An analytical weakly nonlinear model of the Benjamin–Feir instability of a Stokes wave on nonuniform unidirectional current is presented. The model describes evolution of a Stokes wave and its two main sidebands propagating on a slowly varying steady current. In contrast to the models based on versions of the cubic Schrödinger equation, the current variations could be strong, which allows us to examine the blockage and consider substantial variations of the wave numbers and frequencies of interacting waves. The spatial scale of the current variation is assumed to have the same order as the spatial scale of the Benjamin–Feir (BF) instability. The model includes wave action conservation law and nonlinear dispersion relation for each of the wave's triad. The effect of nonuniform current, apart from linear transformation, is in the detuning of the resonant interactions, which strongly affects the nonlinear evolution of the system.

The modulation instability of Stokes waves in nonuniform moving media has special properties. Interaction with countercurrent accelerates the growth of sideband modes on a short spatial scale. An increase in initial wave steepness intensifies the wave energy exchange accompanied by wave breaking dissipation, resulting in asymmetry of sideband modes and a frequency downshift with an energy transfer jump to the lower sideband mode, and depresses the higher sideband and carrier wave. Nonlinear waves may even overpass the blocking barrier produced by strong adverse current. The frequency downshift of the energy peak is permanent and the system does not revert to its initial state. We find reasonable correspondence between the results of model simulations and available experimental results for wave interaction with blocking opposing current. Large transient or freak waves with amplitude and steepness several times those of normal waves may form during temporal nonlinear focusing of the waves accompanied by energy income from sufficiently strong opposing current. We employ the model for the estimation of the maximum amplification of wave amplitudes as a function of opposing current value and compare the result obtained with recently published experimental results and modeling results obtained with the nonlinear Schrödinger equation.

The problem of the interaction of a nonlinear wave with slowly varying current remains an enormous challenge in physical oceanography. In spite of numerous papers devoted to the analysis of the phenomenon, some of the relatively strong effects still await a clear description. The phenomenon can be considered the discrete evolution of the spectrum of the surface wave under the influence of nonuniform adverse current. Experiments conducted by Chavla and Kirby (1998, 2002) and Ma et al. (2010) revealed that sufficiently steep surface waves overpass the barrier of strong opposing current on the lower resonant Benjamin–Feir sideband. These reports highlight that the frequency step of a discrete downshift coincides with the frequency step of modulation instability; i.e., after some distance of wave run, the maximum of the wave spectrum shifts in frequency to the lower sideband. The intensive exchange of wave energy produces a peak spectrum transfer jump, which is accompanied by essential wave breaking dissipation. The spectral characteristics of the initially narrowband nonlinear surface wave packet dramatically change and the spectral width is increased by dispersion induced by the strong nonuniform current.

This paper presents a weakly nonlinear model of the Benjamin–Feir (BF) instability on nonuniform slowly varying current.

The stationary nonlinear Stokes wave is unstable in response to perturbation of two small neighboring sidebands. The initial exponential growth of the two dominant sidebands at the expense of the primary wave gives rise to an intriguing Fermi–Pasta–Ulam recurring phenomenon of the initial state of wave trains. This phenomenon is characterized by a series of modulation–demodulation cycles in which initially uniform wave trains become modulated and then demodulated until they are again uniform (Lake and Yuen, 1978). However, when the initial slope is sufficiently steep, the longtime evolution of the wave train is different. The evolving wave trains experience strong modulations followed by demodulation, but the dominant component is the component at the frequency of the lower sideband of the original carrier. This is the temporary frequency downshift phenomenon. In systematic well-controlled experiments, Tulin and Waseda (1999) analyzed the effect of wave breaking on downshifting, high-frequency discretized energy, and the generation of continuous spectra. Experimental data clearly show that the active breaking process increases the permanent frequency downshift in the latter stages of wave propagation.

The BF instability of Stokes waves and its physical applications have been studied in depth over the last few decades; a long but incomplete list of research is Lo and Mei (1985), Duin (1999), Landrini et al., 1998; Osborne et al. (2000), Trulsen et al. (2000), Janssen (2003), Segur et al. (2005), Shemer et al. (2002), Zakharov et al. (2006), Bridges and Dias (2007), Hwung, Chiang and Hsiao (2007), Chiang and Hwung (2010), Shemer (2010), and Hwung et al. (2011). The latter stages of one cycle of the modulation process have been much less investigated, and many physical phenomena that have been observed experimentally still require extended theoretical analysis.

Modulation instability and the nonlinear interactions of waves are strongly affected by variable horizontal currents. Here, we face another fundamental problem of the mechanics of water waves–interactions with slowly varying current. The effect of opposing current on waves is a problem of practical importance at tidal inlets and river mouths.

Even linear refraction of waves on currents can affect the wave field structure in terms of the direction and magnitude of waves. Waves propagating against an opposing current may have reduced wavelength and increased wave height and steepness.

If the opposing current is sufficiently strong, then the absolute group wave
velocity in the stationary frame will become zero, resulting in the waves
being blocked. This is the most intriguing phenomenon in the problem of
wave–current interaction (Phillips, 1977). The kinematic condition for wave
blocking can be written as

The linear modulation model has a few serious limitations. The most important is that the model predicts the blocking point according to the linear dispersion relation and cannot account for nonlinear dispersive effects. Amplitude dispersion effects can considerably alter the location of wave blocking predicted by linear theory, and nonlinear processes can adversely affect the dynamics of the wave field beyond the blocking point.

Donato et al. (1999), Stocker and Peregrine (1999), and Moreira and Peregrine (2012) conducted fully nonlinear computations to analyze the behavior of a train of water waves in deep water when meeting nonuniform currents, especially in the region where linear solutions become singular. The authors employed spatially periodic domains in numerical study and showed that adverse currents induce wave steepening and breaking. A strong increase in wave steepness is observed within the blocking region, leading to wave breaking, while wave amplitudes decrease beyond this region. The nonlinear wave properties reveal that at least some of the wave energy that builds up within the blocking region can be released in the form of partial reflection (which applies to very gentle waves) and wave breaking (even for small-amplitude waves).

The enhanced nonlinear nature of sideband instabilities in the presence of
strong opposing current has also been confirmed by experimental
observations. Chavla and Kirbi (2002) experimentally showed that the
blockage phenomenon strongly depends on the initial wave steepness; i.e.,
waves are blocked when the initial slope is small (

Wave propagation against nonuniform opposing currents was recently investigated in experiments conducted by Ma et al. (2010). Results confirm that opposing current not only increases the wave steepness but also shortens the wave energy transfer time and accelerates the development of sideband instability. A frequency downshift, even for very small initial steepness, was identified. Because of the frequency downshift, waves are more stable and have the potential to grow higher and propagate more quickly. The ultimate frequency downshift increases with an increase in initial steepness.

The wave modulation instability with coexisting variable current is commonly described theoretically by employing different forms of the modified nonlinear Schrödinger (NLS) equation. Gerber (1987) used the variational principle to derive a cubic Schrödinger equation for a nonuniform medium, limiting to potential theory in one horizontal dimension. Stocker and Peregrine (1999) extended the modified nonlinear NLS equation of Dysthe (1979) to include a prescribed potential current. Hjelmervik and Trulsen (2009) derived an NLS equation that includes waves and currents in two horizontal dimensions allowing weak horizontal shear. The horizontal current velocities are assumed just small enough to avoid collinear blocking and reflection of the waves.

Even though the frequency downshift and other nonlinear phenomena were observed in previous experimental studies on wave–current interactions, the theoretical description of the modulation instability of waves on opposing currents is not yet complete. An interaction of an initially relatively steep wave train with strong current nevertheless may abruptly transfer energy between the resonantly interacting harmonics. Such wave phenomena are beyond the applicability of NLS-type models and await a theoretical description.

Another topic of practical interest in wave–current interaction problems is the appearance of large transient or freak waves with great amplitude and steepness owing to the focusing mechanism (e.g., Peregrine, 1976; Lavrenov, 1998; White and Fornberg, 1998; Kharif and Pelinovsky, 2006; Janssen and Herbers, 2009; Ruban, 2012; Osborne, 2001). Both nonlinear instability and refractive focusing have been identified as mechanisms for extreme-wave generation and these processes are generally concomitant in oceans and potentially act together to create giant waves.

Toffoli et al. (2013) showed experimentally that an initially stable surface
wave can become modulationally unstable and even produce freak or giant
waves when meeting negative horizontal current. Onorato et al. (2011)
suggested an equation for predicting the maximum amplitude

Recently, Ma et al. (2013) experimentally investigated the maximum amplification of the amplitude of a wave on opposing current having variable strength at an intermediate water depth. They mentioned that theoretical values of amplification (Onorato et al., 2011; Toffoli et al., 2013) are essentially overestimated, probably owing to the effects of finite depth and wave breaking.

To address the above-mentioned problems, we present the model of BF instability in the presence of horizontal slowly varying current of variable strength. We analyze the interactions of a nonlinear surface wave with sufficiently strong opposing blocking current and the frequency downshifting phenomenon. The maximum amplification of the amplitude of surface waves is estimated in dependence on relative strength of opposing current. We take into account the dissipation effects due to wave breaking by utilizing the Tulin wave breaking model (Tulin, 1996; Hwung et al., 2011). The results of model simulations are compared with available experimental results and theoretical estimations.

We employ simplified 3-wave model in the presence of significant opposite current. In the meanwhile, the evolution of the wave spectrum in the absence of breaking includes energy exchange between the carrier wave and two main resonant sidebands and spreading of the energy to higher frequencies. Inclusion of higher frequency free waves in the Zakharov, modified Schrödinger or Dysthe equations is crucial, since the asymmetry of the lower and the upper sideband amplitudes at peak modulation in non-breaking case results from that. The temporal spectral downshift has been predicted by computations made by the Dysthe equations (Lo and Mei, 1985; Trulsen and Dysthe, 1990; Hara and Mei, 1991; Dias and Kharif, 1999) for a much higher number of excited waves, the same prediction was also made by simulations of fully nonlinear equations (Tanaka, 1990; Slunyaev and Shrira, 2013). Such a conclusion can be made regarding to developing of modulation instability in calm water.

Nevertheless, the experimental results of Chavla and Kirby (2002) and Ma et al. (2010) on the modulation instability under the influence of adverse current show that energy spectrum is mostly concentrated in the main triad of waves and high-frequency discretized energy spreading is depressed due to the short-wave blocking by the strong enough adverse current. Higher side band modes have also prevailed energy loss during wave breaking (Tulin and Waseda, 1999). That is why we hope that our simplified model still has potentiality to adequately describe some prominent features of wave dynamics on the adverse current.

The paper consists of five sections. General modulation equations are derived in Sect. 2. Section 3 is devoted to stationary nondissipative solutions for adverse and following nonuniform currents and various initial steepness of the surface wave train. We calculate the maximum amplitude amplification in dependence from relative strength of opposing current and compared it with available experimental and theoretical results (Toffoli et al., 2013; Ma et al., 2013). The interaction of steep surface waves with strong adverse current under wave-blocking conditions including wave breaking effects is presented in Sect. 4. Modeling results are compared with the results of a series of experiments conducted by Chavla and Kirby (2002) and Ma et al. (2010). Section 5 summarizes our final conclusions and discussion.

The first set of complete equations that describe short waves propagating over nonuniform currents of much larger scale were given by Longuet-Higgins and Stewart (1964). Wave energy is not conserved, and the concept of “radiation stress” was introduced to describe the average momentum flux in terms that govern the interchange of momentum with the current. In this model, it is also justifiable to neglect the effect of momentum transfer on the form of the surface current because it is an effect of the highest order (Stocker and Peregrine, 1999).

We construct a model of the current effect on the modulation instability of
a nonlinear Stokes wave by making the following assumptions:

Surface waves and current propagate along a common

By

The characteristic spatial scale used in developing the BF instability
of the Stokes wave is

It is assumed that the

The dimensionless set of equations for potential motion of an ideal
incompressible deep-depth fluid with a free surface in the presence of
current

The variables are normalized as

The weakly nonlinear surface wave train is described by a solution to
Eqs. (

We will analyze the surface wave train of a particular form, which describes the development of modulation instability in the presence of current.

For calm water, the initially constant nonlinear Stokes wave with amplitude,
wave number and frequency (

We analyze the problem assuming the wave motion phase

The main kinematic wave parameters

The solution to the problem, uniformly valid for

The free-surface displacement

Only the resonance terms for all three wave modes are included in the
third-order displacement (Eq.

Substitution of the velocity potential (Eq.

The obtained system of Eqs. (

Modulation Eqs. (

Let us analyze the stationary wave solutions of the problem in Eqs. (

Typical behavior of wave instability in the absence of current is presented
in Fig. 1a for a Stokes wave having initial steepness

The development of modulation instability on negative variable current

The region of the most developed instability corresponds to the spatial location of the maximum of the negative current (Fig. 1c). As one can see from Fig. 1b, c, the initial stage of wave-current interaction is characterized by the dominant process absorbing of energy by waves where all three waves grow simultaneously. Initial growth of side band modes (Fig. 1c) leads to a deeper modulated regime. Increasing of wave steepness in turn accelerates instability and finally these two dominate processes alternate. Correspondingly, the triggering of this complicated process essentially depends on the displacement of the current maximum.

Quasi-resonance interaction of waves in the presence of variable current
causes some crucial questions about its detuning properties. Absolute
frequencies for the stationary modulation satisfy the resonance conditions
(Eq.

To clarify this property we present the behavior of phase-shift difference
function

The typical scenario of wave interaction with co-propagating current is
presented in Fig. 1f. The modulation instability is depressed by the
following current

Regimes of modulation presented in Fig. 1a–c demonstrate the strictly symmetrical behavior with respect to the current peak and wave train returns to its initial structure after interaction with current. The modulation equations permit symmetrical solutions for the symmetrical current function, but, outside of interaction zone the nonlinear periodic waves are defined by the boundary conditions and the constant Stokes wave is only one of such possibilities. The symmetrical behavior is typical for a sufficiently slow-varying current. We present in Fig. 1e the example of asymmetrical wave modulation for the same wave initial characteristics as for Fig. 1c and 2 times shorter space scale of the current. At the exit of wave-current interaction zone we mention three waves system with comparable amplitudes and periodic energy transfer.

The increasing strength of the opposing flow (

Nondimensional maximum wave amplitude as a function of

To estimate the possibility of generating large transient waves, we employ
the model and calculate the maximum amplification of the amplitudes of
surface waves generated in still water and then undergo a current quickly
raised to a constant value

Our simulations confirm that initially stable waves in experiments of Toffoli et al. (2013) undergo a modulationally unstable process and wave amplification in the presence of adverse current. Maximum amplification reasonably corresponds to results of experiments at the Tokyo University Tank for moderate strength of current. Maximum of nonlinear focusing in dependence on the value of current is weaker compare to the model of Toffoli et al. (2013).

Experiments of Ma et al. (2013) (Fig. 2b) show that the development of the
modulational instability for a gentle wave and relatively weak adverse
current (

Stokes waves with sufficiently high initial steepness

Dual non-conservative evolution equations for wave energy density

An analysis of fetch laws parameterized by Tulin (1996) reveals that the
rate of energy loss

Tulin and Waseda (1999), through consideration of a multimodal wave system
evolving from a carrier wave and two side bands, showed that energy
downshifting during breaking is determined by the balance between momentum
and dissipation losses, suitably parameterized by the parameter

The sink of energy

The dispersion relations (

Modulation of surface waves by the adverse current

The numerical simulations for initially high steepness waves (

A very weak opposite current

We also performed numerical simulations for the boundary conditions and the form of the variable current obtained in two series of experiments conducted by Chavla and Kirby (2002) and Ma et al. (2010).

Dashed curves show the amplitudes of the waves for primary (Pr),
lower (Lo) and upper (Up) sidebands obtained experimentally by Chavla and
Kirby (2002). The solid lines (

Data for the wave blocking regime in experiments conducted by Chavla and
Kirby (2002) are taken from their Test 6 (Fig. 11). The experimental
results of test 6 and our numerical simulation results are compared in
Fig. 4. A surface wave with initially high steepness (

Dashed curves show amplitudes of the waves for primary (Pr), lower
(Lo) and upper (Up) sidebands obtained from experiments conducted
by Ma et al. (2010). The solid lines (

The model simulations results have distinctive features that agree
reasonably well with the results of experiments:

initial symmetrical growth of the main sidebands with frequencies

asymmetrical growth of sidebands beginning at

energy transfer at very short spatial distances and several increases in
the lower sideband amplitude just on a half meter length

a depressed higher frequency band and primary wave;

an almost permanent increase in the lowest subharmonic along the tank;

sharp accumulation of energy by the lowest subharmonic wave during interaction with increasing opposing current;

final permanent downshifting of the wave energy.

The evolution of a Stokes wave and its two main sidebands on a slowly varying unidirectional steady current gives rise to modulation instability with special properties. Interaction with countercurrent accelerates the growth of sideband modes on much shorter spatial scales. In contrast, wave instability on the following current is sharply depressed. Amplitudes and wave numbers of all waves vary enormously in the presence of strong adverse current. The increasing strength of the opposing flow results in deeper modulation of waves and more frequent mutual oscillations of the waves amplitudes.

Large transient or freak waves with amplitude and steepness several times larger than those of normal waves may form during temporal nonlinear focusing of waves accompanied by energy income from sufficiently strong opposing current. The amplitude of a rough wave strongly depends on the ratio of the current velocity to group velocity.

Interaction of initially steep waves with the strong blocking adverse current results in intensive energy exchange between components and energy downshifting to the lower sideband mode accompanied by active breaking. A more stable long wave with lower frequency can overpass the blocking barrier and accumulate almost all the wave energy of the packet. The frequency downshift of the energy peak is permanent and the system does not revert to its initial state.

The model simulations satisfactorily agree with available experimental data on the instability of waves on blocking adverse current and the generation of rough waves.

We thank V. Shrira and the anonymous referees, especially referee two for useful and stimulating discussions.

The authors would like to thank the Ministry of Science and Technology of Taiwan (Grant supported by NSC 103-2911-I-006 -302) and the Aim for the Top University Project of National Cheng Kung University for their financial support. This study was also supported by RFBR Projects 14-02-003330a, 11-02-00779a. Edited by: V. Shrira Reviewed by: four anonymous referees