2.1. Expression of dispersive and nonlinear coefficients

To obtain the dispersive and nonlinear coefficients $L$ and $N$ of the vor-NLS equation for pure capillary wave trains, we used the expressions we derived for gravity–capillary waves in finite depth and in the presence of vorticity (see equation (2.31) of Hsu et al. (Reference Hsu, Kharif, Abid and Chen2018)) when $\kappa$ goes to infinity. The limit of the dispersive coefficient is obtained easily whereas the limit of the nonlinear coefficient needs to be handled carefully because two terms go to infinity. Fortunately, these two infinite terms cancel each other as is shown in the Appendix.

The NLS equation governing the evolution of the complex envelope of a packet of capillary waves on finite depth in the presence of constant vorticity (vor-NLS) is

(2.8) \begin{equation} ia_{\tau }+L a_{\xi \xi }+N|a|^2 a=0, \end{equation}

with

(2.9) \begin{align} L = \frac {\omega }{k^2(2+X)^3\sigma ^2} & \big[3(1+X)(1+X+X^2)\sigma ^2 -6(1+X)^2\mu \sigma (\sigma ^2-1) \nonumber \\& +\mu ^2(\sigma ^2-1) \big((1+X)^2+(3+2X)\sigma ^2 \big) \big] \end{align}

and

(2.10) \begin{align} N&=-\frac {\omega k^2}{8\sigma ^2} \left [\sigma ^2\frac {5+3X}{2+X}-\frac {P_0+P_1X+P_2X^2+P_3X^3+X^4}{(3-\sigma ^2+3X)(2+X)} +2\sigma (2+X) \right. \nonumber \\[5pt]& \quad \left.\left ( \frac {(1+X)^2-\sigma ^2}{\big(1-\sigma ^2\big)\mu +3\sigma (1+X)} +\frac {1-\sigma ^2}{\mu (\big(1-\sigma ^2\big)+X(2+X))+3\sigma (1+X)} \right )\right ], \end{align}

with $\mu =kh$ , $\sigma =\tanh {\mu }$ , $X=\sigma \varOmega /\omega$ and $P_0=21-10\sigma ^2+\sigma ^4$ , $P_1=42+2\sigma ^2-4\sigma ^4$ , $P_2=30+12\sigma ^2$ , $P_3=9+5\sigma ^2.$

Note that $|X|$ can be considered as a Strouhal number, which is the ratio between the characteristic frequency of the vorticity and that of the capillary waves.

The coefficients $L$ and $N$ given by (2.9) and (2.10) correspond to those of Hsu et al. (Reference Hsu, Kharif, Abid and Chen2018) for $\kappa =k^2 T/(\rho _w g) \rightarrow \infty$ and $\varOmega \neq 0$ and to those of Djordjevic & Redekopp (Reference Djordjevic and Redekopp1977) in finite depth for $\kappa \rightarrow \infty$ and $\varOmega =0$ .

The linear dispersion relation of capillary waves in the presence of constant vorticity is

(2.11) \begin{equation} \omega ^2+\sigma \varOmega \omega -\sigma \frac {k^3 T}{\rho _w}=0. \end{equation}

From (2.11) it is easy to show that $X\gt -1$ .

As emphasised above, we consider a carrier wave propagating from left to right of frequency $\omega$ , phase velocity $c_p$ and group velocity $c_g$ whose expressions are

(2.12) \begin{align}&\qquad\qquad\qquad\qquad\, \omega =-\frac {\sigma \varOmega }{2}+\sqrt {\left(\frac {\sigma \varOmega }{2}\right)^2+\sigma \frac {k^3 T}{\rho _w}},\\[-8pt]\nonumber \end{align}

(2.13) \begin{align}&\qquad\qquad\qquad\qquad c_p=-\frac {\sigma \varOmega }{2k}+\sqrt {\left(\frac {\sigma \varOmega }{2k}\right)^2+\sigma \frac {k T}{\rho _w}},\\[-8pt]\nonumber \end{align}

(2.14) \begin{align}& c_g=-\frac {\varOmega h}{2}\big(1-\sigma ^2\big)+\frac {\big(1-\sigma ^2\big)(\varOmega ^2h\sigma /2+\mu k^2T/\rho _w)+3\sigma k^2T/\rho _w}{\sqrt {(\varOmega \sigma )^2+4\sigma k^3 T/\rho _w}}. \end{align}

2.2. The Wilton ripple phenomenon and resonant triads

The nonlinear coefficient $N$ presents two singularities that one should avoid, either

(2.15) \begin{equation} 3-\sigma ^2+3X=0 \end{equation}

or

(2.16) \begin{equation} \mu (\big(1-\sigma ^2\big)+X(2+X))+3\sigma (1+X)=0. \end{equation}

Equation (2.15) corresponds to the reduction of the second-harmonic resonance condition of gravity–capillary waves propagating at the free surface of an underlying current of constant vorticity when $g$ goes to zero ( $\kappa$ goes to $\infty$ ). This resonance which is also called the Wilton ripple phenomenon occurs when both the fundamental mode and the second harmonic have the same phase velocity, that is, when the phase velocity presents a minimum value. Note that in the absence of positive vorticity ( $\varOmega \lt 0$ ) there is no minimum of phase velocity for pure capillary waves. Martin & Matioc (Reference Martin and Matioc2013) proved rigorously the existence of Wilton ripples on finite depth for gravity–capillary water waves with constant positive and negative values of the vorticity. The fact that the phenomenon of Wilton ripples for pure capillary waves occurs only for waves propagating from left to right on a current of positive vorticity is a consequence of their lemma 2.5. Using a different approach to investigate the Wilton ripple phenomenon of pure capillary waves our result agrees with that of Martin & Matioc (Reference Martin and Matioc2013). Note that their study focused on integer values of the wavenumber. The phase velocity $c_p$ reaches a minimal value when $\varOmega \lt 0$ (positive vorticity) and $k=k_m$ given in deep water by

(2.17) \begin{equation} k_m=\left (\frac {2\rho _w}{T}\varOmega ^2 \right )^{1/3}. \end{equation}

The wavenumber $k$ satisfying $c_p(k)=c_p(2k)$ is

(2.18) \begin{equation} k=\left (\frac {3\rho _w}{4T} \varOmega ^2 \right )^{1/3}. \end{equation}

The second-harmonic resonance corresponds to the following degenerate resonant triad:

(2.19) \begin{align}&\, 2k_1-k_2=0,\\[-10pt]\nonumber \end{align}

(2.20) \begin{align}& 2\omega _1-\omega _2=0, \end{align}

where

(2.21) \begin{align}& \omega _1=-\frac {\varOmega }{2}+\sqrt {\left(\frac {\varOmega }{2}\right)^2 + \frac {T}{\rho _w} k^3},\\[-6pt]\nonumber \end{align}

(2.22) \begin{align}& \omega _2=-\frac {\varOmega }{2}+\sqrt {\left(\frac {\varOmega }{2}\right)^2 + \frac {8T}{\rho _w} k^3}, \end{align}

with $k$ given by (2.18).

Injecting the expression of $k$ in (2.21) and (2.22), we obtain $\omega _1=-3\varOmega /2$ and $\omega _2=-3\varOmega$ .

Constantin & Kartashova (Reference Constantin and Kartashova2009) showed that only positive vorticity ( $\varOmega \lt 0$ ) can trigger the occurrence of three-wave resonant interactions of capillary water waves on an underlying current of constant vorticity. As in Martin & Matioc (Reference Martin and Matioc2013) they considered integer values of the wavenumber. Later on, using the methods of Constantin & Kartashova (Reference Constantin and Kartashova2009), Martin (Reference Martin2017) investigated three-wave resonances in deep water to extend their results to gravity–capillary waves. He demonstrated that three-wave resonances of gravity–capillary waves occur only for positive values of the vorticity.

We have shown that the Wilton ripple phenomenon exists in infinite depth for pure capillary waves travelling on a current of constant positive vorticity. To a given fixed value of the vorticity, $-\varOmega$ , corresponds a resonant real-valued wavenumber given by (2.18); and conversely, to a given fixed real-valued resonant wavenumber, $k$ , corresponds a value of the shear $\varOmega =-\sqrt {4Tk^3/(3\rho _w)}$ . Fixing $k=1$ gives the minimal value of the vorticity found by Constantin & Kartashova (Reference Constantin and Kartashova2009). Note that within the framework of capillary water waves, the vorticity must be strong to trigger the second-harmonic resonance.

The wavenumber given by (2.18) does not satisfy equation (2.15) unless $\sigma =1$ that does not mean that there are no second-harmonic resonances in finite depth within the framework of pure capillary waves.

The second singularity corresponds to the reduction of the short-wave/long-wave resonance of gravity–capillary waves propagating at the free surface of an underlying current of constant vorticity when $g$ goes to zero. Note that this singularity disappears in the nonlinear coefficient $N$ when $\sigma =1$ , that is, in deep water. Nevertheless, this does not mean that there are no resonant triad interactions in deep water.

Equation (2.16) has two real-valued negative roots: one is less than $-1$ and is ruled out because $X\gt -1$ and the other belongs to $(-1,0)$ when $\mu$ ranges from zero to infinity. It is useful to note that whatever the depth, there exists a singular value of $X$ for $-1\lt X\lt 0$ . Constantin & Kartashova (Reference Constantin and Kartashova2009) demonstrated that in deep water, capillary waves on a current of positive vorticity ( $\varOmega \lt 0$ ) may give rise to resonant triads.

We found in finite depth triads of capillary waves satisfying the following resonance conditions:

(2.23) \begin{align}&\,\, k_1-k_2+k_3=0, \end{align}

(2.24) \begin{align}& \omega _1-\omega _2+\omega _3=0, \end{align}

with

(2.25) \begin{equation} \omega _i=-\frac {\sigma \varOmega }{2}+\sqrt {\left(\frac {\sigma \varOmega }{2}\right)^2+\sigma \frac {k_i^3 T}{\rho _w}}, \quad i=1,2,3. \end{equation}

To illustrate this feature of capillary waves on a current of constant positive vorticity in finite depth, three examples are shown in figure 2 which gives pairs of capillary waves of wavenumbers $(k_1,k_2)$ that excite a third wave of wavenumber $k_3$ . To a given value of $k_1$ corresponds a value of $k_2$ and consequently a value of $k_3$ . The three curves present a minimum for the wavenumber $k_2$ .

We can conclude that (2.16) corresponds to resonant triads of capillary waves in finite depth with positive constant vorticity. At any depth, there exists a forbidden positive vorticity value corresponding to a resonant triad.

Figure 2.

Resonant triad in finite depth: a pair of wavenumbers $(k_1, k_2)$ exciting a third wavenumber $k_3=k_2-k_1$ . Here $\mu =1$ (black solid line), $\mu =0.5$ (red solid line), $\mu =0.3$ (blue solid line), with $\varOmega =-1 \,{\textrm s}^{-1}$ and wavenumbers in ${\textrm m}^{-1}$ .

3.1. Effect of viscosity and forcing

The rate of decay of the amplitude of linear waves due to bulk viscosity is $2\nu k^2$ , where $\nu$ is the kinematic viscosity and $k$ the wavenumber. In the following we consider the modulational instability of capillary waves in the presence of dissipation due to viscosity and some forcing due for instance to wind. Note that this problem has been investigated by Kharif et al. (Reference Kharif, Kraenkel, Manna and Thomas2010) for pure gravity waves in deep water. The latter authors used the Miles mechanism to model wind effect.

Equation (2.8) becomes

(3.14) \begin{equation} ia_{\tau }+La_{\xi \xi }+N|a|^2 a=i(\varDelta -2\nu k^2)a, \end{equation}

where $\varDelta \gt 0$ corresponds to a forcing term.

When $\varDelta =0$ , (3.14) is similar to the NLS equation derived rigorously by Dias, Dyachenko & Zakharov (Reference Dias, Dyachenko and Zakharov2008) within the framework of weakly damped water waves (see their equation (40)).

Equation (3.14) admits the following solution:

(3.15) \begin{equation} a_s=a_0\exp ({K\!\tau} ) \exp \left (\frac {iN}{2K}a_0^2\big(e^{2K\!\tau }-1\big) \right ), \end{equation}

where $K=\varDelta -2\nu k^2$ .

Note that $a_s$ is spatially constant.

We superimpose to this solution a small disturbance:

(3.16) \begin{equation} a(\xi ,\tau )=a_s(1+\epsilon a^{\prime}(\xi , \tau )), \end{equation}

with $\epsilon \ll 1$ .

Substituting (3.16) into (3.14) and keeping only terms of order $\epsilon$ and separating the real and imaginary parts of $a^{\prime}(\xi ,\tau )=a^{\prime}_r(\xi ,\tau )+ia^{\prime}_i(\xi ,\tau )$ gives

(3.17) \begin{align}& a^{\prime}_{i\tau }-La^{\prime}_{r\xi \xi }-2N a_0^2 \exp (2K\!\tau ) a^{\prime}_r=0, \end{align}

(3.18) \begin{align}&\qquad\qquad\quad a^{\prime}_{r\tau }+La^{\prime}_{i\xi \xi }=0. \end{align}

Set $a^{\prime}_r(\xi ,\tau )=\mathrm{Re}(a^{\prime}_{r0}(\tau )\exp ( ip\xi ))$ and $a^{\prime}_i(\xi ,\tau )=\mathrm{Im}(a^{\prime}_{i0}(\tau )\exp (ip\xi ))$ , where $p$ is the wavenumber of the perturbation. Then,

(3.19) \begin{align}& \frac {{\textrm d}a^{\prime}_{i0}}{{\textrm d}\tau }+\big(Lp^2-2N a_0^2 \exp (2K\!\tau )\big)a^{\prime}_{r0}=0, \end{align}

(3.20) \begin{align}&\qquad\qquad\quad \frac {{\textrm d}a^{\prime}_{r0}}{{\textrm d}\tau }-Lp^2a^{\prime}_{i0}=0. \end{align}

Combining the two previous equations gives

(3.21) \begin{equation} \frac {{d}^2a^{\prime}_{r0}}{{\textrm d}\tau ^2}+L^2p^4\left (1-\frac {2N}{Lp^2} a_0^2 \exp (2K\!\tau ) \right )a^{\prime}_{r0}=0. \end{equation}

One may distinguish two cases according to whether $K$ is negative or positive.

1. (i) Let $K$ be negative. This case was analysed by Segur, Henderson & Carter (Reference Segur, Henderson and Carter2005) and Kharif et al. (Reference Kharif, Kraenkel, Manna and Thomas2010). In the presence of damping, the modulational instability is stabilised by viscous dissipation. However, note that the modulational instability may grow, but its growth is limited and starts to stabilise after a time $\tau =- ({1}/{2K})\ln ({2Na_0^2}/{Lp^2})$ corresponding to the change of sign of the coefficient of $a^{\prime}_{r0}$ of (3.21). Note that the bandwidth of instability shrinks as the amplitude of the carrier wave diminishes, resulting in stabilisation of unstable modes. The most unstable perturbation starts stabilisation at $\tau =\ln 2/(4\nu k^2)$ . Let us consider a carrier wave of wavelength $\lambda =0.005 \, {\textrm m}$ in deep water. For $X=0$ , stabilising of the most unstable perturbation will start at $\tau \approx 7 \tau _0$ , where $\tau _0$ is the dimensional period of the carrier wave. For $X=0.4$ and $X=-0.4$ , we found that the most unstable perturbation stabilises at $\tau \approx 6 \tau _0$ and $\tau \approx 9 \tau _0$ , respectively. These critical times are more or less confirmed by numerical simulations of dimensionless equation (3.24) (see figure 8) without forcing.

2. (ii) Let $K$ be positive. This case was discussed by Kharif et al. (Reference Kharif, Kraenkel, Manna and Thomas2010) who considered gravity wave trains on the surface of water acted upon by wind and under the influence of viscosity. Herein, a similar situation is considered within the framework of pure capillary wave trains. In this case, the modulational perturbations are initially oscillating for $0 \lt \tau \lt ({1}/{2K})\ln ( {2Na_0^2}/{Lp^2})$ and then start to grow exponentially for $\tau \gt ({1}/{2K})\ln ( {2Na_0^2}/{Lp^2})$ . The bandwidth of instability increases as the amplitude of the carrier wave increases. Consequently, stable modes may become unstable. Let $p$ be a perturbation wavenumber corresponding to a stable state. There is a critical value of the carrier wave amplitude, $a_0$ , for which the bandwidth instability is equal to $p$ . For carrier wave amplitude larger than this critical value, the perturbation becomes unstable. The critical value is given by $\varGamma _i=0$ , that is to say,

   (3.22) \begin{equation} a_{0c}^2=\frac {Lp^2}{2N}. \end{equation}
   The modulational instability will start to grow as soon as the carrier wave amplitude reaches the critical value. When the carrier wave amplitude is set equal to $a_{0c}$ , the instability starts at $\tau =0$ .

Let us consider the case without vorticity ( $X=0$ ) in deep water. The dispersive and nonlinear coefficients of the vor-NLS equation are $L=3\omega /(8k^2)$ and $\omega k^2/16$ . The critical wave steepness of the carrier wave is

(3.23) \begin{align} a_{0c}k=\sqrt {3}\,p/k .\end{align}

This critical value is more or less confirmed by numerical simulations of dimensionless equation (3.24) (see figure 9 a) with forcing, $\tilde {K}=(\tilde {\varDelta }-2/R_e)$ .

For $(X,p/k)=(0.4, 0.72)$ and $(X,p/k)=(-0.4, 0.59)$ we found, in deep water, $a_{0c}k \approx 1$ and $1.5$ , respectively (see figure 9 b,c).

To complete the linear stability analysis, we consider the nonlinear evolution of unstable and stable modes. To do this, we use (3.14) written in dimensionless form with respect to the reference length $1/k$ and reference time $1/\omega$ , where $k$ and $\omega$ are the wavenumber and frequency of the carrier wave, respectively:

(3.24) \begin{equation} i\tilde {a}_{\tilde {\tau }}+\tilde {L}\tilde {a}_{\tilde {\xi } \tilde {\xi }}+\tilde {N}|\tilde {a}|^2 \tilde {a}=i\left(\tilde {\varDelta }-\frac {2}{R_e}\right)\tilde {a}, \end{equation}

where $R_e=\omega /(k^2 \nu )$ .

Within the framework of numerical simulations of the vor-NLS equation, figure 8 shows, in the presence of viscous dissipation only and for several values of the vorticity, the temporal evolution of the amplitude of the carrier wave, $a_0$ , and those of the most unstable sidebands, $a_{-1}$ and $a_{+1}$ , whose curves overlap. At the beginning of the nonlinear interaction, the modulational instability is growing due to a transfer of energy from the carrier wave to the sidebands and then stabilises under the effect of the dissipation (see figure 8 a,b). The case corresponding to figure 8(c) presents a different behaviour because the initial rate of growth of the instability and the rate of damping are of the same order, preventing the instability from developing.

Figure 9 shows, in the presence of forcing assumed to be a constant, the temporal evolution of the amplitude of the carrier wave, $a_0$ , and those of the initially stable sidebands, $a_{-1}$ and $a_{+1}$ , for several values of the vorticity. At the beginning, numerical simulations show that the perturbations are oscillating and then are growing.

Figure 8.

Temporal evolution of the dimensionless amplitudes of the carrier wave (solid blue line) and most unstable sideband modes (solid red line) in the presence of viscous dissipation in deep water: (a) $X=0$ , $R_e \approx 240$ ; (b) $X=0.4$ , $R_e \approx 202$ ; (c) $X=-0.4$ , $R_e \approx 308$ . Herein, $\omega =1$ .

Figure 9.

Temporal evolution of the dimensionless amplitudes of the carrier wave (solid blue line) and stable sideband modes (solid red line) in the presence of weak forcing in deep water: (a) $X=0$ , $p/k=0.7$ ; (b) $X=0.4$ , $p/k=0.72$ ; (c) $X=-0.4$ , $p/k=0.59$ ; $\tilde {K}=3/10\,000$ . Herein, $\omega =1$ .

5.1. Capillary envelope soliton

In two dimensions the NLS equation in deep water reads

(5.1) \begin{equation} ia_{\tau }+L a_{\xi \xi }+M a_{\eta \eta }+N|a|^2 a=0, \end{equation}

with

(5.2) \begin{align}&\qquad L=\frac {3\omega }{k^2} \left (\frac {(1+X)(1+X+X^2)}{(2+X)^3} \right ),\\[-10pt]\nonumber \end{align}

(5.3) \begin{align}& N=\frac {\omega k^2}{24} \left (\frac {3+14X+23X^2+11X^3-3X^4}{(X+1)(3X+2)} \right ),\\[-10pt]\nonumber \end{align}

(5.4) \begin{align}&\qquad\qquad M=\frac {3T}{2\rho _w} \left (\frac {k}{\omega (2+X)} \right ). \end{align}

Here in $X=\varOmega /\omega$ .

Equation (5.1) admits as solution the two-dimensional envelope soliton, $\bar {a}(\xi ,\tau )$ , given by (4.1). We consider the stability of this solution to infinitesimal disturbances:

(5.5) \begin{equation} a(\xi ,\eta , \tau )=\bar {a}(\xi ,\tau )+a^{\prime}(\xi ,\eta ,\tau )\exp \big(iNa_0^2\tau /2\big). \end{equation}

The stability analysis is similar to that used by Abid et al. (Reference Abid, Kharif, Hsu and Chen2019) for gravity–capillary envelope soliton, except that now the coefficients of the NLS equation correspond to pure capillary envelope soliton. Linearisation of (5.1) about $\bar {a}$ with $a^{\prime}=u+iv$ gives

(5.6) \begin{align}& v_{\tau }=Lu_{\xi \xi }+Mu_{\eta \eta }+\Big(3NA^2-Na_0^2/2\Big)u,\\[-10pt]\nonumber \end{align}

(5.7) \begin{align}& u_{\tau }=-Lv_{\xi \xi }-Mv_{\eta \eta }-\Big(NA^2-Na_0^2/2\Big)v, \end{align}

where

(5.8) \begin{align} A=a_0\textrm{sech} \left (a_0\left(\frac {N}{2L}\right)^{1/2} \xi \right ). \end{align}

The system (5.6)–(5.7) of linear partial differential equations with constant coefficients admits solutions of the following form:

(5.9) \begin{align}& u=\hat {u}(\xi )\exp (\textit{iq}\eta +\Gamma \tau ),\\[-10pt]\nonumber \end{align}

(5.10) \begin{align}& v=\hat {v}(\xi )\exp (\textit{iq}\eta +\Gamma \tau ). \end{align}

Substitution of (5.9) and (5.10) into (5.6) and (5.7) gives

(5.11) \begin{align}& \varGamma \hat {v}=L\frac {{d}^2\hat {u}}{{\textrm d}\xi ^2} -Mq^2 \hat {u} +\Big(3NA^2-Na_0^2/2\Big)\hat {u},\\[-10pt]\nonumber \end{align}

(5.12) \begin{align}& \varGamma \hat {u}=-L\frac {{d}^2\hat {v}}{{\textrm d}\xi ^2} +Mq^2 \hat {v} -\Big(NA^2-Na_0^2/2\Big)\hat {v}. \end{align}

The eigenfunction $\hat {v}$ can be eliminated:

(5.13) \begin{equation} \varGamma ^2 \hat {u}=-\big(D_0-Mq^2\big)\big(D_1-Mq^2\big)\hat {u}, \end{equation}

with

(5.14) \begin{align} D_0&=L\frac {{d}^2}{{\textrm d}\xi ^2}+NA^2-Na_0^2/2,\\[-10pt]\nonumber \end{align}

(5.15) \begin{align} D_1&=L\frac {{d}^2}{{\textrm d}\xi ^2}+3NA^2-Na_0^2/2. \end{align}

The eigenfunction $\hat {u} \rightarrow 0$ as $\xi \rightarrow \infty$ (the eigenfunctions are local in $\xi$ ). Note that the non-local states corresponding to the continuum spectrum are stable or marginally stable (Saffman & Yuen Reference Saffman and Yuen1978; Rypdal & Rasmussen Reference Rypdal and Rasmussen1989). Zakharov & Rubenchik (Reference Zakharov and Rubenchik1974) investigated analytically the transverse instability of envelope solitons.

The numerical method used to solve (5.13) is outlined in Appendix A of Abid et al. (Reference Abid, Kharif, Hsu and Chen2019).

Without loss of generality, we set $T=1$ , $\rho _w=1$ and $k=1$ as in Saffman & Yuen (Reference Saffman and Yuen1985) and Tiron & Choi (Reference Tiron and Choi2012). For the sake of clarity, the same symbols are used for the coefficients of the NLS equation. For comparison, we consider the analytic expression of the instability growth rate derived by Rypdal & Rasmussen (Reference Rypdal and Rasmussen1989) and Abid et al. (Reference Abid, Kharif, Hsu and Chen2019), given in dimensionless form by

(5.16) \begin{align}& \varGamma ^2=\frac {32}{\pi ^2}MN q^2\left(1-\frac {M}{3N} q^2\right),\\[-10pt]\nonumber \end{align}

(5.17) \begin{align}&\qquad\quad \varGamma _{max}^2=\frac {24}{\pi ^2}N^2, \end{align}

with

(5.18) \begin{align} q_{max}=\sqrt {\frac {3N}{2M}.} \end{align}

The capillary envelope soliton is unstable to infinitesimal transverse perturbations of wavenumber $q$ satisfying $q^2 \lt 3N/M$ . Figure 12 shows the square of the dimensionless growth rate as a function of the square of the dimensionless transverse wavenumber for different values of the vorticity. The numerical and analytical results are in excellent agreement. The dimensionless growth rate and instability bandwidth increase as the dimensionless vorticity decreases. Note that in deep water

(5.19) \begin{align} \varOmega ^2=\frac {Tk^3}{\rho _w} \frac {X^2}{1+X} \quad \textrm{with} \ X\gt -1, \end{align}

and in dimensionless form

(5.20) \begin{align} \varOmega ^2=\frac {X^2}{1+X}. \end{align}

Figure 12.

Square of the dimensionless growth rate of the transverse unstable mode of capillary envelope soliton in deep water as a function of the square of the dimensionless wavenumber for $X=-0.1$ (stars), $X=0$ (solid line), $X=0.1$ (circles). The dashed lines correspond to the analytical expression given by (5.16).

5.2. Capillary Stokes waves

Equation (5.1) admits as solution the Stokes wave envelope given by (3.1). This solution is unstable to perturbations of longitudinal and transverse wavenumbers $p$ and $q$ if

(5.21) \begin{equation} Lp^2+Mq^2\lt 2Na_0^2. \end{equation}

Using the linear dispersion (2.11) the coefficient $M$ is rewritten as follows:

(5.22) \begin{equation} M=\frac {3\omega }{2k^2}\frac {1+X}{2+X}. \end{equation}

In a more explicit form, condition (5.21) reads

(5.23) \begin{equation} \frac {3(1+X)(1+X+X^2)}{(2+X)^3}\,p^2+\frac {3(1+X)}{2(2+X)}\,q^2\lt \frac {3+14X+23X^2+11X^3-3X^4}{12(1+X)(2+3X)}\,a_0^2k^4. \end{equation}

The instability condition $p^2+2q^2\lt k^4 a_0^2/3$ , established by Chen & Saffman (Reference Chen and Saffman1985), is recovered for $X=0$ .

The square of growth rate of instability is

(5.24) \begin{equation} \varGamma _i^2=\big(Lp^2+Mq^2\big)\big(2Na_0^2-Lp^2-Mq^2\big). \end{equation}

Substituting the expressions of $L$ , $M$ and $N$ into (5.24), we obtain in dimensionless form

(5.25) \begin{align} \frac {\varGamma _i^2}{\omega ^2}&=\frac {3(1+X)}{2+X}\left (\frac {1+X+X^2}{(2+X)^2}\frac {p^2}{k^2}+\frac {q^2}{2k^2}\right )\nonumber \\ & \quad \times \left (\frac {3+14X+23X^2+11X^3-3X^4}{12(1+X)(2+X)}a_0^2k^2 - \frac {3(1+X)}{2+X}\left (\frac {1+X+X^2}{(2+X)^2}\frac {p^2}{k^2}+\frac {q^2}{2k^2}\right )\right ). \end{align}

For $X=0$ we recover the growth rate of instability of Chen & Saffman (Reference Chen and Saffman1985) given by

(5.26) \begin{equation} \varGamma _i=\frac {1}{8}\left(3a_0^2k^3\big(p^2+2q^2\big)-\frac {9}{k}\big(p^2+2q^2\big)^2 \right)^{1/2}. \end{equation}

Note that Chen & Saffman (Reference Chen and Saffman1985) used the relation $\omega ^2=k^3$ .

Figure 13.

Infinite-depth stability diagram in the $(p/k,q/k)$ plane for $a_0k=0.1$ and three values of the vorticity. Blue: $X=0$ ; orange: $X=-0.1$ ; green: $X=0.1$ . Dots correspond to the maximal growth rate of instability.

Figure 14.

Plots for (a) $kh=1$ and (b) $kh=0.5$ . Blue: $X=0$ ; orange: $X=-0.1$ ; green: $X=0.1$ . Dots correspond to the maximal growth rate of instability.

Figure 15.

Dimensionless maximal growth rate of oblique instability as a function of the dimensionless depth, $\mu =kh$ : $X=-0.10$ (orange solid line), $X=0$ (blue solid line), $X=0.10$ (green solid line).

The boundary of the stability diagram in the $(p/k,q/k)$ plane is given by the following ellipse equation:

(5.27) \begin{equation} \frac {(1+X+X^2)}{(2+X)^2}\left(\frac {p}{k}\right)^2+\frac {1}{2}\left(\frac {q}{k}\right)^2=\frac {3+14X+23X^2+11X^3-3X^4}{36(1+X)^2}a_0^2 k^2. \end{equation}

Within the framework of deep water, figure 13 shows the stability diagram in the $(p/k,q/k)$ plane for three values of the dimensionless vorticity. The domain of instability increases as the vorticity decreases. Dots plotted in the figure, located on the $p$ axis, correspond to the dimensionless maximal growth rate of modulational instability. Consequently, the dominant modulational instability of weakly nonlinear capillary waves on deep water is two-dimensional. The wavelength of the dominant modulational instability increases as the vorticity decreases. For $\varOmega =0$ we found $p/k=0.0416$ which is very close to the value obtained by Chen & Saffman (Reference Chen and Saffman1985) in their table 1.

Figure 14 shows the stability diagram in the $(p/k,q/k)$ plane in finite depth for $kh=0.5$ and $kh=1$ and three values of vorticity. The most unstable perturbation is three-dimensional in contrast to two-dimensional perturbations, which are dominant instabilities in deep water. Note that within the framework of gravity waves McLean (Reference McLean1982a ,Reference McLean b ) showed that the most unstable perturbation is two-dimensional in deep water whereas it is three-dimensional in finite depth.

In figure 15 is plotted the dimensionless maximal growth rate as a function of the dispersive parameter $kh$ : the dimensionless maximal growth rate decreases as $kh$ increases.