2.1. Simulation set-up

To obtain the three-dimensional wind field, we perform wall-resolved LES of wind turbulence over prescribed water waves. The LES solves the filtered incompressible Navier–Stokes equations for air motions, and is written

(2.1)\begin{gather} \frac{\partial u_j}{\partial x_j}=0, \end{gather}

(2.2)\begin{gather}\frac{\partial u_j}{\partial t}+\frac{\partial (u_j u_m)}{\partial x_m} ={-}\frac{1}{\rho_a}\frac{\partial p}{\partial x_j} -\frac{\partial\tau_{jm}^d}{\partial x_m}+\nu\frac{\partial^2u_j} {\partial x_m\partial x_m}, \end{gather}

where $x_j (j=1,2,3)=(x,y,z)$ denote the Cartesian coordinates in the streamwise, spanwise and vertical directions, respectively, as illustrated in figure 1, $u_j (j=1,2,3)=(u,v,w)$ are the components of the filtered velocity in the LES at the grid scale, $p$ is the filtered modified pressure, $\tau _{jm}^d$ is the trace-free part of the subgrid-scale (SGS) stress tensor, $\rho _a$ is the density of air and $\nu$ is the air kinematic viscosity. At the water surface, a progressive water wave is imposed as the Dirichlet boundary condition for the airflow, $u_i(z=\eta )= (u_s, v_s, w_s)$, where $\eta$ is the surface-wave elevation and $(u_s, v_s, w_s)$ is the orbital velocity of the wave at the water surface, given as

(2.3)\begin{gather} \eta(x,y,t) = a\sin k(x-ct), \end{gather}

(2.4)\begin{gather}u_s(x,y,t) = akc\sin k(x-ct), \end{gather}

(2.5)\begin{gather}v_s(x,y,t) = 0, \end{gather}

(2.6)\begin{gather}w_s(x,y,t) ={-} akc\cos k(x-ct), \end{gather}

where $a$ is the amplitude of the surface wave, $k=2{\rm \pi} /\lambda$ is its wavenumber, $\lambda$ is its wavelength and $c$ is its phase speed. In (2.3)–(2.6), an Airy wave solution for deep water waves is adopted. In the case of nonlinear waves, such as a Stokes wave, the effect of nonlinearity by the higher harmonics is of $O((ak)^2)$. In the linear analysis of the wave-induced airflow performed in later sections, the $O((ak)^2)$ terms in the governing equations are omitted. As explained in Cao et al. (Reference Cao, Deng and Shen2020), to be consistent, we only consider the dominant Fourier component in the water-wave solution.

Figure 1. Sketch of LES configuration of turbulent wind following fast-propagating water waves. A three-dimensional wind field is driven by a constant velocity $U_0$ applied at the top boundary and flows over a plane propagating surface-wave train. The surface wave propagates in the $x$-direction, with a wavelength $\lambda$ and a phase speed $c$. The Dirichlet boundary condition is employed at the wave surface and periodic boundary conditions are applied in the horizontal directions.

In the simulation, the governing equations (2.1) and (2.2) in the physical space are transformed to the rectangular computational space such that

(2.7a–d)\begin{equation} \tau=t,\quad \xi=x,\quad \psi=y,\quad \zeta=z+g(\zeta)\eta, \quad \mbox{where}\ g(\zeta) = \frac{\zeta}{L_z}-1, \end{equation}

where $(\xi , \psi , \zeta , \tau )$ are the spatial and time coordinates in the computational space, $L_z$ is the mean physical domain height and $g(\zeta )$ denotes the transformation function. The Jacobian matrix corresponding to the spatial derivative transformation in (2.7a–d) is

(2.8)\begin{equation} \boldsymbol{J}=\left[ \begin{array}{ccc} \displaystyle \dfrac{\partial{\xi}}{\partial x} & \displaystyle \dfrac{\partial{\xi}}{\partial y} & \displaystyle \dfrac{\partial{\xi}}{\partial z} \\ \displaystyle \dfrac{\partial{\psi}}{\partial x} & \displaystyle \dfrac{\partial{\psi}}{\partial y} & \displaystyle \dfrac{\partial{\psi}}{\partial z} \\ \displaystyle \dfrac{\partial{\zeta}}{\partial x} & \displaystyle \dfrac{\partial{\zeta}}{\partial y} & \displaystyle \dfrac{\partial{\zeta}}{\partial z} \end{array} \right] =\left[ \begin{array}{cccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \displaystyle \dfrac{g\eta_\xi} {1-g_\zeta \eta} & \displaystyle \dfrac{g\eta_\psi} {1-g_\zeta \eta} & \displaystyle \dfrac{1} {1-g_\zeta \eta} \end{array} \right], \end{equation}

where $g_\zeta = \mathrm {d} g / \mathrm {d} \zeta$. The transformation of time derivative between the computational space and the physical space in (2.7a–d) caused by the surface-wave motion is defined as

(2.9)\begin{equation} \frac{\partial}{\partial t} = \frac{\partial}{\partial \tau} + \frac{\partial{\xi_j}}{\partial t} \frac{\partial}{\partial \xi_j} = \frac{\partial}{\partial \tau} + \frac{\partial{\zeta}}{\partial t} \frac{\partial}{\partial \zeta}, \quad \mbox{where}\ \frac{\partial{\zeta}}{\partial t} = \frac{g\eta_\tau} {1-g_\zeta \eta}. \end{equation}

Then the LES equations (2.1) and (2.2) in the computational space become

(2.10)\begin{gather} J_{pj} \frac{\partial u_j}{\partial \xi_p} = 0, \end{gather}

(2.11)\begin{gather}\frac{\partial u_j}{\partial \tau} + \delta_{p3} \frac{\partial{\zeta}}{\partial t} \frac{\partial u_j}{\partial \xi_p} + J_{lp} \frac{\partial (u_j u_p)}{\partial \xi_l} ={-} \frac{J_{lj} }{\rho_a} \frac{\partial p}{\partial \xi_l} + J_{lp} \frac{\partial \tau_{jp}^d}{\partial \xi_l} + \nu J_{np} \frac{\partial}{\partial{\xi_n}} \left( J_{lp} \frac{\partial u_j}{\partial{\xi_l}} \right), \end{gather}

where $J_{lp}$ is the $(l,p)$ entry of the mapping matrix $\boldsymbol {J}$ and $\delta _{lm}$ is the Kronecker delta. The transformed LES equations (2.10) and (2.11) are discretised and solved in computational space. Specifically, in the $(\xi -\psi )$ plane, a Fourier-series-based pseudo-spectral method is used for discretisation with evenly spaced grid points. In the $\zeta$-direction, a second-order finite-difference method is employed with grid points clustered towards the upper and lower boundaries. To calculate the SGS stress tensor, the dynamic Smagorinsky model is used (Smagorinsky Reference Smagorinsky1963; Germano et al. Reference Germano, Piomelli, Moin and Cabot1991; Lilly Reference Lilly1992). The detailed numerical procedure and extensive validations can be found in our previous studies of turbulent wind–wave interactions (Yang & Shen Reference Yang and Shen2010, Reference Yang and Shen2011; Yang, Meneveau & Shen Reference Yang, Meneveau and Shen2013; Hao & Shen Reference Hao and Shen2019; Cao et al. Reference Cao, Deng and Shen2020).

As sketched in figure 1, the turbulent wind is driven by an external velocity at the top of the simulation domain, $(u, v, w) = (U_0, 0, 0)$; a canonical set-up in the simulations of turbulent airflows over surface waves (e.g. Sullivan et al. Reference Sullivan, McWilliams and Moeng2000; Druzhinin et al. Reference Druzhinin, Troitskaya and Zilitinkevich2012; Cao et al. Reference Cao, Deng and Shen2020). In the present study, the wave age $c/U_0$ varies between $0.1$ and $1.4$, and two wave-steepness values are considered, $ak=0.10$ and $0.15$ (table 1). The wave steepness in the present study is within the range of values adopted in the previous studies of wind over fast waves, e.g. $ak = 0.10$ in Sullivan et al. (Reference Sullivan, McWilliams and Moeng2000), $ak=0.2$ in Hanley & Belcher (Reference Hanley and Belcher2008), $ak=0.10$ and $0.25$ in Yang & Shen (Reference Yang and Shen2010), $ak=0.05\text {--}0.4$ in Jiang et al. (Reference Jiang, Sullivan, Wang, Doyle and Vincent2016) and $ak = 0.10$ in Åkervik & Vartdal (Reference Åkervik and Vartdal2019). The Reynolds number based on the wavelength of the surface wave and the top-driven velocity, $U_0\lambda /\nu$, is 30 000, which is higher than the values 8800–15 000 in the previous DNS of wind over water waves (e.g. Sullivan et al. Reference Sullivan, McWilliams and Moeng2000; Yang & Shen Reference Yang and Shen2010; Druzhinin et al. Reference Druzhinin, Troitskaya and Zilitinkevich2012). The Reynolds number based on the air friction velocity $u_\tau$, i.e. $u_\tau \lambda /\nu$, is obtained a posterior and is summarised in table 1, and is comparable with values in the wall-resolved LES study by Åkervik & Vartdal (Reference Åkervik and Vartdal2019). In the present study, $u_\tau$ is defined using the mean viscous shear stress at the top boundary $\tau _s$, which equals the mean total stress in the streamwise direction at any given height in the wave boundary layer, as $u_\tau =\sqrt {\tau _s/\rho _a}$. Following the previous studies (e.g. Sullivan et al. Reference Sullivan, McWilliams and Moeng2000; Druzhinin et al. Reference Druzhinin, Troitskaya and Zilitinkevich2012; Cao et al. Reference Cao, Deng and Shen2020), a simulation domain of the size $(L_x, L_y, L_z) = (6\lambda , 4\lambda , \lambda )$ is adopted for the wind turbulence. The wind field is discretised in the computational space with $384^2\times 193$ grid points, providing a resolution of $\Delta \xi /(\nu / u_\tau ) < 21$, $\Delta \psi /(\nu / u_\tau ) < 14$ and $\Delta \zeta _{{min}}/(\nu / u_\tau ) < 0.2$, where $\Delta \zeta _{{min}}$ is the minimum grid space in the $\zeta$-direction near the boundaries. The grid resolution in all cases satisfies the requirement of the wall-resolved LES specified by Choi & Moin (Reference Choi and Moin2012). The parameters and resolution of the LES cases in this study are summarised and listed in table 1.

Table 1. List of LES cases for turbulent wind over progressive water waves. The wind field is discretised with $(N_x, N_y, N_z)=(384, 384, 193)$ grid points in the ($\xi , \psi , \zeta$) directions, respectively. The bulk Reynolds number $U_0\lambda /\nu$ is prescribed as 30 000 for all of the wave cases, while $u_\tau$, $u_\tau \lambda /\nu$ and the grid resolution in wall units are obtained a posterior. Here, the superscript ‘$+$’ denotes normalisation by the viscous length scale $\nu /u_\tau$.

4.1. Wave-induced velocity and pressure

In this subsection, we focus on the wave-induced velocity and pressure in the wind field. Figure 4 presents the wave-induced vertical velocity $\tilde {w}$ for different wave ages. It is shown that the structure of $\tilde {w}$ varies substantially with $c/U_0$, indicating that the physical process producing $\tilde {w}$ depends significantly on $c/U_0$. For the slow wave with $(c/U_0, c/u_\tau )=(0.1, 3.46)$ (figure 4a), away from the surface, $\tilde {w}$ is mostly positive above the windward face and negative above the leeward face, without displaying antisymmetry across the wave crest. The $\tilde {w}$ for slow waves ($c \simeq u_\tau$) is produced by the airflow blocked by the windward face of the wave owing to the low wave speed. The airflow goes upward along the wave windward surface and therefore has a positive $\tilde {w}$ there. However, when $c \gg u_\tau$, the wave surface no longer blocks the overlying airflow. As shown, for the intermediate wave with $(c/U_0, c/u_\tau )=(0.4, 15.38)$ (figure 4b), the structure of $\tilde {w}$ is remarkably different from that of the slow wave. Specifically, in the near surface region, $\tilde {w}$ exhibits a nearly antisymmetric spatial structure about the wave crest, namely, positive over the leeward side and negative over the windward side with a comparable magnitude. The antisymmetric distribution of $\tilde {w}$ is also present in the fast wave case with $(c/U_0, c/u_\tau )=(1.2, 54.30)$ (figure 4c), but with a much larger amplitude and extends to a much higher altitude in the airflow.

Figure 4. Wave-induced vertical air velocity $\tilde {w}/(akU_0)$ for the wave conditions: (a) $(c/U_0, c/u_\tau )=(0.1, 3.46)$; (b) $(c/U_0, c/u_\tau )=(0.4, 15.38)$; (c) $(c/U_0, c/u_\tau )= (1.2, 54.30)$. Results for ${ak=0.15}$.

The nearly antisymmetric $\tilde {w}$ in the intermediate (figure 4b) and fast (figure 4c) wave cases indicates that it is associated with the airflow perturbation induced by the vertical wave movement, i.e. $w_s$ (2.6). At the surface, the vertical air velocity satisfies the Dirichlet boundary condition (2.6) and therefore $\tilde {w}(\zeta =0)=\tilde {w}_s=-akc \cos (k\xi )$, with $\tilde {\eta }=a\sin (k\xi )$. Therefore, one can expect that at the wave surface, a strong vertical motion of water wave would initiate a strong upward airflow at the leeward face and downward airflow at the windward face, which is confirmed by the LES result of $\tilde {w}$ shown in figures 4(b) and 4(c). Away from the surface, the airflow perturbation induced by the vertical wave motion weakens gradually and thus the magnitude of $\tilde {w}$ decays with height. The previously described mechanism for the occurrence of $\tilde {w}$ indicates that $\tilde {w}$ has an order of magnitude of $akc$ for intermediate and fast waves. In particular, when the wave speed $c \gtrsim U_0$, the magnitude of $\tilde {w}$ near the wave surface has a magnitude comparable to $akU_0$. Therefore, in figure 4(c) with $(c/U_0, c/u_\tau )=(1.2, 54.30)$, $\tilde {w}/(akU_0)$ has a magnitude of approximately $1$, which is significantly larger than that shown in figures 4(a) and 4(b).

Figure 5 shows the wave-induced streamwise air velocity $\tilde {u}$ for different $c/U_0$. At the low wave age with $(c/U_0, c/u_\tau )=(0.1, 3.46)$ (figure 5a), $\tilde {u}$ has a strong positive value at the windward side near the wave crest, caused by the streamwise velocity acceleration associated with the narrowing of cross-section of the air passage on the windward side of the slow wave. For the intermediate wave with $(c/U_0, c/u_\tau )=(0.4, 15.38)$ (figure 5b), $\tilde {u}$ has a large magnitude only in a very thin region above the surface. For the fast wave with $(c/U_0, c/u_\tau )=(1.2, 54.30)$ (figure 5c), away from the surface, $\tilde {u}$ is nearly in antiphase with the wave elevation, namely, a positive $\tilde {u}$ above the wave trough and a negative $\tilde {u}$ above the wave crest, which appears contrary to the airflow perturbation induced by the streamwise component of the wave kinematics. Specifically, at the wave surface, $\tilde {u}(\zeta =0)=\tilde {u}_s= akc \sin (k\xi )$, with $\tilde {\eta }=a\sin (k\xi )$, according to the Dirichlet boundary condition (2.4). Therefore, the streamwise motion of water wave induces a $\tilde {u}$ in phase with the wave elevation at the water surface (because $akc>0$), which is not exhibited by the structure of $\tilde {u}$ away from the surface in figure 5(c). In fact, for fast waves, the effect of the boundary condition (2.4) on $\tilde {u}$ is limited within a very thin region in the airflow and, thus, is not obvious in figure 5(c) (see more detailed discussions in § 6). Away from the surface, $\tilde {u}$ is mainly produced by the airflow perturbation induced by the vertical wave motion (figure 4c) through mass conservation as

(4.1)\begin{equation} \frac{\partial \tilde{u}}{\partial \xi} ={-} \frac{\partial \tilde{w}}{\partial \zeta} - \frac{\mathrm{d} \langle u \rangle}{\mathrm{d} \zeta} g\tilde{\eta}_{\xi}, \end{equation}

which is derived using (5.4) in § 5 and illustrates how $\tilde {u}$ can be generated by $\tilde {w}$. Under the fast wave condition $c \gtrsim U_0$, at a vertical length scale $l=O(k^{-1})$, $\partial \tilde {w} / \partial \zeta =O(ak^2c)$ and $\mathrm {d} \langle u \rangle / \mathrm {d} \zeta g \tilde {\eta }_{\xi } = O(ak^2u_\tau )$. Equation (4.1) suggests that away from the surface, $\tilde {u}$ is dominated by $\partial \tilde {w} / \partial \zeta$ and therefore $\tilde {u} = O(akc)$. For $c \gtrsim U_0$, the magnitude of $\tilde {u}$ is expected to be comparable to $akU_0$, which is reflected in the result in figure 5(c). In addition, because $\tilde {w}$ has the form of $-\cos (k\xi )$ and decays with $\zeta$ (figure 4c), the dominance of $\partial \tilde {w} / \partial \zeta$ in (4.1) indicates that above the wave, $\tilde {u}$ is in antiphase with wave surface, which is consistent with figure 5(c).

Figure 5. Wave-induced streamwise air velocity $\tilde {u}/(ak U_0)$ for the wave conditions: (a) $(c/U_0, c/u_\tau ) =(0.1, 3.46)$; (b) $(c/U_0, c/u_\tau )=(0.4, 15.38)$; (c) $(c/U_0, c/u_\tau )=(1.2, 54.30)$. Results for $ak=0.15$.

Figure 6 shows the wave-induced air pressure $\tilde {p}$ for different $c/U_0$. At low wave age with $(c/U_0, c/u_\tau )=(0.1, 3.46)$ (figure 6a), $\tilde {p}$ is highly asymmetric about the wave crest, positive at the windward face and negative at the leeward face, which generates a strong form drag on the wave surface to transfer the momentum from the wind to wave. For the intermediate wave with $(c/U_0, c/u_\tau )=(0.4, 15.38)$ (figure 6b), similar to $\tilde {w}$ and $\tilde {u}$, $\tilde {p}$ is significant only within a thin region near the surface. For the fast wave with $(c/U_0, c/u_\tau )=(1.2, 54.30)$ (figure 6c), similar to $\tilde {u}$, $\tilde {p}$ is strong and is nearly in antiphase with the wave surface, owing to the vertical wave motion-induced airflow. To illustrate this point, we obtain the dominant mechanism for producing $\tilde {p}$ using the vertical momentum equation of the wave-induced airflow, which is (detailed derivation is given in § 5.1)

(4.2)\begin{equation} (\langle u \rangle -c )\frac{\partial \tilde{w}}{\partial \xi} \approx{-} \frac{\partial \tilde{p}}{\partial \zeta}, \end{equation}

and the solution follows,

(4.3)\begin{equation} \tilde{p} \vert_\zeta \approx \int_{\zeta}^{\infty} (\langle u \rangle -c )\frac{\partial \tilde{w}}{\partial \xi} \,\mathrm{d} \zeta. \end{equation}

Since for the fast wave, $\tilde {w}$ is nearly in antiphase with the wave slope $\tilde {\eta }_{\xi }$ as shown in figure 4(c) and $\langle u \rangle -c < 0$, (4.3) illustrates that $\tilde {p}$ is nearly in antiphase with the wave elevation. Meanwhile, for $c \gtrsim U_0$, by considering $\tilde {w} = O(akc)$ and $U_0/2 \lesssim |\langle u \rangle -c|$ when $\zeta /\lambda < 0.5$, we obtain that $\tilde {p}=O(ak \rho _a c U_0)$ based on (4.3). This result explains why $\tilde {p}/ak \rho _a U_0^2$ has a magnitude of approximately 1 for $(c/U_0, c/u_\tau )=(1.2, 54.30)$ in figure 6(c).

Figure 6. Wave-induced air pressure $\tilde {p}/(ak\rho _a U_0^2)$ for the wave conditions: (a) $(c/U_0, c/u_\tau )=(0.1, 3.46)$; (b) $(c/U_0, c/u_\tau )=(0.4, 15.38)$; (c) $(c/U_0, c/u_\tau )=(1.2, 54.30)$. Results for $ak=0.15$.

To summarise this subsection, the LES results have shown that the wave-induced airflow is governed by different physical processes between slow waves and fast waves. Specifically, for slow waves, the wave-induced airflow occurs as a result of the blocking effect of the wave surface, while for fast waves it results from the airflow perturbation induced by the vertical motion of the wave. Therefore, when the wave speed $c$ is comparable with, or larger than the outer velocity $U_0$, i.e. $c \gtrsim U_0$, the wave-induced velocity and pressure scale as $\tilde {u}=O(akc)$, $\tilde {w}=O(akc)$ and $\tilde {p}=O(ak \rho _a c U_0)$, respectively, which are significantly stronger than those of slow waves.

4.2. Wave-induced turbulence variance and turbulent shear stress

In this subsection, we examine the spatial structure and magnitude of the wave-induced turbulence variance and turbulent shear stress. Figure 7 presents the distribution of wave-induced turbulence variance $\widetilde {u'u'}+\widetilde {v'v'}+\widetilde {w'w'}=\overline {u'u'} -\langle u'u'\rangle + \overline {v'v'} - \langle v'v' \rangle + \overline {w'w'} - \langle w'w' \rangle$ for different $c/U_0$. In general, $\widetilde {u'u'}+\widetilde {v'v'}+\widetilde {w'w'}$ reflects the wave effects on the turbulence variance in the air, which can, in turn, affect the behaviour of wave-induced airflow as illustrated in the previous studies on wind over slow waves (e.g. Belcher & Hunt Reference Belcher and Hunt1993; Miles Reference Miles1993, Reference Miles1996). As shown in the figure, the slow wave with $(c/U_0, c/u_\tau )=(0.1, 3.46)$ induces a strong positive $\widetilde {u'u'}+\widetilde {v'v'}+\widetilde {w'w'}$ above the leeward face of the wave (figure 7a). Buckley & Veron (Reference Buckley and Veron2016) attribute this enhanced turbulence intensity to sporadic airflow separation past the wave crest. By contrast, the intermediate wave with $(c/U_0, c/u_\tau )=(0.4, 15.38)$ intensifies the turbulence variance at the windward face of the wave, and suppresses the turbulence variance at the leeward face (figure 7b). As the wave propagates faster with $(c/U_0, c/u_\tau )=(1.2, 54.30)$, the regions corresponding to the intensified and suppressed turbulence variance shift farther upstream, such that $\widetilde {u'u'}+\widetilde {v'v'}+\widetilde {w'w'}$ displays a positive value over the wave trough and a negative value over the wave crest, as shown in figure 7(c). Importantly, it is observed that, unlike $\tilde {w}$, $\tilde {u}$ and $\tilde {p}$, the magnitudes of wave-induced turbulence variance normalised by $u_\tau ^2$ are comparable for different $c/U_0$ with a magnitude of $O(1)$, or $\widetilde {u'u'}+\widetilde {v'v'}+\widetilde {w'w'}=O(u_\tau ^2)$. This observation is utilised in § 5.2 to show that for fast waves, the effects of nonlinear forcing on the wave-induced airflow are negligible.

Figure 7. Wave-induced turbulence variance $(\widetilde {u'u'}+\widetilde {v'v'}+\widetilde {w'w'})/u_\tau ^2$ for the wave conditions: (a) $(c/U_0, c/u_\tau )=(0.1, 3.46)$; (b) $(c/U_0, c/u_\tau )=(0.4, 15.38)$; and (c) $(c/U_0, c/u_\tau )=(1.2, 54.30)$. Results for $ak=0.15$.

Figure 8 compares the wave-induced turbulent shear stress $-\widetilde {u'w'} = -\overline {u'w'}-\langle -u'w'\rangle$ for the three wave ages. At the small wave age with $(c/U_0, c/u_\tau )=(0.1, 3.46)$, similar to the turbulence variance (figure 7a), the turbulent shear stress is enhanced above the leeward face of the wave (figure 8a), owing to the effect of intermittent airflow separation downstream of the wave crest (Buckley & Veron Reference Buckley and Veron2016). For the intermediate wave age with $(c/U_0, c/u_\tau )=(0.4, 15.38)$, $-\widetilde {u'w'}$ has a negative value above the windward face and a positive value above the leeward face, with an almost antisymmetric spatial structure across the wave crest (figure 8b). Compared with case $(c/U_0, c/u_\tau )=(0.4, 15.38)$, at the high wave age with $(c/U_0, c/u_\tau )=(1.2, 54.30)$, $-\widetilde {u'w'}$ exhibits a similar spatial distribution, but is limited to a slightly thinner region in the airflow (figure 8c). Similar to $\widetilde {u'u'}+\widetilde {v'v'}+\widetilde {w'w'}$, the LES results suggest that for different wave ages, $-\widetilde {u'w'}=O(u_\tau ^2)$. The previous results demonstrate that, contrary to the wave-induced velocity and pressure, which are significantly stronger in the fast wave case than in the slow wave case, $\widetilde {u'u'}+\widetilde {v'v'}+\widetilde {w'w'}$ and $-\widetilde {u'w'}$ have magnitudes comparable for different wave speeds.

Figure 8. Wave-induced turbulent shear stress $-\widetilde {u'w'}/u_\tau ^2$ for the wave conditions: (a) $(c/U_0, c/u_\tau )= (0.1, 3.46)$; (b) $(c/U_0, c/u_\tau )=(0.4, 15.38)$; and (c) $(c/U_0, c/u_\tau )=(1.2, 54.30)$. Results for ${ak=0.15}$.

To conclude § 4, the LES results indicate that for wind over fast waves, the wave-coherent airflow is dominated by the vertical wave motion-induced air motion, which is much stronger than for wind over slow waves. By contrast, compared with the slow wave, the modulation of turbulence variance and turbulent stress by the fast waves are not strengthened. These results together suggest that the effects of wave-induced turbulent stress on the wave-induced airflow become relatively weaker as the wave propagates faster. In § 5, we develop a linear-analysis framework for the wave effects on the wind, based on which the fast wave-induced airflow is explained theoretically in § 6.

5.1. Non-orthogonal curvilinear model

To capture the wave-induced quantities in the entire region of airflow including the height above the wave trough and below the wave crest, the Cartesian coordinates $(x,y,z)$ can be transformed to the non-orthogonal curvilinear coordinates $(\xi ,\psi ,\zeta )$, as

(5.1a–c)\begin{equation} \xi = x, \quad \psi = y, \quad \zeta = z+g(\zeta)\eta, \end{equation}

where $g(\zeta )$ can be any function that increases monotonically from the wave surface to the top of the simulation domain (Hsu et al. Reference Hsu, Hsu and Street1981). In the present LES, the computational coordinates (2.7a–d) follow the form of (5.1a–c) with a linear mapping function $g = \zeta /L_z-1$. With the general transformation (5.1a–c) and the properties of phase average, the momentum and continuity equations for the wave-induced airflow can be expressed as (Cao et al. Reference Cao, Deng and Shen2020)

(5.2)\begin{align} & (\langle u \rangle -c )\frac{\partial \tilde{u}}{\partial \xi} + (\tilde{w} + (\langle u \rangle -c ) g\tilde{\eta}_{\xi}) \frac{\mathrm{d} \langle u \rangle}{\mathrm{d} \zeta} + \frac{1}{\rho_a} \frac{\partial \tilde{p}}{\partial \xi}\nonumber\\ &\quad = \nu \left(\frac{\partial^2 \tilde{u}}{\partial \zeta^2} - \frac{\partial^2 \tilde{w}}{\partial \xi \partial \zeta}\right) + \nu \frac{\mathrm{d}}{\mathrm{d} \zeta} \left(\frac{\mathrm{d} \langle u \rangle}{\mathrm{d} \zeta} g_\zeta \right) \tilde{\eta} + O((ak)^2) + n.l.f., \end{align}

(5.3)$$\begin{gather} (\langle u \rangle -c )\frac{\partial \tilde{w}}{\partial \xi} + \frac{1}{\rho_a} \frac{\partial \tilde{p}}{\partial \zeta} = \nu \left(\frac{\partial^2 \tilde{w}}{\partial \xi^2} + \frac{\partial^2 \tilde{w}}{\partial \zeta^2}\right) + O((ak)^2) + n.l.f., \end{gather}$$

(5.4)$$\begin{gather}\frac{\partial \tilde{u}}{\partial \xi} + \frac{\partial \tilde{w}}{\partial \zeta} + \frac{\mathrm{d} \langle u \rangle}{\mathrm{d} \zeta} g\tilde{\eta}_{\xi}= 0 + O((ak)^2), \end{gather}$$

where $O((ak)^2)$ denotes the neglected second- and higher-order terms (we neglect the correlations between a wave-induced quantity $\tilde {f}$ and $O(ak)^2$ terms, such as $\tilde {\eta }_\xi ^2\tilde {f}$ compared with $\tilde {f}$ (to capture the dominant flow physics only). Here, $g$ is treated as an $O(1)$ quantity), ‘$n.l.f.$’ represents the neglected nonlinear forcing, namely, the wave-induced fluctuations of the turbulent stress, i.e. $\tilde {\tau }_{jm}$ and the wave-induced fluctuations of the wave-induced stress, i.e. $\tilde {\tau }^w_{jm}$. The effects of $\tilde {\tau }^w_{jm}$ are always small for moderate wave slopes, and the effects of $\tilde {\tau }_{jm}$ are analysed in § 5.2. Note that although (5.2)–(5.4) are on the curvilinear coordinate system, primitive variables in the physical space are used in the equations for more direct illustration of the flow physics.

Next, by eliminating $\tilde {u}$ and $\tilde {p}$, we can reduce the equation system (5.2)–(5.4) to a single equation for $\tilde {w}$ as (Cao et al. Reference Cao, Deng and Shen2020)

(5.5)\begin{align} & -\frac{\nu}{\mathrm{i}k}\left[\frac{\mathrm{d}^4}{\mathrm{d} \zeta^4}-2k^2\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2}+k^4\right]\hat{w}+\left[(\langle u \rangle -c) \left(\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2}-k^2 \right)-\frac{\mathrm{d}^2 \langle u \rangle}{\mathrm{d} \zeta^2} \right]\hat{w}\nonumber\\ &\quad = \nu\hat{\eta}\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2} \left[g \frac{\mathrm{d}^2 \langle u \rangle}{\mathrm{d} \zeta^2} \right] + O((ak)^2) + n.l.f., \end{align}

where $\hat {w}$ is the Fourier coefficient of $\tilde {w}$ (3.2). The curvilinear equation (5.5) is different from the original Orr–Sommerfeld equation (e.g. Lin Reference Lin1955) in that the presence of the wave elevation $\hat {\eta }$ leads to a forcing term on its right-hand side. To evaluate the relative importance of the right-hand side in (5.5) requires information on the dependence of the mean wind velocity profile on the Reynolds number. However, how to model the mean wind velocity profile as a function of Reynolds number and mean stresses in the turbulent-wave boundary layer is still an open question in the research community and should be investigated in the future.

Equation (5.5) can be solved using the boundary conditions imposed by the wave kinematics, including the Dirichlet boundary conditions for $\hat {w}$,

(5.6a,b)\begin{equation} \left. \hat{w} \right\vert_{\zeta=0} = \hat{w}_s, \quad \left. \hat{w}\right\vert_{\zeta=\infty} = 0, \end{equation}

and the Neumann boundary conditions for $\mathrm {d} \hat {w} / \mathrm {d} \zeta$, which can be derived from (5.4) and read

(5.7a,b)\begin{equation} \left.\frac{\mathrm{d} \hat{w}}{\mathrm{d} \zeta}\right\vert_{\zeta=0}={-}\mathrm{i}k\hat{u}_s-\mathrm{i}k\hat{\eta}\left.\frac{\mathrm{d} \langle u \rangle}{\mathrm{d} \zeta} \right\vert_{\zeta=0}, \quad \left.\frac{\mathrm{d} \hat{w}}{\mathrm{d} \zeta}\right\vert_{\zeta=\infty} = 0, \end{equation}

where $\hat {u}_s$ and $\hat {w}_s$ are the Fourier coefficients of the streamwise (2.4) and vertical (2.6) components of wave orbital velocity, respectively. In § 6, (5.5) with $g = \zeta /L_z-1$ is numerically solved using the mean wind velocity $\langle u \rangle$ from the LES, and its results are compared with LES data to illustrate the mechanisms for the fast wave-induced airflow.

To evaluate the effects of coordinate systems, in appendix A we further develop a curvilinear model based on the orthogonal curvilinear coordinates (e.g. Sullivan et al. Reference Sullivan, McWilliams and Moeng2000; Buckley & Veron Reference Buckley and Veron2016; Yousefi & Veron Reference Yousefi and Veron2020; Yousefi, Veron & Buckley Reference Yousefi, Veron and Buckley2020),

(5.8a–c)\begin{equation} \xi = x-\mathrm{i}a\mathrm{e}^{{-}kz}\mathrm{e}^{\mathrm{i}kx}, \quad \psi = y, \quad \zeta = z- a\mathrm{e}^{{-}kz}\mathrm{e}^{\mathrm{i}kx}. \end{equation}

In appendix B, we compare the solutions of the curvilinear equations with different curvilinear coordinates, and show that the mechanisms of fast wave effects on the airflow revealed by the curvilinear models hold the same.

5.2. Scaling analysis of fast wave-induced turbulent stress effects

In this subsection we perform scaling analysis of the effects of fast wave-induced turbulent stress $\tilde {\tau }_{jm}$ on the wave-induced momentum equations (5.2) and (5.3). In § 4, the LES results show that for fast waves, $\tilde {w}=O(akc)$, $\tilde {p}=O(ak \rho _a c U_0)$, $\widetilde {u'u'}+\widetilde {v'v'}+\widetilde {w'w'}=O(u_\tau ^2)$ and $-\widetilde {u'w'} = O(u_\tau ^2)$. Based on these results, it can be further shown that in the curvilinear coordinates, the wave-induced turbulent stress satisfies $\tilde {\tau }_{jm} = O(u_\tau ^2)$, which has been neglected in deriving (5.2) and (5.3). By comparing the magnitude of $\tilde {\tau }_{jm}$ with the stress terms in (5.2) and (5.3), we can evaluate the effects of $\tilde {\tau }_{jm}$ on the fast wave-induced airflow.

To investigate the effects of $\tilde {\tau }_{jm}$, we focus on a vertical length scale $l$ satisfying $kl<1$, because $\widetilde {u'u'}+\widetilde {v'v'}+\widetilde {w'w'}$ (figure 7) and $-\widetilde {u'w'}$ (figure 8) are significant only within this region. To analyse the streamwise momentum equation (5.2), we can reduce it to the following form by using (5.4) to eliminate $\tilde {u}$ as

(5.9)\begin{equation} -(\langle u \rangle -c )\frac{\partial \tilde{w}}{\partial \zeta} + \tilde{w} \frac{\mathrm{d} \langle u \rangle}{\mathrm{d} \zeta} + \frac{1}{\rho_a} \frac{\partial \tilde{p}}{\partial \xi} = \nu \left(\frac{\partial^2 \tilde{u}}{\partial \zeta^2} - \frac{\partial^2 \tilde{w}}{\partial \xi \partial \zeta}\right) + \nu \frac{\mathrm{d}}{\mathrm{d} \zeta} \left(\frac{\mathrm{d} \langle u \rangle}{\mathrm{d} \zeta} g_\zeta \right) \tilde{\eta}. \end{equation}

With the results in § 4, we can obtain the following scaling in (5.9),

(5.10)\begin{gather} \frac{1/\rho_a \partial \tilde{p}/\partial \xi}{-(\langle u \rangle-c) \partial\tilde{w}/\partial\zeta } \sim \frac{ak c U_0 \cdot k}{ak c U_0/l} = kl, \end{gather}

(5.11)\begin{gather}\frac{\tilde{w} \langle u \rangle_\zeta}{-(\langle u \rangle-c) \partial\tilde{w}/\partial\zeta} \sim \frac{akc \langle u \rangle(l) /l}{akcU_0/l} = \frac{\langle u \rangle(l) }{U_0}, \end{gather}

(5.12)\begin{gather}\frac{\partial \tilde{\tau}_{11}/\partial \xi}{-(\langle u \rangle-c) \partial\tilde{w}/\partial\zeta} \sim \frac{u_\tau^2 \cdot k }{akcU_0/l} < \frac{u_\tau^2/l}{akcU_0/l} \sim \frac{\partial \tilde{\tau}_{13}/\partial \zeta}{-(\langle u \rangle-c) \partial\tilde{w}/\partial\zeta} \sim \frac{1}{ak} \frac{u_\tau}{U_0} \frac{u_\tau}{c}, \end{gather}

where $\langle u \rangle (l)$ is the value of $\langle u \rangle$ at $\zeta =l$. Similarly, for the vertical momentum equation (5.3), we have

(5.13)\begin{gather} \frac{(\langle u \rangle -c ) \partial \tilde{w}/\partial \xi}{1/\rho_a \partial \tilde{p}/\partial \zeta} \sim \frac{akcU_0 \cdot k }{akcU_0/l} = kl, \end{gather}

(5.14)\begin{gather}\frac{\partial \tilde{\tau}_{31}/\partial \xi}{1/\rho_a \partial \tilde{p}/\partial \zeta} \sim \frac{u_\tau^2 \cdot k}{akcU_0/l} < \frac{u_\tau^2/l}{akcU_0/l} \sim \frac{\partial \tilde{\tau}_{33}/\partial \zeta}{1/\rho_a \partial \tilde{p}/\partial \zeta} \sim \frac{1}{ak} \frac{u_\tau}{U_0} \frac{u_\tau}{c}. \end{gather}

In the wave surface layer, $ak \simeq 0.1$ and $U_0/u_\tau$ is large. Equations (5.10)–(5.12) illustrate that for high wave age $c/u_\tau$, $\partial \tilde {\tau }_{11}/\partial \xi$ and $\partial \tilde {\tau }_{13}/\partial \zeta$ are significantly smaller than $-(\langle u \rangle -c) \partial \tilde {w}/\partial \zeta$ in the streamwise momentum equation (5.9). Likewise, (5.13) and (5.14) demonstrate that for large $c/u_\tau$, $\partial \tilde {\tau }_{31}/\partial \xi$ and $\tilde {\tau }_{33}/\partial \zeta$ have negligible effects compared with $1/\rho _a \cdot \partial \tilde {p}/\partial \zeta$ in the vertical momentum equation (5.3). If $ak \ll 0.1$, the magnitude of $\tilde {\tau }_{jm}$ should be much smaller than $O(u_\tau ^2)$, and we still expect that its effects are much smaller than the wave-induced advection or pressure. These results suggest that for fast waves, the effects of wave-induced turbulent stress $\tilde {\tau }_{jm}$ on the wave-induced airflow are always much smaller than the dominant forcing in the wave-induced momentum equations, which constitutes the physical grounds for neglecting $\tilde {\tau }_{jm}$, i.e.the nonlinear forcing, in the curvilinear equations. The present scaling analyses support the results of Åkervik & Vartdal (Reference Åkervik and Vartdal2019), who found that for high wave age, the forcing induced by the wave-induced velocity and pressure is mainly balanced by the viscous stress, instead of the turbulent stress, indicating a quasi-laminar flow regime. To further verify this finding, they reproduced the LES results using a linear split system.

Finally, we comment on the effect of Reynolds number on the scaling analyses in this section. The foregoing analyses have shown that the ratio of the magnitude of turbulent stress to the magnitude of inertial forcing induced by the wave-induced velocity or the wave-induced pressure scales with $(u_\tau /U_0)\cdot (u_\tau /c)$ instead of viscosity, as shown by (5.12) and (5.14). Therefore, for high $c/u_\tau$, the trend that the turbulent stress effects become negligible and the inertial forcing will be balanced by either the viscous stress or the wave elevation-induced forcing should hold irrespective of Reynolds number. However, as the Reynolds number increases, $u_\tau /U_0$ decreases and thus the transition to the quasilinear regime might happen at a lower $c/u_\tau$, which is consistent with the conclusions of Åkervik & Vartdal (Reference Åkervik and Vartdal2019).

5.3. Split equations of curvilinear model

In this subsection, we develop split equations for the curvilinear model of the wave effects on airflow. The expressions of the curvilinear model (5.5) and its boundary conditions (5.6a,b) and (5.7a,b), reveal two mechanisms of the impacts of fast waves on the overlying turbulent wind field, namely, the forcing induced by the wave elevation, which is the source term on the right-hand side of (5.5), and the wave orbital velocity, which is shown by the boundary conditions (5.6a,b)–(5.7a,b). To investigate these two mechanisms separately, we can split a curvilinear equation into two separate equations corresponding to the two components of $\hat {w}$, $\hat {w}^k$ and $\hat {w}^f$, respectively,

(5.15)\begin{equation} \hat{w}= \hat{w}^k + \hat{w}^f. \end{equation}

In this equation, $\hat {w}^k$ and $\hat {w}^f$ are the vertical velocity perturbations associated with the wave kinematics and by the forcing caused by the wave elevation, respectively.

Extracted from (5.5), the equation for $\hat {w}^k$ has a zero source term on the right-hand side, but is driven by the wave orbital velocity at the wave surface, which is

(5.16)\begin{equation} -\frac{\nu}{\mathrm{i}k}\left[\frac{\mathrm{d}^4}{\mathrm{d} \zeta^4}-2k^2\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2}+k^4\right]\hat{w}^k+\left[ (\langle u \rangle -c) \left(\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2}-k^2 \right)-\frac{\mathrm{d}^2 \langle u \rangle}{\mathrm{d} \zeta^2} \right]\hat{w}^k=0, \end{equation}

with the boundary conditions

(5.17a,b)\begin{gather} \left. \hat{w}^k \right\vert_{\zeta=0} = \hat{w}_s, \quad \left. \hat{w}^k\right\vert_{\zeta=\infty} = 0, \end{gather}

(5.18a,b)\begin{gather}\left.\frac{\mathrm{d} \hat{w}^k}{\mathrm{d} \zeta}\right\vert_{\zeta=0}={-}\mathrm{i} k\hat{u}_s-\mathrm{i}k\hat{\eta}\left.\left(g\frac{\mathrm{d} \langle u \rangle}{\mathrm{d} \zeta}\right) \right\vert_{\zeta=0}, \quad \left.\frac{\mathrm{d} \hat{w}^k}{\mathrm{d} \zeta}\right\vert_{\zeta=\infty} = 0. \end{gather}

The boundary conditions (5.17a,b) and (5.18a,b) are enforced at $\zeta =0$, because the effects of wave orbital velocity on the airflow are applied on the curved water-wave surface. Therefore, (5.18a,b) implicitly includes some geometrical effects. The extraction of the wave-kinematics effects by (5.16)–(5.18a,b) follows the idea of Åkervik & Vartdal (Reference Åkervik and Vartdal2019), where the particular solution to the kinematics is split from the rest of the response, but with a difference between these two studies. In Åkervik & Vartdal (Reference Åkervik and Vartdal2019), the particular solution to the kinematics does not involve the background shear effect, whereas (5.16)–(5.18a,b) include the background shear effect. If we set the mean wind velocity to be zero, (5.16) reduces to the formulation obtained by Åkervik & Vartdal (Reference Åkervik and Vartdal2019).

The equation for $\hat {w}^f$ has a non-zero source term on the right-hand side, but is not affected by the wave kinematics at the surface, we write

(5.19)\begin{align} & -\frac{\nu}{\mathrm{i}k}\left[\frac{\mathrm{d}^4}{\mathrm{d} \zeta^4}-2k^2\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2}+k^4\right]\hat{w}^f+\left[ (\langle u \rangle -c) \left(\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2}-k^2 \right)-\frac{\mathrm{d}^2 \langle u \rangle}{\mathrm{d} \zeta^2} \right]\hat{w}^f\nonumber\\ &\quad = \nu\hat{\eta}\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2} \left[g \frac{\mathrm{d}^2 \langle u \rangle}{\mathrm{d} \zeta^2} \right], \end{align}

with the homogeneous boundary conditions

(5.20a,b)\begin{gather} \left. \hat{w}^f \right\vert_{\zeta=0} = 0, \quad \left. \hat{w}^f\right\vert_{\zeta=\infty} = 0, \end{gather}

(5.21a,b)\begin{gather}\left.\frac{\mathrm{d} \hat{w}^f}{\mathrm{d} \zeta}\right\vert_{\zeta=0}= 0, \quad \left.\frac{\mathrm{d} \hat{w}^f}{\mathrm{d} \zeta}\right\vert_{\zeta=\infty} = 0. \end{gather}

Here, it is noted that following the previous studies (e.g. Miles Reference Miles1957; Belcher & Hunt Reference Belcher and Hunt1993; Miles Reference Miles1996; Hara & Sullivan Reference Hara and Sullivan2015), to discuss the mechanisms for wave-induced velocity, it is assumed that the mean wind velocity profile can be modelled through the mean flow momentum equation or it can be obtained experimentally. Therefore, we treat the right-hand side of (5.19) as a forcing to drive the wave-induced velocity, rather than a geometric correction to the mean wind velocity derivative given a specific wave-induced velocity forcing. In §§ 6.1 and 6.2, the equations (5.16)–(5.18a,b) and (5.19)–(5.21a,b) are solved separately, and their solutions are compared with the LES data to elucidate the effects of wave kinematics and forcing by the wave elevation on fast wave-induced airflow.

To conclude § 5, we have developed a linear-analysis framework for the fast wave effects on the overlying turbulent wind field. The scaling analysis shows that for high $c/u_\tau$, the nonlinear forcing has secondary effects on the wave-induced airflow compared with the advection induced by $\tilde {w}$ and the gradient of $\tilde {p}$. The curvilinear model of the wave boundary layer reveals two mechanisms for the wave effects on the airflow, namely, the wave kinematics and the forcing induced by the wave elevation, which can be investigated separately by the split curvilinear equations.

6.1. Mechanisms for dominant components of wave-induced airflow

So far, we have demonstrated that for fast waves, $\textrm {Re}[\hat {w}]$, $\textrm {Im}[\hat {u}]$ and $\textrm {Im}[\hat {p}]$ are the dominant components of $|\hat {w}|$, $|\hat {u}|$ and $|\hat {p}|$, respectively. In this subsection, we explain how they are produced. First, we identify the flow dynamics governing the dominant wave-induced airflow. Figure 10 compares the magnitudes and the dominant components of the wave-induced velocity and pressure between the LES results and the solutions of the curvilinear equation (5.5) for four fast waves. The agreement between the LES results and the linear solutions demonstrates that the dominant components of wave-induced airflow are controlled by linear dynamics. As shown, $|\hat {w}|$ and $|\hat {p}|$ have almost the same magnitudes as $\textrm {Re}[\hat {w}]$ and $\textrm {Im}[\hat {p}]$, respectively, throughout the boundary layer, indicating that the overall structures of $\tilde {w}$ and $\tilde {p}$ are determined by $\textrm {Re}[\hat {w}]$ and $\textrm {Im}[\hat {p}]$, respectively. Because $\textrm {Re}[\hat {w}]$ maintains a negative value and $\textrm {Im}[\hat {p}]$ has a positive sign, using (3.2) we can obtain that $\phi _{\tilde {w}\tilde {\eta }} \approx {\rm \pi}/2$ and $\phi _{\tilde {p}\tilde {\eta }} \approx -{\rm \pi}$, which explains the results shown in figures 4(c) and 6(c), respectively. By contrast, $|\hat {u}|$ has a similar magnitude to $\textrm {Im}[\hat {u}]$ only at $\zeta /\lambda \geqslant 0.01$, indicating that $\textrm {Im}[\hat {u}]$ determines the structure of $\tilde {u}$ there. The positive value of $\textrm {Im}[\hat {u}]$ for $\zeta /\lambda \geqslant 0.01$ suggests that $\phi _{\tilde {u}\tilde {\eta }} \approx -{\rm \pi}$. For $\zeta /\lambda < 0.01$, $\phi _{\tilde {u}\tilde {\eta }}$ transits from zero at the surface to $-{\rm \pi}$ farther away, which is caused by the change of the sign of $\textrm {Im}[\hat {u}]$ across this thin region. These results explain figure 9.

Figure 10. Comparison of the magnitudes (a,c,e) and the dominant components (b,d,f) of wave-induced airflow between the LES results and the solutions of the curvilinear equation (5.5) for the fast waves: (a) $|\hat {w}|$; (b) $\textrm {Re}[\hat {w}]$; (c) $|\hat {u}|$; (d) $\textrm {Im}[\hat {u}]$; (e) $|\hat {p}|$; and (f) $\textrm {Im}[\hat {p}]$. Lines are the LES results: solid black line, $(c/U_0, c/u_\tau )=(0.8, 33.61)$; solid cyan line, $(c/U_0, c/u_\tau )=(1.0, 44.64)$; solid blue line, $(c/U_0, c/u_\tau )=(1.2, 54.30)$; solid green line, $(c/U_0, c/u_\tau )=(1.4, 66.67)$. Symbols are the linear solutions: open black box, $(c/U_0, c/u_\tau )=(0.8, 33.61)$; open cyan box, $(c/U_0, c/u_\tau )=(1.0, 44.64)$; open blue box, $(c/U_0, c/u_\tau )=(1.2, 54.30)$; open green box, $(c/U_0, c/u_\tau )=(1.4, 66.67)$. Results for $ak=0.15$.

The quasilinear behaviour of the dominant wave-induced airflow suggests that the underlying mechanisms can be explained using the curvilinear models. The key is to identify the physical processes that generate $\textrm {Re}[\hat {w}]$, which can be elucidated using the split curvilinear equations obtained in § 5.3. As mentioned previously, (5.16) for $\hat {w}^k$ and (5.19) for $\hat {w}^f$ represent the effects of the wave kinematics and the forcing by wave elevation, respectively. Figure 11 compares $\textrm {Re}[\hat {w}^k]$ solved from (5.16), $\textrm {Re}[\hat {w}^f]$ solved from (5.19), and $\textrm {Re}[\hat {w}^k]+\textrm {Re}[\hat {w}^f]$ with the $\textrm {Re}[\hat {w}]$ result from LES for $ak=0.1$ and $0.15$. As shown, for both wave steepnesses, $\textrm {Re}[\hat {w}]$ is contributed to almost entirely by $\textrm {Re}[\hat {w}^k]$ with near zero contribution from $\textrm {Re}[\hat {w}^f]$, indicating that the wave kinematics plays a primary role in perturbing the vertical airflow velocity. To further investigate how the wave kinematics produces $\textrm {Re}[\hat {w}^k]$, we examine the boundary conditions for $\textrm {Re}[\hat {w}^k]$, which can be derived from (5.17a,b) and (5.18a,b), and read

(6.4a,b)\begin{equation} \left.\textrm{Re}[\hat{w}^k]\right\vert_{\zeta=0}={-}\frac{1}{2}akc, \quad \left.\frac{\mathrm{d} \,\textrm{Re}[\hat{w}^k] }{\mathrm{d} \zeta}\right\vert_{\zeta=0} ={-}\frac{1}{2}ak^2c - \frac{1}{2}ak\left.\left(g\frac{\mathrm{d} \langle u \rangle}{\mathrm{d} \zeta}\right) \right\vert_{\zeta=0}.\end{equation}

We find that the vertical component of wave orbital velocity $w_s$ (2.6) initiates a perturbation to the vertical air velocity $\textrm {Re}[\hat {w}^k]$ at the wave surface through the Dirichlet boundary condition (6.4a), and the streamwise component of wave orbital velocity $u_s$ (2.4) only affects the vertical derivative of $\textrm {Re}[\hat {w}^k]$, i.e. $\mathrm {d}\, \textrm {Re}[\hat {w}^k]/\mathrm {d} \zeta$, at the wave surface (6.4b). Away from the wave surface, $\textrm {Re}[\hat {w}^k]$ decays monotonically to zero without changing its sign as shown in figure 11.

Figure 11. Comparison of $\textrm {Re}[\hat {w}^k]$ solved from (5.16), $\textrm {Re}[\hat {w}^f]$ solved from (5.19), $\textrm {Re}[\hat {w}^k]+\textrm {Re}[\hat {w}^f]$ and the $\textrm {Re}[\hat {w}]$ result from the LES for (a) case $(c/U_0, c/u_\tau , ak)=(1.2, 50.42, 0.10)$ and (b) case $(c/U_0, c/u_\tau , ak)=(1.2, 54.30, 0.15)$. Symbols are blue circle, $\textrm {Re}[\hat {w}^k]$; cyan diamond, $\textrm {Re}[\hat {w}^f]$; open green box, $\textrm {Re}[\hat {w}^k]+\textrm {Re}[\hat {w}^f]$; and solid black line, LES result of $\textrm {Re}[\hat {w}]$. Note that $\textrm {Re}[\hat {w}^k]$ and $\textrm {Re}[\hat {w}^k]+\textrm {Re}[\hat {w}^f]$ collapse together.

Based on the results of $\textrm {Re}[\hat {w}]$, the mechanisms for the dominant components of wave-induced streamwise air velocity $\textrm {Im}[\hat {u}]$ and air pressure $\textrm {Im}[\hat {p}]$ can also be explained. Specifically, $\textrm {Im}[\hat {u}]$ occurs as a result of mass conservation in response to the $\textrm {Re}[\hat {w}]$-associated airflow perturbation, which can be obtained from the continuity equation (5.4) as

(6.5)\begin{equation} \textrm{Im}[\hat{u}] = \frac{1}{k} \frac{\mathrm{d}\,\textrm{Re}[\hat{w}]}{\mathrm{d}\zeta}+\frac{1}{2}ag\frac{\mathrm{d} \langle u \rangle}{\mathrm{d}\zeta}. \end{equation}

Equation (6.5) explains that the changing sign of $\textrm {Im}[\hat {u}]$ (figure 10d) and the associated variation of $\phi _{\tilde {u}\tilde {\eta }}$ (figure 9) in the wave boundary layer are because the effects of $u_s$ on $\textrm {Im}[\hat {u}]$, through the Dirichlet boundary condition (2.4), are limited to the proximity of the wave surface and the behaviour of $\textrm {Im}[\hat {u}]$ at higher altitude is determined primarily by the $\textrm {Re}[\hat {w}]$-associated airflow. Meanwhile, $\textrm {Im}[\hat {p}]$ is generated through the momentum conservation of the $\textrm {Re}[\hat {w}]$-associated airflow, which can be seen from the contribution to $\textrm {Im}[\hat {p}]$ derived from (5.3)

(6.6)\begin{equation} \textrm{Im}[\hat{p}] = \int_\zeta^\infty \left[(\langle u \rangle-c)k\,\textrm{Re}[\hat{w}] - \nu \left(k^2 \,\textrm{Im}[\hat{w}]-\frac{\mathrm{d}^2 \,\textrm{Im}[\hat{w}]}{\mathrm{d}\zeta^2}\right)\right]\mathrm{d}\zeta, \end{equation}

where the second term on the right-hand side denotes the viscous stress effect and is much smaller than the first term. Equation (6.6) shows that a strong $\textrm {Re}[\hat {w}]$ leads to a large-magnitude $\textrm {Im}[\hat {p}]$ (figure 10f).

As a summary of this subsection, we have discovered that the dominant components of fast wave-induced velocity and pressure are directly driven by the wave kinematics at the water surface and can be described by the curvilinear models. Based on the split curvilinear equations, we have illustrated that the vertical wave motion ($w_s$) initiates a strong vertical air velocity at the wave surface, which gradually decays away from the wave surface and causes a strong $\textrm {Re}[\hat {w}]$ in the airflow. The $\textrm {Re}[\hat {w}]$ further results in $\textrm {Im}[\hat {u}]$ and $\textrm {Im}[\hat {p}]$ in the air through the conservation of mass and momentum, respectively.

6.2. Mechanisms for weak components of wave-induced airflow

In § 6.1, we have explained the occurrence of $\textrm {Re}[\hat {w}]$, $\textrm {Im}[\hat {u}]$ and $\textrm {Im}[\hat {p}]$, i.e. the dominant components of the fast wave-induced airflow, using the curvilinear equations of the wave boundary layer. In this subsection, we further investigate the physical processes producing the weak components $\textrm {Im}[\hat {w}]$, $\textrm {Re}[\hat {u}]$ and $\textrm {Re}[\hat {p}]$. To show the flow dynamics governing these weak components of wave-induced airflow, figure 12 compares the profiles of $\textrm {Im}[\hat {w}]$, $\textrm {Re}[\hat {u}]$ and $\textrm {Re}[\hat {p}]$ between the LES results and the solutions of the curvilinear equation (5.5) for the fast waves. In all four of the cases shown, the linear solutions agree well with the LES results, suggesting that similar to their strong counterparts, $\textrm {Im}[\hat {w}]$, $\textrm {Re}[\hat {u}]$ and $\textrm {Re}[\hat {p}]$ are also controlled by linear dynamics. For $\tilde {w}$ and $\tilde {p}$, compared with $\textrm {Re}[\hat {w}]$ and $\textrm {Im}[\hat {p}]$ (figure 10b,f), it is found that $|\textrm {Im}[\hat {w}]|<0.04|\textrm {Re}[\hat {w}]|$ and $|\textrm {Re}[\hat {p}]|<0.04|\textrm {Im}[\hat {p}]|$ throughout the boundary layer (figure 12a,c). Therefore, the contributions from $\textrm {Im}[\hat {w}]$ and $\textrm {Re}[\hat {p}]$ to the magnitude, $|\hat {w}|$ and $|\hat {p}|$ (figure 10a,e) and the phase difference, $\phi _{\tilde {w}\tilde {\eta }}$ and $\phi _{\tilde {p}\tilde {\eta }}$, are negligible. However, for $\tilde {u}$, $\textrm {Re}[\hat {u}]$ (figure 12b) is comparable to $\textrm {Im}[\hat {u}]$ (figure 10d) near the surface, $\zeta /\lambda \leqslant 0.01$ and, therefore, $\phi _{\tilde {u}\tilde {\eta }}$ (figure 9) varies significantly across this thin region. Away from the wave surface, $\zeta /\lambda \geqslant 0.01$, $\textrm {Re}[\hat {u}]$ becomes negligible compared with $\textrm {Im}[\hat {u}]$ and correspondingly, $\phi _{\tilde {u}\tilde {\eta }} \approx -{\rm \pi}$ there (figure 9).

Figure 12. Comparison of the profiles of (a) $\textrm {Im}[\hat {w}]$, (b) $\textrm {Re}[\hat {u}]$ and (c) $\textrm {Re}[\hat {p}]$ between the LES results and the solutions of the curvilinear equation (5.5) for the fast waves. Lines are the LES results: solid black line, $(c/U_0, c/u_\tau )=(0.8, 33.61)$; solid cyan line, $(c/U_0, c/u_\tau )=(1.0, 44.64)$; solid blue line, $(c/U_0, c/u_\tau )=(1.2, 54.30)$; solid green line, $(c/U_0, c/u_\tau )=(1.4, 66.67)$. Symbols are the linear solutions: open black box, $(c/U_0, c/u_\tau )=(0.8, 33.61)$; open cyan box, $(c/U_0, c/u_\tau )=(1.0, 44.64)$; open blue box, $(c/U_0, c/u_\tau )=(1.2, 54.30)$; open green box, $(c/U_0, c/u_\tau )=(1.4, 66.67)$. Results for $ak=0.15$.

The quasilinear behaviour of the weak components of fast wave-induced airflow already shown suggests that we can use the split curvilinear equations (5.16) and (5.19) developed in § 5.3 to explain the underlying mechanism. As discussed previously, in (5.16), a strong $\textrm {Re}[\hat {w}^k]$ directly results from the airflow perturbation induced by the vertical wave motion and it almost fully determines $\textrm {Re}[\hat {w}]$, with near-zero contribution from $\textrm {Re}[\hat {w}^f]$ in (5.19). However, this is not the case for $\textrm {Im}[\hat {w}^k]$. To illustrate this point, we first examine the boundary conditions for $\textrm {Im}[\hat {w}^k]$, which can be derived from (5.17a,b) and (5.18a,b), and read

(6.7a,b)\begin{equation} \left.\textrm{Im}[\hat{w}^k]\right\vert_{\zeta=0}=0, \quad \left.\frac{\mathrm{d} \,\textrm{Im}[\hat{w}^k] }{\mathrm{d} \zeta}\right\vert_{\zeta=0} = 0. \end{equation}

Equations (6.7a,b) show that in contrast to $\textrm {Re}[\hat {w}^k]$, $\textrm {Im}[\hat {w}^k]$ has homogeneous boundary conditions, indicating that $\textrm {Im}[\hat {w}^k]$ is not directly driven by the wave kinematics. Meanwhile, an examination of the expression of (5.16) shows that $\textrm {Im}[\hat {w}^k]$ can be forced by $\textrm {Re}[\hat {w}^k]$ and thereby occurs in the boundary layer. This can be seen by taking the imaginary part of (5.16) with respect to its advection term as

(6.8)\begin{equation} \frac{\nu}{k}\left[\frac{\mathrm{d}^4}{\mathrm{d} \zeta^4}-2k^2\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2}+k^4 \right]\textrm{Re}[\hat{w}^k] + \left[(\langle u \rangle -c) \left(\frac{\mathrm{d}^2}{\mathrm{d} \zeta^2}-k^2\right)-\frac{\mathrm{d}^2 \langle u \rangle}{\mathrm{d} \zeta^2}\right]\textrm{Im}[\hat{w}^k] = 0. \end{equation}

Equation (6.8) shows that it is $\textrm {Re}[\hat {w}^k]$ that appears in the viscous term and $\textrm {Im}[\hat {w}^k]$ in the advection term for the two components of $\hat {w}^k$ to be coupled. In other words, because of the wave-induced viscous stress, a strong $\textrm {Re}[\hat {w}^k]$ can lead to the occurrence of $\textrm {Im}[\hat {w}^k]$. In (5.19) for $\hat {w}^f$, it is shown that similar to $\textrm {Re}[\hat {w}^f]$, $\textrm {Im}[\hat {w}^f]$ is also driven by the right-hand-side forcing induced by the wave elevation.

Figure 13 shows the comparison for $\textrm {Im}[\hat {w}^k]$ solved from (5.16), $\textrm {Im}[\hat {w}^f]$ solved from (5.19), $\textrm {Im}[\hat {w}^k]+\textrm {Im}[\hat {w}^f]$ and the $\textrm {Im}[\hat {w}]$ result from the LES for $ak=0.1$ and $0.15$. As shown, for both wave steepnesses, $\textrm {Im}[\hat {w}^k]$ has a magnitude larger than $\textrm {Im}[\hat {w}^f]$. However, the contribution from $\textrm {Im}[\hat {w}^f]$ to $\textrm {Im}[\hat {w}]$ is still appreciable, especially for the higher wave steepness (figure 13b). These results suggest that the small-magnitude $\textrm {Im}[\hat {w}]$ (figure 13a) is mainly generated through the forcing by $\textrm {Re}[\hat {w}]$, but also with an appreciable contribution from the forcing by the wave elevation.

Figure 13. Comparison of $\textrm {Im}[\hat {w}^k]$ solved from (5.16), $\textrm {Im}[\hat {w}^f]$ solved from (5.19), $\textrm {Im}[\hat {w}^k]+\textrm {Im}[\hat {w}^f]$ and the $\textrm {Im}[\hat {w}]$ result from LES for (a) case $(c/U_0, c/u_\tau , ak)=(1.2, 50.42, 0.10)$ and (b) case $(c/U_0, c/u_\tau , ak)=(1.2, 54.30, 0.15)$. Symbols are blue circle, $\textrm {Im}[\hat {w}^k]$; cyan diamond, $\textrm {Im}[\hat {w}^f]$; open green box, $\textrm {Im}[\hat {w}^k]+\textrm {Im}[\hat {w}^f]$; and solid black line, LES result of $\textrm {Im}[\hat {w}]$.

Similar to the dominant components discussed in § 6.1, a weak $\textrm {Im}[\hat {w}]$ can generate $\textrm {Re}[\hat {u}]$ and $\textrm {Re}[\hat {p}]$ through the conservation of mass and momentum of the wave-induced airflow, respectively, as

(6.9)\begin{gather} \textrm{Re}[\hat{u}] ={-}\frac{1}{k} \frac{\mathrm{d}\,\textrm{Im}[\hat{w}]}{\mathrm{d}\zeta}, \end{gather}

(6.10)\begin{gather}\textrm{Re}[\hat{p}]={-}\int_\zeta^\infty \left[ (\langle u \rangle-c)k\,\textrm{Im}[\hat{w}] + \nu \left(k^2 \,\textrm{Re}[\hat{w}]-\frac{\mathrm{d}^2 \,\textrm{Re}[\hat{w}]}{\mathrm{d}\zeta^2} \right) \right] \,\mathrm{d}\zeta, \end{gather}

which can be derived from the continuity equation (5.4) and the vertical momentum equation (5.3), respectively. Equations (6.9) and (6.10) explain the results of figures 12(b) and 12(c), respectively.

As a summary of this subsection, we have shown that the weak components of fast wave-induced airflow can also be described by the curvilinear models. Unlike their strong counterparts, the weak components have homogeneous boundary conditions and are not directly affected by the wave kinematics at the water surface. With the split curvilinear equations, it is discovered that the weak components of wave-induced airflow are forced by their strong counterparts through wave-induced viscous stress and the forcing induced by the wave elevation.

6.3. Mechanisms for wave-coherent stress and form drag

The proceeding subsections explain the occurrence of $\tilde {w}$, $\tilde {u}$ and $\tilde {p}$ in the airflow using the linear-analysis framework. In this subsection, we analyse the contributions from the strong and weak components of the fast wave-induced airflow to the wave-coherent stress $-\tilde {u} \tilde {w}$ in the wind and the form drag on the wave surface. The $-\tilde {u} \tilde {w}$ can be decomposed into a mean component and a fluctuation component as

(6.11)\begin{equation} -\tilde{u} \tilde{w} ={-}\langle \tilde{u}\tilde{w} \rangle -\widetilde{\tilde{u} \tilde{w}} ={-}\langle \tilde{u}\tilde{w} \rangle -\widehat{\tilde{u}\tilde{w}}\mathrm{e}^{\mathrm{i}2k\xi}-\widehat{\tilde{u} \tilde{w}}^*\mathrm{e}^{-\mathrm{i}2k\xi}. \end{equation}

Figure 14 compares the profiles of $-\langle \tilde {u}\tilde {w} \rangle$ and $|\widehat {\tilde {u} \tilde {w}}|$ between the solutions of the viscous curvilinear equation (5.5) and the LES results, with good agreement obtained. Figure 14(a) shows that $-\langle \tilde {u}\tilde {w} \rangle$ is significant only near the surface in the region where $\textrm {Re}[\hat {u}]$ (figure 12b) is appreciable, namely, $\zeta /\lambda \leqslant 0.01$, because $-\langle \tilde {u}\tilde {w} \rangle =-2(\textrm {Re}[\hat {u}]\,\textrm {Re}[\hat {w}]+\textrm {Im}[\hat {u}]\,\textrm {Im}[\hat {w}])$, in which $-2\,\textrm {Re}[\hat {u}]\,\textrm {Re}[\hat {w}]$ plays a dominant role because $|\textrm {Re}[\hat {w}]| \gg |\textrm {Im}[\hat {w}]|$. This result suggests that the weak component of $\tilde {u}$, i.e. $\textrm {Re}[\hat {u}]$, is crucial for generating a mean $-\tilde {u} \tilde {w}$ in the wind, i.e. $-\langle \tilde {u}\tilde {w} \rangle$. Figure 14(b) shows that compared with $-\langle \tilde {u}\tilde {w} \rangle$, $|\widehat {\tilde {u} \tilde {w}}|$ is significant up to a much higher altitude, $\zeta /\lambda \simeq 0.2$, because $|\widehat {\tilde {u} \tilde {w}}|$ is dominated by $-\textrm {Im}[\hat {u}]\,\textrm {Re}[\hat {w}]$ (because $|\textrm {Re}[\hat {w}]| \gg |\textrm {Im}[\hat {w}]|$) and both $\textrm {Im}[\hat {u}]$ and $\textrm {Re}[\hat {w}]$ are significant there. Based on this result, we see that the air velocity perturbations directly induced by the vertical wave motion, namely, $\textrm {Im}[\hat {u}]$ and $\textrm {Re}[\hat {w}]$, cause a strong wave-induced fluctuation of $-\tilde {u} \tilde {w}$, i.e. $-\widetilde {\tilde {u} \tilde {w}}$.

Figure 14. Comparison of (a) $-\langle \tilde {u}\tilde {w} \rangle$ and (b) $|\widehat {\tilde {u}\tilde {w}}|$ between the LES results and the solutions of the curvilinear equation (5.5) for the fast waves. Lines are the LES results: solid black line, $(c/U_0, c/u_\tau )=(0.8, 33.61)$; solid cyan line, $(c/U_0, c/u_\tau )=(1.0, 44.64)$; solid blue line, $(c/U_0, c/u_\tau )=(1.2, 54.30)$; solid green line, $(c/U_0, c/u_\tau )=(1.4, 66.67)$. Symbols are the linear solutions: open black box, $(c/U_0, c/u_\tau )=(0.8, 33.61)$; open cyan box, $(c/U_0, c/u_\tau )=(1.0, 44.64)$; open blue box, $(c/U_0, c/u_\tau )=(1.2, 54.30)$; open green box, $(c/U_0, c/u_\tau )=(1.4, 66.67)$. Results for $ak=0.15$.

The wave-induced velocity results in $-\tilde {u} \tilde {w}$ affecting the momentum transfer within the airflow, and the wave-induced pressure can lead to a form drag on the wave surface, which is the dominant mechanism for the momentum exchange between wind and waves (Belcher & Hunt Reference Belcher and Hunt1998). The form drag $F_p$ is defined as

(6.12)\begin{equation} F_p = \frac{1}{\rho_a u_\tau^2} \int_{0}^{\lambda} \tilde{p}\vert_{\zeta=0} \tilde{\eta}_x \,\mathrm{d} x = \left.\frac{ak \,\textrm{Re}[\hat{p}]}{\rho_a u_\tau^2}\right\vert_{\zeta=0}. \end{equation}

Equation (6.12) shows that $F_p$ can only be induced by the weak component of wave-induced pressure, i.e. $\textrm {Re}[\hat {p}]$. With $F_p$ on the surface, the resultant wave growth-rate parameter $\beta$ is (e.g. Donelan Reference Donelan1999; Li, Xu & Taylor Reference Li, Xu and Taylor2000; Yang et al. Reference Yang, Meneveau and Shen2013)

(6.13)\begin{equation} \beta = \frac{2F_p}{(ak)^2}, \end{equation}

where a positive $\beta$ corresponds to the wave growth, whereas a negative $\beta$ causes the wave to decay.

Figure 15 shows the variation of the wave growth rate parameter $\beta$ as a function of wave age $c/u_\tau$ for all of the fast wave cases in table 1. As shown, the theoretical study of wind over fast wave by Cohen (Reference Cohen1997) and the numerical study using Reynolds-averaged Navier–Stokes (RANS) equations by Mastenbroek (Reference Mastenbroek1996) investigated the range of $c/u_\tau$ up to about $35$. The present study has significantly extended the upper limit of wave age investigated in the literature to $c/u_\tau \approx 67$. The comparison between the LES results and the solutions of the curvilinear equation (5.5) shows good agreement with each other, suggesting that the physical process causing a wind–wave momentum flux at high wave age, $c/u_\tau \gtrsim 33$, is governed by linear dynamics and can be described by the curvilinear equations of wave boundary layer. This result is consistent with the recent finding of Åkervik & Vartdal (Reference Åkervik and Vartdal2019) that as $c/u_\tau$ approaches $36$, the effects of wave-induced viscous stress on the form drag become increasingly more significant, whereas the effects of wave-induced turbulent stress decrease. Åkervik & Vartdal (Reference Åkervik and Vartdal2019) also pointed out that the critical $c/u_\tau$, where the effects of nonlinear forcing on form drag vanish, depend on the Reynolds number. Below the critical $c/u_\tau$, the form drag is mainly induced by the wave-induced turbulent stress, which has been examined by Cohen (Reference Cohen1997) and Mastenbroek (Reference Mastenbroek1996). As a reference, the values of wave growth rate at low wave age collated by Plant (Reference Plant1982) are also included in figure 15. It is shown that the magnitude of wave attenuation rate for $c/u_\tau \gtrsim 45$ is comparable to that of the wave growth rate for $c/u_\tau < 20$, indicating that the strength of wind–wave momentum flux for fast waves can be as large as that for slow waves. This result is reasonable, because for fast waves, although $\textrm {Re}[\hat {p}]$ has a magnitude much smaller than $\textrm {Im}[\hat {p}]$, it is not small compared with $\rho _a u_\tau ^2$ (figure 12c), which is the mean total stress in the wind field. Therefore, the form drag and wave attenuation rate induced by $Re[\hat {p}]$ are not small for fast waves.

Figure 15. Variation of wave growth rate parameter $\beta$ (6.13) as a function of wave age $c/u_\tau$. Data shown are the solutions of (5.5) for $ak=0.10$ (line blue circle) and $ak=0.15$ (line green diamond); the LES results for $ak=0.10$ (line black box) and $ak=0.15$ (line black triangle); the results of simulations using Reynolds-averaged Navier–Stokes (RANS) equations by Mastenbroek (Reference Mastenbroek1996) (dashed line black box); and the results of Cohen (Reference Cohen1997) (dashed line black circle). As a reference, the values of wave growth rate at low wave age collated by Plant (Reference Plant1982) are also included (open black box).

Here, we note that the wave growth rate $\beta$ is more challenging to predict than the wave-induced velocity. Equations (6.12) and (6.13) show that $\beta$ depends on the weak wave-induced pressure at the wave surface, $\textrm {Re}[\hat {p}](\zeta =0)$, which is computed through (6.10). Equation (6.10) reveals two processes generating $\textrm {Re}[\hat {p}](\zeta =0)$, namely, the integration of the advection by $\textrm {Im}[\hat {w}]$ in the boundary layer and the integration of the viscous forcing by $\textrm {Re}[\hat {w}]$. In (6.10), $\textrm {Re}[\hat {w}]$ affects $\textrm {Re}[\hat {p}](\zeta =0)$ mainly through its second-order vertical derivative, i.e. $\mathrm {d}^2\,\textrm {Re}[\hat {w}]/\mathrm {d} \zeta ^2$, near the wave surface. Although the curvilinear model can predict $\textrm {Re}[\hat {w}]$ very well, $\mathrm {d}^2\,\textrm {Re}[\hat {w}]/\mathrm {d} \zeta ^2$ is much more challenging to determine. For higher wave ages, $\textrm {Re}[\hat {w}]$ becomes increasingly stronger than $\textrm {Im}[\hat {w}]$ (figures 10 and 12) and, thus, can affect $\textrm {Re}[\hat {p}](\zeta =0)$ more significantly. Therefore, a slight deviation in $\mathrm {d}^2\,\textrm {Re}[\hat {w}]/\mathrm {d} \zeta ^2$ can cause more obvious discrepancy in $\textrm {Re}[\hat {p}](\zeta =0)$ and thereby $\beta$, for higher wave ages (figure 15). Nevertheless, in the present study, the error of $\beta$ in the curvilinear model for the highest wave age is still below $10\,\%$.

6.4. Effects of wave age on the behaviour of wave-induced airflow

In this subsection, we examine the effects of wave age on the behaviour of the wave-induced air velocity. Figure 16 compares the dominant ($\textrm {Re}[\hat {w}]$) and the weak ($\textrm {Im}[\hat {w}]$) components of the wave-induced vertical velocity for the intermediate wave case $(c/U_0, c/u_\tau , ak)=(0.4, 15.38, 0.15)$ for the solution of (5.16), the solution of (5.19), the summation of the solutions of (5.16) and (5.19) and the LES result. As shown, for $\textrm {Re}[\hat {w}]$ (figure 16a), the summation of the solutions of (5.16) and (5.19) agrees well with the LES result, indicating that the dominant component of the intermediate wave-induced airflow exhibits a quasilinear behaviour and can be described by the curvilinear model. Meanwhile, similar to the fast wave (figure 11), $\textrm {Re}[\hat {w}]$ for the intermediate wave also mainly results from the effects of wave kinematics. By contrast, for $\textrm {Im}[\hat {w}]$ (figure 16b), the summation of the solutions of (5.16) and (5.19) deviates significantly from the LES result, indicating that the weak component of the intermediate wave-induced airflow is not entirely governed by the linear dynamics. For the slow wave case $(c/U_0, c/u_\tau , ak)=(0.1, 3.46, 0.15)$, both $\textrm {Re}[\hat {w}]$ and $\textrm {Im}[\hat {w}]$ cannot be described by the curvilinear model (results are omitted here for space considerations), indicating that the occurrence of the slow wave-induced airflow is a nonlinear process.

Figure 16. Comparison of the dominant ($\textrm {Re}[\hat {w}]$) and the weak ($\textrm {Im}[\hat {w}]$) components of the wave-induced vertical velocity between the split curvilinear equations and the LES result for the intermediate wave case $(c/U_0, c/u_\tau , ak)=(0.4, 15.38, 0.15)$. Symbols are: blue circle, the solution of (5.16); cyan diamond, the solution of (5.19); open green box, the summation of the solutions of (5.16) and (5.19); and solid black line, the LES result.

To conclude § 6, we have demonstrated that the fast wave-induced airflow is governed by quasilinear dynamics and can be described by the curvilinear model. Based on the split curvilinear equations, we have discovered that the dominant components of the wave-induced airflow, namely, $\textrm {Re}[\hat {w}]$, $\textrm {Im}[\hat {u}]$ and $\textrm {Im}[\hat {p}]$, are produced by the airflow perturbations directly induced by the vertical component of wave orbital velocity. On the other hand, the weak components, namely, $\textrm {Im}[\hat {w}]$, $\textrm {Re}[\hat {u}]$ and $\textrm {Re}[\hat {p}]$, are forced by their strong counterparts and by the forcing induced by the wave elevation, and they cause a wave-induced momentum flux in the airflow and a wind–wave momentum exchange at the wave surface. In short, the linear-analysis framework developed in the present study works well in explaining the fast wave-induced airflow and the underlying mechanisms. As wave age decreases, our results show that the wave-induced airflow deviates from the quasilinear behaviour gradually, to become controlled by linear dynamics only for the strong components at intermediate wave age, and totally governed by nonlinear dynamics at slow wave age.