δ13C profiles along growth layers of stalagmites: Comparing theoretical and experimental results

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Geochimica et Cosmochimica Acta 72 (2008) 438–448 www.elsevier.com/locate/gca

d13C profiles along growth layers of stalagmites: Comparing theoretical and experimental results Douchko Romanov b

a,*

, Georg Kaufmann a, Wolfgang Dreybrodt

b

a Institute of Geological Sciences, FU Berlin, Malteserstr. 74-100, Building D, 12249 Berlin, Germany Karst Processes Research Group, Institute of Experimental Physics, University of Bremen, 28359 Bremen, Germany

Received 29 May 2007; accepted in revised form 6 September 2007; available online 1 November 2007

Abstract The isotopic carbon ratio of a calcite-precipitating solution flowing as a water film on the surface of a stalagmite is determined by Rayleigh distillation. It can be calculated, when the HCO3  -concentration of the solution at each surface point of the stalagmite and the fractionation factors are known. A stalagmite growth model based entirely on the physics of laminar flow and the well-known precipitation rates of a supersaturated solution of calcite, without any further assumptions, is employed to obtain the spatial distribution of the HCO3  -concentration, which contributes more than 95% to the dissolved inorganic carbon (DIC). The d13C profiles are calculated along the growth surface of a stalagmite for three cases: (A) isotopic equilibrium of both CO2 outgassing and calcite precipitation; (B) outgassing of CO2 is irreversible but calcite precipitation is in isotopic equilibrium. (C) Both CO2 outgassing and calcite precipitation are irreversible. In all cases the isotopic shift d13C increases from the apex along the distance on a growth surface. In cases A and B, calcite deposited at the apex is in isotopic equilibrium with the solution of the drip water. The difference between d13C at the apex and the end of the growth layer is independent of the stalagmite’s radius, but depends on temperature. For case A, it is about half the value obtained for cases B and C. In case C, the isotopic composition of calcite at the apex equals that of the drip water, but further out it becomes practically identical with that of case B. The growth model has been applied to field data of stalagmite growth, where the thickness and the d13C of calcite precipitated to a glass plate located on the top of a stalagmite have been measured as function of the distance from the drip point. The calculated data are in good agreement to the observed ones and indicate that deposition occurred most likely under conditions B, eventually also C. A sensitivity analysis has been performed, which shows that within the limits of observed external parameters, such as drip rates and partial pressure of carbon dioxide P CO2 in the cave, the results remain valid.  2007 Elsevier Ltd. All rights reserved.

1. INTRODUCTION Stalagmites are now recognized as proxies of environmental signals (McDermott, 2004; McDermott et al., 2006; Fairchild et al., 2006, 2007). They can grow for time spans from 103–105 years, and carry paleoclimatic information, which by use of accurate dating techniques such as Uranium series (Ford and Williams, 2007), can be converted into time series. Oxygen and carbon isotopic signals

*

Corresponding author. Fax: +49 30 838 70729. E-mail address: [email protected] (D. Romanov).

0016-7037/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.gca.2007.09.039

d18O and d 13C, have been the focus of research for decades. Recent studies show that in particular d18O time series are useful to recover paleoclimate temperatures during the past. As an example, Mangini et al. (2005) have shown that the temperatures during the last 2000 years can be reconstructed. The medieval climate optimum and the little ice age have been identified by time series from stalagmites in the Central Alps (Mangini et al., 2005; Vollweiler et al., 2006). Niggemann et al. (2003) have found the existence of sub-Milankovitch cycles during the last 6000 years from a stalagmite of the Atta Cave, Sauerland, Germany. Wang et al. (2001) have reported on oxygen isotope records from five stalagmites of Hulu Cave in China bearing a

d13C profiles along growth layers of stalagmites

remarkable resemblance to oxygen records from Greenland ice cores. These recent advances show the high potential of d 18O time series. However, caveats must be given and an exact interpretation of such data depends on many variables. Especially for isotopic records one has to assume that isotopic fractionation occurs under isotopic equilibrium conditions (Hendy, 1971). When kinetic fractionation operates, speleothems exhibit a positive correlation of d18O and d13C for samples taken along a growth surface and records from such stalagmites should be regarded with care. In view of these problems detailed knowledge on the formation in isotopic signals on stalagmites is needed, first to avoid errors and second to identify other signals, such as drip rates to the apex of a stalagmite, which yield additional information of paleoprecipitation. A first step into this direction has been taken by Mu¨hlinghaus et al. (2007). They have modeled variations in d13C isotope profiles along the growth layer of a stalagmite by use of a multi-box model. From their results they conclude that the interval between two drips to the apex of the stalagmite most significantly determines the variations in the d13C profiles. In this paper we present a new method to model the variations of d13C along growth layers of stalagmites. This model is based entirely on the physics and chemistry of calcite precipitation and isotopic chemistry. It does not need any ad hoc assumptions as used in earlier models on stalagmite growth (Dreybrodt, 1999; Kaufmann, 2003), which served as basis for the work of Mu¨hlinghaus et al. (2007). 2. THEORETICAL BACKGROUND When calcite precipitates from a H2O–CO2–CaCO3 solution in a thin water film to a calcite surface, carbon is transferred from the solution to the solid and as a result of the global chemical reaction 2HCO3  + Ca2+ fi H2O + CO2 + CaCO3, carbon dioxide must degas. This could happen in two different ways. In the first forward and backward reactions for both carbon isotopes are almost equal. The products formed escape from the solution and do not interact with it again. Under such conditions the products are in isotopic equilibrium with the solution, when they are generated. This, however, requires slow deposition rates, whereby the Ca2+-concentration at the surface of the solid must be close to saturation with respect to calcite. Furthermore the CO2 in the surrounding atmosphere must be almost in equilibrium with the CO2 in the solution. The other extreme is irreversible fast precipitation with no backward reaction and consequently fast degassing of CO2, which is diluted into the surrounding atmosphere and lost irreversibly from the solution. Such irreversible processes result in kinetic fractionation, whereas the first produces equilibrium fractionation. In any case the solution has to be regarded as an open system, because the products once formed escape out from the solution and do not react with it again. The evolution of the isotopic ratio is then described by Rayleigh-distillation (Kendall and McDonnell, 1998). For pH values between 8 and 9 the total amount of carbon in

439

the solution within a few percent is represented by the HCO3  ions. Therefore the enrichment in 13C can be obtained by (Usdowski et al., 1979; Mickler et al., 2004)   !a1 HCO3  ðtÞ 13 13   ð1Þ RHCO  ¼ RHCO  ð0Þ  3 3 HCO3  ð0Þ where R13 HCO3  is the isotopic ratio R = [rare_isotope]/[abundant_isotope] = 13C/12C at time t or at time zero, respectively.  a is the fractionation factor taking into account that carbon is lost by CO2-degassing and precipitation of calcite. It takes different values for the two extremes discussed above. This will be discussed later in the text. The concentration [HCO3  ] is related to the concentration of [Ca2+] by  2þ    ¼ 2  HCO3  ð2Þ Ca This is the relaxed equation of charge balance valid for the pH values mentioned above (Dreybrodt, 1988). The precipitation rate F of calcite to the solid in an open system is given by (Buhmann and Dreybrodt, 1985; Dreybrodt et al., 1997): F ¼ a  ðc  ceq Þ;

ð3Þ

with a the kinetic rate coefficient (Table 1). From this one obtains the time evolution of the calcium concentration [Ca2+] = c in a thin sheet of water as:     td ; ð4Þ cðtÞ ¼ ceq þ c0  ceq  exp  a c is the actual concentration of Ca2+ in the solution, c0 the initial concentration of Ca2+, and ceq is the equilibrium concentration with respect to calcite. The time constant is given by d/a, where d is the depth of the water film. Eqs. (1)–(4) are the basic ingredients to calculate the isotopic compositions for calcite deposited to stalagmites. If one knows the time of contact of a parcel of water with the calcite surface and the position on the stalagmite’s surface at this time it is possible to calculate its isotopic composition of HCO3  . The calcite precipitated from this parcel of solution then contains the imprint of the solution’s isotopic composition. To know the position of a selected element of a parcel of water one needs a model of the precipitation rates on the surface of the stalagmite. The first model of stalagmite growth and the evolution of its morphology has been reported by Dreybrodt (1988, 1999) and later by Kaufmann (2003). This model assumes that the stalagmite is covered by a stagnant water film of depth between 0.005 and 0.03 cm, from which calcite is precipitated during the time s between two drops. When a new drop impinges to the apex the entire water film is replaced by the new solution. To account for the fact that the deposition rates must decrease with distance from the axis one uses   l F ðlÞ ¼ a  ðcðtÞ  ceq Þ  exp  ; q ¼ Req ð5Þ q where l is the distance along the actual growth surface from the axis and q turns out to be the equilibrium radius Req of the stalagmite. c(t) is the calcium concentration at time t.

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Table 1 Parameters used in the models Parameter a

Value   0:52 þ 0:04  T c þ 0:004  T 2c  105 ðcm=sÞ

a1 a2 akin 1 akin 2 RVPDB

4.232/Tk + 1.0151 9.483/Tk + 1.02389 1.0 0.988 Æ a2 0.0112372

Comment Rate constant of calcite precipitation, Tc—temperature of the water in degrees centigrade (Baker et al., 1998) aCaCO3 –HCO3  , equilibrium, Tk—temperature in Kelvin (Mu¨hlinghaus et al., 2007) aCOg HCO3  , equilibrium, Tk—temperature in Kelvin (Mu¨hlinghaus et al., 2007) 2 aCaCO3 –HCO3  , kinetic aCOg2 –HCO3  , kinetic

By use of Eqs. (4) and (5), and considering, that growth is always normal to the actual surface of the stalagmite, one finds that under constant conditions the stalagmite attains an equilibrium shape. Its radius Req is related to the kinetic constant a and the flow rate Q = V/s, where V  0.1 cm3 is the volume of a drop. This corresponds to a drop diameter of 6 mm (Curl, 1972). For s < da  300 s this relation is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ð6Þ Req ¼ pas Values of a for various temperatures and film thicknesses d are calculated by the kinetic theory of calcite precipitation (Buhmann and Dreybrodt, 1985) and are listed in Baker et al. (1998). In this approximation, however, which relies on the assumption of Eq. (5), each parcel of solution on the stalagmite experiences an identical evolution of [HCO3  ](t) and therefore the isotopic composition is identical everywhere on its surface. This, however, is not observed. Recently Mu¨hlinghaus et al. (2007) expanded this growth model to overcome this deficiency. They assumed that the new drop of water mixes with the solution on the stalagmite, defining a mixing parameter U1. By this way the calcium concentration c1(0) immediately after impinging of the drop becomes c1 ð0Þ ¼ ð1  U1 Þ  c1 ðsÞ þ U1  cdrop ; 1

ð7Þ

where c1(s) is the concentration of the solution at time s, when a new drop impinges to the apex, and cdrop is the cal1 cium concentration in the drop. When a new drop impinges to the surface the solution moves outwards instantaneously, but stays as a stagnant film until a new drop falls to the apex. A box model has been suggested with mixing parameters Ui. These parameters Ui are calibrated to the shape of corresponding stalagmites created numerically by the model of Dreybrodt. Consequently the concentrations ci(0) become dependent on the distance of box i from the axis, and the stagnant film in the Dreybrodt model is changed to a stagnant film divided into radial boxes with decreasing initial concentrations. It is replaced immediately after each new drop and then stays for the time s until it is reestablished again. During this time Rayleigh distillation is operative in each box i leaving a different isotopic composition of HCO3  in its solution and therefore also in the calcite deposited. The variations along a growth layer show enrichment in 13C and are in the realm of naturally observed val-

ues. These profiles depend on temperature T, the mixing parameter U1 at the apex and most significantly on the drip time s. However, although successful this model still relies on the assumption that precipitation rates of calcite decrease exponentially with distance along the growth surface and on the definition of an arbitrary mixing parameter U1 6 1. A model based entirely on physics and chemistry of stalagmite growth without arbitrary assumptions is required. It is the purpose of this paper to present such a model to approach closer to reality. 3. A PROCESS-ORIENTED APPROACH TO STALAGMITE GROWTH Recently Romanov et al. (2007) have proposed an alternative way to model growth and morphology of stalagmites. In this model a continuously flowing water film replaces the stagnant water film. The results of this new FLOW-model with respect to the stalagmite morphology are equivalent to those from the model of Dreybrodt (1988, 1999) discussed above, provided one replaces the exponential decay of precipitation rates (see Eq. (5)) by a Gaussian decay ! l2 p F ð1Þ ¼ a  ðcðtÞ  ceq Þ exp  2 ; Req ¼ pffiffiffi qg ; ð8Þ qg 2 where qg is a parameter determining the radius Req of the stalagmite. The dependence of Req on a and s(see Eq. (6)) remains valid for s < d/a  300 s. The Gaussian function is obtained from the new FLOW-model with continuously flowing water film by fitting the precipitation rates obtained without any arbitrary assumption to a Gaussian. In a first step we calculate the flow velocity of the water film in laminar flow spreading out radially. At some point on the surface i with distance Ri from the growth axis the flow velocity is given by (Bird et al., 1960) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 q  g  Q  sin ci vi ¼ ð9Þ 2 4  p  R2i  3  g Here, q is the density of water, g the Earth’s acceleration, g the viscosity of water, Q = V/s the continuous flow rate, Ri the distance of point i from the growth axis, and ci the slope angle there (see Fig. 1). We now consider water flowing from the apex of the stalagmite on its way down the steepest slope along the growth

d13C profiles along growth layers of stalagmites

441

Eq. (13) shows that the precipitation rates decrease with distance li. Using a numerical simulation, one can now model the equilibrium shape of a stalagmite. The resulting morphology is almost identical to a stalagmite modeled by use of a stagnant film model with Gaussian decay of the precipitation rates. Fig. 2 shows an example. 4. MODELING d13C IN CALCITE ALONG GROWTH LAYERS

Fig. 1. Geometry of a parcel of fluid spreading radially down the surface of a stalagmite (see text).

We now consider a parcel of water with defined volume, which travels from the apex spreading out radially. From Eq. (14) we can calculate its calcium concentration at each position i with Ri (distance from the growth P axis). From Eqs. (9) and (10) one obtains the time ti ¼ Dtj the parcel of water has needed to arrive at radius Ri. During this time HCO3  and Ca2+ are removed from the solution in this parcel. The evolution of the isotopic composition in time in this solution by Rayleigh distillation is given by Eq. (1). 4.1. Model A: Evolution of d13C under equilibrium conditions

surface. To travel the distance Dli it needs the time Dti. Consequently Dti ¼

Dli vi

ð10Þ

We now consider a radial annulus between the points i and i + 1 with distance (radius) Ri and Ri+1, respectively. On average, the solution flowing radially outwards is in contact to the annulus’ area for the time Dti. Mass balance then requires that the amount of calcite precipitated to the area is equal to the amount of calcium lost from the solution during time Dti: p  ðRi þ Riþ1 Þ  ðliþ1  li Þ  Dti  F ðci Þ ¼ p  ðRi þ Riþ1 Þ  ðliþ1  li Þ  di  ðci  ciþ1 Þ;

ð11Þ

where the left-hand side denotes the amount of calcite precipitated to the area of the annulus with radius (Ri + Ri+1)/ 2 and width (li+1  li) during time Dti. The right hand side is the amount of calcium lost in the solution covering the surface during time Dti. di is the depth of the water film. With Fi = a Æ (ci  ceq) one finds (Romanov et al., 2007)   a  Dti ð12Þ F iþ1 ¼ F i  1  di Expressing Q as V Q ¼ 2  di  p  Ri  vi ¼ s and inserting di in Eq. (12) one finds the recursive relation   2  p  a  Dli  Ri  s ð13Þ F iþ1 ¼ F i  1  V This equation is valid for 0.005 cm < d < 0.04 cm, where a is practically independent of d (Baker et al., 1998; Dreybrodt, 1999). For the concentration one finds     2  p  a  Dli  Ri  s ciþ1 ¼ ci  ceq  1  ð14Þ V

In isotopic equilibrium the fractionation factor is  aeq , where a1 is the fractionation factor a ¼ 12 ða1 þ a2 Þ ¼  between CaCO3 ! HCO3  and a2 that between COgas ! 2 HCO3  , both in isotopic equilibrium. (Mickler et al., 2004; Mu¨hlinghaus et al., 2007). The temperature dependence of a1 and a2 is given in Table 1. d13C in the solid can be obtained by (Salomons and Mook, 1986; Mu¨hlinghaus et al., 2007).   a1  R13 DIC d13 C ¼  1  1000ðpermilleÞ ð15Þ RVPDB with R13 DIC the isotopic ratio of the DIC, and RVPDB the isotopic standard (Table 1). Since for pH-values of about 8, as is the case for water dripping to a stalagmite dissolved inorganic carbon consists almost entirely (>95%) of HCO3  , 13 13 R13 DIC ¼ RHCO3  . To find the variation of d C along a growth layer one has to insert Eq. (1) into Eq. (15). This yields the evolution of R13 HCO3  ðtÞ in the traveling water parcel. Under continuous flow a stationary profile of RHCO3  ðti ðRi ÞÞ evolves and one can write 0 1 !aeq 1    a1  RHCO3  ð0Þ HCO ð t i ðRi ÞÞ 3 13   d CðRi Þ ¼ @   1A RVPDB HCO3  ð0Þ  1000ðpermilleÞ ð16Þ 13

to obtain the variation of d C as a function of Ri, the distance of the point i on the growth surface from the growth axis or alternatively as function of the distance li along the growth layer Pfrom the apex, because li(Ri) is a function of R. ti ðRi Þ ¼ ij¼1 Dtj is the time a parcel needs to flow from the apex to the distance li. 4.2. Model B: Evolution of d13C with kinetic outgassing, but calcite precipitation in isotopic equilibrium So far we have assumed that the reaction converting HCO3  into COgas proceeds in isotopic equilibrium. This, 2

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D. Romanov et al. / Geochimica et Cosmochimica Acta 72 (2008) 438–448

EXP model FLOW model GAUSS model

0.5

Height [m]

0.4

0.3

0.2

0.1

0

0

0.1

0.2 Width [m]

0.3

0.4

Fig. 2. Stalagmite growth. The evolution of a stalagmite at the temperature of 10 C, P CO2 ¼ 4  104 atm and drip time of 30 s modeled by the FLOW-model (full black line) and a stagnant film (full gray line) model with Gaussian decrease of precipitation rates. The shapes are given for every 5000 years. The dotted lines show the corresponding shapes obtained with exponential decrease of precipitation rates. This model was used by Mu¨hlinghaus et al. (2007).

however, is not necessarily true. In case of fast degassing the conversion of HCO3  into CO2 is irreversible following two reactions k 1

HCO3  þ Hþ ! H2 O þ CO2 k 2

HCO3  ! CO2 þ OH k 1

ðAÞ ðBÞ

k 2

and are rate constants. For an irreversible reacHere tion the total rate is then given by (Dreybrodt, 1988)     d HCO3  þ  ¼ HCO3   ðk  1 ½H  þ k 2 Þ dt    ð17Þ ¼ k eff HCO3 This equation is valid for both carbon isotopes, 12C and 13 C. The rate constants, however, differ slightly for the two isotopes. This leads to a change in isotopic ratio given by Eq. (16), where now aeq must be replaced by  1  ; ð18Þ akin ¼  a1 þ akin 2 2 with 13  k 1 ½Hþ  þ k 1 ½H 

akin 2 ¼ 12

13  þ 13 k  k eff 2  ¼ 12  12 k eff þ k2

ð19Þ

12  k 2 are the rate constants for 12C, and 13 k  Here, 12 k  1 and 1 13 and 13 k  for C isotopes. 2 According to Zeebe et al. (1999a,b) the following relations hold: 13  k1 12 k  1

¼ 0:987  a2 ;

13  k2 12 k  2

¼ 0:989  a2

ð20Þ

For pH < 7 reaction (A) is dominant and one can replace  kin k eff ¼ k 1 . Then aðAÞ ¼ 0:987  a2 . At pH > 9 reaction (B)

 exceeds reaction (A) and k Therefore eff ¼ k 2 . akin ¼ 0:989  a . For drip water with pH  8, both reac2 ðBÞ tions are active and the average value akin 2 ðavÞ ¼ 0:988  a2 is appropriate and will be used for later calculations. It should be noted that both factors in Eq. (20) are valid at 24 C. Since no information on their temperature dependence is available these values are used at all temperatures. This might underestimate the temperature dependence of akin 2 ðavÞ, which at lower temperatures can increase fractionation.

4.3. Model C: Evolution of d13C with kinetic outgassing and kinetic precipitation of calcite So far we have assumed that the precipitation of calcite is in isotopic equilibrium with HCO3  . Although most of the isotopic fractionation is due to outgassing of CO2, small effects could arise from kinetic fractionation between HCO3  and CaCO3 (Mickler et al., 2004). Turner (1982) and Clark and Lauriol (1992) have shown that under rapid precipitation of calcite eHCO3  –CaCO3 ¼ 0:35  0:3½permille at 25 C. Similar observations are reported by Michaelis et al. (1985) who found the isotopic composition of DIC in calcite precipitating river identical with that of the calcite precipitated from it and concluded akin 1 ¼ 1 at 10 C. The precipitation rates in such calcite depositing rivers are known (Dreybrodt et al., 1992; Liu et al., 1995; Bono et al., 2001) and are in the order of several times 107 mol/m2 s. Such deposition rates are also characteristic for stalagmites. To include these kinetic effects into our consideration we have replaced a1 by akin 1 ¼ 1 in 1 kin kin Eqs. (16) and (18) using now 12 ðakin 1 þ a2 Þ ¼ 2 ð1 þ a2 Þ.

d13C profiles along growth layers of stalagmites

about one-half of that obtained for kinetic fractionation of outgassing. At fixed temperature in both cases the amplitude D13C = d13Cmax  d13Cmin depends only on the Caconcentration of the drip water and becomes larger with increasing concentration. The two lower panels c and d in Fig. 3, show d 13C as a function of distance l along the surface. The d13C values show a S-shaped change increasing from the apex to the equilibrium radius, where precipitation rates of calcite become zero. Fig. 4 depicts d13C profiles for various temperatures in dependence of R and l for low (a,c) and high (b,d) input concentrations. The drip time is 30 s. The curves exhibit a similar shape as in Fig. 3. However, the amplitudes D13C increase with temperature. This can be seen particularly well for low input concentrations and is less pronounced for high drip water concentrations. The reason for this is the temperature dependence of the calcium equilibrium concentration ceq with respect to the P CO2 in the cave, which has been chosen as P CO2 = 4 · 104 atm in the calculation of the data both in Figs. eq decreases  3 and 4.  Since c  with temperature, the ratio HCO3  si ðReq Þ HCO3  ð0Þ in Eq. (16) becomes smaller for low input concentrations, enhancing the effect of Rayleigh distillation. The additional kinetic effect of calcite precipitation (case C) is illustrated in Fig. 5. The full lines depict the profiles of Fig. 4c and d for kinetic fractionation by outgassing of CO2 (case B), and the dashed lines for equilibrium (case A). The dotted lines show the enrichment in 13 C caused by kinetic fractionation during precipitation (case C). The effect however, is fairly small and noticeable only close to the apex at higher temperatures, as can be seen in the insets.

5. RESULTS 5.1. Profiles of d13C The use of the FLOW-model enables us to calculate d C profiles for any shape with cylindrical symmetry or in other words along any growth layer. In the following we present profiles of d13C along growth layers of stalagmites, which have attained their equilibrium shape. These have been calculated using the stalagmite growth model FLOW to obtain the times Dsi needed for a parcel of radially out flowing water to travel the distance li. These times have been employed in Eq. (16) to obtain d13C for three cases: 13

(A) isotopic equilibrium, using aeq ¼ 12 ða1 þ a2 Þ; (B) kinetic effects of outgassing of CO2, using akin ¼ 12 ða1 þ akin 2 Þ instead of aeq ; (C) for the additional kinetic effect of calcite precipitation, 1 kin kin using akin 1 ¼ 1 and 2 ða1 þ a2 Þ instead of aeq in Eqs. 13 (16) and (18). We use d Cdrop = 10[permille]. Fig. 3 shows the profiles of d13C as a function of the distance R from the growth axis for various drip times s at a fixed temperature of 10 C and for two different calcium concentrations of the drip water: (a) cin = 1.37 mol/m3 and (b) cin = 2.6 mol/m3. The dashed lines depict equilibrium fractionation for both CO2 outgassing and calcite precipitation (case A), whereas the full lines show d 13C for kinetic fractionation by outgassing, but equilibrium fractionation for calcite precipitation (case B). The curves end at R = Req. Req changes with drip time (see Eq. (6)). For equilibrium fractionation (dashed line) the amplitude is

-4

13

-6

5

3

cin = 1.37 mol/m

δ C [permille]

1s-A 1s-B 10 s - A 10 s - B 30 s - A 30 s - B 90 s - A 90 s - B

-2

13

δ C [permille]

0

-8 -10

0

10

20 30 Radius [cm]

40

3

Temperature 10ºC -4

PCO = 4x10 atm 2

-5

0

10

20 30 Radius [cm]

40

50

3

cin = 1.37 mol/m

-4

13

-6

δ C [permille]

5

-2

13

δ C [permille]

cin = 2.6mol/m

0

-10

50

0

-8 -10

443

0

100 200 50 150 Length along profile [cm]

3

cin = 2.6 mol/m

0 -5 -10

0

100 200 50 150 Length along profile [cm]

Fig. 3. d13C profiles for various drip times at T = 10 C along a growth surface of a stalagmite as function of distance R to the growth axis (a,b) and as function of the distance to along the surface (c,d). Panels (a) and (c) are for low input Ca-concentration in the drip water (1.37 mol/m3). Panels (b) and (d) are for high concentration (2.6 mol/m3). P CO2 in the cave is 4 · 104 atm. Dashed lines depict fractionation in isotopic equilibrium (case A). Full lines show the effect of kinetic fractionation by irreversible outgassing of CO2 (case B).

D. Romanov et al. / Geochimica et Cosmochimica Acta 72 (2008) 438–448

1C-A 1C-B 10 C - A 10 C - B 20 C - A 20 C - B 26 C - A 26 C - B

-2 -4

13

-6 -8

5

13

δ C [permille]

0

δ C [permille]

444

3

cin = 1.37 mol/m

-10 5

10 15 Radius [cm]

0 Drip time 30 s -4

PCO = 4x10 atm

-5

20

0

2

0

5

10 15 Radius [cm]

20

5

3

δ C [permille]

cin = 1.37 mol/m

-2 -4

13

-6

13

3

-10 0

δ C [permille]

cin = 2.6mol/m

-8 -10

3

cin = 2.6 mol/m

0 -5 -10

0

10 20 30 40 50 Length along profile [cm]

0

60

10 20 30 40 50 Length along profile [cm]

60

Fig. 4. d13C profiles for various temperatures at a drip time of 30 s along a growth surface of a stalagmite as function of radial distance to the growth axis (a,b) and as function of the distance along the surface (c,d). (a) and (c) are for low-input Ca-concentration in the drip water (1.37 mol/m3). Panels b and d are for high concentration (2.6 mol/m3). P CO2 in the cave is 4 · 104 atm. Dashed lines depict fractionation in isotopic equilibrium (case A). Full lines show the effect of kinetic fractionation by irreversible outgassing of CO2 (case B).

1C-A 1C-B 1C-C 10 C - A 10 C - B 10 C - C 20 C - A 20 C - B 20 C - C 26 C - A 26 C - B 26 C - C

3

0

cin = 1.37 mol/m

-2 -4

4

3

cin = 2.6 mol/m

2

-4

PCO = 4x10 atm

0

2

Drip time = 30 s

-2 -4

-8

-6 -6

-10 -12

δ C [permille]

-10 -8

-14 -16

13

-10

-18 0 -20

-6

-8

13

δ C [permille]

-6

6

0

2

4

6

8

10 20 30 40 50 Length along profile [cm]

-8

-12 -14

-10

-16 -18

10 60

-20

0 0

2

4

6

8

10 20 30 40 50 Length along profile [cm]

10 60

Fig. 5. d13C profiles as function of distance from apex along the growth surface of a stalagmite for various temperatures. Dashed lines, isotopic equilibrium (case A). Full lines, kinetic fractionation by irreversible outgassing of CO2 (see Fig. 4b and c) (case B). Dotted lines depict the additional effect of kinetic rapid precipitation of calcite with a1kin = 1 (case C). Grey shades depict temperature as in Fig. 4. The insets show an enlargement of the profiles for lengths below 10 cm.

5.2. Comparison to experimental data Recently, Mickler et al. (2006) have collected samples of modern calcite precipitated to 10 cm · 10 cm frosted glass plates mounted horizontally to the top of active stalagmites. After 2 years of exposure these samples were

analyzed on a grid of 2 cm · 2 cm for spatial distribution of thickness of the precipitated calcite layer and its d13C as function of the distance from the drip point. The authors also determined the chemistry of the drip water. Concentration of HCO3  was 4.05 ± 0.03 mol/m3, d13C = 12.1 ± 0.4[permille]. Two observations yielded

d13C profiles along growth layers of stalagmites

drip times of 20 to 60 s. Relative humidity was 98,2%, excluding effects of evaporation. P CO2 in the cave was not determined. Therefore we have used P CO2 ¼ 4  104 atm. Although calcite precipitation initially starts at a non-calcite surface by nucleation, after deposition of about 10 lm one expects this surface to be entirely covered by calcite. Then precipitation rates are determined by deposition to calcite. At the observed thicknesses of more than 200 lm one does therefore not expect significant influence of the glass surface. We use data from Fig. 3 in Mickler et al. (2006). We have simulated the growth surface and the d13C profile by employing the FLOW-model for various drip times and P CO2 (for Rayleigh distillation under isotopic equilibrium (case A), kinetic fractionation for irreversible outgassing of CO2 (case B), and irreversible kinetic fractionation (case C) for both outgassing and precipitation of calcite at a temperature of 26 C, the average temperature at the glass plate side). The initial model surface is a cone with 1 m diameter and 0.1 cm height. This is necessary to establish flow away from the apex. Although Mickler used a plane glass plate this is realistic. When a drop hits this surface and uneven water surface will be created with maximum thickness at the apex. Therefore a hydraulic head exists establishing flow. Fig. 6a shows the thickness of deposited calcite for various P CO2 in the cave at a drip time of 30 s. As illustrated by the figure variations of P CO2 from 0.0004 to 0.0016 atm cause only little changes of the profiles. Fig. 6b depicts the thickness of the deposited layer as function of the distance from the drip point for various drip times at P CO2 ¼ 4  104 atm. Best agreement to the experimental points is obtained for s between 40 and 60 s. Within these variations of drip time and P CO2 the agreement to the exper-

a

imental data is remarkable. We have also changed the inclination of the initial surface of the theoretical calculations from a cone with 1 m diameter at its base and 0.1 cm height to 1 cm height. There was no change in the profiles by more than 1%. This shows that choosing a conical initial surface is a good approximation to reality. Fig. 7a depicts d13C for various P CO2 in the cave as a function of distance from the drip point for isotopic equilibrium (case A, dashed lines) and irreversible kinetic (case B, full lines), for various drip times and T = 26 C. Full kinetic fractionation for both outgassing and calcite precipitation (case C) is shown by the dotted lines. There is unsatisfactory agreement for equilibrium fractionation. Using, however, irreversible kinetic fractionation yields excellent agreement to the observed data for s between 60 and 20 s. Fig. 7b shows d13C profiles for various drip times. The experimental data agree closely with the theoretical curves for full kinetics within the variations of drip time and P CO2 . 6. DISCUSSION Any model has its limits and these have to be considered carefully. The FLOW-model idealistically assumes that splashing of the drops does not play an important role. This can happen in the high flow regime and when the fall height of the drop is low. Then a continuously flowing film of water spreads on the surface of the stalagmite, and the FLOW-model is appropriate. This has consequences to the evolution of the isotopic profiles along a growth layer. In any case calcite precipitated at the apex is in equilibrium with the drip water. This is true for both 18O and 13C. In the case of oxygen the buffering action of 18O in water can become active only, when a deviation from equilibrium is present. This, however, does not happen at the apex. The

b

0.2

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0.2

Mickler (2006)

Mickler (2006) 20 s 30 s 40 s 50 s 60 s

-4

4x10 atm -3

1x10 atm -3

1.6x10 atm

0.15

Thickness [cm]

Thickness [cm]

0.15

0.1

0.05

cin = 2.025 mol/m

3

0.1

0.05

cin = 2.025 mol/m

3

Temperature = 26ºC

Temperature = 26ºC

Drip time = 30 s

PCO = 4x10 atm

-4

2

0

0

10 5 15 Length along profile [cm]

20

0

0

10 5 15 Length along profile [cm]

20

Fig. 6. Thickness of deposited calcite layer as a function of distance from the drip point, for the chemical parameters observed by Mickler et al. (2006). See text: (a) for various P CO2 in the cave at a drip time of 30 s; (b) for various drip times at a P CO2 of 4 · 104 atm. The experimental data of Mickler et al. (2006), are plotted as diamonds.

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b

-4

4x10 atm - A

3 2

-4

4x10 atm - B

1

-4

4x10 atm - C -3

1x10 atm - A

20 s - A 20 s - B 20 s - C 30 s - A 30 s - B 30 s - C 40 s - A 40 s - B 40 s - C 50 s - A 50 s - B 50 s - C 60 s - A 60 s - B 60 s - C

0

-3

1x10 atm - B

-1

-3

1x10 atm - C -3

1.6x10 atm - A

-2

-3

1.6x10 atm - B -3

1.6x10 atm - C

13

6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13

δ C [permille]

13

δ C [permille]

a

3

cin = 2.025 mol/m

-3 -4 -5 -6 -7 -8 -9

0

3

-10

cin = 2.025 mol/m

Temperature = 26ºC

-11

Temperature = 26ºC

Drip time = 30 s

-12

PCO = 4x10 atm

2 4 8 10 12 6 Length along profile [cm]

-13

-4

2

0

4 8 12 16 Length along profile [cm]

20

Fig. 7. Profiles of d13C in recent deposited calcite as a function of distance from the drip point. See Fig. 6: (a) for various P CO2 in the cave as in Fig. 6a; (b) for various drip times as in Fig. 6b. Dashed lines depict profiles deposited in isotopic equilibrium (case A). Full lines depict profiles for irreversible outgassing (case B) and dotted lines depict irreversible outgassing and kinetic precipitation of calcite (case C). The experimental data are depicted as diamonds.

consequence of this finding is most important. If the conditions, under which the stalagmite grows match those of the FLOW-model then a correlation between d13C and d18O is still present, but at the apex the calcite has been precipitated in equilibrium with respect to the drip water. This means that the Hendy test is not indispensable, and paleoclimatic records from stalagmites, which exhibit positive correlation between d13C and d18O could still be used. Note that Mickler et al. (2006) have reported, that most of 165 published speleothem stable isotope records show such a positive correlation. Also Fleitmann et al. (2004) have used isotopic records from Southern Oman stalagmites, which show a significant correlation. On the other hand at high drip intervals, s > 600 s, which is also common in caves, a stagnant film will cover the stalagmite. This could be described by the model of Mu¨hlinghaus et al. (2007). Assuming a splash coefficient U1 6 1 will cause fractionation at the apex, because during two drops, Rayleigh distillation is operative also at the apex. Calcite precipitation here will not be in equilibrium with the drip water, but will show a substantial increase of d 13C with decreasing mixing coefficient U1. Furthermore it may be extremely difficult or even impossible to assign a value of U1 to a special stalagmite. As a consequence stalagmites should be selected by criteria, which meet the FLOWmodel. These are large diameters of at least 0.1 m, which ensures a drip interval of s < 50 s at a temperature of 20 C and one of s 6 100 s at 10 C, respectively. For diameters of about 0.3 m drip time is lower by about a factor of 9 (Dreybrodt, 1988). For such diameters the influence of splashing could also be less significant, because drops after touching the surface at low flow height will spread to a smaller part of the stalagmites surface for stalagmites with large diameters.

It should be noted here that the physics of splashing droplets is extremely complex and by no way in any model its influence can be exactly described. One note should be given to the kinetic effects. We assume that the drop pending on a stalactite has already attained equilibrium of its CO2 with respect to the cave atmosphere. This is the first step of outgassing. Later on outgassing of CO2 is a result of CaCO3-precipitation, which is irreversible. Therefore it is most likely that kinetic fractionation factors must be used. This increases the amplitude of d13C in comparison to equilibrium fractionation. The amplitude of d13C depends also on (cin  ceq) as can be seen from Eq. (16). Changes of pCO2 in the cave atmosphere alter ceq and can therefore have influence to the d13C amplitude. This can be seen in Fig. 7. 7. CONCLUSIONS Interpretation of isotopic compositions of calcite from stalagmite needs a sound basis of knowledge on calcite precipitation to the surface of the speleothem. In this paper, we have used the theory of stalagmite growth based entirely on the physics of laminar flowing water sheets and the precipitation rates of calcite from supersaturated H2O–CO2– CaCO3 solutions, without any other arbitrary parameters. With this approach the calcium concentration of the supersaturated solution can be obtained at all points of the stalagmites surface, even if it has not attained its equilibrium shape. From this knowledge it is possible to calculate profiles of d13C along growth surfaces of stalagmites. These are determined by Rayleigh distillation of 13 C–12C in a water parcel flowing down a stalagmite. Such profiles are presented for various drip rates and temperatures. Three approaches are taken: Case A: The reactions

d13C profiles along growth layers of stalagmites

converting HCO3  to COgas 2 and precipitation of CaCO3 are in isotopic equilibrium. Case B: outgassing of CO2 is irreversible, but calcite is precipitated in isotopic equilibrium. Case C: Both CO2 degassing and calcite precipitation proceed irreversibly. It turns out that all profiles similarly reflect the isotopic composition of the drip water at the apex and show enrichment in 13C with distance from it. The total amount of enrichment for irreversible fractionation is higher by a factor of about 2 compared to fractionation in isotopic equilibrium. We have applied our theory to experimental data of calcite deposited to glass plates mounted on top of actively growing stalagmites (Mickler et al., 2006). Excellent agreement is achieved for growth rates and d13C profiles, which show that calcite was precipitated by irreversible fast outgasing of CO2 and kinetic fractionation of HCO3  –CaCO3 . Our theory helps to better understand the influence of temperature, drip time and calcium concentration in the drip water, all depending on climatic conditions and it may be of use unraveling the complicated mechanisms that determine climatic signals hidden in speleothems. ACKNOWLEDGMENTS We thank Dr. Scholz, two anonymous reviewers and the Associate editor Dr. Miryam Bar-Matthews for the time they invested into our manuscript. Their useful comments helped to improve the paper.

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