }_x}/\left(k{u}_{*s}\right)$. Figure 5 includes the three components of the TKE. All quantities in Figs. 3–5 are plotted in the rectangular (ξ–z) coordinate as well as in the mapped (ξ–ζ) coordinate with the vertical axis in a log scale so that the flow fields very close to the wavy surface are magnified. Note that the results for waves following and opposing wind (first and third rows) are almost identical to the results presented in Part I for $c/u_{*s}=\pm 1.4$.

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(left three columns) Normalized phase-averaged flow fields in the ξ–z coordinate and (right three columns) the mapped ξ–ζ coordinate (top two rows) for waves following wind (oblique wind in second row) and (bottom two rows) for waves opposing wind (oblique wind in fourth row). From left to right: turbulent kinetic energy ($\overline{e}/u_{*s}^2$), dissipation rate $\left[\epsilon /\left(k{u}_{*s}^3\right)\right]$, and vorticity magnitude [${\overline{\omega }}_h/\left(k{u}_{*s}\right)$].

Citation: Journal of Physical Oceanography 52, 1; 10.1175/JPO-D-21-0044.1

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(left three columns) Normalized phase-averaged flow fields in the ξ–z coordinate and (right three columns) the mapped ξ–ζ coordinate for (top two rows) waves following wind (oblique wind in second row) and (bottom two rows) waves opposing wind (oblique wind in fourth row). From left to right: vorticity magnitude [${\overline{\omega }}_h/\left(k{u}_{*s}\right)$], cross-wave vorticity [${\overline{\omega }}_y/\left(k{u}_{*s}\right)$], and along-wave vorticity [$\mathrm{-}{\overline{\omega }}_x/\left(k{u}_{*s}\right)$].

Citation: Journal of Physical Oceanography 52, 1; 10.1175/JPO-D-21-0044.1

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(left three panels) Normalized phase-averaged flow fields in the ξ–z coordinate and (right three panels) the mapped ξ–ζ coordinate for (top two rows) waves following wind (oblique wind in second row) and (bottom two rows) waves opposing wind (oblique wind in fourth row). From left to right: the along-wave component ($0.5\overline{u^{\prime }{u}^{\prime }}/u_{*s}^2$), the cross-wave component ($0.5\overline{{\upsilon }^{\prime }{\upsilon }^{\prime }}/u_{*s}^2$), and the vertical component ($0.5\overline{w^{\prime }{w}^{\prime }}/u_{*s}^2$) of TKE.

Citation: Journal of Physical Oceanography 52, 1; 10.1175/JPO-D-21-0044.1

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First, we examine the phase averaged velocity and pressure fields in Fig. 2. In the oblique wind cases with waves following or opposing wind, the along-wave velocity (in the x direction) is reduced, but the overall patterns remain very similar to those of the aligned wind cases (Figs. 2a–d). The cross-wave velocity (in the y direction) is introduced in the oblique cases (Figs. 2e,f) and is much stronger than the along-wave velocity (Figs. 2b,d). We will later show that even in a fixed coordinate, $\langle \overline{\upsilon }\rangle$ is larger than $\langle \overline{u}\rangle$, that is, the mean wind direction is rotated from the x direction by more than 45°. This is expected, at least qualitatively, because waves exert more friction (due to the wave form drag) and wind speed is more reduced in the along-wave (x) direction.

With the oblique wind, both the along-wave and cross-wave velocities are significantly reduced over the leeward face of the wave (Figs. 2b,d–f), suggesting that intermittent airflow separations (or separation-like flows) are modifying the wind fields, as in the aligned wind cases (Figs. 2a,c). The vertical velocity is slightly weaker in the oblique cases, but the patterns remain largely unchanged from the aligned cases (Figs. 2g–j).

As expected, the pressure along the wave phase weakens with oblique wind as less wind forcing is exerted onto the wave shape in the x direction (Figs. 2k–n), resulting in weaker surface pressure and total normal stress than in the aligned cases (Figs. 2A,C). Interestingly, the surface tangential stress in the along-wave direction is even more reduced with oblique winds (Figs. 2B,D), possibly because the horizontally averaged wind vector very close to the surface is rotated by almost 65°–68° from the x direction, as discussed later.

Next, we examine the phase averaged fields of TKE, dissipation rate, and horizontal vorticity magnitude in Fig. 3. For both aligned and oblique cases, regions of enhanced dissipation and vorticity appear to detach from the crest and extend downstream above the dead zone of significantly reduced TKE, dissipation rate, and vorticity on the leeward face of the wave. The enhanced TKE above the crest appears to be advected by the high velocity just above the detached high vorticity layer. These patterns are quite similar between the aligned and oblique wind cases, and suggest that flow is intermittently separating from the crest (or exhibiting separation-like patterns) even in oblique winds.

In general, the opposing and following wave cases in oblique winds are similar, except that the faster relative wind speed in the opposing case tends to limit the vertical extent of the wave induced flow perturbations closer toward the surface (cf. Figs. 3b,f,j,d,h,l). The same trend has been observed in the aligned wind cases in Part I.

One notable difference exists between the oblique and aligned wind cases. The high vorticity and high dissipation regions along the wavy surface are mostly confined near the crest in the aligned wind cases (Figs. 3E,G,I,K). However, these regions extend upstream all the way to the wave trough in the oblique wind cases (Figs. 3F,H,J,L). Consequently, the reduction of dissipation rate and vorticity in the dead zone is muted with the oblique wind.

To shed more light on these features, we next examine the along-wave and cross-wave vorticity components separately in Fig. 4. It is clear that the detached layer of enhanced vorticity is present in both directions (Figs. 4F,I,H,J) in the oblique cases. The crosswind vorticity simply weakens in oblique winds, but its patterns remain very similar between the aligned and oblique cases with reduced vorticity in the dead zone (Figs. 4E–H). In contrast, the along-wave vorticity remains strong along the entire surface, and especially along the windward face and the crest of the wave (Figs. 4I,J). This enhanced along-wind vorticity appears to be well correlated with the enhanced dissipation in the same location (Figs. 3F,H). The enhancement of along-wind vorticity and dissipation rate on the windward side of the crest (between the trough and following crest) may be related to reattachment of the separated oblique wind flow. We will later show that the cross-wave wave-coherent velocity $\tilde{\upsilon }$ is also enhanced there.

One interesting feature for waves opposing oblique winds is the presence of a region of negative cross-wave vorticity near the trough (Fig. 4H). In Part I, we have shown that for waves opposing wind a region of negative cross-wave vorticity near the trough intensifies as waves get faster, and is associated with separation-like detachment of airflow from the surface. In such a region wind shear is negative; that is, the wind speed decreases with height. However, in the case of oblique wind, the mean wind direction is close to 70° (as explained later) and the along-wave vorticity remains strong in the same region. Therefore, the wind speed magnitude increases with height.

In Fig. 5, we separate the TKE into along-wave ($0.5\overline{u^{\prime }{u}^{\prime }}/u_{*s}^2$), cross-wave ($0.5\overline{{\upsilon }^{\prime }{\upsilon }^{\prime }}/u_{*s}^2$), and vertical ($0.5\overline{w^{\prime }{w}^{\prime }}/u_{*s}^2$) components. As expected, in aligned winds the TKE is dominated by the along-wave component (Figs. 5a,c,A,C). One notable exception is a region on the windward side of the crest near the surface, where the cross-wave component is significantly enhanced and is larger than the along-wave component (Figs. 5e,g,E,G). The cause of this enhancement is not clear, but it may be related to the reattachment of separated flow in this area.

In the oblique case, the along-wave component of the TKE is drastically reduced and the cross-wave component is significantly larger. This is consistent with the fact that both mean wind speed and mean wind shear are rotated by more than 45° from the x axis, as discussed later. The enhancement of the cross-wave component on the windward side of the crest is observed in the oblique wind cases as well (Figs. 5f,h,F,H). This enhancement is possibly related to the enhanced along-wind vorticity (Figs. 4I,J) with large $\partial \overline{\upsilon }/\partial z$ in the same area. Finally, the consistently small vertical component suggests that the airflow turbulence is dominated by horizontal velocity variances in the entire wave boundary layer.

b. Instantaneous vorticity fields

To demonstrate the transient character of the flow field over waves in oblique wind, in Fig. 6 we display instantaneous snapshots of horizontal vorticity magnitude [${\omega }_h/\left(k{u}_{*s}\right)$] from a mapped top-down view (in the ξ–y coordinate) at kζ = 0.06 (very close to the surface) and at kζ = 0.40 for waves aligned with wind (left four plots) and waves oblique to wind (right four plots). For these snapshots, we are using the resolved fields and appreciate that very near the surface the total vorticity fluctuations are underestimated. Very close to the surface, the waves strongly modify the horizontal vorticity magnitude. In aligned wind, a pattern of enhanced vorticity is present near the surface along the crest with a sudden reduction downwind where airflow intermittently separates (Figs. 6a,b). The wave signature diminishes with height (Figs. 6c,d), but the enhanced vorticity is now observed further downstream from the crest, suggesting the advected high vorticity due to flow separations and being consistent with Figs. 4A and 4C.

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Normalized instantaneous vorticity magnitude [${\omega }_h/\left(k{u}_{*s}\right)$] fields in the ξ–y coordinate (top) for waves following wind and (bottom) for waves opposing wind in (a)–(d) aligned wind and (e)–(h) oblique wind at heights of kζ = 0.06 in (a), (b), (e), and (f) and kζ = 0.40 in (c), (d), (g), and (h).

Citation: Journal of Physical Oceanography 52, 1; 10.1175/JPO-D-21-0044.1

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In oblique wind near the surface (kζ = 0.06), the areas of enhanced vorticity magnitude are significantly expanded from the crest toward the upwind trough compared with the aligned wind cases, as expected from Figs. 4B and 4D. The region of reduced vorticity just downstream of the crest is narrower. Their pattern is also modified since it is aligned with the oblique mean wind direction. Similar to the aligned case, the signature of advected high vorticity is still visible at kζ = 0.40.

c. Horizontally averaged wind profiles in mapped coordinates

Next, we investigate the vertical profiles of the horizontally averaged wind variables in the mapped (ζ) coordinate, including mean wind speed, mean wind shear, TKE and its components, shown in Fig. 7. In the following subsections profiles for waves following (opposing) aligned wind are shown as thick orange (blue) lines, and profiles for waves following (opposing) oblique wind are shown as thin orange (blue) lines. All the profiles are displayed up to kζ = 2.

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(a),(d),(g),(j) Normalized vertical profiles of horizontally averaged along-wave wind speed ($\langle u\rangle /u_{*s}$), cross-wave wind speed ($\langle \upsilon \rangle /u_{*s}$), wind speed magnitude, and wind speed angle. (b),(e),(h),(k) Normalized vertical profiles of horizontally averaged along-wave wind shear [$\left(\partial \langle u\rangle /\partial \zeta \right)\left(\kappa \zeta /u_{*s}\right)$], cross-wave wind shear [$\left(\partial \langle \upsilon \rangle /\partial \zeta \right)\left(\kappa \zeta /u_{*s}\right)$], wind shear magnitude, and wind shear angle. In (k) dashed lines show angle of horizontally averaged turbulent stress vector $\left(\langle {\tau }_{13}^t\rangle ,\langle {\tau }_{23}^t\rangle \right)$. (c),(f),(i),(l) Normalized vertical profiles of horizontally averaged TKE magnitude ($\langle \overline{e}\rangle /u_{*s}^2$), the along-wave component ($0.5\langle \overline{u^{\prime }{u}^{\prime }}\rangle /u_{*s}^2$), the cross-wave component ($0.5\langle \overline{{\upsilon }^{\prime }{\upsilon }^{\prime }}\rangle /u_{*s}^2$), and the vertical component ($0.5\langle \overline{w^{\prime }{w}^{\prime }}\rangle /u_{*s}^2$) of the TKE. In all panels, profiles for waves following (opposing) aligned wind are shown as thick orange (blue) lines, and profiles for waves following (opposing) oblique wind are shown as thin orange (blue) lines. Gray lines show profiles for flat wall cases (explained in the main text).

Citation: Journal of Physical Oceanography 52, 1; 10.1175/JPO-D-21-0044.1

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The four panels in the top row of Fig. 7 display vertical profiles of the horizontally averaged mean wind speed vector. The x and y components of the wind vector, $\langle u\rangle /u_{*s}$ and $\langle \upsilon \rangle /u_{*s}$, are shown in Figs. 7a and 7d, respectively, while their magnitude, ${\left({\langle u\rangle }^2+{\langle \upsilon \rangle }^2\right)}^{1/2}/u_{*s}$, and angle, θ = arctan(〈υ〉/〈u〉), are shown in Figs. 7g and 7j, respectively. The gray lines in Figs. 7a and 7d represent the flat wall wind speed profiles with a background roughness of kz_ob = 2.70 × 10⁻³, for aligned (thick lines) and oblique (thin lines) cases, for linearly decreasing wind stress (solid) and constant wind stress (dashed) in kζ. In Fig. 7g the thick gray lines apply for both aligned and oblique cases. The thin dashed gray line in Fig. 7j represents the angle of the flat wall wind speed profile, θ = 45°, which is equal to the angle of the wind stress and the imposed pressure gradient.

As discussed in Part I, the wind speed magnitude (Fig. 7g) in the aligned cases (thick orange and blue lines) is significantly reduced from the wind profile for a flat wall (thick gray line) near the top of the domain, indicating that the equivalent surface roughness is increased for both waves following and opposing wind, with the latter having a slightly higher roughness length. However, the wind speed magnitude in the oblique cases (thin orange and blue lines) is very close to that for a flat wall (thick gray line) near the top of the domain for both waves following and opposing the oblique wind, which indicates that the waves do not enhance the equivalent surface roughness above the background roughness (more discussions follow in section 3f).

If we examine the along-wave and cross-wave wind speeds separately (Figs. 7a,d) for the oblique cases, it is clear that both components are significantly modified by the wave. As expected, the along-wave wind speed is reduced from the flat wall profile because of the wave form drag. The along-wave wind profiles for oblique wind are roughly proportional to those for aligned wind, and are reduced by about 25%–30% throughout the wave boundary layer. Interestingly, the cross-wave wind speed is increased near the top of the domain compared to the flat wall wind speed (Fig. 7d). The combined effect of increased cross-wave wind speed and decreased along-wave wind speed yields the almost unchanged wind speed magnitude toward the top.

The opposite wave impacts on the along-wave and cross-wave wind speeds mean that the angle of the wind speed (θ = 52°–54°) is significantly misaligned from the angle of the wind stress (θ = 45°) near the top of the domain (Fig. 7j), even though the wind magnitude is not modified by the wave. The wind speed angle is much larger closer to the surface, falling between θ = 65° and 68°. From about kζ = 0.1–0.4, the angle quickly reduces to about θ ≈ 55°, and reduces more slowly from there to the top at kζ = 2. It is notable that the wind speed angle is strongly dependent on height in the wave boundary layer, and that its misalignment from the total wind stress angle (θ = 45°) appears to persist above the top of the wave boundary layer.

The second row of Fig. 7 displays vertical profiles of the normalized mean wind shear. Figures 7b and 7e show the along-wave and cross-wave shear, $\left(\partial \langle u\rangle /\partial \zeta \right)\left(\kappa \zeta /u_{*s}\right)$ and $\left(\partial \langle \upsilon \rangle /\partial \zeta \right)\left(\kappa \zeta /u_{*s}\right)$, respectively, and Figs. 7h and 7k show the wind shear magnitude and the wind shear angle (solid lines), respectively. The gray lines represent the flat wall wind shear profiles as in the first row.

The shape of vertical profiles of wind shear magnitude (Figs. 7h) is quite similar between the aligned and oblique wind cases, with a reduction near the surface and enhancement at midlevel. However, the wind shear magnitude for the oblique wind is consistently larger throughout the wave boundary layer (compare thin orange line with thick orange line, or thin blue line with thick blue line). This larger wind shear magnitude is responsible for the larger wind speed near the top of the domain (Fig. 7g) and the resulting smaller equivalent roughness length.

The along-wave wind shear profiles (Fig. 7b) for the oblique wind cases are roughly proportional to those for the aligned wind cases (except near the top where they collapse); they are reduced by about 25%–30%. This suggests that the same physical processes (i.e., effects of pressure form drag near the surface and of intermittent airflow separations at midlevel) take place in oblique wind cases, but their impacts are reduced. Near the top of the wave boundary layer, the along-wave wind shear is not much reduced for the oblique cases, and this contributes to the enhanced wind shear magnitude near the top.

In the middle of the wave boundary layer, the cross-wave wind shear profiles (Fig. 7e) for the oblique wind cases are significantly enhanced compared to that for a flat wall case, and this contributes to the enhancement of the wind shear magnitude and far field wind speed as well as the reduced equivalent roughness length. The shear enhancement occurs at lower elevation with waves opposing (compared to following) oblique wind, which is consistent with the earlier observation that waves opposing wind tend to suppress the vertical extent of the wave induced flow perturbations.

The angle of the mean wind shear is plotted in Fig. 7k (solid lines), along with the angle of the horizontally averaged turbulent stress (dashed lines; more discussion on the turbulent stress and momentum budget in the next subsection). Both the wind shear angle and the turbulent stress angle hover around θ = 45° above kζ = 1; that is, the mean wind shear and the turbulent stress are well aligned with the total wind stress there. This suggests that the direction of the mean wind speed gradually approaches the wind stress direction (i.e., the misalignment between wind speed and wind stress gradually disappears) if the constant stress layer is extended upward without a top boundary (i.e., in the open ocean condition).

At midlevel, the wind shear angle oscillates around 45° for waves opposing wind, but it is significantly reduced from 45° at around kζ = 0.30 for waves following wind, associated with the enhancement of the along-wave wind shear (Fig. 7b). Toward the surface, the wind shear angle increases to about θ = 63°.

The profiles of the wind shear angle (solid lines) are generally well correlated with the profiles of the turbulent stress angle (dashed lines). This observation supports the common turbulence closure assumption that the angle of the turbulent wind stress is the same as the angle of the mean wind shear (i.e., the momentum flux is downgradient). However, toward the surface the angle of the turbulent stress is consistently smaller than the angle of the mean wind shear by about 8°–9°.

The bottom row of Fig. 7 displays vertical profiles of the horizontally averaged TKE and its three components. With the oblique wind, the magnitude of TKE remains largely unchanged (Fig. 7c). However, the profiles of along-wave TKE and cross-wave TKE significantly differ between the oblique and aligned cases. In aligned winds, the along-wave TKE component is much larger and is pronounced at midlevel, associated with flow separation (or separation-like) patterns (see also Figs. 4A,C). In oblique winds, the cross-wave TKE component is larger, and both components show modest enhancement at midlevel (see also Figs. 4B,D,F,H). The vertical TKE component remains significantly smaller throughout the wave boundary layer.

d. Horizontally averaged momentum budget in along-wave and cross-wave directions