## Long Term Model Calculations

Values of fluxes in the long-term carbon cycle can be calculated from the fundamental equations for total carbon and 13C mass balance that are stated in the introduction and are repeated here:

+ SmgFmg - SbcFbc - SbgFbg d(5cMc)/dt = SwcFwc + SwgFwg + SmcFmc (1.11)

where

Mc = mass of carbon in the surficial system consisting of the atmosphere, oceans, biosphere, and soils flux from weathering of Ca and Mg carbonates flux from weathering of sedimentary organic matter degassing flux for carbonates from volcanism, metamorphism, and diagenesis degassing flux for organic matter from volcanism, metamor-phism, and diagenesis burial flux of carbonates in sediments burial flux of organic matter in sediments [(13C/12C)/(13C/12C)stnd - 1]1000.

Variants of equations (1.10) and (1.11) have been treated in terms of non-steady-state modeling (e.g., Berner et al., 1983; Wallmann, 2001; Hansen and Wallmann, 2003; Mackenzie et al., 2003; Bergman et al., 2003), where the evolution of both oceanic and atmospheric composition, including Ca, Mg, and other elements in seawater, is tracked over time. However, since the purpose of this book is to discuss the carbon cycle with respect to CO2 and O2, and so as not to overburden the reader with too many mathematical expressions, I discuss only those aspects of the non-steady-state models that directly impact carbon. These are combined with results from steady-state strictly carbon-cycle modeling (Garrels and Lerman, 1984; Berner, 1991, 1994; Kump and Arthur, 1997; Francois and Godderis, 1998; Tajika, 1998; Berner and Kothavala, 2001; Kashiwagi and Shikazono, 2002). For steady state, we have (from chapter 1):

where Fwsi represents the weathering of Ca and Mg silicates with the transfer of carbon from the atmosphere to Ca and Mg carbonates (reaction 1.4).

In all the long-term carbon cycle models, values of carbon fluxes and the concentration of atmospheric CO2 are obtained by combining variants of equations (1.10) and (1.11), with additional expressions for fluxes that include nondimensional weathering and degassing parameters. An example of this approach, taken from steady-state GEOCARB modeling (Berner, 1991, 1994; Berner and Kothavala, 2001), is:

Fwc(t) = fBc(CO2)fBb(CO2) fE(t)fAD(t)fLA(t) kwc C (5.3)

1 wc

1 wg

1 mg

Fbc Fbg 5

where

C, G = global mass of sedimentary carbonate carbon and organic carbon kwc, kwg = rate constants = Fwc(0)/C, Fwg(0)/G for the present earth fBc(CO 2) = fBt(CO2) with different Z values for carbonate dissolution fAD(t) = fo(t)fA(t) fLA(t) = fL(t)fA(t).

Fwsi(0), Fmc(0), and Fmg(0) refer to the present earth, and the other di-mensionless f parameters are defined and discussed in earlier chapters. For convenience, a summary of definitions of all terms is shown in table 5.1. In GEOCARB modeling, the concentration of atmospheric CO2 at each time is calculated by solving equation (5.2) for fB(CO2) = [fBt(CO2)fBb(CO2)] (or for the prevascular plant land surface, fB(CO2) = [fBt(CO2)fBnb(CO2)]) and then by inverting the resulting complex expression to solve for RCO3, the ratio of the mass of atmospheric CO2 at some past time to that for the preindustrial present (280 ppmv). A plot of RCO2 versus fBt(CO2) from GEOCARB III (Berner and Kothavala, 2001) is shown in figure 5.1. The plot shows sudden variations, not seen in the smooth plot of RCO 2 versus the greenhouse parameter fBg(CO2) (see figure 2.5), because of variations over time of GEOG, the effect of changing continent size and distribution on land temperature.

To be able to solve the above equations, it is necessary to track C and G over time. If total crustal carbon is assumed to be constant (in other words, the loss of carbon to the mantle equals its gain from the mantle), then C + G = constant, and dC/dt = Fbc - Fwc - Fmc (5.7)

Analogous expressions for 13C are d(8cC)/dt = Sbc Fbc - SwcFwc - §mcFmc (5.9)

For simplification one can assume that 8c = 8wc = 8mc and 8g = Swg = 8mg. An additional simplification is the assumption that S^ represents the

Table 5.1. Definitions of terms used in GEOCARB long-term carbon cycle modeling.

Fluxes, masses and isotopic composition

Fwc = carbon flux from weathering of Ca and Mg carbonates Fwg = carbon flux from weathering of sedimentary organic matter Fmc = degassing flux from volcanism, metamorphism, and diagenesis of carbonates Fmg = degassing flux from volcanism, metamorphism, and diagenesis of organic matter

Fbc = burial flux of carbonate-C in sediments Fbg = burial flux or organic-C in sediments C = mass of crustal carbonate carbon G = mass of crustal organic carbon kwc = rate constant expressing mass dependence for carbonate weathering kwg = rate constant expressing mass dependence for organic matter weathering swc = s13C of flux from weathering of Ca and Mg carbonates swg = s13C of flux from weathering of sedimentary organic matter smc = s13C of degassing flux from carbonate decomposition due to volcanism, metamorphism, and diagenesis smg = s13C of degassing flux from organic decomposition due to volcanism, metamorphism, and diagenesis sbc = s13C of burial flux of carbonates in sediments (assumed same as s13C of ocean) sbg = s13C of burial flux of organic matter in sediments

### Dimensionless parameters

RCO2 = mass of CO2 in atmosphere (t)/mass of CO2 in atmosphere (0) fR(t) = effect of relief on weathering rate (t)/effect of relief on weathering rate (0) fE(t) = effect of plants on weathering rate (t)/effect of plants on weathering rate (0)

fBb(CO2) = effect of CO 2 on plant-assisted weathering (t)/effect of CO2 on plant-

assisted weathering (0) for silicates and carbonates fBnb(CO2) = direct effect of atmospheric CO2 on weathering rate (t)/direct effect of atmospheric CO2 on weathering rate (0) for silicates and carbonates (applied to period before rise of large land plants) fBg(CO2) = effect of temperature on weathering rate (t)/effect of temperature on weathering rate (0) for silicates due to CO2 greenhouse effect alone fBt(CO2) = effect of temperature on weathering rate (t)/effect of temperature on weathering rate (0) for silicates combining effects of CO2 greenhouse, solar radiation, and paleogeography on temperature fBc(CO2) = effect of temperature on weathering rate (t)/effect of temperature on weathering rate (0) for carbonates combining effects of CO2 greenhouse, solar radiation, and paleogeography on temperature fD(t) = runoff (t) / runoff (0) due to changes in paleogeography fA(t) = land area (t)/land area (0).

fL(t) = fraction of total land area covered by carbonates/the same fraction at present fSR(t) = seafloor area creation rate (t)/seafloor area creation rate (0) fC(t) = effect of carbonate content of subducting oceanic crust on the rate of CO2 degassing (t)/the same effect at present

From Berner (1991, 1994); Berner and Kothavala (2001).

Figure 5.1. Plot of fBt(CO2) versus RCO2. The parameter fBt(CO2) reflects the effect on the rate of silicate weathering and atmospheric CO2 of changes over time in temperature, due to the CO2 greenhouse effect, changes in solar radiation, and changes in mean land temperature accompanying continental drift. RCO2 is the ratio of the mass of atmospheric CO2 at a past time (t) to that at present (assumed to be 280 ppm). The irregularities in the curve (compare with figure 2.5) are due primarily to variations of the GEOG term for mean land temperature.

fBt(C02)

Figure 5.1. Plot of fBt(CO2) versus RCO2. The parameter fBt(CO2) reflects the effect on the rate of silicate weathering and atmospheric CO2 of changes over time in temperature, due to the CO2 greenhouse effect, changes in solar radiation, and changes in mean land temperature accompanying continental drift. RCO2 is the ratio of the mass of atmospheric CO2 at a past time (t) to that at present (assumed to be 280 ppm). The irregularities in the curve (compare with figure 2.5) are due primarily to variations of the GEOG term for mean land temperature.

carbon isotopic composition of the oceans (literally, it is the mean composition of carbon in buried carbonates) and that Sbg - Sbc = A13C, which can be held constant or varied over time.

Combining the above expressions constitutes GEOCARB modeling. By inserting values for present-day masses and fluxes, using the carbon isotopic composition of the oceans plus values of the dimensionless f parameters over the Phanerozoic, and by assuming starting values (at 550 Ma) for fluxes, masses, and isotopic composition, a series of steady-state equations for the past 550 million years is solved at each 1 million year time-step for all fluxes and for RCO2. The assumed starting values are validated only if the present CO2 level is obtained. As values for the various dimensionless parameters are inputted only every 10 million years, with linear interpolation during the time gaps, output results of the modeling are reported on the same basis. This precludes consideration of any phenomenon occurring on a time scale shorter than 10 million years.

The carbon cycle modeling approach of other studies (Kump and Arthur, 1998: Francois and Godderis, 1998; Tajika, 1998; Wallmann, 2001; Kashiwagi and Shikazono, 2002) is fundamentally similar to that for GEOCARB. However, modifications of the fluxes and dimensionless parameters (f) are made, and these modifications are worth discussing in some detail. Kump and Arthur (1997), in a model for the Cenozoic carbon cycle, use the parameter fwr(t) to represent all other factors affecting weathering other than fL(t) and fA(t). In essence they are combining the effects of both climate, fBt(CO2), paleogeographically-affected runoff, fD(t), and mountain uplift/erosion, fR(t), into their fwr(t) parameter. Francois and Godderis (1998), in their model for the Cenozoic, join carbon cycle modeling with modeling of Sr isotopes and greatly simplify weathering such that carbonate weathering, Fwc, and organic matter weathering, Fwg, are assumed to be directly proportional to silicate weathering, Fwsi.

Tajika (1998), in a model for the past 150 Ma, assumes minor loss of carbon to the mantle via subduction and divides the global CO2 degassing flux (Fmc + Fmg) into separate fluxes at mid-ocean ridges, that associated with intraplate (hot-spot) volcanism, and that accompanying metamorphism and volcanism at subduction zones. Mid-ocean ridge and subduction zone degassing is assumed proportional to seafloor spreading rate, fSR(t),whereas for hot-spot degassing over time he introduces a new parameter, fH(t), which varies differently from spreading rate and relies on the data of Larson (1991). Tajika also separates the global weathering flux into that for the Himalayas (including the uplift parameter fR(t) based on Sr isotopes) and that for the rest of the world.

In his carbon cycle model for the past 150 Ma, Wallmann (2001), like Tajika (1998), separates degassing into that at ridges, subduction zones, and hot spots with degassing at ridges and hot spots guided by spreading rate and that at hot spots guided by the data of Larson (1991). Unlike GEOCARB, Wallmann's model emphasizes the submarine weathering of basalt to CaCO3. Other differences are that he considers loss of carbon to the mantle, calculates the carbon isotopic composition of the oceans over time (rather than using equation 1.11), and separates Ca and Mg silicate weathering into young volcanics at convergent margins and silicates from the rest of the world. Kashiwagi and Shikazono (2002), in a model for the Cenozoic, distinguish subduction zone degassing at island arcs from that at back-arc basins, and they, like Wallmann, separate weathering of silicates for the Himalayas from that for the rest of the world.

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