PHYSICS OF FLUIDS
VOLUME 15, NUMBER 12
DECEMBER 2003
Mean drift induced by free and forced dilational waves
Jan Erik Webera) and Kai Haakon Christensenb)
Department of Geosciences, University of Oslo, P.O. Box 1022, Blindern, 0315 Oslo, Norway
共Received 20 March 2003; accepted 3 September 2003; published 29 October 2003兲
The mean drift velocity induced by longitudinal dilational waves in an elastic film is studied
theoretically on the basis of a Lagrangian description of motion. The film is horizontal and situated
at the interface between two viscous fluids. For time-damped dilational waves we let the film 共i兲
move freely with the mean fluid velocity at the interface, and 共ii兲 be kept fixed, i.e., having no mean
motion. In the latter case the mean Lagrangian drift velocity in both fluids becomes oppositely
directed to the wave propagation direction after a very short time. This is due to the fact that a fixed
film initially generates a strong source of negative Eulerian second order mean momentum at the
interface. This effect becomes even more pronounced when we consider forced dilational waves in
a fixed film. Now a suitably arranged shear stress in the upper fluid prevents wave amplitude decay
in the film. Accordingly, the negative mean Eulerian momentum at the interface becomes
independent of time, and the backward drift will propagate deeper and deeper into the lower fluid.
For a no-slip bottom at finite depth we may have a stationary drift solution with negative Lagrangian
drift velocity everywhere in the fluid. © 2003 American Institute of Physics.
关DOI: 10.1063/1.1621867兴
I. INTRODUCTION
parable to the wave period. Since such waves very rapidly
lose their initial mean wave momentum, they are bound to
induce strong mean Eulerian currents in the fluid. It is the
aim of the present paper to study the development of such
currents in a two-layer system subject to various conditions
at the film-covered interface.
The paper is organized as follows. In Sec. II we formulate the problem mathematically, and in Sec. III we calculate
the primary wave field for a two-layer system. We also consider the energy balance for dilational waves in this section,
showing that the energy flux is negligible. Some general
properties of nonlinear, time-damped dilational waves are
discussed in Sec. IV, which also contains explicit expressions
for the nonlinear mean drift when the film is free to move,
and when it is fixed in a mean sense. An analysis of the
nonlinear problem when an external stress is applied to prevent the primary wave field from decaying in time is given in
Sec. V. The waves in this case are referred to as forced dilational waves. Finally, Sec. VI contains a summary and some
concluding remarks.
Thin elastic films on the sea surface lead to an increased
damping of short capillary-gravity waves, as first explained
by Dorrestein,1 and they affect the generation of short waves
by the wind, e.g., Gottifredi and Jameson,2 Saetra.3 Such
films also influence the nonlinear mean drift induced by
capillary-gravity waves, as shown by Weber and Saetra.4 The
key issue here is the existence of longitudinal elastic waves,
or dilational waves, in the film. Sometimes these waves are
referred to as Marangoni waves.5 As far as pure damping is
concerned, maximum damping rate for transverse capillarygravity waves is obtained when the surface wave frequency
nearly coincides with the so-called Marangoni frequency.1,6
Nonlinearly, the existence of an elastic surface film alters the
virtual wave stress at the surface. This stress acts to redistribute the lost mean wave momentum due to damping as a
mean Eulerian current in the fluid.4,7 The works referred to
so far have all been concerned with the influence of an elastic film on various properties of transverse capillary-gravity
waves. However, also studies of purely longitudinal elastic
waves have been reported in the literature. These investigations have basically been designed to study the properties of
surface films, e.g., Mass and Milgram.8
The dilational wave itself is a fascinating phenomenon.
Its characteristics are very different from those usually seen
for waves in geophysics, like ocean swell, which is only
weakly influenced by friction, and therefore can propagate
over long distances with only minor changes in wave amplitude. The dilational wave is nearly critically damped, which
means that its amplitude decays on a time scale that is com-
II. MATHEMATICAL FORMULATION
In the present problem we have two homogeneous, incompressible viscous fluid layers separated by a horizontal
monomolecular layer of surfactant. We consider purely longitudinal elastic waves in the film. Such waves are referred
to as dilational waves. If the upper fluid has much smaller
density and dynamic viscosity than the lower fluid, which is
the case for air above water, the effect of the upper fluid can
be neglected as far as the damping characteristics of the dilational wave is concerned.9 For a slightly lighter, but much
more viscous fluid on top, this fluid will dominate the damping and the nonlinear transfer of mean momentum from
waves to currents. The thickness of the layers is assumed to
a兲
Electronic mail: j.e.weber@geo.uio.no
Electronic mail: k.h.christensen@geo.uio.no
b兲
1070-6631/2003/15(12)/3703/7/$20.00
3703
© 2003 American Institute of Physics
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Phys. Fluids, Vol. 15, No. 12, December 2003
J. E. Weber and K. H. Christensen
be much larger than the wavelength of the problem, and the
horizontal extent is unlimited. A Cartesian coordinate system
is chosen such that the x axis is aligned along the film, and
the z axis is positive upwards. The motion in both layers is
taken to be two-dimensional.
We will use a Lagrangian description of motion for this
problem, e.g., Lamb.10 This description yields directly the
particle drift associated with the wave motion 共the wave
drift兲; see Weber and Saetra4 for a related problem. For twodimensional motion we label each fluid particle with specific
coordinates 共a,c兲. The particle displacement 共x,z兲 and the
pressure p then become functions of the independent variables a, c, and time t. Denoting partial differentiation by
subscripts, the particle velocity and acceleration become
(x t ,z t ) and (x tt ,z tt ), respectively. The transformation from
Eulerian space derivatives of any function f to Lagrangian
derivatives are governed by f x ⫽( f a z c ⫺ f c z a )/(x a z c ⫺x c z a )
and f z ⫽(x a f c ⫺x c f a )/(x a z c ⫺x c z a ), respectively. We here
take 共a,c兲 to be the initial position of the fluid particle.
Strictly speaking, this is only true in an average sense for this
problem. However, this fact does not influence the mean
wave drift to second order in wave amplitude, e.g., Weber.11
The time scale of the present problem is so short that the
effect of the earth’s rotation can safely be neglected. The
equations for the conservation of momentum and volume for
a viscous incompressible fluid can then be written
x tt ⫽⫺ ␳ ⫺1 J 共 p,z 兲 ⫹ v 兵 J 共 J 共 x t ,z 兲 ,z 兲 ⫹J 共 x,J 共 x,x t 兲兲 其 , 共1兲
z tt ⫽⫺ ␳ ⫺1 J 共 x, p 兲 ⫺g⫹ v 兵 J 共 J 共 z t ,z 兲 ,z 兲 ⫹J 共 x,J 共 x,z t 兲兲 其 ,
共2兲
J 共 x,z 兲 ⫽1,
共3兲
where ␳ is the constant density, v is the kinematic viscosity,
and g is the acceleration due to gravity. Furthermore, J is the
Jacobian defined by J( f ,h)⬅ f a h c ⫺ f c h a . Equations 共1兲–共3兲
have been derived in detail by Pierson,12 using a different
notation, and have also been stated in their present form by
Weber and Saetra.4
In a Lagrangian formulation the position of the film covered interface is given by c⫽0 for all times. At the interface
we have a surfactant with concentration ⌫. We here consider
an insoluble film, i.e., there is no exchange of material between the film and the bulk of the bounding fluids. Conservation of film material4 leads to, when the film is horizontal:
x a ⫽⌫ 0 /⌫,
c⫽0,
共4兲
where ⌫ 0 is the concentration equilibrium value. We assume
that the surface tension T and the concentration ⌫ are homotrophic, i.e.,
T⬅T 共 ⌫ 兲 .
共5兲
It appears that a real surface dilational modulus quite accurately predicts the dynamic behavior of surface films.8 The
surface dilational modulus E is defined by
E⫽⫺
dT
,
d 共 ln ⌫ 兲
共6兲
and we take 共the real兲 E to be the only rheological parameter
of our problem. We assume that E is constant, which means
that ⌫ is close to its equilibrium value ⌫ 0 .
We introduce the superscript ( ˆ ) to distinguish the variables of the upper layer from those of the lower layer. For a
freely floating horizontal film with negligible mass, the viscous stress ␶ˆ from the upper fluid must balance the viscous
stress in the lower fluid plus the stress due to the horizontal
change in surface tension of the film. As far as the vertical
stress at the interface is concerned, the assumption of a horizontal film means that the film must be rigid enough to withstand the normal stresses on both sides. Applying 共4兲–共6兲,
the dynamic boundary condition at the interface can be written in Lagrangian form as
␶ˆ ⫽ ␳ v J 共 x,x t 兲 ⫺E
x aa
x 2a
,
c⫽0.
共7兲
In addition, the fluid velocities must both be equal to the film
velocity at the interface, i.e.,
x̂ t ⫽x t ⫽ 共 x t 兲 film ,
c⫽0.
共8兲
Far away from the interface in the vertical direction, we assume that all our variables vanish.
When the surface film supports dilational waves, but is
prevented from having a mean horizontal drift, as in some
laboratory experiments,8 共8兲 leads to a no-slip condition for
the mean Lagrangian velocity. Then an external mean stress
must be added to 共7兲, i.e., the force that must be applied to
prevent the film from sliding along the x axis. Since the flow
field in this case is determined by the no-slip condition, this
force can in turn be calculated from the extended version of
共7兲.
The dependent variables of the problem will be written
as series expansions after an ordering parameter ␧, e.g.,
Pierson12
共 x,z,p 兲 ⫽ 共 a,c,⫺ ␳ gc 兲 ⫹␧ 共 x 共 1 兲 ,z 共 1 兲 ,p 共 1 兲 兲
⫹␧ 2 共 x 共 2 兲 ,z 共 2 兲 ,p 共 2 兲 兲 ⫹¯ ,
共 x̂,ẑ,p̂ 兲 ⫽ 共 a,c,⫺ ␳ˆ gc 兲 ⫹␧ 共 x̂ 共 1 兲 ,ẑ 共 1 兲 ,p̂ 共 1 兲 兲
⫹␧ 2 共 x̂ 共 2 兲 ,ẑ 共 2 兲 ,p̂ 共 2 兲 兲 ⫹¯ .
共9兲
Here g is the acceleration due to gravity. The appropriate
form of the expansion parameter ␧ will be assessed later on.
III. THE PRIMARY WAVE FIELD
By inserting 共9兲 into the governing equations, and equating equal powers of ␧, we get systems of partial differential
equations to solve at each order. The solution to O(␧) determines the linear wave motion, or the primary wave, while
the averaged solution to O(␧ 2 ) yields the nonlinear Lagrangian mean drift.
The primary wave solution may be obtained by separating the wave field into an irrotational part and a rotational
part.10 The procedure is quite simple, and we just give the
results. A similar, linear Eulerian analysis of this problem has
been performed by Lucassen-Reynders and Lucassen.9 The
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Phys. Fluids, Vol. 15, No. 12, December 2003
Mean drift induced by free and forced dilational waves
explicit form of the primary wave field is needed for the
calculation of the nonlinear mean solution, and we state it
here mainly for that reason.
We consider a complex Fourier wave component proportional to exp(i␬a⫹nt), where i is the imaginary unit, ␬ is the
complex wave number in the x direction, and n is the complex time decay rate. The normalized solution in the lower
layer can then be written as
x 共 1 兲 ⫽⫺
z 共 1 兲 ⫽⫺
冋
册
i ␬ ␬ c m mc i ␬ a⫹nt
e ⫺ e e
,
n
␬
共10兲
␬ ␬ c mc i ␬ a⫹nt
,
关 e ⫺e 兴 e
n
共11兲
␳
p 共 1 兲 ⫽ 关共 n 2 ⫹g ␬ 兲 e ␬ c ⫺g ␬ e mc 兴 e i ␬ a⫹nt ,
n
共12兲
where
n
m 2⫽ ␬ 2⫹ .
v
共13兲
x̂
共1兲
冋
册
i ␬ Q ⫺ ␬ c m̂ ⫺m̂c i ␬ a⫹nt
⫽⫺
e
⫺ e
e
,
n
␬
ẑ 共 1 兲 ⫽
共14兲
␬ Q ⫺ ␬ c ⫺m̂c i ␬ a⫹nt
⫺e
,
关e
兴e
n
共15兲
␳ˆ Q
关共 n 2 ⫺g ␬ 兲 e ⫺ ␬ c ⫹g ␬ e ⫺m̂c 兴 e i ␬ a⫹nt ,
n
共16兲
p̂ 共 1 兲 ⫽
where
n
m̂ 2 ⫽ ␬ 2 ⫹ .
v̂
共17兲
In the solutions above we have assumed that the real parts of
m and m̂ are positive. The requirement 共8兲 that the horizontal
velocity must be continuous across the film yields for the
coefficient Q in the upper fluid solution
m⫺ ␬
.
Q⫽
m̂⫺ ␬
共18兲
共19兲
c⫽0.
Inserting from 共10兲 and 共14兲 into 共19兲, we obtain
⫺ ␳ˆ v̂ Q 共 m̂ 2 ⫺ ␬ 2 兲 ⫽ ␳ v共 m 2 ⫺ ␬ 2 兲 ⫹
E␬2
共 m⫺ ␬ 兲 .
n
共20兲
Using 共13兲, 共17兲, 共18兲, and rearranging, we find the dispersion relation9 from 共20兲
冋
n 2 1⫹
册
n ⫽
3
冋 册
E2
* ␬ 4,
␳ˆ 共 m⫺ ␬ 兲
⫽⫺ ␬ 2 共 m⫺ ␬ 兲 E ,
*
␳ 共 m̂⫺ ␬ 兲
共22兲
A 2v
where
A⬅1⫹
冉冊
␳ˆ v̂
␳ v
1/2
共23兲
.
We note that the coefficient A expresses the effect of the
upper fluid in the complex dispersion relation 共22兲. For air
and water, A is very close to one, which means that presence
of air above the film is negligible as far as the propagation
and damping of the dilational wave in the model studied here
is concerned.
It is easily seen from 共22兲 that temporally damped waves
and spatially damped waves have different phase speeds. For
temporal damping we take
n⫽⫺i ␻ ⫺ ␤ ,
共24兲
where ␻ and ␤ are the real frequency and the real damping
rate, respectively, and k is a real wave number. One then
obtains from 共22兲 that
␻⫽
冉 冊
3 1/2 E 2
*
2 A 2v
共21兲
where we have defined E ⫽E/ ␳ . For this problem we as*
sume that 兩 m 兩 Ⰷ 兩 ␬ 兩 , and 兩 m̂ 兩 Ⰷ 兩 ␬ 兩 . The physical implications
1/3
k 4/3,
␤ ⫽3 ⫺1/2␻ ,
共25兲
6
e.g., Dysthe and Rabin for A⫽1. We notice that the wave
here is nearly critically damped, since the damping rate is
comparable to the wave frequency. For spatial damping we
take
n⫽⫺i ␻ ,
␬ ⫽k⫹i ␣ ,
共26兲
k⬎0,
where ␣ is the real spatial attenuation coefficient. In this case
we find
␻ ⫽ 共 8 共 3⫺8 1/2兲兲 1/3
␣ ⫽ 共 2 1/2⫺1 兲 k,
To O(␧) we obtain from 共7兲
␳ˆ v̂ x̂ 共tc1 兲 ⫽ ␳ v x 共tc1 兲 ⫺Ex 共aa1 兲 ,
of these assumptions will be discussed later on. From 共13兲
and 共17兲 we now let m⬇(n/ v ) 1/2 and m̂⬇(n/ v̂ ) 1/2. Then
共21兲 reduces to
␬ ⫽k,
In the upper fluid the corresponding solution becomes
3705
冉 冊
E2
*
2
A v
1/3
k 4/3,
共27兲
see Lucassen5 for A⫽1. Hence the phase speed C⫽ ␻ /k is
different for the two cases. This also means that the group
velocity C g ⫽d ␻ /dk is not well defined for this problem.
This is in contrast to weakly damped capillary-gravity
waves, where the real part of the dispersion relation is the
same for both cases of damping, and where ␤ ⫽C g ␣ , as
shown by Gaster.13
These findings merit a short discussion concerning the
energy transfer in dilational waves. It is well known that the
group velocity is related to the transfer of wave energy in
free nondamped, or weakly damped waves. Since the wave
energy is a second-order quantity in wave amplitude, we
need only consider the linear solutions in this discussion. To
simplify, we neglect the presence of the upper fluid. We mul-
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3706
Phys. Fluids, Vol. 15, No. 12, December 2003
J. E. Weber and K. H. Christensen
tiply the linear horizontal and vertical momentum equations
and z (1)
from 共1兲 and 共2兲 by the real parts of x (1)
t
t , respectively. By integrating in the vertical, and assuming that
(1)
兩 x (1)
t 兩 Ⰷ 兩 z t 兩 , 兩 ⳵ / ⳵ c 兩 Ⰷ 兩 ⳵ / ⳵ a 兩 , we readily find
⳵
⳵t
冕
1 共1兲 2
⳵
共 x t 兲 dc⫹
2
⳵
a
⫺⬁
0
⫽E 共 x 共aa1 兲 x 共t 1 兲 兲 c⫽0 ⫺ v
*
冕
冕
1 共1兲 共1兲
共 p x t 兲 dc
␳
⫺⬁
0
0
⫺⬁
共 x 共tc1 兲 兲 2 dc.
共28兲
IV. THE NONLINEAR WAVE DRIFT
A. Freely drifting film
In the following we consider temporally damped waves.
Then 共24兲 applies to our problem. From 共13兲 and 共17兲 we
obtain for the real and imaginary parts of m and m̂, respectively,
m r ⫽3 ⫺1/4␥ ,
m i ⫽⫺3 1/4␥ ,
m̂ r ⫽3 ⫺1/4␥ˆ ,
m̂ i ⫽⫺3 1/4␥ˆ ,
To obtain the first term on the right-hand side, we have utilized the boundary condition 共19兲, with ␳ˆ ⫽0. For the moment we let an overbar denote an unspecified mean 共in time
or space兲. We then get
⳵
⳵
Ē⫹
F̄⫽W̄⫺D̄,
⳵t
⳵a
共29兲
where
Ē⬅
冕
1 共1兲 2
共 x t 兲 dc
2
⫺⬁
0
is the energy density 共here equal to the kinetic energy per
unit mass兲. Furthermore,
F̄⬅
冕
1 共1兲 共1兲
共 p x t 兲 dc
␳
⫺⬁
2 ␲ A v 2 2 ␲ A 共 v̂ 3 v 兲 1/2
,
.
E
E
␧⫽
␩␻
.
␥
共32兲
*
*
For a typical film value E ⫽30 cm3 s⫺2 , and wave frequen*
cies from 1 to 100 s⫺1, we find from 共25兲 and 共27兲 for air
above water that ␭ is in the range 3–118 cm. This is well
above the thresholds required by 共32兲, where the second limit
on the right-hand side for the same configuration is of the
order 10⫺3 cm.
We can now assess the value of our expansion parameter
␧. If we denote a typical horizontal displacement in the film
by ␩, we must have from 共10兲 and 共14兲 that 兩 ␧x (1) (c⫽0) 兩
⫽ 兩 ␧x̂ (1) (c⫽0) 兩 ⬃ ␩ . Accordingly, for this problem we take
*
is the work done by the elastic film, and
0
⫺⬁
共 x 共tc1 兲 兲 2 dc
u⬅␧ 2 x̄ 共t 2 兲 ,
is the viscous dissipation. Actually, to get correct dimensions, these quantities should have been multiplied by ␧ 2 , as
in 共9兲, but this is irrelevant here. For nondamped, or weakly
damped waves, we have in general that F̄⫽C g Ē. However,
for nearly critically damped dilational waves this is not the
case. Both for temporally damped waves 共averaging over the
wavelength兲 and for spatial damping 共averaging over the
wave period兲 we find that
兩 F̄ 兩 ⰆC g Ē.
共33兲
We solve the nonlinear problem by averaging the variables over one wavelength. This process will be denoted by
an overbar, as used before for the energy discussion in the
previous section. Furthermore, we introduce mean nonlinear,
horizontal drift velocities in the lower and upper layers defined by
W̄⬅E 共 x 共aa1 兲 x 共t 1 兲 兲 c⫽0
冕
1/2
␭Ⰷ
0
is the energy flux,
D̄⬅ v
共31兲
where ␥ ⬅( ␻ /(2 v )) , ␥ˆ ⬅( ␻ /(2 v̂ )) . Our former assumption 兩 m 兩 , 兩 m̂ 兩 Ⰷk, requires that the wavelength ␭
⫽2 ␲ /k must satisfy
1/2
共30兲
This means that the energy flux in free dilational waves is
negligible. Accordingly, the group velocity as a carrier of
wave energy has no meaning here. The rate of change with
time of the energy density for temporally damped waves is
locally determined by the joint action of work by the elastic
film and the viscous dissipation, i.e., ⳵ Ē/ ⳵ t⫽W̄⫺D̄. For
spatially damped waves, ( ⳵ Ē/ ⳵ t⫽0), there is basically a local balance between the work by the elastic film and the
dissipation (W̄⬇D̄).
û⬅␧ 2¯x̂ 共t 2 兲 .
共34兲
By inserting real parts of the primary wave fields 共10兲–共16兲
into 共1兲, and averaging, we obtain to the leading order
u t ⫺ v u cc ⫽⫺2•3 1/2v ␩ 2 k ␻ m r2 e 2m r c⫺2 ␤ t ,
共35兲
û t ⫺ v̂ û cc ⫽⫺2•3 1/2v̂ ␩ 2 k ␻ m̂ r2 e ⫺2m̂ r c⫺2 ␤ t .
共36兲
From the continuity of horizontal stresses 共7兲, we obtain to
O(␧ 2 )
␳ v u c ⫺ ␳ˆ v̂ û c ⫽3 1/2␩ 2 k ␻ 共 ␳ v m r ⫹ ␳ˆ v̂ m̂ r 兲 e ⫺2 ␤ t ,
c⫽0.
共37兲
Continuity of mean horizontal velocities requires
u⫽û,
共38兲
c⫽0.
The variables in the lower and upper layers are assumed to
vanish as c→⫺⬁ and c→⬁, respectively.
As a check on these derivations, we integrate 共35兲 and
共36兲 in the vertical, and apply 共37兲 at the interface. It then
readily follows that
d
dt
冉冕
0
⫺⬁
␳ udc⫹
冕␳ 冊
⬁
0
ˆ ûdc ⫽0,
共39兲
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Phys. Fluids, Vol. 15, No. 12, December 2003
Mean drift induced by free and forced dilational waves
i.e., the total mean horizontal Lagrangian momentum is conserved, as expected, since there are no horizontal external
forces acting at the boundaries of our system.
Particular solutions u p , û p of 共35兲 and 共36兲 are easy to
obtain. We find
u p⫽
3 1/2 2
␩ ␻ ke 2m r c⫺2 ␤ t ,
4
共40兲
û p ⫽
3 1/2 2
␩ ␻ ke ⫺2m̂ r c⫺2 ␤ t .
4
共41兲
Defining the mean wave momentum M as
M⫽
冕
0
⫺⬁
兩 ␧ 2¯x̂ 共t 2 兲 兩 Ⰶ 兩 ␧x̂ 共t 1 兲 兩 ,
c⫽0.
共42兲
From 共10兲 and 共14兲, using 共31兲, we have
兩 ␧x 共t 1 兲 兩 ⬃␧ ␥ ,
兩 ␧x̂ 共t 1 兲 兩 ⬃␧ ␥ ,
c⫽0,
共43兲
using Qm̂⬇m in the expression for the upper fluid. Letting
the particular solutions 共40兲 and 共41兲 represent the secondorder velocities, and utilizing 共33兲, we find from 共42兲 and
共43兲 that
␩ kⰆ1.
共44兲
The relation 共44兲 implies that the amplitude must be at least
one order of magnitude less than the wavelength to assure
convergence of the solutions. Unfortunately, we have no information about the size of the amplitude in the laboratory
experiments of Mass and Milgram.8 Hence, we cannot check
the applicability of our solutions to that problem. Neither can
we estimate the actual size of the drift velocities in their
experiments.
In weakly damped surface waves, the particular solution
of the equation for the mean Lagrangian motion yields the
Stokes drift,14 which is associated with the irrotational part
of the wave field. The present problem is governed by the
effect of viscosity. This is seen from the fact that the wave
field is dominated by its rotational part. Accordingly, the particular solutions 共40兲 and 共41兲 are vorticity solutions, confined to thin vorticity layers at the interface. Since the
vorticity-layer solutions decrease rapidly in time, quasiEulerian currents must evolve in time and space in order to
fulfil the momentum conservation condition 共39兲. Mathematically, these Eulerian mean currents are solutions to the
homogeneous versions of 共35兲 and 共36兲, which we denote by
u h and û h , respectively. This transition of momentum is
achieved by the action of the virtual wave stress ␶ w at the
interface; see Longuet-Higgins7 for a discussion of this concept in connection with surface gravity waves, or Weber and
Førland15 for gravity waves in a two-layer system. The virtual wave stress can here be defined as
␶ w ⬅ ␳ v u hc ⫺ ␳ˆ v̂ û hc ,
c⫽0.
共45兲
From the condition 共37兲 at the interface, utilizing that u
⫽u p ⫹u h and û⫽û p ⫹û h , we obtain that
␶ w⫽
冉
冊
␳ˆ
1 ␳
⫹
␩ 2 ␻ 2 ke ⫺2 ␤ t .
4 m r m̂ r
共46兲
␳ u p dc⫹
冕␳
⬁
共47兲
ˆ û p dc,
0
we readily find from 共40兲, 共41兲, and 共46兲 that
␶ w ⫽⫺
Formally, to ensure convergence of the present approach, we
must require
兩 ␧ 2 x̄ 共t 2 兲 兩 Ⰶ 兩 ␧x 共t 1 兲 兩 ,
3707
d
M.
dt
共48兲
This demonstrates that the virtual wave stress at the interface
acts to transfer mean wave momentum into a mean Eulerian
current. A similar relation have been obtained for weakly
damped surface gravity waves by Weber.16
So far we have considered some general properties related to the conservation of the total mean momentum in
dilational waves. In order to investigate how the Lagrangian
drift current actually varies with time and depth in both layers, we have to solve the governing differential equations
共35兲 and 共36兲, subject to the relevant initial and boundary
conditions. We have already obtained the particular solutions
共40兲 and 共41兲. The solutions to the homogeneous problem
u h ,û h are readily found by applying Laplace transforms.
Transforming the homogeneous parts of 共35兲 and 共36兲, and
the boundary conditions 共37兲 and 共38兲, assuming that u h ,û h
are initially zero, the solutions follows straight away by applying the convolution theorem. The complete mean Lagrangian drift u⫽u p ⫹u h and û⫽û p ⫹û h in the lower and
upper layer can then be written, respectively, as
u⫽
冋
3 1/2 2
␩ ␻ ke ⫺2 ␤ t e 2m r c
4
⫹3 ⫺1/4
û⫽
冉 冊冕
2␻
␲
1/2
t
0
冋
3 1/2 2
␩ ␻ ke ⫺2 ␤ t e ⫺2m̂ r c
4
⫹3 ⫺1/4
冉 冊冕
2␻
␲
1/2
t
0
册
c⬍0, 共49兲
册
c⬎0. 共50兲
exp共 2 ␤ ␰ ⫺c 2 /4v ␰ 兲 d ␰
,
␰ 1/2
exp共 2 ␤ ␰ ⫺c 2 /4v̂ ␰ 兲 d ␰
,
␰ 1/2
In Fig. 1 we have depicted the dimensionless drift velocities
u/( ␩ 2 ␻ k) and û/( ␩ 2 ␻ k) from 共49兲 and 共50兲 as functions of
dimensionless depth kc at various dimensionless times ␻ t,
when the wave number k is 0.5 cm⫺1. In this example we
consider an air–water system with ␳ˆ ⫽1.25⫻10⫺3 g cm⫺3 ,
v̂ ⫽0.14 cm2 s⫺1 , and ␳ ⫽1 g cm⫺3 , v ⫽0.012 cm2 s⫺1 . Furthermore, we have taken E ⫽30 cm3 s⫺2 , which is a typical
*
value for an oleyl alcohol film.8 We realize that because v̂
⬎ v , the drift velocity penetrates further into the air than into
the water. The velocity at c⫽0 is the mean velocity of the
film.
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3708
Phys. Fluids, Vol. 15, No. 12, December 2003
J. E. Weber and K. H. Christensen
FIG. 1. Mean drift velocities u/( ␩ 2 ␻ k) for kc⭐0, and û/( ␩ 2 ␻ k) for kc
⭓0 at times ␻ t⫽1,5,10 induced by free dilational waves in a moving film.
FIG. 2. Mean drift velocities u/( ␩ 2 ␻ k) for kc⭐0, and û/( ␩ 2 ␻ k) for kc
⭓0 at times ␻ t⫽0.1,1,5 induced by free dilational waves in a fixed film.
B. Fixed film
times ␻ t for an air–water system. The values of k and E
*
are the same as in Fig. 1. We note from the figure that the
drift velocity very soon becomes negative in both layers due
to the strong source of negative mean Eulerian momentum at
the interface.
In a laboratory situation, e.g., Mass and Milgram,8 the
film may be prevented from having a mean horizontal motion. In such cases the boundary condition 共8兲 for the mean
motion at the interface reduces to
u⫽û⫽0,
共51兲
c⫽0.
The particular solutions to our problem become the same as
before, i.e., 共40兲, 共41兲. However, the no-slip condition 共51兲
now introduces a strong source of negative momentum at the
interface for the Eulerian part of the flow. We obtain
3 1/2 2
␩ ␻ ke ⫺2 ␤ t ,
u ⫽⫺u ⫽⫺
4
c⫽0,
共52兲
3 1/2 2
␩ ␻ ke ⫺2 ␤ t ,
4
c⫽0.
共53兲
h
p
û h ⫽⫺û p ⫽⫺
By assuming that the homogeneous solutions initially are
zero, as before, and applying Laplace transforms, the complete solutions for this case can be written
冋
û⫽
冕
t
0
册
exp共 2 ␤ ␰ ⫺c 2 /4v ␰ 兲 d ␰
,
␰ 3/2
冋
冕
t
0
册
exp共 2 ␤ ␰ ⫺c /4v̂ ␰ 兲 d ␰
,
␰ 3/2
2
␻⫽
c⬍0,
冉 冊
E2 k4
*
2v
1/3
,
␶ 0 ⫽i ␳␻ .
共58兲
The particular solution is
c⬎0.
The dimensionless drift velocities u/( ␩ ␻ k) and û/( ␩ ␻ k)
from 共54兲 and 共55兲 for a fixed film are plotted in Fig. 2 as
functions of dimensionless depth kc at various dimensionless
2
共57兲
Furthermore, for this problem, we find that m r ⫽⫺m i
⫽( ␻ /2v ) 1/2⫽ ␥ in 共10兲–共12兲. The equation for the mean drift
to second order in the lower fluid now becomes from 共1兲:
u t ⫺ v u cc ⫽⫺6 v ␩ 2 ␻ k ␥ 2 e 2 ␥ c .
共55兲
2
共56兲
Requiring that ␤ ⫽0, we obtain from 共7兲 that
共54兲
3 1/2 2
␩ ␻ ke ⫺2 ␤ t e ⫺2m̂ r c
4
c
⫺
2 共 ␲ v̂ 兲 1/2
Our previous calculations can easily be extended to determine the mean drift induced by nondamped longitudinal
elastic waves. Similar studies of the motion in a viscous fluid
induced by an oscillating plate go back in time; see the discussion by Batchelor.17 To obtain nondamped waves in the
present problem, we may assume that there is a given oscillating horizontal stress in the upper fluid acting on the film.
This stress is chosen such that the damping rate ␤ of the
dilational wave is zero. The resulting nondamped waves will
be termed forced waves. In the boundary condition 共7兲 we
then assume to O(␧):
␶ˆ 共 1 兲 ⫽ ␶ 0 e i 共 ka⫺ ␻ t 兲 .
3 1/2 2
␩ ␻ ke ⫺2 ␤ t e 2m r c
u⫽
4
c
⫹
2 共 ␲ v 兲 1/2
V. FORCED WAVES
u p ⫽ 32 ␩ 2 ␻ ke 2 ␥ c .
共59兲
For a fixed film on top of an infinitely deep fluid, the drift
problem never becomes stationary. By solving the homogeneous problem by Laplace transforms, as before, we obtain
for the total mean Lagrangian drift velocity
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Phys. Fluids, Vol. 15, No. 12, December 2003
Mean drift induced by free and forced dilational waves
FIG. 3. Mean drift velocity u/( ␩ 2 ␻ k) in water as function of depth at times
␻ t⫽1,5,10 induced by forced dilational waves in a fixed film.
冋
冉
冊册
3
c
u⫽ ␩ 2 ␻ k e 2 ␥ c ⫺erfc ⫺
,
2
2 共 v t 兲 1/2
3709
mean velocity induces a strong source of negative Eulerian
mean momentum at the surface. This means that the mean
Lagrangian drift tends to be in the opposite direction of the
waves. For forced waves in a fixed film, this tendency is
strengthened, and the mean drift below the film is negative at
practically all times. Mass and Milgram8 performed laboratory experiments to investigate the dynamic behavior of surfactant films. Longitudinal film waves were generated by a
wave maker, oscillating back and forth at a prescribed frequency. These experiments have much in common with the
present model for forced waves. One important difference is
that the wavelengths generated in the laboratory were considerably longer than the length of the wave trough, which
made the observed motion more like that induced by standing waves. No observations of the mean wave-induced drift
are reported in Ref. 8. This is unfortunate, since such experiments with sufficiently short waves should provide a perfect
setting for observing the backward drift current induced by
progressive longitudinal elastic waves in a fixed film.
ACKNOWLEDGMENTS
c⭐0.
共60兲
In Fig. 3 we have plotted the dimensionless mean drift
u/( ␩ 2 ␻ k) in water from 共60兲 for forced dilational waves in a
fixed film as a function of the dimensionless depth at various
dimensionless times. The parameters for the water and the
film are as in Figs. 1 and 2. Now the source of negative mean
Eulerian momentum at the interface, u h ⫽⫺3 ␩ 2 ␻ k/2, is independent of time. This makes the negative drift more pronounced than in the case of free waves.
In practice, we will have a rigid bottom at a finite distance from the interface where the velocity vanishes. Let this
bottom be placed at c⫽⫺H. If HⰇ1/␥ , we can write the
asymptotic solution of 共58兲 when time tends to infinity as
u⫽ 32 ␩ 2 ␻ k 共 e 2 ␥ c ⫺c/H⫺1 兲 .
共61兲
We note from this expression that the drift velocity is negative everywhere in the fluid.
VI. SUMMARY AND CONCLUDING REMARKS
In this paper we study temporally and spatially damped
dilational film waves. It has been demonstrated5 that the dispersion relation ␻ ⫽ ␻ (k) for the two cases of damping differs by a numerical factor, which suggests an ambiguity as
far as the group velocity is concerned. We show here that the
energy flux in free, strongly damped dilational waves is negligible, which makes the concept of the group velocity as an
energy carrier irrelevant. Hence, a nonunique group velocity
has no consequences for the physics of this problem.
The main aim of the present paper has been to investigate the nonlinear mean Lagrangian drift associated with dilational waves. For a freely drifting film with time-damped
waves, we find that a strong virtual wave stress at the surface
acts to redistribute the initial wave momentum into a Eulerian current. This current is directed in the wave propagation
direction. For a fixed film, the boundary condition on the
During this study K.H.C. was supported by The Research Council of Norway 共NFR兲 through Grant No. 151774/
432. J.E.W. gratefully acknowledges NFR support through
the Strategic University Program ‘‘Modeling of currents and
waves for sea structures.’’
1
R. Dorrestein, ‘‘General linearized theory of the effect of surface films on
water ripples,’’ Proc. K. Ned. Akad. Wet., Ser. B: Phys. Sci. 54, 260
共1951兲.
2
J. C. Gottifredi and G. J. Jameson, ‘‘The suppression of wind-generated
waves by a surface film,’’ J. Fluid Mech. 32, 609 共1968兲.
3
Ø. Saetra, ‘‘Effects of surface film on the linear stability of an air–sea
interface,’’ J. Fluid Mech. 357, 59 共1998兲.
4
J. E. Weber and Ø. Saetra, ‘‘Effect of film elasticity on the drift velocity of
capillary-gravity waves,’’ Phys. Fluids 7, 307 共1995兲.
5
J. Lucassen, ‘‘Longitudinal capillary waves, 1. Theory,’’ Trans. Faraday
Soc. 64, 2221 共1968兲.
6
K. Dysthe and Y. Rabin, ‘‘Damping of short waves by insoluble surface
film,’’ in ONRL Workshop Proceedings-Role of Surfactant Films on the
Interfacial Properties of the Sea Surface, edited by F. L. Herr and J.
Williams 共U.S. Office of Naval Research, London, 1986兲.
7
M. S. Longuet-Higgins, ‘‘A nonlinear mechanism for the generation of sea
waves,’’ Proc. R. Soc. London, Ser. A 311, 371 共1969兲.
8
J. T. Mass and J. H. Milgram, ‘‘Dynamic behavior of natural sea surfactant
films,’’ J. Geophys. Res. 103, 15695 共1998兲.
9
E. H. Lucassen-Reynders and J. Lucassen, ‘‘Properties of capillary
waves,’’ Adv. Colloid Interface Sci. 2, 347 共1969兲.
10
H. Lamb, Hydrodynamics, 6th ed. 共Cambridge University Press, Cambridge, 1932兲.
11
J. E. Weber, ‘‘Attenuated wave-induced drift in a viscous rotating ocean,’’
J. Fluid Mech. 137, 115 共1983兲.
12
W. J. Pierson, ‘‘Perturbation analysis of the Navier–Stokes equations in
Lagrangian form with selected linear solutions,’’ J. Geophys. Res. 67,
3151 共1962兲.
13
M. Gaster, ‘‘A note on the relation between temporally-increasing and
spatially-increasing disturbances in hydrodynamic stability,’’ J. Fluid
Mech. 14, 222 共1962兲.
14
G. G. Stokes, ‘‘On the theory of oscillatory waves,’’ Trans. Cambridge
Philos. Soc. 8, 441 共1847兲.
15
J. E. Weber and E. Førland, ‘‘Effect of air on the drift velocity of water
waves,’’ J. Fluid Mech. 218, 619 共1990兲.
16
J. E. Weber, ‘‘Virtual wave stress and mean drift in spatially damped
surface waves,’’ J. Geophys. Res. 106, 11653 共2001兲.
17
G. K. Batchelor, An Introduction to Fluid Dynamics 共Cambridge University Press, Cambridge, 1967兲.
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