2 Alber’s equation for the wave envelope

In a recent paper, Pal and Dhar [Reference Pal and Dhar27] derived a fourth-order linear shear currents modified NLSE for two-dimensional periodic gravity waves in a finite depth of water using multiscale expansion. In a frame of reference travelling with the group velocity, the fourth-order NLSE in dimensional form on deep water for the wave envelope $B(\xi ,\tau )$ can be written as

(2.1) $$ \begin{align} i\frac{\partial{B}}{\partial\tau}+\beta_{1}\frac{\partial^2B}{\partial\xi^2}+i\beta_{2}\frac{\partial^3B}{\partial\xi^3}=\mu_{1}\vert{B}\vert^2B +i\mu_{2}{\vert{B}\vert}^2\frac{\partial{B}}{\partial\xi}+i\mu_{3}B^{2}\frac{\partial{B}^{*}}{\partial\xi}+\mu_4{B}H\bigg[\frac{\partial}{\partial\xi}\vert{B\vert}^2\bigg], \end{align} $$

where $\xi =x-{c}_gt$ , ${c}_g$ is the group velocity of the carrier wave in dimensional form, $~\tau =t$ , $B^{*}$ is the complex conjugate of B and H is the Hilbert transform operator. The coefficients of equation (2.1) are given in Appendix A.

The linear dispersion relationship connecting the frequency $\Sigma $ and the wave number k is

(2.2) $$ \begin{align} \Sigma^2(1-\overline{v})(1-\overline{v}+\overline{\Omega})=gk, \end{align} $$

where $\overline {v}=v/c$ , is the nondimensional depth uniform current, $c=\Sigma /k$ is the wave velocity, $\overline {\Omega }=\Omega /\Sigma $ is the nondimensional uniform vorticity and g is the gravitational acceleration.

Starting from equation (2.1), we next obtain Alber’s equation. Following Alber [Reference Alber1] and Wigner–Moyal [Reference Moyal22, Reference Wigner30], we introduce the Wigner distribution function $n(\xi ,k,\tau )$ corresponding to the wave amplitude $B(\xi ,\tau )$ given by

$$ \begin{align*} n(\xi,k,\tau)=\frac{1}{2\pi}\int^{+\infty}_{-\infty}\langle{B(\xi+y/2,\tau)B^*(\xi-y/2,\tau)\rangle}e^{-iky}~dy, \end{align*} $$

where $\xi =(\xi _1+\xi _2)/2$ is the average coordinate, $y=\xi _1-\xi _2$ is the spatial separation coordinate and the angle brackets represent an ensemble average.

The wave intensity corresponding to $B(\xi ,\tau )$ can be expressed as

(2.3) $$ \begin{align} \langle{\vert B(\xi,\tau) \vert ^2}\rangle=\int^{+\infty} _{-\infty}n(\xi,k,\tau)~dk. \end{align} $$

To derive the kinetic equation of $n(\xi ,k,\tau )$ , namely, the Wigner–Moyal equation, we take the time derivative of equation (2.1). The nonlinear terms in NLSE will generate the fourth-order correlators

$$ \begin{align*} \langle B_1B_1^{*}B_1B_2^{*}\rangle,\quad \bigg\langle B_1B_1^{*}\frac{\partial B_1}{\partial\xi_1}B_2^{*}\bigg\rangle,\quad \bigg\langle B_1B_1\frac{\partial B_1^{*}}{\partial\xi_1}B_2^{*}\bigg\rangle,\quad \bigg\langle B_1{H}\frac{\partial}{\partial\xi_1}[B_1B_1^*]B_2^*\bigg\rangle ,\end{align*} $$

respectively, where $B_1=B(\xi +y/2)$ and $B_2=B(\xi -y/2)$ . For proceeding with the calculation, a closure that connects second- and fourth-order correlators must be considered (see [Reference Alber1, Reference Halder and Dhar9]). This closure is obtained by considering the quasi-Gaussian approximation

$$ \begin{align*} \begin{aligned} & \langle B_1B_1^{*}B_1B_2^{*}\rangle=2\langle B_1B_2^{*}\rangle\langle \vert B_1\vert^2\rangle,\nonumber \\ & \bigg\langle B_1B_1^{*}\frac{\partial B_1}{\partial\xi_1}B_2^{*}\bigg\rangle =\bigg\langle \frac{\partial B_1}{\partial\xi_1}B_2^{*}\bigg\rangle\langle \vert B_1\vert^2\rangle+\bigg\langle B_1^*\frac{\partial B_1}{\partial\xi_1}\bigg\rangle \langle B_1B_2^{*}\rangle,\nonumber \\ & \bigg\langle B_1B_1\frac{\partial B_1^{*}}{\partial\xi_1}B_2^{*}\bigg\rangle=2\langle B_1B_2^*\rangle\bigg\langle B_1\frac{\partial B_1^*}{\partial\xi_1}\bigg\rangle \nonumber, \\ & \bigg\langle B_1{H}\frac{\partial}{\partial\xi_1}[B_1B_1^*]B_2^*\bigg\rangle=\langle B_1B_2^*\rangle\bigg\langle{H}\frac{\partial}{\partial\xi_1}\vert{B_1}\vert^2\bigg\rangle, \end{aligned} \end{align*} $$

so that the fourth-order correlators can be written as a sum of the products of pairs of second-order correlators.

This practice is familiar for the statistical description of progressive water waves [Reference Zakharov33] and of several other fields, for example plasma physics [Reference Zakharov, Musher and Rubenchik34]. By a standard procedure as described in [Reference Alber1, Reference Onorato, Osborne, Fedele and Serio23] the resulting Alber’s equation takes the form

(2.4) $$ \begin{align} \frac{\partial{n}}{\partial\tau}&+2\beta_{1}k\frac{\partial{n}}{\partial{\xi}}+\beta_2\bigg(\frac{1}{4}\frac{\partial^3{n}}{\partial{\xi}^3}-3k^2\frac{\partial{n}}{\partial{\xi}}\bigg)=4\mu_{1}\sum_{m=0}^{\infty}a_m\frac{\partial^{2m+1}\langle\vert{B}\vert^2\rangle}{\partial\xi^{2m+1}}\frac{\partial^{2m+1}n}{\partial k^{2m+1}} \nonumber \\& +\mu_{2}\bigg[\sum_{m=0}^{\infty}b_m\bigg(\frac{\partial^{2m}\langle\vert{B}\vert^2\rangle}{\partial\xi^{2m}}\frac{\partial^{2m+1}n}{\partial\xi\partial{k^{2m}}}+\frac{\partial^{2m+1}\langle\vert{B}\vert^2\rangle}{\partial\xi^{2m+1}}\frac{\partial^{2m}n}{\partial{k^{2m}}}\bigg)\bigg.\nonumber \\& -2\sum_{m=0}^{\infty}a_m\frac{\partial^{2m+1}\langle\vert{B}\vert^2\rangle}{\partial\xi^{2m+1}}\frac{\partial^{2m+1}(kn)}{\partial k^{2m+1}} \bigg]+2\mu_{3}\sum_{m=0}^{\infty}b_m\frac{\partial^{2m+1}\langle\vert{B}\vert^2\rangle}{\partial\xi^{2m+1}}\frac{\partial^{2m}n}{\partial k^{2m}} \nonumber \\&+\frac{2\mu_4}{\pi}\sum_{m=0}^{\infty}a_m\frac{\partial^{2m+1}}{\partial\xi^{2m+1}}\bigg(\int^{+\infty} _{-\infty}\frac{d\xi^{'}}{\xi^{'}-\xi}\bigg[\frac{\partial}{\partial\xi^{'}}\langle\vert{B(\xi^{'})}\vert^2\rangle\bigg]\bigg)\frac{\partial^{2m+1}n}{\partial k^{2m+1}}, \end{align} $$

where

$$ \begin{align*}a_m=\frac{(-1)^m}{(2m+1)!~2^{2m+1}}\quad \text{and}\quad b_m=\frac{(-1)^m}{(2m)!~2^{2m}}. \end{align*} $$

3 Stability of random wave spectra

Alber [Reference Alber1] stated that a homogeneous spectrum is unstable subject to long-wavelength perturbations if the bandwidth is sufficiently small. So, to emphasize inhomogeneous ensemble wavetrains, let the distribution function $n(\xi ,k,\tau )$ be written as sum of homogeneous envelope spectra and an infinitesimal perturbation given by

(3.1) $$ \begin{align} n(\xi,k,\tau)=n_0(k)+\epsilon n_1(\xi,k,\tau), \end{align} $$

with $n_1(\xi ,k,\tau )\ll n_0(k)$ , and where $\epsilon $ is a slow ordering parameter.

In view of (2.3) and (3.1),

(3.2) $$ \begin{align} \langle{\vert B(\xi,\tau) \vert ^2}\rangle=\langle{\vert B_0 \vert ^2}\rangle+\epsilon\langle{\vert B_1(\xi,\tau) \vert ^2}\rangle, \end{align} $$

where

(3.3) $$ \begin{align} \langle{\vert B_0 \vert ^2}\rangle=\int^{+\infty} _{-\infty}n_0(k)~dk, \quad \langle{\vert B_1(\xi,\tau) \vert ^2}\rangle=\int^{+\infty} _{-\infty}n_1(\xi,k,\tau)~dk. \end{align} $$

Substituting equations (3.1) and (3.2) into equation (2.4) and linearizing, we get the following equation for the perturbation, that is,

(3.4) $$ \begin{align} \frac{\partial{n_1}}{\partial\tau}&+2\beta_{1}k\frac{\partial{n_1}}{\partial{\xi}}+\beta_2\bigg(\frac{1}{4}\frac{\partial^3{n_1}}{\partial{\xi}^3}-3k^2\frac{\partial{n_1}}{\partial{\xi}}\bigg)=4\mu_{1}\sum_{m=0}^{\infty}a_m\frac{\partial^{2m+1}\langle\vert{B_1}\vert^2\rangle}{\partial\xi^{2m+1}}\frac{\partial^{2m+1}n_0}{\partial{k}^{2m+1}}\nonumber \\& +\mu_{2}\bigg[\langle\vert{B_0}\vert^2\rangle\frac{\partial{n_1}}{\partial\xi} +\sum_{m=0}^{\infty}b_m\frac{\partial^{2m+1}\langle\vert{B_1}\vert^2\rangle}{\partial\xi^{2m+1}}\frac{\partial^{2m}n_0}{\partial{k^{2m}}} -2\sum_{m=0}^{\infty}a_m\frac{\partial^{2m+1}\langle\vert{B_1}\vert^2\rangle}{\partial\xi^{2m+1}}\frac{\partial^{2m+1}(kn_0)}{\partial k^{2m+1}} \bigg]\nonumber \\& +2\mu_{3}\sum_{m=0}^{\infty}b_m\frac{\partial^{2m+1}\langle\vert{B_1}\vert^2\rangle}{\partial\xi^{2m+1}}\frac{\partial^{2m}n_0}{\partial k^{2m}}\nonumber \\&+\frac{2\mu_4}{\pi} \sum_{m=0}^{\infty}a_m\frac{\partial^{2m+1}}{\partial\xi^{2m+1}}\bigg(\int^{+\infty} _{-\infty}\frac{d\xi^{'}}{\xi^{'}-\xi}\bigg[\frac{\partial}{\partial\xi^{'}}\langle\vert{B_1(\xi^{'})}\vert^2\rangle\bigg]\bigg)\frac{\partial^{2m+1}n_0}{\partial k^{2m+1}}. \end{align} $$

Considering the Fourier transform of (3.4) defined by

$$ \begin{align*} \overline{n}_1(k,\tau)=\int_{-\infty}^{+\infty}n_1(\xi,k,\tau)e^{-ip\xi}~d\xi, \quad \langle{\vert \overline{B}_1(\tau) \vert ^2}\rangle=\int_{-\infty}^{+\infty}\langle{\vert B_1(\xi,\tau) \vert ^2}\rangle{e^{-ip\xi}}~d\xi, \end{align*} $$

where p is the wave number of perturbation, and taking $\tau $ dependence of $n_1(k,\tau )$ to be of the form $e^{-i{\nu }\tau }$ , we get

(3.5) $$ \begin{align} & [-\nu+2\beta_{1}kp-\beta_2({p^2}/{4}+3k^2)p)]\overline{n}_1=4\mu_{1}\langle\vert{\overline{B}_1}\vert^2\rangle\sum_{m=0}^{\infty}\frac{p^{2m+1}}{(2m+1)!~2^{2m+1}}\frac{\partial^{2m+1}n_0}{\partial k^{2m+1}} \nonumber \\& \quad +\mu_{2}\bigg[p\overline{n}_1\langle\vert{B_0}\vert^2\rangle+p\langle\vert{\overline{B}_1}\vert^2\rangle\sum_{m=0}^{\infty}\frac{p^{2m}}{(2m)!~2^{2m}}\frac{\partial^{2m}n_0}{\partial{k^{2m}}} \bigg.\nonumber \\&\quad \bigg. -2\langle\vert{\overline{B}_1}\vert^2\rangle\sum_{m=0}^{\infty}\frac{p^{2m+1}}{(2m+1)!~2^{2m+1}}\frac{\partial^{2m+1}(kn_0)}{\partial k^{2m+1}} \bigg] +2\mu_{3}p\langle\vert{\overline{B}_1}\vert^2\rangle\sum_{m=0}^{\infty}\frac{p^{2m}}{(2m)!~2^{2m}}\frac{\partial^{2m}n_0}{\partial k^{2m}} \nonumber \\&\quad -4\mu_4\vert p\vert\langle\vert{\overline{B}_1}\vert^2\rangle \sum_{m=0}^{\infty}\frac{p^{2m+1}}{(2m+1)!~2^{2m+1}}\frac{\partial^{2m+1}n_0}{\partial k^{2m+1}}. \end{align} $$

Taking into consideration the following relationships, which are obtained by using the Taylor’s theorem of $n_0(k\pm p/2)$ , that is,

(3.6) $$ \begin{align} & 2\sum_{m=0}^{\infty}\frac{p^{2m}}{(2m)!~2^{2m}}\frac{\partial^{2m}n_0}{\partial{k^{2m}}}=n_0(k+p/2)+n_0(k-p/2), \nonumber \\ & 2\sum_{m=0}^{\infty}\frac{p^{2m+1}}{(2m+1)!~2^{2m+1}}\frac{\partial^{2m+1}n_0}{\partial k^{2m+1}}=n_0(k+p/2)-n_0(k-p/2),\nonumber \\ & 4\sum_{m=0}^{\infty}\frac{p^{2m+1}}{(2m+1)!~2^{2m+1}}\frac{\partial^{2m+1}(kn_0)}{\partial k^{2m+1}}=2k[n_0(k+p/2)-n_0(k-p/2)]\nonumber \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad +p[n_0(k+p/2)+n_0(k-p/2)], \end{align} $$

equation (3.5) becomes

(3.7) $$ \begin{align} [-\nu+f(k)]\overline{n}_1=[g_{+}(k)n_0(k+p/2) +g_{-}(k)n_0(k-p/2)]\langle\vert{\overline{B}_1}\vert^2\rangle, \end{align} $$

where

$$ \begin{align*}f(k)=2\beta_{1}kp-\beta_2({p^2}/{4}+3k^2)p-\mu_{2}p\langle\vert{B_0}\vert^2\rangle,\quad g_{\pm}(k)=\pm2\mu_1\mp\mu_2k+\mu_3p\mp2\mu_{4} \vert p \vert .\end{align*} $$

The Fourier transform of the second relationship of (3.3) with respect to $\xi $ gives

(3.8) $$ \begin{align} \langle\vert\overline{B}_1(\tau)\vert^2\rangle=\int_{-\infty}^{+\infty}\overline{n}_1(k,\tau)~dk. \end{align} $$

Equations (3.7) and (3.8) reduce to

(3.9) $$ \begin{align} 1+\int_{-\infty}^{+\infty}\frac{g_{+}(k)n_0(k+p/2) +g_{-}(k)n_0(k-p/2)}{\nu-f(k)}~dk=0, \end{align} $$

which represents the dispersion relationship for determining the perturbed frequency $\nu $ for the given homogeneous envelope spectrum $n_0(k)$ . Set

(3.10) $$ \begin{align} f(k)=2\beta_{1}kp+\epsilon{\tilde{f}(k)}, \quad g_{\pm}(k)=\pm2\mu_1+\epsilon{\tilde{g}_{\pm}(k)}, \end{align} $$

where

$$ \begin{align*}{\tilde{f}(k)}=-\beta_2({p^2}/{4}+3k^2)p-\mu_{2}p\langle\vert{B_0}\vert^2\rangle,\quad {\tilde{g}_{\pm}(k)}=\mp\mu_2k+\mu_3p\mp2\mu_4\vert p \vert\end{align*} $$

are the fourth-order contributions of the NLSE (2.1).

Inserting (3.10) in equation (3.9) and keeping terms up to $O(\epsilon )$ , the nonlinear dispersion relationship reduces to

(3.11) $$ \begin{align} \begin{aligned} \Phi(\nu)=\epsilon~\Psi(\nu), \end{aligned} \end{align} $$

with

$$ \begin{align*} & \Phi(\nu)=1+2\mu_{1}\int_{-\infty}^{+\infty}\frac{n_0(k+p/2)-n_0(k-p/2)}{\nu-2\beta_{1}kp}~dk, \\ & \Psi(\nu)=2\mu_1\int_{-\infty}^{+\infty}\frac{[n_0(k-p/2)-n_0(k+p/2)]\tilde{f}(k)}{(\nu-2\beta_{1}kp)^2}~dk \\ & \quad \quad \quad -\int_{-\infty}^{+\infty}\frac{[\tilde{g}_{+}(k)n_0(k+p/2)+\tilde{g}_{-}(k)n_0(k-p/2)]}{\nu-2\beta_{1}kp}~dk. \end{align*} $$

For studying the instability of the homogeneous envelope spectrum, we find that the spectrum for $\zeta (x,t)$ is well approximated by the JONSWAP spectrum [Reference Hasselmann et al.12, Reference Komen, Cavaleri, Donelan, Hasselmann, Hasselman and Janssen16]

$$ \begin{align*} P(\Sigma)=\frac{\alpha{g^2}}{\Sigma^5}e^{-({5}/{4})({\Sigma}/{\Sigma_0})^{-4}}\gamma^{\exp[-{(\Sigma-\Sigma_0)^2}/{2\delta^2\Sigma_0^2}]}, \end{align*} $$

where $\alpha $ is a Philips constant, $\gamma $ is a peak enhancement factor, $\delta $ is the spectral bandwidth, $\Sigma _0=\Sigma (k_0)$ and, for the present study, the frequency $\Sigma $ satisfies the linear dispersion relationship (2.2). For a narrowband approximation of the JONSWAP spectrum, $({\Sigma -\Sigma _0})/{\Sigma _0}=r\ll 1$ , and this can be obtained by a second-order Taylor series expansion of $P(r)$ around $r=0$ : that is,

(3.12) $$ \begin{align} P(r)\simeq P(0)+rP^{'}(0)+\frac{r^2}{2}P^{"}(0)\simeq P_0\bigg[1-\frac{P_0^{"}}{2P_0}r^2\bigg]^{-1} \quad\text{if } r\ll1. \end{align} $$

Since $P^{'}(0)=0$ , it is necessary to retain the term up to the second order of r for considering the approximation of $P(r)$ (see [Reference Janssen, Toba and Mitsuyasu15]). Equation (3.12) reduces to the Lorentzian spectrum in wavenumber space given by

$$ \begin{align*}P(k)=\frac{H_s^2}{16\pi}\frac{\sigma}{\sigma^2+(k-k_0)^2}, \end{align*} $$

where

$$ \begin{align*}H_s=4\sqrt{\pi\frac{\alpha{g^2}\gamma\sigma}{e^{5/4}\Sigma_0^5}} \quad \text{and} \quad \sigma=\sqrt{\frac{2\delta^2}{20\delta^2+\ln\gamma}}\ \bigg(\frac{\Sigma_0}{{v}+g/\sqrt{{\Omega}^2+4gk_0}}\bigg). \end{align*} $$

For a symmetric spectrum $P(k)$ of the free surface elevation, the spectrum for a complex envelope B is given by $n_0(k)=4P(k+k_0)$ (see [Reference Crawford, Saffman and Yuen4, Reference Onorato, Osborne, Fedele and Serio23]).

Neglecting $O(\epsilon )$ terms of equation (3.11) and using the expression for $n_{0}(k)$ given by (3.6), the reduced equation $\Phi (\nu )=0$ becomes

(3.13) $$ \begin{align} 1+\frac{\mu_1\sigma{H_s}^2}{2\pi}&\Bigg[\int_{-\infty}^{+\infty}\frac{dp}{(\nu+2\kappa{k}p)\{\sigma^2+(k+{p}/{2})^2)\}} \nonumber \\ & -\int_{-\infty}^{+\infty}\frac{dp}{(\nu+2\kappa{k}p)\{\sigma^2+(k-{p}/{2})^2)\}}\Bigg]=0. \end{align} $$

Now, setting $k\pm {p}/{2}=z$ in (3.13) and integrating the resulting integral by residues at $z=i\sigma $ , we have the nonlinear dispersion relationship

(3.14) $$ \begin{align} \nu=\Big(-2i\kappa\sigma+\sqrt{\kappa^2 p^2-H_s^2\mu_1\kappa}\Big)p, \end{align} $$

where $\kappa =-\beta _{1}$ . The GRI $\nu _i$ , expressed by the imaginary part of $\nu $ of the perturbation corresponding to the positive complex root of equation (3.14), is the following, and can be obtained when $p^2<H_s^2\mu _1/\kappa $ : that is,

(3.15) $$ \begin{align} \nu_i=\Big(\sqrt{H_s^2\mu_1\kappa-\kappa^2 p^2}-2\kappa\sigma \Big)p. \end{align} $$

Note that, for $\sigma \rightarrow 0$ , equation (3.15) represents the BFI.

It is important to mention that the last term of $\nu _i$ in equation (3.15) has a stabilizing (defocusing) effect and plays the same role as the Landau damping in plasma physics [Reference Landau17], that is, a damping of the perturbation. There is a contest between exponential growth and damping of the perturbation which depends on the two parameters, $\alpha $ and $\gamma $ , of the Lorentzian spectrum. For $\sigma>({1}/{2})\sqrt {H_s^2\mu _1/\kappa -p^2}$ , the damping dominates the modulational instability, and the reverse effect to this will occur if $\sigma <({1}/{2})\sqrt {H_s^2\mu _1\kappa -p^2}$ .

Now, we take $\nu =i\nu _i+\epsilon {\nu _1}$ as the root of equation (3.11). Putting this value of $\nu $ into equation (3.11), we readily obtain $\nu _1$ in the lowest order, that is,

$$ \begin{align*}\nu_1=\frac{\Psi(i\nu_i)}{\Phi^{'}(i\nu_i)}, \end{align*} $$

where

$$ \begin{align*}\Phi^{'}(i\nu_i)=-2\mu_1(J_1-J_2),\quad \Psi(i\nu_i)=2\mu_1(J_3-J_4)-(J_5+J_6) , \end{align*} $$

and the expressions for the integrals $J_n(n=1,\dots ,6)$ are given in Appendix B. Following the same procedure, we evaluate the integrals to obtain the expression of the GRI $\Gamma _{i}$ corresponding to the fourth-order results,

(3.16) $$ \begin{align} \Gamma_i=\nu_i+\Im(\nu_1)=\nu_i-\frac{\mu_4\vert p\vert}{2\mu_1}\bigg[\frac{(\nu_i+2\kappa p \sigma)^2+\kappa^2 p^4}{(\nu_i+2\kappa p\sigma)}\bigg]. \end{align} $$

Herein, the last term in the big brackets of (3.16) is obtained from the fourth-order nonlinear term of the right-hand side of equation (2.1) involving the Hilbert transform, and $\nu _{i}$ is given by (3.15).

Figures 1 and 2 show the marginal stability curves in the ( $\gamma ,\alpha $ ) plane for $p\rightarrow 0$ and $p=0.5$ , respectively. Herein, we wish to study the modulational instability of surface gravity waves with random phase spectra. Therefore, in Figures 1 and 2, I indicates the modulational instability region. One can observe from these figures that spectra with larger values of $\gamma $ and $\alpha $ are more likely to show the modulational instability (see [Reference Onorato, Osborne, Fedele and Serio23]). The depth uniform reverse currents can expand the instability region, whereas following currents have the opposite effect. Again, positive vorticity ( $\overline {\Omega }<0$ ) reduces the instability region, whereas negative vorticity ( $\overline {\Omega }>0$ ) increases the instability region. Figure 2 suggests that fourth-order results reduce the instability region compared with third-order results for fixed values of $\overline {\Omega }$ and $\overline {v}$ . As a check, the modulational instability diagram that we obtain is compared in Figure 1 for $\overline {\Omega }=0$ , $\overline {v}=0$ with that found by Onorato et al. [Reference Onorato, Osborne, Fedele and Serio23]. Thus, we can verify that this diagram reproduced exactly. It is to be noted that, for the purpose of finding the effects of modulational instability, we have taken values in the $(\gamma ,\alpha )$ plane that lie far away from the marginal stability curve (see Figure 1).

Figure 1 Instability region in the $(\gamma ,\alpha )$ plane for $p\rightarrow 0$ : (a) $\overline {\Omega }=0$ ; and (b) $\overline {v}=0$ . $\mathbf {{I}}$ and $\mathbf {{S}}$ indicate the instability and stability regions, respectively.

Figure 2 Instability diagram in the $(\gamma ,\alpha )$ plane for $p=0.5$ : (a) $\overline \Omega =0$ ; and (b) $\overline {v}=0$ . Dashed line: third-order results; solid line: fourth-order results. $\mathbf {{I}}$ and $\mathbf {{S}}$ indicate the instability and stability regions, respectively.

In Figures 3 and 4 we have drawn the GRI as a function of p, the wave number of perturbation, for different values of $\overline {\Omega }$ and $\overline {v}$ . These figures suggest that $\Gamma _i$ increases when both $\alpha $ and $\gamma $ increase. These figures also suggest that the fourth-order results produce a diminished GRI. The depth uniform opposing currents considerably increase the growth rate, whereas following currents diminish the modulational instability. Further, the effect of negative vorticity ( $\overline {\Omega }>0$ ) is to enhance the growth rate, whereas for $\overline {\Omega }<0$ we observe a decrease in the growth rate.

Figure 3 Plot of $\Gamma _i$ versus p for $\gamma =3$ and $\alpha =0.03$ : (a) $\overline {\Omega }=0$ ; and (b) $\overline {v}=0$ . Dashed line: third-order results; solid line: fourth-order results.

Figure 4 Plot of $\Gamma _i$ versus p for $\gamma =5$ and $\alpha =0.05$ : (a) $\overline {\Omega }=0$ ; and (b) $\overline {v}=0$ . Dashed line: third-order results; solid line: fourth-order results.

4 The limit of vanishing bandwidth

For vanishing spectral bandwidth $\sigma \rightarrow {0}$ and, in this case, we can rediscover the BFI for deterministic wavetrains. For $\sigma \rightarrow {0}$ , equation (3.16) becomes

(4.1) $$ \begin{align} \Gamma_i=\nu_i-\frac{\kappa\mu_4}{2\nu_i}{H_s^2 p^2\vert p\vert}\quad \text{with} \quad \nu_i=\bigg(\sqrt{H_s^2\mu_1\kappa-\kappa^2 p^2}\bigg)p. \end{align} $$

There is a relationship between the wave amplitude $\overline {a}$ and the wave significant height $H_s$ given by $\overline {a}=H_s/2$ . Employing this relationship in equation (4.1) and replacing $2\overline {a}^2$ by $a^2$ , where $\overline {a}$ and a represent, respectively, the amplitudes of the random and deterministic wavetrains, we get the deterministic growth rate given by

(4.2) $$ \begin{align} \Gamma_i=\big[{2\kappa{a^2}}(\mu_1-\mu_4\vert p \vert)-\kappa^2 p^2\big]^{1/2}p. \end{align} $$

For $\overline {\Omega }=0$ and $\overline {v}=0$ , this expression for $\Gamma _i$ is the same as that of (3.13) of Dysthe [Reference Dysthe7] for the one-dimensional case. In Figure 5, the growth rate $\Gamma _i$ given by (3.16) is compared with the corresponding deterministic growth rate given by (4.2). This figure suggests that the instability growth rate diminishes due to the effect of randomness, which is consistent with the preceding results of Alber [Reference Alber1] and Halder and Dhar [Reference Halder and Dhar8, Reference Halder and Dhar9]. Further, it is observed that the growth rate obtained from fourth-order results is reduced significantly compared with that obtained from third-order results, which is in agreement with the results of Halder and Dhar [Reference Halder and Dhar9]. The physical significance related to fourth-order results can be stated as follows. Dysthe [Reference Dysthe7] stated that a significant improvement in the stability properties can be achieved by considering the effects of fourth-order perturbation. The dominant new effect introduced to the fourth order is the mean flow response to nonuniformities in the radiation stress caused by modulation of a finite amplitude wave.

Figure 5 Plot of $\Gamma _i$ versus p for $\overline {v}=0.4$ , $\overline {\Omega }=0.5$ , $\gamma =3$ and $\alpha =0.03$ . Comparison with the deterministic growth rate. Dashed line: third-order results; solid line: fourth-order results.

5 The BFI

The concept of the BFI in the context of freak waves has been presented for random waves by Janssen [Reference Janssen14] (see also [Reference Onorato, Osborne and Serio24]). For the definition of BFI, he suggested the ratio of the mean square slope to the normalized width of the frequency spectrum. When this parameter is greater than one, the random wave field is unstable, whereas in the opposite case it is stable. From the NLSE, Onorato et al. [Reference Onorato, Osborne, Serio, Cavaleri, Brandini and Stansberg25] also defined the BFI in the context of freak waves.

It is noteworthy that, among the new fourth-order terms of the NLSE (2.1), only the last term involving Hilbert transform contributes to the instability results given by equations (3.16) and (4.1). Therefore, as far as stability properties are considered, it is enough to use the simplified form of the dimensional NLSE (2.1) on deep water [Reference Dysthe7] given by

(5.1) $$ \begin{align} i\frac{\partial{B}}{\partial\tau}+\frac{\Sigma}{k^2}\tilde{\beta}_{1}\frac{\partial^2B}{\partial\xi^2}=\Sigma{k^2}\tilde{\mu}_{1}\vert{B}\vert^2B +\Sigma{k}\tilde{\mu_4}{B}{H}\bigg[\frac{\partial}{\partial\xi}\vert{B\vert}^2\bigg], \end{align} $$

where B is the complex wave envelope and ( $\Sigma ,k$ ) represent, respectively, the carrier frequency and wave number. Here the coefficients appearing in equation (5.1) depend on the parameters $\overline {v}$ and $\overline {\Omega }$ . Following Onorato et al. [Reference Onorato, Osborne, Serio, Cavaleri, Brandini and Stansberg25], we rewrite equation (5.1) by transforming $B^{'}=B/a,~\xi ^{'}=\Delta {k}\xi ,~\tau ^{'}=\Sigma \tilde {\beta }_1(\Delta {k}/k)^2\tau $ given by

(5.2) $$ \begin{align} i\frac{\partial{B}}{\partial\tau}+\frac{\partial^2B}{\partial\xi^2}=\bigg(\frac{ak}{\Delta{k}/k}\bigg)^2\bigg(\frac{\tilde{\mu}_{1}}{\tilde{\beta}_1}\bigg)\vert{B}\vert^2B +\frac{(ak)^2}{\Delta{k}/k}\bigg(\frac{\tilde{\mu_4}}{\tilde{\beta}_1}\bigg)B{H}\bigg[\frac{\partial}{\partial\xi}\vert{B\vert}^2\bigg], \end{align} $$

where primes of $B,~\xi $ and $\tau $ have been omitted and $\Delta {k},a$ denote typical bandwidth and wave amplitude, respectively. According to Onorato et al. [Reference Onorato, Osborne, Serio, Cavaleri, Brandini and Stansberg25], we define the BFI as the square root of the coefficient that multiplies the cubic nonlinear term of equation (5.2). Therefore,

(5.3) $$ \begin{align}\text{BFI}=\frac{ak}{\Delta{k}/k}\sqrt{\frac{\tilde{\mu}_{1}}{\vert\tilde{\beta}_1\vert}}. \end{align} $$

The term $\sqrt {\tilde {\mu }_1/\vert \tilde {\beta }_1\vert }$ on the right-hand side of the BFI given by (5.3) depends on the parameter $\overline {\Omega }$ , the magnitude of the shear. The effect of this term on the BFI against $\overline {\Omega }$ is presented in Figure 6. We observe that, for $\overline {\Omega }>0$ , the BFI increases with an increase of vorticity. Again, as the BFI increases, the nonlinearity also increases. So, we may hope that the number of freak waves is enhanced in the presence of shear currents co-flowing with the waves. It is to be mentioned that this result was first found numerically with the help of numerical simulations of the cubic NLSE in the paper by Onorato et al. [Reference Onorato, Osborne and Bertone26]. Again, for $\overline {\Omega }<0$ , the presence of vorticity diminishes the BFI.

Figure 6 Effect of vorticity on the BFI.

6 Discussion and conclusions

Dysthe [Reference Dysthe7] first stated that a fourth-order NLSE is a good starting point for analysing the stability of surface periodic waves on deep water. Therefore, starting from fourth-order NLSE for surface gravity waves in the presence of linear shear currents, in this paper, we first derive Alber’s equation, and, based on this equation, we then describe the instability of weakly nonlinear waves with random phase spectra. The key outcomes of our study are as follows. The present fourth-order analysis exhibits significant deviations in the instability growth rate compared with the third-order analysis. Also, we find the effect of linear shear currents on the modulational instability properties of weakly nonlinear waves. It is observed that the fourth-order results produce a decrease in the GRI. Further, the effect of randomness is to decrease the GRI, which is consistent with the previous results [Reference Alber1]. For waves on shearing currents, negative vorticity tends to enhance the modulational instability, whereas positive vorticity decreases the instability. Further, for waves moving in the same direction as the uniform current, the current was observed to have a stabilizing effect on the wavetrains and diminishes the instability growth rate. We also recovered the deterministic GRI for vanishing spectral bandwidth.