The effect of a vertically sheared current on rogue properties.

C. Kharif1, S. Viroulet1,∗ and R. Thomas1 1 Aix- Univ., IRPHE CNRS: UMR 7342 - Aix-Marseille Universit - AMU - Ecole Centrale Marseille. (Dated: January 16, 2013) Rogue are unexpected large and spontaneous waves that can occur on the surface of calm and during severe storm. In this study a nonlinear Schrodinger equation governing the complex envelope of a surface water wave train propagating on an uniform vertical shear current is used (Thomas R., Kharif C. & Manna M., 2012). As a result it is shown that the vorticity modifies significantly the properties of rogue wave, namely their lifetime and height. In the presence of a shear current co-flowing with the waves it is shown that the height of the rogue waves is increased whereas the lifetime is decreased. Just the opposite, for a counter-flowing current, the height and lifetime are reduced and increased, respectively.

I. INTRODUCTION nonlinear to three dimensional per- turbations have been studied by Oikawa et al. [4]. Their analysis were illustrated for the case Rogue waves are among the wave naturally of a linear shear. The Benjamin-Feir instability observed by people on the sea surface that repre- of a wave train propagating on positive and neg- sent an inseparable feature of the . Rogue ative shear currents have been studied by Choi waves appear from nowhere, cause danger, and [1]. For a fixed steepness, he found that the en- disappear at once. They may occur on the sur- velope of the modulated wave train grows faster face of a relatively calm sea and not reach very in a positive shear current and slower in a neg- high amplitudes, but still be fatal for and ative one. Thomas et al. [7] using the method crew due to their unexpectedness and abnormal of multiple scales derived a vor-NLS equation in features. A wave is considered as a rogue wave if finite depth when the vorticity is taken into ac- its height Hr is more than twice the significant count. They carried out a stability analysis of height Hs. For a Gaussian sea and a narrow a weakly nonlinear wave train in the presence band spectrum, the significant height Hs = 4σ, of uniform vorticity. They demonstrated that where σ is the standard deviation of the ele- vorticity modifies significantly the modulational vation. Rogue wave can be generated by dif- instability properties of weakly nonlinear plane ferent mechanisms such as wave-current interac- waves, namely the growth rate and nabwidth. tion, geometrical or dispersive focusing, modu- They shown that these plane wave solutions may lation instability (the Benjamin-Feir instability), be linearly stable to in- collison, crossing , etc. In this study dependently of the dimensionless parameter kh. we consider rogue waves due to modulational in- Using the Benjamin-Feir Index (BFI) concept, stability. Several studies have been carried out they demonstrated that the number of rogue on the propagation of surface water waves propa- waves increases in the presence of a shear cur- gating steadily on a rotational current ([2, 5, 6]). rent co-flowing with the wave whereas it is the Few papers have been published on the effect opposite for a counter-flowing current. We con- of a vertical shear current on the Benjamin-Feir sider the vor-NLS equation derived by Thomas instability of a Stokes’ wave train in the pres- et al. [7] to investigate the properties of rogue ence of uniform vorticity. Johnson [3] studied waves in the presence of vertical uniform shear the slow modulation of a harmonic wave on a current, namely their lifetime and amplification. two dimensional flow of arbitrary vorticity. Us- ing the method of multiple scale he obtained the condition of linear stability for a plane nonlin- II. THE VOR-NLS EQUATION ear wave. This condition is verified if the prod- uct of the dispersive and nonlinear terms of the nonlinear Schrodinger equation (NLS equation) Generally in coastal and ocean waters, the ve- is negatif. The instability properties of weakly locity profiles are varying with depth due to the bottom friction and surface stress. For ex- ample, currents may have an important ef- ∗ Electronic address: [email protected] fect on the propagation of waves and wave pack- 2 ets. Surface drift of the water induce by wind can also affect their propagation due to the ve- +∞ locity in the surface layer. The velocity field can φ = ǫjφ , (4) be decomposed into a rotational term and an ir- n X nj rotational term which correspond to the induced j=n wave velocity (the wave motion is assume to be +∞ η = ǫjη , (5) potential). The fluid is inviscid and incompress- n X nj ible. Hence, the Kelvin-Lagrange theorem states j=n that the vorticity is conserved. with ǫ the small parameter corresponding to the wave steepness of the carrier. V =Ωyi + ∇φ(x,y,t) (1) We seek a solution modulated on a slow time scale τ = ǫ2t and slow space scale ξ = ǫ(x − c t) The wave train move along the x axis, the y g axis is oriented upward and gravity downward. 1 2 The water depth h is constant and the bottom is η(x,t)= (ǫa(ξ,τ)exp[i(kx − ωt)] + c.c)+ O(ǫ ) 2 located at y = −h. Ω represents the magnitude (6) of the shear and φ the velocity potential due to With cg the of the carrier the wave. wave. The new system of governing equations becomes : A. Governing equations 2 ǫ φξξ + φyy = 0, −h

2 iaτ + Laξξ + N|a| a = 0 (7) B. The multiple scale analysis where We present briefly the derivation of the NLS equation by using the method of multiple scales ω (for more details see Thomas & al. 2012 [7]). L = µ(1 − σ2)[1 − µσ + (1 − ρ)X] − σρ2 k2σ(2 + X) 2 +∞ −ωk (U + VW ) M = 4 φ = φ exp[in(kx − ωt)], (2) 8(1 + X)(2 + X)σ X n n=−∞ U = 9 − 12σ2 + 13σ4 − 2σ6 + (27 − 18σ2 +∞ + 15σ4)X + (33 − 3σ2 + 4σ4)X2 + (21 + 5σ2)X3 η = η exp[in(kx − ωt)], (3) X n 2 4 5 n=−∞ + (7 + 2σ )X + X V = (1+ X)2(1 + ρ + µΩ)+1+ X − ρσ2 − µσX where k is the wavenumber of the carrier and (1 + X)(2 + X)+ ρ(1 − σ2) ω its frequency. Then φn and ηn are written in W = 2σ3 perturbation series σρ(ρ + µΩ) − µ(1 + X) 3

IV. APPLICATION TO ROGUE WAVES

In the previous section a development to third order and a stability analysis have been performed to investigate the influence of the vor- ticity on the Benjamin-Feir instability of Stoke’s waves (see [7] for more details). In this section we consider the nonlinear evolution of the unsta- ble infinitesimal perturbations within the frame- work of the vor-NLS equation. A series of nu- merical simulations of the vor-NLS equation is FIG. 1: Stability diagram. S: stable, U: unstable. performed for different values of the wave steep- (Thomas et al. (2012) [7]) ness of the carrier wave, the water depth and the vorticity. The nonlinear stability analysis is de- veloped for constant vorticity varying from −0.4 to 0.4 and for kh = ∞ or kh = 2. Due to the lim- with itation of the NLS equation to weakly nonlinear wave trains we choose ak = 0.05 and ak = 0.10. µ = kh The wave number of the carrier wave is k = 10 σ = tanh(kh) and that of the perturbation is l = 1. c ρ = g cp Ω 3.5 Ω = ω 3 X = σΩ

2.5 0 /A max A III. STABILITY ANALYSIS 2

A solution of equation (7) can be expressed 1.5 using a Stoke wave : 1 0 20 40 60 80 100 120 140 160 180 200 t (a) a = a0exp(iNa0τ) (8)

2.8

A linear stability analysis gives the stability 2.6 criterion for a Stoke’s wave : 2.4

2.2

0 2

2 2 /A

L(−2Na0 + l L) ≥ 0 stable (9) max

A 1.8 2 2 L(−2Na0 + l L) < 0 unstable (10) 1.6

1.4

The stability domain is plotted in figure 1 as 1.2 a function of σΩ and kh. 1 0 50 100 150 200 250 300 There are two critical values as a function of t the vorticity and the water depth. Indeed for (b) a value of σΩ < −2/3 which correspond to a kg FIG. 2: Evolution of the normalized maximum ampli- vorticity Ω = −2q 3 , Stoke’s waves are stable. tude as a function of time. In infinite (a) and finite When there is no vorticity (Ω = 0) Stoke’s waves (b) depth. Thick continuous line corresponds to 0 are stable for kh < 1.363. Note that for three di- vorticity, thin continuous one to 0.2 and thin dashed mensional motion there are oblique modulations line to −0.2. even when kh < 1.363. 4

The normalized maximum amplitude evolu- Figure 3 shows the evolution of the lifetime tion as a function of time is shown in figure 2 and normalized maximum amplitude of the en- for three different values of the vorticity (−0.2, velope as a function of the vorticity. In infinite 0 and 0.2). In both, finite and infinite depth, (fig.3.a) and finite depth (fig.3.b) increasing the increasing the vorticity increases the maximum vorticity, increases the maximum amplitude but amplitude of the perturbation but decreases the decreases the lifetime of the rogue wave. In fig- width of the peak. To have a rogue wave event, ure 3.b the Stoke’s wave is stable for values of the lifetime has been evaluated for a amplitude vorticity lower than −0.2. This explain why the higher than two times the initial amplitude of lifetime and maximum amplitude for negative the Stoke’s wave. values of vorticity are lower than those without vorticity, the instability is just high enough to l=1 k=10 ak=0.1 kh=inf be defined as a rogue wave. 2.5 1.3 Lifetime Maximum amplitude 2 1.2

1.5 1.1

Lifetime 1 1 Maximum amplitude 0.5 0.9

0 0.8 −0.4 −0.2 0 0.2 0.4 Vorticity (a)

l=1 k=10 ak=0.1 kh=2 2.5 1.3 Lifetime Maximum amplitude V. CONCLUSION 2 1.2

1.5 1.1

Lifetime 1 1 Using a 1D nonlinear Schrodinger equation in the presence of a vertical shear current of Maximum amplitude 0.5 0.9 non zero constant vorticity in finite and infinite depth, we have shown that the lifetime and max-

0 0.8 imum amplitude of rogue waves are significantly −0.2 −0.1 0 0.1 0.2 0.3 0.4 Vorticity influenced by a vertical shear current. For a (b) counter-flowing current the lifetime of a rogue is increased whereas the height is decreased. The FIG. 3: Evolution of the life time and the maximum opposite situation is observed for a co-flowing amplitude as a function a vorticity in infinite (a) and current. This study is still in progress, to take finite (b) depth. into account the influence of wind by introducing the Miles’ mechanism.

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