A nonlinear Schrödinger equation for gravity-capillary water waves on arbitrary depth with constant vorticity: Part I
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics A nonlinear Schrödinger equation forgravity-capillary water waves on arbitrarydepth with constant vorticity: Part I
H.-C. Hsu , C. Kharif † , M. Abid and Y.-Y. Chen Department of Marine Environment and Engineering, National Sun Yat-sen University,Kaohsiung, 801 Taiwan. Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384, Marseille,France(Received xx; revised xx; accepted xx)
A nonlinear Schrödinger equation for the envelope of two-dimensional gravity-capillarywaves propagating at the free surface of a vertically sheared current of constant vorticityis derived. In this paper we extend to gravity-capillary wave trains the results of Thomas et al. (2012) and complete the stability analysis and stability diagram of Djordjevic& Redekopp (1977) in the presence of vorticity. Vorticity effect on the modulationalinstability of weakly nonlinear gravity-capillary wave packets is investigated. It is shownthat the vorticity modifies significantly the modulational instability of gravity-capillarywave trains, namely the growth rate and instability bandwidth. It is found that therate of growth of modulational instability of short gravity waves influenced by surfacetension behaves like pure gravity waves: (i) in infinite depth, the growth rate is reducedin the presence of positive vorticity and amplified in the presence of negative vorticity,(ii) in finite depth, it is reduced when the vorticity is positive and amplified and finallyreduced when the vorticity is negative. The combined effect of vorticity and surfacetension is to increase the rate of growth of modulational instability of short gravitywaves influenced by surface tension, namely when the vorticity is negative. The rate ofgrowth of modulational instability of capillary waves is amplified by negative vorticityand attenuated by positive vorticity. Stability diagrams are plotted and it is shown thatthey are significantly modified by the introduction of the vorticity.
Keywords : NLS equation, modulational instability, vorticity, surface tension
1. Introduction
Generally, gravity-capillary waves are produced by wind which generates firstly ashear flow in the uppermost layer of the water and consequently these waves propagatein the presence of vorticity. These short waves play an important role in the initialdevelopment of wind waves, contribute to some extent to the sea surface stress andconsequently participate in air-sea momentum transfer. Accurate representation of thesurface stress is important in modelling and forecasting ocean wave dynamics. Further-more, the knowledge of their dynamics at the sea surface is crucial for satellite remotesensing applications.In this paper we consider both the effect of surface tension and vorticity due to a verticallysheared current on the modulational instability of a weakly nonlinear periodic short wavetrains. Recently, Thomas et al. (2012) have derived a nonlinear Schrödinger equation for † Email address for correspondence: kharif@irphe..... a r X i v : . [ phy s i c s . f l u - dyn ] J a n H.-C. Hsu, C. Kharif, M. Abid and Y.-Y. Chen pure gravity water waves on finite depth with constant vorticity. Their main findings were(i) a restabilisation of the modulational instability for waves propagating in the presenceof positve vorticity whatever the depth and (ii) the importance of the nonlinear couplingbetween the mean flow induced by the modulation and the vorticity. One of our aim isto extend Thomas’ investigation to the case of gravity-capillary waves propagating on avertically sheared current.The number of studies on the computation of steadily propagating periodic gravity waveson a vertically sheared current is important. For a review one can refer to the paperby Thomas et al. (2012). On the opposite, investigations devoted to the calculation ofgravity-capillary waves in the presence of horizontal vorticity is rather meagre. One cancite Bratenberg & Brevik (1993) who used a third-order Stokes expansion for periodicgravity-capillary waves travelling on an opposing current and Hsu et al. (2016) whoextended this work to the case of co- and counter-propagating waves. Kang & Broeck(2000) computed periodic and solitary gravity-capillary waves in the presence of constantvorticity on finite depth. They derived analytical solutions for small amplitude wavesand numerical solutions for steeper waves. Wahlen (2006) proved the rigorous existenceof periodic gravity-capillary waves in the presence of constant vorticity.To our knowledge, the unique study concerning the modulational instability of gravity-capillary waves travelling on a verticaly sheared current is that of Hur (2017). Thestability of irrotational gravity-capillary waves has been deeply investigated by severalauthors. Djordjevic & Redekopp (1977) and Hogan (1985) derived nonlinear envelopeequations and considered the modulational instability of periodic gravity-capillary waves.Note that in the gravity-capillary range, three-wave interaction is possible whereasmodulational instability corresponds to a four-wave resonant interaction. The numericalcomputations were extended to capillary waves by Chen & Saffman (1985) and Tiron &Choi (2012). Zhang & Melville (1986) investigated numerically the stability of gravity-capillary waves including, besides the four-wave resonant interaction, three-wave andfive-wave resonant interactions. For a review on stability of irrotational gravity-capillary,one can refer to the review paper by Dias & Kharif (1999).This study is devoted to the modulational instability of weakly nonlinear gravity-capillarywave packets propagating at the surface of a vertically sheared current of finite depth.In section 2, the governing equation are given and the nonlinear Schrödinger equation inthe presence of surface tension and constant vorticity is derived by using a multiple scalemethod. In section 3, the linear stability analysis of a weakly nonlinear wave train iscarried out as a function of the Bond number, the dispersive parameter and the intensityof the vertically sheared current.
2. Derivation of the NLS equation in the presence of surface tensionand vorticity
We consider the modulational instability of weakly nonlinear surface gravity-capillarywave trains in the presence of vorticity. Our investigation is confined to two-dimensionalwater waves propagating in finite depth. Viscosity is disregarded and the fluid is consid-ered incompressible. The geometry configuration is presented in figure 1.We choose an eulerian frame (
Oxyz ) with unit vectors ( ~e x , ~e y , ~e z ). The vector ~e y isoriented upwards so that the gravity is ~g = − g~e y with g >
0. The equation of theundisturbed free surface is y = 0 whereas the disturbed free surface is y = ζ ( x, t ). Thebottom is located at y = − h .The waves are travelling at the surface of a vertically sheared current of constant vorticity. LS equation for gravity-capillary waves with shear Figure 1.
Sketch of the two-dimensional flow.
We consider an underlying current given by ~u = Ωy~e x , so that the fluid velocity reads ~u = ~u + ~ ∇ φ, (2.1)where ∇ φ ( x, y, z, t ) is the wave induced velocity. The waves are potential due to theKelvin theorem which states that vorticity is conserved for a two-dimensional flow of anincompressible and inviscid fluid with external forces deriving from a potential.The potential φ satisfies the Laplace equation ∇ φ = 0 , (2.2)and the Euler equation can be written as follows ~ ∇ ( φ t + 12 u + Pρ w + gy ) = ~u ∧ ~ω, (2.3)with ~ω the vorticity vector along z , P the pressure and ρ w the water density. Subscriptsstand for derivatives in corresponding variables.Using the Cauchy-Riemann relations ψ y = φ x , ψ x = − φ y (2.4)where ψ is the stream function ~u ∧ ~ω = ~ ∇ ( 12 Ω y + Ωψ ) (2.5)The Euler equation (2.3) can be rewritten as follows ~ ∇ ( φ t + 12 φ x + 12 φ y + Ωyφ x + gy − Ωψ + Pρ w ) = 0 (2.6)Spatial integration gives the Bernoulli equation φ t + 12 φ x + 12 φ y + Ωyφ x + gy − Ωψ + Pρ w = C ( t ) (2.7)In the presence of surface tension, T , at the free surface y = ζ ( x, t ) the Laplace law writes P = P a − T ζ xx (1 + ζ x ) / (2.8) H.-C. Hsu, C. Kharif, M. Abid and Y.-Y. Chen where P a is the atmospheric pressure and T surface tension.The dynamic boundary condition at the free surface y = ζ is φ t + 12 φ x + 12 φ y + Ωζφ x + gζ − Ωψ − Tρ w ζ xx (1 + ζ x ) / = 0 (2.9)Witout loss of generality, we set P a = 0 and incorporate C ( t ) into the potential φ .Along with these, we have the kinematic free surface boundary condition ζ t + ζ x ( φ x + Ωy ) − φ y = 0 , y = ζ ( x, t ) (2.10)and the bottom boundary condition φ y = 0 , y = − h (2.11)Following Thomas et al. (2012) we can remove ψ by deriving (2.9) with respect to x andthen using relations (2.4) , keeping in mind that we are dealing with low-steepness waves,and that (2.9) is evaluated in y = ζ , we get the equation φ tx + φ ty ζ x + φ x ( φ xx + φ xy ζ x ) + φ y ( φ xy + φ yy ζ x ) + Ωζ x φ x + Ωζ ( φ xx + φ xy ζ x ) + gζ x + Ω ( φ y − φ x ζ x ) − Tρ w ( ζ xxx − ζ x ζ xxx − ζ xx ζ x ) = 0 , y = ζ ( x, t ) , (2.12)that matches the one first derived in Thomas et al. (2012) for T = 0.Following (Davey & Stewartson 1974), we look for solutions depending on slow variables( ξ, τ ) = ( ε ( x − c g t ) , ε t ) where ε = ak ( ε (cid:28)
1) and a , k and c g are the amplitude,wavenumber and group velocity of the carrier wave, respectively. The system of governingequations becomes ε φ ξξ + φ yy = 0 , − h (cid:54) y (cid:54) ζ ( ξ, τ ) , (2.13) φ y = 0 , y = − h, (2.14) ε ζ τ − εc g ζ ξ + εζ ξ ( εφ ξ + Ω [ y + h ]) − φ y = 0 , y = ζ ( ξ, τ ) , (2.15) ε φ τξ − ε c g ζ ξ + ε φ τy ζ ξ − ε c g φ ξy ζ ξ + ε φ ξ ( φ ξξ + φ ξy ζ ξ )+ εφ y ( φ ξy + φ yy ζ ξ ) + ε Ωζ ξ φ ξ + ε Ωζ ( φ ξξ + φ ξy ζ ξ ) + εgζ ξ + Ω ( φ y − ε φ ξ ζ ξ ) − ε Tρ w ( ζ ξξξ − ε ζ ξ ζ ξξξ − ε ζ ξξ ζ ξ ) = 0 , y = ζ ( ξ, τ ) (2.16)An asymptotic solution to the system (2.13-2.14-2.15-2.16) is sought in the followingform φ = + ∞ X n = −∞ φ n E n , ζ = + ∞ X n = −∞ ζ n E n , (2.17)where E = e i ( kx − ωt ) is a plane wave with ω the frequency of the carrier wave. We imposethat φ − n = ¯ φ n and ζ − n = ¯ ζ n where the bar denotes complex conjugate, so that thefunctions are real. The amplitudes φ n and ζ n are then expanded in a perturbation series LS equation for gravity-capillary waves with shear ε = ak φ n = + ∞ X j = n ε j φ nj , ζ n = + ∞ X j = n ε j ζ nj . (2.18)The terms depending on surface tension occur only at a higher order. The expansions(2.18) are substituted into the system of equations. The linear Laplace equation (2.13)is easier to handle, since solutions can be derived iteratively. Here we will simply writethe first order solution for φ , that is obtained by using the bottom boundary condition(2.14) φ = A ( ξ, τ ) cosh[ k ( y + h )]cosh( kh ) , (2.19)where the slow-varying function A ( ξ, τ ) will be used to express all other terms. Higher-order expansions of the Laplace equation introduce more unknown functions as solutions.Nevertheless, through expansions of the boundary conditions they can be all combinedto A ( ξ, τ ).The evolution of this unknown will depend on the initial condition A ( ξ, ε and E , which allows the expressions for the ζ ij and φ ij to be foundsuccessively.The calculations are somewhat tedious but some steps are of interest. At first, the lineardispersion relation is derived ω + σΩω − σgk (1 + κ ) = 0 , (2.20)where σ = tanh( µ ) with µ = kh and κ = T k ρ w g .The relation between A ( ξ, τ ) and ζ is the following ζ = i ω (1 + X ) g (1 + κ ) A ( ξ, τ ) , (2.21)where X = σΩ/ω From the above dispersion relation we can show easily that
X > −
1. We note that X depends also on the surface tension through ω and its associated dispersion relation. Itis also to be noted that the expression of the mean-flow term, which is important on thedevelopement of the modulational instability, is similar to that of Thomas et al. (2012).Nevertheless, surface tension takes place through the phase velocity c p , the group velocity c g and ω . ( c g ( c g + Ωh ) − gh ) φ ,ξ = ( gσc p (2 ω + σΩ ) + k c g (1 − σ )) | A | , (2.22)and gζ = ( c g + Ωh ) φ ,ξ − k (1 − σ ) | A | . (2.23)Although the expressions are identical to those of Thomas et al. (2012), it should benoted that the surface tension acts through the dispersion relation, affecting ω , c p and c g .It is at the order O ( ε E ) that the nonlinear Schrödinger equation is found for thepotential envelope A , so that iA τ + αA ξξ = γ | A | A, (2.24)where the coefficients depend on ( κ, Ω, kh ). H.-C. Hsu, C. Kharif, M. Abid and Y.-Y. Chen
Then the dispersion coefficient reads α = − ωk σ (2 + X ) (cid:20) σρ + µ X κ ( σ [ σ + µ (1 − σ )] − µ (1 − σ )( ρ − µσ ) X − κ κ α (cid:21) , (2.25)with α = − µ (1 + X )(1 − σ )( µσ −
1) + σ (1 + X )(1 + 2 ρ ) (2.26)+2 (cid:18) σρ + µ (1 − σ ) X − σκ κ (1 + X ) (cid:19) , where ρ = c g /c p is here the ratio of the group velocity to the phase velocity of the carrier.It can be expressed in a concise form ρ = (1 − σ ) µ + (1 + X )( σ + σκ κ ) σ (2 + X ) , (2.27)which depends only on µ, κ, X . The nonlinear coefficient is γ = k ω (1 + X )(2 + X ) (cid:20) − σ κ κ (1 + X ) − κ )(1 − σ )[(1 + X ) − σ ]+ σ (1 + X )(8 + 6 X ) + 1 + Xσ − κ (3 − σ + 3 X ) γ (2.28)+ 2 (1 + X )(2 + X ) + ρ (1 + κ )(1 − σ )(1 + κ )( ρ + µρ Xσ − µ (1+ X ) σ (1+ κ ) ) γ , with γ = 9 − σ + σ + (18 − σ − σ ) X + (15 + 3 σ ) X + (6 + 2 σ ) X + X (2.29)+ κ (cid:2) − σ + σ + (42 + 2 σ − σ ) X + (30 + 12 σ ) X + (9 + 5 σ ) X + X (cid:3) , and finally γ = (1 + κ ) (cid:20) (1 + X ) (1 + ρ + µXσ ) + 1 + X − σ ( ρσ + µX ) (cid:21) − κ (1 + X )(2 + X ) , (2.30)and we can check that these coefficients reduce to those of Djordjevic & Redekopp (1977),or Hogan (1985) in deep water, if Ω = 0 and to those of Thomas et al. (2012) if κ = 0.The last term in brackets of equation (2.28) corresponds to the coupling between themean flow due to the modulation and the vorticity which occurs at third-order. Thiscoupling was found by Thomas et al. (2012) for the case of pure gravity waves and hasan important impact on the stability analysis of progressive wave trains.We can see that in (2.28) there are two possible singularities that one should avoid, either σ − κ (3 − σ + 3 X ) = 0 , (2.31)which corresponds to the first gravity-capillary resonance κ c = σ − σ without vorticity, LS equation for gravity-capillary waves with shear ρ + ρ µXσ − µ (1 + X ) σ (1 + κ ) = 0 , (2.32)which is rewritten as follows c g + gµω X κ X c g − g µσω κ X = 0In the absence of vorticity, the latter condition reduces to c g = gh which matches thelong wave - short wave resonance as shown by Davey & Stewartson (1974) and Djordjevic& Redekopp (1977). In the presence of vorticity and for pure gravity waves the nonlinearcoefficient becomes singular if the following condition is satisfied { µσΩ (2 + X )(1 − σ ) µ + σ (1 + X ) } c g = gh Note that this condition reduces to c g = gh in the absence of vorticity.
3. Stability analysis and results
Let us write ζ in the form ζ = 12 ( (cid:15)ae i ( kx − ωt ) + c.c. ) + O ( (cid:15) )where a = 2 ζ is the envelope of the free surface elevation and c.c. denotes complexconjugation. Using (2.21) the NLS equation (2.24) is rewritten for the complex envelope a ( ξ, τ ) as follows ia τ + αa ξξ = ˜ γ | a | a, (3.1)where ˜ γ = g ω ( 1 + κ X ) γ The nonlinear coefficient ˜ γ can be written in a more compact form˜ γ = ω k σ γ In this section we consider the stability of a Stokes wave solution of the NLS equation(3.1) to infinitesimal disturbances.Equation (3.1) admits the following solution a s ( τ ) = a e − i ˜ γa τ , (3.2)with the initial condition a .We consider infinitesimal perturbations to this solution, in amplitude δ a ( ξ, τ ) and inphase δ w ( ξ, τ )), so that the perturbed solution a s writes a s = a s (1 + δ a ) e iδ w , (3.3)Substituting this expression in the NLS equation (3.1), linearising and separating betweenreal and imaginary parts, yields to a system of linear coupled partial differential equations H.-C. Hsu, C. Kharif, M. Abid and Y.-Y. Chen with constant coefficients. Then, this system admits solutions of the form δ a = δ a e i ( pξ − Γ τ ) ,δ w = δ w e i ( pξ − Γ τ ) , (3.4)The necessary and sufficient condition for the existence of non-trivial solutions is Γ = αp (2˜ γa + αp ) , (3.5)The Stokes wave solution is stable when α (2˜ γa + αp ) (cid:62) α (2˜ γa + αp ) < Γ i = p ( − γαa − α p ) / We set α = ωα /k and ˜ γ = ωk ˜ γ , so that α and ˜ γ are dimensionless functions of µ = kh , X = σΩ/ω and κ only. The growth rate of instability becomes Γ i = ω pk ( − γ α a k − α p ) / (3.6)The maximal growth rate is obtained for p = p − ˜ γ /α a k and its expression is Γ imax = p − ˜ γ /α √− ˜ γ α ω ( a k ) . Note that instability occurs when ˜ γ and α have oppositesign.The growth rate of instability is written in the following dimensionless form Γ i ωa k = ˜ p ( − γ α − α ˜ p ) / (3.7)where ˜ p = p/ ( a k )The dimensionless bandwidth of instability is ∆ ˜ p = p − γ /α and ∆p/k = p − γ /α a k .For κ = 0 and Ω = 0, equation (3.7) gives the rate of growth of Thomas et al. (2012).In figure 2 is plotted the dimensionless maximal growth rate of modulational instabilityof pure gravity waves and gravity waves influenced by surface tension effect ( κ = 0 . Ω for infinite and finite depths. We can observe that combined effect ofsurface tension and vorticity increases significantly the rate of growth of the modulationalinstability of short gravity waves propagating in finite depth and in the presence ofnegative vorticity ( Ω >
0) whereas the effect is insignificant in deep water. For positivevorticity (
Ω <
0) the curves almost coincide in finite depth and deep water as well andthe increase of the rate of growth due to surface tension is of order of κ .For Ω = 0 and κ = 0, equation (2.20) of Djordjevic & Redekopp (1977) becomes ia τ − ω k − κ − κ (1 + κ ) a ξξ = k ω
16 8 + κ + 2 κ (1 − κ )(1 + κ ) | a | a for the envelope of the surface elevation in deep water.The coefficients ˜ γ and α corresponding to this NLS equation are˜ γ = 116 8 + κ + 2 κ (1 − κ )(1 + κ ) , α = −
18 1 − κ − κ (1 + κ ) Consequently, the rate of growth of modulational instability of pure capillary wave trainson infinite depth, obtained for κ → ∞ , is Γ i → ω k (3 a k p − p ) / as κ → ∞ LS equation for gravity-capillary waves with shear Figure 2.
Dimensionless maximal growth rate of modulational instability as a function of Ω infinite depth ( µ = 2) and deep water ( µ = ∞ ). Solid line ( κ = 0 . , µ = 2); Dot-dashed line( κ = 0 . , µ = ∞ ) ; Dotted line ( κ = 0 , µ = 2); Dashed line ( κ = 0 , µ = ∞ ) which can be found in Chen & Saffman (1985). The wavenumber of the fastest-growingmodulational instability is p max = a k / √ ω ( a k ) / κ → ∞ and µ → ∞ )in the presence of vorticity ( Ω = 0). The corresponding analytic expressions of ˜ γ and α are ˜ γ = − X + 23 X + 11 X − X X + 1)(3 X + 2) (3.8) α = 3( X + 1)( X + X + 1)(2 + X ) (3.9)where X = Ω/ω and ω = − Ω/ ± p ( Ω/ + k T /ρ w .Due to high wave frequency of capillaries on deep water we assume | X |(cid:28)
1. Thecoefficients ˜ γ and α becomes˜ γ = −
116 (1 + 136 X ) + O ( X ) (3.10) α = 38 (1 + X O ( X ) (3.11)The rate of growth of modulational instability of capillary waves on deep water in thepresence of vorticity is Γ i = ω p k q a k − p + (8 a k − p ) X + O ( X ) (3.12)and in dimensionless form Γ i ω a k = ˜ p p − p + (8 − p ) X + O ( X ) (3.13)The maximal growth rate of instability is obtained for p = (1 + 5 X/ a k / √ O ( X )and its value is (1+13 X/ ωa k / O ( X ). The bandwidth of modulational instability0 H.-C. Hsu, C. Kharif, M. Abid and Y.-Y. Chen
Figure 3.
Dimensionless growth rate of modulational instability of pure capillary waves in finitedepth ( µ = 2) as a function of the dimensionless wavenumber of the perturbation for severalvalues of Ω . Ω = 0 (solid line); Ω = 2 (dashed line); Ω = − is ∆p = (1 + 5 X/ a k / √ X >
0) than for positive vorticity (
X < Ω . The rate of growth of instability increases as Ω increases as in infinitedepth.The sign of the product α ˜ γ determines the stability of the solution under infinitesimalperturbations. If the product is positive then the solutions are modulationally stable,otherwise they are modulationally unstable and grow exponentially with time. Davey &Stewartson (1974) and Djordjevic & Redekopp (1977) showed that this criterion whichworks for 1 D propagation can be extended to the case of 2 D propagation. In this way, ourstability diagrams could be compared to those of Djordjevic & Redekopp (1977) when Ω = 0. The linear stability analysis only captures the linear part of the instability, andthus its onset. We plot in the ( µ = kh, κ )-plane, for fixed values of the vorticity Ω , theunstable and stable regions. As a check, the instability diagrams we obtain are comparedin Figs. 4 and 5 with the same diagrams obtained by Thomas et al. (2012) for κ = 0 andDjordjevic & Redekopp (1977) for Ω = 0. In that way, we can verify that these limitingcases are reproduced correctly. Following Djordjevic & Redekopp (1977), the boundariesof the unstable regions have been numbered from 1 to 5. Curve 1 crosses the µ -axis atthe point corresponding to restabilisation of the modulational instability. Note that thisfeature holds for two-dimensional water waves. Curve 2 corresponds to vanishing of thedispersive coefficient α and minimum phase velocity ( c g = c p ) whereas along curves 3and 4 the nonlinear coefficient ˜ γ is singular. These singularities define Wilton and longwave/short wave resonances, respectively. Curves 1 and 5 correspond to simple zeros ofthe nonlinear coefficient ˜ γ .Curve 4 has the following asymptote µ = (1 + Ω − r Ω Ω ))( 94 κ −
34 ) , µ (cid:29) , LS equation for gravity-capillary waves with shear Figure 4. ( µ, X )-instability diagram for gravity waves, matching the results of Thomas et al. (2012) (dashed lines). Here, there is no surface tension. The unstable regions are in gray whereasstable regions are in white. For X = 0 (or Ω = 0) the value kh ≈ .
363 is found, below whichthere is no instability.
Figure 5. ( µ, κ )-instability diagram for gravity-capillary waves, matching the results fromDjordjevic & Redekopp (1977) (dashed lines). Here, there is no vorticity. The unstable regionsare in gray whereas stable regions are in white. H.-C. Hsu, C. Kharif, M. Abid and Y.-Y. Chen
Figure 6. ( µ, κ )-instability diagram for Ω = − . Ω = 0. Figure 7. ( µ, κ )-instability diagram for Ω = 0 . Ω = 0. whereas curve 5 has the asymptote µ = 94 (1 + Ω − r Ω Ω ) ) κ + 14 ( −
35 + 3 Ω + 29 Ω √ Ω − Ω √ Ω ) µ (cid:29) , For Ω = 0, the equations of Djordjevic & Redekopp (1977) are redicovered except thatinstead of − / − / κ (cid:29)
1) aremodulationally stable. This feature was emphasized by Djordjevic & Redekopp (1977) inthe absence of vorticity.In figures 6 to 13 the effect of positive and negative vorticity on ( µ = kh, κ ) diagramsis investigated. The curves of Djordjevic & Redekopp (1977) have been plotted to showthe effect of the vorticity. As it can be observed, the vorticty has a significant effect onstability diagrams of gravity-capillary. Very recently, this feature was emphasized by Hur(2017) who proposed a shallow water wave model with constant vorticity and surfacetension, too. Although interesting this model suffers from shortcomings: (i) dispersionis introduced heuristically and is fully linear (ii) nonlinear terms due to surface tensioneffect are ignored (iii) the coupling between nonlinearity and dispersion is not taken intoaccount.As positive vorticity ( Ω <
0) increases, we observe in figures 6, 8, 10 and 12 along the µ -axis in the vicinity of κ = 0 an increase of the region where the Stokes gravity-capillarywave train is modulationally stable. Consequently, gravity waves influenced by surfacetension behave as pure gravity waves (see Thomas et al. (2012)). Nevertheless, a verythin tongue of instability persists, near κ = 0, in the shallow water regime.As the intensity of negative vorticity ( Ω >
0) increases the band of instability along the µ -axis that corresponds to small values of κ becomes narrower, as shown in figures 7, 9,11 and 13. Contrary to the case of positive vorticity, the region of restabilisation alongthe µ -axis does not increase in the vicinity of κ = 0. LS equation for gravity-capillary waves with shear Figure 8.
Same as Fig. 6 for Ω = − Figure 9.
Same as Fig. 7 for Ω = 1. Figure 10.
Same as Fig. 6 for Ω = − . Figure 11.
Same as Fig. 7 for Ω = 1 .
4. Conclusion
A nonlinear Schrödinger equation for capillary-gravity waves in finite depth with alinear shear current has been derived which extends the work of Thomas et al. (2012). Thecombined effect of vorticity and surface tension on modulational instability properties ofweakly nonlinear gravity-capillary and capillary wave trains has been investigated. Theexplicit expressions of the dispersive and nonlinear coefficients are given as a functionof the frequency and wavenumber of the carrier wave, the vorticity, the surface tensionand the depth. The linear stability to modulational perturbations of the Stokes wavesolution of the NLS equation has been carried out. Two kinds of waves have beenespecially investigated that concerns short gravity waves influenced by surface tensionand pure capillary waves. In both cases, vorticity effect is to modify the rate of growthof modulational instability and instability bandwidth. Furthermore, it is shown thatvorticity effect modifies significantly the stability diagrams of the gravity-capillary waves.4
H.-C. Hsu, C. Kharif, M. Abid and Y.-Y. Chen
Figure 12.
Same as Fig. 6 for Ω = − Figure 13.
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Wave Motion8