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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
14
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.
15
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