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Pattern dynamics of vortex ripples in sand: Nonlinear modeling and experimental

validation

Andersen, Ken Haste; Abel, M.; Krug, J.; Ellegaard, C.; Søndergaard, L. R.; Udesen, J.

Published in:

Physical Review Letters

Link to article, DOI:

10.1103/PhysRevLett.88.234302

Publication date:

2002

Document Version

Publisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):

Andersen, K. H., Abel, M., Krug, J., Ellegaard, C., Søndergaard, L. R., & Udesen, J. (2002). Pattern dynamics of

vortex ripples in sand: Nonlinear modeling and experimental validation. Physical Review Letters, 88(23),

234302. https://doi.org/10.1103/PhysRevLett.88.234302

VOLUME 88, N

UMBER 23 PHYSICAL REVIEW LETTERS 10J

UNE 2002

Pattern Dynamics of Vortex Ripples in Sand: Nonlinear Modeling and Experimental Validation

K. H. Andersen,

1

M. Abel,

2

J. Krug,

3

C. Ellegaard,

4

L. R. Søndergaard,

4,5

and J. Udesen

4,5

1

Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

2

Institut für Physik, Universität Potsdam, D-14415 Potsdam, Germany

3

Fachbereich Physik, Universität Essen, D-45117 Essen, Germany

4

Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark

5

Department of Mathematics and Physics, University of Roskilde, Box 260, DK-4000 Roskilde, Denmark

(Received 30 January 2002; published 22 May 2002)

Vortex ripples in sand are studied experimentally in a one-dimensional setup with periodic boundary

conditions. The nonlinear evolution, far from the onset of instability, is analyzed in the framework

of a simple model developed for homogeneous patterns. The interaction function describing the mass

transport between neighboring ripples is extracted from experimental runs using a recently proposed

method for data analysis, and the predictions of the model are compared to the experiment. An analytic

explanation of the wavelength selection mechanism in the model is provided, and the width of the stable

band of ripples is measured.

DOI: 10.1103/PhysRevLett.88.234302 PACS numbers: 45.70.Qj, 47.20.Lz, 47.54. +r

Ever since the establishment of a conceptual frame-

work for pattern formation [1], the description of patterns

formed in sand by the ﬂow of wind or water has posed

a challenge to the community [2–7]. Despite diverse ef-

forts including coupled map models for ripples and dunes

in air [2], stochastic models for ripples in air [3], or con-

tinuum equations based on the symmetries of the problem

[4], theoretical understanding has remained sparse. For ex-

ample, all models display a coarsening of the ripple/dune

pattern, but the coarsening does not terminate at a ﬁnal

selected wavelength, as is frequently observed in nature.

Furthermore, the models are heuristic, and it is not pos-

sible to make a quantitative comparison with experiments.

Here we study

vortex ripples [8], which are created by

an oscillatory water ﬂow, such as that generated near the

sand bed by a surface wave. Vortex ripples have attracted

attention as an example of a nonlinear pattern forming

system with a strongly subcritical ﬁrst bifurcation [6,7,9],

which cannot be described by conventional methods like

amplitude equations [9]. As most other sand patterns, they

display coarsening and saturation at a ﬁnite wavelength.

The approach pursued in this Letter combines a simple

model for the fully developed pattern with a sophisticated

data analysis which allows one to extract the key model

ingredient — the interaction function

f共l兲— directly from

the experimental runs. In this way the validity of the

model can be tested, and additional features required for

the description can be identiﬁed. As far as we know, this

is the ﬁrst quantitative comparison between theory and

experiment for a sand pattern. The basic ideas are general,

and we expect that the theoretical formalism combined

with the experimental analysis can be used for related sand

patterns (e.g., the one studied in [5]), or other strongly

nonlinear systems.

One-dimensional vortex ripples can be created in an an-

nular channel [6,10], ensuring the pattern to be subject to

well-deﬁned periodic boundary conditions. Freely grown

ripples are created from a ﬂat bed by a coarsening process,

which eventually saturates at an equilibrium state, where

the ripple length

˜

l is almost independent of the frequency

n of the driving and proportional to the amplitude of the

oscillation of the plate a [8,11]. Here we are mostly con-

cerned with the stability and evolution of the ripple patterns

themselves, and not with the instability of the ﬂat bed,

which has been discussed elsewhere [9,12]. By creating

a homogeneous ripple pattern, where all ripples have the

same length, and changing amplitude and frequency, the

stability of the pattern is probed. In this way it was found

that there is a stable band of ripples, l

min

,

˜

l,l

max

, for

a given set of driving conditions [7].

Recently, a simple model was proposed, which describes

both the coarsening and saturation of ripples, and repro-

duces the existence of a stable band [9]. In its simplest

form, the change of the length l

j

of the ripple j is a func-

tion of l

j

itself and the lengths of the neighboring ripples:

ᠨ

l

j

苷 2f共l

j21

兲 1 2f共l

j

兲 2 f共l

j11

兲 . (1)

The interaction function f共l兲 describes the transfer of

mass, and consequently length, between neighboring

ripples. Behind each ripple, due to the water oscillation,

a separation vortex forms, which causes an exchange

of mass between the ripples. f共l

j

兲 is the amount of

sand which is taken by a ripple of length l

j

from its

downstream neighbor during one-half of the oscillation

(Fig. 1). At the same time ripple j loses an amount of

mass f共l

j21

兲 due to the action of the upstream ripple.

Adding the reverse process for the second half of the

oscillation results in the model (1) [9]. The purpose of

this Letter is a test of this model through comparison with

experimental data.

Experimental setup.—As shown in Fig. 2, an 11 mm

wide, 15 cm high annular Plexiglas channel of diameter

48.6 cm is ﬁlled with water, A, and glass beads, B, with

a diameter of 250 6 50 mm. In the middle is a conical

234302-1 0031-9007兾02兾88(23)兾234302(4)$20.00 © 2002 The American Physical Society 234302-1

VOLUME 88, N

UMBER 23 PHYSICAL REVIEW LETTERS 10J

UNE 2002

−0.015

−0.01

−0.005

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4

f(

λ

) [cm/period]

λ

/a

f

l

(λ)

f( )λ

f

c

(λ)

f

r

(λ)

FIG. 1. The interaction function for a 苷 6 cm and n 苷

0.6 Hz. The thick gray line is the smooth average of the three

different functions (mean values subtracted). The right limit

of the plot at l

兾a 苷 1.45 is the limit of stability, where new

ripples are created 共l

max

兲. The inset shows a sketch of the

ripples in the part of the oscillation when the ﬂow is from the

left to the right. The interaction function can be interpreted

as the transport of sand in the trough, across the vertical

dashed line.

mirror, C, which is ﬁlmed from above by a CCD camera

with a resolution of 640 3 480 pixels. The channel is

attached to a motor, D, by a 1.5 m long arm E, which

oscillates the channel in an almost sinusoidal fashion.

We create initial conditions with small ripples by oscil-

lating at a small amplitude. The experiment starts when

amplitude and frequency are changed to their desired val-

ues. Throughout this paper, we use only the condition

a 苷 6 cm, n 苷 0.6 Hz. We want to emphasize that there

is nothing special about this condition, the same quali-

tative results are obtained for other parameters. Grow-

ing the ripples from initial conditions with homogeneous

ripples smaller than l

min

results in a ﬁnal number of

19–21 ripples with a length (averaged over 16 realiza-

tions) of

˜

l兾a 苷 1.25. The uncertainty in the length due to

A

B

C

D

C

B

E

FIG. 2. A sketch of the experimental setup seen from above

(left) and from the side (right). The length of the arm E and the

width of the channel are not to scale. See text.

periodicity is around 5%. The images are transformed back

to Cartesian coordinates, and the lengths l

j

are found by

ﬁtting them to triangles with ﬁxed slope (Fig. 3a). From

series of consecutive l

j

, the temporal change,

ᠨ

l

j

, is es-

tablished, and a space-time plot of the ripple evolution is

constructed. Figure 3b shows the evolution starting from

a small initial wavelength, which leads to the annihilation

of ripples. Around each annihilation the ripples have not

been analyzed, as it becomes inaccurate to ﬁt the triangles

during these events.

Measuring the stable band.—By running the experi-

ment with different values of a until a homogeneous

pattern occurs, we create initial conditions with a different

number N of ripples. From this initial pattern, the experi-

ment is started with the above parameters, and run for

10 000 periods. We have created initial conditions with

N

[ 关17; 24兴 and observed that for N . 22 ripples are

annihilated, while for N , 18 one or more new ripples

are created. We therefore conclude that there exists a stable

band N

[ 关17.5; 22.5兴 (for a 苷 6 cm, n 苷 0.6 Hz),

which corresponds to l

min

兾a 苷 1.13 6 0.03 and

l

max

兾a 苷 1.45 6 0.05. Note that this is a more accurate

measurement of the stable band than that conducted in

a

0

100

200

300

400

500

600

700

5 10 15 20

νt

x/

a

0

100

200

300

400

500

600

700

5 10 15 20

νt

x/

a

(a)

(b)

(c)

FIG. 3. (a) An example of an extracted proﬁle (opaque re-

gion) and the ﬁtted triangles with constant slope. The line is

shown above the proﬁle for clarity. (b) Space-time plot of the

experimental evolution of the position of the ripple crests starting

from ripples with lengths 2.5 cm and evolving with a 苷 6 cm

and n 苷 0.6 Hz. (c) A simulation of the model (1) using the

extracted interaction function and the same initial conditions

as above.

234302-2 234302-2

VOLUME 88, N

UMBER 23 PHYSICAL REVIEW LETTERS 10J

UNE 2002

[6,7], as we are always forcing with the same values of

a and n.

Mass transfer model.—The model (1) was originally

presented in Ref. [9], together with a more reﬁned ver-

sion. A stability analysis of the homogeneous state l

j

⬅

¯

l

shows that ripples are stable (unstable) if the derivative

f

0

共

¯

l兲 , 0 共.0兲 . For a convex interaction function, the

lower stability boundary l

min

therefore lies at l

marg

, where

f

0

共l

marg

兲 苷 0. To investigate nonlinear pattern evolution

within this model, Eq. (1) is supplemented by the rule that

ripples which reach zero length are annihilated and re-

moved from the system of equations, while the remaining

ones are relabeled.

Simulations of the model starting from typical initial

conditions [13] in the unstable band l,l

marg

show that

an equilibrium wavelength is reached which is essentially

independent of the initial wavelength, and depends only

on the interaction function f共l兲. To gain some insight

into the selection mechanism, it is useful to recast the

model into potential form by writing it in terms of the

position x

j

of the troughs between the ripples deﬁned by

l

j

苷 x

j

11

2 x

j

. Then

ᠨ

x

j

苷 2≠V 兾≠x

j

, with the potential

V given by

V 苷 2

N

X

j苷1

Z x

j11

2x

j

0

dl f共l兲 , (2)

where N is the number of ripples. It is then plausible to

conjecture that the equilibrium length

˜

l can be found by

minimizing V for homogeneous ripples, under the con-

straint that the total length L

苷 N

˜

l is conserved. This

implies that

˜

l is determined through the Maxwell construc-

tion applied to f:

Z

˜

l

0

dl f共l兲 苷

˜

lf共

˜

l兲 . (3)

Comparison with numerical simulations shows that (3) sys-

tematically overestimates

˜

l, with a better performance the

steeper the stable branch of f. Our interpretation is that

the deterministic dynamics gets stuck in the multitude of

metastable states of (1); recall that

any homogeneous state

with

¯

l.l

marg

is a stable, stationary solution. Since the

mean ripple length can increase only by annihilations, its

evolution freezes once all ripples are in the stable band.

The wavelength predicted by (3) should therefore be an

upper bound on the actual equilibrium wavelength, which

is true in all cases we have considered.

Data analysis.— Given the time series l

j

共t兲 we want to

(i) evaluate how well the model (1) describes the evolution

of the ripples and (ii) extract the interaction function f共l兲.

For the analysis, we write (1) in the more general form

ᠨ

l

j

苷 2f

l

共l

j21

兲 1 2f

c

共l

j

兲 2 f

r

共l

j11

兲 , (4)

where f

l

, f

c

, and f

r

denote the left neighbor, center,

and right neighbor interaction function, respectively. This

yields an additional degree of freedom as the functions are

not required to be equal.

We want to determine the optimal transformations

f

l

共l

j21

兲, f

c

共l

j

兲, f

r

共l

j11

兲 in the sense that they minimize

the error x

2

苷

P

t

关

ᠨ

l

j

共t兲 1 f

l

共t兲 2 2f

c

共t兲 1 f

r

共t兲兴

2

.

The problem is solved numerically by the alternating

conditional expectation value algorithm [14,15]. The

algorithm works by varying f

l,c,r

in the space of all

measurable functions until convergence to the absolute

minimum of x

2

[14]. The results are nonparametric

functions which are given in numerical form by points,

e.g., 关l

j

, f

c

共l

j

兲兴. An upper bound on the error, e.g., on

f

c

, is given by s

f

c

苷 max

l

共df

c

兾dl兲 ?s

l

, with s

l

being

the measurement error at the point l; analogous estimates

apply to f

l

, f

r

. The error in the points on the l axis is

estimated using the above result and equals roughly the

errors in the given values of l.

The quality of each of the resulting functions is given by

the maximum correlation c

l,c,r

of one of the terms with the

sum of all the others [16]. A value of c 苷 1 implies a per-

fect result, lower values indicate either imperfect modeling

or (measurement) noise or both.

Results.—We have performed the analysis on data from

18 realizations of a coarsening process initiated with ap-

proximately 60 small ripples and run with a 苷 6 cm and

n 苷 0.6 Hz. Figure 1 shows the three functions f

l

, f

c

,

and f

r

averaged over the 18 runs. Clearly there is some

noise, but the functions are very similar as expected from

(1). The maximum correlations are found to be C

l

苷 0.66,

C

c

苷 0.88, C

r

苷 0.72. The uncertainty in estimating l

amounts to about 2 mm, with a maximum slope of f of

around 3 3 10

23

共period兲

21

, the absolute error of f is

6 3 10

24

cm兾period. This means that the result does not

ﬂuctuate due to lack of data, rather the model (4) does not

account for all the variation in the ripple evolution. We

will return to this point later.

The maximum of the function lies around l

marg

苷

1.08a, which is a little larger than what was found by

the numerical simulations in [9]. Concerning the shape

of the function, it is interesting that the slope in the

unstable band l,l

marg

is much larger than the slope

in the stable band l.l

marg

. As the slope is a measure

of the time scale of the dynamics, this implies that the

initial coarsening stage is much faster than the equili-

brating stage.

To use the resulting function to integrate (1) numeri-

cally for evaluating (3), information about the smallest

ripples is needed. With a lack of such information,

we have extrapolated linearly to l 苷 0, Fig. 1, dashed

line. Varying the slope of the extrapolated part by

a factor of 2 in either direction does not change the

ﬁnal number of ripples. Figure 3c shows a space-time

plot of the numerical integration. The model predicts

a qualitatively similar behavior as the experiment;

however, the instability of the small ripples develops

slower in the model than in the experiment. If the

extrapolated slope for the interaction function is made

steeper, this instability will evolve faster in the model.

234302-3 234302-3

VOLUME 88, N

UMBER 23 PHYSICAL REVIEW LETTERS 10J

UNE 2002

The model also overestimates the ﬁnal ripple length.

For similar initial conditions and system size, the ﬁnal

number of ripples is typically 18, whereas the experiment

yields 20.

Evaluating the Maxwell construction (3) using the mea-

sured function and extrapolating to account for the larger

ripples, gives

˜

l 苷 2.2a, which lies far outside of the range

of deﬁnition of f共l兲. Evidently, the experimentally deter-

mined f共l兲 belongs to the class of interaction functions

for which the upper bound given by Eq. (3) is not useful.

The upper bound of the stable band, where new ripples

are created, was found to be l

max

苷 1.45a. At this point,

an inﬁnitesimally small ripple, inserted in the trough be-

tween two larger ripples, is able to gain mass from the

neighbors, and thus grows. If the model shall capture this

behavior, Eq. (1) yields that f共l

max

兲 should be smaller

than f共0兲 [9]; otherwise the right-hand side is negative

and a small ripple disappears. The measured interaction

function (cf. Fig. 1), however, suggests the opposite. Even

with the large uncertainty inherent in the extrapolation,

a bend which makes f共1.45a兲 , f共0兲 is hard to imag-

ine. We therefore conclude that the model in the form

(1) is not able to quantitatively predict the creation of new

ripples. The reason for this apparent failure relates to the

model being developed essentially as an expansion around

a homogeneous state [9]. But in the case of a creation, the

state is as inhomogeneous as it can be: a very tiny ripple

ﬂanked by large ripples. To account for this, the interaction

function should be described as a function of the length of

the ripple creating the separation bubble, l

i

, but also of the

length of its neighbors. Ongoing numerical work indicates

that it is most relevant to write f ⬅ f共l

i

, l

i11

兲 where l

i

is the length of the ripples creating the separation bubble,

and l

i11

is the one “touched” by the separation bubble (see

inset of Fig. 1). In fact, the whole coarsening process is

dominated by highly inhomogeneous conﬁgurations, and

this might also be why simulating (1), using the measured

interaction function, does not produce exactly the correct

ﬁnal ripple length.

Conclusions.—We have demonstrated that we can ex-

tract the interaction function f共l兲 from spatiotemporal data

of the evolution of the proﬁle of the sand surface. The

nonlinear data analysis shows its full strength, producing

nonparametric function estimates, where any parametric

approach would have failed. With the interaction function

it is possible to model the evolution of the ripples, in par-

ticular, their coarsening and equilibration.

The prediction from the model of the existence of a

stable band is indeed observed in the experiment. The

lower bound is predicted at l

marg

兾a 苷 1.08 6 0.03, while

the measurement gives l

min

兾a 苷 1.13 6 0.03. In a simi-

lar experiment [6], the lower bound of the stable band was

found to coincide with the ﬁnal ripple length: l

min

苷

˜

l,

whereas we ﬁnd that l

min

,

˜

l. The difference between

the two results might be apparent only, as the number of

ripples in the annulus of [6] was 10 or smaller, giving rise

to an uncertainty in the ripple length on the order of the

difference between l

min

and

˜

l.

The dynamics responsible for the evolution of a 1D

ripple pattern are also relevant for 2D ripple patterns. How-

ever, in the latter case, topological defects will be present

[7], and the ﬁnal length selection might be determined by

the motion of these [9].

It is a pleasure to acknowledge discussions with

T. Bohr, M. van Hecke, and F. Schmidt. J. K. is grateful to

DTU and NBI for gracious hospitality, and to DFG within

SFB237 for support. K. H. A thanks Universität Essen for

hospitality.

[1] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65,

851

(1993), and references therein.

[2] H. Nishimori and N. Ouchi, Phys. Rev. Lett. 71

, 197

(1993).

[3] B. T. Werner and D. T. Gillespie, Phys. Rev. Lett. 71

,

3230– 3233 (1993).

[4] Z. Csahók, C. Misbah, F. Rioual, and A. Valance, Eur.

Phys. J. E 3

, 71 (2000).

[5] A. Betat, V. Frette, and I. Rehberg, Phys. Rev. Lett. 83

,

88– 91 (1999).

[6] A. Stegner and J. E. Wesfreid, Phys. Rev. E 60

, R3487

(1999).

[7] J. L. Hansen, M. v. Hecke, A. Haaning, C. Ellegaard, K. H.

Andersen, T. Bohr, and T. Sams, Nature (London) 410

, 324

(2001); J. L. Hansen, M. v. Hecke, C. Ellegaard, K. H. An-

dersen, T. Bohr, and T. Sams, Phys. Rev. Lett. 87

, 204301

(2001).

[8] R. A. Bagnold, Proc. R. Soc. London A 187

, 1–15 (1946).

[9] K. H. Andersen, M.-L. Chabanol, and M. v. Hecke, Phys.

Rev. E 63

, 066308 (2001).

[10] M. A. Scherer, F. Melo, and M. Marder, Phys. Fluids 11

,

58 (1999).

[11] P. Nielsen, J. Geophys. Res. 86

, 6467 (1981).

[12] P. Blondeaux, J. Fluid Mech. 218

, 1–17 (1990); K. H. An-

dersen, Phys. Fluids 13

, 58 – 64 (2001).

[13] We consider here initial conditions with some amount of

disorder. Different behavior results in the (unrealistic)

case of a perfectly ordered state with a localized pertur-

bation; see J. Krug, Advances in Complex Systems (to be

published).

[14] L. Breiman and J. H. Friedman, J. Am. Stat. Assoc. 80

,

580 –598 (1985).

[15] H. Voss, P. Kolodner, M. Abel, and J. Kurths, Phys. Rev.

Lett. 83

, 3422– 3425 (1999).

[16] For example, for

f

l

: c

l

苷 具f

l

共t兲 ? S共t兲典兾共具 f

2

l

典具S

2

典兲;

S共t兲 苷

ᠨ

l

i

2 2f

c

1 f

r

has zero mean (without loss of

generality).

234302-4 234302-4