REGULAR STALAGMITES: THE THEORY BEHIND THEIR
SHAPE
TEORETIJA RASTI PRAVILNIH STALAGMITOV
Wolfgang DREYBRODT1,2 and Douchko ROMANOV3
Abstract	UCC 551.435.834
Wolfgang Dreybrodt & Douchko Romanov: Regular stalagmites: The theory behind their shape
Under growth conditions constant in time stalagmites grow into an equilibrium shape, which is established, when all points of its surface are shifting by the same vertical distance W0 = 1.2-10''a(cm-ceq)-At(cm/year) during a time interval At(years). Thereby «(t^-c^) is the precipitation rate in molcm'V1, csq is the calcium concentration of the supersatu-rated solution dripping to the apex of the stalagmite, and c^ its equilibrium concentration with respect to calcite and the p^ in the cave atmosphere.
From these ingredients a numerical model of stalagmite growth into an equilibrium shape is presented. In this model one assumes idealistically that the water dripping to the apex flows continuously down the stalagmite, spreading out radially. By simple mass balance one finds that the equilibrium radius is R =Vv7 tcot, where V = 0.1 cm3 is the volume of a drop and t the drip interval. Furthermore numerical modeling reproduces the vertical shifting of the stalagmite's equilibrium shape. Finally an interesting similarity rule is found. If one scales two stalagmites of differing R^ to the same size and chooses their growth axes as common axis and their apexes as common origin, both show identical shapes. In other words regular stalagmites are similar geometrically. This similarity rule is verified by digitizing the shapes of various natural stalagmites with diameters between 5 cm and 20 m. Within small natural variations, the rescaled shapes are identical and close to the shape of the numerical model.
Keywords: stalagmites, morphology, growth rates, scaling law
Izvleček	UDK 551.435.834
Wolfgang Dreybrodt & Douchko Romanov: Teorija rasti pravilnih stalagmitov
V stacionarnih pogojih rasti stalagmiti zrastejo v ravnotežno obliko. Todosežejo, ko se vse točke na površini v določenem časovnem intervalu At premaknejo za enako vertikalno razdaljo W0 = 1.2-109 -o^Ck -c^j-A^cm/year) , pri čemer je a(cm -c^) hitrost odlaganja sige v molcmV, cm je koncentracija kalcija v raztopini, ki kaplja na teme kapnika, c^ ravnotežna koncentracija kalcija glede na kalcit, pœ pa je delni tlak ogljikovega dioksida v jamski atmosferi. Iz teh sestavin je narejen model razvoja stalagmita v ravnotežno obliko. V modelu predpostavimo, da voda odteka iz temena stalagmita enakomerno v radialni smeri. Iz ohranitve mase dobimo ravnotežni radij stalagmita R^ = yjv/7tax, pri čemer je V = 0.1cm3 volumen vodnih kapljic in t interval kapanja. Numerični model napove vertikalno premikanje ravnotežne oblike. Z ustrezno transformacijo koordinat se nazorno izkaže, da so pravilni stalagmiti geometrijsko identični. To v članku prikažemo na izbranih primerih naravnih stalagmitov premerov med 5 cm in 20 m. V okviru majnih odstopanj so umerjene oblike stalagmitov identične z napovedi numeričnega modela.
Ključne besede: stalagmit, morfologija, hitrost rasti, skalna in-varianca.
1	Karst Processes Research Group, Institute of Experimental Physics, University of Bremen, 28359 Bremen, Germany
2	Karst Research Institute ZRC SAZU, Titov trg 2 SI-6230 Postojna
3	Institute of Geological Sciences, FU Berlin, Malteserstr. 74-100, Building D, 12249 Berlin
Received/Prejeto: 21.5.2008
introduction
Stalagmites are ubiquitous in many caves. They grow in a variety of shapes and sizes. Their diameters range from a few centimetres up to several meters. Fig. 1 shows a transect of a small regular stalagmite with a diameter of 4.5 cm (Soubies et al. 2005). The largest stalagmite in the world is reported from Cueva San Martin Infierno in Cuba. This giant stalagmite exhibits a height of about 70 m and 20 m diameter. The photo of this speleothem http://www.goodearthgraphics.com/virtcave/largest.htm is shown in Fig. 2. Both samples, although extremely different in size at a first glance look quite similar. Both shapes can be crudely described as almost regular cylin-ders with a half sphere at their top. Such regular stalagmites sometimes are called candle stalagmites.
Fig. 1: Transect of a small regular stalagmite (Soubies et al. 2005) with a diameter of 4.5 cm. The laminations showgrowth surfaces. The red lines show the distance between various points of two distinct growth surfaces. They are all of the same length.
Franke (1965) was the first who recognized the prin-ciple of their growth. He proposed the following mecha-nism assuming growth conditions constant in time. A drop of solution, supersaturated with respect to calcite falling to the ground spreads out radially when it impinges at the surface. Precipitation rates are maximal at the center and decrease with distance outwards. This way a thin layer of precipitated calcite is created. It has a lenticular shape, thick at the center and thinning out at the edges. The next
layer is build in the same way on the top of the previous one and a vertical candle like shaped stalagmite grows up-wards. After a height of about its diameter its shape on the
Fig. 2: The world's largest stalagmite in Cueva San Infierno, Cuba. Its height is 70 m, its diameter 20 m.
top becomes stable and also the diameter of the stalagmite does not change anymore. In other words the growth rates in the vertical direction are constant everywhere and the shape of the stalagmite is shifted vertically in its growth direction. This is depicted in Fig. 3. Franke suggested this principle from observation on the growth layers of tran-sected stalagmites, such ones as shown in Fig. 1. Drey-brodt (1988,1999) and Romanov et al. (2008a) derived this growth principle from three simple generally valid assumptions by use of numerical modelling, but also by strict mathematical proof. These are:
1)	The point of drop impact and the drip interval are constant in time. This must be true also for temperature.
2)	Growth by deposition of calcite is directed normal to the actual surface of the stalagmite.
Fig. 3: Principie of stalagmite growth: The green lines depict the growth perpendicular to a grow-ing surface during the time span At. They decrease with distance I along the growing surface. The red curves show the shapes of the stalagmite when it starts to grow from a plane surface. Finally an equilibrium shape with radius Rq is attained (black curves). The vertical distance between to growth surfaces is shown by the red arrows with length W0. At, they are all of equal length. See also Fig. 1. The volume between the two utmost layers is composed of rings with radius x, width Ax and height W0. At. Their cross section is a parallelogram with sides Ax and W0. At. See text.
3) Growth rates W(£) decrease monotonically with distance I along the actual surface and become zero at some large but finite distance
Ir
From these assump-tions one can derive also a similarity rule depicted in Fig. 4: Rescaling the shape of a regular stalagmite into dimensionless coordinates
x = x/Req and y = y/Req,
choosing their growth axes as common axis, and their apexes as origin x = 0, y = 0 gives identical shapes, inde-pendent of the diameter Req of the sample. (Romanov et al., 2008a)
In this paper we will first discuss the basic features of stalagmite growth to understand this similarity rule. In the second step we use rescaling of real regular stalagmites from diameters in the range of several cen-timeters up to about 20 m to show its validity.
Fig. 4: Schematical representation of the geometrical similarity of stalagmites: A small stalagmite with 10 cm diameter is scaled by a factor of 1/10. A second one of 2000 cm diameter is scaled by a factor of 1/2000. In the new common coordinate system they are plotted with common growth axis (a). Then the red line is shifted vertically until it merges with the black line (b).
basic principles of stalagmite growth
The most important principle stated already, can be formulated as follows: Under constant growth conditions in time the top of a stalagmite attains an equilib-rium shape, which during further growth is shifted up vertically. The part of the stalagmite below is a vertical cylinder with constant equilibrium radius Req. This is illustrated in Fig. 3. This principle has been proven by Dreybrodt (1988) and Romanov et al. (2008a) as already mentioned in the introduction. We will not repeat this proof here, but we will derive some important consequences from it.
Fig. 3 shows two growth surfaces of a stalagmite in equilibrium, which are separated by some distance W0 • At (cm/year) is the growth rate at the apex and At (year) the time distance when the two layers origi-nated. The total volume V between these two layers is composed of circular rings with radius x and width Ax, all with equal height W0 • At. Summing over all of them yields
V = W0 • At J 2jtxdx = W0JtR'q • At
W0 is the growth rate at the apex in cm/year.
(1)
Precipitation of calcite from thin films of supersatu-rated solutions, as they rest on stalagmites has been in-vestigated theoretically and experimentally by Buhmann and Dreybrodt (1985) and Dreybrodt et al. (1996). These findings have been verified by measuring precipitation rates on stalagmites in relation to the chemical composition of the drip water (Baker and Smart, 1995, Baker et al., 1998, Genty et al. 2001). The precipitation rates R are given by
To convert the precipitation rates R (mol/(cm2 s)) into a growth rate W in cm/year we consider the volume of calcite deposited to 1 cm2 in 1s: This is given by
V8=R.M = r.M = 37.07-R 8 p 2.7
(4)
M is the molecular weight of CaCO3, M = 100 g/mol g
and p = 2.7—r- the specific weight of calcite. To convert cm
this into growth of cm per year one has to multiply by the number NS of seconds in one year and finds 1mol cm-2s-1 = 1.168 1 09 cm/year. Therefore
W = R- —-Ns = 1.168-109a(c-c ) cm/year (5) P
Now the volume Vd deposited to the stalagmite in 1 year (At = 1) must be equal to the volume VL of the amount of calcite lost during this time from the water, which enters with concentration cin at the apex and leaves the stalagmite with equilibrium concentration c .
VL=^"(cta
().M.Ns = ^.(Cta
-c Vl.168-109 (6)
Vd = W0JtR*q = 1.168-109a-(Cin -c^-jtR^
Vdrop is the volume of the drop and t the drip interval. Therefore we find by equating VL= Vd
R = a(c - c ) mol cm 2s
(2)
where a [cm/s] is the rate constant, c [mol/cm3] is the actual calcium concentration in the solution, and ceq its equilibrium concentration with respect to calcite and the partial pressure pC^ of CO2 in the cave atmosphere. a depends on temperature. Values of a in the range be-tween 0°C and 30°C are reported by Baker et al. (1998). These can be fitted by the relation
a = (0.52 + 0.04 • Tc + 0.004 • Tc2) • 10"5 cm/ s
(3)
where Tc is the temperature in centigrade (Romanov et al. 2008b). From this one gets a increase of a factor of ten with increasing temperature from 0°C (a = 5.2 ■ 10-6 cm/s) to 30°C (a = 5.3 ■ 10-5 cm/s).
K =
drop
jtat
(7)
This result relates in a simple way the equilibrium radius to the kinetic constant a and the drip interval t. Note that the radius does not depend on the initial supersaturation (cin-ceq).
As an example one finds with a = 1.3-10-5 cm/s, corresponding to a temperature of 10°C, a drop volume Vdrop = 0.1 cm3, and t = 30 s the equilibrium radius Req = 9 cm, a very realistic number.
This equation also tells us that changes in drip interval t and in kinetic constant a cause changes of the equilibrium radius and therefore also changes to the shape of the stalagmite. Increase in the drip interval t or in a
result in smaller diameters and cone like stalagmites can result. On the other hand decreasing t or a lead to an increase of R,q and club-like shapes arise. This has been shown by numerical simulation first by Dreybrodt (1988, 1999). Kaufmann (2002) and Kaufmann and Dreybrodt (2003) have taken climatic data from ice cores and tried
to find out, how these were imprinted into the shape of stalagmites. They showed that analyzing the shapes of ir-regularly grown stalagmites could give some additional information to the interpretation of climate proxies from oxygen and carbon isotopes.
a model of the equilibrium shape and the scaling rule
In our approach we assume that the water supplied to the apex spreads out to a water film, which expands radially flowing down the stalagmite in laminar flow. This way the stalagmite is covered by a water film of thickness 8 with radial symmetry as shown in Fig. 5.
Fig. 5: A water film with thickness S covers the stalagmite. An an-nulus ofthisfilm between distance l. and ( is shown. The water supply Q flows radially from the apex. At distance t. along the growth surface the water enters into the annulus with concentration c. and leaves it at distance t. , with concentration c. ,.
We now consider an annulus between the points i and i +1 at the surface with radial distance Ri and Ri+1 re-
spectively. The distance between these two points along the surface is li+1 - li=Ali.
Mass balance requires that the amount of calcite deposited to the area of the annulus between points i and i+1 must be equal to the loss of CaC03 in the solution when passing over the annulus.
The mass of calcite lost per second from the solution covering the surface of the annulus is
Ms=Q(Ci-ci+1)
(8)
because the total flow Q = Vdrop/x enters the annulus at R¡ with concentration c¡ and leaves it at Ri+1 with concentration ci+1. M must be equal to the mass Mp of calcite precipitated per second to the surface Sa = 2jt-Al¡-R¡ of the annulus. Note that Ms and Mp in this context are in mol/s.
Mp-^-AI.-R.-a-ic.-c^)	(9)
Equating Mp = Ms one obtains
ci+1 = ci-2jr-a-(ci-ceq)-Ri^	(10)
From this the deposition rates Wi+1 can be calcu-lated by subtracting ceq from both sides of the équation and then multiplying both sides by a
Wi+1=Wi(l-23t-a-Ri^)
(11)
JV	Q
dr°1' = . — eqn. 11 jtax v arc
can be rewritten as
Wi+1=Wr(l-2-^)
(12)
This recursive equation shows growth rates declin-ing with distance li. Rewriting eqn.11 as
w, = wft) =2 < (13)
In the limit of very small Al¡ = dl we obtain dW(l) _ 2Rdl 21dl
W(l) = Rfq £ R^q	(14)
because l(R) > R.
Integrating eqn. 14 yields
WO)sW0-exp(-^-)	(15)
Req
Since the Gauss-function is monotonically deceas-ing with distance l this must be true also for W(l).
Equation 11, which contains only defined parameters and the somewhat idealistic assumption of continu-ous flow spreading out radially can be used for numeri-cal modelling. This has been performed as FLOW-model by Romanov et al. (2008). They employed the following procedure. The actual surface of the stalagmite at time t is approximated by the polygon Pj to P¿ where Pj is the point of drip impact. The surface at later time t+At is obtained by drawing vertical lines with lengths W(l¡)- At from points i. This way the three basic assumptions on stalagmite growth are fulfilled and the new surface is ap-proximated by the polygon P'j to P'n as shown in Fig. 6.
Fig. 6: Consecutive construction of a new growth surface P' P' from an actual surface P1 Pn. The distances P P' depend on the lengths along the actual surface P1 Pn. They decrease with n. P' are the points of drip impact (see text).
Fig. 7: Computer simulation of a stalagmite by the FLOW-model: \op=0.1 cm3, T=30s, a=1.310-5 cm/s (T=10°C). The red lines show the initial state of growth until an equilibrium shape is at-tained (black curves). The growth surfaces are shown every 200 years in the lower part and every 1000 years in the upper part. The vertical lines between the two utmost surfaces are all of equal length W0At. At=1000years
Here we do not give technical détails but we represent the results of the FLOW- model. These are shown in Fig.7. It depicts growth layers of a stalagmite grow-ing on a plane surface. Note that the x and y axes have identical scales. Therefore the picture presents the natural undistorted shape. Clearly the shape of the stalagmite changes in its initial stage of growth (black lines) until an equilibrium shape is attained (red line) from then on this shape is maintained for further growth and the vertical distance between any two equilibrium growth layers remains constant on all points of the surface. The vertical red lines, all of equal lengths between the two upper surfaces indicate this. A further point to be noted is that the equilibrium shape is attained after a growth height close to its diameter.
The similarity rule can be obtained from eqn.12. By introducing dimensionless coordinates X = x/Req and y = y/Req one finds
w^Wi-a-2-ivAî;)
(16)
This, however, means that any regular stalagmite scaled by the factor 1/Req, into this new coordinate system must exhibit a unique shape independent on its former dimensions.
This proves the scaling rule depicted in Fig. 4: Scale the stalagmites in both coordinates by the factor 1/Req and chose their growth axis as y-coordinate and their apex as origin. Then all regular stalagmites have the same shape.
verification of the scaling rule on natural stalagmites
To prove this by observation one has to select regular stalagmites, which show vertical walls about 1 diameter below their apex. Two extreme examples are shown in Figs. 1 and 2.
We have selected six transections of stalagmites, which are exhibited in the Deutsche Archiv für Sinterchronologie (DASC) in the Schillat-cave, located in a limestone quarry close to Hessisch-Oldendorf, Germa-
Fig. 8: Regular small stalagmites from the Deutsche Archiv for Sinterchronologie (DASC) used to test the similarity rule. Regular stalagmites taken from the literature are also shown (see text). a) dasc1, b) dasc2, c) dasc3, d) dasc4, e) dasc9, f) Frisia, g) Genty, h) Bard. The black snd white squares at the base of stalagmites are 1 cm x 1 cm.
ny. These stalagmites exhibit diameters between 4 cm to 8 cm. They are shown in Fig. 8.
Furthermore four stalagmites have been taken from the literature: Bard et al. (2002), Fig. 1, Frisia et al. (2006), Fig. 2, both with diameters of 10 cm, Genty et al. (2006), Fig. 4, with a diameter of 5 cm, and Soubies et al. (2005), Fig. 2a with 4.5 cm diameter. These are also shown in Fig.8. We have digitized suitable growth layers of the surface of those stalagmites and rescaled them. We have also performed a search of regular large stalagmites
in the Internet and found the world largest regular stalagmite with a diameter of 20 m shown in Fig. 2. Other ones have been found on the from Carlsbad Caverns, New Mexico, USA, a column with 4 m diameter (http:// www.fingerlakesbmw.org/visual/az/cccolumn1.jpg) and finally from Lost World Caverns, West Virginia, USA, the Bridal Veil with 2 m diameter, (http://www.wonder-fulwv.com/archives/june01/fea1.cfm). Pictures of these two stalagmites are shown in Fig. 9.
-1.5 -1.0 -0.5 0.0 0.5
Fig. 12: Rescaled shapes of all stalagmites compared to the shape obtained from the FLOW-model and a modified FLOW-model
Fig. 10: Rescaled shapes of three large stalagmites and one small fl°WM. stalagmite. See color legend.
Fig. 11: Rescaled shapes of the stalagmites from Fig. 8. See color legend.
To compare the shapes we have rescaled all stalagmites to the same radius R„, = 1.
eq
In Fig. 10 the rescaled shapes of the stalagmite from Cuba, Column, and the Bridal Veil are shown. Within some natural variation all these shapes are almost the same. We have also added the smallest stalagmite with 4.5 cm from the paper of Soubies et al. (2005), (see Fig. 1). Its shape fits nicely with those of the large stalagmites.
Fig.11 compares the shapes of all small stalagmites from Fig. 1 and Fig. 8, which also prove the similarity rule. Finally, in Fig. 12 we have plotted the complete selection of our samples and compare it to the theoretically obtained rescaled shape of the FLOW-model depicted in Fig. 7.
Although not perfect the theoretical shape is quite close to the natural shapes.
discussion and conclusion
We have shown that the shapes of regular stalagmites are similar from very small equilibrium diameters of several centimetres over three orders of magnitude up to 20 m. This is a rather unexpected result because the hydro-dynamic flow conditions are extremely different. Small stalagmites up to diameters of several 10 cm are fed by drops with drip intervals of about 10 s for a diameter of 30 cm and 200 s for a diameter of about 5 cm. To ob-tain a diameter of 20 m a water supply of 40 cm3/s or 400 drops/s are necessary. Whenever a drop impinges to the surface of a stalagmite splashing will occur and the drop
or the continuous flow of water will cover the central part of the stalagmite's surface by water film, which acts like a reservoir from which outward flow is supplied. In this water film due to mixing, when a new drop impinges, a constant calcium concentration will be maintained and the growth rate is constant within the radius where this water film covers the stalagmite.
To model the shape of this stalagmite requires W,=W0 for all points i with R,<Rc. For R,>Rc=0.33-Req the recursive equation 11 is valid.
The comparison of the equilibrium shapes is shown in Fig. 12. The new shape (black fat line) is exactly what one obtains by cutting the upper part of the idealised stalagmite (obtained from the FLOW - model). The modi-fied model FLOW-M comes closer to the real shapes although it is not perfect. The FLOW- model exhibits a sharper tip.
Concluding we can state that although hydrody-namics are different for small and large stalagmites, they all obey a similarity law, which indicates that the detailed growth mechanisms must be very similar. It may be a topic of future research to incorporate the hydrodynam-ics including surface tension of the water flowing away from the apex into further more realistic growth models.
acknowledgement
We thank Mr. Hartmut Brepohl and the team of the Schillat Cave for their kind and helpful support during our visit to take the pictures of the stalagmites.
references
Baker, A., & Smart, P. L., 1995: Recent flowstone growth rates: field measurements in comparison to theoret-ical prediction.- Chemical Geology 122, 121-128.
Baker, A., Genty, D., Dreybrodt, W., Barnes, W. L., Mock-ler, N. J., & Grapes, J., 1998: Testing theoretically predicted stalagmite growth rate with recent an-nually laminated samples implications for past stalagmite deposition.- Geochimica et Cosmochimica Acta , 62, 393-404
Bard, E., Antonioli, F., & Silenzi, S., 2002: Sea-level dur-ing the penultimate interglacial period based on a submerged stalagmite from Argentarola Cave (Italy).- Earth and Planetary Science Letters 196, 135146.
Buhmann, D., & Dreybrodt, W., 1985: The kinetics of calcite dissolution and precipitation in geologically relevant situations of karst areas: 1. Open system.-Chemical Geology 48, 189-1211.
Dreybrodt, W., 1988: Processes in Karst Systems - phys-ics, chemistry, and geology. Springer. Berlin.
Dreybrodt, W., 1999: Chemical kinetics, speleothem growth and climate.- Boreas 82, 347-356.
Dreybrodt, W., Eisenlohr, L., Madry, B. & Ringer, S., 1997: Precipitation kinetics of calcite in the system: the conversion to CO2 by the slow process H+ + HCO3-^ CO2 + H2O as a rate limiting step.- Geochimica et Cosmochimica Acta 60, 3897-3904.
Frisia, S., Borsato, A., Mangini, A., Spötl, C., Madonia, G., & Sauro, U., 2006: Holocene climate variability in Sicily from a discontinuous stalagmite record and the Mesolithic to Neolithic transition.- Quater-nary Research 66, 388-400.
Franke, H. W., 1965: The theory behind stalagmite shapes.- Studies in Speleology 1, 89-95.
Genty, D., Baker, A., & Vokal, B., 2001: Intra- and inter-annual growth rate of modern stalagmites.- Chem. Geol., 176, 191-212. John Wiley & Sons Ltd.
Genty, D., Blamart, D., Ghaleb, B., Plagnes,V., Caussed,C., Bakalowicz, M., Zouarif, K., Chkir, N., Hellstrom, J., Wainer, K., & Bourges, F., 2006: Timing and dynamics of the last deglaciation from European and North African S13C stalagmite profiles- comparison with Chinese and South Hemisphere stalagmites.-Quaternary Science Reviews, 25, 2118-2142.
Kaufmann, G., 2003: Stalagmite growth and paleocli-mate: the numerical perspective.- Earth and Plan-etary Science Letters., 214, 251-266.
Kaufmann, G., & Dreybrodt, W., 2004: Stalagmite growth and paleoclimate: an inverse approach. - Earth and Planetary Science Letters, 224, 529-545.
Romanov, D., Kaufmann, G., & Dreybrodt, W., 2008 a: Modeling stalagmite growth by first principles of chemistry and physics of calcite precipitation.-Geochimica et Cosmochimica Acta , 72, 423-437.
Romanov, D., Kaufmann, G., & Dreybrodt, W., 2008 b: S13 profiles along growth layers of stalagmites: Comparing theoretical andexperimental results.-Geochimica et Cosmochimica Acta,72, 438-448.
Soubies, F., Seidel, A., Mangin, A., Genty, D., Ronchail, J., Plagnes V., Hirooka, S., & R. Santos, R., 2005: A fifty-year climatic signal in three Holocene stalagmite records from Mato Grosso-Brazil.-Quaternary International 135 (2005) 115-129.