Physica A 282 (2000) 123–136www.elsevier.com/locate/physa

Deterministic KPZ model for stromatolite laminaeM.T. Batchelora , R.V. Burneb , B.I. Henryc;∗, S.D. Wattc

aDepartment of Mathematics, School of Mathematical Sciences, Australian National University,Canberra ACT 0200, Australia

bDepartment of Geology, Australian National University, Canberra ACT 0200, AustraliacDepartment of Applied Mathematics, School of Mathematics, University of New South Wales,

Sydney NSW 2052, Australia

Received 24 January 2000

Abstract

The deterministic variant of the Kardar–Parisi–Zhang equation for the evolution of a growinginterface is used to model patterning produced by successive laminations in certain stromatolites.Algebraic solutions of the model together with numerical simulations are employed to �t modelparameters consistent with digital recordings of individual stromatolite laminae. Numerical valuesfor model parameters, related to lateral growth, vertical growth and surface di�usion, provide aset of indices which may prove useful for classifying di�erent stromatolites. c© 2000 ElsevierScience B.V. All rights reserved.

PACS: 05.40.+j; 68.70.+w

Keywords: Stromatolite laminae; Kardar–Parisi–Zhang equation; Non-equilibrium growth;Morphometry

1. Introduction

One of the classic models for the evolution of the pro�le of a growing interface isthe stochastic Kardar–Parisi–Zhang (KPZ) equation [1],

@h(x; t)@t

= �@2h(x; t)@x2

+�2

(@h(x; t)@x

)2+ �(x; t) : (1)

In this equation, h(x; t) represents the height of the pro�le above a horizontal base-line, �(x; t) represents uncorrelated random noise, the term involving the parameter� represents di�usive surface relaxations, and the term involving the parameter �represents lateral surface growth in a direction normal to the interface. The stochastic

∗ Corresponding author. Tel.: +61-2-9385-7044; fax: +61-2-9385-7123.E-mail address: b.henry@unsw.edu.au (B.I. Henry)

0378-4371/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(00)00077 -7

124 M.T. Batchelor et al. / Physica A 282 (2000) 123–136

KPZ model has been applied to numerous growth problems [2] including, recently, thegrowth of stromatolites [3] where constant upwards vertical growth was also included,without altering the form of the governing equation in a co-moving frame.Stromatolites are laminated structures produced as a result of the environmental

interactions of benthic microbial communities. Processes, such as trapping and bindingof sediment grains by the mat-like microbial community, or nucleation of mineraldeposition on the surfaces of microbes, lead to the accretion of rigid structures thatcan rival coral reefs in scale. Stromatolites are the only macroscopic evidence of lifeprior to the evolution of macroscopic plants and animals [4]. Since original microbesare very rarely preserved within fossil stromatolites, there has been extensive discussionof the criteria that might be useful for distinguishing organically produced stromatolitesfrom similar accretions produced by inorganic means [5,6]. Recently, Grotzinger andRothman [3] have used the stochastic KPZ equation in an inconclusive attempt toaddress this question.A characteristic property of pro�les simulated by the stochastic KPZ equation is that

the pro�le height h(x; t) is a self-a�ne fractal described by a surface roughness expo-nent � = 1

2 and a growth exponent � =13 . Grotzinger and Rothman [3] measured the

surface roughness exponent of stromatolite growth laminae in peak-shaped stromatolitesfrom the Cowles Lake Formation reef complex in Canada and found the same rough-ness exponent � ≈ 1

2 . Since other growth models, for example the Edwards–Wilkinsonequation [7] (which is equivalent to the KPZ model in the absence of surface nor-mal growth, i.e., � = 0) generate the same roughness exponent this work does notunambiguously validate the KPZ equation for modelling stromatolite laminae.In this paper we have investigated a deterministic version of the KPZ equation for

modelling the evolution of smooth stromatolite laminae (with the surface roughnessexponent equal to zero). The evolution of the pro�le height in the deterministic KPZmodel for stromatolite growth is governed by

@h(x; t)@t

= �@2h(x; t)@x2

+ �

(1 +

12

(@h(x; t)@x

)2)+ v : (2)

This di�ers from the standard form of the KPZ equation, Eq. (1), which has beentransformed to a co-moving frame. The constant velocity term in Eq. (2) consists oftwo components; one arising from lateral growth (� 6= 0) and the other from verticalgrowth (v 6= 0).The general solution of the deterministic model, Eq. (2), can be readily obtained by

�rst employing the transformation w(x; t)=exp[(�=2�)h(x; t)] and then using separationof variables. The result is

h(x; t) =2��log{∫ +∞

−∞

dy√4��t

exp[− (x − y)

2

4�t+�2�h0(y)

]}+ (v+ �)t ; (3)

where h0(x) is the initial pro�le height (base of the stromatolite).Other algebraic solutions of the deterministic KPZ model can readily be found

by taking the spatial derivative of the equation to yield Burger’s equation [8] for

M.T. Batchelor et al. / Physica A 282 (2000) 123–136 125

u= @h=@x. In the next two sections we describe some algebraic solutions of the deter-ministic model, �rst without surface tension (Section 2) and then with surface tension(Section 3), and we relate such solutions to patterning formed by successive lamina-tions in stromatolites. A description of the long-term solution, composed of parabolicsegments separated by shocks (discontinuities in @h=@x), is provided from this analysis.Estimates for the relative sizes of the parameters corresponding to the di�erent growthmechanisms described by the model have been made in particular cases by comparingdigital tracings of actual stromatolite laminae with these algebraic solutions and withcomputer simulations of the deterministic KPZ model. A summary and discussion isgiven in Section 4.

2. Laminations without di�usive surface relaxation

If there are no di�usive surface relaxations then solutions can be found to the initialvalue problem

@h(x; t)@t

=�2

(@h(x; t)@x

)2+ v+ � ; (4)

h(x; 0) = h0(x) ; (5)

by using the method of characteristics. We use this method in the following to explorethe evolution of initial pro�les consisting of: (i) a linear segment joining another lin-ear segment; (ii) a linear segment joining a parabolic segment; and (iii) a parabolicsegment joining another parabolic segment. The solutions reveal parabolic smooth-ing where segments initially join to form a convex protrusion and shock propagationfrom regions that initially join to form a concave depression. The asymptotic form ofan initial piecewise linear pro�le thus consists of parabolic segments separated byshocks.From the spatial derivative of the initial value problem, Eqs. (4) and (5), we �rst

construct the new initial value problem

@u@t

− �u@u@x= 0 ; (6)

u(x; 0) = u0(x) ; (7)

where

u(x; t) =@h@x: (8)

Eqs. (6) and (7), de�ne the initial value problem for a �rst-order quasi-linear partialdi�erential equation which can be readily solved, using the method of characteristics[9]. With this approach the solution is given implicitly by

u(x; t) = u0(x + �u(x; t)t) : (9)

126 M.T. Batchelor et al. / Physica A 282 (2000) 123–136

The explicit solution for u as a function of x and t cannot be obtained in general (forarbitrary initial pro�les) from Eq. (9), however the solution at time t can be alwaysconstructed for arbitrary initial pro�les from a characteristic diagram showing lines ofconstant u in the x–t plane. The solution for u at time t is readily found from theselines (characteristics) by following them back to t = 0 where the values of u= u0 areknown. To obtain the equation for the characteristics we note that Eq. (6) represents anequation for a line of constant u with du=dt=0 if dx=dt=−�u. Hence, the characteristicsare given by

x =−�u0t + x0 : (10)

2.1. Linear–linear initial pro�le height

Consider an initial pro�le height

h(x0; 0) =

{h0 − �|x0|; x060

h0 + �|x0|; x0¿0(11)

with the corresponding initial condition

u(x0; 0) =

{�; x0¡ 0 ;

�; x0¿ 0 :(12)

The characteristics are

x(t) =

{−��t − |x0|; x0¡ 0 ;

−��t + |x0|; x0¿ 0 :(13)

Characteristics starting equi-distant from x=0 intersect with |x0|= �(�− �)t if �¡�,but do not intersect if �¿�. We examine these two distinct cases separately:Case (i): �¿�: In this case the two segments join to form a convex protrusion and

from the characteristics, Eq. (13), and the initial conditions, Eq. (12), we deduce thesolution at time t,

u(x; t) =

�; x¡− ��t ;− x�t ; −��t ¡x¡− ��t ;

�; x¿− ��t :(14)

To �nd the pro�le height at the later time t we integrate with respect to x and determinethe spatial constants of integration (functions of time) from the requirement that h(x; t)is a solution of the initial value problem, Eqs. (4) and (5). Thus,

h(x; t) =

�x + �

2�2t + (�+ v)t + h0; x6− ��t ;

− x2

2�t + (�+ v)t + h0; −��t6x6− ��t ;�x + �

2�2t + (�+ v)t + h0; x¿− ��t :

(15)

M.T. Batchelor et al. / Physica A 282 (2000) 123–136 127

Case (ii): �¡�. In this case the two segments join to form a concave depression andthe characteristics from Eq. (10) cross over at x∗(t) = −�(� + �)t=2. This cross-overrepresents the space–time trajectory of a shock wave from the propagation of thediscontinuity in u(x; t). From the characteristics, Eq. (13), and the initial conditions,Eq. (12), we now have

u(x; t) =

{�; x¡vt ;

�; x¿vt ;(16)

where v is the velocity of the shock,

v=−�(�+ �)2

: (17)

After integration with respect to x we obtain

h(x; t) =

{�x + �

2�2t + (�+ v)t + h0; x6vt ;

�x + �2�2t + (�+ v)t + h0; x¿vt :

(18)

Note that the solution in this case can be constructed from the two outer segmentsin the previous case with the cross over value x∗(t) found by matching the twosegments.

2.2. Parabolic–linear initial pro�le height

Consider an initial pro�le height

h(x0; 0) =

{h0 + �a2 − �(−|x0|+ a)2; x060 ;

h0 + �|x0|; x0¿0 ;(19)

where �; h0; a are positive parameters. The corresponding initial condition for u is

u(x0; 0) =

{−2�(−|x0|+ a); x0¡ 0 ;

�; x0¿ 0(20)

and the characteristics are

x(t) =

{2��(−|x0|+ a)t − |x0|; x0¡ 0 ;

−��t + |x0|; x0¿ 0 :(21)

Characteristics starting equi-distant from x = 0 intersect with

|x0|= �t2(2�a+ �1 + ��t

)(22)

if �¿ − 2�a, but do not intersect if �¡ − 2�a. Again we examine the two distinctcases separately.

128 M.T. Batchelor et al. / Physica A 282 (2000) 123–136

Fig. 1. Characteristic diagram for the initial conditions, Eq. (20), corresponding to a parabolic segmentjoining a linear segment at a convex protrusion.

Case (i): �¡−2�a. In this case the two segments join to form a convex protrusion.A representative characteristic diagram is shown in Fig. 1 from which we deduce

u(x; t) =

−2� ( x+a1+2��t

); x¡ 2��at ;

− x�t ; 2��at ¡x¡− ��t ;

�; x¿− ��t(23)

and hence

h(x; t) =

−�((x+a)2

1+2��t

)+ �a2 + h0 + (�+ v)t; x62��at ;

− x2

2�t + h0 + (�+ v)t; 2��at6x6− ��t ;�x + �

2�2t + h0 + (�+ v)t; x¿− ��t :

(24)

Case (ii): �¿−2�a. In this case the two segments join to form a concave depressionand a shock travels along a path x = x∗(t). The solution for u on either side of theshock is readily found by using the characteristics in Eq. (21) to eliminate |x0| in theinitial pro�le, Eq. (20). This yields

u(x; t) =

{−2� ( x+a

1+2��t

); x¡ x∗(t) ;

�; x¿x∗(t) ;(25)

and, after integration over x,

h(x; t) =

−�

((x+a)2

1+2��t

)+ �a2 + h0 + (�+ v)t; x6x∗(t) ;

�x + �2�2t + h0 + (�+ v)t; x¿x∗(t) ;

(26)

where

x∗(t) =− 12�(2�a+ �(1 + 2��t)−

√1 + 2��t(2�a+ �)) (27)

is found by matching the two segments.

M.T. Batchelor et al. / Physica A 282 (2000) 123–136 129

2.3. Parabolic–parabolic initial pro�le height

We now consider the initial pro�le

h(x0; 0) =

{h0 + �a2 − �(−|x0|+ a)2; x060 ;

h0 + �b2 − �(|x0| − b)2; x0¿0 ;(28)

where a; b; �; �; h0 are all positive parameters. In this case using similar methods toabove, we �nd that the evolution of the pro�le is given by

h(x; t) =

−�((x+a)2

1+2��t

)+ �a2 + h0 + (�+ v)t; x6x∗(t) ;

−�((x−b)21+2��t

)+ �b2 + h0 + (�+ v)t; x¿x∗(t) ;

(29)

where

x∗(t) =�a(1 + 2��t) + �b(1 + 2��t)− (�a+ �b)√(1 + 2��t)(1 + 2��t)

−�(1 + 2��t) + �(1 + 2��t)(30)

is the path of the shock.Typical solutions from the above analysis which show parabolic smoothing and shock

propagation are shown for a range of di�erent sets of parameters and initial pro�les inFig. 2.

3. Laminations with di�usive surface relaxations

The analysis of the preceding section revealed that the long-time solution in the ab-sence of di�usive surface relaxations consists of parabolic segments separated by dis-continuities in the spatial derivative of h(x; t). This motivates our search for parabolicsolutions in the case when di�usive surface relaxations are included. Indeed suchparabolic solutions exist. It can be readily con�rmed by substitution that the full de-terministic KPZ model, Eq. (2), has the solution

h(x; t) = A+ (v+ �)t − ��log(2�t − B)− (x − x0)2

2�t − B ; (31)

where A; B; x0 are constants. Without loss of generality, in the following, we take B=0which e�ectively sets the initial time. By comparing this solution and numerical simu-lations with digital tracings of stromatolite laminae it is possible to extract parameterestimates in a KPZ model for the evolution of the stromatolite laminae. The �rst stepin this procedure is to identify successive local parabolic segments in the digitizedlaminae. The points on each segment j are then used to �nd the coe�cients aj; bj; cjin the least-squares best-�t parabola

hj(x) = ajx2 + bjx + cj

=

(cj −

b2j4aj

)− (x + bj=2aj)2

−1=aj : (32)

130 M.T. Batchelor et al. / Physica A 282 (2000) 123–136

Fig. 2. Evolution of the pro�le height from the method of characteristics applied to the deterministic KPZmodel, Eq. (2), with parameters � = 0; � = 1; v = 1 and; (a) an initial linear–linear protrusion, Eq. (11),with � = 1; � = − 1

2 ; (b) an initial parabolic–linear protrusion, Eq. (19), with � =14 ; � = −1; a = 1; (c) an

initial parabolic–linear depression, Eq. (19), with � = 14 ; � = 1; a = 1; and (d) an initial parabolic–parabolic

depression, Eq. (28), with � = 14 ; � = 1; a = 1; b = 1.

We now attempt to relate the measurements of aj; bj; cj to the growth parameters� and v. First, comparing Eq. (31) for B, with Eq. (32) yields

aj =− 12�tj

: (33)

Furthermore, we have

h(x0; tj) ≡ hj(x0) = A− ��log(2�) + (v+ �)tj − �

�log(tj) ;

from which we can derive an expression for the temporal separation tj − t0 betweenthe parabolic segments. Explicitly,

hj(x0)− h0(x0) = (v+ �)tj − ��log(tj)− (v+ �) ;

= (v+ �)tj

(1− �

�(v+ �)log(tj)tj

)− (v+ �) ;

where, for convenience, we have chosen t0 = 1. Considering the case when tj is large,the temporal separation becomes

hj(x0)− h0(x0) = (v+ �)(tj − 1)

M.T. Batchelor et al. / Physica A 282 (2000) 123–136 131

or

tj =hj(x0)− h0(x0)

(v+ �)+ 1: (34)

Combining Eqs. (33) and (34) we obtain the relation

1aj=− 2�

v+ �(hj(x0)− h0(x0))− 2� ; (35)

where

hj(x0) = cj −b2j4aj

:

Hence from the slope m, and intercept b, of the straight line of best �t in a plot of 1=ajversus (hj(x0)− h0(x0)) using data from the experimental �ts of each of the parabolicsegments we deduce

�=−b2; (36)

v=−(m2− 1) b2: (37)

The units for the values of � and v are yet to be speci�ed so that the key result istheir ratio.At this stage of the analysis, � remains a free parameter. In order to obtain estimates

of this parameter we carry out numerical simulations of the KPZ model, Eq. (2), start-ing from an initial pro�le spanning di�erent parabolic segments. In these simulationsthe values of � and v are obtained from Eqs. (36), (37) and � is tuned to providebroad overall agreement between the numerical evolution of the interface pro�le andsuccessive stromatolite laminae in the sample. Our experience (see below where wepresent a detailed analysis for two di�erent samples) is that this admits a limited rangeof acceptable values for �.

3.1. Stromatolite laminae – Marion Lake, Australia

Fig. 3 shows a photograph of smooth laminae in the cross-section of a stromatolitefrom Marion Lake, South Australia. At this locality, described in detail by Von DerBorch et al. [10], stromatolites are widespread across the oor of a former salinelake. In Fig. 4 we have shown a portion of this cross-section (inside the box in Fig. 3)with highlighted lines indicating parabolic segments that were used in the curve �ttingprocedure described above. A plot of 1=aj versus hj(x0)− h0(x0) using data from theexperimental �ts of each of these parabolic segments is shown in Fig. 5. From thestraight line of best �t in this �gure we deduce � ≈ 185:34 and v ≈ 45:65. Finallyin Fig. 6 we compare equal time snapshots of growth from numerical simulationsof the KPZ model starting with the initial pro�le from a digital tracing of the baseof the Marion Lake stromatolite with parameters � = 1; v = 45:65=185:34 ≈ 0:25 and(a) � = 1

4 , (b) � = 1, and (c) � = 4. The simulation with � = � is in best agreement

132 M.T. Batchelor et al. / Physica A 282 (2000) 123–136

Fig. 3. Marion Lake stromatolite laminae.

Fig. 4. Portion of the Marion Lake stromatolite laminae with digitized tracings used in the parabolic curve�tting.

with the stromatolite laminae shown in Fig. 3; the evolving pro�les with the smallervalue of � retain much of the roughness of the original pro�le whereas those with thelarger value of � are too smooth.

3.2. Stromatolite laminae – Ajjers Basin, Algeria

Fig. 7 shows a photograph of a cross-section of a Carboniferous stromatolite sam-ple from the collections of Bertrand-Sarfati obtained from the Ajjers Basin, Algeria[11]. Digitized hand tracings of some of the laminae in this sample are superposed asthick white lines in Fig. 7. From a curve �tting of the central parabolic segment inthe four digitized tracings, using the method described above, we deduce the values

M.T. Batchelor et al. / Physica A 282 (2000) 123–136 133

Fig. 5. A plot of 1=aj versus hj(x0)−h0(x0) using data from the experimental �ts of each of these parabolicsegments together with the straight line of best �t.

Fig. 6. Equal time snapshots of growth from the KPZ model for stromatolite growth, Eq. (2), using parametersobtained by curve �tting parabolic segments in the Marion Lake stromatolite and, (a) �= 1

4 , (b) �= 1, and(c) � = 4.

134 M.T. Batchelor et al. / Physica A 282 (2000) 123–136

Fig. 7. Ajjers Basin stromatolite laminae.

Fig. 8. Equal time snapshots of growth from the KPZ model for stromatolite growth, Eq. (2), using parametersobtained by curve �tting parabolic segments in the Ajjers Basin stromatolite with the initial pro�le as thethird digitized tracing from the bottom in Fig. 7 and with, (a) � = 1

4 , (b) � = 1, and (c) � = 4.

� ≈ 9:42 and v ≈ 2:09. Finally in Fig. 8 we compare equal time snapshots of growthfrom numerical simulations of the KPZ model, Eq. (2), starting with the initial pro-�le from one of the digital tracings of the Ajjers Basin stromatolite with parameters� = 1; v = 2:09=9:42 and (a) �= 1

4 , (b) �= 1, and (c) �= 4. The numerical evolutionof the pro�le is in good agreement with the stromatolite laminae in the case � = �except near the edges of the simulation. We attribute the poorer �t at the boundariesto our simulation being carried out on a �xed width domain with periodic boundaryconditions. Simulations on wedge shaped domains using the radial version of the KPZequation [12] may prove better for this stromatolite. Note again the e�ects of too much,Fig. 8(c), and too little, Fig. 8(a), surface di�usion.

M.T. Batchelor et al. / Physica A 282 (2000) 123–136 135

4. Summary and discussion

In this paper we explored the deterministic variant of the KPZ equation for mod-elling patterning produced by laminations in certain stromatolites. Exact solutions wereobtained from the method of characteristics for parameters consistent with no di�usivesurface relaxations. These solutions revealed parabolic smoothing where segments ofa lamination join at a convex protrusion, and shocks where segments of a laminationjoin at a concave depression. Parabolic solutions were also obtained for the more gen-eral case with di�usive surface relaxations included. We used a parabolic curve �ttingprocedure applied to digital tracings of samples of stromatolite laminae to obtain es-timates of model parameters related to outwards normal growth and upwards verticalgrowth. The range of the third parameter, related to di�usive surface relaxations, wasthen estimated from numerical simulations of the model equation.This work demonstrated that the deterministic KPZ equation provides a useful model

for some stromatolite laminae. Moreover, from �eld measurements of certain stroma-tolite laminae it is possible to make estimates for the various model parameters whichmight then be used in turn as a set of indices for classifying di�erent stromatolites.The deterministic KPZ equation does not permit the modelling of overhangs and

incipient branching that has been observed in many stromatolites. An example of suchpatterning can be seen in the upper most laminae in the Ajjers Basin stromatolite inFig. 7. In future work, we plan to extend the model to accommodate such features.This will enable us to explore the possibility of light sheltering in biotic origins ofstromatolite growth.

Acknowledgements

Janine Bertrand-Sarfati generously provided access to the stromatolite collectionsand laboratory facilities of the University of Montpellier. Alexis Moussine-Pouchkinekindly facilitated the analysis of samples. This work has been supported in part bythe Australian Research Council and by funds from the Kanagawa Foundation of theAustralian Academy of Sciences.

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