# Linear water waves with vorticity: Rotational features and particle paths

**ABSTRACT** We study steady linear gravity waves of small amplitude travelling on a current of constant vorticity. For positive vorticity the situation resembles that of Stokes waves, but if the vorticity is large enough the particle trajectories are affected. For negative vorticity we show that there may appear internal waves and vortices, wherein the particle trajectories are not ellipses.

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**ABSTRACT:**We survey recent results on particle trajectories within steady two-dimensional water waves. Particular emphasis is placed on the linear and exact mathematical theory of periodic and symmetric waves, and the effects of a (possibly rotational) background current. The different results vindicate and detail the classical Stokes drift, and also show the transition of orbits when waves propagate into running water. The classical approximation, depicting the trajectories as closed ellipses, is shown to be a mathematical rarity. 2000 Mathematics Subject Classification. 35Q35, 37N10, 76B15, 76F10.Discrete and Continuous Dynamical Systems-series B - DISCRETE CONTIN DYN SYS-SER B. 01/2009; 12(3). - SourceAvailable from: Mats Ehrnström[Show abstract] [Hide abstract]

**ABSTRACT:**This paper concerns linear standing gravity water waves on finite depth. We obtain qualitative and quantitative understanding of the particle paths within the wave.Journal of Nonlinear Mathematical Physics - J NONLINEAR MATH PHYS. 01/2008; 15. - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]

**ABSTRACT:**In order to obtain quite precise information about the shape of the particle paths below small-amplitude gravity waves travelling on irrotational deep water, analytic solutions of the nonlinear differential equation system describing the particle motion are provided. All these solutions are not closed curves. Some particle trajectories are peakon-like, others can be expressed with the aid of the Jacobi elliptic functions or with the aid of the hyperelliptic functions. Remarks on the stagnation points of the small-amplitude irrotational deep-water waves are also made.Journal of Mathematical Fluid Mechanics 02/2012; 15(1). · 1.42 Impact Factor

Page 1

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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2Preliminaries

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.

3The 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.

3.1The 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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4The 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 ) > 0exactly whenY > 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/2we 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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