Modified nonlinear Schrödinger equation for gravity waves with the influence of wind, currents, and dissipation

Freak waves or rogue waves can endanger ships and offshore structures due to their extraordinary large wave heights.^1,2 Freak waves are often identified as $H>2Hs$⁠, where H is the wave height and H_s is the significant wave height.^3,4 There are several physical mechanisms of generating freak waves:^4–8 MI (modulational instability), wave–current interaction, wind–wave interaction, and so on. The MI of gravity waves can well be modeled by the nonlinear Schrödinger (NLS) equation, and NLS-type equations are often used to study the characteristics of freak waves.^9–11 

In the ocean, waves and currents always coexist, and currents can significantly change wave fields, especially when waves propagate opposite to the currents.¹² Therefore, the influence of currents has received a lot of attention. Gerber¹³ derived a current modified NLS equation and demonstrated that opposing currents can stimulate the growth of MI, and expand the onset criterion. Stocker and Peregrine¹⁴ extended the Dysthe equation¹⁵ to include the effect of depth-uniform currents. Hjelmervik and Trulsen¹⁶ derived a current-modified NLS equation suited for waves on prescribed, stationary collinear currents and found that when the waves meet an opposing jet current, the wave height increases in the center of the jet. Considering waves that approach an opposing current jet, Hjelmervik and Trulsen¹⁷ found that the amount of freak waves is minimum in the center of the jet where the wave heights are largest, while the amount of freak waves is maximum at the sides of the jet where the wave heights are smallest. So, notice in particular that for freak waves on non-uniform currents, the freak wave criterion should in principle be applied to wave data originating from locations with the same current conditions. Based on computations of the modified NLS equation,¹⁶ Onorato et al.¹⁸ found that freak waves can be triggered when a stable wave train enters a region with an opposing current, and demonstrated that the maximum amplitude of the freak wave depends on the ratio between the current velocity and the wave group velocity. Toffoli et al.¹⁹ showed experimentally that a stable wave propagating into a region characterized by an opposite current may become modulationally unstable. This phenomenon was then demonstrated by Ma et al.²⁰ in a well-designed experiment. Toffoli et al.²¹ indicated that the interaction with opposing currents would amplify wave modulation and nonlinear wave dynamics, thus increasing the probability of occurrence of freak waves. Johnson²² derived a NLS equation with coefficients depending on vorticity and studied the slow modulation of the waves over a current with arbitrary vorticity. Li et al.²³ derived a NLS equation for the motion of waves on water of infinite depth in the presence of a uniform velocity shear. Baumstein²⁴ proposed a NLS equation to investigate the effect of piecewise-linear velocity profiles on the MI of gravity waves in infinite depth. Considering the depth-varying current effects, Liao et al.²⁵ derived a NLS equation for the propagation of surface gravity waves on linear shear currents in finite water depth that extended the equation of Thomas et al.²⁶ Curtis et al.²⁷ derived a higher-order nonlinear Schrödinger equation in the presence of shear and surface tension, and the role of surface tension in determining the existence of modulational instability is fully explored. One of the main questions regarding NLS-type equations is the frequency-downshift phenomenon. Frequency downshift means that when the carrier wave loses a significant amount of energy to its lower sidebands. Understanding the mechanism of frequency downshift will help to understand how energy is transmitted in the ocean, and it is possible to improve the prediction ability when the swell approaches the coast, thus affecting all fields from shipping to surfing.²⁸ Carter et al.²⁹ studied the periodic traveling wave solution of the cubic vortical Whitham equation, which is a model for wave motion on a vertically sheared current of constant vorticity in a shallow inviscid fluid, and proved that small-amplitude solutions of the cubic vortex Whitham equation are stable with respect to the modulational instability for larger ranges of vorticity and wavelength than in the vortical Whitham equation. Exact solutions also exist for other simplified model equations with an inviscid fluid of finite depth in the presence of a linear shear current.³⁰ 

Dissipation caused by breaking and friction is also very important in wave evolution and thus also has received significant attention. Kharif et al.³¹ extended the work of Segur et al.³² by deriving a NLS equation for water waves and considered the influence of dissipation damping and wind forcing, and found that for low-frequency waves stronger wind velocities are needed to sustain MI than for high-frequency waves. Onorato and Proment³³ considered the generation of freak waves in the presence of wind forcing or dissipation based on the forced/damped NLS equation, and approximate freak wave solutions of the equation are presented and discussed. They demonstrated that an initially stable (unstable) wave packet could be destabilized (stabilized) by the wind (dissipation). Ma et al.³⁴ performed experiments that confirmed the stabilization theory of Segur et al.³² That is, the dissipation can stabilize the MI; the experiments also revealed that the effect of dissipation depends on perturbation frequencies. Other experimental studies showed that following winds shift the focus point downstream and increase the peak wave amplitude.^35,36 Brunetti et al.³⁷ considered the NLS equation and the Miles mechanism for the wind–wave coupling. Several other studies also investigated dissipation with wind effects on the NLS equations. Brunetti and Kasparian³⁸ considered a wind-forced NLS equation obtained in the potential flow framework and noted that wind forcing gives rise to the enhancement of MI. Dostal et al.³⁹ derived a NLS for the case of wind-forced weakly nonlinear water waves; they showed that breather-type solutions such as the Peregrine breather (PB) solution occur even in strong gusty wind conditions. Li et al.⁴⁰ extended the work of Kharif et al.³¹ deriving a two-dimensional NLS equation describing the evolution of freak waves in water of finite depth under the action of linear shear currents, wind, and dissipation, and focused theoretically on the MI affected by the wind and dissipation.

The effect of wind blowing along the surface is, of course, the main source of energy for the growth of waves, and it generates a rotational current in the water.⁴¹ Dissipation caused by breaking and friction most often coexist with winds. However, there is a lack of quantitative understanding of the combined influence of shear currents, wind, and dissipation. Additionally, the corresponding theoretical consideration of freak waves is also lacking. In this study, a modified nonlinear Schrödinger (MNLS) equation for gravity waves is derived that includes wind, dissipation, and a linear shear current in finite water depth. The horizontal surface current, which is assumed stationary and slowly varying spatially, is more representative of the actual marine environment. Specifically, the focus addresses the influence of wind, dissipation, and shear current on the formation of freak waves and the primary objective of this study.

The paper is organized as follows. In Sec. II, a MNLS equation is derived using the multiple-scale method.⁴² The MI analysis of the MNLS equation is presented in Sec. III. Comparison between the theoretical model and experimental results of wave amplification is given in Sec. IV. Breather solution of the MNLS equation is studied in Sec. V. Finally, the conclusions are presented in Sec. VI.

Assuming that the fluid motion is viscid and that the fluid is incompressible and homogeneous, a two-dimensional Cartesian coordinate (x, z) is adopted to derive the equation. As shown in Fig. 1, the x axis is aligned with the propagation direction of the waves and the z axis is taken vertically upward. p is the pressure, which fluctuates according to the wind, and $ζ(x,t)$ is the free surface elevation, which is a function of the space variable x and the time variable t. Here, the waves are assumed to be propagating steadily on a vertically linear-sheared current, U(x) is assumed slowly varying spatially, and reference is made to a following current and an opposing current when U(x) is along and against the direction of wave propagation, respectively. We also assume that the variation of the water flow U(x) along the space does not affect the strength of the vorticity Ω; that is, Ω is the constant vorticity, which denotes the magnitude of the linear shear. Let a₀ and k be the characteristic amplitude and wavenumber of the surface waves. The current velocities are assumed small enough to avoid collinear reflection of the waves, $U(x)/c=o(ε)$⁠, where c denotes the wave phase velocity and $ε=ka0$ is a small parameter characterizing the steepness of the waves. We will consider waves on the surface of a fluid whose viscosity is small, and define a non-dimensional number $L(k)=υ/g/k3$⁠, where g is the gravitational acceleration, $υ$ is the water kinematic viscosity, and a fluid is of small viscosity if $L(k)≪1$⁠.³¹ For the free surface problem in water, viscous effects are generally weak; in this context, it was shown by Dias et al.⁴³ that the equations governing the fluid's motion can be formulated with the help of potential theory.

We assume that the total velocity field $vtot$ is a superposition of the velocity of a wave field $v$ and a current field $V=(U(x)+Ωz)$ in a Cartesian coordinate system (x, z),¹⁷ 

The Euler equation for the combined wave and current field can be written as follows:

where ρ_w is the density of water, k is the unit vector along Oz, and $∇$ = $(∂/∂x,∂/∂z)$ is the two-dimensional gradient operator. The velocity of the wave field can be represented by a potential, $v=∇φ$⁠.

The continuity equation for the wave field may be written as

It is worth noting that these subscript quantities (e.g., x, z, and t) in the paper represent derivatives of them. Substitution of Eq. (1) into Eq. (2) becomes

where ψ is the stream function, which is related to $φ$ by the Cauchy–Riemann relations:

Equation (4) can be expressed as

where R is the Bernoulli constant, without loss of generality, and R = 0 is used. Hence, the dynamic upper boundary condition can be expressed as

The kinematic upper boundary condition is

The kinematic bottom condition for the assumed impermeable bottom is

By taking the derivative of Eq. (7) and using the Cauchy–Riemann relations to eliminate the stream function, Eq. (7) may be converted into a free-surface boundary condition entirely in terms of $φ$ and ζ. Based on Miles's mechanism,⁶ the fluctuating pressure p depends on the free surface elevation ζ, which defines the wind–wave interactions as follows:^31,37,40,44

where δ is the wind-wave growth rate, normalized by the angular frequency ω of the carrier waves and expressed as a function of the inverse wave age $u*c; u*$ is the wind friction velocity; and χ is an empirical parameter of o(1) for the inverse wave age $u*c$ = 0.2. Conte and Miles⁴⁵ computed the values of χ as a function of $κcu*$ for a logarithmic wind profile. The inverse wave age range $u*c<$ 0.2 embraces most reported wind sea situations, from very old to extremely young, while the range $u*c>$ 0.2 is more relevant to short fetch wave tank conditions. $s=ρaρw$ is the air–water density ratio (⁠$1.2×10−3$⁠), κ = 0.4 is the von Kárm´an constant. Considering the effects of dissipation caused by viscosity on water waves, Eqs. (7) and (8) can be rewritten as^31,40,43

where $υ$ = 10⁻⁶ m² s⁻¹ is the water kinematic viscosity, which can also be treated as a small parameter; here, $υ=ε2υ*$⁠. By taking the derivative of Eq. (12) and using the Cauchy–Riemann relations to eliminate the stream function, condition (12) can be converted into a free surface boundary condition in terms of $φ$ and ζ,

where it is assumed $δ*=ε2δ$⁠. To summarize, in the framework of potential flow theory, the governing equations for two-dimensional surface waves affected by wind, dissipation due to viscosity, and linear shear current on constant water depth are given by Eqs. (3), (9), (12), and (13). The detailed derivation is given in the  Appendix A.

Using the multi-scale method, from the leading-order problem for the first harmonic, we get the linear dispersion relation

where

In the second-order approximation, the corresponding group velocity is

After a very tedious derivation, one finally obtains the MNLS equation

where

and K reflects the effects of wind forcing and dissipation damping, which are related to the linear shear current. The sign of K determines the nature of the perturbation, if $K>$ 0 the wind plays a dominant role, and the waves are magnified. [According to Conte and Miles,⁴⁵ for instance, in deep water, for a carrier wave frequency f = 1 Hz, wave period T = 1 s, ε = 0.1, and χ = 3, when the value of K is in the range of 0 to 0.05, using Eqs. (10) and (18), the correspond $u*$ (∼0–0.26 m s⁻¹).] If $K<$ 0, the waves are decreased. Values of K close to zero correspond to a quasi-equilibrium between wind and damping effects.⁴⁵ The coefficient α of linear term and γ₀ of nonlinear term of the MNLS equation are shown in  Appendix A.

It is noted that under the appropriate assumptions, Eq. (17) agrees with previously derived equations. For example, in deep water, only considering the influence of U(x), Eq. (17) matches the results of Hjelmervik and Trulsen,^16,17 see  Appendix B for the derivation of the equivalence. If only considering vorticities, Eq. (17) agrees with that of Thomas et al. in deep water.²⁶ Without considering linear shear current, Eq. (17) agrees with the results of Kharif et al.³¹ in deep water. Ignoring the influence of the currents, wind, and viscosity, Eqs. (A29) and (A30) agree with the Davey–Stewartson system⁴⁶ and Eq. (A29) becomes the classic NLS equation in finite water depth.^47,48

Using the following dimensionless variable transformations:

Equation (17) can be rewritten as

where

There is an exact solution of Eq. (20) given by

considering small perturbations in amplitude $a′$ and phase $θ′$ of this solution⁴⁹ 

with

where μ₀ and μ₁ are the wave number and frequency of the perturbations, respectively. The dispersion relation for the perturbation is given by

If $α¯2μ02−2α¯γ¯0e2(Kω−u¯ξ2+X)τ<$ 0, the solution becomes imaginary and grows exponentially; therefore, wave trains are unstable. The growth rate of the MI is then

and its maximal value is obtained

for

According to the method adopted by Thomas et al.²⁶ from Eq. (26), the instability bandwidth is

The effect of shear flows on wave instabilities is as follows:²⁵ the following current is found to have a stabilizing effect on the waves and decrease the growth rate of the instability, opposing current increases the growth rate of the instability, and positive vorticity tends to enhance the sideband instabilities, whereas negative vorticity is prone to suppress the instability. Here, we only discuss the effect of dissipation and wind on the wave instability. Let K be negative, this case was analyzed by Segur et al.,³² and it was demonstrated that the MI is weakened by dissipation. Figure 2 shows the effect of K on the normalized growth rate of the MI vs normalized perturbation wave number μ₀ and dimensionless time variable τ in water depth $k0h=10$ according to formula Eq. (26), we select several simple special values of $K/ω$ for calculation, and the numbers on the contour lines indicate the size of the normalized growth rate of the MI. For $K<$ 0, the damping effect plays a dominant role, the unstable domain is decreased, and the modulational perturbation becomes stable after several wave periods, and sufficiently small μ₀ (long perturbation) does not produce instability. In the presence of wind (⁠$K>$ 0), i.e., the wind plays a dominant role, which tends to significantly enhance the MI, the unstable domain is expanded after several wave periods, and shorter disturbances (larger μ₀) can still produce instability. Without dissipation and wind (⁠$K=$ 0) [see Fig. 2(b)], MI does not change with time; that is, MI is independent of time. Figure 3 shows the normalized growth rate of the MI vs normalized perturbation wave number μ₀ and $K/ω$ for various dimensionless time variables τ in water depth $k0h=10$ according to formula Eq. (26). For $K>$ 0, it is obvious that when K increases, that is, the wind intensity increases, the modulation instability also increases rapidly, and the maximum growth rate of the MI grows exponentially with time, and shorter disturbances (larger μ₀) do not produce instability. For $K<$ 0, when $|K|$ increases, that is, the viscous dissipation effect increases, the modulation instability also decreases rapidly; meanwhile, the maximum growth rate of the MI will gradually decrease with time and eventually become stable after several wave periods. Figure 4 shows the normalized instability bandwidth as a function of normalized perturbation wave number and $K/ω$ and dimensionless time variable τ [i.e., Eq. (29)] for several values of $k0h$⁠, and the numbers on the contour lines indicate the size of the normalized instability bandwidth. It can be seen from the figure that the instability bandwidth increases with the increase in water depth. In addition, when $K/ω$ changes from negative to positive (from viscous dissipation to wind action), the instability bandwidth also increases, and the instability bandwidth increases faster in deep water.

Introducing the following approximate transformation relation:⁵⁰ 

we obtain the “time” NLS equation (tNLS)

Here, the following transversally uniform current is chosen, which represents the evolution of a wave train initially in a region of zero current propagating into a stationary current characterized by an entry length Λ:^17,18

where U(x) is the velocity of the current at position x, U₀ is the current at x = 0, and U(x) is the asymptotic value of the current that can be either positive or negative for following or opposing currents, respectively. The wave propagates for several wavelengths before entering the current, the current increases speed (in absolute value) and then adjusts to a constant value U₀, and Λ is the current “build-up” length. Furthermore, Eq. (31) can be reduced to the standard NLS equation

with variable coefficients by applying the following transformations:¹⁸ 

where $ΔU=U(x)−U(0)$⁠. For simplicity, only the physical case of $U(0)=0$ is considered. Primes have been omitted for brevity. The coefficient of the nonlinear term of Eq. (33) increases as waves enter an opposing current up to a certain value. Onorato et al.¹⁸ derived an equation for predicting the maximum amplitude during wave evolution of currents in deep water. They demonstrated that the maximum amplitude of wave depends on the ratio between current velocity U₀ and wave group velocity c_g. A similar equation was derived by Toffoli et al.¹⁹ and Ruban⁵¹ explained that the formula proposed by Onorato et al.¹⁸ does not conserve wave action, and derived a modified equation in deep water. The current study uses Eq. (33) as basis to derive a new equation for predicting the maximum amplitude in finite water depth, which can also consider the following effects of vorticity, dissipation, and wind on waves:

where $E$ is the standard deviation of wave envelope once the current has reached its maximum constant value and N is the number of waves in one group period. Toffoli et al.²¹ presented three independent facilities of laboratory experiments conducted in the wave flume and narrow directional wave basin. Initial signal at the wavemaker consisted of a three-component system: carrier wave of period $T=0.8$ s and two side bands with amplitudes equal to 0.25 times the amplitude of the carrier waves. Wave steepness was ka₀ = 0.064, and the number of waves under the perturbation N was 11, water depth h = 0.75 m, k = 6.29 m⁻¹, and c_g = 0.625 m s⁻¹. Figure 5 shows the maximum normalized amplitude as a function of $U0/cg$ and compared with the theoretical prediction model^19,51 and the experimental results of Toffoli et al.^21,Fig. 5 also indicates that normalized maximum amplitude increases with increasing $|U0|$⁠, thereby confirming the destabilizing effect of the current. In deep-water conditions, Eq. (36) (⁠$K/cg$ = 0) agrees well with Toffoli et al.¹⁹ Quantitatively, Fig. 5 shows that the theoretical model is in good agreement with the experimental results for weak currents (−0.1 $≤U0/cg≤$ 0). If considering dissipation, then K = −2 $k02υ*$ = −0.019 and $K/cg$ = −0.03, while Eq. (36) better predicts the maximum amplification for stronger currents (−0.5 $≤U0/cg≤$ −0.2). This result indicates that the effect of dissipation becomes significant and cannot be disregarded for strong current fields. Furthermore, dissipative effect increases with increasing water velocity. Considering the effect of wind and dissipation, Fig. 6 shows the non-dimensional maximum amplitudes during evolution on different adverse currents, wind, and dissipation. Under the same current conditions (i.e., $U0/cg$ takes a fixed value), when the value of $K/cg$ increases, that is, the wind strength increases, the maximum amplitude of the wave will also significantly increase. In addition, under the action of viscous dissipation, when the intensity of viscous dissipation increases (when the value of $|K/cg|$ increases), the maximum amplitude of waves will significantly decrease. When the value of $K/cg$ is in the range of 0.03 to 0.05, using Eqs. (10) and (18), the corresponding $u*$ is 0.037 to 0.048 m s⁻¹.

We consider the following new variables:

for which Eq. (33) becomes

The PB solution of Eq. (38)^52,53 is as follows:

The solution concentrates the energy of the background into a small region because of the nonlinear properties of the medium. The maximum amplitude of this solution occurs at $x″$ = 0, $t″$ = 0 and is equal to 3. The complete PB solution of Eq. (38) is as follows:

The spatiotemporal evolution of the PB solution of Eq. (40) is shown in Fig. 7 with different K, clearly demonstrating its spatial and temporal localization. It is the spatial and temporal localization characteristics of the PB solution that lead to their association with freak waves and provides an analytic framework for MI in the case of infinite modulation period. Without dissipation and wind (⁠$K=$ 0) [see Fig. 7(b)], the maximum value of wave amplitude reaches three times of the background wave. When $K<$ 0, dissipation will weaken the growth rate of MI according to Sec. III; at this time, as shown in Fig. 7(a), the amplification factor is below 3. On the contrary, when $K>$ 0, wind will enhance modulation instability, and the amplification factor is above 3 [see Figs. 7(c) and 7(d)]. Similar results were found by Onorato and Proment.³³ Fig. 8 shows the PB solution on different depth-uniform currents, and when $U0/cg<$ 0, opposing current will enhance the growth rate of MI according to Sec. IV; at this time, as shown in Figs. 8(c) and 8(d), the amplification factor is above 3. On the contrary, when $U0/cg>$ 0, following current will weaken modulation instability, and the amplification factor is below 3 [see Fig. 8(a)]. The above results show that the freak wave is more easily generated under the condition of opposing current. The influence on the breather of the shear current is shown in Fig. 9. Under uniform opposing current condition (⁠$X=$ 0), the amplification factor of PB exceeds 3 [see Figs. 9(b) and 9(d)]; when $X<$ 0, the modulation instability is further enhanced by shear current, and the amplification factor of PB is also further increased. On the other hand, when $X>$ 0, the vorticity counteracts a part of the effect of water flow, and the shear current weakens a part of the modulation instability, and the amplification factor of PB also slightly decreases [see Figs. 9(a) and 9(d)].

The effects of the shear current, dissipation, and wind on nondimensional maximum amplitudes during the evolution of the PB solution (40) are shown in the following results. As shown in Fig. 10, for the case $K/cg$ = 0, X = 0 (red short dash line), maximum amplitudes are enhanced by an opposing current (amplification factor is above 3) but decreased by a following current (amplification factor is below 3). Figure 10(a) shows that an increase in wind forcing (⁠$K/cg$ > 0) enhances the normalized maximum amplitude. When the value of $K/cg$ is in the range of 0.1 to 0.2, T = 0.8 s, wave steepness ka₀ = 0.064, h = 0.75 m, k = 6.29 m⁻¹, and c_g = 0.625 m s⁻¹, the corresponding $u*$ is from 0.067 to 0.095 m s⁻¹. For a fixed wind strength parameter, in the case of opposing current, negative/positive vorticity aims to increase/decrease the maximum amplitude of the plane wave (all amplification factors are above 3). In the case of a following current, negative/positive vorticity aims to reduce/increase the maximum amplitude of the plane wave (all amplification factors are below 3). Note that the current has a greater effect on the maximum wave height than the vorticity. Figure 10(b) shows that an increase in dissipation (⁠$K/cg$ < 0) decreases the normalized maximum amplitude. For a fixed dissipation parameter, in the case of an opposing current, negative/positive vorticity increases/decreases the maximum amplitude of the plane wave. Overall, wind/dissipation plays a dominant role in causing an increased/reduced maximum envelope. That is, wind is an important factor in generating freak waves.

General solitons on finite background solution of the NLS equation can be written as follows:⁵⁴ 

and the parameter $a$ determines the physical behavior of the solution with $b=8a(1−2a)$ and $c=21−2a$⁠.When the parameter $0<a<0.5$⁠, Eq. (41) describes the Akhmediev breather.⁵⁵ The case of maximal growth rate corresponds to $a=$ 0.25. When $a>$ 0.5, it represents the KM solitons.⁵⁶ When the parameter $a$ → 0.5, the solution describes the PB solution. The complete soliton solution of Eq. (41) is as follows:

Figure 11 shows the Akhmediev breather on different depth-uniform currents, it is evolving periodically along the evolution variable $t″$⁠, and the maximum of this solution at $t″$ = 0 and $x″$ = 0 is $≈$2.41 [with U₀ = 0, see Fig. 11(b)]. The effect of the opposing/following current is to increase/reduce the amplitude of the plane wave. The effects of the shear current, dissipation, and wind on nondimensional maximum amplitudes during the evolution of the Akhmediev breather are similar to PB solution.

In this paper, a cubic nonlinear Schrödinger equation is derived for two-dimensional gravity waves propagating on linear shear currents in finite water depth. The multiple-scale method is used. The manner by which gravity waves are affected by wind, dissipation, and the horizontal surface current (assumed stationary and slowly varying spatially) is presented. Using the derived MNLS equation, MI of gravity wave trains in deep water under wind, and with viscosity considered. The parameter K reflects the effects of wind forcing and damping. For $K<$ 0, the modulational perturbation of waves becomes stable after several wave periods, whereas for $K>$ 0, any modulational perturbation is unstable or becomes unstable after several wave periods, which are consistent with the results of Segur et al.,³² and Onorato and Proment³³ A new equation Eq. (36) is derived for finite water depth, which can also consider the effects of vorticity, dissipation, and wind on waves to predict the maximum amplitude. The equation is confirmed by three independent sets of laboratory experiments of Toffoli et al.,²¹ which were conducted in a wave flume and wave basin. The maximum normalized amplitude is presented as a function of $U0/cg$⁠, and the maximum amplitude increases with increasing $|U0|$⁠. The effect of dissipation is significant and cannot be disregarded for relatively strong current fields. Additionally, the dissipative effect increases with increasing water velocity. Wind also increases the height of waves.

Using variable substitution, the MNLS equation is transformed into a standard NLS equation. An explicit analytical PB solution is found that considers the effects of vorticity, dissipation, and wind. The effect of the wind/dissipation is to increase/reduce the amplitude of the plane wave, and amplification factor is above/below 3. In the case of opposing current, negative/positive vorticity aims to increase/reduce the maximum amplitude of the plane wave. However, in the case of following current, negative/positive vorticity aims to reduce/increase the maximum amplitude of the plane wave. Note that current and wind are important factors in generating freak waves.

In summary, wave–current interactions may be a mechanism of freak wave generation. Evidently, an opposing current and wind forcing accelerate the growth of MI and increase the height of freak waves. Following currents and dissipation have opposite effects. In the calculation and analysis process of this paper, we used the MATLAB software.

This research was supported financially by the National Natural Science Foundation of China (Grant No. 51969005) and Guangxi Science and Technology Department Specific Research Project of Guangxi for Research Bases and Talents (广西科技基地和人才专项) (Guike AD20159016).

The authors have no conflicts to disclose.

Bo Liao: Funding acquisition (equal); Methodology (equal); Writing – original draft (equal). Guohai Dong: Supervision (equal). Yuxiang Ma: Methodology (equal); Supervision (equal); Writing – review & editing (equal). Xiaozhou Ma: Supervision (equal). Marc Perlin: Supervision (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The governing equations are given by Eqs. (3), (9), (12), and (13). The velocity potential at $z=ζ$ as usual a Taylor expansion at z = 0, and the expansions for free surface boundary conditions [Eqs. (12) and (13)] become

An asymptotic set of governing equations are determined in the form of a power of series

where $φn$ and ζ_n are slowly varying functions:

with the following cascade of slower variables in space and time:

Derivatives with respect to x and t must be replaced by

Substituting Eqs. (A4)–(A6) into Laplace equation (3), the perturbed free surface boundary conditions Eqs. (14) and (15), and the bottom condition Eq. (9), and separated by powers of ε. Subsequently, the multiple-scale method⁴² is adopted to derive the evolution equation. The first three orders are collected below. The Laplace equation can be written in the following form:

The kinematic boundary condition on the free surface is as follows:

Bernoulli equation on the free surface Eq. (15) can be written in the following form:

The condition on the bottom can be written in the following form:

We expand $φn$ and ζ_n are expanded into series (the expressions are given in  Appendix B); then substituting the series into Eqs. (A7)–(A10), respectively, and separating different orders of ε and harmonics, one obtains a set of ordinary linear differential equations for $φnm$ and ζ_nm:

The forcing terms F_nm, H_nm, and G_nm contain the lower order quantities with respect to n; for simplicity, their detailed expressions are not written here. In the $o(ε2)$ approximation, the following equation is obtained:

where A denotes the leading-order wave amplitude. Then, F_nm, H_nm, and G_nm are calculated to the third order, and the evolution equation is obtained as follows:

where the zeroth harmonic $φ10$ represents the long-wave potential. Two third-order equations involving $φ10$⁠, ζ₂₀, and A can be deduced from Eqs. (A10) and (A11)

Eliminating ζ₂₀ from these two equations, an equation for $φ10$ and A is obtained as follows:

Substituting Eq. (A17) into Eq. (A18), and considering $φ10$ and A as functions of x₁, y₁, and t₁ only, that is

yielding