NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-255-2017Lagrange form of the nonlinear Schrödinger equation for
low-vorticity waves in deep waterAbrashkinAnatolyPelinovskyEfimpelinovsky@hydro.appl.sci-nnov.ruhttps://orcid.org/0000-0002-5092-0302National Research University Higher School of Economics (HSE),
25/12 Bolshaya Pecherskaya Str., 603155 Nizhny Novgorod, RussiaInstitute of Applied Physics RAS, 46 Ulyanov Str., 603950 Nizhny
Novgorod, RussiaNizhny Novgorod State Technical University n.a. R. Alekseev, 24
Minina
Str., 603950 Nizhny Novgorod, RussiaEfim Pelinovsky (pelinovsky@hydro.appl.sci-nnov.ru)6June201724225526428November201614December201618April201721April2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/24/255/2017/npg-24-255-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/255/2017/npg-24-255-2017.pdf
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 -U′y. At the same time, the
vorticity of wave perturbations Ωn,n≥1, 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=-U′b+∑n≥1εnΩna,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 (Yt→0 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,t2expika0+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-gkAexpika0+ωt0+kb,
which is solved analytically as
p1=-Reiω2-gkkρAexpika0+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.
Quadratic approximation
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-Aa1expika0+ωt0+kb-ik2A2e2kb+iψ1a1]=0,
which is integrated as follows:
w2=ikAψ1-bAa1expika0+ωt0+kb+ψ2+if2,
where ψ2,f2 are the functions of slow coordinates and the
Lagrangian vertical coordinate b, and
f2b=k2A2exp2kb-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=-2k2ωA2exp2kb-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=igAa1-2ωAt1expika0+ωt0+kb+igψ1a1.
Its solution determines the pressure correction:
p2=Re1kgAa1-2ωAt1expika0+ωt0+kb+ρg∫0bψ1a1db+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-w2b‾w1t0a0-+w2t1b-w1bw1t0a1+w1t1a0+w2t0a0++w1a0‾w1t1b+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)AAa1‾e2b+Gbeika0+ωt0+kb=0,Reiw3a0+w3b+Gb+2kψ1t1bω-1Aeika0+ωt0+kbt0+ψ2t1b+ψ1t2b++iωk4kb+5AAa1‾e2kb=-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‾-A‾Aa1e2kb=0,ψ2t1b+ψ1t2b+124kb+5ωkAAa1‾-A‾Aa1e2kb=-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∫-∞bψ1t1e2kb′db′-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=-2iω∂A∂t2+ig∂A∂a2-∂2A∂t12+2ωkψ1t1Aeika0+ωt0+kb++2ω2G2A‾e-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,l≥1. The contribution of the term Ial,tl≠0 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-12iω∂A∂t2-ig∂A∂a2+∂2A∂t12-4ωk2Ae-2kb∫-∞bψ1t1e2kb′db′eika0+ωt0+kb++ρg∫0bψ2a1+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:
2iω∂A∂t2-ig∂A∂a2+∂2A∂t12-4ωk2A∫-∞0ψ1t1e2kbdb=0.
By introducing the “running” coordinate ζ2=a2+cgt2, we can reduce Eq. (38) to a compact form:
i∂A∂a2-kω2∂2A∂t12+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,b′db′-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=U∗sinkx-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
i∂A∂a2-kω2∂2A∂t12-kk2A2+βa2A=0,βa2=4k2ω∫-∞0e2kb∫-∞bΩ2a2,b′db′db.
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+iAexpika+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+∑n≥1εn⋅iAexpika0+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=εΩ∗ψ, 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=1∞εnΩnψ (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=1∞εnΩnb. 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 Ω2≠0; 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=-2exp2kb, 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:
i∂A∂a2-kω2∂2A∂t12-βGk3A2A=0,βG=1+4∫-∞0e2b̃∫-∞b̃Hb̃′db̃′db̃,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∫-∞b̃Hb̃′db̃′db̃=-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,b≤0 and defocusing if ψ1t1>0,b≤0.
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
i∂A∂a2-kω2∂2A∂t12-kβva2A-k31+βwa2A2A=0,βv,wa2=4∫-∞0e2b̃∫-∞b̃ϕv,wa2,b̃′db̃′db̃.
The substitution
A∗=Aexp-ik∫-∞a2βva2da2
reduces Eq. (49) to the NLS equation with a nonuniform multiplier for the
nonlinear term:
i∂A∗∂a2-kω2∂2A∗∂t12-k31+βwa2A∗2A∗=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,t1expika0+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,t1expikX0+ωt0+Oε2,Xl=εlX.
The coordinate a plays the role of X, so the following substitutions are
valid for the Lagrangian approach:
a0→X0;a1→X1;a2→X2.
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.
No data sets were used in this article.
The authors declare that they have no conflict of
interest.
Acknowledgements
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
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