Theory of self-organisation in sorted stone stripes

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Theory of self-organisation in sorted stone stripes

P. Mulheran

To cite this version:

P. Mulheran. Theory of self-organisation in sorted stone stripes. Journal de Physique I, EDP Sciences,

1994, 4 (1), pp.1-5. �10.1051/jp1:1994117�. �jpa-00246880�

Short Communication

Theory of self-organisation in sorted stone stripes

PA. Mulheran

Department of Physics, University of

Reading,

Whiteknights, PO Box 220, Reading, RG6 2AF, U-K-

(Received

18 October 1993, accepted 15

November1993)

Abstract, The dynamics of self-organisation in simulated sorted stripes _are considered. Ba- sic random-walk theory is used to predict that the evolution follows power-law kinetics with the emergence of _an asymptotic scaling state. The simulated patterns confirm these predictions with good correspondence being found between measured and calculated distributions of stripe sizes.

Taking account of the dimensional difference between real and simulated stripes, _a favourable comparison between theory and patterns observed _on alpine slopes is also found. This novel application of scaling theory therefore explains the surprising order found in this natural phe-

nomenon.

In _a recent paper [I] ^{Werner and Hallet studied the evolution of sorted} stone

stripes

on

alpine slopes,

which

provide

_one of the most

striking examples

of

patterned ground

^found in nature.

They

reasoned that the stones _are moved from site to site

by

needle ice

growing

in

freezing conditions,

with

upheaval

of frozen soil domains

preferentially pushing

stones

together. They

modelled this

using

_a cellular automata simulation. When _a

slope

_was included _to bias the stones' motion downhill

they

found that the simulation

self-organised

into

stripes

of stoney

regions

with downhill orientation,

just

as are observed _on the real

slopes. They

commented

that the simulated patterns _coarsen

continuously throughout

time, ^{rather than}

stopping

^when

a dominant

wavelength

_emerges. In this _{paper we} shall extend their

analysis

of the evolution of the pattern ^{and show how the}

self-organisation

_may be understood in terms of

simple

random- walk

theory.

It will be shown that the pattern evolves into _a

scaling

state, where the _average stripe width grows as a power of time and the distribution of widths _is

entirely predictable

from the theory. In

addition,

_once the

dimensionality

of the real _stone domains has been accounted

for,

_our prediction ^will be shown to

give satisfactory

agreement with the field data

reported

in

Ill-

We start with _a brief

description

of the simulation used in this work. A square lattice is

used to present a

plan

view of the

ground,

with stones

represented by

_ones and soil domains

by

_zeros.

Starting

from

completely

random positions, the stones are selected at random and

JOURNAL DE PHYSIQUE I N°1

Fig,

I. Self-organised stone stripes observed in the-simulation after t

= 5000 freezing cycles.

moved

according

to a set of

simple

rules that

depend

_upon the local environment. These rules _are:

(I)

the

probability

of

attempting

_{a move} is

proportional

to the number of _zeros in the

eight

nearest

neighbour sites; (2)

downhill _{moves are} tried

first,

and if these three sites _are

already occupied

the stone can move

sideways

but not

uphill; (3)

the stones have _an attractive interaction and the most

energetically

favourable _move is

always

made. Rule

ii)

models the

preferential

formation of needle ice in soil

domains;

rule

(2)

models

gravity pulling

the stones

downhill;

and

(3)

^{models the} relative

upheaval

of the soil _over the stone domains.

Note how the

gravity

rule

(2)

dominates the _moves in this simulation. We could also define _an effective temperature to bias

against

unfavourable _{moves, as} in

iii,

but _we find that this _{is an}

unnecessary

complication

which does not

dramatically

affect the overall evolution.

In

figure

1 the pattern

produced

in _one such simulation _{on a}

(1000

_x

1000)

lattice is

shown,

after _a time of 5000

freezing cycles

has

elapsed.

One

freezing cycle corresponds

to N trial moves, where N is the number of sites in the lattice. In the

figure

there _are

N/2

stones

occupying

the black

pixels.

Periodic

boundary

conditions have been used.

In the

simulations,

the

preference

for downhill motion

effectively decouples

the horizontal and vertical directions. Once stoney ^{domains have formed from the} disordered initial conditions

they

become

elongated

down the

slope,

whilst in the transverse directions the stones, ^{if free} from _a

stripe,

_are able _to

perform

_a random walk. This _causes the walls of the

stripes

to fluctuate _at random _so that if _a

stripe's

width becomes _zero it is lost from the system, since the

probability

of

nucleating

_new stone

stripes

_is

completely negligible.

This process occurs

by

the downward motion of the terminations

[I],

^which can be identified in

figure

I _as stripes that do not

completely

_{run up} the full

length

of the lattice.

We _{may now} examine the

dynamics

in detail

by considering

the number distribution

f(z, t)

of measured

stripe

widths _z at time t. These measurements _are taken from several

evenly- spaced

traverses _across the simulation lattice, _so that the width of each stripe is measured in several

places.

This is necessary because _a

given stripe

is not of uniform width

along

its whole

length. During

the

subsequent

time step ⁱⁿ the

simulation, f(z, t)

will _{grow or} shrink due _to the

exchange

of stones with stripes ^of size

(z

+

I).

We take these

exchanges

to be random _processes due to the stochastic _nature of the local environment of each

stripe.

A

Taylor

_expansion then

vlt)

_«

).

12)

By changing

to a new time variable

r =

/ _{v(t)dt (3)}

equation ii)

becomes

~~~

^{~~ ~}

~~~~

^~~

' ~~~

where the constant k is

largely

irrelevant to what follows.

Equation (4)

_was used

by

Louat to

study

the

phenomenon

of normal

grain growth

_[2]. The

boundary

condition is that

f(0,T)

= 0, _since the point _z

= 0 acts _{as a} sink for stripes. ^The

required

solution is

~~~'~~

~~~ _~2

~~~~~~~4kT~'

^~~~

where the constant A determines the total number of

stripes

in the system. From

(5)

the variation of the average stripe width with time _can be found. We have

/

co

_Zf(Z,T)dZ

Z(T)

"

~

/

co

_f(Z,T)dZ

0

=

fi,

which

by

equations

(2)

and

(3) gives £(t)

oc t~M This _power law evolution is tested in

figure

2, where the average

stripe

width in the simulation is

given

as a function of time _{on a}

log-log plot.

As _can be _seen, the data follow _a

straight

line with

gradient

_very close to _one quarter as

predicted.

The power law kinetics found above suggest ^{that the} system

obeys scaling. ^Indeed, changing

to the scaled variable _g

=

z/£(T)

in equation

(5),

_we find the width

frequency

distribution

~~~~ ~~~~~~)dZ ^~

0

=

2cg

_exp

(-cg~), (6)

where _c

=

~/4.

In

figure

3 _we compare the

time-independent

distribution

(6)

with those found at five different times in the simulation. For each _case the abscissa is

suitably

scaled

so that the width is

given

in terms of the _mean value at that time. It _can be _seen that the simulation

displays

the

scaling

_property and

closely

follows equation

(6).

The pattern

4 JOURNAL DE PHYSIQUE I N°1

4l Simulation Gradient 1/4

6.0 6.S 70 7.5 8.O 8.5 9.O

In(t)

Fig.

2. Variation of the log of the _average width 2 with the log of elapsed time t, ^{measured from} the simulated patterns.

~ _Ul

l

_t=2000

Ul Z t=3000

A A t=4000

~

n t=5000

FM

i~

Ql

if

o

O-O O.5 O 1.5 2.O 2.5 3.O

scaled width

Fig.

3. The distribution of stripe widths found after time t. The widths _are measured relative to the average value for each time, and compared with equation

(6).

thus remains

statistically

self-similar

throughout

its

evolution,

_except for _very

early

times _as indicated

by

the t ₌ 1000 data _m

figure

3. Here _{we suppose} that the pattern has not

fully

matured

beyond

its disordered

starting

conditions.

Let _{us now} consider observations of real sorted

stripes.

^Werner and Hallet [1]

give

the width distribution measured _on the

slopes

of Mauna Kea. This data is

reproduced

in

figure

4 where

again

the abscissa is scaled to the _average observed width. For

comparison, F(y)

from

equation (6)

is

plotted

_as the broken _curve. It is clear that the field data follows _{a much narrower} and

more

peaked

distribution. An

explanation

^{for this lies} in the fact that the real stone

stripes

are three dimensional

humps

rather than the two dimensional

strips

formed in the simulations.

Therefore the number of stones _m the real stripes _{varies as} the _square of the measured widths, unlike the simulated stripes where the number is

linearly proportional

to the width. However, the basic

equation

of motion

(4)

remains unaltered for this situation

provided

the variable _z

I

O-O O.5 1-O 1.5 2.O 2.5 3.O

scaled width

Fig.

^4. The scaled distribution of stripe widths taken from the field data in [1], compared to

equations (6) and

(7).

represents the cross-sectional _area of the

stripes.

The

scaling

distribution function thus remains

as in

equation (6)

but with _a

correspondingly

_new

interpretation

of the variable _y. To convert

(6)

into the

predicted

distribution of observed widths _we introduce the variable _{w oc}

@.

^Then

the

scaling

distribution becomes

~~~°~

~~~~ ~j

=

4bw~

_exp

(-bw~), (7)

where

=

T(1.25)~

_ensures I ₌ I.

Equation (7)

is

plotted

in

figure

⁴ _as the solid _curve where it is _seen that it is also _narrower and _more

peaked

than

(6).

The agreement with the observations is

adequate

given the absence

of _{any error} bars in the

original

data [1],

although

it appears that the observed

stripes

_are

even more mono-modal than

(7)

suggests. Here _we also note that the

early

data from the

simulation,

_as in

figure 3,

is also _more

peaked

than

expected.

This indicates that the field data would be better

explained

if the real

stripe

pattern ^{could be considered} young in _{some sense.}

Further work _on this

point

would

clearly

be desirable.

In summary, it has been shown how _a

simple application

of basic ideas

concerning

^random walks _can lead to a

good understanding

of the

self-organisation

in _stone

stripes.

The slow

diffusion of _stones between

stripes

_causes the domain walls to fluctuate _at random with stripe elimination

occurring

when _a width becomes _zero. This mechanism leads _{to a} continuous

coarsening

of the pattern with

power-law

^kinetics and _a

predictable asymptotic

state. This

application

of

scaling theory

thus

helps

to

explain

how order _can arise in _a

seemingly

random natural environment. In

particular,

the

theory predicts

that the distribution of widths found

on any

slope

with sorted stripes ^{should follow} the

scaling

function

(7).

It will be

interesting

to test this

powerful prediction against

other sets of field data when

they

become available.

References

[1] ^{Werner B-T- and Hallet} B., Nature 361

(1993)

142.

[2] ^Louat N-P-, Acta Met. 22

(1974)

721.