tters 256 (2007) 258–263www.elsevier.com/locate/epsl

Earth and Planetary Science Le

The dynamics of travertine dams

Ø. Hammer ⁎, D.K. Dysthe, B. Jamtveit

PGP-Physics of Geological Processes, University of Oslo, PO Box 1048 Blindern, 0316 Oslo, Norway

Received 24 April 2006; received in revised form 16 January 2007; accepted 25 January 2007

Available onlin

Editor: H. Elderfield

e 3 February 2007

Abstract

We present a simple, abstract model for travertine dam formation. The simulation uses a depth-averaged hydrodynamic code,coupled with an empirical precipitation model where surface-normal growth rate is proportional to local flow rate. The dynamics ofthe resulting formation, coarsening and migration (progradation) of crests is demonstrated and compared with field observations.The local self-enhancement of crests, due to increasing flow rate, in combination with lateral inhibition due to upstream drowning,implies a classical pattern formation system with a characteristic dam size, however this wavelength is not stable but increasing.The model also reproduces crest lobes, step bunching and other processes that contribute to the complex dynamics of travertinesystems.© 2007 Elsevier B.V. All rights reserved.

Keywords: travertine terraces; hydrodynamics; carbonate precipitation

1. Introduction

Travertine (limestone) dams are common in caves[1], springs [2–4] and rivers [5,6] worldwide, andrepresent one of the most striking examples of geo-logical pattern formation on the Earth's surface (forsimplicity, we use the term travertine rather than tufaalso for cold-water deposits, see [7]). Our geomorpho-logical terms follow [8]. The dams form over a widerange of scales, from millimeters to tens of meters(Fig. 1A, D). Their origin has been poorly understood,but most likely involves a coupling between the pre-

⁎ Corresponding author. Tel.: +47 2285 1658; fax: +47 2285 5101.E-mail addresses: ohammer@nhm.uio.no (Ø. Hammer),

d.k.dysthe@fys.uio.no (D.K. Dysthe), bjorn.jamtveit@fys.uio.no(B. Jamtveit).

0012-821X/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.epsl.2007.01.033

cipitation rate and hydrodynamics [9]. Microbialactivity may also play a role [2]. In a recent work,Goldenfeld et al. [10] developed a cellular automaton-type model for carbonate precipitation in such systems,reproducing domes and terraces. Here we present aminimal model based on shallow-water flow and anempirical positive correlation between the flow velocityand precipitation rate. The resulting self-organizingpattern formation process displays rich and unusualdynamics, consistent with field observations. Damscoarsen with time, crests fold into lobes and migratedownstream with differential rates, resulting in strikingpatterns. This model, in which topography grows ratherthan erodes in response to rapid flow, produces patternsthat are completely different from those generated byflow driven erosion. Our goal is to study a simple,inorganic model and the resulting patterns, based on

259Ø. Hammer et al. / Earth and Planetary Science Letters 256 (2007) 258–263

observed relationships between flow and precipitation.In this way, we can focus on the essentials of the patternformation system, without involving details and uncon-strained processes.

The formation of travertine deposits is drivenprimarily by degassing of carbon dioxide from carbondioxide rich water that has dissolved calcium carbonatein the subsurface. On gradual slopes, a series of dams isformed (Fig. 1A) but well-defined, large dams (asopposed to microterracettes and cascades) form lesseasily on steep slopes with fast, chaotic flow (Fig. 1B).The temporal development of travertine dams can bededuced from cross sections. Excellent sections areavailable in the quarries at Rapolano Terme, Italy [4].The cross sections show development of terraces froman initially smooth surface, downstream step migration,coarsening due to drowning by higher downstreamcrests, and downstream thinning of wedge-shaped layerson the outer crest wall (Fig. 1C). At Rapolano Terme,

Fig. 1. (A) Travertine terraces, Minerva Spring, Yellowstone National ParYellowstone. Immediate precipitation produces a proximal dome and a distalof travertine sequence at Rapolano Terme, Italy. Flow was from right to left.migrate downstream. Drowning events D and crest migration M are indicated.direction from upper right to lower left.

step coarsening eventually leads to the recurrence oflarge-scale topographic smoothness. Although thesespectacular structures clearly result from elevatedprecipitation rates at dam edges (crests) and verticalfaces, the mechanism for this differential rate is lessobvious. Current views favor one or both of the fol-lowing: (1) enhanced precipitation under higher flowvelocity because of thinning of a diffusion-limitingboundary layer [5,9,11]; (2) accelerated degassing ofCO2 due to agitation, pressure drop and shallowing[1,12]. Although these mechanisms are probably im-portant, other processes such as sticking of particles tothe crest [13] possibly enhanced by biofilms, and bal-listic deposition with shadowing effects [14] could alsocontribute. We would like to stress that we do notaddress here the detailed mechanism for differentialprecipitation in these systems. Rather, we accept theempirical relationship between flow velocity and pre-cipitation rate observed by several authors, and also

k, USA. Delayed precipitation gives a flat top. (B) Orange Mound,apron. Large dams do not form on the steep surfaces. (C) Cross sectionFrom an initially smooth surface (bottom), terraces form, coarsen and(D) Small terraces in the Troll springs, Svalbard, Norway. General flow

Fig. 2. (A–C) Days 2000, 4000 and 6500 of the simulation. 20×20 m domain. Lighted from the right. Inflow at the center of upper edge, free outflowalong lower edge. Note the formation of a mound around the inlet, and coarsening of downslope crests. (D) Detailed, oblique view, day 6000, water-filled.

260 Ø. Hammer et al. / Earth and Planetary Science Letters 256 (2007) 258–263

predicted theoretically [5,6,9,11]. Through simulationand comparison with field observations we investigatewhether such a relationship is sufficient to producedams, and, if so, how they grow and evolve in time.

2. Model

A simple computer model based on a linearrelationship between flow velocity and precipitationrate was used to simulate the pattern growth process.The simulation runs in alternate steps: (1) the steadystate water flow is calculated for the current topography;

Fig. 3. Cross sections from the distal apron at simulation time increments of 4rate at outer side of crests, causing downslope step migration, is evident. Ca

and (2) the topography is updated based on the flowvelocity controlled precipitation rate. We used the codeHydro2de [15] for depth-averaged shallow 2D flow. Therate of elevation change is given by

AzAt

¼ Dj2zþmax f juj; að Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jzð Þ2

q

where z(x,y) is the topographic elevation and u(x,y)is the flow velocity. D is a diffusion coefficient forparticle transport, and f is a precipitation scaling factor.

The termffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jzð Þ2

qconverts growth normal to the surface

00 days, illustrating coarsening. Axes in meters. Elevated precipitationses of upstream drowning of dams are also seen.

261Ø. Hammer et al. / Earth and Planetary Science Letters 256 (2007) 258–263

into vertical growth. The influence of flow on precipita-tion is limited by the parameter a, meaning that at highflow rates, precipitation is not rate limited by flowvelocity. This model resembles the KPZ equation [16]which can be interpreted as a general surface-normalgrowth model, except that growth is controlled by theflow velocity and there is no stochastic term. The growthstep is implemented using an explicit scheme on the samegrid as the hydrodynamic model, with grid size h=0.1 m

Fig. 4. (A) Power spectrum of detrended downslope transect in simulation dcorresponding to a period of roughly 8 grid points (80 cm) is significant relatreflects the non-constant amplitude of the cycle. (B) Power spectrum of detrenspacing of 0.28 cm. The dominant spectral peak is found at 0.00704 cycles pereflects a single cycle through the transect.

and time step Δt=1 day. We used D=1.0⁎10−5 m2/day,f=10−3 s/day, and a=10−3 m/day. For |u| ranging from 0to 1 m/s, this gives growth rates from 0 to 1 mm/day,comparable with natural rates (30 cm/yr has been mea-sured locally within the Mammoth Hot Springs, Yellow-stone [3]). The computational domain size is 20 by 20 m(the size of a typical terrace field), with a 10 l/s point-source inflow in the middle of one edge, and free outflowalong the opposite edge. The initial topography is a flat

ay 400. The dominant spectral peak around 0.12 cycles per grid point,ive to a p=0.01 white noise line shown. The splitting of the main peakded contour through a section at Rapolano Terme, Italy, with a sampler sample, giving a period of 39.8 cm. An additional low-frequency peak

262 Ø. Hammer et al. / Earth and Planetary Science Letters 256 (2007) 258–263

slope dropping 2 m from inflow to outflow, with smalluncorrelated initial random perturbations. Bed frictionfollows the Manning law [17] with n=0.033.

3. Results

Fig. 2 shows a typical run. Initially, a mound formsaround the inlet, grading into a distal apron. Thereduction in velocity as flow spreads laterally causesreduced precipitation rate and reduced slope away fromthe source (cf. Fig. 1B). Further downslope, the initialrandom perturbations reorganize into closely spacedsmall scale crests, starting as short, irregular ridges,parallel or oblique to strike. Their initial spacing isinfluenced by the simulation grid size (0.10 m). Waterflows along a ridge until it reaches its end, and thenresumes free downslope flow. Therefore, precipitation isfast along a ridge but also at the ends of ridges, causingthem to elongate. With time, the ridge nuclei straightenand coalesce in a general orientation along strike,resulting in a mature small scale crest. Small crestscoarsen over time, developing into larger crests andponds that flood smaller upstream ponds. On theproximal mound or chimney, dams do not form in thesimulation, because the flow velocity exceeds 1 m/s andis not rate limiting for precipitation (because of themaximum growth rate, a).

Crests migrate downslope without dissolution on theinner side of the terrace, due to the difference inprecipitation rate on the inner and outer sides whichcauses the crest to grow upwards and outwards. Thisprocess is shown clearly in cross sections (Fig. 3). Thefaster growth associated with larger crests results in alarger absolute difference in precipitation rate betweenthe inner and outer side, causing larger crests to migratefaster and overtake smaller downstream terraces. Thisprocess, contributing to coarsening, is reminiscent of‘step bunching’ in crystal growth [18]. Another effect ofcrest migration is an instability causing formation andexpansion of lobate ponds (fingering). In any smallinitial dam, crest migration will proceed in the localdownstream direction, causing outward expansion. Thegeneral slope of the terrain implies larger step size at thedownslope tip of the dam than at the sides, which canproduce a differential crest migration rate and adownslope stretching of the dam. On very steep slopes,fingering is weaker because the gradient is controlled bythe underlying slope. The steepness at the outer face oflarge terraces generally hinders further small scalefingering.

The pattern is stable with respect to perturbation. Ifan incision is made in a crest, flow is diverted away from

the rest of the edge and is concentrated to the notch.Precipitation then ceases at the dry edge, but is amplifiedin the incised channel, filling it until the horizontal rim isrestored. Conversely, placing an object on a crest willdivert flow around it, restoring the crest. Such regenera-tion of shape is likely to contribute to the regularity ofnatural travertine steps even when perturbed by break-age, deposition of large particles, and biological activity.

The morphological complexity makes it difficult todefine and measure dam size. One crude approach is tostudy the distribution of terraces along a one-dimen-sional transect in the general downslope direction. Thetransect will cut dams at random lateral positions, andonly occasionally where the size is maximal. Still, suchanalysis indicates periodicity and hence a dominant damsize. Spectral analysis (Fig. 4A) shows a dominantspectral peak corresponding to a period of about 8 gridcells (80 cm). This periodicity may partly stem fromhomogeneous coarsening of the initial grid-controlledsmall terrace size, but similar periodicity is also found innature (Fig. 4B).

4. Discussion

For better understanding of the basic geometric andhydrodynamic relationships, the model is intentionallyminimal and based on a simple empirical relationship.However, travertine growth rates clearly depend notonly on local flow velocity, but also on larger-scaledownstream development of water chemistry underdegassing and precipitation [19]. A more completesimulation should include reaction, transport anddegassing of chemical species, coupled with a carbonateprecipitation model. In addition, field observationsindicate that surface tension, which is not included inthe present model, is a critical parameter controllingflow in thin films over small terraces. Some of theseeffects are addressed by a new model by Goldenfeldet al. [10].

The flow velocity in natural systems depends in acomplex manner on the surface topology, and it is cor-related with variables such as the water depth. There-fore, the success of our model does not show that theflow velocity is the causal factor controlling precipita-tion rate, since flow velocity and precipitation rate couldboth result from water depth or some other quantity.However, we have shown that a simple model based onthe coupling of precipitation rate and flow velocity issufficient for terrace formation, and we have investigat-ed the resulting pattern formation dynamics. In addition,theoretical and experimental work [1,9,11,12] indicatesthat the flow velocity can indeed act as a direct causal

263Ø. Hammer et al. / Earth and Planetary Science Letters 256 (2007) 258–263

control on precipitation rate, at least under turbulentflow conditions. Other mechanisms may enhance thepositive feedback on step edges and the resultinginstability.

Coarsening is the fundamental dynamic feature.There is no stable characteristic wavelength, like thatfound in ice terrace models [20,21,22]. From fieldobservations, we speculate that a stable wavelength canemerge if surface tension is included in the model [21].However, at any given time the crest distance in thesimulation is relatively constant, as shown by spectralanalysis. Informally, we ascribe this to the competitionbetween two fundamental processes: local self-enhance-ment (positive feedback) of crests through the flowvelocity/precipitation rate coupling, and lateral inhibi-tion by upstream flooding. The inhibition range iscontrolled by step size, as larger steps will flood largerupstream regions. This general situation is a familiarrecipe for self-organization into regular spacing offeatures, for example in reaction–diffusion models [23].Clearly, the range of upstream inhibition of terraceformation by flooding is inversely proportional to slope,giving closer spacing of terraces on steep slopes, acommonly observed feature of travertine deposits.

The precipitation rate law used here is purely phe-nomenological, but easiest to interpret in terms of abioticprocesses. Clearly, biological activity can influence pre-cipitation, e.g. by photosynthetic uptake of CO2 or byforming suitable mechanical substrates [2]. However,field observations clearly indicate that biological activityand the formation of porous travertines with high frac-tions of organic material is more intense inside dams thanon crests. Biological activity may thus be one of the mainfactors controlling water depth inside dams.

Acknowledgements

Hans Amundsen introduced us to the problem andorganized field work in Svalbard. Paul Meakin sug-gested improvements to the manuscript. The work wassupported by the Norwegian Research Council.

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