The nonlinear Schrödinger (NLS) equation describing the propagation of weakly rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. The vorticity effects manifest themselves in a shift of the wave number in the carrier wave and in variation in the coefficient multiplying the nonlinear term. In the case of vorticity dependence on the vertical Lagrangian coordinate only (Gouyon waves), the shift of the wave number and the respective coefficient are constant. When the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally inhomogeneous. There are special cases (e.g., Gerstner waves) in which the vorticity is proportional to the squared wave amplitude and nonlinearity disappears, thus making the equations for wave packet dynamics linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables may be obtained from the Lagrangian solution by simply changing the horizontal coordinates.

Introduction

The nonlinear Schrödinger (NLS) equation was first derived by Zakharov in 1967 (English edition; Zakharov, 1968), who used the Hamiltonian formalism for a description of wave propagation in deep water; see also Benney and Newell (1967). Hasimoto and Ono (1972) and Davey (1972) obtained the same result independently. Like Benney and Newell (1967), they used the method of multiple-scale expansions in Euler coordinates. Yuen and Lake (1975), in turn, derived the NLS equation on the basis of the averaged Lagrangian method. Benney and Roskes (1969) extended those two-dimensional theories to the case of three-dimensional wave perturbations in a finite-depth fluid and obtained equations that are now known as the Davey–Stewartson equations. In this particular case, the equation proves the existence of the transverse instability of a plane wave, which is much stronger than a longitudinal one. This circumstance diminishes the role and meaning of the NLS equation for sea applications. Meanwhile, the one-dimensional NLS equation has been successfully tested many times in laboratory wave tanks, and natural observations have been compared with numerical calculations in the framework of this equation.

In all the cited papers, wave motion was considered to be potential. However, wave formation and propagation frequently occur against the background of a shear flow possessing vorticity. Wave train modulations upon arbitrary vertically sheared currents were studied by Benney and Maslowe (1975). Using the method of multiple scales, Johnson (1976) examined the slow modulation of a harmonic wave moving at the surface of an arbitrary shear flow with a velocity profile Uy, where y is the vertical coordinate. He derived the NLS equation with coefficients that depend in a complicated way on a shear flow (Johnson, 1976). Oikawa et al. (1987) considered the properties of instability in weakly nonlinear three-dimensional wave packets in the presence of a shear flow. Their simultaneous equations reduce to the known NLS equation for the case of purely two-dimensional wave evolution. Li et al. (1987) and Baumstein (1998) studied the modulation instability of the Stokes wave train and derived an NLS equation for a uniform shear flow in deep water when Uy=Ω0y and Ωz=Ω0 is constant vorticity (z is the horizontal coordinate normal to the x,y plane of the flow; the wave propagates in the x direction).

Thomas et al. (2012) generalized their results for a finite-depth fluid and confirmed that a linear shear flow may significantly modify the stability properties of weakly nonlinear Stokes waves. In particular, for the waves propagating in the direction of the flow, the Benjamin–Feir (modulational) instability can vanish in the presence of positive vorticity (Ω0<0) for any depth.

In the traditional Eulerian approach to the propagation of weakly nonlinear waves against the background current, a shear flow determines vorticity in a zero approximation. Depending on the flow profile Uy, it may be arbitrary and equal to -Uy. At the same time, the vorticity of wave perturbations Ωn,n1, i.e., the vorticity in the first and subsequent approximations in the wave steepness parameter ε=kA0 (k is wave number and A0 is wave amplitude), depends on its form. In Eulerian coordinates, the vorticity of wave perturbations is a function not only of y, but of x and t variables as well. Plane waves on a shear flow with a linear vertical profile are regarded to be an exception (Li et al., 1987; Baumstein, 1998; Thomas et al., 2012). For such waves the vorticity is constant in a zero approximation, and all the vorticities in wave perturbations are equal to zero. For an arbitrary vertical profile of the shear flow (Johnson, 1976), expressions for the functions Ωn can hardly be predicted even qualitatively.

The Lagrangian method allows for the application of a different approach. In the plane flow, the vorticity of fluid particles is preserved and can be expressed via Lagrangian coordinates only. Thus, not only the vertical profile of the shear flow defining the vorticity in a zero approximation, but also the expressions for the vorticity of the following orders of smallness can be arbitrary. The expression for the vorticity is written in the form Ωa,b=-Ub+n1εnΩna,b, where a,b are the horizontal and vertical Lagrangian coordinates, respectively, Ub is the vertical profile of the shear flow, and the particular conditions for defining the Ωn functions can be found. For the given shear flow, this approach allows for the study of wave perturbations under the most general law of the distribution of vorticities Ωn. In this paper, we do not consider shear flow and vorticity in the linear approximation (U=0;Ω1=0), whereas vorticity in the quadratic approximation is an arbitrary function. This corresponds to the rotational flow proportional to ε2. We can define both the shear flow and the localized vortex.

The dynamics of plane wave trains on the background flows with arbitrary low vorticity have not been studied before. The idea to study wave trains with quadratic (with respect to the wave steepness parameter) vorticity was realized earlier for the spatial problems in the Euler variables. Hjelmervik and Trulsen (2009) derived the NLS equation for vorticity distribution as ΩyΩyωω=Oε2,Ωx,ΩzΩx,Ωzωω=Oε3, where ω is the wave frequency. The vertical vorticity of wave perturbations exceeds the other two vorticity components by a factor of 10. This vorticity distribution corresponds to the low (of order ε) velocity of the horizontally inhomogeneous shear flow. Hjelmervik and Trulsen (2009) used the NLS equation to study the statistics of rogue waves on narrow current jets, and Onorato et al. (2011) used that equation to study the opposite flow rogue waves. The effect of low vorticity (ε2 order of magnitude) in the paper by Hjelmervik and Trulsen (2009) is reflected in the NLS equation. This fact, like the NLS nonlinear term for plane potential waves, may be attributed to the presence of an average current nonuniform over the fluid depth.

Colin et al. (1995) considered the evolution of three-dimensional vortex disturbances in a finite-depth fluid for a different type of vorticity distribution: Ωy=0,Ωx,ΩzΩx,Ωzωω=Oε2. They reduced the problem to a solution for the Davey–Stewartson equations by means of the multiple-scale expansion method in Eulerian variables. In this case, vorticity components are calculated after the solution to the problem. Similarly to the traditional Eulerian approach (Johnson, 1976), the form of the quadratic vorticity distribution is very special and does not cover all of its numerous possible distributions.

In this paper, we consider the plane problem of nonlinear wave packets propagating in an ideal incompressible fluid with the following form of vorticity distribution: ΩzΩzωω=Oε2. In contrast to Hjelmervik and Trulsen (2009), Onorato et al. (2011), and Colin et al. (1996), the flow is two-dimensional (Ωx=Ωy=0). The propagation of a packet of potential waves gives rise to a weak counterflow underneath the free water surface with velocity proportional to the square of the wave steepness (McIntyre, 1982). In the considered problem, this potential flow is superimposed with the rotational one of the same order of magnitude. This results in the appearance of an additional term in the NLS equation and in a change of the coefficient in the nonlinear term. So, the difference from the NLS solutions derived for a strictly potential fluid motion was revealed.

The examination is made in the Lagrangian variables. The Lagrangian variables are rarely used in fluid mechanics because of a more complex type of nonlinear equation in Lagrangian form. However, when considering the vortex-induced oscillations of a free fluid surface, the Lagrangian approach has two major advantages. First, unlike the Euler description method, the shape of the free surface is known and determined by the condition of the equality to zero (b=0) of the vertical Lagrangian coordinate. Second, the vortical motion of liquid particles is confined within the plane and is a function of Lagrangian variables Ωz=Ωza,b, so the type of vorticity distribution in the fluid can be preset. The Eulerian approach does not allow this. In this case, the second-order vorticity is defined as a known function of Lagrangian variables.

Here, hydrodynamic equations are solved in Lagrangian form through the multiple-scale expansion method. A nonlinear Schrödinger equation with variable coefficients is derived. Possible ways of reducing it to the NLS equation with constant coefficients are studied.

The paper is organized as follows. Section 2 describes the Lagrangian approach to studying wave oscillations at the free surface of a fluid. The zero of the Lagrangian vertical coordinate corresponds to the free surface, thus simplifying the formulation of the pressure boundary conditions. The specific feature of the proposed approach is the introduction of a complex coordinate of a fluid particle trajectory. In Sect. 3, a nonlinear evolution equation is derived on the basis of the method of multiple-scale expansion. Different solutions to the NLS equation adequately describing various examples of vortex waves are considered in Sect. 4. The transform from of the Lagrangian coordinates to the Euler description of the solutions to the NLS equation is shown in Sect. 5. Section 6 summarizes the obtained results.

Basic equations in Lagrangian coordinates

Consider the propagation of a packet of gravity surface waves in a rotational infinitely deep fluid. Two-dimensional hydrodynamic equations of an incompressible inviscid fluid in Lagrangian coordinates have the following form (Lamb, 1932; Abrashkin and Yakubovich, 2006; Bennett, 2006): DX,YDa,b=X,Y=1,XttXa+Ytt+gYa=-1ρpa,XttXb+Ytt+gYb=-1ρpb, where X,Y are the horizontal and vertical Cartesian coordinates, a,b are the horizontal and vertical Lagrangian coordinates of fluid particles, t is time, ρ is fluid density, p is pressure, g is acceleration due to gravity, and the subscripts mean differentiation with respect to the corresponding variable. The square brackets denote the Jacobian. The b axis is directed upwards, and b=0 corresponds to the free surface. Equation (1) is a volume conservation equation. Equations (2) and (3) are momentum equations. The geometry of the problem is presented in Fig. 1.

Problem geometry: vx is the average current.

By making use of cross differentiation, it is possible to exclude pressure and obtain the condition of the conservation of vorticity along the trajectory (Lamb, 1932; Abrashkin and Yakubovich, 2006; Bennett, 2006): XtaXb+YtaYb-XtbXa-YtbYa=Ωa,b. This equation is equivalent to the momentum Eqs. (2) and (3) but involves the explicit vorticity of liquid particles, Ω, which in the case of two-dimensional flows is the function of Lagrangian coordinates only.

We introduce a complex coordinate of a fluid particle trajectory W=X+iYW=X-iY, where the overline means complex conjugation. In the new variables, Eqs. (1) and (4) take the form W,W=-2i,ReWt,W=Ωa,b. After simple algebraic manipulations, Eqs. (2) and (3) reduce to the following single equation: Wtt=-ig+iρ-1p,W. Equations (5) and (6) will be further used to find the coordinates of the complex trajectories of fluid particles, and Eq. (7) determines the pressure of the fluid. The boundary conditions are the non-flowing condition at the bottom (Yt0 at b-) and constant pressure at the free surface (at b=0).

The Lagrangian coordinates mark the position of fluid particles. In the Eulerian description, the displacement of the free surface Ys(X,t) is calculated in an explicit form, but in the Lagrangian description it is defined parametrically by the following equalities: Ys(a,t)=Y(a,b=0,t) and Xs(a,t)=X(a,b=0,t), where the Lagrangian horizontal coordinate a plays the role of a parameter. Its value along the free surface b=0 varies in the -; range. In Lagrangian coordinates, the function Ys(a,t) defines the displacement of the free surface.

Derivation of evolution equation

Let us represent the function W using the multiple-scale method in the following form: W=a0+ib+wal,b,tl,al=εla,tl=εlt;l=0,1,2, where ε is the small parameter of wave steepness. All unknown functions and the given vorticity can be represented as a series in this parameter: w=n=1εnwn;p=p0-ρgb+n=1εnpn;Ω=n=1εnΩna,b. In the formula for the pressure, the term with hydrostatic pressure is selected, and p0 is the constant atmospheric pressure at the fluid surface. The representations (8) and (9) are substituted into Eqs. (5)–(7).

Linear approximation

In a first approximation in the small parameter, we have the following system of equations: Imiw1a0+w1b=0,Reiw1a0+w1bt0=-Ω1,w1t0t0+ρ-1p1a0+ip1b=igw1a0. The solution satisfying the continuity Eq. (10) and the equation of the conservation of vorticity (11) describes a monochromatic wave (for definiteness, we consider the wave propagating to the left) and the average horizontal current: w1=Aa1,a2,t1,t2exp⁡ika0+ωt0+kb+ψ1a1,a2,b,t1,t2,Ω1=0. Here, A is the complex amplitude of the wave, ω is its frequency, and k is the wave number. The function ψ1 is real and will be found in the next approximation.

The substitution of solution (13) into Eq. (12) yields the equation for the pressure, ρ-1p1a0+ip1b=ω2-gkAexp⁡ika0+ωt0+kb, which is solved analytically as p1=-Reiω2-gkkρAexp⁡ika0+ωt0+kb+C1a1,a2,t1,t2, where C1 is an arbitrary function. The boundary condition at the free surface is p1b=0=0, which leads to ω2=gk and C1=0. Thus, in the first approximation the pressure correction p1 is equal to zero.

The equations of the second order of the perturbation theory can be written as follows: Imiw2a0+w2b+iw1a1-w1a1w1b=0,Reiw2t0a0+w2t0b+iw1t0a1+w1t1a0-w1t0a0w1b+w1t1b+w1t0bw1a0=-Ω2,w2t0t0+ρ-1p2a0+ip2b=igw2a0+wa1-2w1t1t0. By substituting expression (13) for w1 into Eq. (16), we obtain Im[iw2a0+w2b-ikψ1bA-Aa1exp⁡ika0+ωt0+kb-ik2A2e2kb+iψ1a1]=0, which is integrated as follows: w2=ikAψ1-bAa1exp⁡ika0+ωt0+kb+ψ2+if2, where ψ2,f2 are the functions of slow coordinates and the Lagrangian vertical coordinate b, and f2b=k2A2exp⁡2kb-ψ1a1, where ψ2 is an arbitrary real function. It will be determined in a solution in the cubic approximation.

When Eqs. (13) and (20) are substituted into Eq. (17), the sum of the terms containing the exponential factor becomes equal to zero, and the remaining terms satisfy the equation ψ1t1b=-2k2ωA2exp⁡2kb-Ω2. The expression for the function ψ1 can be found by simple integration. It should be emphasized that the vorticity in the second approximation, which is part of Eq. (22), is an arbitrary function of slow horizontal and vertical Lagrangian coordinates so that Ω2=Ω2a1,a2,b.

Taking into account the solutions in the first two approximations, we can write Eq. (18) as ρ-1p2a0+ip2b=igAa1-2ωAt1exp⁡ika0+ωt0+kb+igψ1a1. Its solution determines the pressure correction: p2=Re1kgAa1-2ωAt1exp⁡ika0+ωt0+kb+ρg0bψ1a1db+C2a1,a2,t1,t2. The integration limits in the penultimate term are chosen so that this integral term equals zero at the free surface. Due to the boundary condition for pressure, (p2b=0=0),C2=0, and At1-cgAa1=0;cg=g2ω=12gk, where cg is the group velocity of wave propagation in deep water, which in this approximation is independent of fluid vorticity. As expected, in this approximation the wave moves with group velocity cg to the left (the “minus” sign in Eq. 24).

Cubic approximation

The equation of continuity and the condition of the conservation of vorticity in the third approximation are written in the form Imiw2a0+w3b+iw1a2+w2a1+w2a0-w1a1+w2a2w1b-w1a0w2b=0,Reiw3t0a0+w3t0b+iw1t2a0+w1t1a1+w1t0a2+w2t1a0+w2t0a1+w1t2b-w2bw1t0a0-+w2t1b-w1bw1t0a1+w1t1a0+w2t0a0++w1a0w1t1b+w2t0b+w1t0bw1a1+w2a0=-Ω3. We substitute the solutions in the first and second approximations into the simultaneous equations: Imiw3a0+w3b+iψ1a2+ψ2a1+2k(kb+1)AAa1e2b+Gbeika0+ωt0+kb=0,Reiw3a0+w3b+Gb+2kψ1t1bω-1Aeika0+ωt0+kbt0+ψ2t1b+ψ1t2b++iωk4kb+5AAa1e2kb=-Ω3,G=ibAa2+b22Aa1a1-kb+1ψ1Aa1-ikψ2+kf2-k22ψ12A. We seek a solution for the third approximation in the following form: w3=G1-Geika0+ωt0+kb+G2e-ika0+ωt0+kb+ψ3+if3, where G1,G2,ψ3, and f3 are functions of slow coordinates and b. By substituting this expression into Eqs. (28) and (29), we immediately find f3b+ψ2a1+ψ1a2+k(kb+1)AAa1-AAa1e2kb=0,ψ2t1b+ψ1t2b+124kb+5ωkAAa1-AAa1e2kb=-Ω3. The function ψ2 according to Eq. (33) is determined by known solutions for A and ψ1 and by the given distribution of Ω3. The expression for the function f3 is then derived from Eq. (32). These functions determine the horizontal and vertical average motion, respectively. But in this approximation they are not included in the evolution equation for the wave envelope. The function ψ3 will be found in the next approximation.

When solving Eqs. (28) and (29), we found G1=-kω-1ψ1t1A,G2=kω-12ke-2kb-bψ1t1e2kbdb-ψ1t1A. These relationships should be substituted into Eq. (7), which in this approximation has the form w3t0t0-igw3a0=iρ-1ip2a1+p3a0-p3b-p2bw1a0+ρgw1a2+w2a1--2w1t2t0-w1t1t1-2w2t0t1. Taking into account Eqs. (13), (20), (24), (31), and (34), we rewrite it as follows: ρ-1p3a0+ip3b=-2iωAt2+igAa2-2At12+2ωkψ1t1Aeika0+ωt0+kb++2ω2G2Ae-ika0+ωt0+kb+igψ2a1+ψ1a2+I,I=-gf2a1-b0ψ1a1a1db-ψt1t1. By virtue of the relationships (21), (22), and (25), the derivative of I along the vertical Lagrangian coordinate is zero (Ib=0), so I is the only function of the slow coordinates and time - al,tl,l1. The contribution of the term Ial,tl0 to the pressure is complex, so it demands I=0.

The solution to Eq. (36) yields the expression for the pressure perturbation in the third approximation: p3ρ=Reik-12iωAt2-igAa2+2At12-4ωk2Ae-2kb-bψ1t1e2kbdbeika0+ωt0+kb++ρg0bψ2a1+ψ1a2db. In Eq. (37), the integration limits for the second integral term have been preset to satisfy the boundary condition at the free surface (the pressure p3 should turn to zero). Then the factor before the exponent should be equal to zero: 2iωAt2-igAa2+2At12-4ωk2A-0ψ1t1e2kbdb=0. By introducing the “running” coordinate ζ2=a2+cgt2, we can reduce Eq. (38) to a compact form: iAa2-kω22At12+4k3Aω-0ψ1t1e2kbdb=0. Further, it will be shown that the variables in Eqs. (38) and (39) have been chosen so that they could be easily reduced (under particular assumptions) to the classical NLS equation.

The explicit form of the function ψ1t1 is found by integrating Eq. (22): ψ1t1=-kωA2e2kb--bΩ2a2,bdb-Ua2,t1. This expression includes three terms. All of them describe a certain component of the average current. The first one is proportional to the square of the amplitude modulus and describes the classical potential drift of fluid particles (see Henderson et al., 1999, for example). The second one is caused by the presence of low vorticity in the fluid. Finally, the third item, including Ua2,t1, describes an additional potential flow. It appears in the integration of Eq. (22) over the vertical coordinate b and will evidently not disappear in the case of A=0. This is a certain external flow that is chosen depending on a specific problem. Note that a term of that kind arises in the Eulerian description of potential wave oscillations of the free surface as well. In the paper by Stocker and Peregrine (1999), U=Usin⁡kx-ωt was chosen and interpreted as a harmonically changing surface current induced by an internal wave. We shall further take U=0.

After the substitution of Eq. (40), Eq. (39) may be written in the final form iAa2-kω22At12-kk2A2+βa2A=0,βa2=4k2ω-0e2kb-bΩ2a2,bdbdb. It is the nonlinear Schrödinger equation for the packet of surface gravity waves propagating in the fluid with vorticity distribution Ω=ε2Ω2a2,b. The function Ω2a2,b determining flow vorticity may be an arbitrary function setting the initial distribution of vorticity. On integrating it twice, we find the vortex component of the average current, which is in no way related to the average current induced by the potential wave.

Examples of the waves

Let us consider some special cases following from Eq. (41).

Potential waves

In this case, Ω2=0 and Eq. (41) becomes the classical nonlinear Schrödinger equation for waves in deep water. Three kinds of analytical solutions to the NLS equation are usually discussed regarding water waves. The first one is the Peregrine breather propagated in space and time (Peregrine, 1983). This wave may be considered as a long wave limit of a breather, which is a pulsating mode of infinite wavelength (Grimshaw et al., 2010). The two others are the Akhmediev breather, which is the solution periodic in space and localized in time (Akhmediev et al., 1985), and the Kuznetsov–Ma breather, which is the solution periodic in time and localized in space (Kuznetsov, 1977; Ma, 1979). Both of the latter solutions evolve against the background of an unperturbed sine wave.

Gerstner waves

The exact Gerstner solution in complex form is written as (Lamb, 1932; Abrashkin and Yakubovich, 2006; Bennett, 2006) W=a+ib+iAexp⁡ika+ωt+kb. It describes a stationary traveling rotational wave with a trochoidal profile. Its dispersion characteristic coincides with the dispersion of linear waves in deep water ω2=gk. The fluid particles move in circles and there is no drift current.

Equation (42) is the exact solution to the problem. Following Eqs. (8) and (9), the Gerstner wave should be written as W=a0+ib+n1εniAexp⁡ika0+ωt0+kb. All of the functions wn in Eqs. (8) and (9) have the same form. To derive the vorticity of the Gerstner wave, Eq. (43) should be substituted into Eq. (6). Then one can find that in the linear approximation, the Gerstner wave is potential (Ω1=0), but in the quadratic approximation it possesses vorticity: Ω2Gerstner=-2ωk2A2e2kb.

For this type of vorticity distribution, the sum of the first two terms in the parentheses in Eq. (41) is equal to zero. From the physical point of view, this is due to the fact that the average current induced by the vorticity compensates exactly for the potential drift. The packet of weakly nonlinear Gerstner waves in this approximation is not affected by their nonlinearity, and the effect of the modulation instability for the Gerstner wave does not occur.

Generally speaking, this result is quite obvious. As there is no particle drift in the Gerstner wave, the function ψ1 equals zero. So, the multiplier of the wave amplitude in Eqs. (38) and (39) may be neglected without finding the vorticity of the Gerster wave.

Let us consider some particular consequences of the obtained result. For the irrotational (Ω2=0) stationary (A=A=const) wave, Eq. (40) for the velocity of the drifting flow takes the form ψ1t1=-ωkA2e2kb. It coincides with the expression for the Stokes drift in Lagrangian coordinates (in the Eulerian variables the profile of the Stokes current may be obtained by the substitution of b for y). Thus, our result may be interpreted as a compensation of the Stokes drift by the shear flow induced by the Gerstner wave in a quadratic approximation. This conclusion is also fair in the “differential” formulation for vorticities. From Eq. (22), it follows that the vorticity of the Stokes drift equals the vorticity of the Gerstner wave with the inverse sign.

The absence of a nonlinear term in the NLS equation for the Gerstner waves obtained here in the Lagrangian formulation is a robust result and should appear in the Euler description as well. This follows from the famous Lighthill criterion for the modulation instability because the dispersion relation for the Gerstner wave is linear and does not include terms proportional to the wave amplitude.

Gouyon waves

As shown by Dubreil-Jacotin (1934), the Gerstner wave is a special case of a wide class of stationary waves with vorticity Ω=εΩψ, where Ω is an arbitrary function and ψ is a stream function. Those results were later developed by Gouyon (1958), who explicitly represented the vorticity in the form of a power series Ω=n=1εnΩnψ (see also the monograph by Sretensky, 1977).

When a plane steady flow is considered in the Lagrangian variables, the stream lines ψ coincide with the isolines of the Lagrangian vertical coordinate b (Abrashkin and Yakubovich, 2006; Bennett, 2006). We consider a steady-state wave at the surface of indefinitely deep water. Assume that there is no undisturbed shear current, but the wave disturbances have vorticity. Then, the formula for the vorticity is written as Ω=n=1εnΩnb. Here we will refer to the steady-state waves propagating in such a low-vorticity fluid as Gouyon waves. The properties of the Gouyon wave for the first two approximations were studied by Abrashkin and Zen'kovich (1990) in the Lagrangian description.

In our case, Ω1=0 and Ω20; assuming the function Ω2 to be independent of the coordinate a, we can describe the Gouyon waves. The vorticity Ω2 depends on the coordinate b only and has the following form: Ω2Goyuon=ωk2A2Hkb, where Hkb is an arbitrary function. In the case of Hkb=-2exp⁡2kb, the vorticities of the Gerstner and Gouyon waves in the quadratic approximation coincide (compare Eqs. 44 and 46). In the considered approximation, the Gouyon wave generalizes the Gerstner wave. From Eq. (22), it follows that the function ψt1 is equal to zero only when the vorticity of the Gouyon wave is equal to the vorticity of the Gerstner wave. Except for this case, the average current ψt1 will always be present in the modulated Gouyon waves.

The substitution of the ratio (46) into Eq. (41) yields the NLS equation for the modulated Gouyon wave: iAa2-kω22At12-βGk3A2A=0,βG=1+4-0e2b̃-b̃Hb̃db̃db̃,b̃=kb, where b̃ is a dimensionless vertical coordinate. The coefficient of the nonlinear term in the NLS equation varies when the wave vorticity is taken into account. For the Gerstner wave it may be equal to zero like for the Gouyon wave when the following condition is satisfied: -0e2b̃-b̃Hb̃db̃db̃=-14. Clearly, an infinite number of distributions of the vorticity H(b̃) meeting this condition are possible. However, the realization of one of them seems hardly probable. In the real ocean, distributions of the vorticity with a certain sign of βG are more likely to be implemented. Its negative values correspond to the defocusing NLS equation and the positive ones are related to the focusing NLS equation. In the latter case, the maximum value of the increment and the width of the modulation instability zone of a uniform train of vortex waves vary depending on the value of βG.

Equations (39) and (47) will be focusing for ψ1t1<0,b0 and defocusing if ψ1t1>0,b0. The case of the sign–variable function ψ1t1requires additional research. From the physical viewpoint, the sign of this function is defined by the ratio of the velocity of the Stokes drift (45) to the velocity of the current induced by the vorticity (the integral term in Eq. 40). For ψ1t1<0, the Stokes drift either dominates over a vortex current or both of them have the same direction. When ψ1t1>0, the vortex current dominates over the counter Stokes drift. In the case of the sign variable ψ1t1, the ratio of these currents varies at different vertical levels, thereby requiring a direct calculation of βG.

Waves with inhomogeneous vorticity distribution along both coordinates

Neither a vorticity expression nor methods of its definition were discussed when deriving the NLS equation. Sections 4.2 and 4.3 are devoted to the problems of the Gerstner and Gouyon waves; the vorticity was set to be proportional to a square modulus of the wave amplitude. Note that waves can propagate against the background of some vortex current, for example, the localized vortex. In this case, the vorticity may be presented in the form Ω2a2,b=ωϕva2,b+k2A2ϕwa2,b, where the function ωϕv defines the vorticity of the background vortex current and the function ωk2A2ϕw defines the vorticity of waves. In the most general case, both functions depend on the horizontal Lagrangian coordinate as well. Then, Eq. (41) takes the form iAa2-kω22At12-kβva2A-k31+βwa2A2A=0,βv,wa2=4-0e2b̃-b̃ϕv,wa2,b̃db̃db̃. The substitution A=Aexp⁡-ik-a2βva2da2 reduces Eq. (49) to the NLS equation with a nonuniform multiplier for the nonlinear term: iAa2-kω22At12-k31+βwa2A2A=0. Let us consider the propagation of the Gouyon wave when βw=const=βG-1, and Eq. (51) turns into the classical NLS equation (47). As shown in Sect. 4.3, it describes the modulated Gouyon waves. Therefore, based on the substitution of Eq. (50) one can conclude that the propagation of the Gouyon waves against the background of the nonuniform vortex current results in the variation in the wave number of the carrier wave. For βw=0, Eq. (51) describes the propagation of a packet of potential waves against the background of the nonuniform weakly vortical current. The specific features of the wave propagation related to the variable βw require special investigation.

On the equivalence of Lagrangian and Eulerian approaches

Consider the correlation between the Eulerian and the Lagrangian description of wave packets. To obtain the value for the elevation of the free surface we substitute the expressions (8), (9), and (13) and b=0 into the equation for Y=ImW written in the following form: YL=εImAa2,t1exp⁡ika0+ωt0, where Aa2,t1 is the solution to Eq. (41). This expression defines the wave profile in Lagrangian coordinates. To rewrite this equation in the Eulerian variables, it is necessary to define a via X. From the relation (8), it follows that X=a+εRew1+n=2εn-1wn=a+Oε, and the elevation of the free surface in the Eulerian variables YE will be written as YE=εImAX2,t1exp⁡ikX0+ωt0+Oε2,Xl=εlX. The coordinate a plays the role of X, so the following substitutions are valid for the Lagrangian approach: a0X0;a1X1;a2X2. This result may be called an “equivalence principle” between the Lagrange and the Euler descriptions for solutions in the linear approximation. This principle is valid for both the potential and rotational waves.

To express the solution to Eq. (41) in the Eulerian variables, it is necessary to use the equivalence principle and to replace the horizontal Lagrangian coordinate a2 with the X2 coordinate. So, there are no discrepancies between the Eulerian and the Lagrangian estimations of the NLS equation for the free surface elevation.

Taking this into account, we can conclude that the result will be the same in the Eulerian description if the vorticity Ω2 is a function of the x,y coordinates. So, when studying the wave packet dynamics in the vortical liquid in the Eulerian variables, it is necessary to replace (for example, in Eqs. 41 and 51) the horizontal Lagrangian coordinate with the Eulerian one.

Equation (47) can also be derived in Eulerian variables. The key idea is to take into consideration a weak shear flow. This approach is similar to the method used in the paper by Hjelmervik and Trulsen (2009), where the wave propagates along a weak horizontal shear current. Shrira and Slunyaev (2014) used this technique to study trapped waves in a uniform jet stream. They derived the NLS equation for a single mode. Later, Slunyaev (2016) generalized the result to the case of a vortex jet flow. Our result was obtained with a weak vertical shear flow taken into account. In particular, to describe modulated Guyon waves, the Johnson approach (1976) should be modified, assuming a shear flow of the order of epsilon.

The solutions to the considered problem in the Lagrange and the Euler forms in the quadratic and cubic approximations differ from each other. To obtain a full solution in the Lagrange form, one should find the functions ψ1,ψ2,ψ3,f2, and f3. This problem should be considered within a special study.

Conclusions

We have derived the vortex-modified nonlinear Schrödinger equation using the method of multiple-scale expansions in the Lagrangian variables. The fluid vorticity Ω is specified as an arbitrary function of the Lagrangian coordinates, which is quadratic in the small parameter of the wave steepness. The calculations have been performed by introducing a complex coordinate of the fluid particle trajectory.

The nonlinear evolution equation for the wave packet in the form of the nonlinear Schrödinger equation has been derived as well. From the mathematical viewpoint, the novelty of this equation is related to the emergence of a new term proportional to the envelope amplitude and the variance of the coefficient of the nonlinear term. If the vorticity depends on the vertical Lagrangian coordinate only (Gouyon waves), this coefficient is constant. There are special cases when the coefficient of the nonlinear term equals zero and the resulting nonlinearity disappears. The Gerstner wave belongs to the latter case. Another effect revealed in the present study is the relation of the vorticity to the wave number shift in the carrier wave. This shift is constant for the modulated Gouyon wave. If the vorticity depends on both Lagrangian coordinates, the shift of the wave number is horizontally inhomogeneous. It is shown that the solution to the NLS equation for weakly rotational waves in the Eulerian variables may be obtained from the Lagrangian solution with an ordinary change in the horizontal coordinates.