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.

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

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 (

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

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

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

Colin et al. (1995) considered the evolution of three-dimensional vortex
disturbances in a finite-depth fluid for a different type of vorticity
distribution:

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:

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 (

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.

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):

Problem geometry:

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):

We introduce a complex coordinate of a fluid particle trajectory

The Lagrangian coordinates mark the position of fluid particles. In the
Eulerian description, the displacement of the free surface

Let us represent the function

In a first approximation in the small parameter, we have the following system
of equations:

The substitution of solution (13) into Eq. (12) yields the equation for the
pressure,

The equations of the second order of the perturbation theory can be written
as follows:

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

Taking into account the solutions in the first two approximations, we can
write Eq. (18) as

The equation of continuity and the condition of the conservation of vorticity in
the third approximation are written in the form

When solving Eqs. (28) and (29), we found

The solution to Eq. (36) yields the expression for the pressure perturbation
in the third approximation:

The explicit form of the function

After the substitution of Eq. (40), Eq. (39) may be written in the final
form

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

In this case,

The exact Gerstner solution in complex form is written as (Lamb, 1932;
Abrashkin and Yakubovich, 2006; Bennett, 2006)

Equation (42) is the exact solution to the problem. Following Eqs. (8) and
(9), the Gerstner wave should be written as

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

Let us consider some particular consequences of the obtained result. For the
irrotational (

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.

As shown by Dubreil-Jacotin (1934), the Gerstner wave is a special
case of a wide class of stationary waves with vorticity

When a plane steady flow is considered in the Lagrangian variables, the
stream lines

In our case,

The substitution of the ratio (46) into Eq. (41) yields the NLS equation for
the modulated Gouyon wave:

Equations (39) and (47) will be focusing for

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

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

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

Taking this into account, we can conclude that the result will be the same in
the Eulerian description if the vorticity

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

We have derived the vortex-modified nonlinear Schrödinger equation using
the method of multiple-scale expansions in the Lagrangian variables. The fluid
vorticity

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.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

E. Pelinovsky appreciates the support obtained from the RNF under grant 16-17-00041. The authors wish to thank the editor, Roger Grimshaw, the reviewers for their very useful comments, and Nadezhda Krivatkina for providing English corrections.Edited by: R. Grimshaw Reviewed by: two anonymous referees