Appendix Climate Model Description


These calculations, and those that follow in the next section, assume a balance between a land weathering sink for carbon dioxide and a metamorphic and juvenile outgassing source. The organic carbon burial sink is ignored. Organic carbon burial is irrelevant to abiotic models. As a first approximation, surface temperatures on Earth through the Archean are computed, ignoring possible variations in net organic carbon burial in the past.

Another carbon sink has been proposed—namely, the reaction of carbon dioxide with oceanic basaltic crust (see discussion in chapter 2), particularly in the Archean (Walker 1983, 1985; Staudigel et al. 1989; Veizer et al. 1989a, 1989b). Neodymium isotopes in Archean and Proterozoic sediments apparently support the notion that Archean ocean chemistry was dominated by the reaction with basalt, presumably at the ridges ( Jacobsen and Pimentel-Klose 1988a, 1988b; for a contrary view see Alibert and McCulloch 1990). However, a thermal barrier to subduction of carbonate sediment or reaction product with basaltic crust (Des Marais 1985) may simply contribute an equivalent volcanic carbon outgassing source to the postulated sink, without affecting geochemical modeling of a land-weathering sink balancing the total assumed volcanic/metamorphic outgassing source, provided the variation of the latter parallels the fraction balancing the land-weathering sink (Berner 1990b).

First recall the equation given in chapter 6:

An equation for the calculation ofpast surface temperatures (global mean) is derived for different assumed values of the ratio of biotic enhancements to that of the past (BR). In general,

where Bt is the biotic enhancement at time t; (Pb, TJ and (Pt, Tt) are the abiotic atmospheric pCO2 and temperature and biotically enhanced values, respectively; AT = Tb — Tt; and the reference state is at time t and not the present as in equation (1). As before, To is taken as 288°K, and assumed values are £ = 0.056, 7 = 0.017, and e<^T> < 2.


(A/A„)(V/V) (Pab/P)a e«) e~<(T-1 -T) (3) B/ B = -

where the numerator is simply from equation (1). Reducing to

Time B.P. (billion years)


Growth of continental land area (mass) versus time; A/Ao is the ratio of land area at time t to present. Curves b and c correspond to our models b and c, respectively, described in text. Curves (1) and (2) are from limiting tectonic/geochemical models of mean age of sediments and continents of Allegre and Jaupart (1985); curve J is from Jacobsen's (1988) inversion of Sm-Nd mass balance for the depleted mantle-continental crust system. Curve L, the linear growth of continental area through time, with A = 0 at 3.8 Ga, is taken as a lower limit to A/Ao as a function of age. Substantial early growth of the continents has received recent support from Bowring and Haush (1995), but a vigorous controversy is now in progress on this issue.

FIGURE 8-12.

Carbon outgassing rate relative to present rate (V/V) versus time (B.P.) for models a, b, and c.

FIGURE 8-12.

Carbon outgassing rate relative to present rate (V/V) versus time (B.P.) for models a, b, and c.

Ao, A and Vo, V are defined here as the present and past continental land areas and volcanic/metamorphic outgassing rates (of carbon dioxide), respectively (figures 8-11 and 8-12). The following functions for volcanic outgassing, V, and land area, A, variation are assumed:

Vparallels the decrease in radioactive heat generation in the Earth to present, where t is time (BP) in billions of years. We couple the rates of continental land area growth and volcanic outgassing:

The rationale for assuming linkage of land area and outgassing rate is that both outgassing of carbon dioxide and continental crust generation are presumably a function of subduction and juvenile outgassing rates. Furthermore, intracontinental growth via underplating results in both outgassing and land area increases through uplift. This parameterization also simplifies

TABLE 8-3.

Parameters for Assumed Variation of Land Area (A) and Carbon Dioxide Outgassing Rate (V) as a Function of Age (t)

Model a (constant) b (preferred) c (upper limit)

(V/V„)(AJA)at t = 3.8 Ga 1 x 1 = 1 3 x 4 = 12 8 x 10 = 80

0.375 0.289

0.129 0.547

model calculations. In any case, an alternative parameterization of land area as a function of time (Allegre and Jaupart 1985; Jacobsen 1988; see figure 8-11) consistent with the isotopic evolution of the crust/mantle system, gives similar model results for computed surface temperatures as a function oftime (see next section). The variation of land area and outgassing rate as a function of time follows from models a, b, and c in table 8-3, with model a representing a constant V and A, model b the preferred intermediate variation, and model c an upper limit to Vand A since 3.8 Ga.

The relationship between surface temperature Tt and P,, pCO2 at any time t, was computed from two different greenhouse functions:

(a) For Pt < 0.03 bar, an updated version of Kasting and Ackerman's (1986) function, given in equation form by Caldeira (personal communication), with Tt = f (P, St), where St is the relative solar flux at time t:

(7) Tt = 138.114 - 73.179(p) - 73.960(p2) + 56.048(p3)

+ 405.836(St ) + 595.774( p )( St ) + 385.004(p2)(St)

- 548.963(p2)(St2) + 461.125( p3)(St2) + 129.545(St3)

The maximum error is 1.95 K, the root mean square error is 0.65 K.

(8) St = (1 + 0.0835t)-1D~2, where p is the distance of the Earth to the Sun (AU) [St as an f (t ) from Caldeira and Kasting 1992b]. (D values different from 1 AU will be pertinent to modeling in chapter 11).

(b) For 0.03 > Pt ^ 0.0003 bar, a slightly modified function from Walker et al. (1981) (T = 288°K rather than 285°K, present T = 255°K rather than 253°K):

(9) T = 2 T + 4.6 (PP/P )0364 - 226.4, where T is the effective radiating temperature of the Earth (°K; no greenhouse effect) at time t. The following relation for T from Kasting (1987) is used:

At low global mean temperatures (TJ, the use of Tas above rather than the actual latitudinal distribution of temperature will exaggerate the feedback between climate and weathering rate because a much smaller contribution now comes from higher latitudes (White and Blum 1995), and latitudinal differences in temperature likely decrease with higher Tm. For model climates at higher Tm atmospheric pCO2 levels than present values, the choice of Tm instead of the actual latitudinal distribution of temperature and rainfall probably becomes more realistic because latitudinal differences decline as Tm increases and may actually disappear at Tm equal to about 30°C (Hoffert, personal communication). Using the present continental latitudinal values of temperature and runoff (or precipitation) gives a much lower computed weathering rate (about 7%) compared with that derived from the Tm model, with the latitudinal distribution of runoff responsible for this difference (computed using temperature and runoff from Francois and Walker 1992, land area for 5° latitudinal strips from Sverdrup et al. 1942 with data from Kossinna 1921). However, the computed rate for actual latitudinal distribution of land, temperature, and runoff may well underestimate the actual silicate weathering sink because it does not take into account the silicate denudation rates by latitude. Another approach to this problem is as follows.

Assume using the present Tm instead of the actual latitudinal temperature variation and runoff pattern to get the present global weathering rate (Wn). Substituting Tm in the previously derived equation:

TABLE 8-4. Computed Model Results

Atm. pCO2

Time B.P

Required BR

T (°C)
























Because the computed Wn is greater than the actual global weathering rate, a lower Po should be used to compensate, i.e., to bring the computed Wn down to the actual weathering rate. Using this hypothetical Po will give higher computed BR estimates.


A value of a = 0.3 is assumed, corresponding to abiotic weathering of CaMg silicates in an open system (Schwartzman and Volk 1989; Berner 1992, 1993); in a closed or nearly closed system, a most likely is > 0.3 (table 8-4). The choice of a, (P (probable minimum*), and greenhouse function (pCO2 is the probable minimum because cloud feedbacks are assumed absent) all make the computed BR values lower limits. The normalization to the present global mean temperature rather than to the actual present latitudinal distribution of temperature and runoff also probably makes the computed BR values lower limits (see previous discussion on the normalization problem). Probable higher impact-derived regolith, volcanic contribution to weathering, and the postulated sea floor weathering carbon sink (Francois and Walker 1992) relative to the present have the same effect on on the limit to Br. Two possible factors would make the computed BR values too high: (a) reducing gas greenhouses in the Archean [although the (Pt/Po)a term

*Regarding the eP(AT) term, the more precise formulation using [(1/288) — (1/T)] instead of a AT term, as well as the low activation energy (E) assumed (12.5 kcal/ mole), give somewhat lower eP(AT) values than for Brady and Carrol's (1994) lower limit on E (11.5 kcal/mole). Estimates of E are > 11.5 kcal/molefor CaMg silicates for most experiments near pH = 7 (Brantley and Chen 1995). Furthermore, E tends to increase as pH drops from the neutral range (Casey and Sposito 1992), again making eP(AT) values lower limits for the assumed p (= 0.056) for high pCO2.

would drop, only modest increases in the assumed a and P values would give the same or greater computed BR values; see also previous discussion].

(b) greater net organic carbon burial now relative to the past (Des Marais et al. 1992). The present ratio is about 5.4 (data from Berner 1994) if V corresponds to volcanic/metamorphic flux only (weathering of kerogen is then substracted from organic C burial; see discussion in chapter 1). Let R = ratio of silicate weathering to net organic C sink, assume R = 5.4 now. If all the carbon sink with respect to the atmosphere/ocean was silicate weather-

mga^.5 then ( V>/ V3.5 GaXilicate sink = 0.84 ( Vo/V3.5 Ga)xotal, since the (silicate weathering C sink/Total C sink)now = 0.84. Hence, under the above assumptions, the maximum reduction of BR at 3.5 Ga is only 16% lower than the value computed without consideration of the influence of the organic C sink.

The effects of assumed model ratios of VJV and A/Ao being too low, as a result of the neglect of degassing from subduction of Cenozoic deposits of pelagic CaCO3 (Volk 1989a; Caldeira 1991) and early continental crust formation (Bowring and Housh 1995) results in the computed BR values again being minimal. On the other hand, pelagic CaCO3 deposition also may have occurred in the Archean (Grotzinger 1994). Subduction of exogenic carbon into the mantle and its reappearance in volcanic outgassing (Hauri et al. 1993) has probably decreased the variation of VJV ratios over time. From the observed carbon and helium isotopic compositions and CO2/3He ratios of volcanic gases, Sano and Williams (1996) estimated that 60% of the present degassed carbon flux comes from subducted sediment carbon (mainly carbonate).

A rough material balance of recent pelagic carbonate deposition and sub-duction/decarbonation fluxes indicates that about one third of pelagic carbonate is now degassed, implying carbon loss to the mantle (Caldeira 1991). This conclusion is supported by a recent study (Nishio et al. 1998). Furthermore, it is possible that no exogenic carbonate reservoir existed 3.8 billion years ago as a result of low pH oceans and rainwater. A shift from the contribution to outgassing of juvenile to metamorphic carbon (in subduction zones) from the Precambrian to now would tend to stabilize the flux as a function of time. Clearly, a greater understanding of the carbon geodynamic cycle is needed before results from modeling the carbonate-silicate cycle back into the Precambrian can be accepted as definitive. Nevertheless, useful limits can be obtained with modeling.

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