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J. R. Soc. Interface
doi:10.1098/rsif.2011.0516
Published online
Pattern formation in stromatolites:
insights from mathematical modelling
R. Cuerno1, C. Escudero2,*, J. M. Garcı́a-Ruiz3 and M. A. Herrero4
1
Departamento de Matemáticas and Grupo Interdisciplinar de Sistemas Complejos (GISC),
Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
2
Departamento de Economı́a Cuantitativa and Instituto de Ciencias Matemáticas
(CSIC-UAM-UCM-UC3M), Universidad Autónoma de Madrid, Ciudad Universitaria
de Cantoblanco, 28049 Madrid, Spain
3
Laboratorio de Estudios Cristalográficos, CSIC, 18100 Armilla, Granada, Spain
4
IMI and Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad
Complutense de Madrid, 28040 Madrid, Spain
To this day, computer models for stromatolite formation have made substantial use of
the Kardar– Parisi – Zhang (KPZ) equation. Oddly enough, these studies yielded mutually
exclusive conclusions about the biotic or abiotic origin of such structures. We show in this
paper that, at our current state of knowledge, a purely biotic origin for stromatolites can
neither be proved nor disproved by means of a KPZ-based model. What can be shown, however, is that whatever their (biotic or abiotic) origin might be, some morphologies found in
actual stromatolite structures (e.g. overhangs) cannot be formed as a consequence of a process
modelled exclusively in terms of the KPZ equation and acting over sufficiently large times.
This suggests the need to search for alternative mathematical approaches to model these
structures, some of which are discussed in this paper.
Keywords: pattern formation; stromatolites; Kardar –Parisi – Zhang equation;
origin of life
1. INTRODUCTION
biological reactions [14]. To this day, the so-called molecular fossils are the most solid evidence for life, but the
technology available for their recognition can only be
applied when the geological context is well-known and
therefore the singenicity of the fossils can be assured
[15,16]. All these concerns and limitations have made
the laminated, accretionary structures called stromatolites important pieces in the search for life’s infancy
[17,18]. Indeed, as some stromatolites are thought to
date back as far as about 3.5 Ga [9,19–22], their possible
biotic origin would have deep consequences for our current views on how, and when, life started on the Earth.
The relevance of stromatolites in the context of the
search for life’s origin originates in their complex
shape and structure, which in modern extant examples
is, undoubtedly, the result of a cooperative microbiological behaviour [23– 27]. They appear as sedimentary
laminated structures whose sizes range from a few millimetres for an individual to a kilometre for reefs made of
many individuals, and displaying corrugated surfaces
and sometimes columnar shapes with a rich variety of
patterns (e.g. [17,22,23] and figure 1). But, interestingly, stromatolites cannot be considered as proper
fossils. Like termite mounds, see figure 2, stromatolites
being currently formed are not actual biological organisms, but petrological structures built by living
organisms and rarely preserve detectable organismal
remnants of their putative builders or users. However,
while the complexity of termite mounds structures
The detection of the oldest remnants of life on the Earth
is an important scientific problem. Indeed, it is crucial to
set the timing of life in the only planet where matter is
known to have self-organized to yield complex shapes
and behaviour. The search for remnants of the earliest
life is conducted along four different paths, namely: (i)
the identification of carbonaceous decorated microstructures displaying shapes characteristic of primitive life and
unable to form solely by mineral precipitation [1,2]; (ii)
the search for reduced carbon or sulphur with an isotopic
signature characteristic of life [3,4]; (iii) the detection of
organic molecules that may be exclusively derived from
biochemical degradation of once living organisms [5,6];
and (iv) the study of sedimentary structures, named
stromatolites [7–10] that are thought to be built by
microbial communities. Each of these approaches meets
considerable difficulties [11,12]. For instance, very old
microstructures with complex shapes are not easily
distinguished, neither morphologically nor chemically,
from self-ordered mineral assemblies [13]. On the other
hand, the isotopic signatures of carbon compounds
formed by purely mineral reactions such as Fischer–
Tropp cycles are very similar to those appearing in
*Author for correspondence ([email protected]).
Electronic supplementary material is available at http://dx.doi.org/
10.1098/rsif.2011.0516 or via http://rsif.royalsocietypublishing.org.
Received 2 August 2011
Accepted 14 September 2011
1
This journal is q 2011 The Royal Society
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2 Pattern formation in stromatolites
R. Cuerno et al.
(a)
(b)
(c)
(d)
Figure 1. Stromatolitic pattern from (a), (b) ca 2.7 Ga Tumbiana formation; (c) ca 1.8 Ga Duck Creek Dolomite; (d ) ca 3.49 Ga
Dresser formation.
sedimentation) and the influence (if any) of the growth
of mats on these processes. Moreover, analyses and simulation of the corresponding models should shed light into
the manner in which stromatolites are formed. To the
best of our knowledge, mathematical models of stromatolite formation have been proposed and discussed only
recently [32–34]. The choice of equations selected to
model their formation basically reduces to the so-called
Kardar–Parisi–Zhang (KPZ) stochastic equation [35]
@h
l
¼ nr2 h þ jrhj2 þ hðx; tÞ þ F;
2
@t
Figure 2. In the biotic scenario, ancient stromatolites are
viewed not as biological entities, but as the result of biological
activity from organisms that are no longer present, and from
which no biological record remains. The situation would be
much like a termite mound, as depicted above, once insects
had left.
rules out a non-biotic origin, the much simpler structures of stromatolites require that the possibility of a
non-biological origin has to be explored, particularly
for the case of many Archean and some Proterozoic
stromatolites.
Biotic or abiotic in origin, stromatolite pattern formation needs to be better understood. By this, we mean
the issue of ascertaining the way in which the threedimensional topographies and textures of these rock
structures are built up. Modern stromatolites provide
the only dynamic models available for experimentation
[28–31]. Such limitations emphasize the importance of
insights gained from mathematical models of stromatolite morphogenesis. To provide useful information,
mathematical models should be able to represent those
physical processes likely to be involved in stromatolite
formation (mainly associated with precipitation and
J. R. Soc. Interface
ð1:1Þ
which provides a classical model to describe the growth of
an interface, whose position is identified by a height function h, separating a growing medium which advances into
a receding one under the effect of random fluctuations h
[35–37]. The KPZ equation has been postulated for many
different types of systems, from polymers to vortex lines,
domain walls, thin films, to biophysical systems. In our
context, the interface position h is identified with the
boundaries of the laminae displayed in stromatolites.
For instance, in Grotzinger & Rothman [32], the authors
propose that a model based on a KPZ equation is able to
reproduce some morphological features (fractal dimension, power spectra, . . .) observed in some stromatolite
specimens and they claim a possible purely abiotic
origin for stromatolite formation. On the other hand, a
particular limit of the deterministic KPZ equation
(where noise effects are ignored) has been studied
[33,34], where a biotic origin for stromatolite structures
is proposed. The use of similar equations to support
mutually exclusive conclusions raises at once the question: what features of stromatolite formation can
actually be inferred from the analysis of simple mathematical models of which KPZ is an example? In this
paper, we claim that, at our current state of knowledge,
a purely biotic origin for stromatolites cannot be proved
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Pattern formation in stromatolites
(nor disproved) by means of a KPZ-based mathematical
model. The reason is that each term in equation (1.1) can
be shown to correspond to a physical process (diffusion,
convection, sedimentation, precipitation . . .) which can
be of a biotic or abiotic nature, both cases being indistinguishable in the absence of further information. What
can be shown, however, is that whatever their (biotic or
abiotic) origin may be, some stromatolite morphologies
cannot be formed as a consequence of a process modelled
exclusively in terms of equation (1.1) and acting over
sufficiently large times.
We conclude this introduction by describing the plan
of this paper. We first recall in §2 some basic facts concerning the KPZ equation. In particular, the physical
meaning of each term in that equation is described. We
then address in §3, the main issue in this work. More precisely, we compare the behaviour of solutions to KPZ
with the laminar morphologies observed in some stromatolite samples. In particular, it is observed that some of
the former are not compatible with a formation process
basically described by a KPZ equation. Section 4 contains a discussion on the relevance of KPZ in our
current context, and some directions to model stromatolite formation are suggested to gain insight into
stromatolite pattern formation. Finally, a number of
technical results related to the physical derivation of
KPZ in the context of kinetic roughening, as well as on
the mathematical properties of that equation, are gathered in electronic supplementary material, appendices
A and B at the end of the paper.
2. THE MEANING OF KARDAR –
PARISI – ZHANG EQUATION
The KPZ equation (1.1) is a classical model for kinetic
roughening. This last term denotes the general problem
of describing how microscopic fluctuations, which are
present in virtually any interface displacement process,
eventually develop into large-scale behaviours with
universal, observable properties [37]. As mentioned,
in equation (1.1), h ¼ h(x,t) denotes the position of an
interface separating an advancing medium which is
invading a receding one; x [ Rd represents a space
coordinate in a d-dimensional space (d 1), t stands
for time, n , l and F are positive parameters to be discussed below; and h(x,t) is a Gaussian noise
characterized by its two first moments
khðx; tÞl ¼ 0
ð2:1Þ
R. Cuerno et al. 3
be any point in Rd and t can be any positive real
number. While particularly convenient for mathematical treatment, continuous equations are generally
understood as suitable approximations to discrete
ones. The latter are characterized by being formulated
in discrete domains, with given time and space unit
lengths Dt and Dx.
Modelling particular aspects of stromatolite formation by means of the KPZ equation raises at once a
number of questions, as for instance:
(i) How can equation (1.1) be derived from first
principles, and which physical assumptions
need to be considered to that end?
(ii) What is the physical meaning of parameters n, l
and F in equation (1.1), and how can they be
related to experimental measurements?
(iii) Which types of behaviour do solutions to equation (1.1) display, and what relation (if any)
exists between such behaviours and stromatolite
morphologies.
The mathematically oriented reader will find a discussion on points (i) and (ii) (respectively, (iii)) above in
electronic supplementary material, appendix A (respectively, in electronic supplementary material, appendix
B). Here, we merely describe the main conclusions that
result from electronic supplementary material, appendix
A concerning the physical meaning of the terms arising
in the KPZ equation.
To begin with, equation (1.1) represents an approximation, obtained under a number of simplifying
hypotheses, to a large class of non-equilibrium processes. These are characterized by the coexistence of
two different media (or phases), one of which is advancing into the other. Both media are separated by a
comparatively thin, moving interface. The location of
such interface at any space coordinate x and time t is
given by a function h(x,t), sometimes termed the interface height, whose evolution in time is governed by
equation (1.1). Parameter l represents the interface velocity along its normal direction. Such value is thus
assumed to be constant in time. On the other hand,
n . 0 is the (material) diffusion coefficient along the
interface and F is an external forcing term that has
dimensions of velocity. Denoting by [u] the dimensions
of a quantity u, we thus have
½l ¼
length
l
¼
time
t
and
½n ¼
ðlengthÞ2 l 2
¼ :
time
t
ð2:3Þ
and
khðx; tÞhðx 0 ; t 0 Þl ¼ D dðx x 0 Þdðt t 0 Þ;
ð2:2Þ
where D is a positive parameter known as the noise
delta function, so that
amplitude and d(.) is Dirac’s
Ð
d(x) ¼ 0 for x = 0, and Rd dðxÞdx ¼ 1, where d 1 is
the dimension of the space where the previous integral
is considered. Conditions (2.1) and (2.2) are usually
rephrased by stating that the noise term h(x,t) in
equation (1.1) has zero mean, a variance given by D
and correlations are considered negligible.
At this juncture, we should stress that equation (1.1)
is a continuous equation, by which we mean that x can
J. R. Soc. Interface
On its turn, the effect of noise on the interface
motion is encoded in the term h(x,t) whose properties
(zero mean noise, amplitude given by D . 0) have
been recalled before. As a matter of fact, in the particular setting discussed in electronic supplementary
material, appendix A, n, l and F can be related to
some material properties of the interface as the interfacial tension s and the effective interface mobility
G (that is the interface speed per unit of bending
force measured along the normal direction to the
interface), as well as to an external driving potential
(necessary for the interface motion), m0 by means of
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4 Pattern formation in stromatolites
R. Cuerno et al.
the relations
n ¼ sG
and F ¼ l ¼ Gm0
ð2:4Þ
hence
½s ¼
energy
ml
¼ force ¼ 2 ;
length
t
and ½G ¼
lt
:
m
½m ¼ ½m0 ¼
½s
l
ð2:5Þ
Note that equation (2.3) is consistent with equations
(2.4) and (2.5). In view of our previous discussion, we
may say that equation (1.1) describes interface motion
under the action of four superposing effects. These are
represented by the four terms on the right-hand side
(r.h.s.) of equation (1.1) and correspond to diffusion,
growth, external forcing and noise. Diffusion at the
interface is characterized by the diffusion coefficient n
given in equation (2.4), whereas the mean interface
velocity in equation (2.4), which is the key player in the
interface motion, is given by l. Except for the noise
term h(x,t), no additional effect is considered to be relevant for interface motion. At this modelling level, the
particular physico-chemical properties of any given process under consideration are contained in the material
parameters s and G, the external potential at work m0,
and the noise term h(x,t). In the absence of further specifications, it is impossible to assign to any of them a biotic
or abiotic origin. Therefore, the use of the KPZ equation
in itself does not provide any hint about the biotic (or
abiotic) nature of the process being modelled.
3. A TEST FOR MATHEMATICAL
MODELLING: STROMATOLITE BAND
INTERFACES
In this section, we address the issue of discussing which
morphological features (if any) of stromatolites can be
efficiently modelled by means of the KPZ equation
(1.1). Following earlier studies [32– 34], we shall concentrate on the formation of laminae, and more precisely on
that of their separating interfaces, which constitute a
characteristic feature of stromatolites. To be more
specific, we will select a sample recently collected from
the 2.724 Ga Tumbiana formation in the Fortescue
Group (Australia), to be referred henceforth as the
Andalusian Hill (AH) specimen, a picture of which is
shown in figure 3.
A central point in our analysis consists of ascertaining whether the band interfaces observed in the
previous figure can be somehow associated with the
sequential profiles of a function h(x,t) which solves
equation (1.1) for a suitable choice of the parameters
n, l and D therein. This is the basic assumption made
[32,33]. For instance, in an earlier study [32] authors
assert that, under suitable assumptions on the equation
parameters ‘numerical solutions of equation (1.1) (our
notation) yield interfaces that evolve to a statistical
steady state with fractal dimension Df (our notation)
ffi 2.6 independent of the initial conditions. Assuming
that our measurement of Df ffi 2.5 is not significantly
different from the theoretical prediction of Df ffi 2.6,
J. R. Soc. Interface
Figure 3. A stromatolitic sample from 2.7 Ga Tumbiana formation (Andalusian Hill). The arrow indicates the location
of an overhang that cannot be described by the KPZ equation.
our theoretical description is consistent with our
quantitative measurements’.
We have quoted in some detail the previous excerpt
from the study of Grotzinger & Rothman [32] as it
clearly illustrates an implicit assumption often made
at the modelling stage. More precisely, it is widely
assumed that the bands actually observed in rock
samples correspond to what is called in mathematics
an asymptotic pattern. By this, we mean a surface
morphology to which any solution resembles, irrespective of its initial profile, after a sufficiently large time
has elapsed. However, before an asymptotic pattern
(assumed to exist) unfolds, solutions go through what
is called a transient period. During such a period,
solutions may display features quite different from
their asymptotic morphology. Incidentally, the transient period may be quite long and there seems to be
no general technique to a priori estimate its duration.
Actually, when looking for asymptotic patterns in
equations similar to equation (1.1), serious mathematical difficulties are encountered which reflect underlying,
deep scientific problems. In particular, and as a consequence of the presence of noise terms like h(x,t) in
these equations, no general existence theory for solutions of nonlinear stochastic partial differential
equations is available mainly for high space dimensions d (cf. [38]). This fact notwithstanding a wealth
of results, mainly numerical simulations, have been
obtained for equation (1.1) [37,39,40], a solution to
the one-dimensional case having been achieved only
very recently (see [41] and references therein). Although
results suggested by these simulations are not always
compatible among themselves, altogether they provide
significant information about processes governed by
the KPZ equation, as will be noticed below. On the
other hand, if the noise term h(x,t) in equation (1.1)
is dropped, a convenient existence theory is at hand
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Pattern formation in stromatolites
for the deterministic equation thus obtained [42,43]. Furthermore, in these papers, a complete characterization of
the solutions to this equation is presented, and the conditions under which non-uniqueness takes place are found.
Let us consider the question of describing what morphologies can be identified, for sufficiently large times,
in the interface height h(x,t) of a process governed by
equation (1.1). As it turns out, in order to obtain a meaningful answer, the previous question needs to be carefully
reformulated. First of all, different morphologies may be
obtained, each of them being an appropriate approximation over different periods of time. The latter may be
quite large, reaching up to the scale of 104 –106 years
characteristic of carbonate platform deposition. Such
different transient periods are separated by crossover
times, for which only rough estimates can be provided.
Moreover, as a consequence of the stochastic nature
of equation (1.1), the asymptotic morphologies for
which a theory exists correspond to some average interface properties, but not to the interface height itself.
Fortunately enough, these interface-related quantities
on which patterns can be shown to unfold have a clear
morphological meaning as discussed below.
Notice that the term F in equation (3.4) can be dispensed with upon replacing h by h 2 Ft. For equation
(3.4), it is known that, after some regularization [37]
W ðtÞ t 1=2
N
X
¼ 1
hðxi ; tÞ;
hðtÞ
N i¼1
@h
¼ nr2 h þ hðx; tÞ:
@t
N
1X
2
ðhðxi ; tÞ hðtÞÞ
:
N i¼1
t1C J. R. Soc. Interface
ðDxÞ2
;
n
ð3:7Þ
where Dx is the length scale corresponding to the discrete process of which equation (3.7) represents a
continuous approximation. Equation (3.6) is linear
and can be explicitly solved. It can be shown that for
such equation
W ðtÞ t 1=4
as time increases:
ð3:8Þ
Finally, when t . t1C becomes large enough, all
terms in equation (1.1) become relevant, and the full
equation (1.1) needs to be considered. Transition from
EW equation (3.6) to the KPZ equation (1.1) takes
place at a second crossover time t2C, which is such
that [36]
t2C n5 D 2 l4 :
ð3:2Þ
ð3:3Þ
Then for sufficiently small times, the corresponding solution is expected to remain almost flat as well, so that the
first and second terms on the r.h.s. of equation (1.1) are
negligible, and equation (1.1) can be effectively replaced
by the so-called random deposition (RD) equation
@h
¼ hðx; tÞ þ F:
@t
ð3:6Þ
ð3:9Þ
For convenience of the reader, let us compute t2C in a
hypothetical situation, where
Therefore, W(t) represents the mean quadratic deviation of the interface height around its average value, a
measure of the roughness of the interface itself. Suppose
now that at t ¼ 0, the interface is flat so that
hðx; 0Þ ¼ 0:
ð3:5Þ
The transition from the range of validity of equation
(3.4) to that of equation (3.6) occurs at a first crossover
time, t1C that can be estimated as follows
ð3:1Þ
and the interface width W(t) is defined as follows:
W ðtÞ2 ¼
as t increases;
where W(t) f (t) for suitable times t is to be understood as follows: there exist positive constants C1, C2
such that C1f (t) W(t) C2f (t) in the range of times
under consideration.
As time passes, solutions to equations (1.1) and (3.3)
begin to develop non-flat profiles, so that the terms of
equation (1.1) which were discarded to obtain equation
(3.4) become relevant. In view of the assumptions made
in electronic supplementary material, appendix A to
derive equation (1.1), the new term to be retained is the
first on the r.h.s. of equation (1.1). One thus obtains
the so-called Edwards–Wilkinson (EW) equation
3.1. Evolution of the interface width in Kardar –
Parisi– Zhang equation: crossover times
Consider first the simpler idealized case d ¼ 1, in which
the interface position depends on a single spatial coordinate, and on time. Then solutions are known to develop
different features as time passes, as recalled next.
Bearing in mind the morphologies actually observed in
stromatolite samples, we shall focus herein on a quantity
associated with solutions of equation (1.1), the interface
width W(t), which is defined as follows. Consider a
medium of length L . 0, divided into N equal intervals
Ii (1 i N), each having a size Dx. Let h(xi,t) be the
average value of h(x,t) on each interval Ii. Then, the
is given by
average value of h(x,t) at time t, hðtÞ
R. Cuerno et al. 5
ð3:4Þ
l 10 cm yr1
n 106 cm2 s1 ;
and
D 103 cm3 yr1 ;
then equation (3.9) yields t2C 1023 years, that is, a
few hours. This fact is consistent with the assumptions
made that correspond to a fast-growing process. On the
other hand, for the same value of n as before, taking
l 1 cm yr1
and D 1 cm3 yr1 ;
gives t2C 107 years. Note the strong dependence of
formula (3.19) on the values of the parameters involved.
Once the full equation (1.1) has been developed, the
interface width W(t) continues to exhibit a power-like
growth first.
W ðtÞ t 1=3 ;
ð3:10Þ
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6 Pattern formation in stromatolites
R. Cuerno et al.
2.5
ts
log W (arb. units)
2.0
1.5
t2C
1.0
t1C
0.5
0
1
2
3
4
5
log t (arb. units)
6
7
8
Figure 4. Time evolution of the surface width for the KPZ equation in dimension d ¼ 1. The slopes of the straight lines are 1/2,
1/4, 1/3, left to right. Crossover times t1C, t2C and saturation time ts are indicated by arrows.
to eventually saturate to an L-dependent value
W La ; L1=2 :
50
ð3:11Þ
40
30
This happens for times t . ts, where ts is the
saturation time [37]
ts Lz ; L3=2 :
10
ð3:12Þ
The set of parameters a, z and b ¼ a/z (whose
meaning will be explained in electronic supplementary
material, appendix B) are called the critical exponents
for the KPZ equation in one space dimension. They
are also said to characterize the dynamical scaling
corresponding to that equation. The dependence of
W(t) on the subsequent crossover times is sketched
in figure 4.
To conclude this paragraph, we remark that results
similar to those previously described continue to hold
when, instead of equation (3.3), we consider initial data
that are close enough to a flat profile. For instance, this
is the case when we take h(x,0) in the form of a small oscillation with small amplitude, a situation appropriate as a
starting point for stromatolite formation.
3.2. Interface patterns in different transient
regions
We next provide some simulations to illustrate the morphologies that can be generated by means of KPZ-type
equations. Consider first the earliest stages of a process
governed by equation (1.1) with h(x,0) being a smallamplitude sinusoidal function. As observed before,
that transient is governed by an RD equation (3.4). A
plot of the unfolding profiles is given in figure 5.
Concerning our previous figure, some remarks are in
order. First, some memory of the initial condition is preserved during the times considered, although the initial
profile is increasingly distorted owing to stochastic
J. R. Soc. Interface
h 20
0
–10
1 3 5 7 9 11 13 15 17 19 21 23
x
Figure 5. Simulated profiles at different times for solutions
of the RD equation for which the initial condition is a
small-amplitude sinusoidal profile. Here F ¼ 10 and D ¼ 1.
Snapshots correspond to times: dark blue, t ¼ 0; red, t ¼ 1;
green, t ¼ 2; purple, t ¼ 3; light blue, t ¼ 4.
effects. At the technical level, what is shown in
figure 5 is a (explicit) realization of the stochastic
process that can be shown to provide a solution to the
problem under consideration. Such realization can be
shown to be discontinuous almost everywhere, although
the plots actually given above have been made continuous for ease of visualization. Namely, in the numerical
computation, we have discretized the equation in space,
and have solved it on a finite number of nodes. Then,
we have plotted the values of the solution in the different
nodes, linking the values of first neighbours with
straight lines.
We next consider the full KPZ equation (1.1) with
same initial value as before, and simulate the solution
profiles in each of the subsequent regimes (RD, EW and
KPZ). Without loss of generality, we may set F ¼ 0, as
this amounts to replacing h(x,t) by (h(x,t) 2 Ft) there.
The corresponding results are illustrated in figure 6.
Figure 6a displays the evolution starting out from an
initial sinusoidal shape of small amplitude, while
figure 6b shows snapshots of the dynamics corresponding
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Pattern formation in stromatolites
(a)
R. Cuerno et al. 7
(a) 1.5
30
1.0
h(x)
h(x)
20
0.5
10
0
–0.5
(b)
(b)
1.0
h(x)
h(x)
40
0.5
20
0
0
0
100
300
400
500
x
Figure 6. (a) Simulation of the KPZ equation with sinusoidal
initial data corresponding to the RD transient (t ¼ 0.01,
t ¼ 0.1), EW transient (t ¼ 1) and the full KPZ stage (t ¼
100, 1000). Parameter values: l ¼ 3, g ¼ 1, D ¼ 1. (b) Simulation for a different initial sinusoidal data, whose amplitude
has been taken of the same order as that of the profile for
the full KPZ regime. Parameter values are as in (a): black,
t ¼ 0; red, t ¼ 0.01; green, t ¼ 0.1; blue, t ¼ 1; brown, t ¼
100; purple, t ¼ 1000.
to an initial sinusoidal shape with the same wavelength,
but with a larger amplitude that is comparable with
the roughness of the stationary state. Comparing the
two panels, we see that the long time morphologies
(e.g. t ¼ 100, 1000 in the plots) are disordered and
rough, independent of the initial condition, while there
are intermediate times (e.g. t ¼ 1) at which the periodicity of the initial condition may still exist if its
amplitude was large enough (figure 6b).
We now consider the case of the deterministic
equation obtained when the term h(x,t) in equation
(1.1) is dropped and the constant forcing term is
rescaled upon the transformation h ! (h 2 Ft).
@h
l
¼ nr2 h þ jrhj2 :
2
@t
ð3:13Þ
We show in figure 7, how an initially sinusoidal
profile evolves as time passes, and consider also the situation where the initial value itself is of a stochastic
nature. Notice that in both cases considered in
figure 7, the corresponding solution approaches towards
a nearly constant profile as time passes.
To conclude this paragraph, we next consider a particular case of the deterministic KPZ equation (3.12),
which has been used [33,34] to support the biotic
J. R. Soc. Interface
0
200
100
200
300
400
500
x
Figure 7. (a) Solution profiles for the deterministic KPZ
equation when l ¼ 10, n ¼ 1. (b) Profiles corresponding to
the same equation and same values of l and n in the case
where h(x,0) is an uncorrelated white noise of amplitude
D ¼ 0.01 (black, t ¼ 0; pink, t ¼ 11.7).
8
6
h
4
2
5
10
x
15
20
Figure 8. Evolution of a sinusoidal profile under equation (20).
Notice the existence of an initial transient period, where the
solution remains as smooth as its initial value (snapshots 1
and 2) after which cusps begin to unfold at the local
maxima of the solution (snapshots 3 and 4). The profile will
eventually converge to zero everywhere for a sufficiently
long time. The initial profile is h(x,0) ¼ cos( px/5), F ¼ 2
and n ¼ 1/2. Snapshots correspond to times t ¼ 0, 1, 2, 3, 4.
origin of a class of stromatolites. More precisely, we
shall deal with the following equation
@h
¼ njrhj2 þ F;
@t
ð3:14Þ
where F . 0 is constant, and h(x,0) is a prescribed sinusoid. Figure 8 provides information on the
behaviour in time of the corresponding solutions.
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8 Pattern formation in stromatolites
R. Cuerno et al.
It is interesting to compare figure 8 with figure 1 in
the study of Batchelor et al. [33]. In both cases, one
makes use of explicit formulae to represent the solutions. However, the authors in [33] are able to derive
their plots through formulae (2.1) and (2.2) therein
because they consider initial values with a single maximum. In our case, the initial value has several maxima,
so that characteristics originating from them may intersect in time, which eventually results in loss of
regularity and cusp formation. Therefore, the coexistence of several adjacent paraboloid-like structures as
those represented in figure 2 in the study of Batchelor
et al. [34] can be maintained only for sufficiently small
times (see electronic supplementary material, appendix
B for further details).
3.3. Is Kardar –Parisi– Zhang a suitable model of
stromatolite growth?
Consider again the AH specimen in figure 3. One readily
sees there a system of three undulations that run
from bottom to top, the one on the left eventually
undergoing a tip-splitting process. It is not a priori
clear whether a precise wavelength can be associated
with such undulations, as their number is too small to
perform a reliable statistical analysis. This could be
done, however, if larger samples displaying about 10–
20 of such shapes were available. In any case, if such
a wavelength could be identified, this would exclude
that band formation had occurred at times corresponding to the last asymptotic regime for the KPZ
equation. The reason is that the large-time behaviour
of solutions to equation (1.1) is known to be incompatible with the persistence of a characteristic length [37],
as illustrated in figure 6a. However, characteristic
lengths can be preserved for comparatively long times,
as seen in figure 6b. For instance, they are compatible
with the first time regime (t , ts in equation (3.12)),
which in practice may last for a long while. Note for
instance that, as observed before, ts . t2C, and the
latter quantity might be of the order of 10 Myr under
assumptions on l and D as those discussed in our
previous paragraph.
Another point to be noticed is that neither overhangs (as that visible on the right of the first
undulation at the bottom left in figure 3) nor increasingly steep interface profiles (the rightmost system in
the same figure) are compatible with the asymptotic
KPZ regime [37].
A final remark in this paragraph is concerned with
the effect of noise on the KPZ equation. This is
known to be crucial for non-zero patterns to evolve
from an initially flat interface h(x,0) ¼ 0. Indeed, if we
drop the noise term h(x,t) in equation (1.1), then the
only solution in the whole space of the resulting equation corresponding to an initial value h(x,0) ¼ 0 is
h(x,t) ¼ 0. Moreover, in that case, solutions corresponding to data which are not too large (see electronic
supplementary material, appendix B for details) are
known to approach flat profiles as time goes to infinity.
Therefore, the patterns depicted earlier [33,34] either
develop from sufficiently large initial values, or will
eventually be damped out as time passes.
J. R. Soc. Interface
3.4. More on kinetic roughening: beyond the
Kardar – Parisi –Zhang equation
We have already illustrated some of the problems that
arise when using equation (1.1) to model stromatolite
morphologies. We should add in this respect that, to
this day, there are very few experiments on rough interfaces where the scaling behaviour corresponding to the
KPZ equation had been unambiguously observed [44].
While the reasons for that discrepancy remain still
open for discussion, partial explanations thereof can
be provided. Some of them are related to the manner
in which equation (1.1) is obtained. For instance, the
general arguments on symmetry assumptions to be
recalled in electronic supplementary material, appendix
A are fully meaningful at the largest time and space
scales only. For smaller scales, however, mechanisms
that are not compatible with the KPZ formalism (arising for instance from specific microscopic details) may
play a significant role. This may result in the onset
of a KPZ scaling being delayed, and even completely
hindered for practical purposes. To illustrate this
point, a specific example is shortly discussed below.
Consider a growth problem in which particles diffuse
within a vapour phase until they reach a solid substrate
onto which they attach through chemical reactions,
forming an aggregate with a rough surface. If the chemical reactions resulting in that attachment are not too
fast, it has been shown that the aggregate interface
evolves according to the noisy Kuramoto– Shivashinsky
equation [45]
@h
l
¼ nr2 h K r4 h þ jrhj2 þ h:
2
@t
ð3:15Þ
Here n, K and l are positive constants and h is as in
equation (1.1).
As in the KPZ case, equation (3.15) is symmetric
under reflection of the space coordinates, x ! 2x, while
not having up-down symmetry (h ! 2h) either. Based
on standard arguments on universality [37], this might
lead to expect a similar dynamical behaviour as described
by both equations. However, the sign of the diffusion
term in equation (3.15) is the opposite to that in the
KPZ equation (1.1). This introduces important differences in the behaviour of solutions to both equations,
particularly at small time scales. For instance, solutions
to equation (3.15) may develop a periodic pattern of
cells for times t . 0. This morphological instability is
certainly compatible with the presence of undulatory
patterns as those already discussed for the (AH) specimen. In the case of equation (3.15), such a periodic
pattern subsequently evolves in a chaotic manner to
eventually yield, for a suitable parameter range, a full
KPZ scaling for sufficiently large times [45,46]. Thus,
KPZ scaling can be viewed as an emergent property of
equation (3.15), although that equation itself is strongly
different from equation (1.1). Moreover, for some (finite)
system sizes, and some parameter choices in equation
(3.15), the asymptotic KPZ scaling may not be observable at all in equation (3.15) [44]. Incidentally, this
seems to be the case in many experimental systems
where diffusive instabilities seem to be a norm rather
than an exception [44].
Downloaded from http://rsif.royalsocietypublishing.org/ on June 16, 2017
Pattern formation in stromatolites
R. Cuerno et al. 9
and in the small gradient limit (jr hj 1), @ reduces to
"
2 2 #
@2h @2h
@ h
@
;
ð3:18Þ
@x 2 @y 2
@x@y
2.0
1.5
h
1.0
0.5
0.2
0.4
0.6
0.8
1.0
which is precisely the drift in equation (3.16). Thus, the
dynamics of equation (3.16) favours the growth of patterns with positive Gaussian curvature (as is the case of
holes and mounds) with respect to those with negative
Gaussian curvature (as saddles).
s
Figure 9. One-dimensional cut of the two-dimensional
numerical solution of equation (3.16) shown along the
24
(red line). The variable
x ¼ y-axis
at time
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi t ¼ 3.05 10
s ¼ x 2 þ y 2 = 2 parametrizes this axis. The values of the
parameters are l ¼ 1 and n ¼ 0.1. We have assumed periodic
boundary conditions and the symmetric initial condition h0 ¼
sin(2px)sin(2py) (green line). One can clearly see how the
mass is displaced from the saddle points to the top of the
hills as a consequence of the nonlinear instability mechanism.
As a matter of fact, morphological instabilities may
be driven by mechanisms of nature other than diffusive,
and noise is not required to sustain them. A simple
example is provided by the nonlinear equation
"
2 2 #
@h
@2 h @2 h
@ h
K r4 h;
¼n
@t
@x 2 @y 2
@x@y
ð3:16Þ
which has been recently studied in the context of
non-equilibrium growth [47,48]. Within the theory of
surface growth, the second, linear term on the r.h.s.
of equation (3.16) is usually associated with surface diffusion: it models how newly deposited particles diffuse
along the growing surface. The nonlinear term in
equation (3.16) is, as we explain in the following, tightly
related to the surface Gaussian curvature (which is an
intrinsic measure of the curvature of the surface).
Furthermore, it is one of the simplest nonlinearities
compatible with the symmetry requirements detailed
in electronic supplementary material, appendix A1.
Figure 9 presents a one-dimensional cut along the x ¼
y-axis of a two-dimensional numerical simulation of
equation (3.16). What this figure shows is mass displacement from saddle points to the top of the hills by means of
a nonlinear instability mechanism. A complementary
effect, displacement of mass from saddle points to the
bottom of the valleys, can be observed along the perpendicular axis. This sort of behaviour requires the action of
a nonlinearity. Indeed, a linear instability would promote
the growth of both holes and mounds. That phenomenology can be explained by means of a simple geometric
argument as follows. The surface Gaussian curvature @
is given by
"
2 2 #
@2h @2h
@ h
2
@¼
2
@x @y
@x@y
"
2 2 #1
@h
@h
þ
;
1þ
@x
@y
J. R. Soc. Interface
ð3:17Þ
4. DISCUSSION
In this work, we have addressed some aspects of pattern
formation in stromatolites. In particular, we have been
concerned with the question of whether the interfaces
appearing on such structures can be modelled by
means of the KPZ equation (1.1). This issue has been
raised as a tool to prove (or disprove) the biogenicity
of such structures, and consequently on the way in
which life was started in our planet.
We have focused into a particular aspect of the modelling question described before. More precisely, an
effort has been made to clarify the following points:
(i) what does any term in the KPZ equation represents
in terms of physical mechanisms involved? (ii) What
conclusions can be drawn from comparison of the properties of solutions to the KPZ equation with the profiles
( particularly band interfaces) observed in actual
stromatolite samples, and in particular in the AH specimen depicted in figure 3. A pressing motivation to
ascertain (i) and (ii) is given by the fact that [32,33]
different versions of the KPZ equation have been used
to sustain seemingly contradicting views, namely a
possibly abiotic (in the first case) and biotic (in the
second one) origin of stromatolites. We now summarize
our results, and put forward some research directions
that can be pursued to understand pattern formation
in such structures.
To begin with, we have observed that KPZ is a stochastic partial differential equation. This simple model
has been largely used to describe the evolution of an interface separating a medium which advances into another.
Three important aspects to retain such formulation are
that (i) KPZ is a continuous equation, therefore obtained
by means of an approximation process to a discrete
phenomenon, (ii) noise effects are crucial, and (iii) at
any given time, the interface is not considered to be in
equilibrium: by assumption, there is an external force
that drives the growth problem.
Interestingly enough, and in spite of the generality of
the arguments leading to KPZ derivation (some of
which are recalled in electronic supplementary material,
appendix A), a first obstacle to be reckoned with is its
scarce experimental relevance. Indeed, very few processes have been clearly shown to be governed by the
KPZ equation. This admittedly inconvenient aspect
has been linked to the very generality of the modelling
assumptions leading to equation (1.1). For instance, it
has been suggested [44] that in any particular case
considered, the microscopic aspects of the underlying
dynamics could have a strong impact on the dynamics
of that process, and that those might have been ignored
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10
Pattern formation in stromatolites
R. Cuerno et al.
(at least in part) when deriving the continuous equation
(1.1). We should bear in mind that any modelling
equation as that of equation (1.1) is arrived at under
a number of simplifying assumptions, some of which
may be quite heavy (or utterly unacceptable) for our
current problem.
While the previous remark has been specifically tailored to the KPZ equation, the following one is of a
general nature. When using any particular equation
for modelling purposes, one has to pay particular attention to the physical meaning of the various terms and
coefficients retained. These encode information about
the mechanisms that are assumed to be mainly, if not
solely, responsible for the process we intend to model.
However, rarely (if ever) do such terms and coefficients
provide information about the biotic or abiotic origin of
the forces that set such terms in action.
Consider for instance, the first term on the r.h.s. in
equation (1.1). As remarked in §2, it can be interpreted
as a diffusion term, and the corresponding coefficient
has dimensions of a diffusion coefficient. One of the
many possible ways in that such a term can be obtained
is represented by formula (2.4), which provides a link
between diffusivity and two material properties of the
moving interface (interfacial tension and effective mobility). However, there is no way to clarify at that stage
whether such physical mechanisms are the result of a
biotic or abiotic process. What is more, no clear link
between the underlying microscopic mechanisms and
the averaged, effective coefficients appearing in
equation (2.4) has yet been established. Clearly,
equation (2.4) provides some insight into the nature
of the laws at work at the interface. The knowledge
thus obtained is, however, too scarce to draw any conclusion, not only on the biogenicity question, but also
on the nature of the chemistry leading to interface formation. For this reason, we are left in almost complete
ignorance of the physical forces that shaped the interface and determined its motion. To gain insight about
such issues, additional information is needed. For
instance, a direction that we consider as promising is
that of deriving the effective coefficients appearing in
equation (2.4) from the actual grain properties
(volume fraction, composition, mechanisms of formation, and so on) of the structures that are present
in an actual interface (that is, in the boundary of a
band) of a living young stromatolite sample. This
requires going beyond the representation of the interface as a comparatively thin line, to take instead into
account its internal structure. Any significant result in
that direction will be instrumental to improve our
understanding on the way these bands were formed.
Concerning the remaining terms in equation (1.1),
the second one in its r.h.s. has a particularly precise
meaning. It postulates that interface velocity is nearly
constant, a fact that might be perhaps compared with
geological estimates (assumed available) on band formation. Finally, the presence of noise is crucial to
produce non-trivial patterns out of nearly flat initial
profiles. This makes a strong case for stochastic
equations versus deterministic ones as models to
describe stratified structures arising out of gentle sand
ripples. The latter is widely assumed to be a starting
J. R. Soc. Interface
point for currently forming stromatolites in shallow
water environments.
A point to be stressed is that the KPZ equation can
be shown to be incompatible with some particular interface morphologies. We have remarked on this issue in
§3. In particular, we have noticed that overhangs and
steep profiles are incompatible with solutions to KPZ.
This is not a reason to exclude altogether KPZ from
that scenario, but certainly hints at least at the presence of additional mechanisms at work. We have also
noticed that, in our view, wave-like structures as that
apparent in the AH specimen deserve further study.
Indeed, if such undulations were common (a fact that
the size of AH sample does not allow to conclude) the
corresponding wavelength could possibly be identified.
In that case, KPZ should be discarded as a guide to
describe band formation in comparatively long times.
Other equations, which take into account different driving mechanisms should then be considered, a fact that
has been briefly discussed in §3.4.
A highly relevant question is often ignored at the
modelling stage: it is widely assumed that, out of a
manifold variety of patterns that a given equation can
sustain, only a few of them will be preserved in the
long run. In some cases, such robust, observable
patterns unfold very quickly. However, for a given
equation, different time regimes may occur, separated
by suitable crossover times, and the behaviour of solutions might differ sharply from one region to
another. We have addressed that issue in §3.1 where
we have recalled that any of such time regions may be
quite long, and that a given pattern might have been
formed, and then stabilized by some additional mechanisms changing the sedimentary and/or precipitation
conditions, such as stronger sedimentation owing to
local storms or transgressive conditions, changes of pH
owing to hydrothermal activity, or fast evaporation
inducing evaporitic precipitation. In this way, the
observed pattern may look incompatible with the behaviour which eventually sets in for very long times.
The conclusion that emerges is that a given process
(for instance, band formation in stromatolites) might
have occurred according to a, say, KPZ mechanism,
and still display morphologies that are incompatible
with the behaviour of KPZ solutions in the long run
(as time goes to infinity). This in turn raises another
research direction: identifying the crossover times of a
given equation in terms of measurable quantities. In
our case, a possible strategy could be to relate such
crossover times to the grain properties mentioned
before. Incidentally, a remark made earlier [32] (see
fourth line after formula (2.2) therein) may be related
to the consideration of any of such crossover times,
although the precise form of their assumption may
not agree with estimates (3.7), (3.9) and (3.11) in §3.
Summing up, we believe that linking interface evolution with internal interface structure, searching for
wave-like regularities in larger samples than those
examined so far, and estimating the actual size of crossover times (as well as relating such times with material
properties of stromatolites) are particular issues whose
study might be helpful to obtain suitable interface
lamina formation models for stromatolite samples.
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Pattern formation in stromatolites R. Cuerno et al.
Gaining insight into such issues, as well as on the general question of lithification and on how accretion
proceeds [17], will improve our current knowledge of
pattern formation in such geological structures.
We thank Martin van Kranendok and Malcolm Walter for
their critical reading of the manuscript. We would also like
to acknowledge the two anonymous referees for their
detailed and insightful remarks on our original manuscript.
This work has been supported in part by MICINN grant
nos FIS2009-12964-C05-01 (R.C.), CGL2010-12099-E
(J.M.G.R.) and MTM2008-01867 (M.A.H.).
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