2 Basic equations

Consider the equation of the undisturbed free surface as the $y=0$ plane. Here, we take two-dimensional Cartesian coordinates $x,y$ in which the x-axis is directed towards the direction of propagation of the waves and the y-axis is oriented upwards. We suppose that the fluid motion is incompressible and nonviscous and the waves are moving steadily on a vertical linear shear current, which can be separated into a depth uniform current v and a uniform vorticity $-\omega $ . Let $y=\eta (x,t)$ be the equation of the free surface in the perturbed state. There is a potential function $\phi (x,y,t)$ for which the total velocity $\textbf {u}$ of the fluid flow can be represented as

$$ \begin{align*} \begin{aligned} \textbf{u}(x,y)=(v+\omega y)\textbf{i}+\nabla\phi(x,y,t), \end{aligned} \end{align*} $$

where v is the speed of the linear shear current along the direction of propagation of the waves at the free surface and $\textbf {i}$ represents the unit vector along the x-axis. As the vorticity $-\omega $ of the basic flow is constant, the velocity field induced by a two-dimensional perturbation must be irrotational due to Kelvin’s theorem [Reference Thomas, Kharif and Manna39].

As the perturbation is supposed to be potential, the perturbed velocity potential $\phi $ and the stream function $\psi $ of the fluid satisfy the two-dimensional Laplace equations as follows:

(2.1) $$ \begin{align} \nabla^2\phi=0, \quad \nabla^2\psi=0, \quad \text{in} \, -d<y<\eta(x,t), \end{align} $$

in which $\phi $ , $\psi $ are connected by the Cauchy–Riemann relations

$$ \begin{align*} \begin{aligned} \frac{\partial\phi}{\partial x}=\frac{\partial\psi}{\partial y},\quad \frac{\partial\phi}{\partial y}=-\frac{\partial\psi}{\partial x}. \end{aligned} \end{align*} $$

The kinematic free surface boundary condition is

(2.2) $$ \begin{align} \begin{aligned} \frac{\partial\phi}{\partial y}-\frac{\partial\eta}{\partial t}-v\frac{\partial\eta}{\partial x}=\bigg(\frac{\partial\phi}{\partial x}+\omega\eta\bigg)\frac{\partial\eta}{\partial x}, \quad \text{on} \, y=\eta(x,t). \end{aligned} \end{align} $$

The dynamic surface boundary condition is given by

(2.3) $$ \begin{align} \begin{aligned} \frac{\partial\phi}{\partial t}+v\frac{\partial\phi}{\partial x}-\omega\psi+g\eta=-\frac{1}{2}(\nabla\phi)^{2}-\omega\eta\frac{\partial\phi}{\partial x} +\bigg(\frac{T}{\rho}\bigg)\frac{\eta_{xx}}{(1+\eta_x^{2})^{3/2}},\quad \text{on } y=\eta(x,t), \end{aligned} \end{align} $$

where T and $\rho $ denote the surface tension coefficient and density of bulk water, respectively.

Also at the bottom, $\phi $ and $\psi $ satisfy the following boundary conditions:

(2.4) $$ \begin{align} \begin{aligned} \frac{\partial\phi}{\partial y}=0, \quad \psi=0, \quad \text{on } y=-d. \end{aligned} \end{align} $$

We consider the solutions of equations (2.1)–(2.4) as follows:

(2.5) $$ \begin{align} \begin{aligned} G=G_0+\displaystyle\sum_{m=1}^{\infty}[G_m\exp\{ i m(k x-\sigma t) \}+\text{c}.\text{c}.], \end{aligned} \end{align} $$

where G stands for $\phi (x,y,t),~\psi (x,y,t)$ and the free surface elevation $\eta (x,t)$ , k and $\sigma $ represent carrier wavenumber and frequency respectively and c.c. means complex conjugate. Here $\phi _{0},~\phi _{m},~\phi _m^{*},~\psi _{0},~\psi _{m},~\psi _m^{*}~(m=1,2,\ldots )$ are slowly varying functions of $x_1=\epsilon x,~t_1=\epsilon t$ , $y~;~\eta _{0},~\eta _{m},~\eta _m^{*}~(m=1,2, \ldots )$ are functions of $x_{1},~t_{1}$ . Here, $\epsilon $ is a slow ordering parameter which measures the weakness of nonlinearity and $0<\epsilon <<1$ .

The linear dispersion relation to determine the frequency $\sigma $ of the carrier wave is

$$ \begin{align*} \begin{aligned} f(\sigma,k)\equiv\sigma^{2}(1-\overline{v})(1-\overline{v}+\beta)-g k\mu(1+\kappa)=0, \end{aligned} \end{align*} $$

where $\overline {v}=v/c,~c=\sigma /k$ , the velocity of the carrier wave, $\beta =\mu \overline {\omega },~\overline {\omega }=\omega /\sigma , \mu =\tanh {p},~p=k d,~\kappa =T k^{2}/\rho g$ .

The group velocity $c_g$ becomes

$$ \begin{align*} c_{g}&=c\{(1-\overline{v})^{2}p(1-\mu^{2})/\mu+(1-\overline{v})(1-\overline{v}+\beta)(1+3\kappa)/(1+\kappa) \nonumber \\ &\quad +\overline{v}(2-2\overline{v}+\beta)\}(2-2\overline{v}+\beta)^{-1}. \end{align*} $$

3 Derivation of evolution equation using multiple scale method

In this section, we develop an NLEE accurate up to fourth order for narrow banded GCWs in the case of finite water depth, and discuss the two types of singularity.

On substituting the expansions in equation (2.5) into equation (2.1) and using bottom conditions in equation (2.4), we get the required solutions for $\phi _{m},~\psi _m, (m=1,2)$ as follows:

(3.1) $$ \begin{align} \begin{aligned} &\phi_m=\frac{\cosh{((y+d)K_{m})}}{\cosh{(d K_m)}}C_{m},\\[3pt] &\psi_m=\frac{\sinh{((y+d)K_{m})}}{\cosh{(d K_m)}}D_{m}, \end{aligned} \end{align} $$

where $K_{m}=k m-i\epsilon ({\partial }/{\partial x_1})$ and $C_{m},~D_{m}$ are functions of $x_1$ and $t_1$ . Next, for $m=0$ ,

(3.2) $$ \begin{align} \begin{aligned} &\overline{\phi}_0=\frac{\cosh{((y+d)\epsilon\overline{k})}}{\cosh{(\epsilon d\overline{k})}}\overline{C}_0,\\ &\overline{\psi}_0=\frac{\sinh{((y+d)\epsilon\overline{k})}}{\cosh{(\epsilon d\overline{k})}}\overline{D}_0, \end{aligned} \end{align} $$

in which $\overline {C}_0,~\overline {D}_0$ are functions of $\overline {k},~\overline {\omega }_1$ , and $\overline {\phi }_0,~\overline {\psi }_0$ are Fourier transforms of $\phi _{0},~\psi _0$ , respectively, given by

$$ \begin{align*} \begin{aligned} (\overline{\phi}_0,\overline{\psi}_0)=\frac{1}{2\pi}\iint^\infty _{-\infty} \mathrm{({\phi_0}, {\psi_0})}~\mathrm{exp}[-i (\overline{k} x_1-\overline{\omega}_{1}t_1)]\,{d}x_1\, {d}t_1. \end{aligned} \end{align*} $$

Inserting the expansions in equation (2.5) into the Taylor’s expanded form of equations (2.2) and (2.3) about $y=0$ and then equating coefficients of $\mathrm {exp}~i m(k x-\sigma t)$ , for $m=1,2,0$ on both sides, we obtain three sets of equations into each of which we substitute the solutions for $\phi _{m},~\psi _m$ given by equations (3.1)–(3.2). For the purpose of solving these equations, we take the expansions as follows:

(3.3) $$ \begin{align} H_m=\displaystyle\sum_{n=1}^{\infty}\epsilon^n H_{m n}~(m=0,1),\quad H_2=\displaystyle\sum_{n=2}^{\infty}\epsilon^n H_{2n}, \end{align} $$

where $H_p$ stands for $C_p,~D_p$ and $\eta _p~(p=0,1,2)$ .

Inserting equation (3.3) into the three sets of equations and then equating coefficients of several powers of $\epsilon $ , we get a sequence of equations. From the first (lowest) and second-order equations of the three sets, we get solutions for $(C_{11},~C_{12})$ , $(C_{22},~\eta _{22},~C_{23},~\eta _{23})$ and $(C_{01},~\eta _{01},~C_{02},~\eta _{02})$ , respectively.

The equation corresponding to equation (2.3) of the first set of equations can be put in the following convenient form:

(3.4) $$ \begin{align} \begin{aligned} f(\sigma_{1},K_{1})\eta_{1}=-i\{(\sigma_{1}-K_{1}v)+\omega\mu\}a_{1}-K_{1}\mu b_1, \end{aligned} \end{align} $$

where $\sigma _{1}=\sigma +i\epsilon ({\partial }/{\partial t_1}),~K_{1}=k-i\epsilon ({\partial }/{\partial x_1})$ and $a_1,~b_1$ are obtained from nonlinear terms.

We retain terms up to fourth-order $O(\epsilon ^{4})$ and insert solutions for different perturbed quantities arising on the right-hand side of equation (3.4). Finally, applying the transformations

(3.5) $$ \begin{align} \xi=\epsilon(x-c_{g}t),\quad \tau=\epsilon^2 t, \end{align} $$

and taking $\eta =\eta _1=\epsilon \eta _{11}+\epsilon ^{2}\eta _{12}$ , we get the fourth-order NLEE as follows:

(3.6) $$ \begin{align} i\frac{\partial\eta}{\partial\tau}+\alpha_{1}\frac{\partial^2\eta}{\partial\xi^2}+i\alpha_2\frac{\partial^3\eta}{\partial\xi^3}&=\mu_{1}\mid\eta\mid^{2}\eta+i\mu_{2}\mid\eta\mid^{2} \frac{\partial\eta}{\partial\xi}+i\mu_{3}\eta^{2}\frac{\partial\eta^{*}}{\partial\xi}\nonumber \\ &\quad +\mu_{4}\eta\frac{\partial}{\partial\xi}F^{-1}\bigg[\frac{F\frac{\partial}{\partial\xi}(\mid\eta\mid^{2})}{\overline{k}\tanh{(\epsilon\overline{k}p)}}\bigg], \end{align} $$

where F represents the spatial Fourier transform. The coefficients $\alpha _{1},~\alpha _{2}$ and $\mu _i (i=1,2,3,4)$ are given in Appendix A.

We have applied the scaling transformations $\eta ^{\prime}=2k\eta ,~\xi ^{\prime}=k\xi $ , $\tau ^{\prime}=\sigma \tau $ in equation (3.6) and then dropped the primes. The nonlinear spatio-temporal evolution of weakly nonlinear GCWs can be described by the NLEE in equation (3.6), provided the wave steepness is small $(\ll 1)$ and the spectral bandwidth is narrow $(\ll 1)$ .

Note that the derivation of the NLEE correct to fourth order involves some algebra. To render the results convincing, it is useful to compare with other results.

We can check that the coefficients $\alpha _{1}$ and $\mu _1$ corresponding to cubic NLEE reduce to those of Liao et al. [Reference Liao, Dong, Ma and Gao22] for $\kappa =0$ and to those of Hsu et al. [Reference Hsu, Kharif, Abid and Chen16] for $\overline {v}=0$ .

In the expression for $\mu _1$ , due to the presence of the factor

$$ \begin{align*}\mu^{2}(1-\overline{v})-\kappa\{{(3-\mu^2)(1-\overline{v})+3\beta}\}\end{align*} $$

in its denominator resulting from the expression for $\eta _{22}$ , the NLEE in equation (3.6) does not remain valid when $\kappa $ satisfies equation (3.7). The value of $\kappa $ for which the singularity occurs is

(3.7) $$ \begin{align} \kappa=\frac{\mu^{2}(1-\overline{v})}{(3-\mu^2)(1-\overline{v})+3\beta}. \end{align} $$

Herein, the speeds c of the carrier wave and second harmonic wave coincide (for clarification, see [Reference McGoldrick28]), resulting in the phenomenon known as second harmonic resonance.

This resonance, also called capillary-gravity resonance, was first pointed out for $\beta =0,~\overline {v}=0$ and for deep water by Wilton [Reference Wilton41], where $\kappa =1/2$ . The physical significance of the value of $\kappa =1/2$ was explained by Harrison [Reference Harrison12] for deep water. He argued that the effect of nonlinearity on GCWs is completely different depending on whether $\kappa $ is greater than or less than $1/2$ . The effect of nonlinearity (higher harmonics) for $\kappa>1/2$ is to distort the wave profile, so that the crests are flattened and the troughs are sharpened. Wave profiles of this kind are known as pure capillary waves. A nonlinear effect reverse to this is observed when $\kappa <1/2$ and profiles of this kind are called gravity waves.

We also get a second possible singularity related to the long/short wave resonance, in which the group velocity of the short wave is equal to the phase velocity of the long wave. The last term within the brackets of the expression for $\mu _1$ (see Appendix A) corresponds to the coupling between the wave induced mean flow response and the vorticity which occurs at third order, and this coupling has a significant impact on modulational instability. This term of $\mu _1$ is found to be singular when

$$ \begin{align*} (\gamma-\overline{v})(\gamma+\overline{\omega}p)=\frac{(1-\overline{v})(1-\overline{v}+\beta)p}{\mu(1+\kappa)}, \end{align*} $$

where $\gamma =c_{g}/c$ , and may be expressed as

(3.8) $$ \begin{align} c_g(c_{g}+\omega d)=g d \quad \text{for } \, \overline{v}=0. \end{align} $$

Equation (3.8) reduces to $c_g^{2}=g d$ in the absence of vorticity, which corresponds to a long wave resonance found by Davey and Stewartson [Reference Davey and Stewartson8], and Djordjevic and Redekopp [Reference Djordjevic and Redekopp10].

We have plotted the critical surface tension coefficient $\kappa $ given by equation (3.7) as a function $kd$ in Figure 1 for some values of $\overline {v}$ and $\overline {\omega }$ . It is found that the value of $\kappa $ decreases with the increase of negative vorticity $(\overline {\omega }>0)$ and eventually, the influence of capillarity is lost. Again, the value of $\kappa $ increases with the increase of positive vorticity $(\overline {\omega }<0)$ , and the influence of capillarity is expected to become important. Further, we observe that for a fixed value of negative vorticity, the depth uniform following current decreases the value of $\kappa $ , while the reverse current increases it. The effect reverse to this is found in the case of positive vorticity. The curve drawn in Figure 1(a) for $\overline {v}=0$ and $\overline {\omega }=0$ is identical to the curve in figure 2(a) of Hsu et al. [Reference Hsu, Francius, Montalvo and Kharif15].

Figure 1 Behaviour of $\kappa $ against $kd$ : (a) $\overline {v}=0$ and different values of $\overline {\omega }$ ; (b) $\overline {\omega }=0.3$ and different values of $\overline {v}$ ; (c) $\overline {\omega }=-0.3$ and different values of $\overline {v}$ .

4 Evolution equation for deep water and nonlinear coupling

Herein, we first reduce the evolution equation for deep water, and then discuss the coupling due to nonlinearity between the wave-induced mean flow term and the vorticity, which occurs at both third and fourth orders. The infinite depth approximation is $\mu =\tanh {p}\rightarrow 1$ . Further, if $\tanh {(\epsilon \overline {k}p)}\rightarrow 1 $ (with $p=k d$ ) is used in the last term of equation (3.6), then it takes the form

in which H is the Hilbert transform operator given by

$$ \begin{align*} H[\Gamma(\xi)]=\frac{1}{\pi}P\int^\infty _{-\infty}\frac{\Gamma(\xi^{\prime})}{\xi^{\prime}-\xi}\,{d}\xi^{\prime}. \end{align*} $$

Now equation (3.6) for deep water can be written as

(4.1)

where the coefficients for deep water are given in Appendix B. Without vorticity and depth uniform current $\overline {v}$ , these coefficients agree with those of Hogan [Reference Hogan14].

At third order, the coefficient of equation (4.1) contains two terms. The first term is obtained from the dispersion relation of GCWs propagating on a linear shear current, and the second term arises from the nonlinear coupling between the wave-induced mean flow response and the vorticity. Without vorticity, it is to be noted that this coupling disappears. Again, for $\beta =0$ and $\overline {v}=0$ , the third-order nonlinear coefficient vanishes when $\omega =-2/3$ and equation (4.1) contains only fourth-order nonlinear terms. At fourth order, the coefficient arises from a nonlinear coupling between wave-induced current and the wave field, and this coupling is still present when $\omega =0$ , as found a long time ago by Dysthe [Reference Dysthe11]. The significant effect introduced to fourth order is the wave-induced mean flow response to nonuniformities in the radiation stress caused by modulation of the finite amplitude wavetrain, as reported by Dysthe [Reference Dysthe11].

5 Modulational instability analysis and results

On the basis of the NLEE for deep water, we investigate here the influence of linear shear current on the modulational instability and obtain the nonlinear dispersion relation and the instability condition.

The solution for the uniform wavetrain of equation (4.1) is

$$ \begin{align*} \eta=\eta_{0}\,\mathrm{exp}(i\Delta\sigma\tau), \end{align*} $$

where $\eta _0$ is the wave steepness and the nonlinear frequency shift $\Delta \sigma $ is given by

We next introduce the following small perturbation in the above solution:

(5.1) $$ \begin{align} \eta=\eta_{0}[1+\eta^{\prime}(\xi,\eta)]\,\mathrm{exp}(i\Delta\sigma\tau). \end{align} $$

We substitute equation (5.1) in equation (4.1), linearize with respect to $\eta ^{\prime}$ and $\eta ^{*}{'}$ , and separate the equations after setting $\eta ^{\prime}=\eta _r^{\prime}+i\eta _i^{\prime}$ , where $\eta _r^{\prime}$ and $\eta _i^{\prime}$ are real. Then we take the Fourier transform of these equations defined by

$$ \begin{align*} (\overline{\eta}_r^{\prime},\overline{\eta}_i^{\prime})=\int_{-\infty}^{\infty} (\eta_r^{\prime},\eta_i^{\prime})\,\mathrm{exp}(-i\lambda\xi)\,{d}\xi, \end{align*} $$

and obtain two equations

and

Assuming $\tau $ -dependence of $\eta _r^{\prime}$ and $\eta _i^{\prime}$ to be of the form , we get the nonlinear dispersion relation as follows:

(5.2)

where are the perturbed frequency and wave number, respectively.

For instability,

(5.3)

and the growth rate then becomes

(5.4)

where indicates the imaginary part of .

From equation (5.3), the bandwidth of instability becomes

Again, , which is the real part of at marginal stability, gives

If the condition in equation (5.3) is satisfied, the maximum growth rate takes the form

(5.5)

which occurs for wavenumber of perturbation $\lambda _m$

Further, , the real part of corresponding to $\lambda _m$ takes the form

(5.6)

Physically, we observe two types of nonlinear interaction influencing the results. First, the relative signs of the frequency dispersion term and the nonlinear term of equation (4.1) that occur at third order govern the overall modulational instability properties of the solution. It is important to note that the key element is the sign of the product . Second, the corrections to the modulational instability that occur at fourth order come from the nonlinear interaction between the induced mean flow term and the frequency dispersion term . Further, significance has been attached to the term of equation (4.1). We observe from equation (5.2) that it gives the real $O(\eta _0^{2})$ correction to the frequency of very long plane perturbation to the wave train.

Figure 2 Plot of modulational instability growth rate against $\lambda $ for $\overline {v}=0$ and several values of $\overline {\omega }$ and $\kappa $ : (a) $\eta _{0}=0.1$ ; (b) $\eta _{0}=0.2$ .

Figures 2 and 3 exhibit the influence of vorticity, depth uniform current and capillarity from the fourth-order result on the growth rate of instability given by equation (5.4) in deep water. Herein, the instability growth rate is found to be considerably changed by the magnitude and sign of the current shear. As observed from Figure 2, the current shear for $\overline {\omega }>0$ tends to increase the growth rate, whereas the current shear for $\overline {\omega }<0$ has the reverse effect. Figure 3 exhibits that depth uniform reverse currents can spread out the onset criterion and considerably increase the growth rate, whereas following currents decrease the growth rate. Furthermore, the effect of capillarity depresses the growth rate giving a stabilizing influence up to a certain value of $\lambda $ and the growth rate is shown to increase with the increase of $\eta _0$ .

Figure 3 Plot of as a function of $\lambda $ for $\overline {\omega }=0$ and several values of $\overline {v}$ and $\kappa $ : (a) $\eta _{0}=0.1$ ; (b) $\eta _{0}=0.2$ .

Using equation (5.5), the maximum growth rate of instability has been drawn in Figures 4 and 5 against $\eta _0$ for $\overline {v}=0$ and $\overline {\omega }=0,0.5$ respectively. It is observed from Figure 4 that the maximum growth rate obtained from fourth-order results first increases with $\eta _0$ and then it diminishes, whereas the maximum growth rate obtained from third-order results increases steadily with $\eta _0$ . Further, the maximum growth rate increases with $\overline {\omega }$ and it decreases with the increase of depth uniform current $\overline {v}$ . The curve corresponding to $\overline {v}=0,~\overline {\omega }=0,~\kappa =0$ is the same as the curve found in figure 2 of Dysthe [Reference Dysthe11], and he reported that equation (5.5) for $\overline {v}=0,~\overline {\omega }=0,\kappa =0$ is considerably close to the exact findings of Longuet-Higgins [Reference Longuet-Higgins24, Reference Longuet-Higgins and Stewart25] for $\eta _0<0.3$ . Thus, we get an excellent agreement with the findings obtained by Longuet-Higgins [Reference Longuet-Higgins24, Reference Longuet-Higgins and Stewart25].

Figure 4 Plot of against $\eta _0$ for $\overline {v}=0$ and different values of $\overline {\omega }$ and $\kappa $ : (a) fourth-order result; (b) third-order result.

Figure 5 Plot of as a function of $\eta _0$ for several values of $\overline {v}$ and $\kappa $ : (a) $\overline {\omega }=0$ ; (b) $\overline {\omega }=0.5$ .

The plot of given by equation (5.6) has been depicted against $\eta _0$ , as shown in Figure 6. From Figure 6(a), we find that the curve obtained from the results for $\overline {v}=0,~\overline {\omega }=0~\text {and}~\kappa =0$ is the same as that obtained in figure 3 of Dysthe [Reference Dysthe11]. This particular curve, as indicated by Dysthe, is in good agreement with the curve found from the exact findings of Longuet-Higgins [Reference Longuet-Higgins24, Reference Longuet-Higgins and Stewart25], and the experimental results of Lake and Yuen [Reference Yuen and Lake42] and Benjamin and Feir [Reference Benjamin and Feir1] for $\eta _0<0.2$ .

Figure 6 Plot of as a function of $\eta _0$ : (a) $\overline {v}=0$ ; (b) $\overline {\omega }=0$ .

5.1 Instability growth rate and bandwidth for finite depth

Note that the last term of equation (3.6) contains a term $\tanh {(\epsilon \overline {k}p)}$ . In accordance with Brinch-Nielsen and Jonsson [Reference Brinch-Nielsen and Jonsson4], the arbitrary water depth supposition is $\tanh {(\epsilon \overline {k}p)}=\epsilon \overline {k}k d$ , and they have noted that fourth-order terms of equation (3.6) do not contribute to the expression for $\mathrm {Im}(\Omega )$ , where $\Omega $ represents the perturbed frequency for finite water depth.

To obtain the growth rate $\mathrm {I m}(\Omega )$ for finite depth, we replace and by $\alpha _{1}$ and $\mu _{1}$ , respectively, in equation (5.4), where $\alpha _{1}$ and $\mu _{1}$ are the coefficients of third-order dispersive and nonlinear terms, respectively, of equation (3.6). Thus, at third order, the normalized growth rate becomes

(5.7) $$ \begin{align} \frac{\mathrm{I m}(\Omega)}{\eta_0^{2}}=\frac{\lambda}{\eta_0}\sqrt{-\alpha_{1}\bigg\{\alpha_{1}\bigg(\frac{\lambda}{\eta_0} \bigg)^{2}+2\mu_{1}\bigg\}}, \end{align} $$

and the instability bandwidth is

(5.8) $$ \begin{align} \lambda=\sqrt{\frac{-2\mu_1}{\alpha_1}}\eta_0. \end{align} $$

Further, the maximal growth rate of instability $G_m$ becomes

(5.9) $$ \begin{align} G_{m}=|\mu_{1}|\eta_0^{2}. \end{align} $$

For $\overline {v}=0$ , the expressions of $\mathrm {I m}(\Omega )$ and $G_m$ reduce to the corresponding expressions of Hsu et al. [Reference Hsu, Kharif, Abid and Chen16].

The instability diagrams are shown in Figure 7 as a function of the parameters $\beta $ and $kd$ for $\kappa =0$ and $0.035$ . For $\kappa =0$ , this diagram is identical to figure 3 of Thomas et al. [Reference Thomas, Kharif and Manna39]. The critical water depth $kd_{\mathrm {crit}}$ of $kd$ for $\beta =0$ has the well-known value of 1.363, above which the instability prevails. For finite depth, the condition $\alpha _{1}\mu _{1}<0$ corresponds to modulational instability. For $\overline {\omega }=-2/3$ , $\alpha _{1}\mu _1$ alters sign and, as a consequence, the nature of stability changes. Hence, in deep water, there is no instability when $-1<\overline {\omega }\leq -2/3$ .

Figure 7 Instability diagrams in the $(\beta ,kd)$ plane for $\overline {v}=0$ : (a) $\kappa =0$ ; (b) $\kappa =0.035$ . The unstable regions are in blue while the stable regions are in white (colour available online).

Stable and unstable regions are plotted in Figure 8 for $\omega =0,~\overline {v}=0$ . The red and black curves (online) are due to the singularities in the nonlinear coefficient $\mu _1$ . For $\omega =0,~\overline {v}=0$ , results for instability boundaries are presented by Djordjevic and Redekopp [Reference Djordjevic and Redekopp10], and exhibited in figure 6 of Hsu et al. [Reference Hsu, Kharif, Abid and Chen16], and our stability diagram matches the findings of Djordjevic and Redekopp [Reference Djordjevic and Redekopp10]. Thus, we can check that this limiting case is again produced correctly.

Figure 8 Instability diagram in the $(kd,\kappa )$ plane for $\omega =0,~\overline {v}=0$ . The unstable regions are in cyan while the stable regions are in white. The red curve is due to the singularity in $\mu _1$ obtained from the second harmonic resonance given by equation (3.7) and the black curve is due to the singularity in $\mu _1$ obtained from the long wave resonance condition given by equation (3.8) (colour available online).

The effect of negative vorticity $(\omega>0)$ is shown in Figure 9 for $\overline {v}=0$ . The red and black curves correspond to the singularities of $\mu _1$ . It is found that the vorticity has a considerable effect on the instability diagram of GCWs. With the increase of $\omega $ , the instability band along the $kd$ -axis that corresponds to small values of $\kappa $ becomes narrower.

Figure 9 Instability diagrams in the $(kd,\kappa )$ plane for $\overline {v}=0$ and several values of $\omega $ . The unstable regions are in cyan while the stable regions are in white (colour available online).

In Figures 10 and 11, the growth rate $\mathrm {Im}(\Omega )/\eta _0^{2}$ given by equation (5.7) against $\lambda /\eta _0$ has been plotted for $\overline {v}=0$ , and some values of $\overline {\omega }$ and $\kappa $ for finite depth and deep water, respectively. From Figure 10, one can observe that for finite depth, the growth rate increases notably due to the combined effect of vorticity and capillarity when $\overline {\omega }>0$ , consistent with the results of Hsu et al. [Reference Hsu, Kharif, Abid and Chen16]. As observed in these figures for $\kappa =0$ , as shown by Liao et al. [Reference Liao, Dong, Ma and Gao22] and Thomas et al. [Reference Thomas, Kharif and Manna39], the current shear for $\overline {\omega }>0$ tends to significantly enhance the modulational instabilities, whereas the current shear for $\overline {\omega }<0$ has the adverse effect. Again, the growth rate increases with the increase of water depth.

Figure 10 Plot of $\mathrm {Im}(\Omega )/\eta _0^{2}$ as a function of $\lambda /\eta _0$ for $\overline {v}=0$ and several values of $\overline {\omega }$ and $\kappa $ : (a) $kd=1.5$ ; (b) $kd=2$ .

Figure 11 Plot of $\mathrm {Im}(\Omega )/\eta _0^{2}$ as a function of $\lambda /\eta _0$ in deep water for $\overline {v}=0$ and several values $\overline {\omega }$ and $\kappa $ .

The plots of $\mathrm {Im}(\Omega )/\eta _0^{2}$ as a function of $\lambda /\eta _0$ for $\overline {\omega }=0~\text {and}~0.5$ in finite depth are shown in Figures 12 and 13, respectively. It is found that the growth rate increases with the increase of depth uniform opposing current.

Figure 12 Plot of $\mathrm {Im}(\Omega )/\eta _0^{2}$ as a function of $\lambda /\eta _0$ for $\overline {\omega }=0$ and several values of $\overline {v}$ and $\kappa $ : (a) $kd=1.5$ ; (b) $kd=2$ .

Figure 13 Plot of $\mathrm {Im}(\Omega )/\eta _0^{2}$ as a function of $\lambda /\eta _0$ for $\overline {\omega }=0.5$ and several values of $\overline {v}$ and $\kappa $ : (a) $kd=1.5$ ; (b) $kd=2$ .

Figure 14 shows that in deep water, depth uniform reverse currents significantly increase the growth rate, whereas following currents decrease the modulational instability, consistent with the findings in figure 4(b) of Liao et al. [Reference Liao, Dong, Ma and Gao22] for $\overline {\omega }=0,~\kappa ~=~0$ .

Figure 14 Plot of $\mathrm {Im}(\Omega )/\eta _0^{2}$ as a function of $\lambda /\eta _0$ in deep water for $\overline {\omega }=0$ and several values $\overline {v}$ and $\kappa $ .

In Figure 15, the growth rate $\mathrm {Im}(\Omega )/\eta _0^{2}$ in quiescent water has been drawn at different water depths $kd$ and two values of $\kappa $ . The curve for $kd=1.37$ indicates that the instability vanishes as $kd$ tends to $1.363$ , which is compatible with the celebrated classical theory. It is observed that the growth rate increases with water depth, compatible with the previous results of Ma et al. [Reference Ma, Ma, Perlin and Dong26] and Sedletsky [Reference Sedletsky36].

Figure 15 Plot of $\mathrm {Im}(\Omega )/\eta _0^{2}$ as a function of $\lambda /\eta _0$ at several water depths $kd$ for $\overline {v}=0,~\overline {\omega }=0$ and $\kappa =0,~0.035$ . BFIs indicates the Benjamin–Feir instability [Reference Liao, Dong, Ma and Gao22] in deep water.

The maximum growth rate $G_m$ given by equation (5.9) at different water depths $kd$ against $\eta _0$ is plotted for $\overline {v}=0$ and $\overline {\omega }=0$ in Figures 16 and 17, respectively. From Figure 16, we find that the maximum growth rate increases with $\overline {\omega }$ and Figure 17 shows that depth uniform reverse currents increase the maximum growth rate, whereas following currents decrease the growth rate.

Figure 16 Plot of $G_m$ against $\eta _0$ for $\overline {v}=0$ and some values of $\overline {\omega }$ and $\kappa $ : (a) $kd=1.5$ ; (b) $kd=2$ .

Figure 17 Plot of $G_m$ against $\eta _0$ for $\overline {\omega }=0$ and some values of $\overline {v}$ and $\kappa $ : (a) $kd=2$ ; (b) $kd=2.5$ .

Next, in Figures 18 and 19, the ratio of the maximum growth rate $G_m$ to its value without shear currents is drawn against $\overline {\omega }$ and $\overline {v}$ , respectively. In Figure 18, graphs are drawn for $\overline {\omega }>-2/3$ and for several values of $kd$ and $\kappa $ . It is found that for finite depth, the influence of $\overline {\omega }$ is to diminish the maximum growth rate when $-2/3<\overline {\omega }<0$ , while for $\overline {\omega }>0$ , growth rate first increases with $\overline {\omega }$ and then its value diminishes. For deep water and for $kd=3.14$ , it increases steadily with $\overline {\omega }>0$ . The influence of capillarity is to increase the maximum growth rate for both finite depth and deep water when $\overline {\omega }>0$ . Furthermore, in Figure 19, we observe first an increase and afterward a decrease of the maximum growth rate for different values of water depth $kd$ .

Figure 18 Plot of $G_{m}/G_{0m}$ against $\overline {\omega }$ for $\overline {v}=0$ and some values of $kd$ and $\kappa $ . Here, $\mathrm {G}_{0m}$ represents the maximum growth rate when the shear currents are absent.

Figure 19 Plot of $G_{m}/G_{0m}$ against $\overline {v}$ for $\overline {\omega }=0$ , $\kappa =0$ and some values of $kd$ . Here, $\mathrm {G}_{0m}$ represents the maximum growth rate when the shear currents are absent.

Figures 20 and 21 show the behaviour of the normalized maximum growth rate of instability against $kd$ . In these cases, the normalization is performed by taking the ratio of the maximum growth rate to its value when $kd\rightarrow \infty $ . We found from Figure 20 that for $\overline {\omega }\approx 0$ , the critical value $kd$ related to re-stabilization is very close to $1.363$ . Herein, the maximum growth rate increases with the water depth $kd>1.363$ , but diminishes with $\mid \overline {\omega }\mid $ , compatible with the findings of Thomas et al. [Reference Thomas, Kharif and Manna39]. Again, Figure 21 shows that depth uniform reverse current increases the maximum growth rate, while following current decreases the growth rate.

Figure 20 Plot of $G_{m}/G_{0m}$ against $kd$ for $\overline {v}=0$ and some values of $\overline {\omega }$ and $\kappa $ . Here, $\mathrm {G}_{0m}$ represents the maximum growth rate when $kd\rightarrow \infty $ .

Figure 21 Plot of $G_{m}/G_{0m}$ against $kd$ for $\overline {\omega }=0$ and some values of $\overline {v}$ and $\kappa $ . Here, $\mathrm {G}_{0m}$ represents the maximum growth rate when $kd\rightarrow \infty $ .

The ratio of the normalized instability bandwidth to its value in the absence of shear currents as a function of $\overline {\omega }$ and $\overline {v}$ has been plotted respectively in Figures 22 and 23. It is found that for finite depth, the bandwidth of instability $\mathrm {BW}$ first increases and then its value decreases with $\overline {\omega }$ , while for deep water, its value increases steadily with $\overline {\omega }$ . Moreover, for $\overline {\omega }>0$ , the influence of capillarity shows an increase in the bandwidth for both finite depth and deep water. As found in Figure 22, as described by Thomas et al. [Reference Thomas, Kharif and Manna39], our findings without surface tension are in good agreement with the exact numerical results of Oikawa et al. [Reference Oikawa, Chow and Benney30]. From Figure 23, we also observe that the instability bandwidth increases with the increase of $\overline {v}$ but decreases with the increase of $kd$ .

Figure 22 Normalized bandwidth of instability against $\overline {\omega }$ for $\overline {v}=0$ and several values of $kd$ and $\kappa $ .

Figure 23 Plot of BW against $\overline {v}$ for $\overline {\omega }=0$ and several values of $kd$ and $\kappa $ .

5.2 Benjamin–Feir index

The concept of the Benjamin–Feir index (BFI) in connection with the random waves was started by Janssen [Reference Janssen18] and then elaborated by Onorato et al. [Reference Onorato, Osborne, Serio and Bertone32]. The ratio of the mean square slope to the normalized width of the spectrum is considered as the definition of the BFI. Onorato et al. [Reference Onorato, Osborne, Serio, Cavaleri, Brandini and Stansberg33] defined the BFI as

$$ \begin{align*} \mathrm{BFI}=\frac{\eta_{0}}{\Delta k}\sqrt{\bigg\lvert\frac{\mu_1}{\alpha_1}\bigg\rvert}. \end{align*} $$

Further, the BFI for deep water and for $\overline {\omega }=0,~\kappa =0$ becomes

$$ \begin{align*} \mathrm{BFI}_{0}=\frac{4\eta_{0}}{\Delta k}, \end{align*} $$

where $\Delta k$ means a typical spectral bandwidth. So the normalized BFI takes the form

(5.10) $$ \begin{align} \mathrm{R}=\frac{\mathrm{BFI}}{\mathrm{BFI}_0}=\frac{1}{4}\sqrt{\bigg\lvert\frac{\mu_1}{\alpha_1}\bigg\rvert}. \end{align} $$

Onorato et al. [Reference Onorato, Osborne, Serio, Cavaleri, Brandini and Stansberg33] have reported the influence of water depth on the BFI in their figure 1. Thomas et al. [Reference Thomas, Kharif and Manna39] have also described the influence of water depth and current shear on the BFI. Herein, we have given attention to the influence of both capillarity and depth uniform current on the BFI. To measure the influence of capillarity on the BFI, we have taken $\overline {\omega }=0$ , $\overline {v}=0$ in equation (5.10) and then we have portrayed in Figure 24(a) the ratio $\mathrm {R}$ of the BFI in the presence of capillarity to its value without capillarity in deep water against $kd$ for several values of $\kappa $ . It is found that the BFI increases with water depth $kd$ for a fixed value of $\kappa $ and also the BFI increases with the increase of surface tension $\kappa $ . For $\kappa =0$ , our results are in good agreement with those of Onorato et al. [Reference Onorato, Osborne, Serio, Cavaleri, Brandini and Stansberg33]. The normalized BFI diagram we have obtained is compared in Figure 24(b) with that obtained in figures 11 and 12 of Thomas et al. [Reference Thomas, Kharif and Manna39] for $\overline {v}=0$ , $\kappa =0$ . In that way, we can check that this limiting case is again produced accurately. Further, to measure the influence of depth uniform current on the BFI, we have set $\overline {\omega }=0$ , $\kappa =0$ in equation (5.10) and then we have drawn in Figure 24(c) the ratio $\mathrm {R}$ of the BFI with depth uniform current to its value without current as a function of $kd$ . It is found that the BFI increases with $kd$ for a fixed value of $\overline {v}$ but it decreases with the increase of $\overline {v}$ .

Figure 24 Normalized BFI as a function of $kd$ : (a) $\overline {v}=0$ , $\overline {\omega }=0$ and $\kappa =0,~0.035$ ; (b) $\overline {v}=0$ , $\kappa =0$ and $\overline {\omega }=-0.3,~0,~1,~2$ ; (c) $\overline {\omega }=0$ , $\kappa =0$ and $\overline {v}=0,~0.2,~0.25,~0.3$ .