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Linear water waves with vorticity: rotational
features and particle paths
Mats Ehrnstr¨ om
Department of Mathematics, Lund University, PO Box 118, 221 00 Lund, Sweden.
Gabriele Villari
Dipartimento di Matematica, Viale Morgagni 67/A, 50134 Firenze, Italy.
Abstract
Steady linear gravity waves of small amplitude travelling on a current of constant
vorticity are found. For negative vorticity we show the appearance of internal waves
and vortices, wherein the particle trajectories are not any more closed ellipses. For
positive vorticity the situation resembles that of Stokes waves, but for large vorticity
the trajectories are affected.
Key words: Steady water waves, Vorticity, Particle paths, Trajectories, Phase
portrait
1991 MSC: 35Q35, 76B15, 37N10
1 Introduction
The subject of this paper are periodic gravity water waves travelling with
constant shape and speed. Such wave-trains are an everyday observation and,
typically, one gets the impression that the water is moving along with the
wave. In general, this is not so. Rather, the individual fluid particles display a
motion quite different from that of the wave itself. While for irrotational waves,
recent studies have enlightened the situation, we investigate the situation of
waves propagating on a rotational current, so that there is a non-vanishing
curl within the velocity field.
Email addresses: mats.ehrnstrom@math.lu.se (Mats Ehrnstr¨ om),
villari@math.unifi.it (Gabriele Villari).
Preprint submitted to5 February 2008
arXiv:0712.0608v1 [math-ph] 4 Dec 2007
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For irrotational waves, there is a classical first approximation showing that the
fluid particles move in ellipses, back and forth as the wave propagates above
them. This can be found in classical [1,2,3] as well as modern [4,5] text books,
and it is consistent with the only known explicit solutions for gravity water
waves: the Gerstner wave [6,7] for deep water, and the edge wave solution for a
flat beach [8], both with a depth-varying vorticity. A formal physical argument
involving a balance between opposing forces was used in [9] to get a similar
result without the use of irrotationality. There are also experimental evidence
supporting this picture. Those include photographs [4,10,3] and movie films
[11].
However, as anyone having used bottle post would guess, there are other find-
ings. Even in [9], where it is asserted that the orbits are elliptic, and where the
photographs and movie films are referenced, the author notes that “I am not
aware of any measurements that show that the particle orbits of shallow water
waves are indeed ellipses.” In fact it was observed already in the 19th century
that there seems to be a forward mass drift [12], so that the average motion
of an average fluid particle is along with the wave. This phenomenon can be
seen by making a second approximation of the governing equations, and it is
known as Stokes drift (see also [13,14]). In [15,16] it was deduced that for steep
waves the orbits deviate from simple ellipses. There is also mathematical evi-
dence uniformly showing that a more thorough study of the equations yields
non-closed orbits with a slight forward drift. Those include investigations of
the precise orbits of the linearized system [17,18,19], and two recent papers
on exact Stokes waves [20,21] (steady irrotational and periodic gravity waves
which are symmetric and monotone between trough and crest). The relation
between such results and experimental data is discussed in [20], where it is
argued that the ellipses – at least near the bottom – are approximations of
the exact trajectories.
While many situations are adequately modelled by irrotational flows – e.g.
waves propagating into still water – there are situations when such a math-
ematical model is insufficient. Tidal flow is a well-known example when con-
stant vorticity is an appropriate model [22], a fact confirmed by experimental
studies [23]. This is one reason why, recently, the interest for exact water
waves with vorticity has increased. At this point existence [24], variational
characterization [25], uniqueness [26,27], symmetry [28], and a unique contin-
uation principle [29] for finite depth steady gravity waves with vorticity are
established. There is also a theory for deep-water waves [30,31], as well as for
capillary and capillary-gravity waves [32,33]. However, due to the intricacy of
the problem, studies of the governing equations for water waves are extremely
difficult. In-depth analyses are very rare. To gain insight into qualitative fea-
tures of flows with vorticity Ko and Strauss recently performed a numerical
study [34] extending earlier work by DaSilva and Peregrine. We will pursue a
different approach. Notice that the intuitive notion of vorticity is captured in
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what happens when one pulls the tap out of a bath tub. It should therefore
come as no surprize that the particle paths of waves travelling upon a rota-
tional current deviate from those in the case of waves without vorticity. That
is the main result of this paper. More precisely, we make a first attempt at
understanding the particle trajectories by deducing a linear system for con-
stant vorticity. Here, linearity means that the waves are small perturbations
of shear flows, hence of small amplitude. The system obtained is solvable in
the sense of closed expressions, and thus it is possible to make a phase portrait
study of the steady wave.
It is found that for positive vorticity, the steady wave resembles that of the
irrotational situation [17,18], though the physical particle paths behave differ-
ently if the size of the vorticity is large enough. For negative vorticity, however,
we show the existence of a steady periodic surface wave containing an internal
wave as well as a vortex, or so called cats-eye (cf. [35] and [36, Ex. 2.4]). For
unit depth this situation occurs if the absolute size of the negative vorticity
exceeds the wave speed, while in the opposite situation both the steady wave
and the physical particle trajectories resemble the irrotational case. When the
size of the negative vorticity exceeds the wave speed the particle trajectories
of the internal wave behave in the same manner as in the irrotational case –
nearly closed ellipses with a forward drift – but within the vortex and the sur-
face wave the particles are moving mainly forward. This indicates that such a
wave may be unordinary or unstable, since measurements show that for waves
not near breaking or spilling the speed of an individual particle is generally
considerably less than that of the wave itself [2]. Such a situation is excluded
in [20,21], and our result is therefore not in contrast to those investigations.
An interesting feature of the phase portrait for negative vorticity is that it
captures the almost ideal picture of what vorticity is. It furthermore indicates
that in the case of large negative vorticity the governing equations allow for
travelling waves very different from the classical Stokes waves (see [37] for
a good reference of that subject). Finding those waves with analytic tools
could prove difficult; so far the existence results [24,31,38] for steady waves
with vorticity rely on the assumption that no particle moves as fast as the
wave itself. This study suggests that the presence of vorticity – even when it
is constant – changes the particle trajectories in a qualitative way, that this
change depends on the size of the vorticity, and that it applies less to particles
near the bottom.
The disposition is as follows. Section 2 gives the mathematical background for
the water wave problem, while in Section 3 we deduce the linearization and its
solution. The main findings are presented in Section 4, and the implications
for the particle trajectories in Section 5. In Section 6 we give a brief summary
and discussion of our results.
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2 Preliminaries
The waves that one typically sees propagating on the surface of the sea are
locally approximately periodic and two-dimensional (that is, the motion is
identical in any direction parallel to the crest line). Therefore – for a descrip-
tion of these waves propagating over a flat bed – it suffices to consider a cross
section of the flow that is perpendicular to the crest line. Choose Cartesian
coordinates (x,y) with the y-axis pointing vertically upwards and the x-axis
being the direction of wave propagation, while the origin lies on the flat bed
below the crest. Let (u(t,x,y), v(t,x,y)) be the velocity field of the flow, let
h > 0 be the depth below the mean water level y = h, and let y = h + η(t,x)
be the water’s free surface. We assume that gravity is the restoring force once
a disturbance was created, neglecting the effects of surface tension. Homo-
geneity (constant density) is a physically reasonable assumption for gravity
waves [2], and it implies the equation of mass conservation
ux+ vy= 0 (2.1a)
throughout the fluid. Appropriate for gravity waves is the assumption of in-
viscid flow [2], so that the equation of motion is Euler’s equation
where P(t,x,y) denotes the pressure and g is the gravitational constant of
acceleration. The free surface decouples the motion of the water from that of
the air so that, ignoring surface tension, the dynamic boundary condition
ut+ uux+ vuy= −Px,
vt+ uvx+ vvy= −Py− g,
(2.1b)
P = P0
ony = h + η(t,x), (2.1c)
must hold, where P0is the constant atmospheric pressure [5] . Moreover, since
the same particles always form the free surface, we have the kinematic bound-
ary condition
v = ηt+ uηx
The fact that water cannot penetrate the rigid bed at y = 0 yields the kine-
matic boundary condition
ony = η(t,x). (2.1d)
v = 0ony = 0. (2.1e)
The vorticity, ω, of the flow is captured by the curl,
vx− uy= ω. (2.1f)
We now introduce a non-dimensionalization of the variables. As above, h is the
average height above the bottom, and we let a denote the typical amplitude,
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and λ the typical wavelength. It is reasonable – and fruitful – to take√gh as the
scale of the horizontal velocity. That is the approximate speed of irrotational
long waves [5]. We shall use c to denote the wave speed, and we let
c ?→
c
√gh
be the starting point of the non-dimensionalization. We then make the trans-
formations
√ght
λ
x ?→x
λ,
y ?→y
h,
t ?→
,u ?→
u
√gh,
v ?→
λv
h√gh,
η ?→η
a.
Having made these transformations, define furthermore a new pressure func-
tion p = p(t,x,y) by the equality
P ≡ P0+ gh(1 − y) + ghp.
Here P0is the constant atmospheric pressure, and gh(1−y) is the hydrostatic
pressure distribution, describing the pressure change within a stationary fluid.
The new variable p thus measures the pressure perturbation induced by a
passing wave. It turns out that the natural scale for the vorticity is
we thus map
?h
The water wave problem (2.1) then transforms into the equations
?
h/g and
ω ?→
gω.
ux+ vy= 0, (2.2a)
(2.2b)ut+ uux+ vuy= −px,
vt+ uvx+ vvy= −λ2
h2
λ2vx− uy= ω,
h2py, (2.2c)
(2.2d)
valid in the fluid domain 0 < y < 1 +a
hη, and
v =a
p =a
h(ηt+ uηx),
hη,
(2.2e)
(2.2f)
valid at the surface y = 1 +a
(2.1e) on the flat bed y = 0. Here appear naturally the parameters
hη, in conjunction with the boundary condition
ε ≡a
h,
δ ≡h
λ,
called the amplitude parameter, and the shallowness parameter, respectively.
Since the shallowness parameter is a measure of the length of the wave com-
pared to the depth, small δ models long waves or, equivalently, shallow water
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waves. The amplitude parameter measures the relative size of the wave, so
small ε is customarily used to model a small disturbance of the underlying
flow. We now set out to study steady (travelling) waves, and will therefore
assume that the equations (2.1) have a space-time dependence of the form
x − ct in the original variables, corresponding to λ(x − ct) in the equations
(2.2). The change of variables
(x,y) ?→ (x − ct,y)
yields the problem
ux+ vy= 0, (2.3a)
(2.3b)(u − c)ux+ vuy= −px,
(u − c)vx+ vvy= −py
δ2vx− uy= ω,
δ2, (2.3c)
(2.3d)
valid in the fluid domain 0 < y < 1 + εη,
v = ε(u − c)ηx,
p = εη,
(2.3e)
(2.3f)
valid at the surface y = 1 + εη, and
v = 0 (2.3g)
along the flat bed y = 0.
3 The linearization
To enable the study of explicit solutions, we shall linearize around a laminar
– though rotational – flow. Such shear flows are characterized by the flat
surface, y = 1, corresponding to η = 0, so insertion of this into (2.3) yields
the one-parameter family of solutions,
U(y) ≡ U(y;s) ≡ s −
?y
0
ω(y)dy,
with η = 0, p = 0, v = 0. We now write a general solution as a perturbation
of such a solution U, i.e.
u = U + ε˜ u,v = ε˜ v,p = ε˜ p. (3.1)
We know from the exact theory of water waves that such solutions exist at
the points where the non-trivial solutions bifurcate from the curve of trivial
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flows [24]. Remember that small ε corresponds to waves whose amplitude is
small in comparison with the depth. Since the surface is described by 1 + εη,
η should thus be of unit size. Dropping the tildes, we obtain
ux+ vy= 0, (3.2a)
(3.2b)(U − c)ux+ vUy+ ε(vuy+ uux) = −px,
(U − c)vx+ ε(vvy+ uvx) = −py
δ2, (3.2c)
valid in the fluid domain 0 < y < 1 + εη,
v = (U − c + εu)ηx,
p = η,
(3.2d)
(3.2e)
valid at the surface y = 1 + εη, and
v = 0(3.2f)
on the flat bed y = 0. The corresponding linearized problem is valid in the
sense that its solution satisfies the exact equations except for an error whose
size can be expressed as a square of the size of the linear solution. The lin-
earization is attained by formally letting ε → 0, and it is given by
ux+ vy= 0, (3.3a)
(3.3b)(U − c)ux+ vUy= −px,
(U − c)vx= −py
δ2, (3.3c)
valid for 0 < y < 1, and
v = (U − c)ηx,
p = η,
(3.3d)
(3.3e)
valid for y = 1. In order to explicitly solve this problem we restrict ourselves
to the simplest possible class of vorticities, i.e. when ω(y) = ω ∈ R is constant.
It then follows that
U(y;s) = −ωy + s.
Looking for separable solutions we make the ansatz η(x) = cos(2πx) (note
that the original wavelength λ and the original amplitude a have both been
non-dimensionalized to unit length). The solution of (3.3) is then given by
u(x,y)
v(x,y)
p(x,y)
= 2δπC cos(2πx)cosh(2πδy),
= 2πC sin(2πx)sinh(2πδy),
= C cos(2πx)
?
2πδ(c − s + ωy)cosh(2πδy) − ω sinh(2πδy)
?
(3.4)
,
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where
C ≡c − s + ω
sinh(2πδ),
and c,δ,h,s,ω must satisfy the relation
(c − s + ω)
?
2πδ(c − s + ω)coth(2πδ) − ω
?
= 1 (3.5)
This indicates that the properties of the wave are adjusted to fit the rotational
character of the underlying flow. Note in (3.4) that while the horizontal and
vertical velocities are given by straightforward expressions, the complexity of
the pressure has drastically increased compared to the irrotational case [17,18].
Remember that this solution is a small disturbance of the original shear flow,
according to (3.1). For small ε, we thus have an approximate solution to (2.3).
To normalize the reference frame Stokes made a now commonly accepted pro-
posal. In the case of irrotational flow he required that the horizontal velocity
should have a vanishing mean over a period. Stokes’ definition of the wave
speed unfortunately cannot be directly translated to waves with vorticity (see
[24]). In the setting of waves with vorticity we propose the requirement
?1
0
u(x,0)dx = 0, (3.6)
a “Stokes’ condition” at the bottom. This is consistent with deep-water waves
(cf. [39]), and for U(y;s) it results in s = 0. As we shall see in subsection 3.1
this indeed seems to be the natural choice of s, since this and only this choice
recovers the well established bound√gh for the wave speed. This is also the
choice made in [22]. We emphazise that (3.6) is only a convention for fixing
the reference frame; without such a reference it is however meaningless to e.g.
discuss whether physical particle paths are closed or not.
The corresponding approximation to the original system (2.1) is
u(t,x,y)= −ωy +a(f+khω)
=
= P0+ g(h − y) +a(f+khω)
×
= h + acos(kx − ft).
sinh(kh)cos(kx − ft)cosh(ky),
sinh(kh)sin(kx − ft)sinh(ky),
k sinh(kh)cos(kx − ft)
?
v(t,x,y)
a(f+khω)
P(t,x,y)
(f + kωy)cosh(ky) − ω sinh(ky)
?
,
η(t,x)
(3.7)
Here
k ≡2π
λ
andf ≡2πc
λ
are the wave number and the frequency, respectively. The size of the distur-
bance is proportional to a in the whole quadruple (η,u,v,p), so this solution
satisfies the exact equation with an error which is O(a2) as a → 0. Concern-
ing the uniform validity of the approximation procedure, leading to the linear
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system, a closer look at the asymptotic expression indicates that this solution
is uniformly valid for
−∞ < x − ct < ∞
asε → 0,
while for the vorticity we have uniform validity in the region
εω = o(1)asε → 0.
A rigorous confirmation of this requires a detailed analysis similar to that
presented in [40,41], but is outside the scope of our paper.
3.1 The dispersion relation
The identity (3.5) can be stated in the physical variables as the dispersion
relation
?
2k
valid for linearized small amplitude gravity waves on a sheared current of
constant vorticity. Note that s√gh − hω is the surface velocity of the trivial
solution U(y;s) stated in the physical variables. The equation (3.8) is the
general version of the dispersion relation presented in [24, Section 3.3]. The
authors consider waves of wavelength 2π, whence k = 1. They also require that
u < c, and that the relative mass flux is held constant along the bifurcation
curve for which the linearization is the first approximation. They found the
dispersion relation
c−s
gh+hω =
1
?
ω tanh(kh) ±
?
4gk tanh(kh) + ω2tanh2(kh)
?
, (3.8)
c − u∗
0=1
2
?
ω tanh(h) +
?
4g tanh(h) + ω2tanh2(h)
?
,
where u∗
case of (3.8) the problem to uniquely determine c from k, h, and ω is related
to the fact that the requirement u < c is necessary for the theory developed
in [24], while in our linear theory, ω and s can be chosen as to violate that
assumption. E.g., when s = 0 and hω < −c it is easy to see from (3.7) that for
waves of small amplitude a << 1 the horizontal velocity u exceeds the speed of
the wave, at least at the surface where u ≈ −hω > c. The sign in front of the
square root depends on the sign of c−s√gh+hω. It is immediate from (3.8)
that this expression is bounded away from 0. Positivity corresponds to the
case dealt with in [24], and in that case the existence of exact solutions is well
established. Our investigation indicates that there might also be branches of
exact solutions fulfilling the opposite requirement u > c, and as shall be seen
below, in that case it is possible that c is negative so that there are leftgoing
waves on a rightgoing current. In [24] it is assumed that c > 0.
0is the surface velocity of the trivial solution. In the more general
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If c − s√gh + hω is positive, and the vorticity is positive as well, we get a
uniform bound for the speed of the wave. Let
α ≡tanhhk
hk
∈ (0,1).
Then
c
h−s
?g
h=1
2
ω(α − 2) +
?
4gα
h
+ ω2α2
=
2
?
ω2(α − 1) +gα
(2 − α)ω +
h
?
?4gα
h+ ω2α2<
?gα
h,
meaning that
c <
?tanhkh
kh
+ s
??
gh < (1 + s)
?
gh
If instead c − s√gh + hω < 0 and ω < 0, the same argument gives that
c > −(1 + s)
?
gh.
These calculations vindicate the choice of s = 0, since in that case we recover
the classical critical speed√gh.
Another comment is here in place. In [24, Section 3.3] the authors show that
for positive vorticity, local bifurcation from shear flows requires additional
restrictions on the relative mass flux. Again the problem is related to the re-
quirement that u < c, and the reason can be seen directly from their dispersion
relation stating that
?
As ω → −∞ it forces s → −∞ to guarantee that U(y;s) < c for all y ∈ [0,h].
If on the other hand ω → ∞, the inequality (3.9) admits that s → ∞. But
for s big enough, U(0;s) = s√gh > c. Since we allow also u > c there is no
corresponding restriction for positive ω.
c − s
gh + hω > 0(3.9)
To summarize, we have proved
Theorem 3.1 For a linear gravity wave on a linear current U(y;0) = −ωy
we have
c ?= −hω,
and the dispersion relation is given by (3.8) with s = 0, where the square root
is positive (negative) according as c + hw is positive (negative). In particular,
if the speed and the vorticity are of the same sign, then
|c| <
?
gh.
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4 The phase portraits for right-going waves
In this section we study a cross-section of the steady solution for a right-going
wave. This corresponds to a phase-portrait analysis of the ODE-system in
steady variables with c > 0. Since
?
we find that the particle paths are described by the system
˙ x(t), ˙ y(t)
?
=
?
u(x(t),y(t),t),v(x(t),y(t),t)
?
˙ x(t)
˙ y(t)
= −ωy + Acos(kx − ft)cosh(ky)
= Asin(kx − ft)sinh(ky)
(4.1)
where
A ≡a(f + khω)
sinh(kh)
(4.2)
is proportional to the small amplitude parameter a. In order to study the
exact linearised system, let us rewrite (4.1) once more via the transformation
x(t) ?→ X(t) ≡ kx(t) − ft,y(t) ?→ Y (t) ≡ ky(t),(4.3)
yielding
˙X(t)
˙Y (t)
= Ak cos(X)cosh(Y ) − ωY − f
= Ak sin(X)sinh(Y )
(4.4)
Remember that the obtained wave is a perturbation of amplitude size, and
thus the constant A (which includes a) should always be considered very
small in relation to ω and f. Changing sign of A corresponds to the mapping
X ?→ X + π, so we might as well consider A > 0. Since we now study only
right-going waves for which c > 0, for positive vorticity A will always be
positive by (4.2). For large enough negative vorticity, −ω > c/h, the original
A is however negative, meaning that the phase portrait will be translated by
π in the horizontal direction. This is important for the following reason: the
presumed surface
h + acos(X)
attains its maximum at X = 0. Thus the crest for c + hω > 0 is at X = 0 in
our phase portraits, but at X = π for c + hω < 0.
4.1 The case of positive vorticity
Lemma 4.1 The phase portrait for the irrotational case is given by Figure 1,
where the physically realistic wave corresponds to the area of bounded trajec-
tories.
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!02!
Figure 1. The phase portrait for positive and zero vorticity.
Remark 4.2 The details of this are given in [18] and a similar investigation
is pursued in [17]. We therefore give only the main phase plane arguments for
Figure 1. Analytic details can be found in the just mentioned papers.
PROOF. Symmetry and periodicity of (4.4) allow for considering only Ω =
[0,π] × [0,∞). In this region the 0-isocline,˙Y = 0, is given by the boundary
∂Ω, i.e. X = 0, X = π, and Y = 0. Within Ω holds˙Y > 0. The ∞-isocline,
˙X = 0, is the graph of a smooth and convex function
γ(X) = cosh−1
?
f
Ak cosX
?
,X ∈ [0,π/2),
with γ(X) → ∞ as X ? π/2. Here and elsewhere in this paper cosh−1denotes
the positive branch of the pre-image of cosh. We have
˙X(X,Y ) > 0 exactly when Y > γ(X),
whence˙X < 0 for X ∈ [π/2,π] as well as below γ(X).
The only critical point is thus given by P ≡
tory intersecting X = 0 below P can be followed backwards in time below
γ(X) until it reaches X = π. For any trajectory intersecting γ(X) the same
argument holds. Hence there exists a separatrix separating the two different
types of trajectories, and connecting X = π with P.
?
0,cosh−1(f/Ak)
?
. Any trajec-
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Any trajectory intersecting X = 0 above P can be followed forward in time
above γ(X) and is thus unbounded. Any trajectory intersecting γ(X) can in
the same way be followed forward in time above γ(X) and is likewise un-
bounded. There thus exists a second separatrix, unbounded as well, going
out from P above γ(X) and separating the trajectories intersecting X = 0
from those intersecting γ(X). By mirror symmetry around X = 0 the ciritical
point P must be a saddle point, and the phase portrait is complete. The last
proposition of Lemma 4.1 is the only reasonable physical interpretation.
Theorem 4.3 For positive vorticity, ω > 0, the properties of the phase por-
trait are the same as for the irrotational case, ω = 0.
PROOF. The proof is based on what we call the comparison principle, i.e.
by comparing the phase portrait for ω > 0 with that for ω = 0. Now changing
ω does not affect the 0-isoclines. The change of˙X induced by adding the term
ωY is
˙Xω>0<˙Xω=0,
at any fixed point in the phase plane with Y > 0 (where the subsripts denote
the two different phase-portraits). Hence the velocity field is conserved wher-
ever˙X < 0 in the portrait for ω = 0, and we need only check what happens
with the ∞-isocline (which encloses all the points where˙X > 0).
For any fixed X ∈ (−π/2,π/2) and ω ≥ 0, the function
(4.5)
ϕ(Y ) = Ak cosX coshY − ωY − f, Y > 0, (4.6)
is convex, satisfying ϕ(0) < 0 and ϕ(Y ) → ∞ as Y → ∞, whence it has a
exactly one zero in (0,∞). It is moreover decreasing in ω, so that if ω increases
the solution Y of ϕ(Y ) = 0 increases. This means that the ∞-isocline for
ω > 0 remains practically the same as in the irrotational case: it is a convex
graph lying above the one for ω = 0. Just as before there is no ∞-isocline for
X ∈ (π/2,π) since there ϕ(Y ) < 0.
Remark 4.4 We remark that according to (4.5) the wave flattens out as ω
increases. In view of the scale X = k(x − ct) this is the same as saying that
large positive vorticities allow only for large wavelengths.
4.2The case of negative vorticity
Theorem 4.5 For negative vorticity and small amplitude a << 1 the prop-
erties of the phase portrait are given by Figure 2. For hω > −c the crest is at
X = 0, while for hω < −c the crest is at X = π. In particular, the steady wave
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for hω < −c contains from bottom and up: an internal wave propagating left-
wards, a vortex enclosed by two critical layers, and a surface wave propagating
rightwards.
Remark 4.6 In all essential parts this resembles the Kelvin–Stuart cat’s-eye
flow, which is a particular steady solution of the two-dimensional Euler equa-
tions [36, Ex 2.4]. It arises when studying strong shear layers (which in our
case means large constant negative vorticity).
2!0!
Figure 2. The phase portrait for negative vorticity.
In order to handle this we need to investigate the ∞-isocline for ω > 0. Recall
that A = A(a) depends linearly on the amplitude (see (4.2)).
Lemma 4.7 For negative vorticity ω < 0, if a > 0 is small enough so that
ω
αsinh−1
?ω
α
?
−
?
1 +
?ω
α
?2
−f
α
(4.7)
is positive for α ≡ Ak, then the ∞-isocline of (4.4) for X ∈ [0,π] consists of
two disjoint parts:
(1) the graph of an increasing function Y1(X) defined for X ∈ [0,π], and
(2) the graph of a decreasing function Y2(X) defined in (π/2,π].
We have Y1(X) < Y2(X) → ∞ as X ? π/2, and for any δ > 0 there exist
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Page 15
Y∗> 0 and a possible smaller a such that the slope satisfies
0 <∂Y
∂X<δπ
inR ≡ [0,π] × [Y∗,Y∗+ δ].
PROOF. Just as before ϕ(Y ) as in (4.6) is convex for X ∈ (−π/2,π/2) with
ϕ(0) < 0. However, as
X → π/2 we now haveY → −f/ω
along the ∞-isocline ϕ(Y ) = 0. According to the Implicit Function Theorem
[42, Theorem I.1.1] the curve can be continued across this point into X ∈
(π/2,π]. There cosX < 0, and consequently ϕ(Y ) is now concave with ϕ(0) <
0, ˙ ϕ(0) > 0, and ϕ(Y ) → −∞ as Y → ∞. The function ϕ(Y ) attains its
global maximum when
Y = sinh−1
?
ω
Ak cosX
?
> 0,X ∈ (π/2,π].
Thus the equation ϕ(Y ) = 0 has no, one, or two solutions according as (4.7)
is negative, vanishing, or positive, for α ≡ −Ak cosX.
It is easy to see that if α > 0 is small enough this expression is positive,
while it becomes negative for large α. In view of that α vanishes as X ?
π/2, we see that at X = π/2 a new branch of the ∞-isocline appears from
Y = +∞. Keeping in mind that ϕ(Y ) is concave for X ∈ (π/2,π), where cosX
is decreasing, it follows that the upper branch of ϕ(Y ;X) = 0 is decreasing
as a parametrization Y (X) while the lower branch is increasing in the same
manner.
Depending on the relation between A, ω, and f, it may be that the two
branches both reach X = π separately, that they unite exactly there, or that
they unite for some X < π, where they cease to exist. However, if A is small
enough in relation to |ω| and f, (4.7) guarantees that both branches of the
∞-isocline exist as individual curves throughout X ∈ (π/2,π].
For the final assertion, remember that the slope is given by
∂Y
∂X=
Ak sinX sinhY
Ak cosX coshY − ωY − f.
(4.8)
Fix Y∗with −ωY∗> (1 + f + δ). Since A → 0 as a → 0 there exists a0such
that for any a < a0the inequality Ak cosh(Y∗+ δ) < δ/π holds. In view of
that sinhξ < coshξ this proves the lemma.
We are now ready to give the proof of Theorem 4.5.
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Page 16
PROOF. By periodicity and horizontal mirror symmetry it is enough to
consider Ω ≡ [0,π] × [0,∞) (remember that Y = 0 is the bed). The first critial
point is
P0≡ (0,Y )
while the second and third critical points are
whereAk coshY − ωY − f = 0,
P1≡ (π,Y1) andP2≡ (π,Y2),
where Y1and Y2are, in order of appearance, the smallest and largest solutions
of
Ak coshY + ωY + f = 0.
In Ω holds˙Y > 0, while the sign of ˙X is negative below Y1(X) and above
Y2(X), respectively positive elsewhere in Ω. This follows from Lemma 4.7,
and can be confirmed by considering ∂Y˙X for a fixed X.
The system (4.4) admits a Hamitonian,
H(X,Y ) ≡ Ak cosX sinhY −1
2ωY2− fY,(4.9)
with
˙X = ∂YH,
˙Y = −∂XH,
for which the trajectories of (4.4) are level curves. To determine the nature of
the critical points we study the Hessian,
D2H = −Ak
cosX sinhY sinX coshY
sinX coshY
− cosX sinhY +
ω
Ak
.
At P0, where X = 0, we immediately get that there exists one positive and
one negative eigenvalue, whence the Morse lemma [43] guarantees that P0is
a saddle point. Insertion of P2= (π,Y2(π)) yields
D2H(P2) = Ak
sinhY2(π)0
0
− sinhY2(π) −
ω
Ak
.(4.10)
Now remember that P2is the point where the function ϕ(Y ), for X = π, at-
tains its second zero. This happens when the derivative ϕ?(Y ) = −Ak sinhY −
ω < 0, and consequently also P2is a saddle point.
That P1is a center can be seen in the following way: D2H(P1) differs from
(4.10) only in that Y2(π) is substituted for Y1(π). Since P1is the point of the
first zero of ϕ(Y ) for X = π, it follows that there ϕ?(Y ) = −Ak sinhY −ω > 0,
16
Page 17
whence the Hessian is a diagonal positive definite matrix. According to the
Morse lemma there exists a chart (x,y): R2→ R2such that
H(X,Y ) = H(P1) + x2(X,Y ) + y2(X,Y )
in a neighbourhood of P1. (An alternative and efficient way is a phase por-
trait argument using the symmetry, which ensures that any trajectory that
intersects X = π twice is closed.)
P0being a saddle point, there is a separatrix γ−
two qualitatively different trajectory behaviours1– which can be followed
backwards in time from P0below Y1(X), and a separatrix γ+
followed forward in time from P0above Y1(X). By the direction of the velocity
field, γ−
1– i.e. a trajectory separating
2 which can be
1connects P0with X = π below P1.
In the same manner there are separatrices γ−
P2, and since γ+
unbounded trajectories starting from X = π above P2. The separatrix γ−
be followed backward from P2below Y2(X).
3and γ+
4leaving the saddle point
4 lies above Y2(X) it is unbounded and encloses a family of
3can
Now, according to the last part of Lemma 4.7, there is a family of trajectories,
{F}, starting from X = 0 above P0 and reaching X = π in finite time in
between P1and P2. Following γ−
{F}. In general, starting from Y2(X), following any trajectory backwards, we
find that it must intersect X = 0 above {F}. Since ˙X|Y2(X) = 0 the same
trajectory starting from Y2(X) is contained above Y2(X) and is unbounded.
The family {F} also guarantees that γ+
but below P2. The phase portrait is thus complete.
3 we therefore must intersect X = 0 above
2 connects P0with X = π above P1
Remark 4.8 In case (4.7) does not hold to be positive one might still pur-
sue the analysis, finding a fluid region where the situation is the same as for
positive and vanishing vorticity. However, above the surface the situation is
radically different and, for fixed amplitude and other parameters, indicates a
transition between negative and positive vorticity. In particular, there exists an
ω < 0 for which a bifurcation takes place: a second critical point appears, and
as ω decreases it immediately gives birth to a third critical point. Since our
linearized model presupposes that the amplitude is small, we do not investigate
this transition further. Some of the main features are given, without proof, in
Figure 3 below.
1The use of the word separatrix is somewhat ambigous. In our case, however, the
geometrical definition corresponds to the analytic notion of the stable and unstable
manifolds which are defined by H(X,Y ) = H(Pi), i = 0,2, and whose existence
follow from by the Implicit Function Theorem [42, Theorem I.1.1].
17
Page 18
2!0!2!0!
Figure 3. The bifurcation of the phase portrait.
5 The physical particle paths
In this section we shall investigate how from the behaviour of the trajectories
(X(t),Y (t)) we might infer the motion of the physical particles (x(t),y(t)).
Remember that the relation between those two pairs are given by (4.3).
Before moving on we recall from [17,18] that in the case of irrotational linear
waves, the particles were found to move in almost closed orbits, with a slight
but positive forward drift. This is in line with the classical Stokes drift [12]
according to which there is a forward mass drift. It is also consistent with
results for the exact equations [20,21], showing that within regular Stokes
waves – which are irrotational – no orbits are closed.
We emphasize that the appearance of vorticity radically changes the picture.
In particular, it will be shown that both when the vorticity is negative and
satisfies c + hω < 0, and in the case of large positive vorticity, there does not
exist a single pattern for all the fluid particles. Rather, different layers of the
fluid behave in qualitatively different ways, with some layers moving constantly
in one direction. For the same reason, it is hard to state any transparent results
other than Theorem 5.3 stating that for small amplitude waves on a current of
large positive vorticity there are indeed closed orbits, and Theorem 5.5 which
asserts that for negative vorticity all fluid particles display a forward drift.
Figure 4 however shows the main features for negative vorticity, as well as a
possible situation when the vorticity is positive and very large.
Lemma 5.1 Particles near the flat bed y = 0 have a forward (rightward)
drift.
PROOF. Consider the time τ that it takes for a particle (X(t),0) with
18
Page 19
X(0) = π to reach X(τ) = −π. We have
?−π
dX
?π/2
τ =
π
dt(X)
dX =
?π
?
−π
dX
f − Ak cosX
1
f − Ak cosX+=
−π/2
1
f + Ak cosX
?π/2
?
dX
= 2f
−π/2
dX
f2− (Ak cosX)2>2π
f.
We assert that this holds also near the bed: for any fixed X we may differen-
tiate˙X(X,Y ) with respect to Y , obtaining
∂Y˙X = Ak cosX sinhY − ω.
By continuity, there exists δ(ε) > 0 such that |˙X(X,Y )−˙X(X,0)| < ε when-
ever 0 < Y < δ, uniformly for X ∈ R. If we thus consider τ for a trajectory
intersecting X = 0 at level Y ∈ (0,δ) we may choose ε arbitrarily small so to
obtain τ > 2π/f.
Now a closed physical trajectory implies y(T) = y(0) for some T > 0 so that
Y (T) = Y (0) in view of (4.3). It follows from the phase portraits that for
trajectories close enough to the bed, this forces
X(T) − X(0) = −2πn,
for some n ∈ N. By periodicity, T/n = τ is the time it takes the trajectory
X(t) to pass from X = π to X = −π (which any trajectory near the bottom
does). From (5.1) we infer that τ = 2π/f.
meaning 0 = x(T) − x(0) = fT − 2πn, (5.1)
Remark 5.2 We also see from this reasoning that if τ > 2π/f, then the
particle will be to right of its original position, and contrariwise.
5.1 The case of positive vorticity
Theorem 5.3 If the vorticity ω > 0 is large enough, and the amplitude small
enough, then there particles moving constantly forward as well as particles
moving constantly backward. In particular there are closed orbits.
PROOF. By Lemma 5.1 the particles nearby the flat bed y = 0 display a
slight forward drift. In principle, they behave as in the irrotational case (see
[17,18]).
19
Page 20
By continuity˙X(X,Y ) can be made arbitrarily small, uniformly for all (X,Y ),
close enough to the critical point P0. Thus for the trajectories near P0the time
τ as in Lemma 5.1 can be made arbitraily large, and hence there is a forward
drift ˙ x > 0 for the corresponding physical particles.
In between those two layers something different might happen. Fix Y∗, δ > 0,
and choose 0 < a ? 1 small enough such that
Ak cosh(Y∗+ δ) < δ.
Then choose ω ? 1 such that ωY∗− δ > π, and such that the solution Y0of
Ak coshY0− ωY0− f = 0 satisfies Y0> Y∗+ δ (cf. (4.6) and the paragraph
following it). Then the slope given by (4.8) satisfies
|∂Y/∂X| < δ/π,
so that the trajectory for which Y (0) = Y∗+ δ remains in [Y∗,Y∗+ δ] where
˙X < −π − f. Hence
˙ x(t) < 0,
for all t for the physical particle and there is a constant backward drift. Since
the physical surface is given by Y = k(h + acosX) we can adjust our choices
so that Y∗+ δ < k(h − a) guarantees that the orbits we consider are indeed
within the fluid domain.
Using continuity once more we find that for large positive vorticity and small
amplitude, there do exist closed physical orbits.
Remark 5.4 Since the physical crest might lie below the critical point P0we
can only be sure that there is at least one (infinitessimally thin) layer of closed
orbits. However, this appears to happen only for small a and large ω, and even
so it does not have to affect more than one single trajectory in the (X,Y )-
plane. The particle paths are depicted to the left in Figure 4.
5.2 The case of negative vorticity
In the case of irrotational linear waves, it was found in [17,18] that all fluid
particles display a forward drift. This is confirmed for negative vorticity, and
the proof of Theorem 5.5 also shows the situation in the three different layers
of the fluid. A schematic picture of this can be found to the right in Figure 4.
Recall that for c + hω > 0 all of the fluid domain lies beneath the lowest
separatrix, so that the situation is the same as for zero vorticity. For c+hω < 0
the fluid domain streches above the vortex so that the picture is quite different
from irrotational waves.
20
Page 21
Theorem 5.5 For negative vorticity, all the fluid particles have a forward
drift.
PROOF. For reference, consider Figure 2. We treat separately
i) the interior wave (beneath the lowest separatrix),
ii) the vortex (between the first and second separatricies from bottom and up),
and
iii) the surface wave (between the second and third separatricies from bottom
and up).
The case i). Any trajectory (X(t),Y (t)) in this region passes X = kπ, k ∈ Z.
We may thus consider τ ≡
takes for the particle to travel from X(0) = π to X(τ) = −π. Again, for any
fixed X,
?−π
π
dt
dXdX as in Lemma 5.1, τ being the time it
∂Y˙X = Ak cosX sinhY − ω > −ω − Ak sinhY = ϕ?(Y )
with the notation of (4.6). Since ϕ?has its only zero at the level of P1, it
follows that ∂Y˙X > 0 in the interior wave. Since˙X < 0 at Y = 0, we deduce
that
?π
τ =
−π
dX
−˙X
>
?π
−π
dX
−˙X|Y =0
>2π
f,
so that – according to Remark 5.2 – the physical particle path describes a
forward motion.
The case ii). Any trajectory (X(t),Y (t)) within the vortex is bounded and
passes X(0) = π, whence
x(t) = x(0) +ft + B(t)
k
, where |B(t)| ≤ π.
In particular, at P1we have˙X = 0, so that the physical particle moves straight
forward according to x(t) = (π + ft)/k, for all t > 0.
The case iii). We need only observe that whenever ˙X is positive so is ˙ x =
(˙X + f)/k, whence all trajectories above the 0-isocline connecting P0and P1
correspond to fluid particles moving constantly forward.
21
Page 22
!
X(0) =
!
SS
S
S
CC
C
X(0) =
Figure 4. To the left the physical particle paths for small-amplitude waves with
large positive vorticity is depicted. The black arrows describe what happens to
some typical particles in time, while the axis marked S correspond to the sepa-
ratrix of Figure 1, and the C is the critical point in the same figure. Note that
depending on the amplitude and the vorticity, the surface of the physical wave need
not correspond to the uppermost arrow. Theorem 5.3 however guarantees that for
large enough vorticity and small enough amplitude the surface lies strictly above
the closed particle path (the first circle from bottom and up), so that there are par-
ticles with a mean backward drift as well as the opposite. To the right we see the
particle paths when the vorticity is negative such that c+hω < 0. This corresponds
to Figure 2 with the same notation as above. Near the bottom we have a mean
forward drift with nearly closed ellipses, but within the vortex of Figure 2 we see
a drastic change of behaviour with a constant forward drift. This is retained even
above the separatrix separating the vortex from the surface wave.
6 Summary and discussion
We have deduced and investigated the closed solutions of linear gravity water
waves on a linearly sheared current (constant vorticity). Such linear waves
satisfy the exact governing equations with an error of magnitude a2, where a
is the amplitude of the wave. The main purpose has been to understand how
the presence of vorticity influences the particle paths. While in the irrotational
case all the particles describe nearly closed ellipses with a slight forward drift,
we have found that vorticity might change the picture. For positive vorticity
the situation is very much the same as in the irrotational case, but for large
22
Page 23
enough vorticity and small enough waves, there are closed orbits within the
fluid domain. For negative vorticity exceeding the wave speed sufficiently much
all the particles describe a forward drift, but the nearly closed ellipses can be
found only in an interior wave near the flat bed.
It seems that all waves of constant vorticity are qualitatively though not quan-
titatively the same unless we accept the speed of individual particle to exceed
the speed of the wave. Then appears waves with interior vortices. So far there
is no corresponding exact theory of such rotational waves, since all work has
focused on regular waves not near breaking and without stagnation points.
When discussing particle paths it is important to remember that the question
of closed orbits is valid in relation to some reference speed. For irrotational
waves Stokes required that the average horizontal velocity should vanish. For
waves with vorticity we propose that the same requirement at the bottom is
the most sensible counterpart of Stokes’ definition. This is supported by the
fact that only for that choice we recover the classical critical wave speed√gh.
While interesting in its own right, the investigation pursued here might have
further implications for the numerous and well-known model equations for
water waves, e.g. the Kortweg–deVries, Camassa–Holm, and Benjamin–Bona–
Mahony equations. They all describe the surface – or nearly so – of the wave.
Though reasonable for irrotational waves, findings on uniqueness for rotational
waves indicate the same as our investigation: beneath two identical surfaces
there might be considerable different fluid motions (see Figures 1 and 2). Apart
from the trivial case of a flat surface there are so far no known exact examples
of this possible phenomenon, but if true it might motivate a new understanding
of in what sense the established model equations model the fluid behaviour.
Indeed vorticity, even when constant, is a major determining factor of the fluid
motion, and it should as such be considered highly important in the study of
water waves.
Acknowledgement The questions and remarks by an anonymous referee
considerably aided in improving the manuscript. The authors are also thankful
to Adrian Constantin for helpful comments and suggestions.
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