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Wind effect on the evolution of two obliquely interacting random wave trains in deep water
Wave Motion 89 (2019) 14–27
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Wind effect on the evolution of two obliquely interacting random wave trains in deep water ∗
Sumana Kundu a , Suma Debsarma b , , K.P. Das b a b
Salkia Mrigendra Dutta Smriti Balika Vidyapith (High), Salkia, Howrah 711106, India Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata 700009, India
highlights • • • • •
Spectral transport equations are derived for a pair of random surface wavetrains. In the model the presence of uniform wind flow is taken into account. Stability analysis is performed for a pair of Gaussian wave spectra. The effect of randomness is to reduce the growth rate of instability slightly. Growth rate of instability is much higher than for a single wavetrain.
article
info
Article history: Received 31 July 2018 Received in revised form 27 February 2019 Accepted 28 February 2019 Available online 8 March 2019 Keywords: Crossing seas Instability Randomness Spectral transport equation Wind effect
a b s t r a c t A pair of coupled nonlinear evolution equations for the spectral functions are derived in a situation of crossing sea states characterized by two narrowband Gaussian random surface wave systems in the presence of uniform wind flow. These two equations are employed to perform stability analysis of two initially homogeneous wave spectra subject to infinitesimal perturbations. It is found that the results of the stability analysis remain qualitatively similar with the corresponding deterministic situation. The notable difference is that the growth rate of instability reduces slightly due to the effect of randomness. But, it is much higher than the growth rate of a single wave system. As the spectral bandwidth increases the growth rate of instability decreases. © 2019 Published by Elsevier B.V.
1. Introduction In the analysis of ship accidents due to bad weather conditions Toffoli et al. [1] observed that a large number of ship accidents occurred in a situation of crossing seas. So, they pointed out that crossing seas should be considered as possibly dangerous situation. Recently, there has been increased interest in the evolution of nonlinear surface water waves in crossing seas characterized by two obliquely interacting wave systems. Some of the investigations (Onorato et al. [2], Shukla et al. [3], Laine-Pearson [4], Gramstad and Trulsen [4], Brunetti and Kasparian [5] and many others) are from the deterministic point of view with emphasis on the stability properties of two obliquely interacting uniform wave systems. It is found that the growth rate of instability of two obliquely interacting uniform wave systems is much higher than that for a single wave system (see [2–4,6]). ∗ Corresponding author. E-mail addresses: [email protected] (S. Kundu), [email protected] (S. Debsarma), [email protected] (K.P. Das). https://doi.org/10.1016/j.wavemoti.2019.02.008 0165-2125/© 2019 Published by Elsevier B.V.
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
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Some other investigations are based upon either numerical simulations or laboratory experiments. By performing numerical simulations Ruban [7] showed how the formation of freak waves depends on the relative positions of two spectral maxima in the Fourier plane. To investigate the statistical properties of the surface wave envelope in a situation of crossing sea states Toffoli et al. [8] conducted an experiment in a very large wave basin by mechanically generating two wave systems which were forced to propagate along two different directions. They found that the number of extreme events depends upon the angle between the two wave systems. As a support of this experimental finding Toffoli et al. [8] carried out also a numerical simulation. The action of wind blowing over a water body is to produce variations in pressure and surface shear. Also, it can generate rotational current in the water. After the publications of Philips [9] and Miles [10] about the theory of surface wave generation by wind, many researchers (Bliven et al. [11], Waseda and Tulin [12], Peirson and Garcia [13], Janssen [14], Tian et al. [15], Galchenko et al. [16], Brunetti et al. [17], Chabchoub et al. [18], Buckley and Veron [19] and many others) have made notable contributions towards the understanding of wind–wave interactions, generation of waves by wind, wind effect on modulational instability, wave growth by wind etc. Kharif et al. [20] considered the direct effect of strong wind on the extreme wave events. To observe the influence of wind on extreme wave events due to spatio-temporal focusing Kharif et al. [20] conducted experiments in a large wind–wave tank. They also carried out numerical simulations for extreme wave events caused by modulational instability. They found that the height and duration of the extreme waves get increased in the presence of wind. To study the statistical properties of wind-generated waves Toffoli et al. [21] conducted a wind-sea experiment in an annular flume, over which a constant and quasihomogeneous wind flows. This is the first study showing emergence of rogue waves in wind-generated wave fields. White and Fornberg [22] considered a regular ocean swell that traverses a region of deep water with random current fluctuations. With the assumption that the current can focus wave action into a caustic region just like an optical lens, they showed the formation of freak waves at random locations. They verified the theory with Monte Carlo simulations. Toffoli et al. [23] carried out two experiments in two different laboratories and observed that a stable wave propagating into a region characterized by an opposite current may become modulationally unstable and result in formation of a rogue wave. To investigate the role of opposing currents in changing the statistical properties of unidirectional and directional mechanically generated random wave fields Toffoli et al. [24] performed three independent laboratory experiments. The results demonstrate the amplification of modulational instability in weakly unstable regular wave packets due to interaction with an opposing current. This amplification depends on the ratio of current speed to group velocity and non-uniformity of the current. It is observed that opposing currents induce a sharp and rapid transition from weakly to strongly non-Gaussian properties which is associated with an increase in the probability of occurrence of extreme waves. As the wind flows over the water surface it produces randomness at the water surface. One way of studying the effect of randomness on the stability properties of surface wave trains is through the application of a transport equation for the ensemble averaged two point space correlation function with the assumption that the complex wave amplitude is random function of space variables. When Fourier transform of this correlation equation is taken with respect to spatial separation variables another equation is obtained which govern the evolution of the wave power spectral density function. This spectral equation is useful for studying stability of an initially homogeneous wave spectrum to small wave number perturbations. This technique is used by Alber [25], Crawford et al. [26] in the field of nonlinear water waves. Similar technique was used by Wigner [27], Leaf [28], Hasegawa [29] in the fields of plasma physics and quantum mechanics. Toffoli et al. [30] studied the effect of the second order interaction in directionally spread sea states on the statistical properties of surface gravity waves. Observation of Onorato et al. [31] is that freak waves may appear with a higher probability if the angle of interaction between two non-collinear wave packets is between 10◦ and 30◦ . In support of this they presented numerical simulations of two long crested wave systems characterized each by a JONSWAP spectrum with random phases. In the research article of Bitner-Gregersen and Toffoli [32] we find a discussion on some rogue-waveprone crossing seas and their probabilities of occurrence in the ocean. Their analysis is supported also by Monte Carlo simulations. One work earlier in the literature (Masson [33]) studied the effect of nonlinear interactions between swell and wind sea on a stationary wave spectrum. Masson [33] showed that as a result of nonlinear coupling swell decays at a rate that decreases rapidly as the swell frequency moves away from the peak frequency of the short waves. In the frequency range just below the peak frequency of the short waves swell grows at the expense of the local sea. In this investigation we obtain a pair of spectral transport equations in a situation of crossing sea states characterized by two obliquely interacting initially homogeneous random wave systems in presence of uniform wind flow. Using these two equations we carry out stability analysis of a pair of obliquely interacting Gaussian random wave systems. We observe that the effect of randomness is to reduce the growth rate of instability slightly. But it is still higher than that for a single wave system. It is observed that the growth rate of instability increases with the increase in the wind flow velocity, which is similar to the deterministic case. It is also found that the growth rate of instability decreases with the increase in spectral bandwidth. It is observed that the growth rate of instability of one wave system increases as the mean square wave steepness of the second wave system increases. The paper is organized as follows: In Section 2 we derive spectral transport equations for a pair of obliquely interacting random wavetrains; in Section 3 we present the stability analysis; in Section 4 we report a summary of the important findings of stability analysis.
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2. Spectral transport equations We consider a crossing sea states situation in the presence of uniform wind flow. Such a situation can occur in reality when the wave configuration is composed by two obliquely interacting swells and wind starts blowing uniformly. A third wave system (wind-generated waves) may appear in the direction of the wind. For the sake of simplicity we do not take into account the third wave system in the present model. Senapati et al. [34] considered a two fluid domain, namely air–water model both extending up to infinity. In the atmospheric medium wind flows with uniform velocity V along x-direction. In the water medium two surface gravity wave packets interact obliquely. This model is an approximation of reality. The nonlinear evolution equations obtained by Senapati et al. [34] in a situation of crossing sea states in the presence of uniform wind flow are the following i
∂ζ1 ∂ζ1 ∂ζ1 ∂ 2 ζ1 ∂ 2 ζ1 ∂ 2 ζ1 + iβ1 + iβ2 + β3 2 + β4 + β5 2 = λ1 ζ12 ζ1∗ + µ1 ζ1 ζ2 ζ2∗ ∂τ ∂ x1 ∂ y1 ∂ x1 ∂ y1 ∂ x1 ∂ y1
(2.1)
i
∂ζ2 ∂ζ2 ∂ζ2 ∂ 2 ζ2 ∂ 2 ζ2 ∂ 2 ζ2 + iβ1 − iβ2 + β3 2 − β4 + β5 2 = λ1 ζ22 ζ2∗ + µ1 ζ2 ζ1 ζ1∗ ∂τ ∂ x1 ∂ y1 ∂ x1 ∂ y1 ∂ x1 ∂ y1
(2.2)
In Eqs. (2.1) and (2.2), ζ1 , ζ2 are the complex wave envelopes of two wave systems having carrier wave numbers (k, l) and (k, −l) respectively. x-axis is taken along the direction of wind flow and the two wave systems are propagating making equal angle θ with the direction of wind flow. x1 , y1 and τ are the slow spatio-temporal variables defined by x1 = ϵ x, y1 = ϵ y, τ = ϵ t, where ϵ is a small ordering parameter. The coefficients βi ’s, λ1 and µ1 are given in the Appendix. It is found in Senapati et al. [34] that there are two possible modes of wave propagation, namely, positive and negative modes according as the wave frequency is ω = ω+ or ω = ω− , where ω± are given by
ω± =
rVk ±
√
(1 − r 2 ) − rk2 V 2
(2.3)
1+r
Here V is the uniform velocity of wind blowing parallel to x1 direction. Eqs. (2.1)–(2.3) are in dimensionless form with the dimensionless variables defined below √ k l x∗1 = kc x1 , y∗1 = kc y1 , t1∗ = t1 gkc , k∗ = , l∗ = , kc kc
√ ω∗ = ω/ gkc ,
V∗ = V
√
kc /g ,
ζ1∗ = kc ζ1 ,
ζ2∗ = kc ζ2 ,
kc =
√
k2 + l2
For convenience the asterisks have been dropped out. The evolution equations (2.1)–(2.2) are correct up to third order in the wave steepness and they remain valid if the wind flow velocity V is less than a critical velocity Vc given by Vc =
1 k
√
1 − r2 r
(2.4)
Here r(< 1) is the density ratio between air and water. The coefficients βi ’s, λ1 and µ1 appearing in Eqs. (2.1) and (2.2) depend on the wind velocity V and density ratio r. Setting V = 0 and r = 0 one can recover the evolution equations studied by Onorato et al. [2]. We now assume that ζ1 (x1 , y1 , τ ) and ζ2 (x1 , y1 , τ ) are random functions of ξ⃗1 = (x1 , y1 ) and τ . For the two waves undergoing weak nonlinear interactions we define two point space correlation functions as follows
ρ1 (ξ⃗1 , ξ⃗2 , τ ) = ⟨ζ1 (ξ⃗1 , τ )ζ1∗ (ξ⃗2 , τ )⟩ ρ2 (ξ⃗1 , ξ⃗2 , τ ) = ⟨ζ2 (ξ⃗1 , τ )ζ2∗ (ξ⃗2 , τ )⟩
(2.5)
where ξ⃗2 = (x2 , y2 ) is another space point, the superscript (∗) denotes the complex conjugate and the angle brackets denote an ensemble average. ρ1 and ρ2 are, in fact, second order statistical moments. To derive equations for the slow variation of ρ1 and ρ2 we consider equations (2.1)–(2.2) at the point ξ⃗2 also. We first multiply equation (2.1) at the point ξ⃗1 by ζ1∗ (ξ⃗2 , τ ). Then, we consider complex conjugate of Eq. (2.1) at the point ξ⃗2 and multiply it by ζ1 (ξ⃗1 , τ ). Subtracting the second equation from the first one and then taking ensemble average we get the following equation: i
( ∂ ( ∂ ( ∂2 ∂ρ1 ∂ ) ∂ ) ∂2 ) + iβ1 + ρ1 + i β 2 + ρ1 + β3 − ρ1 ∂τ ∂ x1 ∂ x2 ∂ y1 ∂ y2 ∂ x22 ∂ x22 ( ∂2 ( ∂2 ∂2 ) ∂2 ) + β4 − ρ1 + β5 − ρ1 ∂ x1 ∂ y1 ∂ x2 ∂ y2 ∂ y21 ∂ y22 = λ1 ⟨ζ12 (ξ⃗1 )ζ1∗ (ξ⃗1 )ζ1∗ (ξ⃗2 )⟩ − λ1 ⟨ζ1∗ 2 (ξ⃗2 )ζ1 (ξ⃗2 )ζ1 (ξ⃗1 )⟩ + µ1 ⟨ζ1 (ξ⃗1 )ζ2 (ξ⃗1 )ζ2∗ (ξ⃗1 )ζ1∗ (ξ⃗2 )⟩ − µ1 ⟨ζ1∗ (ξ⃗2 )ζ2∗ (ξ⃗2 )ζ2 (ξ⃗2 )ζ1 (ξ⃗1 )⟩
(2.6)
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In Eq. (2.6), ζ1 (ξ⃗1 , τ ), ζ1 (ξ⃗2 , τ ), ζ2 (ξ⃗1 , τ ) and ζ2 (ξ⃗2 , τ ) are written simply ζ1 (ξ⃗1 ), ζ1 (ξ⃗2 ), ζ2 (ξ⃗1 ) and ζ2 (ξ⃗2 ) respectively for the sake of simplicity of notation. We now introduce the average coordinates 1
1 (x1 + x2 ), Y = (y1 + y2 ) 2 2 and the spatial separation coordinates X =
r x = x1 − x2 ,
(2.7)
ry = y1 − y2
(2.8)
In terms of the newly introduced variables the left hand side of Eq. (2.6) becomes i
( ∂2 ∂ρ1 ∂ρ1 ∂ 2 ρ1 ∂ 2 ρ1 ∂ρ1 ∂2 ) + i β1 + iβ2 + 2β3 + β4 + ρ1 + 2β5 ∂τ ∂X ∂Y ∂ X ∂ rx ∂ X ∂ ry ∂ Y ∂ rx ∂ Y ∂ ry
(2.9)
⃗ τ ) and ζ2 (ξ, ⃗ τ ), where ξ⃗ = (X , Y ) follow initially Gaussian random process. We assume also that the We assume that ζ1 (ξ, evolving random statistical amplitude fields retain the same Gaussian statistical properties [Benney and Saffman [35]]. Using the property of Gaussian statistics, the fourth order correlation terms on the right hand side of Eq. (2.6) can be written down in terms of the products of pairs of second order correlations as follows ⟨ζ1 (ξ⃗1 )ζ1∗ (ξ⃗2 )ζ1 (ξ⃗1 )ζ1∗ (ξ⃗1 )⟩ = 2⟨ζ1 (ξ⃗1 )ζ1∗ (ξ⃗2 )⟩⟨ζ1 (ξ⃗1 )ζ1∗ (ξ⃗1 )⟩ = 2ρ1 a¯ 21 (ξ⃗1 )
(2.10)
where ¯ ξ⃗ = ⟨ζ1 (ξ⃗1 )ζ1∗ (ξ⃗1 )⟩ is the ensemble averaged mean square amplitude of the first wave system. For the second wave system it is a¯ 22 (ξ⃗1 ) = ⟨ζ2 (ξ⃗1 )ζ2∗ (ξ⃗1 )⟩. In terms of the coordinates defined in (2.7) and (2.8), ξ⃗1 and ξ⃗2 can be put as follows a21 ( 1 )
1
ξ⃗1 = (x1 , y1 ) = ξ⃗ + ⃗r , 2
1
ξ⃗2 = (x2 , y2 ) = ξ⃗ − ⃗r 2
(2.11)
where ξ⃗ = (X , Y ) and ⃗r = (rx , ry ). Considering Taylor series expansion of a¯ 21 (ξ⃗1 ) about the point ξ⃗ we get the following
(1 ∂ ) (1 ∂ 1 ∂ ) 2 a¯ 1 (ξ⃗) = 2ρ1 exp ⃗r . + ry a¯ 2 (ξ⃗) ⟨ζ1 (ξ⃗1 )ζ1∗ (ξ⃗2 )ζ1 (ξ⃗1 )ζ1∗ (ξ⃗1 )⟩ = 2ρ1 exp rx 2 ∂X 2 ∂Y 2 ∂ ξ⃗ 1
(2.12)
Treating the remaining three terms on the right hand side of Eq. (2.6) we can rewrite Eq. (2.6) as follows i
( ∂2 ∂ρ1 ∂ρ1 ∂ρ1 ∂ 2 ρ1 ∂ 2 ρ1 ∂2 ) + i β1 + iβ2 + 2β3 + β4 + ρ1 + 2β5 ∂τ ∂X ∂Y ∂ X ∂ rx ∂ X ∂ ry ∂ Y ∂ rx ∂ Y ∂ ry (1 ∂ ) (1 ∂ ) 2 2 = 4λ1 ρ1 sinh ⃗r . a¯ + 2µ1 ρ1 sinh ⃗r . a¯ 2 ∂ ξ⃗ 1 2 ∂ ξ⃗ 2
(2.13)
Eq. (2.13) is the evolution equation for the correlation function ρ1 . We now take Fourier transform of Eq. (2.13) with respect to spatial separation coordinates defined below F1 (P⃗ , ξ⃗ , τ ) =
F2 (P⃗ , ξ⃗ , τ ) =
∫ ∫
1 (2π )2
( ) 1 1 ⃗ ⃗ ⃗ ρ1 ξ + ⃗r , ξ − ⃗r , τ e−iP .⃗r d⃗r 2
−∞
∫ ∫
1
∞
(2π )2
∞
2
) ( 1 1 ⃗ ⃗ ⃗ ⃗ ⃗ ρ2 ξ + r , ξ − r , τ e−iP .⃗r d⃗r 2
−∞
2
(2.14)
where d⃗r = drx dry and P⃗ = (Px , Py ) is the Fourier wave number conjugate to the spatial separation coordinates ⃗r . F1 (P⃗ , ξ⃗ , τ ) and F2 (P⃗ , ξ⃗ , τ ) are the wave envelope power spectral density functions of the first and second wave system respectively. Clearly, Fourier inversions of Eqs. (2.14) are given by ∞
( ) ∫ ∫ 1 1 ρ1 ξ⃗ + ⃗r , ξ⃗ − ⃗r , τ = 2
2
⃗
∞
( ) ∫ ∫ 1 1 ρ2 ξ⃗ + ⃗r , ξ⃗ − ⃗r , τ = 2
2
∫ ∫
∞
F1 (P⃗ , ξ⃗ , τ )dP⃗ −∞
( ) ρ2 ξ⃗ , ξ⃗ , τ =
∫ ∫
∞
F2 (P⃗ , ξ⃗ , τ )dP⃗ −∞
F2 (P⃗ , ξ⃗ , τ )eiP .⃗r dP⃗ ⃗
−∞
⃗ in Eq. (2.15), we get If we set ⃗r = 0
( ) ρ1 ξ⃗ , ξ⃗ , τ =
F1 (P⃗ , ξ⃗ , τ )eiP .⃗r dP⃗
−∞
(2.15)
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S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
⃗ τ) = a¯ 21 (ξ,
∫ ∫
∞
F1 (P⃗ , ξ⃗ , τ )dP⃗ −∞
a¯ 22 (ξ⃗, τ ) =
∫ ∫
∞
F2 (P⃗ , ξ⃗ , τ )dP⃗
(2.16)
−∞
Thus, Fourier transform of Eq. (2.13) results in the following equation
∂ F1 ∂ F1 ∂ F1 ∂ F1 ∂ F1 ∂ F1 ∂ F1 + β1 + β2 + 2β3 Px + β4 P y + β4 Px + 2β5 Py ∂τ ∂X ∂Y ∂X ∂X ∂Y ∂Y = 4λ1 sin
( 1 ∂2 ) ( 1 ∂2 ) F1 a¯ 21 + 2µ1 sin F1 a¯ 22 2 ∂ ξ⃗ ∂ P⃗ 2 ∂ ξ⃗ ∂ P⃗
For the right hand side terms of Eq. (2.17) the operator sin
(2.17)
(1
∂2
2 ∂ ξ⃗ ∂ P⃗
)
stands for
( 1 ∂2 ( 1 ∂2 ) 1 ∂2 ) ≡ sin + 2 ∂ ξ⃗ ∂ P⃗ 2 ∂ X ∂ Px 2 ∂ Y ∂ Py
sin
where ∂∂X , ∂∂Y operate on a¯ 21 only and ∂∂P , ∂∂P operate on F1 only. Eq. (2.17) is the desired spectral transport equation for x y the first wave system. Next, starting with Eq. (2.2) and proceeding in a similar manner as described by Eqs. (2.5)–(2.17) we get the following spectral transport equation for the second wave system
∂ F2 ∂ F2 ∂ F2 ∂ F2 ∂ F2 ∂ F2 ∂ F2 + β1 − β2 + 2β3 Px − β4 P y − β4 Px + 2β5 Py ∂τ ∂X ∂Y ∂X ∂X ∂Y ∂Y ( 1 ∂2 ) ( 1 ∂2 ) 2 2 = 4λ1 sin F2 a¯ 2 + 2µ1 sin F2 a¯ 1 2 ∂ ξ⃗ ∂ P⃗ 2 ∂ ξ⃗ ∂ P⃗
(2.18)
Eqs. (2.17) and (2.18) form a coupled system of nonlinear spectral transport equations in a situation of crossing sea states. Retaining all the nonlinear terms in the sine-operator expansion on the right hand side of Eqs. (2.17) and (2.18) we have made stability analysis in the next section. 3. Stability analysis For a solution of the nonlinear spectral transport Eqs. (2.17) and (2.18), which is independent of ξ⃗ , the governing equations reduce to the following
∂ F2 =0 ∂τ
∂ F1 = 0, ∂τ
(3.1)
Solutions of Eqs. (3.1) are of the following form (0)
⃗, F1 = F1 (P)
(0)
⃗ F2 = F2 (P)
(3.2)
which can be considered as the random counterpart of uniform amplitude Stokes wave trains in the deterministic theory (0) ⃗ (0) ⃗ (0) ⃗ (0) ⃗ of crossing seas. We assume that F1 (P) and F2 (P) satisfy Gaussian properties. Thus F1 (P) and F2 (P) together represent a homogeneous crossing seas with Gaussian properties. To study the stability of homogeneous solution (3.2) we introduce the following infinitesimal perturbation (0)
(1)
(0)
(1)
⃗ + ϵ F (P⃗ , ξ⃗ , τ ) F1 = F1 (P) 1 ⃗ + ϵ F (P⃗ , ξ⃗ , τ ) F2 = F2 (P) 2
(3.3)
where ϵ is a small ordering parameter. Substituting (3.3) in (2.16) we get a¯ 21 (ξ⃗ , τ ) = a¯ 210 + ϵ a¯ 211 (ξ⃗ , τ ) a¯ 22 (ξ⃗ , τ ) = a¯ 220 + ϵ a¯ 221 (ξ⃗ , τ )
(3.4)
where a¯ 210 =
∫ ∫
∞ (0)
⃗ P⃗ , a¯ 220 = F1 (P)d −∞
∫ ∫
∞ (0)
⃗ P⃗ F2 (P)d −∞
(3.5)
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
a¯ 211 (ξ⃗ , τ ) =
∫ ∫
19
∞ (1)
F1 (P⃗ , ξ⃗ , τ )dP⃗ −∞
a¯ 221 (ξ⃗ , τ ) =
∫ ∫
∞ (1)
F2 (P⃗ , ξ⃗ , τ )dP⃗
(3.6)
−∞
Here a¯ 210 , a¯ 220 denote mean square wave steepness of the first and second wave systems respectively. Now, substituting Eqs. (3.3) and (3.4) in Eqs. (2.17) and (2.18) and then linearizing we get
∂ F1(1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) + β1 1 + β2 1 + 2β3 Px 1 + β4 Py 1 + β4 Px 1 + 2β5 Py 1 ∂τ ∂X ∂Y ∂X ∂X ∂Y ∂Y ( 1 ∂2 ) ( 1 ∂2 ) (0) (0) = 4λ1 sin F a¯ 2 + 2µ1 sin F a¯ 2 2 ∂ ξ⃗ ∂ P⃗ 1 11 2 ∂ ξ⃗ ∂ P⃗ 1 21
(3.7)
∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F2(1) + β1 2 − β2 2 + 2β3 Px 2 − β4 Py 2 − β4 Px 2 + 2β5 Py 2 ∂τ ∂X ∂Y ∂X ∂X ∂Y ∂Y ( 1 ∂2 ) ( 1 ∂2 ) (0) 2 (0) 2 = 4λ1 sin F a¯ + 2µ1 sin F a¯ 2 ∂ ξ⃗ ∂ P⃗ 2 21 2 ∂ ξ⃗ ∂ P⃗ 2 11 (1)
(3.8)
(1)
We now assume space–time dependence of F1 , F2 , a¯ 211 , a¯ 221 in the following form (1)
⃗ exp[i(Lx X + Ly Y − Ω τ )] F1 (P⃗ , ξ⃗ , τ ) = f1 (P) (1)
⃗ exp[i(Lx X + Ly Y − Ω τ )] F2 (P⃗ , ξ⃗ , τ ) = f2 (P) a¯ 211 (ξ⃗ , τ ) = α1 exp[i(Lx X + Ly Y − Ω τ )] a¯ 221 (ξ⃗ , τ ) = α2 exp[i(Lx X + Ly Y − Ω τ )]
(3.9)
where ⃗L = (Lx , Ly ) is perturbation wave number vector; Ω is the perturbed frequency; α1 and α2 are two constants. Substituting (3.9) in (3.7) and (3.8) and then equating coefficients of exp[i(Lx X + Ly Y − Ω τ )] we get
[ ⃗ ⃗ ] ⃗ ]f1 (P) ⃗ = (2λ1 α1 + µ1 α2 ) F (0) (P⃗ + L ) − F (0) (P⃗ − L ) [−Ω + G+ (P) 1 1
(3.10)
[ ⃗ ⃗ ] ⃗ ]f2 (P) ⃗ = (2λ1 α2 + µ1 α1 ) F (0) (P⃗ + L ) − F (0) (P⃗ − L ) [−Ω − G− (P) 2 2
(3.11)
2
2
2
2
where
⃗ = β1 Lx ± β2 Ly + 2β3 Px Lx ± β4 Py Lx ± β4 Px Ly + 2β5 Py Ly G± (P)
(3.12)
By (3.6) and (3.9) we get the following relationships ∞
∫ ∫
∫ ∫
⃗ P⃗ = α1 , f1 (P)d −∞
∞
⃗ P⃗ = α2 f2 (P)d
(3.13)
−∞
⃗ and f2 (P) ⃗ as given by Eqs. (3.10) and (3.11) respectively in Eq. (3.13) we get the following equations Inserting f1 (P) (2λ1 I1 + 1)α1 + µ1 I1 α2 = 0
µ1 I2 α1 + (2λ1 I2 + 1)α2 = 0
(3.14)
where
∫ ∫
∞
I1 = −∞
∫ ∫
∞
I2 = −∞
⃗
⃗
⃗
⃗
[F1(0) (P⃗ + 2L ) − F1(0) (P⃗ − 2L )] dP⃗ ⃗ Ω − G+ (P) [F2(0) (P⃗ + 2L ) − F2(0) (P⃗ − 2L )] dP⃗ ⃗ Ω − G− (P)
(3.15)
20
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
For non-trivial solution of the system of equations in (3.14), I1 and I2 must satisfy the following equation (4λ21 − µ21 )I1 I2 + 2λ1 (I1 + I2 ) + 1 = 0
(3.16)
The nonlinear integral equation (3.16) is the desired dispersion relation. (0) ⃗ (0) ⃗ We now assume two dimensional normal spectrum F1 (P) and F2 (P) as given below (0)
⃗ = F1 (P)
(0)
⃗ = F2 (P)
a¯ 210 2π σ 2
[ P2 + P2 ] x y 2σ 2
exp −
[ P2 + P2 ] x y 2σ 2
a¯ 220
exp −
(3.17) 2π σ 2 where σ is the spectral spread about the carrier wavenumber. σ specifies the degree of randomness. For convenience, we introduce the following two vectors K⃗1 = (2β3 Lx + β4 Ly )xˆ + (β4 Lx + 2β5 Ly )yˆ K⃗2 = (2β3 Lx − β4 Ly )xˆ + (−β4 Lx + 2β5 Ly )yˆ
(3.18)
⃗ can then be written as follows where xˆ and yˆ are unit vectors along x and y directions respectively. G± (P) ⃗ = β1 Lx + β2 Ly + P .K⃗1 = β1 Lx + β2 Ly + P1 |K⃗1 | G+ (P) ⃗ = β1 Lx − β2 Ly + P .K⃗2 = β1 Lx − β2 Ly + P2 |K⃗2 | G− (P)
(3.19)
where P1 and P2 are the components of P⃗ along K⃗1 and K⃗2 respectively given by (2β3 Lx + β4 Ly )Px + (β4 Lx + 2β5 Ly )Py P1 = 1 [(2β3 Lx + β4 Ly )2 + (β4 Lx + 2β5 Ly )2 ] 2 P2 =
(2β3 Lx − β4 Ly )Px + (−β4 Lx + 2β5 Ly )Py 1
[(2β3 Lx − β4 Ly )2 + (−β4 Lx + 2β5 Ly )2 ] 2
(3.20)
Now, components of P⃗ in the directions normal to K⃗1 and K⃗2 are given respectively by K⃗1 )
(
P1′ = P⃗ · zˆ ×
|K⃗1 | K⃗2 )
(
P2′ = P⃗ · zˆ ×
|K⃗2 |
=
=
1
|K⃗1 | 1
|K⃗2 |
[(2β3 Lx + β4 Ly )Py − (β4 Lx + 2β5 Ly )Px ]
[(2β3 Lx − β4 Ly )Py − (−β4 Lx − 2β5 Ly )Px ]
(3.21)
where zˆ unit vector along z-direction. Clearly, any double integration with respective to Px and Py is equivalent to integration with respect to P1 , P1′ and also with respective to P2 , P2′ . Now, P1′ appears solely in the numerator of I1 . ⃗
(0)
Therefore, the integration of F1 (P⃗ ± 2L ) over P1′ reduces the double integral to a single integral over P1 with the wave number component ν1 = (Lx , Ly ) ·
∫
∞
I1 = −∞
[F¯1(0) (P1 +
ν1 2
K⃗1
|K⃗1 |
as parameter. Thus,
(0)
) − F¯1 (P1 −
ν1 2
)]
Ω − (β1 Lx + β2 Ly + P1 |K⃗1 |)
dP1
(3.22)
where (0)
F¯1 (P1 ) =
∫
∞ (0)
⃗ 1′ F1 (P)dP
(3.23)
−∞
Similarly, I2 becomes
∫
∞
I2 = −∞
[F¯2(0) (P2 +
ν2 2
(0)
) − F¯2 (P2 −
ν2 2
)]
Ω − (β1 Lx − β2 Ly + P2 |K⃗2 |)
dP2
(3.24)
where
ν2 = (Lx , Ly ) ·
K⃗2
|K⃗2 |
, F¯2(0) (P2 ) =
∫
∞
−∞
(0)
⃗ 2′ F2 (P)dP
(3.25)
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27 ′
′
(0)
21
(0)
⃗ F (P) ⃗ as follows Since Px2 + Py2 = P12 + P12 = P22 + P22 we can write F1 (P), 2 exp −
2π σ 2
[ P 2 + P ′2 ] 2 2 2σ 2
a¯ 220
(0)
⃗ = F2 (P)
[ P 2 + P ′2 ] 1 1 2σ 2
a¯ 210
(0)
⃗ = F1 (P)
exp −
2π σ 2
(3.26)
Thus,
[ P2 ] a¯ 2 (0) F¯1 (P1 ) = √ 10 exp − 1 2 2σ 2π σ [ P2 ] a¯ 2 (0) F¯2 (P2 ) = √ 20 exp − 2 2 2σ 2π σ
(3.27)
Finally, using suitable transformation, we can rewrite I1 , I2 as follows
I1 = −
I2 = −
ia¯ 210
σ |K⃗1 | ia¯ 220
σ |K⃗2 |
√ π . [w(Ω1(+) ) − w(Ω1(−) )] 2
√ π . [w(Ω2(+) ) − w(Ω2(−) )]
(3.28)
2
where w (z) is the complex integral function defined by Abramowitz and Stegun [36] as follows
w(z) = (±)
π
2
∞
e−u
−∞
z−u
∫
i
du, Im(z) > 0
(3.29)
(±)
appearing in Eq. (3.28) are given below
(±)
= √
Ω1 , Ω2
Ω1
(±)
Ω2
= √
1 2σ |K⃗1 | 1 2σ |K⃗2 |
˙[Ω − β1 Lx − β2 Ly ± β3 L2x ± β4 Lx Ly ± β5 L2y ]
˙[Ω − β1 Lx + β2 Ly ± β3 L2x ∓ β4 Lx Ly ± β5 L2y ]
(3.30)
The nonlinear dispersion relation (3.16) can also be rewritten as follows AI1′ I2′ + B1 I1′ + B2 I2′ + 1 = 0
(3.31)
where A=
I1′ =
(4λ21 − µ21 )a¯ 210 a¯ 220 2π σ 2 |K⃗1 ||K⃗2 |
∫
−∞
I2′ =
∫
2
∞
e−u (+)
Ω1
−∞
∫
−u
e−u (+)
Ω2
∞
du − −∞
2
∞
2λ1 a¯ 210
, B1 = √
∫
−u
e−u (−)
Ω1
∞
du − −∞
2λ1 a¯ 220 B2 = √ 2πσ |K⃗1 | 2πσ |K⃗2 |
(−)
2
du
−u
e−u
Ω2
(3.32)
2
du
(3.33)
−u (±)
(±)
In the limit of vanishing bandwidth of both the wave systems, σ → 0. In this case, Ω1 , Ω2 expansion of w (z) as z → ∞, is given by i
1
3
w(z) ≈ √ z −1 [1 + z −2 + z −4 + . . . ...] 2 4 π
→ ∞. Now, asymptotic
(3.34)
If we consider approximation of (3.34) correct up to order z −2 , then i
w(z) ≈ √ z −1 π
(3.35)
22
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
Substituting the approximation (3.35) in (3.16) [or in (3.31)] we obtain the following equation a¯ 210 a¯ 220 2σ 2 |K⃗1 ||K⃗2 |
(4λ21 − µ21 )
2λ1 a¯ 210 ( 1
+√
2σ |K⃗1 | Ω1
(+)
( 1
−
(+) Ω1
−
1 ) (−)
Ω1
1 )( 1 (−) Ω1
(+) Ω2
−
1 )
2λ1 a¯ 220 ( 1
+√
(−)
Ω2
2σ |K⃗2 | Ω2
(+)
−
1 ) (−)
Ω2
+1=0
(3.36)
On simplification of Eq. (3.36) we find that this equation matches with Eq. (2.14) of Senapati et al. [37] at the lowest order when 2a¯ 210 and 2a¯ 220 are replaced by their deterministic counterparts a210 and a220 respectively. (±)
For unidirectional perturbations along x-direction, Ly = 0. Then Ω1
(±)
and Ω2
becomes
Ω − β 1 Lx + β √ = Ω2(+) = √ 2σ Lx 4β32 + β42 2 3 Lx
(+)
Ω1
Ω − β1 Lx − β3 L2x √ = Ω2(−) = √ 2σ Lx 4β32 + β42
(−)
Ω1
(3.37)
Thus, I1′ = I2′ . Eq. (3.31) then becomes ′
AI12 + (B1 + B2 )I1′ + 1 = 0
(3.38)
Solving Eq. (3.38) we get
−(B1 + B2 ) ±
I1′ =
√
(B1 + B2 )2 − 4A
(3.39)
2A
It is easy to check that I1′ is real. Setting Ω = β1 Lx + Ωr + iΩi , Ωr , Ωi being real, we can rewrite I1′ as follows I1′
α
∞
∫ =
−∞
Ωr + β3 L2x − α u − iΩi −u2 e du − (Ωr + β3 L2x − α u)2 + Ωi2
∫
∞ −∞
Ωr − β3 L2x − α u − iΩi −u2 e du (Ωr − β3 L2x − α u)2 + Ωi2
(3.40)
where
α=
√
2σ Lx
√
4β32 + β42
The requirement that I1′ to be real demands that
Ωr = 0
(3.41)
Eq. (3.40) then becomes I1′ = 2
∫
∞ −∞
p−u (p − u)2 + q2
2
e−u du = 2π Im[w (p + iq)]
(3.42)
where p=
β3 L2x , α
q=
Ωi , α
and w (p + iq) can be calculated in terms of complementary error function as follows 2
w(p + iq) = e−(p+iq) erfc[−i(p + iq)],
for q > 0
(3.43)
In Fig. 1 we have plotted growth rate of instability Gr against perturbation wavenumber Lx for different values of θ and taking a¯ 10 = .1, a¯ 20 = .12. It is observed that Gr decreases as θ increases in the range 0◦ < θ < θc . But, in the range θc < θ < 90◦ , Gr increases as θ increases, where θc is a critical value of θ depending upon V and r. The reason for this will be clear if we look at the Eqs. (3.32), (3.39) and (3.42). These equations show that for a given set of values of σ , a¯ 10 and a¯ 20 the growth rate of instability Gr depends upon β3 , β4 , λ1 and µ1 each of which varies with θ . In the range 0◦ < θ < 90◦ the coefficients λ1 , 4λ21 − µ21 , α are all positive and β4 is negative. Consequently, the coefficients A, B1 , B2 remain positive. Only β3 changes sign from negative value to positive value at θc . A plot of β3 is shown in Fig. 2 for V = 5, 10, 15. In Fig. 3 we have plotted growth rate of instability Gr against perturbation wavenumber Lx for different values of wind flow velocity V taking σ = 0.1 and θ = 22.5◦ , 30◦ . We observe that Gr increases as V increases. In Fig. 4 we have plotted growth rate of instability Gr against perturbation wavenumber Lx considering variation both in θ and V and taking σ = 0.15, a¯ 10 = a¯ 20 = 0.1. This figure gives an idea of the effect of the angle on the growth rate of instability for two different values of wind velocity: V = 5, 12.
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
23
Fig. 1. Growth rate of instability Gr against perturbation wavenumber Lx for different values of θ taking a¯ 10 = .1 and a¯ 20 = .12.
Fig. 2. Plot of β3 for different values of V .
In the left hand side figure of Fig. 5 we have shown variation in Gr of the first wave packet with respect to the change in a¯ 20 of the second wave system taking σ = 0.1 and θ = 22.5◦ and V = 15. We find that Gr increases with the increase in the value of a¯ 20 . In the right hand side figure of Fig. 5 we get a comparison between the growth rate of instability in a situation of crossing seas and the growth rate of instability of a single wave packet both in presence and in absence of wind. Comparing dotted curve with solid line curve and comparing dashed curve with dashed–dotted curve we find that the growth rate of instability of two obliquely interacting waves is higher than that for a single wave system both in presence and in absence of wind flow. Again comparing dashed curve with solid line curve and comparing dashed–dotted curve with dotted curve we note that the effect of uniform wind flow is to increase growth rate of instability for the case of a single wave train as well as for the case of two obliquely interacting uniform wave trains. Fig. 6 is a plot of Gr against Lx for different values of σ when θ = 22.5◦ , V = 15 and θ = 12◦ , V = 10. We observe that Gr decreases as the value of σ increases. In Fig. 7 we have compared growth rate of instability Gr determined by Eq. (3.39) with the corresponding deterministic growth rate. We find that the growth rate of instability decreases slightly due to the effect of randomness, but it is much higher than the growth rate for a single wave system.
24
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
Fig. 3. Growth rate of instability Gr against perturbation wavenumber Lx for different values of wind velocity V taking a¯ 10 = .1 and a¯ 20 = .1.
Fig. 4. Growth rate of instability Gr against perturbation wavenumber Lx for different values of θ and V taking σ = .15 and a¯ 10 = .1 and a¯ 20 = .1.
4. Conclusion To examine the stability properties of a pair of obliquely interacting random wave trains we have considered a very simple ocean model with two obliquely interacting wave systems in the presence of wind blowing uniformly. Strictly speaking our model is not appropriate for real wind sea situation. We have derived a pair of transport equations for the spectral functions of the two wavetrains. Using these two equations we have performed stability analysis for an initially homogeneous solution satisfying Gaussian properties. In the stability analysis presented here we have not considered directional distribution about the mean wave direction although it plays an important role in wave modeling and nonlinear interactions are sensitive to the directional distribution of energy. We have obtained nonlinear dispersion relation in the form of an integral equation. Finally, we have graphically presented the growth rate of instability against unidirectional perturbation wave number. We have found that the growth rate of instability increases as the wind flow velocity increases. When the two wave trains interact making an angle less than certain critical value the growth rate of instability decreases
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
25
Fig. 5. Growth rate of instability Gr against perturbation wavenumber Lx : Left—for different values of a¯ 20 taking a¯ 10 = .1, Right—a comparison with growth rate of instability of a single wave system.
Fig. 6. Growth rate of instability Gr against perturbation wavenumber Lx for different values of σ taking a¯ 10 = .1 and a¯ 20 = .12.
as the angle between them increases. But we see the exactly reverse result when the angle between the two interacting wavetrains is greater than the critical value. We have also found that the growth rate of instability decreases with the increase of spectral bandwidth. As the mean square wave steepness of one wave system increases the growth rate of instability of the second wave system increases. It is found that the consideration of randomness has a slight stabilizing influence on the growth rate of instability in a situation of crossing sea states although it is much greater than the growth rate for a single wave system. The ocean model studied here is merely a toy model. For a more realistic situation one should take into account the presence of third wave system generated by wind flow. It is also of interest to see the evolution when the wind forces one of the two wave systems.
26
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
Fig. 7. Growth rate of instability Gr against perturbation wavenumber Lx taking a¯ 10 = .1 : a comparison with the deterministic growth rate.
Acknowledgments Authors are thankful to the reviewers for their suggestions to improve this report. Appendix. Coefficients of evolution equations
β1 = β2 = β3 =
β4 = β5 = λ1 = µ1 =
2rV ω − 2rkV 2 + k(1 − r) 2(1 + r)ω − 2rkV l(1 − r) 2(1 + r)ω − 2rkV [ (1 + r){2rV ω − 2rkV 2 + k(1 − r)}2 2rV {2rV ω − 2rkV 2 + k(1 − r)} + − 2 {2(1 + r)ω − 2rkV } 2{2(1 + r)ω − 2rkV } 2] (1 − r)l − rV 2 + /{2(1 + r)ω − 2rkV } 2 [ 2l(1 − r 2 ){2rV ω − 2rkV 2 + k(1 − r)} ] − − (1 − r)kl /{2(1 + r)ω − 2rkV } {2(1 + r)ω − 2rkV }2 [ (1 − r)k2 ] (1 − r)2 (1 + r)l2 + /{2(1 + r)ω − 2rkV } − 2 {2(1 + r)ω − 2rkV } 2 [ ] 4 2(1 − r) + {ω2 − r(ω − Vk)2 }2 /{2(1 + r)ω − 2rkV } f1
[ 8k f2
{ω2 − r(ω − Vk)2 }2 (−k2 − 2l2 + 2k)2 −
4l4 1−r
{ω2 − r(ω − Vk)2 }2
− 4k2 (2k − 1){ω2 + r(ω − Vk)2 } ] + 4(2k2 − l2 ){ω2 + r(ω − Vk)2 } /{2(1 + r)ω − 2rkV } f1 = D(2ω, 2k, 2l) f2 = D(2ω, 2k, 0)
√
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Wind effect on the evolution of two obliquely interacting random wave trains in deep water ∗
Sumana Kundu a , Suma Debsarma b , , K.P. Das b a b
Salkia Mrigendra Dutta Smriti Balika Vidyapith (High), Salkia, Howrah 711106, India Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata 700009, India
highlights • • • • •
Spectral transport equations are derived for a pair of random surface wavetrains. In the model the presence of uniform wind flow is taken into account. Stability analysis is performed for a pair of Gaussian wave spectra. The effect of randomness is to reduce the growth rate of instability slightly. Growth rate of instability is much higher than for a single wavetrain.
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Article history: Received 31 July 2018 Received in revised form 27 February 2019 Accepted 28 February 2019 Available online 8 March 2019 Keywords: Crossing seas Instability Randomness Spectral transport equation Wind effect
a b s t r a c t A pair of coupled nonlinear evolution equations for the spectral functions are derived in a situation of crossing sea states characterized by two narrowband Gaussian random surface wave systems in the presence of uniform wind flow. These two equations are employed to perform stability analysis of two initially homogeneous wave spectra subject to infinitesimal perturbations. It is found that the results of the stability analysis remain qualitatively similar with the corresponding deterministic situation. The notable difference is that the growth rate of instability reduces slightly due to the effect of randomness. But, it is much higher than the growth rate of a single wave system. As the spectral bandwidth increases the growth rate of instability decreases. © 2019 Published by Elsevier B.V.
1. Introduction In the analysis of ship accidents due to bad weather conditions Toffoli et al. [1] observed that a large number of ship accidents occurred in a situation of crossing seas. So, they pointed out that crossing seas should be considered as possibly dangerous situation. Recently, there has been increased interest in the evolution of nonlinear surface water waves in crossing seas characterized by two obliquely interacting wave systems. Some of the investigations (Onorato et al. [2], Shukla et al. [3], Laine-Pearson [4], Gramstad and Trulsen [4], Brunetti and Kasparian [5] and many others) are from the deterministic point of view with emphasis on the stability properties of two obliquely interacting uniform wave systems. It is found that the growth rate of instability of two obliquely interacting uniform wave systems is much higher than that for a single wave system (see [2–4,6]). ∗ Corresponding author. E-mail addresses: [email protected] (S. Kundu), [email protected] (S. Debsarma), [email protected] (K.P. Das). https://doi.org/10.1016/j.wavemoti.2019.02.008 0165-2125/© 2019 Published by Elsevier B.V.
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
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Some other investigations are based upon either numerical simulations or laboratory experiments. By performing numerical simulations Ruban [7] showed how the formation of freak waves depends on the relative positions of two spectral maxima in the Fourier plane. To investigate the statistical properties of the surface wave envelope in a situation of crossing sea states Toffoli et al. [8] conducted an experiment in a very large wave basin by mechanically generating two wave systems which were forced to propagate along two different directions. They found that the number of extreme events depends upon the angle between the two wave systems. As a support of this experimental finding Toffoli et al. [8] carried out also a numerical simulation. The action of wind blowing over a water body is to produce variations in pressure and surface shear. Also, it can generate rotational current in the water. After the publications of Philips [9] and Miles [10] about the theory of surface wave generation by wind, many researchers (Bliven et al. [11], Waseda and Tulin [12], Peirson and Garcia [13], Janssen [14], Tian et al. [15], Galchenko et al. [16], Brunetti et al. [17], Chabchoub et al. [18], Buckley and Veron [19] and many others) have made notable contributions towards the understanding of wind–wave interactions, generation of waves by wind, wind effect on modulational instability, wave growth by wind etc. Kharif et al. [20] considered the direct effect of strong wind on the extreme wave events. To observe the influence of wind on extreme wave events due to spatio-temporal focusing Kharif et al. [20] conducted experiments in a large wind–wave tank. They also carried out numerical simulations for extreme wave events caused by modulational instability. They found that the height and duration of the extreme waves get increased in the presence of wind. To study the statistical properties of wind-generated waves Toffoli et al. [21] conducted a wind-sea experiment in an annular flume, over which a constant and quasihomogeneous wind flows. This is the first study showing emergence of rogue waves in wind-generated wave fields. White and Fornberg [22] considered a regular ocean swell that traverses a region of deep water with random current fluctuations. With the assumption that the current can focus wave action into a caustic region just like an optical lens, they showed the formation of freak waves at random locations. They verified the theory with Monte Carlo simulations. Toffoli et al. [23] carried out two experiments in two different laboratories and observed that a stable wave propagating into a region characterized by an opposite current may become modulationally unstable and result in formation of a rogue wave. To investigate the role of opposing currents in changing the statistical properties of unidirectional and directional mechanically generated random wave fields Toffoli et al. [24] performed three independent laboratory experiments. The results demonstrate the amplification of modulational instability in weakly unstable regular wave packets due to interaction with an opposing current. This amplification depends on the ratio of current speed to group velocity and non-uniformity of the current. It is observed that opposing currents induce a sharp and rapid transition from weakly to strongly non-Gaussian properties which is associated with an increase in the probability of occurrence of extreme waves. As the wind flows over the water surface it produces randomness at the water surface. One way of studying the effect of randomness on the stability properties of surface wave trains is through the application of a transport equation for the ensemble averaged two point space correlation function with the assumption that the complex wave amplitude is random function of space variables. When Fourier transform of this correlation equation is taken with respect to spatial separation variables another equation is obtained which govern the evolution of the wave power spectral density function. This spectral equation is useful for studying stability of an initially homogeneous wave spectrum to small wave number perturbations. This technique is used by Alber [25], Crawford et al. [26] in the field of nonlinear water waves. Similar technique was used by Wigner [27], Leaf [28], Hasegawa [29] in the fields of plasma physics and quantum mechanics. Toffoli et al. [30] studied the effect of the second order interaction in directionally spread sea states on the statistical properties of surface gravity waves. Observation of Onorato et al. [31] is that freak waves may appear with a higher probability if the angle of interaction between two non-collinear wave packets is between 10◦ and 30◦ . In support of this they presented numerical simulations of two long crested wave systems characterized each by a JONSWAP spectrum with random phases. In the research article of Bitner-Gregersen and Toffoli [32] we find a discussion on some rogue-waveprone crossing seas and their probabilities of occurrence in the ocean. Their analysis is supported also by Monte Carlo simulations. One work earlier in the literature (Masson [33]) studied the effect of nonlinear interactions between swell and wind sea on a stationary wave spectrum. Masson [33] showed that as a result of nonlinear coupling swell decays at a rate that decreases rapidly as the swell frequency moves away from the peak frequency of the short waves. In the frequency range just below the peak frequency of the short waves swell grows at the expense of the local sea. In this investigation we obtain a pair of spectral transport equations in a situation of crossing sea states characterized by two obliquely interacting initially homogeneous random wave systems in presence of uniform wind flow. Using these two equations we carry out stability analysis of a pair of obliquely interacting Gaussian random wave systems. We observe that the effect of randomness is to reduce the growth rate of instability slightly. But it is still higher than that for a single wave system. It is observed that the growth rate of instability increases with the increase in the wind flow velocity, which is similar to the deterministic case. It is also found that the growth rate of instability decreases with the increase in spectral bandwidth. It is observed that the growth rate of instability of one wave system increases as the mean square wave steepness of the second wave system increases. The paper is organized as follows: In Section 2 we derive spectral transport equations for a pair of obliquely interacting random wavetrains; in Section 3 we present the stability analysis; in Section 4 we report a summary of the important findings of stability analysis.
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2. Spectral transport equations We consider a crossing sea states situation in the presence of uniform wind flow. Such a situation can occur in reality when the wave configuration is composed by two obliquely interacting swells and wind starts blowing uniformly. A third wave system (wind-generated waves) may appear in the direction of the wind. For the sake of simplicity we do not take into account the third wave system in the present model. Senapati et al. [34] considered a two fluid domain, namely air–water model both extending up to infinity. In the atmospheric medium wind flows with uniform velocity V along x-direction. In the water medium two surface gravity wave packets interact obliquely. This model is an approximation of reality. The nonlinear evolution equations obtained by Senapati et al. [34] in a situation of crossing sea states in the presence of uniform wind flow are the following i
∂ζ1 ∂ζ1 ∂ζ1 ∂ 2 ζ1 ∂ 2 ζ1 ∂ 2 ζ1 + iβ1 + iβ2 + β3 2 + β4 + β5 2 = λ1 ζ12 ζ1∗ + µ1 ζ1 ζ2 ζ2∗ ∂τ ∂ x1 ∂ y1 ∂ x1 ∂ y1 ∂ x1 ∂ y1
(2.1)
i
∂ζ2 ∂ζ2 ∂ζ2 ∂ 2 ζ2 ∂ 2 ζ2 ∂ 2 ζ2 + iβ1 − iβ2 + β3 2 − β4 + β5 2 = λ1 ζ22 ζ2∗ + µ1 ζ2 ζ1 ζ1∗ ∂τ ∂ x1 ∂ y1 ∂ x1 ∂ y1 ∂ x1 ∂ y1
(2.2)
In Eqs. (2.1) and (2.2), ζ1 , ζ2 are the complex wave envelopes of two wave systems having carrier wave numbers (k, l) and (k, −l) respectively. x-axis is taken along the direction of wind flow and the two wave systems are propagating making equal angle θ with the direction of wind flow. x1 , y1 and τ are the slow spatio-temporal variables defined by x1 = ϵ x, y1 = ϵ y, τ = ϵ t, where ϵ is a small ordering parameter. The coefficients βi ’s, λ1 and µ1 are given in the Appendix. It is found in Senapati et al. [34] that there are two possible modes of wave propagation, namely, positive and negative modes according as the wave frequency is ω = ω+ or ω = ω− , where ω± are given by
ω± =
rVk ±
√
(1 − r 2 ) − rk2 V 2
(2.3)
1+r
Here V is the uniform velocity of wind blowing parallel to x1 direction. Eqs. (2.1)–(2.3) are in dimensionless form with the dimensionless variables defined below √ k l x∗1 = kc x1 , y∗1 = kc y1 , t1∗ = t1 gkc , k∗ = , l∗ = , kc kc
√ ω∗ = ω/ gkc ,
V∗ = V
√
kc /g ,
ζ1∗ = kc ζ1 ,
ζ2∗ = kc ζ2 ,
kc =
√
k2 + l2
For convenience the asterisks have been dropped out. The evolution equations (2.1)–(2.2) are correct up to third order in the wave steepness and they remain valid if the wind flow velocity V is less than a critical velocity Vc given by Vc =
1 k
√
1 − r2 r
(2.4)
Here r(< 1) is the density ratio between air and water. The coefficients βi ’s, λ1 and µ1 appearing in Eqs. (2.1) and (2.2) depend on the wind velocity V and density ratio r. Setting V = 0 and r = 0 one can recover the evolution equations studied by Onorato et al. [2]. We now assume that ζ1 (x1 , y1 , τ ) and ζ2 (x1 , y1 , τ ) are random functions of ξ⃗1 = (x1 , y1 ) and τ . For the two waves undergoing weak nonlinear interactions we define two point space correlation functions as follows
ρ1 (ξ⃗1 , ξ⃗2 , τ ) = ⟨ζ1 (ξ⃗1 , τ )ζ1∗ (ξ⃗2 , τ )⟩ ρ2 (ξ⃗1 , ξ⃗2 , τ ) = ⟨ζ2 (ξ⃗1 , τ )ζ2∗ (ξ⃗2 , τ )⟩
(2.5)
where ξ⃗2 = (x2 , y2 ) is another space point, the superscript (∗) denotes the complex conjugate and the angle brackets denote an ensemble average. ρ1 and ρ2 are, in fact, second order statistical moments. To derive equations for the slow variation of ρ1 and ρ2 we consider equations (2.1)–(2.2) at the point ξ⃗2 also. We first multiply equation (2.1) at the point ξ⃗1 by ζ1∗ (ξ⃗2 , τ ). Then, we consider complex conjugate of Eq. (2.1) at the point ξ⃗2 and multiply it by ζ1 (ξ⃗1 , τ ). Subtracting the second equation from the first one and then taking ensemble average we get the following equation: i
( ∂ ( ∂ ( ∂2 ∂ρ1 ∂ ) ∂ ) ∂2 ) + iβ1 + ρ1 + i β 2 + ρ1 + β3 − ρ1 ∂τ ∂ x1 ∂ x2 ∂ y1 ∂ y2 ∂ x22 ∂ x22 ( ∂2 ( ∂2 ∂2 ) ∂2 ) + β4 − ρ1 + β5 − ρ1 ∂ x1 ∂ y1 ∂ x2 ∂ y2 ∂ y21 ∂ y22 = λ1 ⟨ζ12 (ξ⃗1 )ζ1∗ (ξ⃗1 )ζ1∗ (ξ⃗2 )⟩ − λ1 ⟨ζ1∗ 2 (ξ⃗2 )ζ1 (ξ⃗2 )ζ1 (ξ⃗1 )⟩ + µ1 ⟨ζ1 (ξ⃗1 )ζ2 (ξ⃗1 )ζ2∗ (ξ⃗1 )ζ1∗ (ξ⃗2 )⟩ − µ1 ⟨ζ1∗ (ξ⃗2 )ζ2∗ (ξ⃗2 )ζ2 (ξ⃗2 )ζ1 (ξ⃗1 )⟩
(2.6)
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In Eq. (2.6), ζ1 (ξ⃗1 , τ ), ζ1 (ξ⃗2 , τ ), ζ2 (ξ⃗1 , τ ) and ζ2 (ξ⃗2 , τ ) are written simply ζ1 (ξ⃗1 ), ζ1 (ξ⃗2 ), ζ2 (ξ⃗1 ) and ζ2 (ξ⃗2 ) respectively for the sake of simplicity of notation. We now introduce the average coordinates 1
1 (x1 + x2 ), Y = (y1 + y2 ) 2 2 and the spatial separation coordinates X =
r x = x1 − x2 ,
(2.7)
ry = y1 − y2
(2.8)
In terms of the newly introduced variables the left hand side of Eq. (2.6) becomes i
( ∂2 ∂ρ1 ∂ρ1 ∂ 2 ρ1 ∂ 2 ρ1 ∂ρ1 ∂2 ) + i β1 + iβ2 + 2β3 + β4 + ρ1 + 2β5 ∂τ ∂X ∂Y ∂ X ∂ rx ∂ X ∂ ry ∂ Y ∂ rx ∂ Y ∂ ry
(2.9)
⃗ τ ) and ζ2 (ξ, ⃗ τ ), where ξ⃗ = (X , Y ) follow initially Gaussian random process. We assume also that the We assume that ζ1 (ξ, evolving random statistical amplitude fields retain the same Gaussian statistical properties [Benney and Saffman [35]]. Using the property of Gaussian statistics, the fourth order correlation terms on the right hand side of Eq. (2.6) can be written down in terms of the products of pairs of second order correlations as follows ⟨ζ1 (ξ⃗1 )ζ1∗ (ξ⃗2 )ζ1 (ξ⃗1 )ζ1∗ (ξ⃗1 )⟩ = 2⟨ζ1 (ξ⃗1 )ζ1∗ (ξ⃗2 )⟩⟨ζ1 (ξ⃗1 )ζ1∗ (ξ⃗1 )⟩ = 2ρ1 a¯ 21 (ξ⃗1 )
(2.10)
where ¯ ξ⃗ = ⟨ζ1 (ξ⃗1 )ζ1∗ (ξ⃗1 )⟩ is the ensemble averaged mean square amplitude of the first wave system. For the second wave system it is a¯ 22 (ξ⃗1 ) = ⟨ζ2 (ξ⃗1 )ζ2∗ (ξ⃗1 )⟩. In terms of the coordinates defined in (2.7) and (2.8), ξ⃗1 and ξ⃗2 can be put as follows a21 ( 1 )
1
ξ⃗1 = (x1 , y1 ) = ξ⃗ + ⃗r , 2
1
ξ⃗2 = (x2 , y2 ) = ξ⃗ − ⃗r 2
(2.11)
where ξ⃗ = (X , Y ) and ⃗r = (rx , ry ). Considering Taylor series expansion of a¯ 21 (ξ⃗1 ) about the point ξ⃗ we get the following
(1 ∂ ) (1 ∂ 1 ∂ ) 2 a¯ 1 (ξ⃗) = 2ρ1 exp ⃗r . + ry a¯ 2 (ξ⃗) ⟨ζ1 (ξ⃗1 )ζ1∗ (ξ⃗2 )ζ1 (ξ⃗1 )ζ1∗ (ξ⃗1 )⟩ = 2ρ1 exp rx 2 ∂X 2 ∂Y 2 ∂ ξ⃗ 1
(2.12)
Treating the remaining three terms on the right hand side of Eq. (2.6) we can rewrite Eq. (2.6) as follows i
( ∂2 ∂ρ1 ∂ρ1 ∂ρ1 ∂ 2 ρ1 ∂ 2 ρ1 ∂2 ) + i β1 + iβ2 + 2β3 + β4 + ρ1 + 2β5 ∂τ ∂X ∂Y ∂ X ∂ rx ∂ X ∂ ry ∂ Y ∂ rx ∂ Y ∂ ry (1 ∂ ) (1 ∂ ) 2 2 = 4λ1 ρ1 sinh ⃗r . a¯ + 2µ1 ρ1 sinh ⃗r . a¯ 2 ∂ ξ⃗ 1 2 ∂ ξ⃗ 2
(2.13)
Eq. (2.13) is the evolution equation for the correlation function ρ1 . We now take Fourier transform of Eq. (2.13) with respect to spatial separation coordinates defined below F1 (P⃗ , ξ⃗ , τ ) =
F2 (P⃗ , ξ⃗ , τ ) =
∫ ∫
1 (2π )2
( ) 1 1 ⃗ ⃗ ⃗ ρ1 ξ + ⃗r , ξ − ⃗r , τ e−iP .⃗r d⃗r 2
−∞
∫ ∫
1
∞
(2π )2
∞
2
) ( 1 1 ⃗ ⃗ ⃗ ⃗ ⃗ ρ2 ξ + r , ξ − r , τ e−iP .⃗r d⃗r 2
−∞
2
(2.14)
where d⃗r = drx dry and P⃗ = (Px , Py ) is the Fourier wave number conjugate to the spatial separation coordinates ⃗r . F1 (P⃗ , ξ⃗ , τ ) and F2 (P⃗ , ξ⃗ , τ ) are the wave envelope power spectral density functions of the first and second wave system respectively. Clearly, Fourier inversions of Eqs. (2.14) are given by ∞
( ) ∫ ∫ 1 1 ρ1 ξ⃗ + ⃗r , ξ⃗ − ⃗r , τ = 2
2
⃗
∞
( ) ∫ ∫ 1 1 ρ2 ξ⃗ + ⃗r , ξ⃗ − ⃗r , τ = 2
2
∫ ∫
∞
F1 (P⃗ , ξ⃗ , τ )dP⃗ −∞
( ) ρ2 ξ⃗ , ξ⃗ , τ =
∫ ∫
∞
F2 (P⃗ , ξ⃗ , τ )dP⃗ −∞
F2 (P⃗ , ξ⃗ , τ )eiP .⃗r dP⃗ ⃗
−∞
⃗ in Eq. (2.15), we get If we set ⃗r = 0
( ) ρ1 ξ⃗ , ξ⃗ , τ =
F1 (P⃗ , ξ⃗ , τ )eiP .⃗r dP⃗
−∞
(2.15)
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S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
⃗ τ) = a¯ 21 (ξ,
∫ ∫
∞
F1 (P⃗ , ξ⃗ , τ )dP⃗ −∞
a¯ 22 (ξ⃗, τ ) =
∫ ∫
∞
F2 (P⃗ , ξ⃗ , τ )dP⃗
(2.16)
−∞
Thus, Fourier transform of Eq. (2.13) results in the following equation
∂ F1 ∂ F1 ∂ F1 ∂ F1 ∂ F1 ∂ F1 ∂ F1 + β1 + β2 + 2β3 Px + β4 P y + β4 Px + 2β5 Py ∂τ ∂X ∂Y ∂X ∂X ∂Y ∂Y = 4λ1 sin
( 1 ∂2 ) ( 1 ∂2 ) F1 a¯ 21 + 2µ1 sin F1 a¯ 22 2 ∂ ξ⃗ ∂ P⃗ 2 ∂ ξ⃗ ∂ P⃗
For the right hand side terms of Eq. (2.17) the operator sin
(2.17)
(1
∂2
2 ∂ ξ⃗ ∂ P⃗
)
stands for
( 1 ∂2 ( 1 ∂2 ) 1 ∂2 ) ≡ sin + 2 ∂ ξ⃗ ∂ P⃗ 2 ∂ X ∂ Px 2 ∂ Y ∂ Py
sin
where ∂∂X , ∂∂Y operate on a¯ 21 only and ∂∂P , ∂∂P operate on F1 only. Eq. (2.17) is the desired spectral transport equation for x y the first wave system. Next, starting with Eq. (2.2) and proceeding in a similar manner as described by Eqs. (2.5)–(2.17) we get the following spectral transport equation for the second wave system
∂ F2 ∂ F2 ∂ F2 ∂ F2 ∂ F2 ∂ F2 ∂ F2 + β1 − β2 + 2β3 Px − β4 P y − β4 Px + 2β5 Py ∂τ ∂X ∂Y ∂X ∂X ∂Y ∂Y ( 1 ∂2 ) ( 1 ∂2 ) 2 2 = 4λ1 sin F2 a¯ 2 + 2µ1 sin F2 a¯ 1 2 ∂ ξ⃗ ∂ P⃗ 2 ∂ ξ⃗ ∂ P⃗
(2.18)
Eqs. (2.17) and (2.18) form a coupled system of nonlinear spectral transport equations in a situation of crossing sea states. Retaining all the nonlinear terms in the sine-operator expansion on the right hand side of Eqs. (2.17) and (2.18) we have made stability analysis in the next section. 3. Stability analysis For a solution of the nonlinear spectral transport Eqs. (2.17) and (2.18), which is independent of ξ⃗ , the governing equations reduce to the following
∂ F2 =0 ∂τ
∂ F1 = 0, ∂τ
(3.1)
Solutions of Eqs. (3.1) are of the following form (0)
⃗, F1 = F1 (P)
(0)
⃗ F2 = F2 (P)
(3.2)
which can be considered as the random counterpart of uniform amplitude Stokes wave trains in the deterministic theory (0) ⃗ (0) ⃗ (0) ⃗ (0) ⃗ of crossing seas. We assume that F1 (P) and F2 (P) satisfy Gaussian properties. Thus F1 (P) and F2 (P) together represent a homogeneous crossing seas with Gaussian properties. To study the stability of homogeneous solution (3.2) we introduce the following infinitesimal perturbation (0)
(1)
(0)
(1)
⃗ + ϵ F (P⃗ , ξ⃗ , τ ) F1 = F1 (P) 1 ⃗ + ϵ F (P⃗ , ξ⃗ , τ ) F2 = F2 (P) 2
(3.3)
where ϵ is a small ordering parameter. Substituting (3.3) in (2.16) we get a¯ 21 (ξ⃗ , τ ) = a¯ 210 + ϵ a¯ 211 (ξ⃗ , τ ) a¯ 22 (ξ⃗ , τ ) = a¯ 220 + ϵ a¯ 221 (ξ⃗ , τ )
(3.4)
where a¯ 210 =
∫ ∫
∞ (0)
⃗ P⃗ , a¯ 220 = F1 (P)d −∞
∫ ∫
∞ (0)
⃗ P⃗ F2 (P)d −∞
(3.5)
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
a¯ 211 (ξ⃗ , τ ) =
∫ ∫
19
∞ (1)
F1 (P⃗ , ξ⃗ , τ )dP⃗ −∞
a¯ 221 (ξ⃗ , τ ) =
∫ ∫
∞ (1)
F2 (P⃗ , ξ⃗ , τ )dP⃗
(3.6)
−∞
Here a¯ 210 , a¯ 220 denote mean square wave steepness of the first and second wave systems respectively. Now, substituting Eqs. (3.3) and (3.4) in Eqs. (2.17) and (2.18) and then linearizing we get
∂ F1(1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) + β1 1 + β2 1 + 2β3 Px 1 + β4 Py 1 + β4 Px 1 + 2β5 Py 1 ∂τ ∂X ∂Y ∂X ∂X ∂Y ∂Y ( 1 ∂2 ) ( 1 ∂2 ) (0) (0) = 4λ1 sin F a¯ 2 + 2µ1 sin F a¯ 2 2 ∂ ξ⃗ ∂ P⃗ 1 11 2 ∂ ξ⃗ ∂ P⃗ 1 21
(3.7)
∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F (1) ∂ F2(1) + β1 2 − β2 2 + 2β3 Px 2 − β4 Py 2 − β4 Px 2 + 2β5 Py 2 ∂τ ∂X ∂Y ∂X ∂X ∂Y ∂Y ( 1 ∂2 ) ( 1 ∂2 ) (0) 2 (0) 2 = 4λ1 sin F a¯ + 2µ1 sin F a¯ 2 ∂ ξ⃗ ∂ P⃗ 2 21 2 ∂ ξ⃗ ∂ P⃗ 2 11 (1)
(3.8)
(1)
We now assume space–time dependence of F1 , F2 , a¯ 211 , a¯ 221 in the following form (1)
⃗ exp[i(Lx X + Ly Y − Ω τ )] F1 (P⃗ , ξ⃗ , τ ) = f1 (P) (1)
⃗ exp[i(Lx X + Ly Y − Ω τ )] F2 (P⃗ , ξ⃗ , τ ) = f2 (P) a¯ 211 (ξ⃗ , τ ) = α1 exp[i(Lx X + Ly Y − Ω τ )] a¯ 221 (ξ⃗ , τ ) = α2 exp[i(Lx X + Ly Y − Ω τ )]
(3.9)
where ⃗L = (Lx , Ly ) is perturbation wave number vector; Ω is the perturbed frequency; α1 and α2 are two constants. Substituting (3.9) in (3.7) and (3.8) and then equating coefficients of exp[i(Lx X + Ly Y − Ω τ )] we get
[ ⃗ ⃗ ] ⃗ ]f1 (P) ⃗ = (2λ1 α1 + µ1 α2 ) F (0) (P⃗ + L ) − F (0) (P⃗ − L ) [−Ω + G+ (P) 1 1
(3.10)
[ ⃗ ⃗ ] ⃗ ]f2 (P) ⃗ = (2λ1 α2 + µ1 α1 ) F (0) (P⃗ + L ) − F (0) (P⃗ − L ) [−Ω − G− (P) 2 2
(3.11)
2
2
2
2
where
⃗ = β1 Lx ± β2 Ly + 2β3 Px Lx ± β4 Py Lx ± β4 Px Ly + 2β5 Py Ly G± (P)
(3.12)
By (3.6) and (3.9) we get the following relationships ∞
∫ ∫
∫ ∫
⃗ P⃗ = α1 , f1 (P)d −∞
∞
⃗ P⃗ = α2 f2 (P)d
(3.13)
−∞
⃗ and f2 (P) ⃗ as given by Eqs. (3.10) and (3.11) respectively in Eq. (3.13) we get the following equations Inserting f1 (P) (2λ1 I1 + 1)α1 + µ1 I1 α2 = 0
µ1 I2 α1 + (2λ1 I2 + 1)α2 = 0
(3.14)
where
∫ ∫
∞
I1 = −∞
∫ ∫
∞
I2 = −∞
⃗
⃗
⃗
⃗
[F1(0) (P⃗ + 2L ) − F1(0) (P⃗ − 2L )] dP⃗ ⃗ Ω − G+ (P) [F2(0) (P⃗ + 2L ) − F2(0) (P⃗ − 2L )] dP⃗ ⃗ Ω − G− (P)
(3.15)
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S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
For non-trivial solution of the system of equations in (3.14), I1 and I2 must satisfy the following equation (4λ21 − µ21 )I1 I2 + 2λ1 (I1 + I2 ) + 1 = 0
(3.16)
The nonlinear integral equation (3.16) is the desired dispersion relation. (0) ⃗ (0) ⃗ We now assume two dimensional normal spectrum F1 (P) and F2 (P) as given below (0)
⃗ = F1 (P)
(0)
⃗ = F2 (P)
a¯ 210 2π σ 2
[ P2 + P2 ] x y 2σ 2
exp −
[ P2 + P2 ] x y 2σ 2
a¯ 220
exp −
(3.17) 2π σ 2 where σ is the spectral spread about the carrier wavenumber. σ specifies the degree of randomness. For convenience, we introduce the following two vectors K⃗1 = (2β3 Lx + β4 Ly )xˆ + (β4 Lx + 2β5 Ly )yˆ K⃗2 = (2β3 Lx − β4 Ly )xˆ + (−β4 Lx + 2β5 Ly )yˆ
(3.18)
⃗ can then be written as follows where xˆ and yˆ are unit vectors along x and y directions respectively. G± (P) ⃗ = β1 Lx + β2 Ly + P .K⃗1 = β1 Lx + β2 Ly + P1 |K⃗1 | G+ (P) ⃗ = β1 Lx − β2 Ly + P .K⃗2 = β1 Lx − β2 Ly + P2 |K⃗2 | G− (P)
(3.19)
where P1 and P2 are the components of P⃗ along K⃗1 and K⃗2 respectively given by (2β3 Lx + β4 Ly )Px + (β4 Lx + 2β5 Ly )Py P1 = 1 [(2β3 Lx + β4 Ly )2 + (β4 Lx + 2β5 Ly )2 ] 2 P2 =
(2β3 Lx − β4 Ly )Px + (−β4 Lx + 2β5 Ly )Py 1
[(2β3 Lx − β4 Ly )2 + (−β4 Lx + 2β5 Ly )2 ] 2
(3.20)
Now, components of P⃗ in the directions normal to K⃗1 and K⃗2 are given respectively by K⃗1 )
(
P1′ = P⃗ · zˆ ×
|K⃗1 | K⃗2 )
(
P2′ = P⃗ · zˆ ×
|K⃗2 |
=
=
1
|K⃗1 | 1
|K⃗2 |
[(2β3 Lx + β4 Ly )Py − (β4 Lx + 2β5 Ly )Px ]
[(2β3 Lx − β4 Ly )Py − (−β4 Lx − 2β5 Ly )Px ]
(3.21)
where zˆ unit vector along z-direction. Clearly, any double integration with respective to Px and Py is equivalent to integration with respect to P1 , P1′ and also with respective to P2 , P2′ . Now, P1′ appears solely in the numerator of I1 . ⃗
(0)
Therefore, the integration of F1 (P⃗ ± 2L ) over P1′ reduces the double integral to a single integral over P1 with the wave number component ν1 = (Lx , Ly ) ·
∫
∞
I1 = −∞
[F¯1(0) (P1 +
ν1 2
K⃗1
|K⃗1 |
as parameter. Thus,
(0)
) − F¯1 (P1 −
ν1 2
)]
Ω − (β1 Lx + β2 Ly + P1 |K⃗1 |)
dP1
(3.22)
where (0)
F¯1 (P1 ) =
∫
∞ (0)
⃗ 1′ F1 (P)dP
(3.23)
−∞
Similarly, I2 becomes
∫
∞
I2 = −∞
[F¯2(0) (P2 +
ν2 2
(0)
) − F¯2 (P2 −
ν2 2
)]
Ω − (β1 Lx − β2 Ly + P2 |K⃗2 |)
dP2
(3.24)
where
ν2 = (Lx , Ly ) ·
K⃗2
|K⃗2 |
, F¯2(0) (P2 ) =
∫
∞
−∞
(0)
⃗ 2′ F2 (P)dP
(3.25)
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27 ′
′
(0)
21
(0)
⃗ F (P) ⃗ as follows Since Px2 + Py2 = P12 + P12 = P22 + P22 we can write F1 (P), 2 exp −
2π σ 2
[ P 2 + P ′2 ] 2 2 2σ 2
a¯ 220
(0)
⃗ = F2 (P)
[ P 2 + P ′2 ] 1 1 2σ 2
a¯ 210
(0)
⃗ = F1 (P)
exp −
2π σ 2
(3.26)
Thus,
[ P2 ] a¯ 2 (0) F¯1 (P1 ) = √ 10 exp − 1 2 2σ 2π σ [ P2 ] a¯ 2 (0) F¯2 (P2 ) = √ 20 exp − 2 2 2σ 2π σ
(3.27)
Finally, using suitable transformation, we can rewrite I1 , I2 as follows
I1 = −
I2 = −
ia¯ 210
σ |K⃗1 | ia¯ 220
σ |K⃗2 |
√ π . [w(Ω1(+) ) − w(Ω1(−) )] 2
√ π . [w(Ω2(+) ) − w(Ω2(−) )]
(3.28)
2
where w (z) is the complex integral function defined by Abramowitz and Stegun [36] as follows
w(z) = (±)
π
2
∞
e−u
−∞
z−u
∫
i
du, Im(z) > 0
(3.29)
(±)
appearing in Eq. (3.28) are given below
(±)
= √
Ω1 , Ω2
Ω1
(±)
Ω2
= √
1 2σ |K⃗1 | 1 2σ |K⃗2 |
˙[Ω − β1 Lx − β2 Ly ± β3 L2x ± β4 Lx Ly ± β5 L2y ]
˙[Ω − β1 Lx + β2 Ly ± β3 L2x ∓ β4 Lx Ly ± β5 L2y ]
(3.30)
The nonlinear dispersion relation (3.16) can also be rewritten as follows AI1′ I2′ + B1 I1′ + B2 I2′ + 1 = 0
(3.31)
where A=
I1′ =
(4λ21 − µ21 )a¯ 210 a¯ 220 2π σ 2 |K⃗1 ||K⃗2 |
∫
−∞
I2′ =
∫
2
∞
e−u (+)
Ω1
−∞
∫
−u
e−u (+)
Ω2
∞
du − −∞
2
∞
2λ1 a¯ 210
, B1 = √
∫
−u
e−u (−)
Ω1
∞
du − −∞
2λ1 a¯ 220 B2 = √ 2πσ |K⃗1 | 2πσ |K⃗2 |
(−)
2
du
−u
e−u
Ω2
(3.32)
2
du
(3.33)
−u (±)
(±)
In the limit of vanishing bandwidth of both the wave systems, σ → 0. In this case, Ω1 , Ω2 expansion of w (z) as z → ∞, is given by i
1
3
w(z) ≈ √ z −1 [1 + z −2 + z −4 + . . . ...] 2 4 π
→ ∞. Now, asymptotic
(3.34)
If we consider approximation of (3.34) correct up to order z −2 , then i
w(z) ≈ √ z −1 π
(3.35)
22
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
Substituting the approximation (3.35) in (3.16) [or in (3.31)] we obtain the following equation a¯ 210 a¯ 220 2σ 2 |K⃗1 ||K⃗2 |
(4λ21 − µ21 )
2λ1 a¯ 210 ( 1
+√
2σ |K⃗1 | Ω1
(+)
( 1
−
(+) Ω1
−
1 ) (−)
Ω1
1 )( 1 (−) Ω1
(+) Ω2
−
1 )
2λ1 a¯ 220 ( 1
+√
(−)
Ω2
2σ |K⃗2 | Ω2
(+)
−
1 ) (−)
Ω2
+1=0
(3.36)
On simplification of Eq. (3.36) we find that this equation matches with Eq. (2.14) of Senapati et al. [37] at the lowest order when 2a¯ 210 and 2a¯ 220 are replaced by their deterministic counterparts a210 and a220 respectively. (±)
For unidirectional perturbations along x-direction, Ly = 0. Then Ω1
(±)
and Ω2
becomes
Ω − β 1 Lx + β √ = Ω2(+) = √ 2σ Lx 4β32 + β42 2 3 Lx
(+)
Ω1
Ω − β1 Lx − β3 L2x √ = Ω2(−) = √ 2σ Lx 4β32 + β42
(−)
Ω1
(3.37)
Thus, I1′ = I2′ . Eq. (3.31) then becomes ′
AI12 + (B1 + B2 )I1′ + 1 = 0
(3.38)
Solving Eq. (3.38) we get
−(B1 + B2 ) ±
I1′ =
√
(B1 + B2 )2 − 4A
(3.39)
2A
It is easy to check that I1′ is real. Setting Ω = β1 Lx + Ωr + iΩi , Ωr , Ωi being real, we can rewrite I1′ as follows I1′
α
∞
∫ =
−∞
Ωr + β3 L2x − α u − iΩi −u2 e du − (Ωr + β3 L2x − α u)2 + Ωi2
∫
∞ −∞
Ωr − β3 L2x − α u − iΩi −u2 e du (Ωr − β3 L2x − α u)2 + Ωi2
(3.40)
where
α=
√
2σ Lx
√
4β32 + β42
The requirement that I1′ to be real demands that
Ωr = 0
(3.41)
Eq. (3.40) then becomes I1′ = 2
∫
∞ −∞
p−u (p − u)2 + q2
2
e−u du = 2π Im[w (p + iq)]
(3.42)
where p=
β3 L2x , α
q=
Ωi , α
and w (p + iq) can be calculated in terms of complementary error function as follows 2
w(p + iq) = e−(p+iq) erfc[−i(p + iq)],
for q > 0
(3.43)
In Fig. 1 we have plotted growth rate of instability Gr against perturbation wavenumber Lx for different values of θ and taking a¯ 10 = .1, a¯ 20 = .12. It is observed that Gr decreases as θ increases in the range 0◦ < θ < θc . But, in the range θc < θ < 90◦ , Gr increases as θ increases, where θc is a critical value of θ depending upon V and r. The reason for this will be clear if we look at the Eqs. (3.32), (3.39) and (3.42). These equations show that for a given set of values of σ , a¯ 10 and a¯ 20 the growth rate of instability Gr depends upon β3 , β4 , λ1 and µ1 each of which varies with θ . In the range 0◦ < θ < 90◦ the coefficients λ1 , 4λ21 − µ21 , α are all positive and β4 is negative. Consequently, the coefficients A, B1 , B2 remain positive. Only β3 changes sign from negative value to positive value at θc . A plot of β3 is shown in Fig. 2 for V = 5, 10, 15. In Fig. 3 we have plotted growth rate of instability Gr against perturbation wavenumber Lx for different values of wind flow velocity V taking σ = 0.1 and θ = 22.5◦ , 30◦ . We observe that Gr increases as V increases. In Fig. 4 we have plotted growth rate of instability Gr against perturbation wavenumber Lx considering variation both in θ and V and taking σ = 0.15, a¯ 10 = a¯ 20 = 0.1. This figure gives an idea of the effect of the angle on the growth rate of instability for two different values of wind velocity: V = 5, 12.
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
23
Fig. 1. Growth rate of instability Gr against perturbation wavenumber Lx for different values of θ taking a¯ 10 = .1 and a¯ 20 = .12.
Fig. 2. Plot of β3 for different values of V .
In the left hand side figure of Fig. 5 we have shown variation in Gr of the first wave packet with respect to the change in a¯ 20 of the second wave system taking σ = 0.1 and θ = 22.5◦ and V = 15. We find that Gr increases with the increase in the value of a¯ 20 . In the right hand side figure of Fig. 5 we get a comparison between the growth rate of instability in a situation of crossing seas and the growth rate of instability of a single wave packet both in presence and in absence of wind. Comparing dotted curve with solid line curve and comparing dashed curve with dashed–dotted curve we find that the growth rate of instability of two obliquely interacting waves is higher than that for a single wave system both in presence and in absence of wind flow. Again comparing dashed curve with solid line curve and comparing dashed–dotted curve with dotted curve we note that the effect of uniform wind flow is to increase growth rate of instability for the case of a single wave train as well as for the case of two obliquely interacting uniform wave trains. Fig. 6 is a plot of Gr against Lx for different values of σ when θ = 22.5◦ , V = 15 and θ = 12◦ , V = 10. We observe that Gr decreases as the value of σ increases. In Fig. 7 we have compared growth rate of instability Gr determined by Eq. (3.39) with the corresponding deterministic growth rate. We find that the growth rate of instability decreases slightly due to the effect of randomness, but it is much higher than the growth rate for a single wave system.
24
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
Fig. 3. Growth rate of instability Gr against perturbation wavenumber Lx for different values of wind velocity V taking a¯ 10 = .1 and a¯ 20 = .1.
Fig. 4. Growth rate of instability Gr against perturbation wavenumber Lx for different values of θ and V taking σ = .15 and a¯ 10 = .1 and a¯ 20 = .1.
4. Conclusion To examine the stability properties of a pair of obliquely interacting random wave trains we have considered a very simple ocean model with two obliquely interacting wave systems in the presence of wind blowing uniformly. Strictly speaking our model is not appropriate for real wind sea situation. We have derived a pair of transport equations for the spectral functions of the two wavetrains. Using these two equations we have performed stability analysis for an initially homogeneous solution satisfying Gaussian properties. In the stability analysis presented here we have not considered directional distribution about the mean wave direction although it plays an important role in wave modeling and nonlinear interactions are sensitive to the directional distribution of energy. We have obtained nonlinear dispersion relation in the form of an integral equation. Finally, we have graphically presented the growth rate of instability against unidirectional perturbation wave number. We have found that the growth rate of instability increases as the wind flow velocity increases. When the two wave trains interact making an angle less than certain critical value the growth rate of instability decreases
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
25
Fig. 5. Growth rate of instability Gr against perturbation wavenumber Lx : Left—for different values of a¯ 20 taking a¯ 10 = .1, Right—a comparison with growth rate of instability of a single wave system.
Fig. 6. Growth rate of instability Gr against perturbation wavenumber Lx for different values of σ taking a¯ 10 = .1 and a¯ 20 = .12.
as the angle between them increases. But we see the exactly reverse result when the angle between the two interacting wavetrains is greater than the critical value. We have also found that the growth rate of instability decreases with the increase of spectral bandwidth. As the mean square wave steepness of one wave system increases the growth rate of instability of the second wave system increases. It is found that the consideration of randomness has a slight stabilizing influence on the growth rate of instability in a situation of crossing sea states although it is much greater than the growth rate for a single wave system. The ocean model studied here is merely a toy model. For a more realistic situation one should take into account the presence of third wave system generated by wind flow. It is also of interest to see the evolution when the wind forces one of the two wave systems.
26
S. Kundu, S. Debsarma and K.P. Das / Wave Motion 89 (2019) 14–27
Fig. 7. Growth rate of instability Gr against perturbation wavenumber Lx taking a¯ 10 = .1 : a comparison with the deterministic growth rate.
Acknowledgments Authors are thankful to the reviewers for their suggestions to improve this report. Appendix. Coefficients of evolution equations
β1 = β2 = β3 =
β4 = β5 = λ1 = µ1 =
2rV ω − 2rkV 2 + k(1 − r) 2(1 + r)ω − 2rkV l(1 − r) 2(1 + r)ω − 2rkV [ (1 + r){2rV ω − 2rkV 2 + k(1 − r)}2 2rV {2rV ω − 2rkV 2 + k(1 − r)} + − 2 {2(1 + r)ω − 2rkV } 2{2(1 + r)ω − 2rkV } 2] (1 − r)l − rV 2 + /{2(1 + r)ω − 2rkV } 2 [ 2l(1 − r 2 ){2rV ω − 2rkV 2 + k(1 − r)} ] − − (1 − r)kl /{2(1 + r)ω − 2rkV } {2(1 + r)ω − 2rkV }2 [ (1 − r)k2 ] (1 − r)2 (1 + r)l2 + /{2(1 + r)ω − 2rkV } − 2 {2(1 + r)ω − 2rkV } 2 [ ] 4 2(1 − r) + {ω2 − r(ω − Vk)2 }2 /{2(1 + r)ω − 2rkV } f1
[ 8k f2
{ω2 − r(ω − Vk)2 }2 (−k2 − 2l2 + 2k)2 −
4l4 1−r
{ω2 − r(ω − Vk)2 }2
− 4k2 (2k − 1){ω2 + r(ω − Vk)2 } ] + 4(2k2 − l2 ){ω2 + r(ω − Vk)2 } /{2(1 + r)ω − 2rkV } f1 = D(2ω, 2k, 2l) f2 = D(2ω, 2k, 0)
√
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