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arXiv:1111.3637v1 [math.AP] 15 Nov 2011

Particle paths in small amplitude solitary waves with negative vorticity

Ling-Jun Wang

College of Science, Wuhan University of Science and Technology, Wuhan 430081, China

Abstract We investigate the particle trajectories in solitary waves with vorticity, where the vorticity is

assumed to be negative and decrease with depth. We show that the individual particle moves in a similar

way as that in the irrotational case if the underlying laminar flow is favorable, that is, the flow is moving in

the same direction as the wave propagation throughout the fluid, and show that if the underlying current

is not favorable, some particles in a sufficiently small solitary wave move to the opposite direction of wave

propagation along a path with a single loop or hump .

Keywords:

2010 MSC: 35J65, 34C05, 76B15

particle paths, solitary water waves, negative vorticity

1. Introduction

The water-wave problem concerns the gravity-driven flow of a perfect fluid of unit density; the flow

is bounded below by a rigid horizontal bottom {Y = −d} and above by a free surface {Y = η(X,t)}, where

η depends upon the horizontal spatial coordinate X and time t. Steady waves are waves which propagate

from left to right with constant speed c and without change of shape, so that η(X,t) = η(X − ct). Solitary

waves are steady waves which have the property that η(X − ct) → 0 as X − ct → ±∞. We consider in this

paper the particle trajectories in a fluid as a solitary wave propagates on the free surface, assuming that the

flow admits a negative vorticity decreasing with depth.

There have been a series of works concentrating on the study of solitary waves, in the setting of both

irrotational flows [1, 2, 3, 6] etc., and rotational flows which become active only in the last few years. One

of the interests in the above works is the description of individual particle path in the fluid. In irrotational

flow, particle paths underneath a solitary wave are investigated in [4, 6] , both for the smooth solitary wave

and the solitary wave of greatest height for which the crest is a stagnation point. It was shown in [6] that

in an irrotational solitary water wave, each particle is transported in the wave direction but slower than the

wave speed; as the solitary wave propagates, all particles located ahead of the wave crest are lifted while

those behind have a downward motion, with the particle trajectory having asymptotically the same height

above the flat bed. For rotational flow, some results on particle paths under periodic waves are obtained

(cf. [10, 9, 15] for instance), following a series of works on the corresponding results in irrotational case (see

[5, 8, 12] etc).

Recently, rigorous existence results on small-amplitude solitary waves with arbitrary vorticity distri-

bution were obtained in [13] and in [11], using generalized implicit function theorem of Nash-Moser type

and spatial dynamics method, respectively. The solitary waves established in [13, 11] are of elevation and

decays exponentially to a horizontal laminar flow far up- and downstream. The study of solitary waves

of large-amplitude with an arbitrary distribution of vorticity remains open. So this arises the question of

particle paths in a rotational small-amplitude solitary wave. Following the pattern in [6] for irrotational case,

we prove in this work the corresponding results on particle paths in rotational solitary waves by using the

Email address: wanglingjun@wust.edu.cn (Ling-Jun Wang)

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properties of solutions obtained in [13]. We consider in this work only solitary waves with negative vorticity,

however with modifications our arguments can be applied also for positive vorticity.

Precisely we show that if the vorticity is negative and increasing from bottom to surface of the flow,

and if the underlying current is favorable, i.e., is moving throughout the fluid in the same direction as the

wave, then as time goes on the particle moves similarly as in the irrotational case [6]. We also show that

if the underlying current is not favorable, then in solitary waves with sufficiently small amplitude, some

particles above the flat bed move to the opposite direction of wave propagation along a path with a single

loop or a single hump. Note that this single-loop kind of path does not exist in the irrotational case. In [13],

as in most of the works on waves with vorticity, the author considered only waves that are not near breaking

or stagnation, i.e., the speed of an individual fluid particle is far less than that of the wave itself throughout

the fluid domain. We consider only such waves as well. We do not consider the case of wave with stagnation

which is however studied in [6] in the irrotational case.

The paper is organized as follows. In Section 2 we present the mathematical formulation for the

solitary waves and recall from [13] some useful properties of the established solitary wave solutions. Section

3 contains some conclusions on the vertical and horizontal velocity that are relevant for our purposes. The

main result is presented and proved in Section 4. In the final section we give two examples which lie in our

settings.

2. Preliminaries

We first describe the governing equations for rotational solitary water waves and then recall some

properties available on their solitary wave solutions.

2.1. The governing equations for rotational solitary water waves.

Choose Cartesian coordinates (X,Y ) so that the horizontal X-axis is in the direction of wave prop-

agation and the Y -axis points upwards. Consider steady waves traveling at constant speed c > 0. In the

frame of reference moving with the wave, which is equivalent to the change of variables (X −ct,Y ) ?→ (x,y),

we use

Ωη= {(x,y) ∈ R2: −d < y < η(x)},

to denote the stationary fluid domain and (u(x,y),v(x,y)) to denote denote the velocity field, and define

the stream function ψ(x,y) by ψ(x,η(x)) = 0 and

0 < d < ∞,

ψy= u − c,ψx= −v. (1)

Consider also only waves that are not near breaking or stagnation, so that ψy(x,y) ≤ −δ < 0 in¯Ωηfor some

δ > 0, which implies that the vorticity ω = vx− uyis globally a function of the stream function ψ, denoted

by γ(ψ); see [7]. The solitary-wave problem is then, for given p0< 0 and γ ∈ C1([0,−p0];R), to find a real

parameter λ, a domain Ωηand a function ψ ∈ C2(¯Ωη) such that

ψy< 0,(x,y) ∈¯Ωη,

(x,y) ∈ Ωη,

y = η(x),

(2)

△ψ = −γ(ψ),

|∇ψ|2+ 2gy = λ,

(3)

(4)

ψ = 0,y = η(x),(5)

ψ = −p0,

η(x) → 0

y = −d,

as|x| → ∞,

(6)

(7)

ψx(x,y) → 0as |x| → ∞ uniformly for y.(8)

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Here g > 0 is the gravitational constant of acceleration,

p0=

?η(x)

−d

ψy(x,y) dy < 0

is the relative mass flux (independent of x), and the boundary conditions (7) and (8) express that the wave

profile approaches a constant level of depth and the flow is almost horizontal in the far field, respectively.

Moreover we require that the nontrivial solitary wave is of positive elevation, i.e., η(x) > 0 for all x ∈ R. It

is therefore symmetric about its single crest and admits a strictly monotone wave profile on either side of

this crest (see [14]). Assuming the wave crest is located at x = 0, the solitary-wave problem (2)-(8) is thus

supplemented with the symmetry and monotonicity conditions

ψ(−x,y) = ψ(x,y),η(−x) = η(x), and η′(x) < 0 for x > 0. (9)

We refer to [13, 7] for more details on the derivation of the system (2)-(8).

2.2. Rotational solitary water waves.

We collect some properties of the solitary wave solutions established in [13]. Let

Γ(p) =

?p

0γ(−s)ds and Γmin= min

p∈[p0,0]Γ(p) ≤ 0. (10)

Given p0< 0 and γ ∈ C1([0,−p0];R), for each λ ∈ (−2Γmin,∞) the system (2)-(8) admits a trivial solution

pair (η(x),Ψ(y)) defined on¯Ω0, where η(x) ≡ 0, the stream function Ψ(y) is x-independent and is the inverse

of the function

y(Ψ) =

?−Ψ

p0

dp

?λ + 2Γ(p)− d,

and the fluid domain

Ω0= {(x,y) ∈ R2: −d < y < 0},d =

?0

p0

dp

?λ + 2Γ(p).

The corresponding relative horizontal velocity and vertical velocity are thus given by

U(y) − c = Ψy(y) = −

?

λ + 2Γ(−Ψ(y)),V (x,y) = −Ψx(y) ≡ 0. (11)

Note that Ψ(0) = 0 and Ψ(−d) = −p0. Thus

U(0) = c −

√

λ andU(−d) = c −

?

λ + 2Γ(p0).

Throughout this paper, we adopt the terminology in [8] to say this underlying trivial flow is favorable if

U(y) ≥ 0 for all y ∈ [−d,0], is adverse if U(y) < 0 for all y ∈ [−d,0], and is mixed if U(y) changes sign. Note

that favorable flow moves in the same direction as the wave propagation (i.e., to the right), while adverse

flow moves in the opposite direction of the wave propagation.

To ensure the existence of nontrivial small amplitude solitary waves, the parameter λ must be chosen

to satisfy λ > λcbut close to λc, where λc> −2Γminis the unique solution of

?0

p0

dp

(λc+ 2Γ(p))3/2=1

g.

(12)

For each such a given λ and for given p0 < 0, it was shown in [13] that there exists a nontrivial small

amplitude solitary-wave solution pair (η(x),ψ(x,y)) to (2)-(8) defined on¯Ωη, with η(x) satisfying

|η(x)| + |η′(x)| + |η′′(x)| ≤ (λ − λc)r for all x ∈ R

(13)

and for some constant r > 0 independent of λ, and the corresponding horizontal velocity satisfying the

following properties:

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(P1) u(x,η(x)) → c −√λ = U(0),

(P2) u(x,y) → U(y)

as |x| → ∞;

uniformly for as |x| → ∞y.

The first property is obvious since by (4), (7) and (8), we have (u(x,η(x)) − c)2= ψ2

|x| → ∞, which equivalently gives (P1) as we have assumed u(x,y) − c = ψy(x,y) < 0 in¯Ωη, while the

property (P2) can be deduced from the construction of solitary wave solutions; see [13]. Indeed for λ = λc+ε

with ε > 0, there exists a function wε(q,p), whose derivatives with respect to (q,p) up to order 2 tend to 0

uniformly for p as |q| → ∞, such that the horizontal velocity is determined by

y(x,η(x)) → λ as

u(x,y) = c −

1

(λ + 2Γ(−ψ(x,y))−1/2+ εwε

p(√εx,−ψ(x,y))

,

where wε

p(q,p) denotes differentiation in the p-variable.

We conclude this section by recalling some properties of streamlines. Due to ψy< 0 throughout¯Ωη,

we have that for all p ∈ [−p0,0] the streamline

{(x,y) : ψ(x,y) = −p}

is a smooth curve y = σp(x). Note that

σ0(x) = η(x),σp0(x) = −dand σ′

p(x) = −ψx(x,σp(x))

ψy(x,σp(x)). (14)

Observing the fact that

that for each fixed y0∈ [0,η(0)] the streamline y = σp(x) with p = −ψ(0,y0), passing through the point

(0,y0), has an asymptote y = l(y0) as |x| → ∞, with l(η(0)) = 0 and l(−d) = −d.

1

|ψy|=

1

c−uis bounded and that ψx(x,y) → 0 uniformly for y as |x| → ∞, we deduce

3. Vertical and horizontal velocity

We first divide the fluid domain Ωηinto two components

Ω−= {(x,y) ∈ R2: x < 0,0 < y < η(x)} andΩ+= {(x,y) ∈ R2: x > 0,0 < y < η(x)},

and denote their boundaries by

S−= {(x,y) ∈ R2: x < 0,y = η(x)},B−= {(x,y) ∈ R2: x < 0,y = −d},

respectively

S+= {(x,y) ∈ R2: x > 0,y = η(x)},

Lemma 3.1. Suppose that γ′(p) ≤ 0 for all p ∈ [0,|p0|]. Then

B+= {(x,y) ∈ R2: x > 0,y = −d}.

(a) v(x,−d) = 0 for all x ∈ R, and v(0,y) = 0 for y ∈ [−d,η(0)] ;

(b) v(x,y) < 0 if (x,y) ∈ Ω−∪ S−, and v(x,y) > 0 if (x,y) ∈ Ω+∪ S+;

(c) vy(x,−d) < 0 if x < 0, and vy(x,−d) > 0 if x > 0;

(d) vx(0,y) > 0 for y ∈ (−d,η(0)).

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Proof. Since v = −ψx, the first result follows from (6) and the symmetry property ψ(−x,y) = ψ(x,y) in (9).

Next we only prove the lemma for x > 0 and the results for x < 0 can be proved similarly. Differen-

tiating (5) with respect to x gives v = −ψx= ψyη′(x), from which follows v > 0 for (x,y) ∈ S+as ψy< 0

and η′(x) < 0 for x > 0 in view of (2) and (9). To prove v > 0 in Ω+, we assume first on the contrary that

there exists a point (x0,y0) ∈ Ω+such that v(x0,y0) = −ε < 0. Then we can find a bounded domain

Ω+,k= {(x,y) ∈ R : 0 < x < k,−d < y < η(x)},k ∈ R+,

such that (x0,y0) ∈ Ω+,k. Moreover in view of (8) we can choose k sufficiently large so that v(k,y) > −ε

for y ∈ [−d,η(k)]. This means that v attains its minimum at the interior point (x0,y0) of Ω+,k, which

contradicts to the strong maximum principle applied to v on the domain¯Ω+,k, as v satisfies △v+γ′(ψ)v = 0

with γ′(p) ≤ 0 by differentiating (3). Therefore we have v ≥ 0 in Ω+. If v = 0 at a point (x0,y0) of Ω+.

Again we can choose a bounded domain Ω+,ˆkcontaining (x0,y0). Then v ≥ 0 on the boundary of Ω+,ˆk. We

can thus apply the strong maximum principle on Ω+,ˆkonce more to conclude that v ≡ 0 on¯Ω+,ˆk, which

contradicts v > 0 on the half surface S+. This proves v > 0 in Ω+.

Since v attains its minimum in¯Ω+on the half bed B+and on the crest line x = 0, Hopf’s maximum

principle ensures that vy> 0 on B+and vx> 0 on {(0,y);−d < y < η(0)}, completing the proof.

The above lemma combined with (14) and the fact that ψy< 0 gives

Corollary 3.2. The streamline y = σp(x) with p ∈ (p0,0] satisfies that σ′

for x > 0.

p(x) > 0 for x < 0 and σ′

p(x) < 0

Lemma 3.3. Suppose that γ(p),γ′(p) ≤ 0 for all p ∈ [0,|p0|], that 0 < λ − λc< ε with ε small such that

nontrivial solitary wave exists, and that c ≥

?λ + 2Γ(p0). Then u(x,y) > 0 for x ∈¯Ωη.

Proof. We assume throughout this proof that γ(p) ?≡ 0 for p ∈ [0,|p0|], since we already know that for

γ(p) ≡ 0, i.e., the irrotational case, u(x,y) > 0 in¯Ωη; see [6]. Recall that U(y) is the horizontal velocity

of the trivial laminar flow. Thus U′(y) = Ψyy(x,y) = −γ(Ψ(y)) ≥ 0, and there exists an interval I such

that the strict inequality holds for y ∈ I since γ(p) ?≡ 0 by assumption. This combined with (11) and the

assumption c ≥

?

It was proved in [16] that if γ(p) ≤ 0 for all p ∈ [0,|p0|], then

d

dxu(x,η(x)) ≥ 0

?λ + 2Γ(p0) gives

U(y) ≥ U(−d) = c −

λ + 2Γ(p0) ≥ 0 for y ∈ [−d,0], andU(0) > 0.

for x < 0, and

d

dxu(x,η(x)) ≤ 0 for x > 0.(15)

In other words, along the free surface u increases from x = −∞ to the crest x = 0, and thereafter it is

decreasing. On the bottom B = {(x,y) ∈ R2: y = −d}, we have, in view of ux= −vy= ψxyand Lemma

3.1-(c),

ux(x,−d) > 0for x < 0, andux(x,−d) < 0 for x > 0. (16)

Then the fact that U(−d) ≥ 0 and U(0) > 0 together with the monotonicity properties (15) and (16) yields

u > 0 on the free surface and on the bottom. Differentiating (3) with respect to y yields that u satisfies

△u + γ′(ψ)u = γ′(ψ)c ≤ 0.

Finally considering U(y) ≥ 0 for y ∈ [−d,0] and the properties (P1)-(P2), we can deduce u > 0 in¯Ωη by

using the strong maximum principle as in the proof of Lemma 3.1-(b), completing the proof.

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