Wave-induced drift in the presence of a
flexible surface cover
Kai Håkon Christensen
Dissertation for the degree of
Philosophiae Doctor
Department of Geosciences, University of Oslo
March 2005
Preface
The present thesis, entitled “Wave-induced drift in the presence of a flexible
surface cover”, is submitted in partial fulfillment of the requirements for the
degree Philosophiae Doctor. It consists of the following papers:
Paper 1:
“Mean drift induced by free and forced dilational waves”, by
Weber, J.E. and Christensen, K.H., Phys. Fluids, 15(12), 3703-3709 (2003).
Paper 2:
“Transient and steady drift currents in waves damped by sur-
factants”, by Christensen, K.H., Phys. Fluids, 17(4), 9 pages (2005).
Paper 3:
“Drift of an inextensible sheet caused by surface waves”, by
Christensen, K.H. and Weber, J.E., Environ. Fluid Mech., in press.
Paper 4:
“Wave-induced drift of large floating sheets”, by Christensen,
K.H. and Weber, J.E., submitted to Geophys. Astro. Fluid Dyn.
The research presented in this thesis has been supported by the
Research Council of Norway through Grant No. 151774/432.
My sincerest gratitude goes to my supervisor Jan Erik Weber. There
are many dead-end roads an unexperienced young scientist may walk down,
and many traps to avoid. By his advices and skillful guidance I was at no
time at loss.
Special thanks goes to my assistant supervisors Göran Broström and
Øyvind Sætra for much help and stimulating discussions. I would also like
to express my sincere gratitude to everyone at the Dept. of Oceanography,
Earth Sciences Centre, Gothenburg University for their hospitality.
Finally, I would like to thank my wife Sofia and my close family for
their continuous encouragement and support.
1
Introduction
The surface of any still body of water is seldom “clean”, that is, the materials
that constitute the surface are rarely of the exact same chemical composition as the fluid beneath the surface. Under calm conditions surface-active
materials can accumulate and form thin films. Such films change the dynamics of the surface, inhibit wind-wave growth (Gottifredi and Jameson, 1968;
Saetra, 1998), and are effective in damping the capillary-gravity waves with
wavelengths typically of the order 1-10 cm (e.g. Lucassen-Reynders and
Lucassen, 1969). In much the same way, a cover of sea ice is very effective in
damping oceanic swell with wavelengths typically of the order 100-1000 m
(e.g. Squire et al., 1995). Enhanced damping of the waves causes changes in
the wave drift, because the mean momentum lost by the waves is ultimately
transferred to a mean current in the waves (e.g. Longuet-Higgins, 1969;
Weber, 2001)
This thesis is a compilation of four papers on the effect of a flexible
surface cover on the drift velocities in progressive deep-water waves. The
papers treat quite different aspects, from the drift in short dilational waves
on film-covered surfaces to the wave drift of large floating objects, for example ice floes in the marginal ice zone. Still, the basic mechanisms are
the same, and therefore a brief introduction to the mathematical modeling
of wave drift, relevant for this thesis, will be given. Modifications of the
wave growth/damping rates are first-order effects of a surface cover, while
modifications of the mean drift velocities in the waves are second-order effects. The wave drift is, however, intimately connected to the first-order
wave dynamics, and we will use this as a starting point.
1.1
Effect of a surface cover on waves
The fundamental properties of deep-water capillary-gravity waves are readily obtained from linear theory assuming that the motion is irrotational, i.e.
1
neglecting the fluid viscosity. Requiring that the pressure at the surface
balances the normal stress due to surface tension one obtains the dispersion
equation. The dispersion equation connects the wave period to the wavelength, and thus one obtains the phase and group velocities of the waves.
In the first approximation, individual fluid “particles” in irrotational waves
orbit in circles, and the radius of each circle decreases exponentially with
depth at a rate inversely proportional to the wavelength.
In a real fluid there is always a thin oscillatory surface boundary
layer (SBL) where the motion is influenced by friction. If there is some form
of cover on the surface, the fluid particles at the surface are no longer free to
move in circular orbits. Consider a simple example with the surface covered
by an inelastic film. The fluid adjacent to the film is forced to move up and
down, without the periodic horizontal motion of waves on a film-free surface.
Just below the SBL the particle trajectories will be the same as for the filmfree case. The corresponding strong shear in the SBL leads to increased
viscous dissipation, and hence the waves lose energy and are damped at a
higher rate than waves on a film-free surface (e.g. Lamb, 1932).
In general, thin films found at sea are viscoelastic, not inextensible.
The elastic properties of the film are usually expressed in terms of the elastic
modulus E. The elastic modulus is connected to the surface tension T , and
expresses the resistance against a change in the area of a surface element.
The area A of a surface element is inversely proportional to the concentration
of film material, Γ, and the elastic modulus is therefore defined as
E=
dT
dT
.
=−
d(ln Γ)
dA/A
(1)
Two different wave modes can be excited on a surface covered by
an elastic film. In addition to the capillary-gravity waves, the surface also
supports dilational waves (Lucassen, 1968). Dilational waves are, unlike
capillary-gravity waves, not influenced by gravity or the static surface tension: A local expansion of the surface leads to an elastic force that strives to
2
bring back the concentration of film material to equilibrium. This creates
a horizontally sheared flow in the SBL, which subsequently will compress
the surface yielding an elastic force in the opposite direction, and so on.
This system is highly dissipative and dilational waves are almost critically
damped. Analysing the coupled system of capillary-gravity and dilational
waves one finds that the capillary-gravity waves provide a forcing term in
the dilational wave dispersion equation (Ermakov, 2003). In consequence, it
is not possible to generate capillary-gravity waves without at the same time
forcing the excitation of dilational waves. The opposite statement is not
true, i.e. dilational waves can be generated without generating capillarygravity waves.
The first consistent treatment on the effect of an inextensible film on
capillary-gravity waves is due to Lamb (1932), and later Dorrestein (1951)
showed that the wave damping rate for elastic films can be twice that for
inextensible films. In both cases the damping rate is an order of magnitude
larger than in the film-free case. That elastic films are more effective in
damping waves than inelastic films is because of the increased shear in the
SBL due to the excitation of dilational waves (e.g. Dysthe and Rabin, 1986).
Experiments have confirmed the theoretical damping rates of both dilational
and capillary-gravity waves for films formed by various surface-active materials, including such surfactants as commonly found at sea (e.g. Mass and
Milgram, 1998).
1.2
The wave drift
For unidirectional, monochromatic capillary-gravity waves the equation for
the horizontal Lagrangian drift velocity u in the direction of wave propagation takes the form (e.g. paper 2)
Lu = F (c)e−2(αa+βt) ,
3
(2)
where L is a linear differential operator. Furthermore, (a, c) are the horizontal and vertical coordinates, respectively, t is time, α is the spatial damping
coefficient, and β is the temporal damping coefficient. F (c) is a function
derived from the first-order wave solutions (for details on the separation of
the flow field into a first-order wave field and a second-order drift current,
see for example paper 2).
The drift velocity can be split into three parts:
u = u(S) + u(ν) + u(h) .
(3)
Here u(S) + u(ν) is the particular solution, and u(h) is the homogeneous
solution of (2). Since F (c) is known, both u(S) and u(ν) are determined by
the first-order solutions.
The first part on the righthand-side of (3) is the Stokes drift (Stokes,
1847), which is the mean drift in irrotational waves:
u(S) = (ζ0 k)2 Cp e2kc e−2(αa+βt) ,
(4)
where ζ0 is the wave amplitude, k is the wave number, and Cp is the phase
velocity of the waves. The Stokes drift is an inherent part of the waves,
hence if the waves attenuate completely, the Stokes drift contribution in
(3) becomes zero. The second part, u(ν) , represents a steady streaming restricted to the SBL and generally depends on the structure of the oscillatory
motion in the SBL. The third part, u(h) , develops in response to the initial
and boundary conditions that apply to the drift current (Longuet-Higgins,
1953). Neglecting the horizontal gradients, the equation for u(h) is (e.g. paper 2)
(h)
ut
− νu(h)
cc = 0,
(5)
where ν is the kinematic viscosity. Equation (5) shows that this part of the
drift current is a result of viscous diffusion of momentum. For u(h) to grow,
both time and a source of momentum are required.
4
We will consider two different examples of the drift in capillarygravity waves: (i) a freely drifting thin film, and, (ii) a stationary thin film.
In both cases we assume that the surface is given by c = 0, and the water
depth is infinite.
In case (i), a dynamic boundary condition must be applied at the
surface. In the absence of external stresses, e.g. wind, the appropriate
boundary condition is (Weber and Førland, 1989; Weber and Saetra, 1995)
ρνuc = 0,
c = 0,
(6)
assuming that the film has negligible mass. We may rewrite (6) as
ρνu(h)
c = τw ,
c = 0,
(S)
(7)
(ν)
introducing the virtual wave stress τw = −ρνuc −ρνuc
1969). Since
u(S)
and
u(ν)
(Longuet-Higgins,
indirectly are obtained from the first-order so-
lutions, τw can be considered as known prior to solving the complete drift
problem. The virtual wave stress provides a convenient way to describe how
the momentum is transferred from the wave motion to the drift current. Assuming that the motion initially is irrotational, the total mean momentum
in the waves is
0
−∞
ρu(S) dc = M0 .
(8)
If no external forces act on the fluid this momentum cannot be lost. For
temporally damped waves the following relation holds for both a clean surface (Longuet-Higgins, 1969), and a surface covered by an inextensible or
elastic film (Weber and Førland, 1989; Weber and Saetra, 1995):
∞
τw dt = M0 .
(9)
0
Equations (7) and (9) demonstrate that all the initial mean wave momentum
will be transferred to the transient part of the drift current u(h) by the action
of the virtual wave stress. Thus the final drift velocity will be very different
from the initial Stokes drift.
5
In case (ii), where the film is stationary, the appropriate boundary
condition is kinematic, that is,
u = 0,
c = 0.
(10)
The boundary condition (10) may be rewritten as
u(h) = −u0 ,
c = 0,
(11)
where u0 = u(S) (c = 0) + u(ν) (c = 0). By the same arguments as before,
the value of u0 can be considered as known once the first order solutions are
found. The total drift velocity will also in this case become very different
from the initial Stokes drift (paper 2). Because an external force is needed
to keep the film stationary, the relation (9) does not hold. However, from
the full expression for u one can calculate the force required to keep the film
from drifting (e.g. Foss, 2000), and theoretical results have been experimentally verified for both inextensible and elastic films (Kang and Lee, 1995;
Gushchin and Ermakov, 2003).
2
Summary of papers
Paper 1: Weber, J. E. and Christensen, K. H., Mean drift induced
by free and forced dilational waves, Phys. Fluids, 15(12), 37033709 (2003).
In this paper the drift in dilational waves on the interface between two
viscous fluids is studied. Weber and Førland (1989) examined the effect
of an inextensible film on the drift in capillary-gravity waves. Allowing
the film to be elastic, Weber and Saetra (1995) later showed that the drift
in capillary-gravity waves is strongly dependent on the value of the elastic
modulus. The difference in the results for inextensible and elastic films,
which clearly was due the the excitation of dilational waves in the latter
case, was the motivation for the study of the drift in pure dilational waves.
6
First in the paper some basic properties of dilational waves are examined.
Using energy considerations it is shown that dilational waves do not carry
energy, there is a local balance between the elastic work done by the film
and viscous dissipation. The wave-induced drift is investigated for three
different cases: (i) temporally damped waves and freely drifting film, (ii)
temporally damped waves and fixed film (i.e. stationary in a mean sense),
and, (iii) undamped/forced waves and fixed film. There is no Stokes drift
in dilational waves and, except for the case of forced dilational waves, the
drift currents in both fluids rapidly attenuate. For freely drifting films the
drift currents are in the wave propagation direction, while for fixed films the
drift currents are shown to be directed opposite to the wave propagation
direction.
Paper 2: Christensen, K. H., Transient and steady drift currents in
waves damped by surfactants, Phys. Fluids, 17(4), 9 pages (2005).
In this paper the drift in spatially damped capillary-gravity waves on a
surface covered by an elastic film is studied. The main aim of the paper is
to present theoretical results that can be tested in laboratory experiments.
Earlier theoretical studies treat freely drifting films (e.g. Craik, 1982; Weber
and Førland, 1989; Weber and Saetra, 1995), and the results apply more to
large slicks on the ocean surface than to a laboratory situation. In essence,
the mathematical formulation in paper 2 describes the experimental setup of
Mass and Milgram (1998), where the film covers the whole surface of a wave
tank that is closed in both ends. As a consequence the film must become
stationary in a mean sense. The wave field is a superposition of capillarygravity waves and forced dilational waves, and the results show that for
certain values of the elastic modulus the drift current in the upper part of
the water column is directed opposite to the wave propagation direction.
7
Paper 3: Christensen, K. H. and Weber, J. E., Drift of an inextensible sheet caused by surface waves, Environ. Fluid Mech., in
press.
Experimental studies of freely drifting films are few, and unfortunately most
of them lack important information needed to compare the experimental
data with existing theory. An exception is the study by Law (1999) of the
drift of thin plastic sheets subjected to gravity waves in a large wave tank.
The theoretical results of Weber (1987) are applied to such plastic sheets,
and a comparison is made with drift data from Law’s experiments. In several
studies, including the study by Law, it is assumed a priori that the sheets
attain a steady drift velocity. It is pointed out in paper 3 that the source of
mean momentum is unlimited because the waves are continuously generated
(Weber, 2001). For this reason the drift velocity increases in time, and data
on the time development of the drift is needed. In some experiments the
observed drift shows an almost linear dependence on the wave steepness, in
contrast to a quadratic dependence as predicted by theory. In paper 3 a
possible explanation for this discrepancy is given in a short discussion on
the experimental procedure.
Paper 4: Christensen, K. H. and Weber, J. E., Wave-induced drift
of large floating sheets, submitted to Geophys. Astro. Fluid Dyn.
While the three previous papers deal with short waves/ripples and thin films,
the aim of this paper is to investigate the wave-induced drift of very large
floating objects. The horizontal extent of the floating object is assumed
much larger than the thickness, and such objects are referred to as sheets.
The sheet is assumed flexible, so that long surface waves may propagate
through it. Being damped by the sheet, the waves yield up a portion of
their momentum, which results in a mean drift of the sheet. The sheet and
the drift current in the water beneath are influenced by the earth’s rotation.
8
Both constant and depth varying eddy viscosities are used to model the
turbulence in the water. The solutions for the drift velocities of the sheet
and in the water are shown to be sensitive to the formulation of the eddy
viscosity.
3
Discussion and concluding remarks
In order to resolve the dynamics in the SBL one needs to use some sort of
particle following coordinates, and in all the papers a Lagrangian description of motion is used. In the Lagrangian frame of reference, the surface
boundary conditions are easily formulated as we continuously “track” the
particles at the surface. As opposed to using Eulerian curvilinear coordinates
(e.g. Longuet-Higgins, 1953), it is also straightforward to include spatial or
temporal damping of the waves, allowing for a more consistent treatment
of the dynamics. There are nevertheless nontrivial problems connected to
the use of a Lagrangian description of motion. For example, two particles
that initially are close will eventually drift far apart, which renders terms
containing spatial derivatives meaningless. It appears however, in spite of
the problems connected to the use of particle following coordinates, that the
results are robust in the sense that they capture the physics well. Equations (5) and (7) do model the process of diffusion of second-order mean
momentum from the SBL, as described by Longuet-Higgins (1953). Also,
theoretical results for the wave-induced mean stress that acts on a surface
cover have been experimentally verified, as discussed in Section 1.2. The
author would rather point to the need for experimental data. As pointed
out in papers 2 and 3, experimental studies are few, and many studies lack
essential information needed to compare the experimental data with theory.
It is by comparison with experiments that we really can revise our theories,
and in that way increase our understanding of the physics involved.
9
References
A. D. D. Craik. The drift velocity of water waves. J. Fluid Mech., 116:
187–205, 1982.
R. Dorrestein. General linearized theory of the effect of surface films on
water ripples. Proc. K. Ned. Akad. Wet., Ser. B: Phys. Sci., B54:260–
272,350–356, 1951.
K. Dysthe and Y. Rabin. Damping of short waves by insoluble surface film.
In F. L. Herr and J. Williams, editors, ONRL Workshop Proceedings Role of Surfactant Films on the Interfacial Properties of the Sea Surface.
U.S. Office of Naval Research, London, 1986.
S. A. Ermakov. Resonance damping of gravity-capillary waves on the water surface covered with a surface-active film. Izvestiya, Atmos. Oceanic
Phys., 39(5):691–696, 2003.
M. Foss. Wave damping and momentum transfer. Dr. thesis, University of
Tromsø, Tromsø, Norway, 2000.
J. C. Gottifredi and G. J. Jameson. The suppression of wind-generated
waves by a surface film. J. Fluid Mech., 32:607–618, 1968.
L. A. Gushchin and S. A. Ermakov. Laboratory study of surfactant redistribution in the flow field induced by surface waves. Izvestiya, Atmos.
Oceanic Phys., 40(2):244–249, 2003.
H. K. Kang and C. M. Lee. Steady streaming of viscous surface layer in
waves. J. Mar. Sci. Technol., 1:3–12, 1995.
H. Lamb. Hydrodynamics. Cambridge University Press, 6th edition, 1932.
Adrian W. K. Law. Wave-induced surface drift of an inextensible thin film.
Ocean Engrg., 26:1145–1168, 1999.
10
M. S. Longuet-Higgins. Mass transport in water waves. Philos. Trans. R.
Soc. London, Ser. A, 245:535–581, 1953.
M. S. Longuet-Higgins. A nonlinear mechanism for the generation of sea
waves. Proc. R. Soc. Lond. A, 311:371–389, 1969.
J. Lucassen. Longitudinal capillary waves. Trans. Faraday Society, 64:2221–
2235, 1968.
E. H. Lucassen-Reynders and J. Lucassen. Properties of capillary waves.
Advan. Colloid Interface Sci., 2:347–395, 1969.
J. T. Mass and J. H. Milgram. Dynamic behavior of natural sea surfactant
films. J. Geophys. Res., 103(C8):15,695–15,715, 1998.
Ø. Saetra. Effects of surface film on the linear stability of an air-sea interface.
J. Fluid Mech., 357:59–81, 1998.
V. A. Squire, J. P. Dugan, P. Wadhams, P. J. Rottier, and A. K. Liu. Of
ocean waves and sea ice. Annu. Rev. Fluid. Mech., 27:115–168, 1995.
G. G. Stokes. On the theory of oscillatory waves. Trans. Camb. Phil. Soc.,
8:441, 1847.
J. E. Weber. Wave attenuation and wave drift in the marginal ice zone. J.
Phys. Oceanogr., 17(12):2351–2361, 1987.
J. E. Weber. Virtual wave stress and mean drift in spatially damped surface
waves. J. Geophys. Res., 106(C6):11653–11657, 2001.
J. E. Weber and E. Førland. Effect of an insoluble surface film on the drift
velocity of capillary-gravity waves. J. Phys. Oceanogr., 19(7):952–961,
1989.
J. E. Weber and Ø. Saetra. Effect of film elasticity on the drift velocity of
capillary-gravity waves. Phys. Fluids, 7(2):307–314, 1995.
11
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
3704
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
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
冋
册
i ␬ ␬ c m mc i ␬ a⫹nt
x 共 1 兲 ⫽⫺
e ⫺ e e
,
n
␬
共10兲
␬
z 共 1 兲 ⫽⫺ 关 e ␬ c ⫺e mc 兴 e i ␬ a⫹nt ,
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 ⫽⫺
ẑ 共 1 兲 ⫽
冋
册
i ␬ Q ⫺ ␬ c m̂ ⫺m̂c i ␬ a⫹nt
e
⫺ e
e
,
n
␬
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
Q⫽
m⫺ ␬
.
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
e.g., Dysthe and Rabin6 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 Lucassen 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 mul5
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
0
⫺⬁ 2
共 x 共t 1 兲 兲 2 dc⫹
⳵
⳵a
⫽E 共 x 共aa1 兲 x 共t 1 兲 兲 c⫽0 ⫺ v
*
冕
冕
0
1
⫺⬁ ␳
0
⫺⬁
共 p 共 1 兲 x 共t 1 兲 兲 dc
共 x 共tc1 兲 兲 2 dc.
共28兲
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
0
共 x 共t 1 兲 兲 2 dc
⫺⬁ 2
is the energy density 共here equal to the kinetic energy per
unit mass兲. Furthermore,
F̄⬅
冕
1
0
共 p 共 1 兲 x 共t 1 兲 兲 dc
⫺⬁ ␳
W̄⬅E 共 x 共aa1 兲 x 共t 1 兲 兲 c⫽0
*
is the work done by the elastic film, and
冕
0
⫺⬁
共 x 共tc1 兲 兲 2 dc
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 ,
共31兲
where ␥ ⬅( ␻ /(2 v )) 1/2, ␥ˆ ⬅( ␻ /(2 v̂ )) 1/2. Our former assumption 兩 m 兩 , 兩 m̂ 兩 Ⰷk, requires that the wavelength ␭
⫽2 ␲ /k must satisfy
␭Ⰷ
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
共33兲
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.
A. Freely drifting film
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
is the energy flux,
D̄⬅ v
IV. THE NONLINEAR WAVE DRIFT
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 ,
û t ⫺ v̂ û cc ⫽⫺2•3 v̂ ␩
1/2
2
k ␻ m̂ r2 e ⫺2m̂ r c⫺2 ␤ t .
共35兲
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
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
1 ␳
␳ˆ
⫹
␩ 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.
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
u h ⫽⫺u p ⫽⫺
3 1/2 2
␩ ␻ ke ⫺2 ␤ t ,
4
c⫽0,
共52兲
û h ⫽⫺û p ⫽⫺
3 1/2 2
␩ ␻ ke ⫺2 ␤ t ,
4
c⫽0.
共53兲
By assuming that the homogeneous solutions initially are
zero, as before, and applying Laplace transforms, the complete solutions for this case can be written
u⫽
c
2 共 ␲ v 兲 1/2
冕
t
0
册
exp共 2 ␤ ␰ ⫺c 2 /4v ␰ 兲 d ␰
,
␰ 3/2
冋
c
⫺
2 共 ␲ v̂ 兲 1/2
冕
0
册
exp共 2 ␤ ␰ ⫺c /4v̂ ␰ 兲 d ␰
,
␰ 3/2
2
共56兲
Requiring that ␤ ⫽0, we obtain from 共7兲 that
␻⫽
c⬍0,
共54兲
3 1/2 2
␩ ␻ ke ⫺2 ␤ t e ⫺2m̂ r c
4
t
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
4
⫹
û⫽
冋
V. FORCED WAVES
冉 冊
E2 k4
*
2v
1/3
,
␶ 0 ⫽i ␳␻ .
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58兲
The particular solution is
c⬎0.
共55兲
The dimensionless drift velocities u/( ␩ 2 ␻ k) and û/( ␩ 2 ␻ 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
u p ⫽ 23 ␩ 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
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.
PHYSICS OF FLUIDS 17, 042102 共2005兲
Transient and steady drift currents in waves damped by surfactants
Kai Haakon Christensena兲
Department of Geosciences, University of Oslo, P.O. Box 1022, Blindern, N-0315 Oslo, Norway
共Received 24 March 2004; accepted 18 January 2005; published online 15 March 2005兲
In this paper we study the Lagrangian mean drift induced by spatially damped capillary-gravity
waves on a surface covered by an elastic film. The analysis is developed with regard to a typical
laboratory setup, and explicit solutions for both transient and steady horizontal drift velocities are
given. We consider a situation where the film covers the entire surface and is prevented from drifting
away, e.g., by a film barrier. The drift below an inextensible film resembles the drift under an ice
cover, with a jetlike current in the wave propagation direction just below the surface. If the film is
elastic the solution changes drastically. For certain values of the film elasticity parameter the mean
flow is in the direction opposite to that of wave propagation in the upper part of the water column.
© 2005 American Institute of Physics. 关DOI: 10.1063/1.1872112兴
I. INTRODUCTION
Surface-active materials can form large slicks on the
ocean surface that influence the formation and evolution of
capillary-gravity waves. Such surface films both inhibit wind
wave growth1,2 and cause rapid damping of existing
waves.3,4 The reason for this is the film’s ability to restrain
tangential motion and the accompanying strong shear in the
surface boundary layer. Nonlinearly, surface films cause enhanced transfer of momentum from the waves to a mean
motion.
A marked maximum in the transverse wave damping rate
for a finite value of film elasticity is connected to the excitation of dilational waves, which are characterized by alternating compression and dilation of the surface.5 During the
course of the present study it became clear that we can regard the film and the surface boundary layer as a solid
stretched elastic membrane. The capillary-gravity waves
force the excitation of longitudinal 共i.e., the dilational兲 waves
in this membrane,6 and the frequency for which we find
maximum damping of the transverse waves corresponds to
the natural frequency of longitudinal waves in the membrane. As these findings provide valuable physical insight of
the problem, we will discuss the linear system in some detail
prior to the nonlinear analysis.
Vorticity will diffuse from the surface boundary layer
into the lower parts of the fluid, and a mean Eulerian drift
current develops.7 Previous studies8,9 have shown that both
elastic and inextensible films significantly modify the drift
currents in capillary-gravity waves. Unfortunately, there is a
lack of experimental data, especially for elastic films. Furthermore, the above mentioned theoretical studies treat freely
drifting films, hence the results apply to large slicks on the
ocean surface rather than to a typical laboratory situation. It
is the main aim of this paper to present theoretical results
that can be validated in experiments. For this reason we focus on the part of the drift current that reaches below the thin
surface boundary layer and thus easily can be measured dia兲
Electronic mail: k.h.christensen@geo.uio.no
1070-6631/2005/17共4兲/042102/9/$22.50
rectly. We use a Lagrangian description of motion and
present both a transient and a time independent solution.
The outline of the paper is as follows: In Sec. II we
formulate the problem mathematically. The first-order periodic solutions are presented in Sec. III, and in Sec. III A we
discuss some properties of the waves. In Sec. IV we derive
the equations for the wave drift. Analytical solutions are
found for the horizontal drift velocity: a transient solution is
presented in Sec. IV A, while a steady-state solution is presented in Sec. IV B. In Sec. IV C we discuss how the waves
may affect the film cover in view of the assumptions the
analysis rests upon. Finally, Sec. V contains a summary and
some concluding remarks.
II. MATHEMATICAL FORMULATION
We consider monochromatic waves on the surface of a
viscous, incompressible fluid of density ␳. An insoluble
monomolecular film covers the entire surface. The depth is
such that the deep-water limit applies, i.e., at least larger than
half a wavelength. The fluid motion is two dimensional, and
we use a Cartesian coordinate system with the x axis aligned
along the undisturbed surface and with the z axis positive
upwards.
Changes in the area of a material surface 共film兲 element
lead to local variations in surface tension, and we must allow
for this in our boundary conditions. Denoting the Eulerian
fluid velocities by 共U , W兲, pressure by P, and the surface
elevation by ␨, the horizontal and vertical dynamic boundary
conditions at the surface, correct to the second order,
become9
␳␯共Uz + Wx兲 + P␨x − 2␳␯Ux␨x + ␴␨xx␨x − ␴x − ␴z␨x
= ␶ˆ − ␴ˆ ␨x ,
共1兲
− P + 2␳␯Wz − ␳␯共Uz + Wx兲␨x − ␴␨xx − ␴x␨x = ␴ˆ + ␶ˆ ␨x ,
共2兲
respectively. Here ␯ is the kinematic viscosity of the fluid, ␴
is the surface tension, and ␶ˆ and ␴ˆ constitute external tan-
17, 042102-1
© 2005 American Institute of Physics
042102-2
Phys. Fluids 17, 042102 共2005兲
Kai Haakon Christensen
gential and normal stresses acting on the surface. Furthermore, we have let subscripts denote partial differentiation.
Our main aim is to investigate the mean drift velocities
induced by the wave motion, and these are directly obtained
by our choice of a Lagrangian description of motion. We
label each fluid particle with coordinates 共a , c兲, chosen as the
initial position when the fluid is at rest. Our dependent variables become the particle displacement 共x , z兲 and the pressure p in the vicinity of the particle, while our independent
variables are 共a , c兲 and time, t. Spatial derivatives in the
Eulerian description may be transformed into derivatives in
Lagrangian independent coordinates by
currents if the waves are unaffected. Furthermore, we take E
to be constant, which means that ⌫ is close to ⌫0. The validity of this assumption is discussed in Sec. IV C 1. The possible role of the surface shear viscosity in stabilizing the film
cover is discussed in Sec. IV C 2.
To facilitate the calculations we make use of series expansions of the dependent variables after an ordering parameter ⑀ 共e.g., Pierson10兲:
x = a + ⑀x共1兲 + ⑀2x共2兲 + ¯ ,
共10兲
z = c + ⑀z共1兲 + ⑀2z共2兲 + ¯ ,
共11兲
f x = J共f,z兲/J共x,z兲 = 共f azc − f cza兲/共xazc − xcza兲,
共3兲
p = − ␳gc + ⑀ p共1兲 + ⑀2 p共2兲 + ¯ ,
共12兲
f z = J共x, f兲/J共x,z兲 = 共xa f c − xc f a兲/共xazc − xcza兲,
共4兲
␴ = T + ⑀␴共1兲 + ⑀2␴共2兲 + ¯ ,
共13兲
⌫ = ⌫0 + ⑀⌫共1兲 + ⑀2⌫共2兲 + ¯ .
共14兲
introducing the Jacobian J. An advantage of using the Lagrangian description is that the free surface is given by c
= 0 at all times. For a similar analysis in Eulerian description
one would have to use curvilinear coordinates.7 The velocity
and acceleration of a fluid particle are simply 共xt , zt兲 and
共xtt , ztt兲. Because the time scale of the present problem is
much less than an inertial period we disregard Coriolis
forces. The equations for conservation of momentum and
volume then read3,10
␳xtt = − J共p,z兲 + ␳␯兵J关J共xt,z兲,z兴 + J关x,J共x,xt兲兴其,
共5兲
␳ztt = − J共x,p兲 − ␳g + ␳␯兵J关J共zt,z兲,z兴 + J关x,J共x,zt兲兴其,
共6兲
J共x,z兲 = 1.
共7兲
Here g is the acceleration due to gravity. We note that by 共7兲
the denominators in 共3兲 and 共4兲 equal unity.
For insoluble films, as we consider here, there is no exchange of film material with the fluid, and we must require
that the total amount of film material at the surface is conserved, hence9
共x2a + z2a兲1/2 = ⌫0/⌫,
c = 0.
共8兲
Here ⌫ is the concentration of film material and ⌫0 the concentration equilibrium value. Surface tension variations will
be related to the fluid motion through the dilational modulus
of the film, defined by
E=−
d␴
.
d共ln ⌫兲
共9兲
The real part of E describes the elastic properties of the film,
while the surface viscosity is incorporated in the imaginary
part 共e.g., Hansen and Ahmad11兲. One distinguishes between
the surface dilational and shear viscosity, which are associated to resistance against changes in area and shape of a
surface element, respectively. For insoluble or sparingly
soluble monolayers we have in general12 Re共E兲 Ⰷ Im共E兲, and
we will take E to be real. Experimental evidence shows that
effects of the surface viscosity are negligible as far as the
wave dynamics are concerned.13 As will become clear later,
the nonlinear drift depends on the wave motion, hence the
surface viscosity should not influence the induction of drift
Using 共10兲–共14兲, we substitute the variables in the governing
equations and collect terms of the same order in ⑀. The equations to O共⑀兲 yield the periodical motion or the primary
wave, whereas the equations to O共⑀2兲 yield the particle drift.
It should be noted that ⑀ is dimensional, and for waves of
amplitude ␨0, wave frequency ␻, and wave number k, we
have ⑀ = ␨0␻ / k. The equivalent nondimensional form is the
wave steepness ␨0k which is assumed to be a small quantity.9
Finally, in order to obtain a consistent set of equations,
we must expand the external stresses in series:
␶ˆ = ⑀␶ˆ 共1兲 + ⑀2␶ˆ 共2兲 + ¯ ,
共15兲
␴ˆ = ⑀␴ˆ 共1兲 + ⑀2␴ˆ 共2兲 + ¯ .
共16兲
III. LINEAR MOTION: TRANSVERSE AND DILATIONAL
WAVES
The linear problem is extensively studied and we will
not go into details on the derivation of the first order solutions 共e.g., Lucassen and Lucassen-Reynders14兲. They are
nonetheless needed to derive the equations to O共⑀2兲 and will
therefore be stated below. The solutions are obtained by
separating the flow field into an irrotational and a rotational
part, representing a capillary-gravity and a dilational wave,
respectively, as further discussed in Sec. III A. For a progressive wave component the solutions are given by the real
parts of15
x共1兲 = −
冉
冊
m
i ␬ ␬c
e − iB emc ei␬a+nt ,
␬
n
共17兲
␬
z共1兲 = − 共e␬c − iBemc兲ei␬a+nt ,
n
共18兲
␳
p共1兲 = 关共n2 + g␬兲e␬c − ig␬Bemc兴ei␬a+nt .
n
共19兲
We consider spatially damped waves, so we take n = −i␻,
with the wave frequency ␻ real and positive. Furthermore,
we let ␬ = k + i␣, where both the wave number k and the
spatial damping coefficient ␣ are real and positive. B is a
042102-3
Phys. Fluids 17, 042102 共2005兲
Transient and steady drift currents in waves
complex coefficient to be determined from the boundary
conditions. The solutions 共17兲–共19兲 are based on the assumption that the waves are sufficiently short, i.e., with vanishing
motion near the bottom. The value of m reflects the vertical
scale of the surface boundary layer and is given by
m2 = ␬2 − i␻/␯ .
共20兲
We have approximately
m = 共1 − i兲␥ ,
共21兲
where ␥ = 共␻ / 2␯兲1/2 is the inverse boundary layer thickness.
It follows from 共20兲 that this approximation is valid for
k / ␥ Ⰶ 1. For wavelengths between 1 and 10 cm the value of
k / ␥ is between 0.02 and 0.08.
From 共8兲 and 共9兲 we obtain to O共⑀兲
␴共1兲 = Ex共1兲
a .
共22兲
Using 共22兲, the dynamic boundary conditions 共1兲 and 共2兲
become
共1兲
共1兲
␳␯共x共1兲
tc + zta 兲 = Exaa ,
共1兲
− p共1兲 + 2␳␯z共1兲
tc = Tzaa ,
c = 0,
c = 0.
共23兲
共24兲
We have here set ␶ˆ 共1兲 = ␴ˆ 共1兲 = 0, assuming that there are no
external stresses acting on the surface to this order, i.e., no
periodic stresses due to the presence of air above the film.
Inserting the solutions 共17兲 and 共18兲 into 共23兲, we find that B
is given by
B=
冉
冊
␬ ␬/␥ + iE
,
␥ 1 − 共1 − i兲E
共25兲
where E is a nondimensional elasticity parameter defined by
E=
Ek2␥
.
␳␻2
共26兲
The coefficient B is small, ranging from O共k2 / ␥2兲 for a clean
surface to O共k / ␥兲 for a film covered surface. The real and
imaginary parts of B will in the following be denoted Br and
Bi, respectively. Because both k and ␥ are fixed for a given
wave frequency ␻, E is a linear measure of film elasticity.
Using 共18兲, 共19兲, and 共24兲 we obtain the dispersion relation
and the damping coefficient14
␻2 = ␻20共1 + Bi兲,
冉
␻20 = gk + 共T/␳兲k3 ,
冊
␣ 1 C p k E2 k 2共1 − E兲
=
+
.
F
k 2 Cg ␥ F ␥
共27兲
⍀ = 共E/␳d兲1/2k.
共29兲
Here d = ␥ is the boundary layer thickness. By 共29兲 we may
write E = ⍀2 / ␻2. The physical significance of E will be discussed in the following section.
−1
A. Resonant behavior of the two wave modes
Ermakov6 has argued that we can regard the irrotational
part of the wave field as a capillary-gravity wave and the
rotational part as a dilational wave which is excited by the
capillary-gravity wave. We denote the irrotational part by 共 ˜兲
and the rotational part by 共 ˆ兲 so that ⑀x共1兲 = x̃ + x̂ and ⑀z共1兲
= z̃ + ẑ. In the following subscript 0 denotes surface values,
and x0 = x̃0 + x̂0 is the total horizontal surface displacement.
Inserting for ⑀ we obtain to the leading order from 共17兲 and
共18兲
x̃ = ␨0ekcei␪ ,
共30兲
z̃ = − i␨0ekcei␪ ,
共31兲
x̂ = 共␨0/F兲关E共1 − 2E兲 − iE兴e共1−i兲␥cei␪ ,
共32兲
ẑ = − 共k/␥兲共␨0/F兲关E2 + iE共1 − E兲兴e共1−i兲␥cei␪ ,
共33兲
where ␪ = ␬a − ␻t. We see that all the information on how the
film influences the wave motion is incorporated in the parameter E. In the boundary layer there is a balance between
inertia and viscous forces, so x̂ satisfies
x̂tt = ␯x̂tcc .
共34兲
Furthermore, we note that ẑ is small, which means that the
shape of the surface is virtually unaltered for any value of E.
The horizontal surface displacement has a monotonically increasing phase lag for increasing E, and in the inextensible
limit we have x0共E → ⬁兲 / x0共E = 0兲 = exp关−i共3␲ / 4兲兴.
A curious feature of dilational waves is that the phase
velocity differs by a numerical factor for temporally and spatially damped waves, hence the group velocity is not well
defined 共Lucassen,5 Weber and Christensen17兲. Weber and
Christensen considered forced dilational waves, i.e., when a
fluctuating horizontal stress applied at the surface prevents
wave amplitude decay. For such forced dilational waves the
phase velocity differs from both temporally and spatially
damped waves, in fact, using 共29兲, the phase velocity can
then be written as
C p = ⍀/k.
共35兲
6
共28兲
Here F = 1 − 2E + 2E2, C p = ␻ / k is the phase velocity, and Cg
= ⳵␻ / ⳵k is the group velocity of the waves. It can be shown
that the spatial damping coefficient is related to the temporal
damping coefficient ␤ 共e.g., Weber and Saetra9兲 by ␤ = Cg␣,
in accordance with the results of Gaster.16 For not too low
values of E the damping coefficient is proportional to E2 / F,
hence ␣ has a maximum for E = 1 at a frequency ⍀ given by
As pointed out by Ermakov, the transverse waves provide a
forcing term in the horizontal boundary condition 共23兲,
which yields the dilational wave dispersion relation 共e.g.,
Dysthe and Rabin18兲. This term plays the same role as the
stress term included by Weber and Christensen, allowing for
no or weak damping of the dilational waves. Weber and
Christensen also showed that the energy flux in dilational
waves is negligible, there is rather a local balance between
the work done by the elastic film and viscous dissipation. As
a consequence the energy provided by the capillary-gravity
waves to sustain the dilational waves is lost during the wave
042102-4
Phys. Fluids 17, 042102 共2005兲
Kai Haakon Christensen
cycle. Because the effect of film elasticity on transverse
wave dispersion is insignificant, the dilational waves will
have to adopt k and ␻ from 共27兲. Accordingly, the damping
maximum occurs when the frequency of the transverse
waves coincides with the natural frequency of forced dilational waves, as given by 共29兲.
Now, define the mean boundary layer displacement
X̂ =
1
d
冕
0
共36兲
x̂dc.
in w should therefore come from the potential part of the
flow. Then wc ⬃ kw, and using 共40兲 we find w ⬃ 共␣ / k兲u Ⰶ u.
We therefore neglect w in 共39兲, and calculate ⌸ from the
reduced equation. In doing this we restrict ourselves to consider the drift some distance from the end walls where the
motion is mainly horizontal. To the leading order we obtain
␣
⌸a = − 2␯␥2 e2kc−2␣a + Q.
k
共41兲
−⬁
From 共34兲 and 共36兲 we obtain dX̂tt = ␯共x̂tc兲c=0. To the leading
order the horizontal boundary condition 共23兲 then reads
X̂tt + ⍀2x0 = 0.
IV. THE NONLINEAR WAVE DRIFT
We derive the equations for the mean drift velocities by
inserting the real parts of the first-order solutions 共17兲–共19兲
into 共5兲–共7兲, and collecting all terms of O共⑀2兲. Because we
consider spatially damped waves we average the secondorder equations over a wave period, denoting average values
by an overbar. Defining nondimensional drift velocities
共2兲
2
共u , w兲 = 共⑀2 / uS兲共x̄共2兲
t , z̄t 兲, where uS = ␨0␻k is the surface value
of the classic inviscid Stokes drift, we obtain
再冉
ut − ␯ⵜL2 u = − ⌸a − ␯k2 4 1 +
冊
␣ ␥2 2kc
e
2k k2
␥4 2
␥3
共B + B2i 兲e2␥c − 4 3 关共Br + Bi兲cos ␥c
k4 r
k
冎
− 共Br − Bi兲sin ␥c兴e␥c e−2␣a ,
再
wt − ␯ⵜL2 w = − ⌸c + 2␯␥2 e2kc −
␥2
k2
共38兲
冎
⫻共Br cos ␥c + Bi sin ␥c兲e␥c e−2␣a ,
ua + wc = 0.
再
共37兲
It follows that the mean boundary layer displacement is in
phase with the total horizontal displacement of the film, a
result which to the author’s knowledge has not been presented in the literature before. The film and the mean boundary layer displacement X̂ act together as a solid elastic membrane so that the boundary layer responds instantaneously to
changes in the position of the surface. This situation is unlike
that of oscillatory motion near rigid boundaries 共e.g., Schlichting and Gersten19兲, and arises because the film is advected
by the capillary-gravity waves. The phase velocity 共35兲 is
exactly that of longitudinal waves in an elastic membrane of
width d and elastic modulus E.
+6
In 共41兲 a new variable Q, independent of c, is added to simulate the effect of a sloping surface. Using 共25兲 and 共41兲, the
equation for the mean horizontal drift velocity becomes
共39兲
共40兲
Here ⵜL2 = ⳵2 / ⳵a2 + ⳵2 / ⳵c2, and ⌸ = 共␻ / ␳k3兲共p̄共2兲 + ␳gz̄共2兲兲 is the
共nondimensional兲 dynamic pressure to O共⑀2兲.
From 共38兲 and 共40兲 we obtain wc = 2␣u. At the surface
the mean vertical velocity must vanish, hence the boundary
layer terms in w should be negligible. The main contribution
ut − ␯ucc = − Q − ␯k2 4e2kc + 6
⫻
冉
E
F
cos ␥c +
␥ 2 E 2 2␥c
␥2
e −4 2
k2 F
k
E共1 − 2E兲
F
冊
冎
sin ␥c e␥c e−2␣a .
共42兲
In 共42兲 terms of O共k / ␥兲 compared to other terms are neglected, and the equation is therefore not valid for a film-free
surface.
The first term inside the curly brackets on the right-hand
side of 共42兲 yields the Stokes drift. The other two terms are
mainly second-order corrections to the frictional force, i.e.,
␳␯Uzz in Eulerian description. Keeping in mind the subdivision of the wave field into a transverse and a dilational wave
关Eqs. 共30兲–共33兲兴, we realize that the first of these terms represents nonlinear effects of the dilational wave, while the
second represents interaction between the transverse and the
dilational wave. A careful retracing of the originating terms
in 共5兲 is not necessary, it suffices to note the difference in the
factor of the exponents 共we have used e共k+␥兲c ⬇ e␥c兲. If we
neglect the contributions from the transverse wave on the
right-hand side of 共42兲 we are left with the e2␥c term, which
originates from a forced dilational wave. The frequency of
such a wave is given by 共29兲, thus E = 1. Letting Q = 0 and
␣ → 0, Eq. 共42兲 becomes
ut − ␯ucc = − 6␯␥2e2␥c ,
共43兲
which is precisely the equation for the drift induced by
forced dilational waves.17 Letting E → ⬁, as for inextensible
films, Eq. 共42兲 becomes equal to the equation for the mean
drift under an ice cover.20,21
So far we have not made any assumptions about the
conditions at the surface, and 共42兲 is valid for both stagnant
and freely drifting films. We will assume that the film is
fixed, i.e., prevented from drifting away by a film barrier as
in some laboratory situations.13,22
A. Transient solution
For the time development of the drift current we will
assume that the period of time in consideration is so short
that the drift current never reaches the bottom. Accordingly,
we must require
042102-5
Phys. Fluids 17, 042102 共2005兲
Transient and steady drift currents in waves
c → − ⬁.
u = 0,
共44兲
In a tank of finite length, a return current driven by a pressure force 共Q兲 will develop after some time. Ünlüata and
Mei23 provide an estimate for the time scale Tr required for
the development of the return current. For a tank of length L
we have Tr ⬃ L / uS, or equivalently, ␻Tr ⬃ 共L / ␨0兲 / 共␨0k兲. With
L = 100 cm and a wave steepness of ␨0k = 0.1, we find that
␻Tr ⬃ O共104兲. For the present we consider the onset of the
motion limited to a time scale ␻t = O共102兲, say, so we will
neglect Q in 共42兲. A time independent solution with Q nonzero is presented in Sec. IV B.
It proves convenient to separate u into three parts,
u = u共S兲 + u共␯兲 + u共h兲 .
共45兲
The first two terms on the right-hand side of 共45兲 are given
by the particular solution of 共42兲, with u共S兲 representing the
inviscid Stokes drift, and u共␯兲 a vorticity solution confined to
the surface boundary layer. The homogeneous solution u共h兲
represents a transient quasi-Eulerian mean current resulting
from diffusion of vorticity from the surface 共e.g., Craik24兲.
With Q = 0, we find from 共42兲
u共S兲 = e2kc−2␣a ,
共46兲
3 E 2␥c−2␣a 2E
e
+ 共关1 − 2E兴cos ␥c − sin ␥c兲e␥c−2␣a .
2F
F
FIG. 1. Transient solution u from 共50兲, ␻t = 100, k = 2␲ cm−1, and ␻R
= 10␲.
+
2
u 共␯兲 =
共47兲
The complete particular solution of 共42兲 is then u共p兲 = u共S兲
+ u 共␯兲.
The wave motion does not start from rest, and it is therefore not obvious what initial condition we should apply for u.
One may assume that the first-order motion and the associated inviscid Stokes drift are established at any point after a
few waves have passed by. Initially the surfactant is evenly
distributed and there are no stresses at the surface opposing
the drift. Thus the film is drifting freely and we take u equal
to the inviscid Stokes drift 共e.g., Weber and Saetra9兲:
u = u共S兲,
共48兲
t = 0.
The wave-induced mean stress on the film will compress it
toward the film barrier, resulting in a higher concentration of
film material away from the wave maker. In this way a surface tension gradient develops. After some time the surface
tension gradient will be large enough to balance the viscous
stress on the film, which becomes stagnant. The time required to reach this equilibrium depends on the physical
properties of the particular surfactant in question. We will
model this process by letting the surface drift velocity decay
in time so that
u = e−t/Re−2␣a,
c = 0,
再
冑
u = u共p兲 − e−2␣a u共p兲
0 erfc共− c/2 ␯t兲
−
ce−t/R
2冑␯␲
2E
␲F
冕
t
0
冕冋
⬁
0
exp关− 共c2/4␯␰兲 + 共␰/R兲兴
d␰
␰3/2
3E
4共␰ + 2␻兲
冎
+
⫻e−␰t sin共c冑␰/␯兲d␰ ,
共1 − 2E兲␰ + ␻
共 ␰ 2 + ␻ 2兲
册
共50兲
where u共p兲
0 is the surface value of the particular solution.
In Figs. 1 and 2 vertical profiles of u from 共50兲 are
shown using k = 2␲ and 1 cm−1, ␻t = 100, and ␻R = 10␲. We
have set a = 0 in these and all subsequent figures. Figures 3–5
contain contour plots of u for k = 2␲ cm−1, showing the time
development for ␻t = 1 – 100, using the same value of R as in
Figs. 1 and 2. The relaxation time R is chosen such that the
surface drift is nonzero throughout the time interval.
The most striking feature is that the drift beneath the
boundary layer is in the direction opposite to that of wave
propagation for E = 0.5. In order to explain this result we look
共49兲
and where the relaxation time R is specific to the film. Equation 共49兲 implies that the film drifts a distance proportional to
uSR before a steady state is reached.
The solution for u共h兲 can be found by applying Laplace
transforms. From 共42兲, 共44兲, 共48兲, and 共49兲, we obtain for the
total drift velocity
FIG. 2. Transient solution u from 共50兲, ␻t = 100, k = 1 cm−1, and ␻R = 10␲.
042102-6
Phys. Fluids 17, 042102 共2005兲
Kai Haakon Christensen
FIG. 3. Contour plot of u from 共50兲, showing the initial development for
␻t = 1 – 100. E = 0.5, k = 2␲ cm−1, and ␻R = 10␲.
FIG. 5. Same as Fig. 3, but with E → ⬁.
at the solution and boundary condition after the film has
become stagnant. At large times t Ⰷ R, the boundary condition 共49兲 becomes
face layer, the film is still compressed against the far end of
the tank. The Lagrangian shear stress acting on the film is
−␳␯uc共c = 0兲, which is a positive quantity for all E and t.
The particular solution u共p兲 represents the direct effect of
the waves, and because u共p兲 is completely determined by the
O共⑀兲 solutions, it has a strong dependence on E. The major
difference in Eq. 共42兲 between elastic and inextensible films
lies in the term representing interaction between the transverse and the dilational waves 共i.e., the term proportional to
e␥c兲. The phase lag between the horizontal and vertical surface displacements, representing the dilational and transverse
waves, is determined by E, as described in Sec. III A. Nonlinearly, the value of E then determines the transfer of momentum from periodic motion to a mean flow by altering the
interaction between the two wave modes.
The surface value of u共p兲 is of special significance, as it
determines the magnitude and direction of the transient solution u共h兲. From 共46兲 and 共47兲 we find 共setting a = 0兲
u = 0,
共51兲
c = 0,
and for the homogeneous solution u
共h兲
we find from 共50兲,
冑
u共h兲 ⬇ − u共p兲
0 erfc共− c/2 ␯t兲.
共52兲
Equation 共52兲 describes the evolution of a drift current which
may penetrate deep below the surface layer. This downward
flux of momentum stems from the need to balance the contribution from the particular solution in 共51兲, i.e., u共h兲
= −u共p兲 at the surface. Physically, the force that keeps the film
from drifting 共from the film barrier兲 is transferred to the bulk
fluid and momentum diffuses into deeper layers by friction.
In contrast, the particular solution is insignificant below the
wave penetration depth at all times. The transient part of the
drift thus develops in response to the no-slip condition 共51兲,
and the drift below the surface layer is against the wave
共p兲
propagation direction only when u共p兲
0 is positive. When u0 is
negative the transient response is a drift in the wave propagation direction. Even with a backward drift below the sur-
u共p兲
0 =
2 − E2
2F
.
共53兲
冑
It follows that u共p兲
0 changes sign for E = 2 共F ⬎ 0 for all E兲.
The right-hand side of 共53兲 has a sharp peak with a maximum value of 1.78 for E = 0.44, and a limiting value of −0.25
for E → ⬁. If the elastic modulus of the film is E
= 20 mN m−1, which is a typical value for many films, we
find that E increases from 0.5 to 冑2 as the wavelength is
shortened from 8.2 to 3.5 cm.
B. Steady solution
In practice the wave tank is closed in both ends, and
there is a rigid bottom at c = −h. The boundary condition at
the bottom is
u = 0,
FIG. 4. Same as Fig. 3, but with E = 1.
c = − h.
共54兲
The value of Q in 共42兲 is determined by demanding that the
net volume flux is zero, i.e.,
042102-7
Phys. Fluids 17, 042102 共2005兲
Transient and steady drift currents in waves
FIG. 6. Steady-state solution u from 共56兲, k = 2␲ cm−1.
冕
0
共55兲
udc = 0.
−h
In a viscous fluid a steady state will be reached after some
time. Neglecting the time dependence in 共42兲, applying 共51兲,
共54兲, and 共55兲, we obtain
冋冉 冊 冉 冊册
冋 冉 冊 冉 冊 册 冋 冉 冊 冉 冊册
c
h
u = u共p兲 + 6
− 3
c
h
2
2
+4
+
c
h
U共p兲
h
c
c
+ 1 u共p兲
0 − 3
h
h
2
+2
c
h
u共p兲
b .
共56兲
0 共p兲
共p兲
共p兲
Here u共p兲
b = u 共c = −h兲 and U = 兰−hu dc. The value of Q is
given by
Q=
冉
冊
6 U共p兲
共p兲
− u共p兲
2
.
0 − ub
h2
h
共57兲
The solution 共56兲 consists of the particular solution u共p兲, and
a combination of the quasi-Eulerian drift in the upper part of
the water column and the return current below.
In Figs. 6 and 7 vertical profiles of u from 共56兲 are
shown for k = 2␲ and 1 cm−1, and h = 10 cm. As discussed in
Sec. IV A, the drift below the boundary layer is determined
by the value of u共p兲
0 . In Figs. 6 and 7 we see that the total
transport for E = 0.5 is by far larger than for both E = 1 and
E → ⬁. But now we see that a backward drift in the upper
layer is obtained for E = 1 as well, as is the case for all values
of E ⬍ 冑2, in accordance with the results in Sec. IV A. The
profiles intersect at almost the same point, this is because the
vanishes for
factor in front of the dominating term u共p兲
0
c = −h / 3. The maximum value of u for E = 0.5 is higher for
k = 2␲ cm−1 than for k = 1 cm−1. This is explained by noting
that u共p兲 depends on k and ␥, whereas the other terms in 共56兲
depend on the channel depth h, and that the vertical axis in
Figs. 6 and 7 is dimensional. Below the wave penetration
depth the profile is basically given by the third term on the
right-hand side of 共56兲, which is proportional to u共p兲
0 , and the
absolute value of this term increases monotonically from c
= −h / 3 to 0. Closer to the surface the particular solution u共p兲
becomes more important. As we approach the surface where
兩c / h兩 Ⰶ 1, we have u ⬇ u共p兲 − u共p兲
0 → 0.
The solution for an inextensible film is quite different
from the solution for elastic films. In comparison we have
共p兲
u共p兲
0 共E → ⬁兲 / u0 共E = 0.5兲 = −1 / 7. The magnitude of the drift in
the interior is therefore much smaller in the limit E → ⬁ than
for E = 0.5, but now the boundary layer terms in the particular
solution u共p兲 give rise to a jetlike current in the direction of
wave propagation just below the surface. This is similar to
the findings of Weber20 and Melsom,21 who considered spatially damped gravity waves in the marginal ice zone.
C. Effect of the waves on the film cover
In addition to the local compression and dilation of the
film, the waves may influence the film in other ways. Two
aspects of importance for this study, spatially varying surface
tension and self-organized motion in the film, are discussed
below.
1. Spatially varying surface tension
The waves will compress the film toward one end of the
tank due to the wave-induced stress on the film. Because the
waves are rapidly damped, this stress will vary along the
tank and cause the surfactant concentration, and hence surface tension, to vary accordingly.22 We have assumed that the
elastic modulus E is approximately constant, but this may
not be the case if the total change in surface tension along the
tank is too large. As we already have expressions for both the
oscillatory and secondary mean motion, we can calculate the
change in surface tension from the appropriate form of the
horizontal boundary condition 共1兲. To O共⑀2兲 we obtain
冕
L
␳␯uSuc共c = 0兲da = ⌬␴ˆ − uS⌬w.
共58兲
0
FIG. 7. Steady-state solution u from 共56兲, k = 1 cm−1.
Here ␴ˆ = ⑀2¯␴共2兲, so that ⌬␴ˆ = ␴ˆ 共a = 0兲 − ␴ˆ 共a = L兲 denotes the
total drop in surface tension along the tank. The waveinduced stress 共acting on the fluid兲, is given by the integrand
on the left-hand side of 共58兲, and becomes to the leading
order,25
042102-8
Phys. Fluids 17, 042102 共2005兲
Kai Haakon Christensen
␳␯uSuc共c = 0兲 = − ␳␯uS共E2/F兲␥e−2␣a .
共59兲
The above result is experimentally verified for inextensible
films by Kang and Lee,26 and for elastic films by Gushchin
and Ermakov.22
Let us suppose the tank is so long that the waves are
virtually extinct before they reach the far end of the tank.
With E = 30 mN m−1 and wavelengths from 5 to 7 cm, it
would suffice with a tank length of less than 1 m in order to
reduce the amplitude by 95%. Using 共59兲, and neglecting the
vertical velocities, the total drop in surface tension becomes
⌬␴ˆ =
␳ u SC g
.
2k
共60兲
From plots of E versus ␴ we can find acceptable limits for
the total drop in surface tension so that the value of E will be
approximately constant. As an example we consider oleyl
alcohol which is commonly used in laboratory studies. For a
mean surface tension ␴0 = 69 mN m−1 共or alternatively a film
pressure ␴water − ␴0 = 4 mN m−1兲, the value of E is
30 mN m−1. Maximum damping 共E = 1兲 occurs for a wavelength of 6.55 cm. If the initial amplitude of the waves is
␨0 = 1 mm, which is quite small, the drop in surface tension
becomes ⌬␴ = 2.93 mN m−1. Judging from Fig. 10 of Mass
and Milgram,13 the change in film elasticity would be ⌬E
= Emax − Emin ⬇ 9 mN m−1. It follows that the elastic modulus
would vary as much as 30% along the tank, which is definitely unacceptable. By reducing the amplitude to ␨0
= 0.5 mm, we find that ⌬E ⬍ 2 mN m−1. It must be
emphasized that the above example applies to oleyl alcohol
only, whose value of E has a strong dependence on film
pressure/surface tension. For most surfactants, the value of
the elastic modulus becomes gradually more insensitive to
changes in surface tension for increasingly dense films, i.e.,
higher concentration of film material.
and an unsteady Reynolds ridge in generating such motion is
unclear. No recirculation of film material is reported by Mass
and Milgram.
In the portion of the film near the ridge the surfactant
concentration will be low, and resistance against surface
shear is probably small. Surface-active substances that are
solid in bulk phase show a higher resistance to shear than
substances that are liquid,30 and the latter is used in all but
one of the above studies. Vogel and Hirsa examined monolayers prepared from both liquid and solid bulk. Furthermore, the fluid velocities were quite high in Mockros and
Krone’s, Scott’s, and Vogel and Hirsa’s experiments, of the
order of 10 cm s−1. In Gushchin and Ermakov’s experiments
the amplitudes of the incoming waves at the ridge predict a
Stokes drift between 1 and 2 cm s−1 共␨0 = 2.5– 3.5 mm, and a
wavelength of 10 cm兲. It appears from their results that the
film was stagnant in the parts of the film where the amplitudes were reduced to about 1 mm. Gushchin and Ermakov,
studying a nonlinear effect, did not use small amplitude
waves, and the recirculation patterns appeared in all the experimental runs. It is therefore not possible to say if the
recirculation patterns would have disappeared for some
lower wave amplitude, ␨0 ⬍ 1 mm, say 共Ermakov, personal
communication, 2004兲.
A fundamental assumption in the present study is that
the film covers the entire surface, hence there is no ridge. It
appears that with a sufficiently dense film layer and low
wave amplitudes, self-organized motion in the film can be
avoided. Self-organized motion may also be prevented by
using surface-active substances that are solid in bulk phase,
and therefore have a higher surface shear viscosity.
V. SUMMARY AND CONCLUDING REMARKS
2. Self-organized motion
When an obstacle is placed on the surface of steady
flowing water, surface-active substances will be trapped on
the upstream side and form a film. The upstream front of the
film is generally known as the Reynolds ridge.27 In laboratory experiments it is often observed that film material tend
to recirculate in regularly spaced channels 共usually two channels兲 oriented along the bulk flow 共e.g., Mockros and
Krone,28 Scott,29 and Vogel and Hirsa30兲. The recirculatory
surface flow is usually an order of magnitude less than the
bulk flow.30 Recently, Gushchin and Ermakov22 investigated
the effect of waves in compressing a surface film toward a
film barrier in a set of experiments much similar to the ones
mentioned above. Gushchin and Ermakov observed recirculation patterns within the film in the part nearest to the Reynolds ridge, where the wave amplitudes were highest and the
film least dense. Such recirculation of surface-active material
must be avoided for the boundary condition 共51兲 to hold.
Common for all the above mentioned studies is that only a
part of the surface is covered by the film, in contrast to, e.g.,
Mass and Milgram’s experiments. The role of the sidewalls
The main aim of this paper has been to investigate the
effect of surface films on the mean drift in spatially damped
capillary-gravity waves. It has been shown that the existing
linear theory agrees well with experiment for a variety of
surfactants, but there is a lack of experimental data on the
nonlinear wave-induced drift. The presented theory has been
developed with a typical laboratory situation in mind, in contrast to previous studies. The horizontal drift velocity is
shown to be strongly dependent on the value of the elasticity
parameter E. An intriguing result is that the horizontal drift in
the upper layer of the water column is directed opposite to
the direction of wave propagation for values of E ⬍ 冑2. For
inextensible films, the drift is similar to the wave-induced
drift under an ice cover, with a jetlike current in the wave
propagation direction just below the surface. For all values
of E the viscous stresses on the film act in the wave propagation direction, compressing the film against the film barrier. It is concluded that the wave amplitudes must be kept
small, and the film layer dense, to ensure that the elastic
modulus of the film remains approximately constant. Using
small amplitude waves and a dense film is also desirable in
order to prevent self-organized motion in the film.
042102-9
ACKNOWLEDGMENTS
This study was supported by The Research Council of
Norway through Grant No. 151774/432. The author wish to
thank Professor Jan Erik Weber for continuous support and
guidance throughout the study. The author also wish to thank
Professor Kristian Dysthe for pointing him to the experiments on the Reynolds ridge, and Dr. Stanislav Ermakov for
providing detailed information on the experiments in
Ref. 22.
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K. Dysthe and Y. Rabin, “Damping of short waves by insoluble surface
film,” 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.
19
H. Schlichting and K. Gersten, Boundary-Layer Theory, 8th ed. 共Springer,
New York, 2000兲.
20
J. E. Weber, “Wave attenuation and wave drift in the marginal ice zone,”
J. Phys. Oceanogr. 17, 2351 共1987兲.
21
A. Melsom, “Wave-induced roll motion beneath an ice cover,” J. Phys.
Oceanogr. 22, 19 共1992兲.
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L. A. Gushchin and S. A. Ermakov, “Laboratory study of surfactant redistribution in the flow field induced by surface waves,” Izvestiya, Atmos.
Oceanic Phys. 40, 244 共2003兲.
23
Ü. Ünlüata and C. C. Mei, “Mass transport in water waves,” J. Geophys.
Res. 75, 7611 共1970兲.
24
A. D. D. Craik, “The drift velocity of water waves,” J. Fluid Mech. 116,
187 共1982兲.
25
M. Foss, “Wave damping and momentum transfer,” Doctoral thesis, University of Tromsø, Tromsø, Norway, 2000.
26
H. K. Kang and C. M. Lee, “Steady streaming of viscous surface layer in
waves,” J. Marine Sci. Technol. 1, 3 共1995兲.
27
J. F. Harper and J. N. Dixon, “The leading edge of a surface film on
contaminated flowing water,” Fifth Australasian Conference on Hydraulics
and Fluid Mechanics, 1974.
28
L. F. Mockros and R. B. Krone, “Hydrodynamic effects on an interfacial
film,” Science 161, 361 共1968兲.
29
J. C. Scott, “Flow beneath a stagnant film on water: The Reynolds ridge,”
J. Fluid Mech. 116, 283 共1982兲.
30
M. J. Vogel and A. H. Hirsa, “Concentration measurements downstream of
an insoluble monolayer front,” J. Fluid Mech. 472, 283 共2002兲.
Drift of an inextensible sheet caused by surface waves
K. H. Christensen (k.h.christensen@geo.uio.no) and J. E. Weber
Department of Geosciences, University of Oslo, Oslo, Norway
Abstract. Inextensible films are often used to simulate surface-active material
as commonly found at sea. It is important to understand the mechanism behind
wave-induced transport of surfactants with regard to e.g. oil spills in coastal areas.
In this paper we compare theory, based on a Lagrangian description of motion,
with observations of the wave-induced drift of thin inextensible plastic sheets in a
controlled laboratory experiment. It is found that the analytical solution is able to
reproduce the observed drift. In a laboratory situation with continuously generated,
spatially damped waves, the drift velocity increases in time. Hence earlier theoretical
treatments in which a steady state is assumed predict too low values of the drift
velocity. The need for data on the time development of the drift is pointed out.
Keywords: Wave drift, surface film, surface waves, oil spill, laboratory experiments.
1. Introduction
Objects floating on the sea surface in general possess a mean drift
velocity caused by the joint action of wind, currents and waves. The
role of waves is perhaps the least investigated, and if the object in
question is a slick of surfactant, the problem becomes even more complicated because the wavefield is modified by the slick in the whole area
it covers. The slick acts to restrict motion tangential to the surface
and strong shear is produced in the viscous surface boundary layer,
which ultimately leads to more rapid damping of the waves [1–3]. The
momentum lost as oscillatory motion is regained as an Eulerian mean
current by the diffusion of vorticity from the boundary layer [4]. This
Eulerian mean current comes in addition to the inviscid Stokes drift
[5], and the slick, being advected by the mean currents, may therefore
have a drift velocity significantly higher than that predicted by Stokes
theory.
A Lagrangian description of motion is particularly well suited for
studying the wave-induced drift. It allows for a simple description of the
free surface, and directly yields the particle drift. Although theoretical
results for the wave-induced drift of inextensible and elastic surface
films exist [6, 7], a validation of these results is difficult due to a lack of
experimental data. Field studies cannot be used for validation purposes,
as both the effects of wind and an irregular wavefield are not accounted
for in the analyses. Furthermore, one of the basic assumptions in the
above-mentioned theoretical studies is that the film cover is continuous,
c 2005 Kluwer Academic Publishers. Printed in the Netherlands.
EFM.tex; 19/05/2005; 11:22; p.1
2
Christensen and Weber
whereas the few conducted laboratory studies has focused on the drift
of films of limited size (with width and length just a fraction of the
wave length, say) which cannot modify the waves to any extent [8, 9].
In other laboratory studies the absence of important information, for
example the duration time of the experiment, makes a comparison between experiment and theory impossible [10–12]. In a typical laboratory
experiment, waves are generated in one end of a wave tank, and the
surface film is let free to drift. The waves will attenuate in space rather
than time, and since momentum is continuously put into the system,
the source of Eulerian mean momentum is unlimited. Accordingly, the
Eulerian mean current will increase in time [13].
A notable exception among the existing laboratory studies is the
study by Law [14] of the wave-induced drift of thin rectangular polypropylene sheets. Three requirements for comparing Law’s data with existing
theory are fulfilled: (i) the sheets were large enough to modify the
waves, justifying the use of the mathematical model, (ii) all the necessary wave parameters were reported, (iii) the approximate duration
time of the experiments can be deduced. Law also presented a theoretical analysis, based on Phillips [15]. However, assuming that the
vorticity is confined to the surface boundary layer, Law’s analysis does
not capture the time development of the drift current. The observed
drift velocities are in general found to be higher than predicted by
Law’s theory.
In this paper we compare the experimental results of Law with
the theoretical results of Weber [16]. The outline of the paper is as
follows: In section 2 we present an analytical solution for inextensible
films of finite length. In section 3 we compare the solution with drift
data from Law, while section 4 contains some concluding remarks and
recommendations for future laboratory studies.
2. Analytical solution in Lagrangian description
We simplify and take the problem to be two-dimensional. The waves
are monochromatic and continuously generated. An inextensible, freely
drifting film of length L covers a part of the water surface, which is
otherwise free from any surfactants. The waves enter the film with an
amplitude ζ0 and are damped as they propagate through the film. The
surface is assumed to be free from any external stresses, thus we neglect
the presence of air above the film. The validity of this assumption will
be discussed later on.
We use a Lagrangian description of motion. The spatial coordinates
(a, c) denote the fluid particles initial horizontal and vertical position,
EFM.tex; 19/05/2005; 11:22; p.2
Wave-drift of an inextensible sheet
3
respectively, in a Cartesian coordinate system. The vertical axis is
positive upwards and the free surface is given by c = 0 at any time,
t. The dependent variables are expanded in perturbation series and
inserted in the equations of motion and continuity [17]. The parameter
of expansion is taken proportional to the wave slope ζ 0 k, where k is the
wave number. We will need to pursue our calculations to O((ζ 0 k)2 ) to
obtain the wave-induced drift.
A detailed derivation of the equations for the drift velocities in
spatially damped waves can be found in Weber [16]. Although Weber
considered water covered by grease ice, the formulation is applicable for
water covered by an inextensible film (see [6] for the case of temporally
damped waves and inextensible film, and e.g. [18] for both spatially
and temporally damped waves in a film-free rotating ocean). For deepwater gravity waves with wave frequency ω, the equation for the mean
horizontal drift velocity u, in a non-rotating system, is
ut − νucc = −uS νk2 {4e2kc
γ2
γ2
+3 2 e2γc + 4 2 eγc sin γc}e−2αa .
k
k
(1)
2
Here ν is the kinematic viscosity,
u S = ζ0 ωk is the surface value of
the inviscid Stokes drift, γ = ω/(2ν) is the inverse boundary layer
thickness, and α = k 2 /(2γ) is the spatial attenuation coefficient for
inextensible films. Furthermore, we have let subscripts denote partial
differentiation. The derivation of (1) is based on the assumption that
the boundary layer width is much smaller than the wave length, that
is k/γ 1, a requirement that is very well fulfilled for gravity waves.
For wave lengths of 1.5 meters the corresponding boundary layer width
is approximately 0.6 millimeters, which means that k/γ = O(10 −3 ).
It proves convenient to separate u into three parts,
u = u(S) + u(ν) + u(E) .
(2)
The first two parts on the righthand side of (2) is given by the particular
solution of (1), with u(S) representing the inviscid Stokes drift and
u(ν) a vorticity solution confined to the surface boundary layer. The
homogeneous solution of (1) represents the transient Eulerian mean
current, u(E) , resulting from diffusion of vorticity from the surface. We
find from (1):
u(S) = uS e2kc−2αa ,
3 2γc
e − 2eγc cos γc e−2αa .
u(ν) = uS
4
(3)
(4)
With negligible mass, the film cannot support any net stress acting
on it. In the absence of any external stresses, the appropriate boundary
EFM.tex; 19/05/2005; 11:22; p.3
4
Christensen and Weber
condition at the surface is therefore that the mean Lagrangian shear
stress is zero [4, 7]:
ρνuc = 0 , c = 0.
(5)
Here ρ is the fluid density.
For continuously generated waves and infinite water depth the source
of second-order mean momentum is unlimited. The waves constantly
produce vorticity in the boundary layer, and the resulting mean stress
this vorticity exerts on the surface must be balanced in order for (5)
to be fulfilled. The balancing stress is due to the Eulerian mean current and is formally known as the virtual wave stress, τ w [19]. In our
Lagrangian framework the virtual wave stress is defined as
(ν)
τw = ρνu(E)
c (c = 0) = −ρνuc (c = 0),
(6)
where the last equality follows from (2) and (5). Strictly speaking, the
contribution from the Stokes drift should be included in (6), but this
contribution is of O(k/γ) compared to the contribution from u (ν) and
therefore neglected. From (4) and (6) we obtain
1
τw = ρνγuS e−2αa .
2
(7)
Now consider a situation where the sheet is not allowed to drift freely,
but is kept stagnant by a resisting force. The force F required to keep
a sheet of surface area A from drifting can be written [16]
F =−
A
τw dA.
(8)
Experiments by Kang and Lee [11] show that there is excellent agreement between equation (8) and the measured wave-induced drag on an
inextensible sheet kept stagnant.
We may now formulate the problem for u(E) . From (1)-(6), assuming
that the depth is such that the drift velocity never reaches the bottom,
we find that the time development of the mean Eulerian drift current
is determined by [13]
(E)
ut
u(E)
c
(E)
u
= νu(E)
cc ,
τw
=
,
ρν
→ 0
,
(9)
c = 0,
(10)
c → −∞.
(11)
It follows from (7) and (10) that τw provides a time independent source
of Eulerian mean momentum.
The derivation of (1) is based on the assumption that the wavefield is immediately established. The Stokes drift is an inherent part
EFM.tex; 19/05/2005; 11:22; p.4
5
Wave-drift of an inextensible sheet
of irrotational wave motion, hence a straightforward choice of initial
condition for u is
u = u(S) , t = 0.
(12)
Using (2) and (12), the corresponding initial condition for u (E) becomes
u(E) = −u(ν)
,
t = 0.
(13)
The equations (9)-(11) and (13) now completely determine the Eulerian
mean current.
We denote average values for a film of length L by using capital
letters, i.e.
1 L
u da.
(14)
U≡
L 0
The solution to (9)-(11), applying (13), can be found by the method of
Laplace transforms. For a film of finite length we obtain
U
(E)
= AuS
+
−
1
2
(2ω)1/2
π
ω
2π
1/2 t
exp{−c2 /(4νξ)}
∞
0
√
0
ξ+ω
ξ 1/2 (ξ 2 + ω 2 )
ξ
dξ
3
e−ξt cos(cξ 1/2 /ν 1/2 )dξ ,
4ξ 1/2 (ξ + 2ω)
(15)
where A = (1 − exp{−2αL})/(2αL). The coefficient A expresses the
dependence on the relative amplitude damping over the film. For short
films or weak damping we have A close to unity, and for very long films
or heavily damped waves the value of A approaches zero. The solution
(15) describes an accelerating Couette type flow, driven by the virtual
wave stress acting on the surface. At each depth the velocity increases
in time by the downward diffusion of momentum. The Stokes drift and
boundary layer solution are then added to (15) to obtain the total
velocity profile. The drift velocity of the film equals the surface value
of U , from (3), (4) and (15) we obtain
U0 = U (c = 0) = AuS
t/T − 1/4
ξ+ω
(2ω)1/2 ∞
π
ξ 1/2 (ξ 2 + ω 2 )
0
3
− 1/2
e−ξt dξ ,
4ξ (ξ + 2ω)
+
(16)
where T = 2π/ω is the wave period.
EFM.tex; 19/05/2005; 11:22; p.5
6
Christensen and Weber
The derivation of (16) is based on the assumption that the film is
so long that it can actually modify the waves. Short films with L λ,
where λ is the wave length, will not be able to dampen the waves to
any extent, in which case equation (1) will not hold. Such films will
rather be advected by the drift current as induced by waves on clean
surfaces. Even with a clean surface the drift velocity can exceed the
classic inviscid Stokes drift due to the damping effect of air [13]. In our
case the effect of the film will dominate. An inextensible film is very
effective in suppressing surface waves, for waves with a period of one
second the loss in amplitude is 40% in 100 meters. Without any film
(but including the damping effect of air), the amplitude reduction is
1.3%. The virtual wave stress τw from (7) is of O(γ/k) larger than the
virtual wave stress in the case of air over a clean surface, justifying our
assumption that the effect of air can be neglected.
3. Comparison with laboratory experiments
3.1. Description of the experiments
Law [14] used a wave tank 45 meters long, 1.6 meters wide and with a
water depth of 3 meters. Rectangular pieces of 0.08 millimeter thick
polypropylene, 1.2 meters wide and of varying length, were placed
transversally centered on the surface initially at rest. Upon generating
waves with a period T = 1 second, the subsequent drift of the sheets
was recorded by a video camera. The corresponding wave length would
be that of deep water waves, with λ = 1.56 meters, and a phase velocity
λ/T = 1.56 m/s.
3.1.1. Experiment duration time
Data collection were restricted to the time interval determined by two
criteria: (i) after at least five waves had passed, and, (ii) before the reflected waves arrived back at the measurement area. The experiments in
general lasted 20-60 seconds from the onset of the wave maker, but these
figures are approximate since there were variations between each experimental run (Law, private communication, 2004). Because it would take
a few wave cycles before the wave maker produced uniform waves, and
since the waves would take some time to reach the measurement area,
the corresponding time the sheets were subjected to the waves should
be approximately 10-50 seconds. Given the wave period of one second,
the first criterion for data collection imply that the shortest possible
time for which a sheet were allowed to drift would be t = 5 seconds. A
longest possible drift time t = 50 seconds is consistent with the time
EFM.tex; 19/05/2005; 11:22; p.6
Wave-drift of an inextensible sheet
7
required for a wave to propagate from the measurement area, reflect at
the far end of the wave tank, and propagate back to the measurement
area.
3.1.2. Length of sheet
The ratio between sheet length and wave length were varied from L/λ =
0.2 to 3 in Law’s experiments, though only results for the sheets with
L/λ ≥ 0.8 are presented. The value of the coefficient A is between 1
to 0.98 for L/λ between 0 and 3, with λ = 1.56 meters. Hence for
the sheets used by Law we can safely set A = 1. The drift velocity
should therefore be practically the same for any two films with L/λ
between some value (L/λ)min and 3, as long as the dynamics can be
considered as well described by equation (1). The pertinent question is
then for which minimum value of (L/λ)min we can expect the solution
(16) to become valid? Law states that the dependence of sheet length
becomes less significant for L/λ > 0.8. Kang and Lee [12] conducted
a series of experiments, much similar to those of Law, but do not give
sufficient details on the experiments for us to use them in this study.
However, they report that the drift velocity seems to become more or
less independent on the sheet length for L/λ > 1.5. For the comparison
we will use the data for the longer sheets with L/λ = 2, 2.5 and 3.
3.2. Results
Considering the experimental setup, computing the drift velocity from
the observed displacement of the sheet requires some form of time averaging, where the distance covered by the sheet is measured in a certain
time interval. This time interval may be chosen differently for each
sheet. If the time interval is short, the computed drift velocity would
resemble the instantaneous drift velocity. According to our solution
(16), the sheets accelerate at all times and no steady state occurs. In
consequence, if the displacement of the sheet is measured at the beginning of the experiment, a lower value for the drift velocity is expected
compared to the case where the displacement is measured towards the
end of the experiment. If the time interval is long, the computed drift
velocity should in any case lie between the lower and upper bounds
defined by the instantaneous drift velocity at the times given by the
shortest and longest experiment duration time, respectively. Another
aspect related to the required time averaging is discussed in section 3.3.
The observed drift velocities from Law’s experiments are shown in
Figure 1, with the solution from (16) for t = 5, 10 and 50 seconds,
using A = 1. Note that the drift velocities are in general several times
higher than predicted by Stokes theory. All but two data points are
EFM.tex; 19/05/2005; 11:22; p.7
8
Christensen and Weber
within the theoretical bounds defined by the solution (16) evaluated
at t = 5 seconds and 50 seconds. There is considerable spread in the
experimental data, which is to be expected since the duration times
varied over a relatively large interval. What is most important is that
the solution (16) predicts such high drift velocities as observed in the
experiments. It is clear that one has to take into account the transient
part of the drift current to obtain realistic values. The observed drift
velocities are often 1.5-2 times higher than predicted by the steady
state solutions of both Law and Phillips [15].
In another set of experiments, Wong and Law [9] investigated the
drift of sheets of elliptical shape. Our theoretical results are not applicable since these sheets were relatively small compared to the waves.
However, Wong and Law observed that the longer sheets (L/λ > 0.65)
accelerated, and in some experimental runs, the drift velocity was found
to be increasing even after the reflected waves theoretically interfered
with the incident waves. The energy of the reflected waves was small
due to an absorbent beach at the far end of the wave tank, but the point
is that the sheets did accelerate during the whole experimental run, and
did not attain a steady drift velocity. According to our solution (16),
the acceleration will decrease after some time. The relative increase in
drift speed between 40 and 50 wave periods after the onset of motion
is approximately ten percent. In a laboratory situation, such a small
velocity increase after an initial large acceleration may appear as a
steady state, keeping in mind that the mean drift velocity comes in
addition to the orbital velocity of the waves.
3.3. Apparent linearity in experimental data
Our solution (16) predicts that U will increase quadratically with the
wave steepness. However, in some experimental studies, including the
study by Law, the observed drift velocities of long sheets show an almost
linear growth with ζ0 k (e.g. Kang and Lee [12]). These observations do
not necessarily contradict our result (16), but may be explained as artifacts of the experimental procedure. Alofs and Reisbig [8] describe how
they computed the drift velocity based on the observed displacement of
the film, a method which one may assume is used by others: The film is
placed a fixed distance D1 from the measurement area, initially at rest,
and the wave maker is started. The film drifts into the measurement
area, of fixed length D2 , after a time t1 , and leaves the measurement
area after a time t2 (both t1 and t2 are measured from the onset of the
waves). An average drift velocity can then be computed as
Uav = D2 /(t2 − t1 ).
(17)
EFM.tex; 19/05/2005; 11:22; p.8
9
Wave-drift of an inextensible sheet
6
5
3
0
U (cm/s)
4
2
L/O = 2
L/O = 2.5
L/O = 3
t=5s
t = 10 s
t = 50 s
1
0
0
1
2
3
u (cm/s)
4
5
6
S
Figure 1. Observed drift velocities of inextensible plastic sheets from Law [14] and
theoretical predictions from (16). uS = ζ02 ωk is the surface value of the classic
inviscid Stokes drift.
Given values of D1 and D2 one can compute t1 and t2 , and hence
Uav , from the theoretical solution (16), and check whether the result is
quadratic in ζ0 k or not. The full expression in (16) is somewhat difficult
to manipulate, but after a short while (t ∼ T ) we have approximately
U = AuS
t/T .
(18)
Using (18) as the theoretical drift velocity, and letting A = 1 as before,
we obtain
Uav = (ζ0 k)4/3 B1 B2 ,
(19)
where
B1 =
2ω 3
9πk 2
1/3
,
B2 =
D2
2/3
(D1 + D2 )2/3 − D1
.
(20)
We note from (19) that Uav becomes proportional to (ζ0 k)4/3 , hence if
one uses the measuring technique of Alofs and Reisbig, one should
expect to find an almost linear growth of U with increasing wave
steepness. While B1 depends on the wave characteristics, B 2 will vary
depending on the setup of the experiment. It should be noted that
EFM.tex; 19/05/2005; 11:22; p.9
10
Christensen and Weber
6
5
U0 (cm/s)
4
3
2
L/O = 2
L/O = 2.5
L/O = 3
Eq. (19)
1
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
]0 k
Figure 2. Observed drift velocities of inextensible plastic sheets from Law [14] and
the theoretical prediction from (19). Here we have used D1 = 30 cm and D2 = 10
cm.
B1 varies slowly with the wave frequency, for gravity waves we have
B1 ∼ ω −1/3 . Thus if a narrow range of wave frequencies is used in a
particular set of experiments, the product B 1 B2 will be approximately
constant.
The measuring technique used by Law differs from that of Alofs and
Reisbig in the sense that the values of D1 and D2 both would vary
between the experimental runs (Law, private communication, 2004). It
is tempting, however, to compare (19) to Law’s data, choosing reasonable values for D1 and D2 . A comparison with other, similar studies
has not been possible due to lack of detail on the experimental setup.
Figure 2 shows the observed drift velocities from Law’s experiments
and the theoretical solution from (19) using D 1 = 30 centimeters and
D2 = 10 centimeters. These values for the drift distances are consistent
with the duration of the experiments and the observed drift velocities.
It seems that equation (19) describes well the trend for increasing wave
steepness.
EFM.tex; 19/05/2005; 11:22; p.10
Wave-drift of an inextensible sheet
11
4. Concluding remarks
In this paper we have compared the theoretical analysis of Weber [16]
with the observed wave-induced drift of inextensible sheets as reported
by Law [14]. The time-dependent solution we present predicts the high
drift velocities observed in the experiments. This is in contrast to the
steady-state solutions of Law and Phillips [15] which underestimate
the drift. Since most experimental data, like those reported by Law,
are given as time averages, it has not been possible to compare the
predicted time development of the drift with observations. Data on
the temporal variation of the wave-induced surface drift is desirable,
and we strongly recommend that this is taken into account in future
laboratory studies.
Acknowledgments
The authors wish to thank Dr. Adrian W. K. Law for providing more
detailed information on his experiments. K. H. Christensen gratefully
acknowledges financial support from The Research Council of Norway
through Grant No. 151774/432.
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Address for Offprints: Kai H. Christensen
Department of Geosciences
Section for Meteorology and Oceanography
P.O. Box 1022, Blindern
N-0315 Oslo
Norway
EFM.tex; 19/05/2005; 11:22; p.12
Wave-induced drift of large floating sheets
K. H. Christensen∗ and J. E. Weber
University of Oslo, Norway
In this paper we study the wave-induced drift of large, flexible
shallow floating objects, referred to as sheets. When surface waves
propagate through a sheet, they provide a mean stress on the
sheet, resulting in a mean drift. In response, the sheet generates
an Ekman current. The drift velocity of the sheet is determined
by (i) the wave-induced stress, (ii) the viscous stress due to the
Ekman current, and (iii) the Coriolis force. The sheet velocity
and the current beneath the sheet are determined for constant
and depth varying eddy viscosities.
Keywords: Wave drift; Swell; Sea ice; Oil spill; Ekman veering
∗
Corresponding author
1
1
Introduction
Most theoretical studies on the drift of large and shallow floating objects,
for example ice floes, are of empirical nature with parameterized momentum
fluxes between the atmosphere and the floating object, and between the
floating object and the ocean (e.g. [1, 2]). The role of surface waves in relation
to the drift of such objects is usually neglected. Waves do not affect the drift
of for example sea ice on a large scale, because they are very effectively
damped by an ice cover. In some situations, however, surface waves may be
important, e.g. when incoming swell from a distant storm enters the marginal
ice zone under otherwise calm conditions with negligible wind.
Large ice floes, dense pack ice, brash ice, heavy crude oil spills, and
similar large floating objects have a large surface area but shallow draft,
and may be considered as large inextensible sheets that offer little resistance
against the vertical motion of the waves. It is the main purpose of this paper
to study the isolated effect of surface waves on the drift of such objects,
hereafter referred to as sheets.
The paper is organized as follows: In section 2 we formulate the
problem mathematically, section 2.1 treats the motion in the water, while
section 2.2 treats the motion of the floating object. In section 3.1 we derive
2
analytical solutions for the steady drift in the water and of the sheet, using a
constant eddy viscosity to model the turbulence in the ocean. These solutions
are used to discuss the qualitative behaviour of the system. In section 3.2
we present transient and time independent numerical solutions for the case
of a vertically varying eddy viscosity. The main results are summarised in
section 4.
2
Mathematical formulation
We consider a viscous ocean of infinite depth and constant density ρ. On
the surface there is a floating sheet, which has a horizontal extent larger
than the length of the dominating waves. We assume that the sheet offers
little resistance to bending, but cannot easily be compressed or expanded in
the direction tangential to the surface. Thus surface waves may propagate
through the sheet, while the overall geometry remains unchanged.
We use a Cartesian coordinate system (x, y, z), with the x- and yaxes aligned along the undisturbed interface between the water and the sheet.
The z-axis is taken positive upwards. The system rotates about the z-axis
with an angular velocity f /2, where f is the Coriolis parameter. We use
3
a Lagrangian description of motion. The spatial coordinates (a, b, c) serve
as labels for the individual fluid particles, and the dependent variables are
functions of (a, b, c) and time, t. With our particle-following coordinates, the
interface between the water and the floating object is given by c = 0 at all
times.
The dependent variables are expanded in perturbation series and
inserted in the equations of motion and continuity [3]. The parameter of
expansion is taken proportional to the wave slope ζ0 k, where ζ0 is the wave
amplitude and k is the wave number. We will need to pursue our calculations
to O((ζ0 k)2 ) to obtain the wave induced drift. A detailed derivation of the
equations for the drift velocities under a stationary ice cover due to spatially
damped waves can be found in Weber [4].
The motion in the ocean’s interior is turbulent, and we will model the
turbulence by using an eddy viscosity assumption. As discussed elsewhere
[5, 6], one will need to use different eddy viscosity formulations for the periodic wave motion and the mean wave-induced drift. The wave solutions are
obtained using a constant eddy viscosity, see for example Weber [4], where it
is shown that the theoretical damping rates obtained using a constant eddy
viscosity formulation agree well with observations in ice-covered seas. We
4
denote this constant value of the eddy viscosity ν̃. In the drift equation we
allow the eddy viscosity ν to vary with the vertical coordinate c. It should be
emphasised that our aim is to make a qualitative investigation on how waves
contribute to the drift of large sheets in general, and not to study oceanic
turbulence. We therefore apply fairly simple models for the eddy viscosity
distribution with depth.
2.1
Drift velocities in the water
In the marginal ice zone, the ice acts as a low-pass filter for waves, and short
ripples and high frequency wind waves are rapidly damped out (e.g. [7]). For
the freely floating sheets studied here we consider relatively short gravity
waves with a frequency ω such that ω f. Then the direct effect of the
earth’s rotation on the waves is negligible [8]. Waves falling into this category
are long wind waves and swell with a wavelength λ typically of the order 102
m.
The periodic wave solutions are obtained using a no-slip condition
at the interface between the water and the sheet. For this reason the solutions should apply to all large floating objects that are sufficiently rigid
to withstand periodic compression and expansion tangential to the surface.
5
The dispersion equation is
ω 2 = gk,
(1)
where g is the acceleration due to gravity. The influence of the floating object
on wave dispersion is neglected, see for example Liu and Mollo-Christensen
[9], who show that the difference between (1) and the dispersion equation for
waves in compact sea ice can be significant for wavelengths shorter than 100
m and an ice thickness of more than 0.5 m.
We consider monochromatic and unidirectional waves of initial amplitude ζ0 . For unidirectional waves the vertical velocities are negligible
[10], and we will only solve for the horizontal velocity components. For
convenience we use a complex representation of the drift velocities so that
w = u + iv, where (u, v) are the drift velocities in the water in the x- and
y-directions, respectively. We will neglect the waves reflected from the edge
of the sheet and only consider the transmitted waves (e..g. [11]). For waves
propagating in the direction of positive x the drift equation is [4, 12]
−wt + (νwc )c − if w = uS ν̃k 2 e−2αa 4e2kc
+4
γ 2 γc
γ 2 2γc e
sin
γc
+
3
e
.
k2
k2
(2)
Here γ −1 = (2ν̃/ω)1/2 is the thickness of the surface boundary layer, α =
6
k 2 /(2γ) is the spatial damping coefficient of the waves, and uS = ζ02 ωk is the
surface value of the inviscid Stokes drift. We will not yet assign a specific
functional form to the eddy viscosity, but we assume that ν is a function of
the vertical coordinate c only. Furthermore, in (2) the horizontal gradients
have been neglected, that is, we have assumed that ∂/∂a, ∂/∂b ∂/∂c.
A general solution of (2) consists of a particular plus a homogeneous
solution, which we denote as w(p) and w(h) , respectively. The particular solution represents the direct effect of the waves, combining the inviscid Stokes
drift and the steady streaming in the surface boundary layer. The homogeneous solution represents a transient response to the wave-induced drag on
the sheet, and is determined by the initial and boundary conditions.
For an infinitely deep ocean we must require that
w→0 ,
c → ∞.
(3)
The boundary condition at the interface depends on the type of floating
object. The motion of sheets with a non-negligible mass is governed by a
separate equation, and we need to solve both (2) as well as the equation of
motion for the sheet. At the interface the velocities are continuous, hence
w = w(p) + w(h) = W
7
,
c = 0,
(4)
where W = U +iV is the sheet velocity. We do not consider any effects of local
winds, i.e. we neglect the mean wind stress and the wind-wave generation
within the sheet. For sheets of negligible mass, i.e. monomolecular films,
we must require that the stresses that act on both sides of the sheet are
balanced. The appropriate boundary condition at the surface then becomes
[13]
ρνwc = 0 ,
2.2
c = 0.
(5)
Drift velocity of the floating sheet
For simplicity we consider a rectangular sheet of dimensions L × B × H in
the x-, y-, and z-directions, respectively. The density is ρ̂ and the mass is
m = ρ̂LBH. We assume that L ∼ B H. An analysis similar to that given
here, but developed for much smaller plastic sheets used in laboratory experiments has been carried out by Christensen and Weber [14]. The theoretical
drift velocities agreed well with observations for sheets from approximately
twice as long as the wavelength and longer. A possible explanation for the
agreement between theory and experiment even for such short sheets is that
they generate an equal oscillatory boundary layer, and hence equal shear
stresses in this layer, as longer sheets do. If so, then we may expect the
8
present analysis to hold for floating objects with L, B > 2λ.
In line with the scope of this paper, we neglect the effects of local
winds, mean sloping surface, internal friction, and edge-induced drag as far
as the motion of the sheet is concerned. Accordingly, the sheet motion is
governed by
mWt = Fw + FE + FC .
(6)
The forces on the righthand side of (6) are: Fw , viscous drag due to the
waves; FE , viscous drag due to the Ekman current; FC , Coriolis force. These
forcing terms can be written
Fw = −ρν0 B
x(0)
FE = −ρν0 B
x(L)
x(L)
x(0)
FC = −imf W,
wc(p) (c = 0)dx,
(7a)
wc(h) (c = 0)dx,
(7b)
(7c)
where ν0 = ν(c = 0). Because the sheet is taken to be inextensible we have
dx = da at c = 0 and the integration limits become a = 0, L. Note that for
sheets of negligible mass FC = 0, and (6) reduces to equation (5) integrated
over the area of the floating object.
9
3
3.1
Results
Constant eddy viscosity
We consider first the steady wave-induced drift obtained for the case of a
constant eddy viscosity. Although this situation may be somewhat unrealistic, the simplicity of the problem allows us to find analytical solutions, which
we can use to examine the behaviour of the system.
Taking ν = ν̃, the particular solution of (2) becomes [4]
w(p) = uS e−2αa
1 + 2iq
3 2γc c/S
γc
e
e
−
2e
cos
γc
+
.
1 + 4q 2
4
(8)
Here q = S 2 /D2 , where D = (2ν̃/f )1/2 here defines the thickness of the
Ekman layer, and S = 1/(2k) is the Stokes depth. The above result follows
if one assumes that the drift in the surface boundary layer is not affected by
the earth’s rotation. We consider steady state drift, i.e. W = const. From
(4) we obtain
w(h) = W − w(p)
,
c = 0.
(9)
A solution of the homogeneous version of (2), subject to (3) and (9) is
(p)
w(h) = (W − w0 )e(1+i)c/D ,
(p)
where w0 = w(p) (c = 0).
10
(10)
The forces (7a) and (7b) can be calculated from (8) and (10). From
(6) we then obtain
ρ̂H 1
W 1 + (1 + i)
= uS L̂γD(1 − Q)(1 − i).
ρD
4
(11)
The coefficient L̂ = (1 − exp[−2αL])/(2αL) lies between 0 and 1, and is a
measure of how effectively the waves are damped by the sheet. In (11) we
have defined
2 1 + 2iq
(1 + i) 1 − 8iq + 20q 2
Q=
+
.
γL 1 + 4q 2
γD
2(1 + 4q 2 )
(12)
In general |Q| 1. Defining the ratio between the wave frequency and the
Coriolis parameter ∆2f = ω/f = (γD)2 , the ratio between the mass of the
sheet and the mass of the Ekman layer ∆m = (ρ̂H)/(ρD), the drift velocity
in (11) can be written
W =
uS ∆f L̂(1 − Q) 1
−
i(1
+
2∆
)
.
m
4(1 + 2∆m + 2∆2m )
(13)
The sheet is accelerated by the wave-induced drag Fw , which acts
in the wave propagation direction, and the sheet in turn induces an Ekman
current w(h) . There is no qualitative difference between w(h) and the traditional wind-driven Ekman current [15], and the surface drift is deflected to
the right (northern hemisphere) of the wave propagation direction. It is clear
11
that the horizontal extent of the sheet has to be much larger than the depth
of the Ekman layer for the solutions to be physically realistic, i.e. L, B D.
The drift speed |W | from (13) exceeds the surface value of the inviscid
Stokes drift by an order of magnitude. We have |W |/uS ∼ γ/(1/D) =
∆f , expressing the balance between the wave-induced stress and the stress
induced by the Ekman current. Provided that the wave steepness ζ0 k is kept
constant, the product uS ∆f increases for increasing wavelengths, that is,
longer waves produce higher drift velocities. The drift velocity also increases
with the wavelength for sheets of fixed length L because the relative damping
of the wave amplitude, represented by L̂, becomes smaller. Furthermore,
increasing the value of ∆m yields lower drift speeds. The deflection angle
depends on the mass of the floating object, increasing linearly with ∆m , due
to the action of the Coriolis force FC .
The particular solution w(p) decays in the direction of wave propagation. On the other hand the sheet has a constant drift velocity, and as a
result the Ekman current induced by it will vary from one end of the sheet
to the other. The surface value of the Ekman current relative to the sheet is
(p)
w(h) (c = 0) − W = −w0 = uS e−2αa
12
5
4
−
1 + 2iq .
1 + 4q 2
(14)
3.2
Variable eddy viscosity
For the numerical simulations we have chosen two different vertical profiles
of the eddy viscosity, see figure 1. The eddy viscosity ν (a) is the same as
used by Melsom [12]. The second profile ν (b) is more like that suggested by
McPhee and Martinson [16], with a linear increase with depth in the upper
few meters, and a constant high value below. Both profiles are similar in the
surface layer, and we have chosen ν (a) (c = 0) = ν (b) (c = 0) = ν̃ = 5 cm2 /s.
[Insert figure 1 about here]
The time development of the surface drift for the first 24 pendulum
hours after the onset of the waves is shown in figure 2. The sheet thickness is
H = 0.5 m, length L = 1000 m, and density ρ̂ = 900 kg/m3 . The wavelength
is λ = 200 m, and the wave amplitude is ζ0 = 1 m. The density of the water
is ρ = 1028 kg/m3 , and the Coriolis parameter f = 1.4 × 10−4 s−1 . The
motion is started from rest.
[Insert figure 2 about here]
The sheet quickly attains a drift velocity of the order 10 cm/s, and
the maximum velocity in the direction of wave propagation is obtained after
13
approximately 3 pendulum hours. The surface value of the inviscid Stokes
drift is uS = 1.72 cm/s. The transition to a steady state is more rapid for
the viscosity profile ν (b) . Physically, the distribution of the eddy viscosity
ν (a) corresponds to an intermediate layer of very viscous fluid, and this layer
slides on top of the less viscous fluid below. The distribution ν (b) corresponds
to a very viscous fluid at all depths except for the upper few meters, and the
strong shear at all levels in the water column accelerate the transition to a
steady state.
Figure 3 shows a hodograph of the drift velocities in the water after a
steady state has been reached, and figure 4 shows the corresponding velocity
profiles. The parameters are the same as in figure 2. We notice that the
maximum drift velocity is attained below the interface between the floating
object and the water, in accordance with the results of Weber and Saetra
[17]. The drift velocity decays more monotonically with depth when using
ν (b) than using ν (a) . In the latter case the drift velocity is quite high even
at 10 m depth, but diminishes at 20 m depth, cf. figure 1. The thickness
of the Ekman layer becomes larger when using ν (b) compared to using ν (a) .
The ratio between the mass of the floe and the mass of the Ekman layer,
∆m , is therefore smaller. As discussed in section 3.1, the drift speed will
14
increase, and the deflection angle decrease, if the ratio ∆m decreases, which
is precisely what can be seen in figure 3.
[Insert figure 3 about here]
[Insert figure 4 about here]
Effective values of ν̃ may be calculated by matching the damping
coefficient α with observations [4]. In the marginal ice zone the values vary
from 2-1500 cm2 /s [11]. The steady state drift speed of the sheet obtained
using ν (b) is approximately 15 cm/s. Using a similar eddy viscosity profile
with a surface value of 50 cm2 /s, i.e. a tenfold increase, the drift speed
becomes approximately 25 cm/s.
In traditional wind-driven Ekman flow, there is a balance between
the force due to the wind stress on the surface and the Coriolis force acting
on the water column. Because the Coriolis force always acts perpendicular
to the velocity vector, the vertically integrated mass transport is at right
angles to the wind stress, regardless of the turbulent stresses in the water
[15]. Also in our case the vertically integrated mass transport is to the right
of the wave-induced stress, but we need to include the mass transport, or
equivalently, the momentum, of the sheet to obtain this result. Because the
15
eddy viscosities ν (a) and ν (b) are similar in the surface layer, the difference in
the wave-induced drag Fw is insignificant. The total mass transport obtained
is therefore the same.
[Insert figure 5 about here]
In figure 5 we show hodographs for a ‘thick’ sheet with H = 1 m and
ρ̂ = 900 kg/m3 , compared to a ‘thin’ sheet of negligible mass. Both sheets
are of length L = 1000 m, and the viscosity profile ν (b) has been used. The
wave parameters are the same as used in figure 2. As discussed above, the
mass transport is equal in both cases, because the wave-drag Fw is the same.
The difference in the mass transport in the water therefore corresponds to
the momentum of the thick sheet, i.e.
0
ρ
(wthin − wthick )dc = ρ̂HWthick .
(15)
−∞
4
Summary
The wave-induced drift of large floating sheets that are influenced by the
earth’s rotation has been examined. After the onset of the waves, the drift
speed of the sheet quickly exceeds the surface value of the Stokes drift, uS .
There is a clear analogy to wind-driven drift, and the drift is deflected to
16
the right (northern hemisphere) of the wave-induced stress, which acts in
the wave propagation direction. The drift speed and the deflection angle
both depend on the ratio between the mass of the sheet and the mass of the
Ekman layer. Because the Ekman layer depth is related to the turbulence in
the oceanic surface layer, the results are sensitive to the formulation of the
eddy viscosity.
Acknowledgements
K. H. Christensen gratefully acknowledges financial support from the Research Council of Norway (NFR) through Grant No. 151774/432. K. H.
Christensen would also like to thank Professors Göran Björk, Anders Omstedt and Anders Stigebrandt for helpful discussions.
References
[1] A. S. Thorndike and R. Colony, J. Geophys. Res. 87 5845 (1982).
[2] M. Leppäranta, The Drift of Sea Ice (Springer-Praxis, Chichester, 2005).
[3] W. J. Pierson, J. Geophys. Res. 67 3151 (1962).
[4] J. E. Weber, J. Phys. Oceanogr. 17 2351 (1987).
17
[5] A. D. Jenkins, D. Hydrogr. Z. 42 133 (1989).
[6] J. E. Weber and A. Melsom, J. Phys. Oceanogr. 23 193 (1993).
[7] V. A. Squire, in Physics of Ice-covered Seas, Vol. 1, edited by
M. Leppäranta (Helsinki University Press, Helsinki, 1998).
[8] R. T. Pollard, J. Geophys. Res. 75 5895 (1970).
[9] A. K. Liu and E. Mollo-Christensen, J. Phys. Oceanogr. 18 1702 (1988).
[10] A. Melsom, J. Phys. Oceanogr. 22 19 (1992).
[11] V. A. Squire, J. P. Dugan, P. Wadhams, P. J. Rottier, and A. K. Liu,
Annu. Rev. Fluid. Mech. 27 115 (1995).
[12] A. Melsom, Ann. Geophys. 11 78 (1993).
[13] J. E. Weber and E. Førland, J. Phys. Oceanogr. 19 952 (1989).
[14] K. H. Christensen and J. E. Weber, Environ. Fluid Mech. (in press,
2005).
[15] V. W. Ekman, Ark. Mat. Astron. Fys. 2 1 (1905).
[16] M. G. McPhee and D. G. Martinson, Science 263 218 (1994).
[17] J. E. Weber and Ø. Saetra, Phys. Fluids 7 307 (1995).
18
Figure captions
1. Eddy viscosity profiles. Full line, ν (a) ; broken line, ν (b) .
2. Time development of the nondimensional drift velocity (U, V )/uS of the
sheet. Full line, result obtained using ν (a) ; broken line, result obtained
using ν (b) . Time in pendulum hours are indicated by crosses. Steady
state drift velocities indicated by five-pointed stars.
3. Hodograph of the steady nondimensional drift velocities (u, v)/uS in the
water. Full line, result obtained using ν (a) ; broken line, result obtained
using ν (b) . Water depth in meters are indicated on the graphs.
4. Profiles of the steady drift velocities in the water. Figure (a), result
obtained using ν (a) ; figure (b), result obtained using ν (b) .
5. Hodograph of the steady nondimensional drift velocities (u, v)/uS in
the water for a ‘thick’ sheet (full line), and a ‘thin’ sheet (broken line).
The eddy viscosity ν (b) has been used. Water depth in meters are
indicated on the graphs.
19
0
Depth [m]
−5
−10
(a)
−15
(b)
−20
−25
0
1
2
3
4
2
5
ν [m /s]
Figure 1.
20
6
7
8
9
−3
x 10
0
−1
1
1
−2
12
V/uS
−3
24
12
−4
24
3
3
−5
−6
6
−7
−8
0
6
1
2
3
4
5
U/uS
Figure 2.
21
6
7
8
9
10
1
−20
0
−20
−1
−10
v/uS
−2
−10
−3
−5
−4
−2
−5
−2
0
−5
0
−6
−2
−1
0
1
2
3
u/u
S
Figure 3.
22
4
5
6
7
8
(a)
0
Depth [m]
−5
v
−5
u
−10
−10
−15
−15
−20
−20
−25
−25
−30
−30
−35
−35
−40
−40
−45
−45
−50
−10
−5
0
(b)
0
5
Drift velocity [cm/s]
−50
10
Figure 4.
23
v
−10
u
−5
0
5
Drift velocity [cm/s]
10
1
0
−20
−20
−1
−10
−10
v/uS
−2
−3
−5
−5
−2
−4
0
−2
−5
0
−6
−1
0
1
2
3
u/u
4
S
Figure 5.
24
5
6
7
8
9