Particle paths in small amplitude solitary waves with negative vorticity
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 .
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Article: Symmetry of solitary waves
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ABSTRACT: It is shown that all supercritical solitary wave solutions to the equations for water waves are symmetric, and monotone on either side of the crest. The proof is based on the Alexandrov method of moving planes. Further a priori estimates, and asymptotic decay properties of solutions are derivedCommunications in Partial Differential Equations 01/1988; 13(5):603633. · 1.19 Impact Factor  SourceAvailable from: Adrian Constantin[Show abstract] [Hide abstract]
ABSTRACT: Analyzing a free boundary problem for harmonic functions in an infinite planar domain, we prove that in a 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 it experience a downward motion, with the particle trajectory having asymptotically the same height above the flat bed.Bulletin of the American Mathematical Society 08/2007; 44(3). · 1.17 Impact Factor  Archive for Rational Mechanics and Analysis 02/1981; 76(1):995. · 2.02 Impact Factor
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arXiv:1111.3637v1 [math.AP] 15 Nov 2011
Particle paths in small amplitude solitary waves with negative vorticity
LingJun 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 waterwave problem concerns the gravitydriven 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 smallamplitude solitary waves with arbitrary vorticity distri
bution were obtained in [13] and in [11], using generalized implicit function theorem of NashMoser 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 largeamplitude with an arbitrary distribution of vorticity remains open. So this arises the question of
particle paths in a rotational smallamplitude 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 (LingJun 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 singleloop 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 Xaxis 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 solitarywave 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),
∇ψ2+ 2gy = λ,
(3)
(4)
ψ = 0,y = η(x), (5)
ψ = −p0,
η(x) → 0
y = −d,
asx → ∞,
(6)
(7)
ψx(x,y) → 0as x → ∞ uniformly for y.(8)
2
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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 solitarywave 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)dsand 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 xindependent 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 solitarywave 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)rfor all x ∈ R
(13)
and for some constant r > 0 independent of λ, and the corresponding horizontal velocity satisfying the
following properties:
3
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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 pvariable.
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) < 0for 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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