Download On rotational water waves with surface tension

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Phil. Trans. R. Soc. A (2007) 365, 2215–2225
doi:10.1098/rsta.2007.2003
Published online 13 March 2007
On rotational water waves with
surface tension
B Y E RIK W AHLÉN *
Department of Mathematics, Lund University, PO Box 118,
22100 Lund, Sweden
The purpose of this paper is to present some recent advances in the study of water waves
with vorticity and surface tension. These are periodic, two-dimensional waves over a flat
bottom and the surface profiles are symmetric and monotone between crest and trough.
The proofs rely on bifurcation theory.
Keywords: water waves; vorticity; surface tension
1. Introduction
It is a matter of common experience that many waves observed on the surface of
a body of water can be roughly described by a periodic wave train, propagating
steadily without change of form and with no variation along the crests. We
consider two-dimensional wave trains propagating over water flowing on a
current, over a flat, impermeable bed. A current here means a water flow with a
flat surface and it is specified by vorticity. For example, a current which is
uniform with depth is described by zero vorticity, while constant non-zero
vorticity describes a linearly sheared current and so on. Ignoring viscosity, the
restoring forces acting on these waves are due to gravity and surface tension.
Historically, mathematical studies of water waves have mainly been restricted
to irrotational flows. Using bifurcation theory, a rich set of periodic wave trains
has been found. When surface tension was absent, global continua were found,
containing in their closure limiting waves with a sharp peak at the crests (Toland
1996). The crests are stagnation points, meaning that the vertical velocity
component is zero while the horizontal velocity component equals the speed of the
wave. At the other extreme, in the absence of gravity, explicit solution formulae
were found for the case of infinite depth by Crapper (1957) and for the case of finite
depth by Kinnersley (1976). For large amplitudes, these waves are overhanging,
with highly rounded crests. As the amplitude increases, a limiting shape is
reached, where the wave profile becomes self-intersecting, enclosing an air bubble
at the trough. In the presence of both surface tension and gravity, research has
mostly been confined to small-amplitude waves. It was formally observed by
Wilton (1915) that the combination of gravity and surface tension causes a
resonance to occur between two waves with a wavelength ratio 1 : 2, and that this
*[email protected]
One contribution of 13 to a Theme Issue ‘Water waves’.
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E. Wahlén
creates mixed solutions, having two crests per period. The existence of such
solutions, and other mixed solutions of different wavelength ratios, was rigorously
confirmed later (e.g. Reeder & Shinbrot 1981a,b; Jones & Toland 1985, 1986;
Jones 1989). The mathematical theory of large-amplitude waves remains largely
unknown. Numerical studies indicate that there are limiting waves of two kinds:
those with self-intersections, as in the pure capillary case, and those with contact
points between the surface and the bottom (Okamoto & Shoji 2001).
While the irrotational setting is regarded as appropriate for waves travelling
into still water (Johnson 1997), there are many situations in which it is necessary
to take vorticity into account. For example, in any region where the wind is
blowing, vorticity is generated (Da Silva & Peregrine 1988). Moreover, a realistic
description of tidal flows is given by constant vorticity (Da Silva & Peregrine
1988), and the outflowing current at the mouth of an estuary generally exhibits a
non-uniform vorticity distribution (Swan et al. 2001).
In the last few years, there has been a growing amount of interest in rotational
waves. Constantin & Strauss (2004) proved the existence of global continua of
finite-depth gravity waves with a general vorticity. These continua contain
waves with horizontal velocity arbitrarily close to the propagation speed. Other
areas of interest have been symmetry properties of these waves (Okamoto &
Shoji 2001; Constantin & Escher 2004a, b), uniqueness issues (Kalisch 2004;
Ehrnstöm 2005), as well as variational formulations (Constantin et al. 2006).
These recent investigations have exclusively been restricted to pure gravity
waves, ignoring the effects of surface tension.
Wahlén (2006b) recently considered the analogue of Kinnersley’s capillary waves,
and mathematically proved the existence of small-amplitude solutions for an
arbitrary vorticity distribution using a method similar to Constantin & Strauss
(2004). In Wahlén (2006a) the focus was shifted to capillary–gravity waves. Again,
small-amplitude solutions for an arbitrary vorticity were found using bifurcation
theory. The solutions found in both of these papers have one crest and one trough
per period, and are monotonic between crest and trough. The existence of mixed
waves remains an open question in the presence of vorticity, although Wahlén’s
(2006a) results indicate that such solutions could indeed exist. The aim of this paper
is to present the results of Wahlén (2006a, b) to the scientific community in this field.
Note that there is an earlier existence proof for capillary and capillary–gravity waves
with vorticity using a different approach due to Zeidler (1973). It is the opinion of the
author that the approach used by Wahlén (2006a,b) is somewhat simpler.
2. Mathematical formulation
Let us consider two-dimensional waves propagating over water with a flat bed. In
its undisturbed state, the equation for the flat surface is yZ0 and the flat bottom
is given by yZKd for some dO0. The x -variable represents the direction of
propagation and the wavelength is 2p/k, where kO0 is the wavenumber. In the
presence of waves, the free surface is represented by yZh(t, x) andÐ we choose a
2p=k
coordinate system so that the mean water level is zero, i.e. 0 hdx Z 0.
Homogeneity (constant density) is a good approximation for water and yields the
equation of mass conservation
ux C vy Z 0:
ð2:1Þ
Phil. Trans. R. Soc. A (2007)
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Rotational water waves with surface tension
The equation of motion is Euler’s equation
8
1
>
ut C uux C vuy ZK Px ;
>
>
<
r
1
>
>
>
: vt C uvx C vvy ZK r Py Kg;
2217
ð2:2Þ
where P(t, x, y) denotes the pressure; g is the gravitational constant; and r is the
density. The boundary conditions for capillary–gravity waves are (cf. Johnson
1997) the dynamic boundary condition
hxx
on y Z hðt; xÞ;
ð2:3Þ
P Z P0 Ks
3=2
ð1 C h2x Þ
where P0 is the constant atmospheric pressure and sO0 is the coefficient of
surface tension, as well as the kinematic boundary conditions
v Z ht C uhx
on y Z hðt; xÞ;
ð2:4Þ
and
v Z0
on y ZKd:
ð2:5Þ
By making the change of variables P 1 P=r and s1 s=r, we can assume
that rZ1.
For steady symmetric waves travelling at a speed cO0, i.e. the space–time
dependence of the free surface, the pressure and the velocity field is of the form
(xKct). In a reference frame moving at a speed c, the wave becomes fixed and we
can describe the problem in terms of the relative stream function j(x, y), defined
by jxZKv and jyZuKc throughout the fluid and byÐ jZ0 on the free surface. On
hðxÞ
the flat bed, j is constant and its value is p0 Z Kd ðuKcÞdy. Owing to the
symmetry assumption, j is even in the x -variable. Previous study indicates that
for waves not near the spilling or breaking state, the propagation speed c of the
surface wave is considerably higher than the horizontal velocity u of each
individual water particle (Lighthill 1978), so that u!c. Under this condition, the
vorticity uZvxKu y is a function of j, uZg(j). In other words, DjZKg(j).
Introduce the function
ð
p
GðpÞ Z
gðKsÞds;
0
p0 % p% 0;
and let
ðcKuÞ2 C v 2
C gy C P KGðKjÞ;
2
where E is the total mechanical energy which is constant throughout the fluid
according to Bernoulli’s law. The dynamic boundary condition is equivalent to
the condition
hxx
jVjj2 C 2gðy C dÞK2s
Z Q;
3=2
ð1 C h2x Þ
EZ
where QZ2(EKP0Cgd ).
The main difficulty in this problem is that h is not known a priori. For this
purpose, we make a change of variables following Dubreil-Jacotin (1934). Since j
is constant on the free surface and the bottom, and strictly decreasing as a
Phil. Trans. R. Soc. A (2007)
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E. Wahlén
function of y, we choose the new variables qZx and pZKj(x, y). Introducing the
height function h(q, p)ZyCd, we obtain the following formulation of the
capillary–gravity problem:
8
1 C h 2q hpp K2hp hq hpq C h 2p hqq ZKgðKpÞh 3p in ½0; 2p=k !½p0 ; 0;
>
>
>
>
>
<
h 2p hqq
ð2:6Þ
2
2
1
C
h
C
ð2ghKQÞh
K2s
Z 0 on p Z 0;
>
q
p
>
2 3=2
>
1 C hq
>
>
:
h Z 0 on p Z p0 ;
where h is 2p/k-periodic and even in the q-variable.
3. The existence of steady waves
The construction of the solutions relies on the Crandall–Rabinowitz local
bifurcation theorem (Crandall & Rabinowitz 1971). The first step is identifying
the underlying current associated with the vorticity function g. In terms of the
formulation (2.6), this corresponds to a trivial solution, by which we mean a
solution independent of q (in particular, the surface is flat). Fixing the propagation
speed c, we find that there is a one-parameter family of trivial solutions, H( p; l),
parameterized
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by the square root of the relative horizontal velocity at the surface,
lZ cKU ð0Þ, where U is the horizontal velocity component corresponding to the
trivial solution H. The parameter l ranges through the interval (K2Gmin,N),
where Gmin is the minimum value of the function G. The depth d and the parameter
Q depend on l.
Theorem 3.1 (Pure capillary waves; Wahlén 2006b). Given the relative mass
flux p0!0, the vorticity function g2C a[0, jp0j] and the wave speed cO0, there
exists for every sufficiently large wavenumber kO0 a curve CZ fðu; v; hÞg 3
1Ca
2Ca
C 1Ca
per ðUh Þ !C per ðUh Þ !C per ðRÞ of small-amplitude periodic waves with wavelength 2p/k.
The solutions are regular and symmetric, by which we mean (i) u and h are
even, while v is odd and (ii) h has one maximum (crest) and one minimum
(trough) per period and it is strictly monotone between the crest and the trough.
If g%0, then there exists such a curve for every positive wavenumber k.
Theorem 3.2 (Capillary–gravity waves; Wahlén 2006a). Given the relative
mass flux p0!0, the vorticity function g2C a[0, jp0j] and the wave speed cO0,
there exists for every sufficiently large wavenumber kO0 a curve C 3C 1Ca
per ðUh Þ!
2Ca
C 1Ca
ðU
Þ
!C
ðRÞ
of
small-amplitude
regular
and
symmetric
periodic
waves
h
per
per
with wavelength 2p/k.
If sRs0, where s0 is given by equation (3.3), then there exists a solution curve
for every positive wavenumber k.
Remark 3.1. Although s is more or less constant for a water–air interface at a
certain temperature, the value of s0 depends upon the vorticity function and p0.
The condition sRs0 can therefore be seen as a condition on the vorticity and
relative mass flux for water at a certain temperature.
Phil. Trans. R. Soc. A (2007)
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Rotational water waves with surface tension
m
–2Gmin
–
l
2
0
m0
Figure 1. Negative eigenvalues of equation (3.2) for gZ0.
When applying the Crandall–Rabinowitz theorem, one is led to study the
linearization of equation (2.6) around a trivial solution. This takes the form
8
>
fa 3 ðp; lÞwp gp C aðp; lÞwqq Z 0 in ½0; 2p=k !½p0 ; 0;
>
<
ð3:1Þ
a3 ð0; lÞwp C swqq Kgw Z 0 on p Z 0;
>
>
:
w Z 0 on p Z p0 ;
with
w even and 2p/k-periodic. The function a is given explicitly by aðp; lÞZ
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lC 2GðpÞ, and is related to the trivial solution H by aK1( p; l)ZHp( p; l). We
remark that the second-order boundary condition on the top is rather unusual.
Nevertheless, a theory for such problems does exist (Luo & Trudinger 1991) and
was applied by Wahlén (2006a, b).
We are looking for points l such that the linearized problem (3.1) has nontrivial solutions. Expanding in a cosine series leads to the spectral problem
8
Kða 3 ðp; lÞu 0 Þ 0 Z maðp; lÞu; p0 ! p! 0;
>
>
<
ð3:2Þ
Ka3 ð0; lÞu 0 ð0Þ C guð0Þ Z msuð0Þ;
>
>
:
uðp0 Þ Z 0:
If this problem has an eigenvalue mZKk2n2, n2Z for some lZl, then w( p, q)Z
u( p)cos(knq), where u is the corresponding eigenfunction, solves equation (3.1).
Thus, we are particularly interested in the negative eigenvalues of equation (3.2).
A description of the negative spectrum is given in lemmas 3.1 (figure 1) and 3.2
(figure 2). We begin with the pure capillary case.
Lemma 3.1 (Wahlén 2006b). If gZ0, then there exists for every lOK2Gmin
a unique non-positive eigenvalue m0(l). The function m0(l) is strictly decreasing
with liml/Nm0(l)ZKN and liml/K2Gmin m0 ðlÞZKk 20 % 0.
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E. Wahlén
If kOk 0, we find that there is a unique l such that m0(l)ZKk2. Furthermore,
for this value of l, this is the unique negative eigenvalue of equation (3.2), and in
particular the solution space of the linearized problem (3.1) is one-dimensional. It
can be shown that all the conditions of the Crandall–Rabinowitz theorem are
satisfied, and thus bifurcation occurs at l. In a neighbourhood of the bifurcation
point, the non-trivial solutions inherit the nodal pattern of the linearized solution
u( p)cos(kq). If g%0, then it can be shown that k 0Z0 so that there are waves for
any wavenumber (Wahlén 2005).
In order to explain the situation in the capillary–gravity case, we introduce the
parameters l0 and s0 determined by
ð0
a
p0
K3
1
ðp; l0 Þdp Z
g
and s0 Z g
2
ð0
p0
ð s
2
K3
aðp; l0 Þ
a ðs; l0 Þds dp:
ð3:3Þ
p0
Lemma 3.2 (Wahlén 2006a). If gO0 and sRs0, then there are no non-positive
eigenvalues for K2Gmin ! l! l 0 . For every lRl0, there is a unique non-positive
eigenvalue, m0(l), with m00 ðlÞ! 0, m0(l0)Z0 and liml/Nm0(l)ZKN.
If s!s0, then there is a l1RK2Gmin so that the following holds. For every l!l1
(this set can be empty), there exists no non-positive eigenvalues. For l1!l%l0, there
exists two non-positive eigenvalues m0(l)!m1(l)%0. For lOl0, there is only the
non-positive eigenvalue m0(l). The function m0 is strictly decreasing in lOl1 with
liml/Nm0(l)ZKN, while m1 is strictly increasing in l1!l%l0 with m1(l0)Z0. If
l1RK2Gmin, then there is exactly one non-positive eigenvalue for lZl1.
When sRs0, it is clear that for every kO0 there exists a lOl0 such that
m0(l)ZKk2 is the unique negative eigenvalue at l. Again, the Crandall–
Rabinowitz theorem can be applied.
When s!s0, the situation is as before for lOl0, that is, for every k2Ojm0(l0)j,
there exists a unique lOl0 such that equation (3.2) has the unique negative
eigenvalue Kk2 at l.
For any
l1 andffi
In the region l!l0, the situation is more complicated.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi l between
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l0, there are now two negative eigenvalues. Let k 0 Z jm0 ðl Þj and k1 Z jm1 ðl Þj.
If k 0 and k 1 are rationally dependent, say k 0Znk and k 1Zmk for some kO0 and
some n, m2Z, then the solution space of equation (3.1) in a space of 2p/k-periodic
functions is two-dimensional. If k 0 is not an integer multiple of k 1 (i.e. m does not
divide n), then there are waves of wavenumbers k 0 and k 1 bifurcating at l.
We simply apply the Crandall–Rabinowitz theorem after restricting the period to
2p/k 0 and 2p/k 1, respectively. However, if k 0 is an integer multiple of k 1, then the
Crandall–Rabinowitz theorem only yields the existence of waves of the higher
wavenumber k 0. Restricting to 2p/k 1-periodic functions does not reduce the
dimension of the solution space.
It can be expected that the solution set close to these double bifurcation points
is in fact much larger, containing ‘mixed’ solutions which are neither 2p/k 0- nor
2p/k 1-periodic (Jones & Toland 1985, 1986; Jones 1989). The existence of such
solutions requires more sophisticated techniques.
Note that any sufficiently large ratio n/m can occur since m0(l)/m1(l)/N
as l/l0. When l1OK2Gmin, we can in fact get any ratio since m0(l)/m1(l)/1
as l/l0.
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Rotational water waves with surface tension
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4. Some examples
(a ) Irrotational flow
We now consider the special case of irrotational flow gZ0. The underlying
current then has the form (U, 0), where U is a constant. The bifurcation points
are solutions l of the equation
k2s C g
kjp0 j
tanh pffiffiffiffiffi :
l Z
ð4:1Þ
k
l
If gZ0, then there exists a unique solution l for each kO0, and l is an increasing
function of k. When gO0, the numbers l0 and s0 defined in equation (3.3) are given by
l0Z(gjp0j)2/3 and s0Z1/3(gjp0j4)1/3. If sRs0, then there is again a unique solution of
equation (4.1), which is increasing in k. If s!s0, then there is still a unique
solution for each k, but it may not be increasing in k. It is possible that for both
k 0O0 and k1O0, the same solution l can be obtained. If k 0 and k 1 are rationally
dependent, then l is a double bifurcation point.
If we write equation (4.1) in terms of the depth d and the speed of the wave
relative to the underlying uniform current U , we obtain
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2s C g
cKU Z
tanhðkdÞ:
k
This is the dispersion relation for linearized small-amplitude waves. Double
bifurcation points occur if there are two rationally dependent wavenumbers k 0
and k 1 giving the same ‘eigenspeed’ cKU . In principle, there exist periodic
waves of minimal period 2p/k for every speed given by the above dispersion
relation. The only restriction being that we have not proved the existence of
waves of minimal period 2p/k 1 when k 0Znk 1 for some n2Z.
The dispersion relation shows that the relative wave speed is an increasing
function of the depth. In the limiting cases gZ0 and sZ0, two opposite phenomena
occur. In the first case, the relative wave speed is an increasing function of the
wavenumber, i.e. shorter waves travel faster. In the second case, the wave speed is a
decreasing function of the wavenumber, so that longer waves travel faster. When
both gravity and surface tension are present, the wave speed is an increasing
function of k when s/gd 2R1/3; otherwise, it is decreasing for small values of k,
increasing for large values of k and has a unique minimum. In the latter case, the
dispersion relation is qualitatively the same as for pure gravity waves for long waves
and the same as for pure capillary waves for short waves. The minimum point of the
wave speed as a function of k has a finite limit as d/N, and under typical physical
conditions it can be seen that this corresponds to a wavelength of approximately
1.7 cm (Crapper 1984). In this sense, it can be seen that if the water is shallow or if
the waves are very short, surface tension is in some sense the dominating force, while
in deeper water the force of gravity dominates for most wavelengths.
(b ) Constant vorticity
In the case of a constant vorticity gs0, the underlying current has the form
(U0Kgy, 0), where U0 is the horizontal velocity component at the surface. We
cannot give an explicit formula for l0 and s0. However, we can calculate the
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2222
E. Wahlén
(a) m
l0
–2Gmin
l
m0
(b) m
l0
–2Gmin
l
m1
m0
(c) m
m1
–2Gmin
l1
l0
l
m0
Figure 2. (a) Negative eigenvalues for gO0 and sRs0. (b) Negative eigenvalues for gO0, s!s0 and
l1ZK2Gmin. (c) Negative eigenvalues for gO0, s!s0 and l1OK2Gmin.
following dispersion relation in terms of the depth d and the speed of the trivial
flow at the surface U0 :
g
cKU0 Z tanhðkdÞ C
2k
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2s C g
g2
tanhðkdÞ C 2 tanh2 ðkdÞ:
k
4k
As in the irrotational case, the relative wave speed is an increasing function of
the depth. Furthermore, it can be seen that it is an increasing function of g.
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Rotational water waves with surface tension
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5. Discussion
There are several unanswered questions about water waves with surface tension
and vorticity. In this paper, we have considered only the waves on water of finite
depth. However, in the case of gravity waves, there is an explicit family of deepwater solutions with a particular non-vanishing vorticity (cf. Gerstner 1809; see
also Constantin 2001b), and recently Hur (2006) used global bifurcation theory to
construct large-amplitude deep-water waves for a large class of vorticity functions.
Using a combination of the methods of Wahlén (2006a, b) and Hur (2006), it
should be possible to construct families of small-amplitude deep-water waves with
vorticity and surface tension. Note that the existence proof of Zeidler (1973) also
allows for infinite depth.
Another interesting direction is the study of solitary waves. In the irrotational
case, there is a vast literature on capillary–gravity solitary waves (Kirchgässner
1988; Amick & Kirchgässner 1989; Iooss & Kirchgässner 1990, 1992; Sachs 1991;
Iooss & Pérouème 1993; Buffoni et al. 1996; Buffoni & Groves 1999). On the other
hand, Hur (in press) has recently extended the Beale (1977) construction of
small-amplitude gravity solitary waves to include vorticity. Groves & Wahlén
(2006) used the spatial dynamics methods (Kirchgässner 1982; Mielke 1991) to
study the solitary capillary–gravity water waves with vorticity.
Finally, in order to fully understand the fluid motion, it is important to have a
qualitative picture of the particle trajectories. The leading-order analysis of the
linearized Stokes problem (periodic irrotational gravity waves) suggests that all
particles move on closed orbits. However, in the recent paper by Constantin &
Villari (in press), it was proved that there are no closed orbits for the linearized
problem. Furthermore, a recent investigation by Constantin (2006) shows that
this is also the case for the fully nonlinear problem. The particle trajectories for
Crapper’s and Kinnersley’s waves were calculated by Hogan (1984, 1986); see also
the numerical study of irrotational capillary–gravity waves by Hogan (1985).
Again, the particle paths do not close up. On the other hand, the presence of
vorticity can lead to closed orbits in some cases. In particular, the water waves
discovered by Gerstner (1809) and some related edge waves (Constantin 2001a)
have the property of all particles moving on circles. A natural direction would
therefore be to make a rigorous study of the particle paths of irrotational
capillary–gravity waves and of rotational waves, with and without surface tension.
The author is grateful to the organizers of the 2005 Wave Motion programme at the Mittag-Leffler
Institute. The author would also like to thank the referee for several useful suggestions.
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