NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus GmbHGöttingen, Germany10.5194/npg-22-313-2015An analytical model of the evolution of a Stokes wave and its two
Benjamin–Feir sidebands on nonuniform unidirectional currentShuganI. V.ishugan@rambler.ruHwungH. H.YangR. Y.International Wave Dynamics Research Center, National Cheng Kung University, Tainan City, TaiwanProkhorov General Physics Institute, Russian Academy of Sciences, Moscow, RussiaI. V. Shugan (ishugan@rambler.ru)21May20152233133247November20145December201421April201529April2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/22/313/2015/npg-22-313-2015.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/22/313/2015/npg-22-313-2015.pdf
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.
Introduction
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
Cg+U(X)→0,
where Cg is the intrinsic group velocity of waves in a moving
frame and U(X) is slowly varying horizontal current, with X being the
horizontal coordinate in the direction of wave propagation. Waves
propagating against opposing current are stopped if the magnitude of the
current, in the direction of wave propagation, exceeds the group velocity of
the oncoming waves. This characteristic feature of wave blocking has drawn
the interest of oceanographers and coastal engineers alike for its ability
to be used as signature patterns of underlying large-scale motion (e.g.,
fresh water plumes and internal waves) and for the navigational hazard it
poses. Smith (1975), Peregrine (1976), and Lavrenov (1998) analyzed
refraction/reflection around a blocking region and obtained a uniformly
valid linearized solution, including a short reflecting wave.
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 (ak<0.16, where a
and k are the wave amplitude and wave number, respectively). When the slope
is sufficiently steep (ak>0.22), the behavior of waves is
principally different; i.e., waves are blocked only partly and
frequency-downshifted waves overpass the blocking barrier. The lower
sideband mode may dramatically increase; i.e., the amplitude rises several
times within a distance of a few wavelengths.
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 Amax
during the wave evolution of currents in deep water. Their numerical results
revealed that the maximum amplitude of the freak wave depends on U/Cg,
where U is the velocity of the current and Cg is the group velocity of
the wave packet.
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.
Modulation equations for one-dimensional interaction
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 x-direction.
By ac, kc, and ωc we denote the characteristic
amplitude, wave number, and angular frequency of the surface waves. We use a
small conventional average wave steepness parameter; ε=ackc≪1.
The characteristic spatial scale used in developing the BF instability
of the Stokes wave is lc/ε2, where lc=2π/kc
is the typical wavelength of surface waves (Benjamin and Feir, 1967). We
consider slowly varying current U(x) with horizontal length scale L of the
same order: L=O(lc/ε2).
It is assumed that the U(x) dependence is due to the inhomogeneity of
the bottom profile h(x), which is sufficiently deep so that the deep-water
regime for surface waves is ensured; i.e., exp(-2kch)≪1. The
characteristic current length L at which the function U(x) varies noticeably is
assumed to be much larger than the depth of the fluid, h(x)≪L. Under
these conditions, shallow water model for the current description may be
adopted: U(x)h(x) is assumed to be approximately constant, and the vertical
component of the steady velocity field on the surface z=η(x) can be
neglected. Correspondingly, it follows from the Bernoulli time-independent
equation that the surface displacement induced by the current is small
(Ruban, 2012). Such a situation can occur, for example, near river mouths or
in tidal/ebb currents.
In all following equations, variables and sizes are scaled according to the
above assumptions, and made dimensionless using the characteristic length
and timescales of the wave field.
The dimensionless set of equations for potential motion of an ideal
incompressible deep-depth fluid with a free surface in the presence of
current U(x)is given by the Laplace equation
φxx+φzz=0,-h(x)<z<εη(x,t).
The boundary conditions at the free surface are
-η=φt+Uφx+ε12(φx2+φz2),z=εη(x,t),ηt+Uηx+ϵφxηx=φz,z=ϵη(x,t),
and those at the bottom are
φ→0,z=-h(x).
Here, φ(x,z,t) is the velocity potential, η(x,t) is the
free-surface displacement, z is the vertical coordinate directed upward and t is time.
The variables are normalized as
φ=acgkcφ′=εgkc3φ′,η=acη′=εkcη′,t=1gkct′,z=z′kc,x=x′kc,U(Kx)=U′(K/kcx′)cp=U′(ε2x′)cp,
where g is acceleration due to gravity, K=2π/L, and cp is the
phase speed of the carrier wave, but the primes are omitted in Eqs. ()–().
Note that normalization () explicitly specifies the principal
scales of sought functions φ and η.
The weakly nonlinear surface wave train is described by a solution to
Eqs. ()–(), expanded into a Stokes series in terms of ε.
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 (a1, k1, σ1) is unstable in
response to a perturbation in the form of a pair of small waves with similar
frequencies and wavenumbers: a superharmonic wave (a2, k2=k1+Δk,
σ2=σ1+Δσ) and subharmonic wave
(a0, k0=k1-Δk, σ0=σ1-Δσ).
For most unstable modes, Δσ/σ1=ε and
Δk/k1=2ε, where ε=a1k1 is the
initial steepness of the Stokes wave (Benjamin and Feir, 1967). This is the
BF or modulation instability of the Stokes wave. Kinematic resonance
conditions for waves in the presence of slowly varying current are the same,
with one important particularity that intrinsic wave numbers and frequencies
of waves in the moving frame are variable and modulated by the current.
We analyze the problem assuming the wave motion phase θi=θi(x,t) exists for each wave in the presence of a slowly varying current
U(x), and we define the local wave number ki and absolute observed
frequency ωi as
ki=θix,ωi=σi+kiU=-θit,i=0,1,2.
For stationary modulation, the intrinsic frequency σi and wave
number ki for each wave slowly change in the presence of variable
current, but the resonance condition
2ω1≈ω0+ω2
remains valid throughout the region of wave propagation owing to the
stationary value of the absolute frequency for each harmonic.
The main kinematic wave parameters (σi,ki) together with the
first-order velocity potential amplitudes, φi, are considered
further as slowly varying functions with typical scale, O(ε-1), longer than the primary wavelength and period (Whitham, 1974):
φi=φi(εx,εt),ki=ki(εx,εt),σi=σi(εx,εt).
On this basis, we attempt to recover the effects of slowly varying current
and nonlinear wave dispersion (having the same order) additional to the
Stokes term with the order of wave steepness squared.
The solution to the problem, uniformly valid for O(ε3), is
found by a two-scale expansion with the differentiation:
∂∂t=-∑(σi+kiU)∂∂θi+ε∂∂T,∂∂x=∑ki∂∂θi+ε∂∂X,T=εt,X=εx.
Substitution of the wave velocity potential in its linear form,
φ=∑i=0i=2φiekizsinθi,
satisfies the Laplace Eq. () to the first order of ε owing
to Eq.
() and gives the additional terms of the second orderO(ε2):
ε(2kiφiX+kiXφi+2kikiXφiz)ekizcosθi+…=0.
To satisfy the Laplace equation to second order, Yuen and Lake (1982),
Shugan and Voliak (1998), and Hwung et al. (2009) suggested an
additional phase-shifted term with a linear and quadratic z correction in the
representation of the potential function φ:
φ=∑i=0i=2φiekizsinθi-εφiXz+kiXφi2z2ekizcosθi+….
Exponential decaying of the wave's amplitude with increasing -z is
accompanied by a second-order subsurface jet owing to slow horizontal
variations in the wave number and amplitude of the wave packet.
The free-surface displacement η=η(x,t) is also sought as an
asymptotic series:
η=η0+εη1+ε2η2+…,
where η0, η1, and η2 are O() functions to be
determined. Using Eqs. () and () subject to the dynamic boundary
condition (Eq. ), we find the components of the free-surface displacement:
η0=∑i=0i=2σiφicosθi,η1=-∑i=0i=2(φiTsinθi+UφiXsinθi)+∑i=0i=2φi2ki2cos[2θi]/2+∑i=0i=2∑j≠ij=2(σi-σj)2σiσjφiφjcos[θi-θj]/2+∑i=0i=2∑j≠ij=2(σi+σj)2σiσjφiφjcos[θi+θj]/2,-8η2=∑i=02φiσi2(3φi2σi5+∑j≠i(2σj2+σi2)(2σj-σi))cos[θi]-φ0φ12σ0σ12(σ02+2σ12)(σ02-4σ0σ1+2σ12)cos[θ2+ϕ]-φ12φ2σ12σ2(σ22+2σ12)(σ22-4σ2σ1+2σ12)cos[θ0+ϕ]-2φ0φ1φ2σ0σ1σ2(σ02+σ12+σ22)(σ02-2σ0σ1+σ12-2σ1σ2+σ22)cos[θ1-ϕ],
where ϕ is a slowly varying phase-shift difference: ϕ=2θ1-θ0-θ2.
Only the resonance terms for all three wave modes are included in the
third-order displacement (Eq. ).
Substitution of the velocity potential (Eq. ) and displacement (Eqs. 13–) into
the kinematics boundary condition (Eq. ) gives, after much routine algebra,
relationships between the modulation characteristics of the resonant wave:
σ02=k0+ε2σ03(φ02σ05+φ12σ13(2σ12-σ0σ1+σ02)+φ22σ23(2σ22-σ0σ2+σ02))+ε2φ12φ2σ13σ22φ0(2σ13-2σ12σ2+2σ1σ22-σ23)cos[ϕ];σ22=k2+ε2σ23(φ22σ25+φ02σ03(2σ02-σ0σ2+σ22)+φ12σ13(2σ12-σ1σ2+σ22))+ε2φ12φ0σ13σ02φ2(2σ13-2σ0σ12+2σ02σ1-σ03)cos[ϕ];σ12=k1+ε2σ13(φ12σ15+φ02σ03(2σ02-σ0σ1+σ12)+φ22σ23(σ12-σ1σ2+2σ22))+ε2φ0φ2σ0σ12σ2(σ04-σ03σ1-σ0σ1(σ1-σ2)2+σ02(σ12-σ1σ2+2σ22)+σ2(-σ13+σ12σ2-σ1σ22+σ23))cos[ϕ],[φ02σ0]T+[(U(X)+12σ0)φ02σ0]X=εφ12φ2φ0σ13σ22(2σ13-2σ12σ2+2σ1σ22-σ23)sin[ϕ];[φ22σ2]T+[(U(X)+12σ2)φ22σ2]X=εφ12φ2φ0σ02σ13(2σ13-2σ0σ12+2σ02σ1-σ03)sin[ϕ];[φ12σ1]T+[(U(X)+12σ1)φ12σ1]X=-εφ12φ2φ0σ0σ12σ2σ04-σ03σ1-σ0σ1(σ1-σ2)2++σ02(σ12-σ1σ2+2σ22)-σ2(σ13-σ12σ2+σ1σ22-σ23)sin[ϕ].
The formulas () represent the “intrinsic” dispersion relations of the
nonlinear wave for each of the harmonics in the presence of current, U(X).
Equation () yields the known wave action law with the energy exchange terms on the
right side of the equations.
The obtained system of Eqs. () and () in the absence of current is similar
to classical Zakharov equations for discrete wave interactions (Stiassnie and Shemer, 2005; Mei et al.,
2009); it has a strong symmetry with respect to indexes zero and two. The main
property of the derived modulation equations is the variability of
interaction coefficients in the presence of nonuniform current.
Modulation Eqs. () and () are closed by the equations of wave phase
conservation that follow from Eq. () as the compatibility condition (Phillips,
1977):
kiT+(σi+kiU)X=0,i=0,1,2.
The derived set of nine modulation Eqs. ()–() form the complete system
for nine unknown functions (ki,σi,φi,i=0,1,2).
Nondissipative stationary wave modulations
Let us analyze the stationary wave solutions of the problem in Eqs. ()–()
supposing that all unknown functions depend on the single coordinate X. Then,
after integrating Eq. (), we have the conservation law for the absolute
frequency of each wave:
σi+kiU=ωi=const,i=0,1,2.
The wave action laws for resonant components take the form
[(U+12σ0)ϕ02σ0]X=εϕ12ϕ2ϕ0σ13σ22(2σ13-2σ12σ2+2σ1σ22-σ23)Sin[φ][(U+12σ2)ϕ22σ2]X=εϕ12ϕ2ϕ0σ13σ02(2σ13-2σ0σ12+2σ02σ1-σ03)Sin[φ][(U+12σ1)ϕ12σ1]X=-εϕ12ϕ2ϕ0σ0σ12σ2(σ04-σ03σ1-σ0σ1(σ1-σ2)2+σ02(σ12-σ1σ2+2σ22)-σ2(σ13-σ12σ2+σ1σ22-σ23))Sin[φ].
To perform the qualitative analysis of the stationary problem, we suggest
the law of wave action conservation flux in a slowly moving media as
analogue of the three Manley–Rowe dependent integrals:
U+12σ2ϕ22σ2+U+12σ0ϕ02σ0+U+12σ1ϕ12σ1=const;12U+12σ1ϕ12σ1+U+12σ0ϕ02σ0=const;12U+12σ1ϕ12σ1+U+12σ2ϕ22σ2=const;U+12σ0ϕ02σ0-U+12σ2ϕ22σ2=const.
These integrals follow from the system (Eq. ) with acceptable
accuracy O(ε4)for the stationary regime of modulation. The
second and third relations here clearly show that the wave action flux of
the side bands can grow up at the expense of the main carrier wave flux. The
last relationship manifests the almost identical behavior of the main
sidebands for the problem of their generation due to the Benjamin–Feir
instability.
(a) BF instability without current. (b) and (c) modulation of surface
waves by adverse current U=U0Sechε2(x-x0), (U0=-0.15); (b)x0=200 and (c)x0=400. (d) phase
difference function φ[X]=2θ1[X]-θ0[X]-θ2[X], θ1[0]=0;θ0[0]=θ2[0]=-π/4; (e)
modulation of surface waves by adverse current U=U0Sech2ε2(x-x0), (U0=-0.2), (f) modulation
instability for following current (U0=0.16,x0=400). (g) and (h)
functions of wave amplitude and wave number, respectively, for U0=-0.2.
(I), (II), (III) amplitude envelopes of the carrier, superharmonic and
subharmonic waves, respectively. (IV) linear solution for the carrier
envelope. The initial steepness of the carrier wave is ε=0.1,
side band initial amplitudes are equal to 0.1ε. (i) Relative
distortion of the linear dispersion relation for the case (g), (I) –
carrier, (II) – higher side band.
Typical behavior of wave instability in the absence of current is presented
in Fig. 1a for a Stokes wave having initial steepness ε=0.1. Two
initially negligible side bands (II) and (III) exponentially grow at the
expense of the main Stokes wave (I), and after saturation, the wave system
reverts to its initial state, which is the Fermi–Pasta–Ulam recurrence
phenomenon. One can see here also the characteristic spatial scale for the
developing of modulation instability O(1/(kcε2)).
The development of modulation instability on negative variable current
U=U0Sechε2(x-200),U0=-0.1, is
presented in Fig. 1b. The modulation instability develops far more quickly
on opposing current and reaches deeper stages of modulation. The energetic
process is described as follows. The basic Stokes wave (I) absorbs energy
from the counter current U and its steepness increases. This in turn
accelerates the wave instability; there is a corresponding increase in
energy flow to the most unstable sideband modes (II) and (III). Wave
refraction by current is acting simultaneously with nonlinear wave
interactions. The linear modulation model (Gargett and Hughes, 1972; Lewis
et al., 1974) assumes the independent variations of harmonics with current
and gives much larger maximum amplitude of the carrier wave (IV). It just
adsorbs energy from the adverse 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. ) for the entire region of interaction. But the local wave numbers and
intrinsic frequencies are substantially variable due to current effects and
nonlinearity. Maybe the almost resonance conditions are totally destroyed
in this case due to large detuning (Shrira and Slunyaev, 2014)?
To clarify this property we present the behavior of phase-shift difference
function φ[X]=2θ1-θ0-θ2 (Fig. 1d)
corresponding to waves modulation shown in Fig. 1c. Intensity of nonlinear
energy transfer is mostly defined by this function together with wave
amplitudes (see Eqs. , ). The results look rather surprising – several
strong phase jumps take place with corresponding changes of the wave
energy flux direction. Nevertheless, we see an intensive energy
exchange in the entire interaction zone. Quasi-resonant conditions are
satisfied locally in space with a relatively small detuning factor.
Qualitatively similar behavior of phase-shift function we found also for
other regimes of wave modulation.
The typical scenario of wave interaction with co-propagating current is
presented in Fig. 1f. The modulation instability is depressed by the
following current U(X)>0 and the resonant sideband modes develop at almost
two time's longer distance in comparison with the evolution without 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 (U0=-0.2) results in
deeper modulation of waves and more frequent mutual oscillations of the
amplitudes (Fig. 1g). There are essential oscillations of wave-number
functions of the sideband modes (II) and (III) (Fig. 1h) owing to the
nonlinear dispersion properties of waves. We mention also that the wave
number of the carrier wave in the linear model (IV) is much higher than that
in the nonlinear model (I). The width of the wave-number spectrum of the
wave train in the nonlinear model locally increases to almost twice that of the
initial width. To give an idea about the strength of nonlinearity, we present
in Fig. 1i the relative distortion of the linear dispersion relation for
different modes. As one can see the effect of nonlinearity for the carrier
(at maximum is about 10 %) is much less compared to side bands (at peak is
more than 30 %). The main impact of nonlinearity comes from the amplitude
Stokes dispersion.
Nondimensional maximum wave amplitude as a function of
-U/Cg, where Cg is the group velocity of the carrier
wave and E1/2 is the local standard deviation of the wave envelope.
(a) experiments conducted by Toffoli et al. (2013) for carrier wave
of period T=0.8s (wavelength λ≅1 m), initial steepness
k1a1=0.063, and frequency difference Δω/ω1=1/11, initial amplitudes of side bands equal to 0.25 times the amplitude of
the carrier wave. Solid dots show measurements made using a flume at Tokyo
University and squares show results obtained at Plymouth University. Line (I)
shows the prediction made using Eq. (2) (Toffoli et al., 2013) while line (II)
shows the model prediction. (b) case T11 in Ma et al. (2013) for
carrier-wave frequency ω1=1 Hz, initial steepness k1a1=0.115, initial amplitudes of side bands equal to 0.05 times the amplitude
of the carrier wave and frequency difference Δω/ω1=0.44a1k1. Solid dots show measurements. Line (I) shows the model
prediction, while line (II) shows the prediction made using Eq. ()
(Toffoli et al., 2013).
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 -U. The boundary conditions for the unperturbed
waves were taken from experiments conducted by Toffoli et al. (2013) and Ma
et al. (2013). Results of the calculations are presented in Fig. 2a and b.
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 (U/Cg∼-0.1) (see the first experimental point) is limited
due to the presence of dissipation. Our model simulations are in good
agreement with the experimental values for the moderate values of adverse
current -U/Cg∼0.2-0.4. Results of Toffoli et al. (2013) notably
overestimate the maximum of wave amplification.
Wave propagation under the blocking conditions of strong adverse current
Stokes waves with sufficiently high initial steepness ε under
the impact of strong blocking adverse current (U(X)<-Cg) will
inevitably reach the breaking threshold for the steepness of water waves. We
employ the adjusted weakly nonlinear dissipative model of Tulin (1996),
Tulin and Li (1999) and Hwung et al. (2011) to describe the effect of
breaking on the dynamics of the water wave.
Dual non-conservative evolution equations for wave energy density E=1/2gη2 and wave momentum M=E/c=E/(ω/k), or, correspondingly, for energy
E and celerity c were rigorously derived (Tulin and Li, 1999) using the
variational approach: a modified Hamiltonian principle involving the
modulating wave Lagrangian plus a work function representing the
nonconservative effects of wave breaking. It was also shown that these dual
equations correspond to the complex NLS equation, as modified by the
non-conservative effects, i.e. to energy and dispersion equations. Wave
breaking effects were characterized by the energy dissipation rate Db
and momentum loss rate Mb.
An analysis of fetch laws parameterized by Tulin (1996) reveals that the
rate of energy loss Db due to breaking is of fourth order of the wave
amplitude:
Db/E=ωDη2k2,
and D=O(10-1)is a small empirical constant. Momentum-loss rate Mbwas quantified in terms of energy dissipation rate Dband parameterized
in such a way that
cMb=(1+γ)Db,
where γ is empirical coefficient, which is varied in the range γ=0.4-0.7. Strong plunging breaking corresponds to γ=0.4 and weak
to γ=0.7. In all our numerical simulations was chosen γ=0.4,D=0.1.
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 γ:
∂∂tE0-E2=γDb/(δω/ω),
where E0,E2 – energy densities of the sub and super harmonics,
respectively. The parametric value γ was found to be positive γ>0 and providing the long term downshifting.
The sink of energy Db and momentum Mb due to wave breaking leads
to additional terms at the right sides of the wave energy Eq. () and
dispersive Eq. () for each of the waves. Tulin (1996) suggested
using sink terms along the entire path of wave interaction with the wind.
The wave dissipation function for the adjusted model (Hwung et al.,
2011)
includes also the wave steepness threshold function
Hε∑σiϕikiAS-1,
where H is the Heaviside unit step function and AS is the threshold value
of the combined wave steepness ε∑σiϕiki,
is applied to calculate energy and momentum losses in high steepness
zones. In our computations the threshold AS=0.32 was chosen.
The dispersion relations () and wave energy laws () including break
dissipation take the form
σ02=k0+ε2σ03(φ02σ05+φ12σ13(2σ12-σ0σ1+σ02)+φ22σ23(2σ22-σ0σ2+σ02))+ε2φ12φ2σ13σ22φ0(2σ13-2σ12σ2+2σ1σ22-σ23)Cos[ϕ]+ε2H[χ]DSin[ϕ]φ12φ2/φ0k04+8γε2X(φ02+φ12+φ22)-4γφ12φ2/φ0Sin[ϕ],σ22=k2+ε2σ23(φ22σ25+φ02σ03(2σ02-σ0σ2+σ22)+φ12σ13(2σ12-σ1σ2+σ22))+ε2φ12φ0σ13σ02φ2(2σ13-2σ0σ12+2σ02σ1-σ03)Cos[ϕ]+ε2H[χ]DSin[ϕ]φ12φ0/φ2k24+8γε2X(φ02+φ12+φ22)+4γφ12φ0/φ2Sin[ϕ],σ12=k1+ε2σ13(φ12σ15+φ02σ03(2σ02-σ0σ1+σ12)+φ22σ23(σ12-σ1σ2+2σ22))+ε2φ0φ2σ0σ12σ2(σ04-σ03σ1-σ0σ1(σ1-σ2)2+σ02(σ12-σ1σ2+2σ22)+σ2(-σ13+σ12σ2-σ1σ22+σ23))Cos[ϕ]+ε2H[χ]D-2Sin[ϕ]φ2φ0k14+8γε2X(φ02+φ12+φ22)+4γφ2φ0Sin[ϕ],
where χ=ε∑σiϕiki/AS-1, and
[(U(X)+12σ0)φ02σ0]X=εφ12φ2φ0σ13σ22(2σ13-2σ12σ2+2σ1σ22-σ23)Sin[ϕ]ε+H[χ]D-k04φ12φ2φ0Cos[ϕ]-k04φ02(φ02+2φ12+2φ22)+4γφ02k04(φ12+φ22+φ12φ2/φ0Cos[ϕ]),[(U(X)+12σ2)φ22σ2]X=εφ12φ2φ0σ13σ02(2σ13-2σ0σ12+2σ02σ1-σ03)Sin[ϕ]+εH[χ]D-k24φ12φ2φ0Cos[ϕ]-k24φ22(φ22+2φ02+2φ12)+4γφ22k24(φ12+φ02+φ12φ0/φ2Cos[ϕ]),[(U(X)+12σ1)φ12σ1]X=-εφ12φ2φ0σ0σ12σ2(σ04-σ03σ1-σ0σ1(σ1-σ2)2+σ02(σ12-σ1σ2+2σ22)-σ2(σ13-σ12σ2+σ1σ22-σ23))Sin[ϕ]+εH[χ]D-2k14φ12φ2φ0Cos[ϕ]-k14φ12(φ12+2φ02+2φ22)+4γφ12k14((φ22-φ02)+φ2φ0Cos[ϕ]).
Wave breaking leads to permanent (not temporal) frequency downshifting at a
rate controlled by breaking process. A crucial aspect here is the
cooperation of dissipation and near-neighbor energy transfer in the
discretized spectrum acting together.
Modulation of surface waves by the adverse current U=U0x.
(a)U0=-2.5 10-4, (b)U0=-5 10-4, (c)U0=-10-3. (I), (II), (III)
– amplitude envelopes of the carrier, subharmonic and superharmonic waves,
respectively. Initial wave steepness ε=0.25, side bands
amplitudes equal to 0.05 times the amplitude of the carrier.
The numerical simulations for initially high steepness waves (ε=0.25) propagation with wave breaking dissipation is presented in Fig. 3a–c.
We calculate the amplitudes of surface waves on linearly increasing
opposing current U(x)=-U0x with different strengthU0. The most
unstable regime was tested for frequency space Δω±/ω1∼ε, initial side bands amplitudes equal to 0.05
times the amplitude of the carrier wave and most effective initial phases
θ1(0)=0,θ0(0)=θ2(0)=-π/4.
A very weak opposite current U0=2.5 10-4 (Fig. 3a) has a
pure impact on wave behavior: it finally results in almost bichromatic
wave train with two dominant waves: carrier and lower side band. Frequency
downshift here is not clearly seen. Two times stronger current case
with U0=5×10-4 is presented in Fig. 3b. We note some
tendency to final energy downshift to the lower side band. Really strong
permanent downshift with total domination of the lower side band is seen for
two times more strong current U0=10-3 (Fig. 3c).
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 (A1,A0,A2) are wave
amplitudes calculated in modeling. (U-U0)/C is the no-dimensional
variable current, where C is the initial phase speed of the carrier
wave, U0=-0.32m/s;k1=4.71/m,C=1.44m/s,T=1.2s.
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 (A1k1=0.296) and
period T=1.2 s meets adverse current with increasing amplitude.
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 (A1,A0,A2) are the
wave amplitudes calculated in modeling. (U-U0)/(4C) shows the
no-dimensional variable current, where C is the initial phase speed of the
primary wave, U0=-0.25m/s;k1=4.1/m,C=1.56m/s,T=1s.
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 f0=0.688 Hz, f2=0.978 Hz at distances up to k1x<-2;
asymmetrical growth of sidebands beginning at (k1x≈-2) and
downshifting of energy to the lower sideband;
energy transfer at very short spatial distances and several increases in
the lower sideband amplitude just on a half meter length k1x∈(-2,0);
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.
Modulation evolution of breaking waves in experiments of Ma et al. (2010)
for the most intriguing case 3 are presented in Fig. 5 together with the
results of our numerical computations. A primary wave with period T=1s and
steepness A1k1=0.18 meets linearly increasing opposing current
that finally exceeds the threshold to be a linear blocking barrier for the
primary wave U(x)<-1/4C. In experiments, sideband frequencies arose
ubiquitously from the background noise of the flume. In numerical
simulations, the sidebands were slightly seeded at frequencies corresponding
to the most unstable modes. The wave-breaking region in this experimental
case ranged from k1x=52 to k1x=72. The lower sideband amplitude
grew with increasing distance at the expense of the primary wave, while
there was little change in the higher sideband energy. There was an
effective frequency downshift following initial breaking (k1x=56). The
modeling results agree reasonably well with the experimental data.
Conclusions
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.
Acknowledgements
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
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