3.1. Wavenumber modulation

Solving analytically for the wavenumber requires dropping the spatial derivatives of $k$ in (2.5) and assuming homogeneous $g$

(3.1) \begin{equation} \dfrac {\partial k}{\partial t} = - k \dfrac {\partial U}{\partial x}. \end{equation}

Although Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021) pointed out that the solution by Longuet-Higgins & Stewart (Reference Longuet-Higgins and Stewart1960) requires assuming homogeneous group speed of the short-wave field, in fact, it also requires assuming no horizontal propagation and advection of the short waves. We combine (2.8) and (3.1) and integrate in time to get

(3.2) \begin{equation} \frac {\widetilde {k}}{k} = e^{\varepsilon _L \cos {\psi } e^{\varepsilon _L \cos {\psi }}}. \end{equation}

Notice that the Taylor expansion of (3.2) to the first order recovers the original solution by Longuet-Higgins & Stewart (Reference Longuet-Higgins and Stewart1960)

(3.3) \begin{equation} \frac {\widetilde {k}}{k} = 1 + \varepsilon _L \cos {\psi }. \end{equation}

Coming from the conservation of crests, as opposed to the velocity potential superposition, the solution by Longuet-Higgins & Stewart (Reference Longuet-Higgins and Stewart1960) requires two approximations: first, evaluating the long-wave velocity at $z = 0$ rather than $z = \eta _L$ ; and second, truncating the Taylor expansion series beyond ${\mathcal{O}}(\varepsilon _L)$ . These approximations underestimate the short-wave modulation magnitude (figure. 2). At the wave crest, (3.2) yields the wavenumber modulation that is 16.9 %, 38.5 %, 66.4 % and 104 % larger than that of Longuet-Higgins & Stewart (Reference Longuet-Higgins and Stewart1960) for $\varepsilon _L = 0.1, 0.2, 0.3, 0.4$ , respectively. Although not immediately obvious from the expressions, the modulated wavenumber (3.3) is conserved across the long-wave phase, whereas the higher-order solution (3.2) is not. This is because combining the orbital velocities evaluated at $z=\eta _L$ and the linearised conservation of wave crests in (2.5) creates excess short-wave wavenumber. The solution of (2.5) must be conservative across the long-wave phase, as we will see later from the numerical simulations. The steady analytical solutions for modulation of short-wave effective gravity, amplitude, steepness, intrinsic frequency and phase speed are also shown in figure 2, for reference, however, their formulations appear in the following subsections. Although the analytical solutions by Longuet-Higgins & Stewart (Reference Longuet-Higgins and Stewart1960) have been updated by the numerical results of Longuet-Higgins (Reference Longuet-Higgins1987), they are here nevertheless relevant to use as reference for the proposed analytical solutions. This comparison allows the quantification of solely the effect of evaluating the long-wave properties at the surface rather than at the mean water level.

Figure 2.

Steady analytical solutions for the modulation of short-wave (a) wavenumber, (b) gravitational acceleration, (c) amplitude, (d) steepness, (e) intrinsic frequency and (f) intrinsic phase speed as functions of long-wave phase for $\varepsilon _L = 0.1$ , based on Longuet-Higgins & Stewart (Reference Longuet-Higgins and Stewart1960) (L-HS 1960, black) and this paper (blue). Long-wave crest and trough are located at $\psi = 0$ and $\psi = \pi$ , respectively.

3.2. Gravity modulation

Three distinct effects contribute to the modulation of the effective gravity of short waves in the presence of long waves. First, long waves induce orbital (centripetal) accelerations on their surface, so the short waves ride in an accelerated reference frame and experience an effective gravity that is lower at the crests and higher in the troughs (Longuet-Higgins Reference Longuet-Higgins1986, Reference Longuet-Higgins1987). This is an ${\mathcal{O}}(\varepsilon _L)$ effect. Second, it is important to distinguish between the Eulerian and Lagrangian effective gravities (Longuet-Higgins Reference Longuet-Higgins1986). The Eulerian gravity is that which would be measured by a fixed wave staff, for example, and features sharp and strong minima at the crests and broad and weak maxima in the troughs. In contrast, the Lagrangian gravity is that which is experienced by a short-wave group that propagates with its own group speed and is advected by the ambient, long-wave-induced velocity. The Lagrangian velocity is overall smoother and more attenuated compared with its Eulerian counterpart because the local slope as experienced by the travelling short-wave group is gentler. The two gravities are related by

(3.4) \begin{equation} \widetilde {g_l} = g + \frac {{\textrm d}W}{{\textrm d}t} = g + \frac {\partial W}{\partial t} + \left (C_g + U\right ) \frac {\partial W}{\partial x} = \widetilde {g_e} + \left (C_g + U\right ) \frac {\partial W}{\partial x}, \end{equation}

where $l$ and $e$ subscripts serve to denote ‘Lagrangian’ and ‘Eulerian’, respectively. Although Longuet-Higgins (Reference Longuet-Higgins1986) thoroughly described the differences between the Eulerian and Lagrangian effective gravities, he did so for a fully nonlinear irrotational gravity wave, which can be computed numerically but not analytically. Note also that, when evaluating the Lagrangian effective gravity, Longuet-Higgins (Reference Longuet-Higgins1987) considered the orbital motion of the long wave but not the propagation speed of the short-wave group, which can exceed the ambient velocity $U$ depending on the properties of the short and long waves considered. Group speed of the short waves aside, the Lagrangian correction to the gravity being an advective term is ${\mathcal{O}}(\varepsilon _L^2)$ . Third, the projection of the long-wave-induced horizontal acceleration in the curvilinear coordinate system contributes small but non-negligible effects on the short-wave effective gravity (Phillips Reference Phillips1981; Zhang & Melville Reference Zhang and Melville1990). These three effects are completely described by projecting the gravitational acceleration vector from the coordinate that is perpendicular to the mean water surface ( $z=0$ ) to that which is perpendicular to the long-wave surface ( $z=\eta _L$ ), and likewise for the long-wave orbital accelerations

(3.5) \begin{equation} \frac {\widetilde {g}}{g} = \cos {\alpha } + \frac {1}{g} \dfrac {{\textrm d}W_{z=\eta _L}}{{\textrm d}t} \cos {\alpha } + \frac {1}{g} \dfrac {{\textrm d}U_{z=\eta _L}}{{\textrm d}t} \sin {\alpha }, \end{equation}

(3.6) \begin{equation} \alpha = \tan ^{-1}{\dfrac {\partial \eta }{\partial x}}, \end{equation}

where $\alpha$ is the local slope of the long-wave surface. For short waves on a linear long wave, the Eulerian gravity modulation is

(3.7) \begin{equation} \frac {\widetilde {g_e}}{g} = \frac { 1 - \varepsilon _L \cos {\psi } e^{\varepsilon _L \cos {\psi }} \left [ 1 + \left (\varepsilon _L \sin {\psi }\right )^2 e^{\varepsilon _L \cos {\psi }} \right ] } {\sqrt {\left (\varepsilon _L \sin {\psi }\right )^2 + 1}}. \end{equation}

The same derivation can be carried out for the Lagrangian gravity modulation following (3.4)–(3.6), and is omitted here for brevity. Curvilinear effects on effective gravity can be removed altogether by setting $\alpha = 0$ . In that case, (3.7) simplifies to

(3.8) \begin{equation} \frac {\widetilde {g_e}}{g} = 1 - \varepsilon _L \cos {\psi } e^{\varepsilon _L \cos {\psi }}. \end{equation}

If we evaluate the orbital accelerations at the mean water level $z=0$ , this further simplifies to

(3.9) \begin{equation} \frac {\widetilde {g_e}}{g} = 1 - \varepsilon _L \cos {\psi }, \end{equation}

which is the form used by Longuet-Higgins & Stewart (Reference Longuet-Higgins and Stewart1960) and Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021), for example. The Lagrangian correction in (3.4) is

(3.10) \begin{equation} U \frac {\partial W}{\partial x} = g \varepsilon _L^2 \cos ^2{\psi } e^{2 \varepsilon _L \cos {\psi }} + {\mathcal{O}}\left(\varepsilon _L^3\right), \end{equation}

and so it becomes important only for large $\varepsilon _L$ .

The general form of the gravity modulation (3.5) is convenient because it allows us to easily attribute the modulation to different processes. It also makes it straightforward to evaluate the gravity modulation for different long-wave forms, by evaluating the orbital accelerations and the local slope for each wave form. For a third-order Stokes wave, for example, the elevation is

(3.11) \begin{equation} \eta _{St} = a_L \left [ \cos {\psi } + \dfrac {1}{2} \varepsilon _L \cos {2\psi } + \varepsilon _L^2 \left ( \dfrac {3}{8} \cos {3\psi } - \dfrac {1}{16} \cos {\psi } \right ) \right ] + {\mathcal{O}}\left(\varepsilon _L^4\right). \end{equation}

Without considering the curvilinear effects in (3.5), the gravitational acceleration at the surface of a Stokes wave is, to the third order

(3.12) \begin{equation}\begin{aligned} \frac {\widetilde {g}}{g} &= \!\left \{ 1 - \varepsilon _L e^{k \eta _{St}}\! \left [ \cos {\psi } - \varepsilon _L \sin {\psi } \!\left ( \sin {\psi } + \varepsilon _L \sin {2\psi } - \dfrac {1}{16} \varepsilon _L^2 \sin {\psi } + \dfrac {9}{8} \varepsilon _L^2 \sin {3\psi }\! \right )\! \right ]\! \right \}.\\[6pt] \end{aligned}\end{equation}

The equivalent expression that includes the curvilinear effects can be derived by evaluating the velocities and the local slope $\alpha$ at $z = \eta _{St}$ , and inserting them into (3.5)

(3.13) \begin{equation} \frac {\widetilde {g}}{g} = \cos {\alpha _{St}} + \dfrac {{\textrm d}W_{z=\eta _{St}}}{{\textrm d}t} \cos {\alpha _{St}} + \dfrac {{\textrm d}U_{z=\eta _{St}}}{{\textrm d}t} \sin {\alpha _{St}}, \end{equation}

(3.14) \begin{equation} \alpha _{St} = \tan ^{-1}{\dfrac {\partial \eta _{St}}{\partial x}}. \end{equation}

Let us now examine the contributions of the different terms to the short-wave effective-gravity modulation. Figure 3(a) shows the effective-gravity modulation of short waves as a function of the long-wave phase for $\varepsilon _L = 0.3$ , for a variety of the gravity modulation forms. Although quite steep, this value of $\varepsilon _L$ allows easier visualisation of the modulation differences. To ${\mathcal{O}}(\varepsilon _L)$ , the gravity modulation is out of phase with the long-wave elevation, with the short waves experiencing a lower effective gravity on the long-wave crests and a higher effective gravity in the troughs. Evaluating the orbital accelerations at $z=\eta$ and the Lagrangian gravity correction each have an ${\mathcal{O}}(\varepsilon _L^2)$ contribution but of opposite signs. The former amplifies the gravity modulation while the latter attenuates it. The curvilinear effects are significant at the front and rear faces of the long wave but not near the crests and troughs. The long-wave surface slope has a ${\mathcal{O}}(\varepsilon _L^2)$ effect. Naturally, the slope is exactly zero at the crests and the troughs, and its effects are largest at the front and rear faces of the long wave, where the covariance between the local slope and the horizontal accelerations is largest. Evaluating the velocities and accelerations at the surface of a third-order Stokes wave causes a correction mostly at the faces of the long wave. Finally, numerically computing the surface velocities and accelerations of a fully nonlinear wave allows us to exclude uncertainties associated with the truncation errors of the Stokes expansion at first and third orders in $\varepsilon _L$ . At this steepness, the fully nonlinear solution is marginally different from that of the third-order Stokes wave (see also Appendix B for the errors in surface velocity estimates of the linear and third-order Stokes approximations).

Figure 3.

Analytical solutions for the effective gravitational acceleration modulation by long waves, as functions of the long-wave phase, for $\varepsilon _L = 0.3$ . Black is for Eulerian gravity of a linear wave evaluated at $z=0$ ; blue is the same as black but evaluated at $z=\eta$ ; orange is the same as blue but for Lagrangian gravity; green is the same as orange but in a curvilinear reference frame; red is the same as green but for a third-order Stokes wave; and purple is the same as red but for a fully nonlinear wave. The elevation and surface velocities for the fully nonlinear wave are computed following Clamond & Dutykh (Reference Clamond and Dutykh2018). Long-wave crest and trough are located at $\psi = 0$ and $\psi = \pi$ , respectively.

It is instructive to also examine the minimum values of the effective gravity as functions of long-wave steepness $\varepsilon _L$ (figure 4), as the minima occur at or near the long-wave crests where the short-wave modulation is largest. Without the Lagrangian correction to the effective gravity, the gravity modulation is overestimated; for accelerations evaluated at the surface of a linear wave, the maximum gravity reduction is 60 % at $\varepsilon _L = 0.4$ . The Lagrangian correction attenuates the gravity modulation, making the gravity reduction at the crests $\approx 25\, \%$ . As the slope effects on the effective gravity exist only away from crests and troughs, they may be neglected in the steady solutions where modulation is typically evaluated at the long-wave crests. Here, the slope effect causes a difference in the minimum effective gravity for $\varepsilon _L \gtrsim 0.3$ . The effective-gravity reduction at the crests in the case of a fully nonlinear wave is $\approx$ 10 %, 19 %, 27 % and 40 %, for $\varepsilon _L$ of 0.1, 0.2, 0.3 and 0.4, respectively. These are similar to the fully nonlinear numerical estimates of Longuet-Higgins (Reference Longuet-Higgins1986, Reference Longuet-Higgins1987).

Figure 4.

As figure 3 but showing the minimum short-wave gravity modulation as a function of long-wave steepness $\varepsilon _L$ .

3.3. Amplitude and steepness modulation

The modulation of short-wave amplitude can be derived in a similar way as we did for the wavenumber, except that here we linearise the wave action balance (2.6) and integrate it in time

(3.15) \begin{equation} \frac {\widetilde {N}}{N} = e^{\varepsilon _L \cos {\psi } e^{\varepsilon _L \cos {\psi }}}. \end{equation}

Since wave action is energy divided by the intrinsic frequency $\sigma$ , and energy scales with $ga^2$ , the modulation of the amplitude is related to the modulations of gravity, wavenumber and action

(3.16) \begin{equation} \dfrac {\widetilde {a}}{a} = \sqrt { \dfrac {g}{\widetilde {g}} \dfrac {\widetilde {\sigma }}{\sigma } \dfrac {\widetilde {N}}{N}} = \left ( \dfrac {\widetilde {k}}{k} \right )^{\frac {1}{4}} \left ( \dfrac {\widetilde {N}}{N} \right )^{\frac {1}{2}} \left ( \dfrac {\widetilde {g}}{g} \right )^{-\frac {1}{4}}. \end{equation}

To ${\mathcal{O}}(\varepsilon _L)$ , and using (3.2) and (3.15), the above simplifies to the result of Longuet-Higgins & Stewart (Reference Longuet-Higgins and Stewart1960)

(3.17) \begin{equation} \dfrac {\widetilde {a}}{a} = \left ( \dfrac {\widetilde {g}}{g} \right )^{-\frac {1}{4}} \left ( 1 + \varepsilon _L \right )^{\frac {3}{4}} \approx \frac {(1 + \varepsilon _L)^{\frac {3}{4}}}{(1 - \varepsilon _L)^{\frac {1}{4}}} \approx 1 + \varepsilon _L. \end{equation}

The amplitude and steepness modulation of the short waves as a function of the long-wave phase for $\varepsilon _L = 0.1$ are compared with the Longuet-Higgins & Stewart (Reference Longuet-Higgins and Stewart1960) solutions in figures 2(c) and 2(d), respectively. Also shown are the modulation of the short-wave intrinsic frequency and phase speed in panels (e) and (f), respectively. As the wavenumber and gravity modulations are both ${\mathcal{O}}(\varepsilon _L)$ and out of phase, the intrinsic frequency modulation is ${\mathcal{O}}(\varepsilon _L^2)$ , positive on the long-wave faces and negative on the crests and troughs. For the short-wave phase speed modulation, the wavenumber and gravity modulations work in tandem and cause the short-wave phase speed to be reduced on the long-wave crests and increased in the troughs.

Finally, the short-wave steepness modulation follows from (3.16)

(3.18) \begin{equation} \dfrac {\widetilde {ak}}{ak} = \left ( \dfrac {\widetilde {k}}{k} \right )^{\frac {5}{4}} \left ( \dfrac {\widetilde {N}}{N} \right )^{\frac {1}{2}} \left ( \dfrac {\widetilde {g}}{g} \right )^{-\frac {1}{4}}. \end{equation}

The short-wave steepness modulation at the long-wave crests is thus largely controlled by the wavenumber modulation (5/8 or 62.5 %), followed by the convergence of wave action (1/4 or 25 %), and by least amount by the effective-gravity reduction (1/8 or 12.5 %) (figure 5). Equation (3.18) and figure 5 demonstrate that more than half of the steepness modulation can be captured by the wavenumber modulation alone. Further, considering the wavenumber and wave action modulations, while neglecting the gravity modulation, captures $\approx$ 88 % of the steepness modulation at the long-wave crests. This result may be useful for the interpretation of remote sensing products that resolve some but not all of the short-wave modulation effects. Finally, the relative contributions of different modulation factors remain mostly constant as $\varepsilon _L$ increases, with only a mild decrease of the effective-gravity modulation contribution, and mild increases of the wavenumber and wave action modulation contributions.

Figure 5.

Contributions of short-wave wavenumber, action and effective-gravity modulations to the steepness modulation, as functions of the long-wave steepness $\varepsilon _L$ . Panel (a) shows each modulation factor (colour) and their product (black), and panel (b) shows their relative contributions in percentage, calculated as the ratio of the logarithm of each modulation factor to the logarithm of the product of all modulations.

3.4. Limits to the wave action conservation

Following the variational approach by Whitham (Reference Whitham1965), Bretherton & Garrett (Reference Bretherton and Garrett1968) showed that the wave action is conserved for small-amplitude (linear) waves that propagate in slowly varying media. Longuet-Higgins (Reference Longuet-Higgins1987) correctly pointed out both requirements – that both the medium and the short-wave train must be slowly varying – for the wave action conservation to hold. However, he incorrectly stated that there is no explicit restriction on the steepness of the long wave, and that it appears necessary to only assume small-amplitude short waves and significant scale separation ( $k/k_L$ ). The restriction on $\varepsilon _L$ becomes apparent when we recognise that the long wave-induced orbital velocity scales with $\varepsilon _L$ , and that the divergence of this velocity is the dominant term in (2.5) and (2.6). Here, we show that in the general case both the homogeneity and the stationarity of a short-wave quantity $q$ depend on the long-wave steepness $\varepsilon _L$ .

The homogeneity and stationarity of a scalar quantity $q$ can be expressed as conditions on the instantaneous fractional rate of change of $q$ being much smaller than the short-wave inverse scales

(3.19) \begin{equation} \frac {1}{q} \frac {\partial q}{\partial x} \ll k, \end{equation}

(3.20) \begin{equation} \frac {1}{q} \frac {\partial q}{\partial t} \ll \sigma. \end{equation}

The slowly varying conditions on the medium (in our case, the long-wave orbital velocity U) are then

(3.21) \begin{equation} \frac {1}{U} \frac {\partial U}{\partial x} \ll \widetilde {k}, \end{equation}

(3.22) \begin{equation} \frac {1}{U} \frac {\partial U}{\partial t} \ll \widetilde {\sigma }, \end{equation}

respectively. For a linear long wave, these conditions are equivalent to the scale separation between the long and short waves

(3.23) \begin{equation} k_L \ll \widetilde {k}, \end{equation}

(3.24) \begin{equation} \sigma _L \ll \widetilde {\sigma }. \end{equation}

Bretherton & Garrett (Reference Bretherton and Garrett1968) also require that the medium is nearly uniform over the vertical scales of motion of the short waves

(3.25) \begin{equation} \frac {1}{U} \frac {\partial U}{\partial z} \ll \widetilde {k}, \end{equation}

which, for a long deep-water wave, is equivalent to (3.23). Alternatively, if we use the second Froude number criterion following Ellingsen (Reference Ellingsen2014) and quantify the shear length under the long-wave surface as

(3.26) \begin{equation} l_S = \frac {g}{S^2} \approx \frac {g}{\varepsilon _L^2 \omega _L^2}, \end{equation}

we arrive at an additional requirement

(3.27) \begin{equation} \frac {k}{k_L} \gg \varepsilon _L^2, \end{equation}

which relates the scale separation to the long-wave steepness.

Now, to address the variation of short-wave properties, we apply (3.19)–(3.20) to the long-wave phase-dependent wavenumber, gravitational acceleration and wave action, and define the homogeneity and stationarity as

(3.28) \begin{equation} H_q = 1 - \left | \frac {1}{qk} \frac {\partial q}{\partial x} \right |, \end{equation}

(3.29) \begin{equation} S_q = 1 - \left | \frac {1}{q\sigma } \frac {\partial q}{\partial t} \right |. \end{equation}

With the above definitions, we consider $q$ to be homogeneous and stationary as $H_q \rightarrow 1$ and $S_q \rightarrow 1$ , respectively.

For the linearised modulation solutions (3.2), (3.9) and (3.15), the respective homogeneity expressions are

(3.30) \begin{equation} H_{k,N} = 1 - \frac {k_L}{k} \left | \frac {\varepsilon _L \sin {\psi }}{\left (1 + \varepsilon _L \cos {\psi }\right )^2} \right |, \end{equation}

(3.31) \begin{equation} H_{g} = 1 - \frac {k_L}{k} \left | \frac {\varepsilon _L \sin {\psi }}{\left (1 - \varepsilon _L^2 \cos ^2{\psi }\right )} \right |, \end{equation}

(3.32) \begin{equation} S_{k,N} = 1 - \frac {\sigma _L}{\sigma } \left | \frac {\varepsilon _L \sin {\psi }}{\left (1 + \varepsilon _L \cos {\psi }\right ) \sqrt {1 - \varepsilon _L^2 \cos ^2{\psi }}} \right |, \end{equation}

(3.33) \begin{equation} S_{g} = 1 - \frac {\sigma _L}{\sigma } \left | \frac {\varepsilon _L \sin {\psi }}{\left (1 - \varepsilon _L \cos {\psi }\right ) \sqrt {1 - \varepsilon _L^2 \cos ^2{\psi }}} \right |. \end{equation}

The steady, linearised solutions thus depend on both the wavenumber ratio $k_L/k$ (homogeneity) or the frequency ratio $\sigma _L/\sigma$ (stationarity) and the long-wave steepness $\varepsilon _L$ .

Figure 6 shows the homogeneities and stationarities of the short-wave wavenumber and effective gravity based on the steady solutions and their derived criteria as minima of (3.19)–(3.20). Stationarity is a stronger requirement than homogeneity. At the lowest scale separation considered ( $k/k_L = 10$ ), short-wave wavenumber, action and effective gravity are all strongly homogeneous ( $H_{k,N,g} \gt 0.99$ ) for $\varepsilon _L \lt 0.1$ . However, the solutions are only strongly stationary ( $S_{k,N,g} \gt 0.99$ ) at a high scale separation of $k/k_L = 100$ for $\varepsilon _L \lt 0.1$ . If we consider weakly stationary conditions as $S_{k,N,g} \gt 0.9$ , then the solutions satisfy the requirements for the wave action conservation for $\varepsilon _L \lt 0.3$ at $k/k_L = 10$ , and for $\varepsilon _L \lt 0.4$ at $k/k_L \gt 20$ . Although Bretherton & Garrett (Reference Bretherton and Garrett1968) state that the wave action balance is valid to ${\mathcal{O}}(k_L/k)$ , applying the general criteria (3.19)–(3.20) yields an error of ${\mathcal{O}}(\varepsilon _L k_L/k)$ . We will revisit these criteria based on the numerical solutions of nonlinear wave action and crest conservation equations in the next section. For the asymptotic limits of wave action conservation in terms of the wave short-wave steepness, see Appendix A.3.

Figure 6.

Homogeneity (left) and stationarity (right) of the short-wave wavenumber (a), (b) and effective gravity (c), (d) as a function of the long-wave steepness $\varepsilon _L$ and the wavenumber ratio $k/k_L$ , based on the linearised solutions (3.30)–(3.33). Solid and dashed lines highlight the 0.99 and 0.90 values, respectively.

4.1. Model description

To quantify the contribution of the nonlinear terms in (2.5) and (2.6), and to evaluate the steadiness assumption of the analytical solutions, we now proceed to numerically integrate the full set of crest and action conservation equations. Equations (2.5) and (2.6) are discretised using second-order central finite difference in space and integrated in time using the fourth-order Runge–Kutta method (Butcher & Wanner Reference Butcher and Wanner1996). The space is divided into 128 grid points. This is effectively the same equation set and numerical configuration as those of Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021), except that their long-wave orbital velocities are evaluated at $z=0$ instead of $z=\eta _L$ , thus neglecting the Stokes drift induced by long waves on short-wave groups (Stokes Reference Stokes1847; van den Bremer & Breivik Reference van den Bremer and Breivik2018; Monismith Reference Monismith2020). Their gravity modulation also does not consider the nonlinear, Lagrangian or slope effects, however, this difference does not qualitatively affect the results. Another difference is the choice of the numerical scheme for spatial differences, which in their case is the more sophisticated 4th order Monotonic Upstream-centered Scheme for Conservation Laws scheme (Kurganov & Tadmor Reference Kurganov and Tadmor2000). Although the second-order central finite difference is not appropriate for many numerical problems, in our case it is sufficient because it is conservative and the fields that we compute derivatives of are smooth and continuous. The maximum relative error of the centred finite difference in this model is $\approx 0.066\, \%$ . The wavenumber and wave action are conservative over time within ${\mathcal{O}}(10^{-7})$ and ${\mathcal{O}}(10^{-5})$ relative error, respectively. We show in the next subsection that our numerical solutions for infinite long-wave trains are qualitatively equivalent to those of Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021).

The numerical equations here are integrated in a fixed reference frame with periodic boundary conditions, rather than that moving with the long-wave phase speed. Either approach produces equivalent modulation results, however, a fixed reference frame allows for a more intuitive interpretation of the results. This difference is important to keep in mind when comparing the results of Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021) and those presented here; in their figure 1 the long-wave phase is fixed and the short waves are moving leftward with the speed of $C_{pL} - C_g - U$ (neglecting group speed inhomogeneity), whereas in the figures that follow, the long wave is moving rightward with its phase speed $C_{pL}$ and the short waves (that is, their action) are moving rightward with the speed of $C_g + U$ (neglecting group speed inhomogeneity). The crest and action conservation equations are integrated in the curvilinear coordinate system in which the horizontal coordinate follows the long-wave surface, akin to Zhang & Melville (Reference Zhang and Melville1990).

Figure 7.

Comparison of numerical solutions (orange) of wavenumber (a), (b), (c), amplitude (d), (e), (f) and steepness (g), (h), (i) modulation with their analytical solutions (blue), for $\varepsilon _L$ = 0.1, 0.2 and 0.4.

4.2. Comparison with analytical solutions

The first set of simulations is performed for the long-wave steepness of $\varepsilon _L = 0.1$ , $0.2$ and $0.4$ (figure 7). The long-waves are initialised from rest ( $a_L = 0$ ) and gradually ramped up to their target steepness over 5 long-wave periods to allow for a gentle increase of the long-wave forcing on the short waves (the importance of which we discuss in more detail in the next subsection). At $\varepsilon _L = 0.1$ , the numerical solutions for the wavenumber, amplitude and steepness modulation are all remarkably similar to the analytical solutions (figure 7 a,d,g). This suggests that for low $\varepsilon _L$ the steady analytical solution is a reasonable approximation for the nonlinear crest-action solutions. The numerical solutions diverge more notably from the analytical ones at moderate $\varepsilon _L = 0.2$ (figure 7 b,e,h), and are considerably different at very high $\varepsilon _L = 0.4$ (figure 7 c,f,i). These differences suggest that at high $\varepsilon _L$ the inhomogeneity of gravitational acceleration and group speed, otherwise ignored in the steady solutions, become important.

Figure 8.

Maximum wavenumber modulation as a function of long-wave steepness $\varepsilon _L$ . Black line and circles are for the analytical and numerical solutions from Longuet-Higgins & Stewart (Reference Longuet-Higgins and Stewart1960) and Longuet-Higgins (Reference Longuet-Higgins1987), respectively. Blue line is based on the analytical solutions from this paper. Green and orange lines are for the numerical solutions from this paper using the linear and third-order Stokes approximations of long waves, respectively. Red line is for the fully nonlinear gravity wave based on Clamond & Dutykh (Reference Clamond and Dutykh2018).

Figure 9.

As figure 8 but for the amplitude modulation.

Figure 10.

As figure 8 but for the steepness modulation.

The maximum modulations of the short-wave wavenumber, amplitude and steepness from the analytical and numerical solutions, are shown in figures 8, 9 and 10, respectively, as functions of the long-wave steepness $\varepsilon _L$ . Overall, the numerical solutions, based on the linear and the third-order Stokes long waves alike, begin to diverge from the analytical solution at $\varepsilon _L \approx 0.15$ . The numerical solution using the third-order Stokes long wave is quantitatively very similar to the fully nonlinear solutions of Longuet-Higgins (Reference Longuet-Higgins1987) in wavenumber modulation, however, with somewhat lower amplitude and steepness modulations. Using the velocity and gravity properties of a fully nonlinear wave, based on the approach by Clamond & Dutykh (Reference Clamond and Dutykh2018), the numerical solutions diverge from those of the third-order Stokes wave only at very large steepnesses ( $\varepsilon _L \gt 0.3$ ). Using a fully nonlinear form of the long-wave thus does not collapse the difference between the solutions here and those of Longuet-Higgins (Reference Longuet-Higgins1987). For example, at $\varepsilon _L = 0.4$ , the steepness modulation in our solution is 6.6, whereas Longuet-Higgins (Reference Longuet-Higgins1987) finds it to be 7.8. A possible reason for this difference is that the solutions by Longuet-Higgins (Reference Longuet-Higgins1987) are stationary (see his § 2), whereas in our numerical model we solve the prognostic nonlinear equations for the short-wave wavenumber and action. As we will see in the next section, and as was suggested by Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021), they are often, but not always, mostly steady, and can be strongly unsteady in special cases. Nevertheless, the overall steepness modulation between the two numerical approaches are remarkably similar, and either solution would cause the short waves of moderate steepness to break (Banner & Peregrine Reference Banner and Peregrine1993). Specifically, where the two modulation solutions differ ( $\varepsilon _L \gtrsim 0.3$ ), the steepness modulation approaches and exceeds 3. Tripling the short-wave steepness of even $\varepsilon = 0.1$ would bring the wave train near the observed breaking thresholds of $\varepsilon \gtrsim 0.32$ (Perlin, Choi & Tian Reference Perlin, Choi and Tian2013), although recent measurements show evidence of significantly higher breaking thresholds (Toffoli et al. Reference Toffoli, Babanin, Onorato and Waseda2010; McAllister et al. Reference McAllister, Draycott, Calvert, Davey, Dias and van den Bremer2024), even exceeding the geometric Stokes limit of 0.44. Nevertheless, in the cases where the short waves would survive the modulation and persist over multiple long-wave periods, the two solutions are effectively equivalent.

Figure 11.

Homogeneity (a), (c), (e) and stationarity (b), (d), (f) of the short-wave wavenumber (a), (b), action (c), (d) and effective gravity (e), (f) as a function of the long-wave steepness $\varepsilon _L$ and the wavenumber ratio $k/k_L$ , based on the numerical solutions of the full wave crest and action conservation equations. Note that the colour ranges are different than those in figure 6.

As the numerical solutions are for the nonlinear wave action and crest conservation equations, it is important to re-evaluate the homogeneity and stationarity requirements for the wave action conservation to remain valid. Figure 11 shows the homogeneities and stationarities of the short-wave wavenumber, action and effective gravity as functions of the long-wave steepness $\varepsilon _L$ and the wavenumber ratio $k/k_L$ , based on the numerical simulations. Although homogeneity is slightly lower ( $H_{k,N,g} \approx 0.92$ for $k/k_L = 10$ and $\varepsilon _L = 0.4$ ) compared with the analytical solutions, the stationarity is significantly reduced. Following the same criteria for strong ( $S_{k,N,g} \gt 0.99$ ) and weak ( $S_{k,N,g} \gt 0.9$ ) stationarity as in § 3.4, the numerical solutions are only weakly stationary for $\varepsilon _L \lt 0.21$ at $k/k_L = 10$ , and for $\varepsilon _L \lt 0.36$ at $k/k_L = 100$ . This provides a quantitative guidance on when the short-wave action can be considered to be conserved in the presence of longer waves.

4.3. Unsteady growth in Pereux et al. (Reference Peureux, Ardhuin and Guimarães2021)

The key result of Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021) is that of unsteady steepening of short waves when a homogeneous field of short waves is forced by an infinite long-wave train. The steepening is driven by the progressive increase in the action of short waves (as opposed to their wavenumber), and is centred on the short-wave group that begins its journey in the trough of the long wave. This is because the unmodulated short waves that begin their journey in the trough first enter the convergence region before reaching the divergence region. Inversely, the short waves that begin around the crest of the long wave first enter the divergence region and will thus have their action and wavenumber decreased. We reproduce this behaviour in figure 12, which shows the evolution of the short-wave action, wavenumber and steepness modulation (here defined as the change of a quantity relative to its initial value). The long waves have $\varepsilon _L = 0.1$ and $k_L = 1$ , and the short waves have $k = 10$ . After 10 long-wave periods, the short-wave action is approximately doubled, consistent with Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021). Another important property of this solution is that the short-wave group that experiences peak amplification (attenuation) are not locked-in to the long-wave crests (troughs), as they are in the prior steady solutions (Longuet-Higgins & Stewart Reference Longuet-Higgins and Stewart1960; Longuet-Higgins Reference Longuet-Higgins1987; Zhang & Melville Reference Zhang and Melville1990).

Figure 12.

Changes in the short-wave action (a), wavenumber (b) and steepness (c) relative to their initial values as functions of initial long-wave phase and time. Here, $k_L = 1$ , $k = 10$ , $\varepsilon _L = 0.1$ . The black contour follows the value of 1, which corresponds to no change from the initial values.

Why does this unsteady growth of short-wave action occur? Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021) explain that the change in advection velocity due to the modulation of the short-wave group speed yields an additional amplification of the short-wave action. That is, the inhomogeneity term $N \partial C_g / \partial x$ is responsible for the instability. However, the instability only occurs if the short waves are initialised as a uniform field. If they are initialised instead as a periodic function of the long-wave phase such that their wavenumber (and optionally, action) is higher on the crests and lower in the troughs, akin to the prior steady solutions, the instability vanishes and the short-wave modulation pattern locks-in to the long-wave crests.

Figure 13.

Same as figure 12 but with a linear ramp applied to $a_L$ during the initial five long-wave periods. Notice the change in the colour range.

Why is it, then, that the solution for short-wave modulation is not only quantitatively, but also qualitatively, different depending on the short-wave initial conditions? It may at first seem that there is something special about the uniform initial conditions that makes its solutions grow unsteadily and indefinitely. Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021) described this scenario as ‘…the sudden appearance of a long-wave perturbation in the middle of a homogeneous short wave field’. However, it is not obvious whether such a scenario is common as waves tend to disperse into groups, and the time scales of incoming swell fronts are much longer than those of short-wave generation by wind. Exceptional situations with such a sudden onset of swell may occur around complex coastlines or in shadows behind islands, where wind-generated ripples may propagate from an otherwise calm sea state into the path of a crossing swell. Within the limits of the modulation theory used here, such short waves could quickly steepen and break.

To test whether the sudden onset of long-wave forcing causes the unsteady growth, we repeat the numerical simulation of figure 12, except for a linear ramp applied to $a_L$ (and consequently, the other long-wave properties including the orbital velocities and the local steepness) for the first five periods: $a_L(t) = min(a_L, a_L t / (5 T_L ))$ , where $T_L$ is the long-wave period. This result is shown in figure 13. Note that the linear ramp is strictly a numerical modelling technique to test the sensitivity of the solution to the sudden forcing at the beginning of the simulation, and is not intended to represent a physical process. The key distinction that arises with the introduction of the ramp is that the modulation pattern of short waves is now locked-in to the long-wave crests, and their indefinite steepening vanishes. The stabilisation effect of the long-wave ramp is more readily assessed in figure 14, which shows the maximum short-wave steepness modulation as a function of time over 30 long-wave periods, for the infinite long-wave train case and the case of applying a linear ramp. Comparing the wave action propagation ( $C_g \partial N / \partial x$ ) and inhomogeneity ( $N \partial C_g / \partial x$ ) tendencies reveals that, after only one long-wave period, the propagation and inhomogeneity tendencies are unbalanced and amplify the steepening of short waves (figure 15). Peureux et al. (Reference Peureux, Ardhuin and Guimarães2021) correctly pointed out that the unsteady growth of short-wave steepness occurs due to the resonance of the inhomogeneity tendency with the short-wave action perturbation. However, in the case of a gentle ramp-up of the long-wave forcing, the propagation and inhomogeneity tendencies balance each other and allow the short-wave steepening to lock-in to the long-wave crests.

Figure 14.

Maximum short-wave steepness modulation as a function of time in case of infinite long-wave train case (blue) and the same case but with a linear ramp during the initial five long-wave periods (orange). The dashed black line corresponds to the short-wave steepness of 0.4 at which most waves are expected to break.

Figure 15.

Propagation ( $C_g {\partial N}/{\partial x}$ ) (blue) and inhomogeneity ( $N {\partial C_g}/{\partial x}$ ) (orange) tendencies and their sum (green) for the short-wave action after one long-wave period, in the case of (a) an infinitely long-wave train and (b) a linear ramp of long-wave amplitude. Dashed line is the long-wave elevation and the dotted line is the short-wave action modulation minus 1.

4.4. Modulation by long waves of varying amplitude

We now proceed to examine the numerical solutions of short-wave modulation by long waves whose amplitude varies with time. The amplitude variation aims to mimic the behaviour of long-wave groups, a phenomenon that is ubiquitous in the ocean due to the dispersive nature of surface gravity waves, as well as due to the fact that wind-generated waves evolve to have broad-banded spectra. For simplicity of implementation and interpretation, the wave group here is simulated as a non-dispersive train of long waves whose amplitude scales with $\sin [\pi t / (n T_L)]$ , where $n$ is the number of long waves in the group. The elevation of the long waves thus gradually ramps up from rest to $a_L$ at $t = n T_L / 2$ , after which it gradually decreases toward zero. In this simulation, $k/k_L = 10$ and $\varepsilon _L = 0.1$ as in the previous two simulations. The modulations of short-wave wavenumber, amplitude and steepness in response to such long-wave group is shown in figure 16. Like in the case of a linear ramp, the modulation pattern here is locked in to the long-wave crest and peaks at approximately 1.2 for short-wave steepness of 0.1. After the peak of the long-wave group passes, the short-wave modulation recedes back toward the resting state.

Figure 16.

Same as figure 12 but for a long-wave group, causing the long-wave amplitude to gradually increase and peak at $a_L = 0.1$ after five long-wave periods, and then conversely decay back to a calm sea state.