11.2.1 Steady State and Residence Time

The rate of change of C mass in reservoir i, dM_i/dt, is given by

$\frac{{\mathit{dM}}_i}{\mathit{dt}}=\sum J_{\mathit{ji}}-\sum J_{\mathit{ij}},$(11.1)

where the first term on the right represents the sum of all mass flows of C from reservoirs j into i and the second term right represents the sum of all mass flows of C out of reservoir i to reservoirs j. More simply, (11.1) expressed as total inputs J_in and outputs J_out from a reservoir M, such as the entire exogenic system M_ex (ocean + atmosphere + biosphere), is given by

$\frac{{\mathit{dM}}_{\mathit{ex}}}{\mathit{dt}}=J_{\mathit{\text{in}}}-J_{\mathit{out}}.$(11.2)

To completely model the CO₂ content of the atmosphere (and not just the total exogenic system), both alkalinity and total C fluxes must be tracked, as the stoichiometry of these fluxes can vary considerably and their relative balance determines the proportion of exogenic C partitioned into the atmosphere. The treatment here, by ignoring this complexity, captures the first-order behavior of the system. J_in would then represent all inputs of C (volcanic and metamorphic degassing) and J_out the total C drawdown by silicate weathering (followed by carbonate burial) and organic C burial. J_out must scale with the amount of C in the system; after all, if there is no C in the exogenic system, then the output must be zero. At the most basic level, we can adopt a linear scaling, such that

where k_out is the sum of all rate constants (s^–1) describing the efficiency of different pathways by which C is sequestered from the exogenic system

Analogous to radioactive decay, k_out represents the probability that an atom of C is removed from the system per unit time. Because the output rate depends on the amount of C in the system, k_out represents a negative feedback, which essentially counteracts any change in C in the system (see LasagaReference Lasaga²⁹). As we will discuss later, k_out represents a complex function of many processes. For example, C removal by silicate weathering and carbonate precipitation might be enhanced with greater water precipitation and temperature, both of which increase when atmospheric CO₂ increases.Reference Walker, Hays and Kasting² The net effect is a negative feedback (Figure 11.4).

Figure 11.4

Feedback loop describing the silicate weathering feedback acting on the exogenic C system. Arrows correspond to transfer functions, with positive symbols indicating positive feedback and negative symbols representing negative feedback. Increases in pCO₂ increase temperature, which leads to enhanced precipitation and runoff, accelerating chemical weathering rates and drawdown of CO₂. Similarly, an increase in pCO₂ increases seafloor weathering rates, increasing drawdown of CO₂. Tectonics and mantle dynamics drive the entire system by: (1) increasing erosion rates during mountain building, thereby increasing the availability of weatherable substrate and drawdown rates of CO₂; and/or (2) increasing magmatic or metamorphic inputs of CO₂ into the exogenic system. Whether tectonics enhance degassing or drawdown will depend on the nature of the tectonic activity.

J_in must also scale with the mass of C from which the flux derives. If most of J_in derives from volcanic degassing, then J_in = k_inM_m, where M_m represents the mass of C in the mantle. If we assume that the mass of C in the mantle is much larger compared to that in the exogenic system, then J_in is approximately constant and (11.2) becomes a first-order differential equation:

$\frac{{\mathit{dM}}_{\mathit{ex}}}{\mathit{dt}}=J_{\mathit{\text{in}}}-k_{\mathit{out}}{M}_{\mathit{ex}},$(11.5)

with the following solution:

$M_{\mathit{ex}}\left(t\right)=\frac{J_{\mathit{\text{in}}}}{k_{\mathit{out}}}+\left(M_{\mathit{ex}}^o-\frac{J_{\mathit{\text{in}}}}{k_{\mathit{out}}}\right)exp\left(-k_{\mathit{out}}t\right).$(11.6)

Here, $M_{\mathit{ex}}^o$ represents an initial perturbation to the amount of C in the exogenic system. Equation (11.6) shows that the system will return exponentially to a steady state, given by

$M_{\mathit{ex}}^{\infty }=J_{\mathit{\text{in}}}/k_{\mathit{out}}.$(11.7)

At steady state, inputs and outputs are balanced, hence the C content of the exogenic system does not evolve (steady state does not mean thermodynamic equilibrium). The time for the system to relax back to steady state after a perturbation (response time) is roughly the e-fold timescale of the system; that is:

For a linear feedback, the response time is equivalent to the residence time of C in the system (Figure 11.1d). The residence time represents the average time a C atom remains in the exogenic system when the system is at steady state, dM_ex∕dt = 0. At steady state, J_in = J_out, such that the residence time of C in the system is given by

${\tau }_{\mathit{ex}}^r=\frac{M_{\mathit{ex}}}{J_{\mathit{\text{in}}}}=\frac{M_{\mathit{ex}}}{J_{\mathit{out}}}=\frac{M_{\mathit{ex}}}{k_{\mathit{out}}{M}_{\mathit{ex}}}=\frac{1}{k_{\mathit{out}}}.$(11.9)

When negative feedbacks operate, a system perturbed suddenly, such as by a pulse of CO₂ outgassing, will eventually return to a steady state, defined by the ratio of the external forcings (e.g. the input) to the efficiency of the negative feedback, k_out (11.6). On timescales longer than the response time of the system, the C in the system should evolve toward a steady state, after which the only processes that can change the C content in the exogenic system would be temporal changes in the forcings, such as changes in magmatic or metamorphic activity, or changes in the efficiency of carbonate precipitation, which may relate to changes in tectonics and the efficiency of silicate weathering.

The response time for C in the exogenic system has been determined in two ways. One approach is to take the mass of C in the exogenic system and divide it by the measured inputs or outputs to obtain residence times. This approach is only valid if the measured inputs or outputs truly represent steady-state conditions. Another approach is to measure the relaxation time of a perturbed system, with one example being the C isotope excursion observed at the Paleocene–Eocene thermal maximum (PETM).Reference Zachos, Dickens and Zeebe⁴ Both approaches suggest that C in the exogenic system has a response time between 10 and 100 ky (Figure 11.1c and d).Reference Zachos, Dickens and Zeebe⁴ Thus, on My timescales, the C content of the exogenic system should be close to steady state. However, the fact that atmospheric CO₂ nevertheless changes on My and Gy timescales means that tectonic/magmatic forcings or the global efficiency of the negative feedbacks must change with time as Earth evolves. In other words, Earth’s exogenic system is characterized by time-varying steady states or a baseline C content and climate. On short timescales, the exogenic system is not strongly influenced by the deep Earth, but on My timescales, the total budget of the exogenic C is controlled by the deep Earth through degassing and weathering.

11.2.2 Climatic Drivers versus Negative Feedbacks

The most important feature of negative-feedback systems is that steady state, if given enough time, will eventually be attained. Without a negative feedback, where the output does not depend on the amount of C in the exogenic system, a system is unlikely to attain steady state. In such a system, the C content of the exogenic system would rise indefinitely if inputs exceeded outputs or eventually decline to zero if outputs exceeded inputs. Earth’s climate has changed through time, but liquid water appears to have been present for almost all of Earth’s history, implying that inputs and outputs must be balanced on long timescales, which in turn implies that C in the exogenic system and climate is controlled by a negative-feedback mechanism. Any imbalances in inputs and outputs must be transient and occur on timescales shorter than the response time of exogenic C to perturbations (Figure 11.1d).Reference Berner and Caldeira²⁷

Baseline or steady-state C in the exogenic system is driven by the inputs and modulated by the efficiency of the outputs. The greater the kinetic rate constant for C removal from the exogenic system, the more sensitive the negative feedback. In a steady-state system, this concept is illustrated well in a graph of C input or output versus total C in the exogenic system.Reference Caves, Jost, Lau and Maher^30, Reference Kump, Arthur and Ruddiman³¹ For a linear system, the sensitivity of the negative feedback is the slope of the output as shown in Figure 11.5. Because the input of C from the deep Earth does not depend on the amount of exogenic C, the output function is represented by a horizontal line in Figure 11.5a. The intersection between the two lines defines the steady state. For a given sensitivity, k, the only way to change steady-state exogenic C is to change inputs. If inputs are instead held constant, exogenic C can only change by changing the strength (slope) of the weathering feedback. If exogenic C were to suddenly decrease below the system’s steady state, inputs would temporarily exceed outputs, returning exogenic C contents back to steady-state values. Conversely, if exogenic C contents were to suddenly increase, outputs would then exceed inputs, eventually returning the system to steady state. These concepts are shown in Figure 11.5b by keeping track of the intersection of the input and output functions. If the system were to become more efficient at C drawdown (higher k), such as if Earth was characterized by a global increase in mountain building and erosion, steady-state exogenic C would decrease. If drawdown efficiency were to decrease, such as through decreased weatherability of continents, steady-state exogenic C would increase without any change in the input (Figure 11.5b). Most importantly, if the negative-feedback efficiency is strong, then large changes in the inputs are required to change exogenic C, which is to say that the exogenic C content is buffered. This is why Earth’s climate, while variable over time, has rarely veered to extreme climatic conditions.

Figure 11.5

Inputs J_in and outputs J_out as a function of pCO₂. Geologic inputs are assumed not to depend on exogenic C and are thus represented as horizontal lines. Output rate depends on pCO₂, with the slope k representing the sensitivity of the negative weathering feedback. The intersection between J_in and J_out represents steady state, where J_in = J_out. The pCO₂ at this intersection represents steady-state pCO₂. In (a), we show that steady-state pCO₂ increases by increasing J_in while holding the weathering feedback constant (red arrows from states 1 to 2 show an increase in steady-state pCO₂). In (b), we hold inputs constant but decrease the sensitivity of the weathering feedback. This causes steady-state pCO₂ to increase without any change in the input.

It is often debated whether Earth’s climate and exogenic C is input or sink driven. This debate is not about whether inputs or outputs are in balance because, on long timescales, they are in balance.Reference Zeebe and Caldeira³² The crux of this debate is whether changes in the C content of the exogenic system are controlled by changes in input or changes in the efficiency of the negative feedback (Figure 11.5). The former involve changes in magmatic, metamorphic, or weathering-related outgassing rates, while the latter involve changing the efficiency of weathering and organic C burial.

11.2.3 When Systems Transition to New Steady States

Despite the overall stability of Earth’s climate, there are examples when Earth or other planets have veered toward extreme climatic states. The Snowball climate of the Neoproterozoic, when Earth is thought to have frozen over, is an example of runaway cooling.Reference Hoffman, Kaufman, Halverson and Schrag⁸ Mars represents an example of a permanent icehouse, with surface temperatures of –55°C and no active hydrologic cycle.Reference Lodders and Fegley^33, Reference Ingersoll³⁴ Permanent greenhouse characterizes Venus, where surface temperatures are 460°C and the atmosphere is made of 96% CO₂.Reference Lodders and Fegley^33, Reference Ingersoll³⁴ Earth itself may have undergone short-lived excursions to hothouse conditions. These include the PETM and the various hyperthermals around that time.Reference Zachos, Dickens and Zeebe^4, Reference Dickens, O’Neil, Rea and Owen^35, Reference Zachos, Pagani, Sloan, Thomas and Billups³⁶

In a linear system, runaway processes cannot happen. However, negative-feedback mechanisms need not scale linearly with exogenic C content. For example, if the rate-limiting step for silicate weathering and precipitation of carbonates becomes the rate at which fresh new rock is exposed to the atmosphere, one could envision a “threshold” of atmospheric CO₂ content above which further increases in CO₂ do not result in increased weathering. Similarly, a scenario could be envisioned in which surface temperatures drop to a threshold below which weathering kinetics become negligible, and because surface temperature is in part controlled by the amount of greenhouse gases, this translates to a threshold of atmospheric CO₂ below which weathering rates are insensitive to CO₂. Thus, the functional form for the negative feedback might involve low-sensitivity regimes at low and high atmospheric CO₂ separated by a sensitive, quasi-linear regime at intermediate CO₂ contents (Figure 11.6). For a scenario in which C inputs to the exogenic system do not depend on atmospheric CO₂, one can track how steady-state exogenic C varies as inputs vary. From Figure 11.6, we see that when inputs rise toward an upper threshold, steady-state exogenic C contents can rise uncontrollably, leading to a runaway greenhouse. If inputs decrease below some lower threshold, steady-state exogenic C contents will plummet, resulting in runaway icehouse conditions. Crossing these thresholds may move the planet to a different steady state, where a new type of negative feedback operates (Figure 11.7). Understanding the functional form of global weathering feedbacks is thus critical to understanding where these thresholds lie in terms of exogenic C.

Figure 11.6

(a) Input J_in and output J_out versus pCO₂. The conceptual model is identical to that of Figure 11.5, but the functional form of the weathering feedback J_out is nonlinear, consisting of low-sensitivity regimes at low pCO₂ and high pCO₂. These low-sensitivity regimes are referred to here as kinetically limited and substrate-limited, respectively. At low pCO₂, temperatures are low and kinetics of dissolution are sluggish. At high pCO₂, reaction is limited by the availability of new substrate for weathering. Between these two low-sensitivity regimes lies a quasi-linear high-sensitivity regime, where climate is stable. Intersections of J_in (thin horizontal lines) with J_out (thick bold line) are denoted by red circles. (b) Steady-state pCO₂ in (a) is plotted versus J_in. When inputs rise to some threshold (e.g. from states 1 to 2), further increases in J_in will lead to rapid rises in pCO₂. Conversely, when inputs drop below some threshold (states 1 to 3), further decreases in J_in will lead to rapid declines of pCO₂. Hothouse and icehouse excursions are controlled by how close the system’s baseline climate is to the threshold.

Figure 11.7

(a) and (b) are schematic diagrams that are conceptually identical to those of Figure 11.6, except that two different functional forms of the weathering feedback J_in are shown. The two different weathering feedback mechanisms are assumed to operate at different pCO₂ values such that the system can transition into a fundamentally new state after it crosses a pCO₂ threshold (e.g. from states 1 to 2 to 3).

11.3.1 Modern and Primitive Mantle Reservoirs

Excluding the core, more than 99.99% of all C on Earth is in the mantle and crust, although exact quantities are highly uncertain.Reference Dasgupta and Hirschmann¹⁸ Estimates of the C budget of the uppermost mantle are indirectly calculated from the C content of mid-ocean ridge basalts (MORBs), with C itself inferred by bootstrapping to other elements, such as Nb, Ba, H, or noble gases.Reference Saal, Hauri, Langmuir and Perfit^37–Reference Marty and Jambon³⁹ These studies have resulted in estimates of 16–300 ppm C for the MORB source mantle, typically assumed to represent the upper mantle (Figure 11.3). However, reconstructing mantle C contents is faced with the challenge of correcting for degassing of C and other elements. Tucker et al.Reference Tucker, Mukhopadhyay and Gonnermann¹⁹ and Hauri et al. (Chapter 9 of this book) provide the most recent and rigorous reconstructions of the MORB source region and arrive at 30 ppm C (Table 11.1). This estimate agrees well with previous estimates of ~15–50 ppm C for the MORB source mantle that used undegassed or minimally degassed MORB CO₂ contents and CO₂/Ba, CO₂/Nb, and H/C ratios (e.g. Refs. Reference Saal, Hauri, Langmuir and Perfit37, Reference Hirschmann and Dasgupta38, Reference Michael and Graham40, and Reference Rosenthal, Hauri and Hirschmann41). A homogeneous upper mantle (<670 km; mass of 1.06 × 10²⁴ kg⁴²) with a MORB source C content would contain ~3.2 × 10⁷ Gton C (Figure 11.8). If the MORB source was representative of the entire mantle down to the core–mantle boundary (4.0 × 10²⁴ kg), the whole mantle would contain 1.2 × 10⁸ Gton C.

Table 11.1Reservoirs of carbon

BSE = bulk silicate Earth; DM = depleted mantle; ROL = recycled oceanic lithosphere.

Figure 11.8

(a) Estimates of total C content (Gton C) in various exogenic and endogenic reservoirs. See Table 11.1 and text for details. Estimates of the continental lithospheric mantle and bulk silicate Earth (BSE) are from this study. Downward-pointing arrows denote that these estimates represent maximum bounds. Depleted mantle volume corresponds to mantle above 670 km with C concentration equal to that inferred for the MORB mantle source. Lower mantle corresponds to mantle between 670 km and the core–mantle boundary with a C concentration equivalent to BSE. (b) Graphical representation of C fluxes (Gton C/y) with inputs as positive values and outputs as negative values. Silicate weathering, organic C burial, and seafloor weathering are shown in yellow. Inputs of CO₂ via weathering of organic C and carbonates are combined (negative bars), but such weathering is thought to be balanced by rapid re-precipitation of carbonate and burial of organic carbon (positive bars). Endogenic outgassing is shown in green, while subduction is shown in blue. Degassing through Mt. Etna and Cretaceous continental arcs is shown in order to illustrate the importance of carbonate-intersecting continental arcs. Vertical arrow points to anthropogenic production of CO₂ through fossil fuel burning and cement production (10 Gton C/y).

Of interest is the C content of the bulk silicate Earth (BSE), which is the primordial mantle prior to extraction of the continents, oceans, and atmosphere (Figure 11.9a). The present mantle represents what remains of the mantle after degassing and silicate differentiation, which we refer to as the Depleted Mantle (DM); the composition of the DM is assumed to be represented by the MORB source mantle.Reference Hofmann^43, Reference Zindler and Hart⁴⁴ Thus, the BSE can be calculated by re-homogenizing the atmosphere, oceans, and continents back into the DM, but this is challenging if the C contents of all of these reservoirs and their sizes are not known well. A common approach, used for refractory lithophile elements, is to extrapolate magmatic differentiation trends to chondritic ratios, but this cannot be done for C because of its volatility during nebular condensation/planetary accretion and siderophile nature during core formation (e.g. Ref. Reference Chi, Dasgupta, Duncan and Shimizu45).

Figure 11.9

(a) Cartoon showing how C concentration in BSE is determined. BSE is the hypothetical primordial mantle composition after combining the DM, continental crust (CC), continental lithospheric mantle (CLM), and all other reservoirs, such as recycled oceanic lithosphere (ROL), as well as oceans, atmosphere, etc. (latter not shown because of small size). (b) C content of BSE as constrained by relative depletion of Ba and U in DM. Horizontal axis represents the relative efficiency k by which C is retained (during melting) or recycled (during subduction) into the mantle compared to Ba or U; k is likely larger than 1. (c) Missing fraction of C from the DM after subtraction of C in continental crust (DM is assumed to be the size of the upper mantle down to 670 km). Thick black lines correspond to estimates based on C/Ba-constrained BSE and thin blues lines are based on C/U-constrained BSE. Two sets of lines for each color are denoted and correspond to calculations using the crustal budgets of C from WedepohlReference Wedepohl⁵⁸ and Gao et al.Reference Gao⁵⁹ This missing C can be stored in the CLM or as recycled components in the mantle. (d) Maximum concentration of C in CLM, assuming all missing C is the CLM.

Instead, we infer the C of BSE from elements that behave geochemically like C but are refractory in a cosmochemical sense. For example, during mantle melting, C is thought to behave like Ba, a refractory element.Reference Tucker, Mukhopadhyay and Gonnermann^19, Reference Rosenthal, Hauri and Hirschmann⁴¹ The depletion of Ba in the DM (0.563 ppmReference Workman and Hart⁴⁶) relative to the BSE (6.6 ppmReference McDonough and Sun⁴⁷) – that is, Ba_DM/Ba_BSE – is ~0.085. If Ba and C also behave similarly during subduction, then the relative depletion of C in the MORB source is equivalent to that of Ba. The BSE concentration of C is then calculated by dividing the C concentration of the DM by the Ba depletion factor (C_BSE = C_DM/(Ba_DM/Ba_BSE)), yielding ~350 ppm C for the BSE (Figure 11.9b). If C does not behave identically to Ba, then a correction factor must be applied, C_BSE = C_DM/(k_C/Ba × Ba_DM/Ba_BSE), where k_C/Ba represents the efficiency by which C is returned to the mantle relative to Ba (or the relative efficiency by which C is retained or recycled into the mantle relative to Ba). If k_C/Ba > 1, C is preferentially subducted deep into the mantle relative to Ba and our estimates of C in the BSE are hard maximum bounds. If k_C/Ba < 1, Ba is preferentially recycled and our estimates are minimum bounds (Figure 11.9b). It seems likely that Ba, mostly concentrated in slab sediments, is more efficiently removed from the subducting slab given the extreme enrichments of Ba in arc lavas.Reference Elliott, Plank, Zindler, White and Bourdon⁴⁸ While some of the carbonate in subducting sediments might also be removed efficiently,Reference Kelemen and Manning⁴⁹ organic C and carbonate in the oceanic crust are likely retained,Reference Duncan and Dasgupta^50, Reference Dasgupta⁵¹ so it seems likely that k_C/Ba > 1, and thus C_BSE < 350 ppm. If C during melting behaves more like U, a refractory element with a DM depletion factor of ~0.16 relative to BSE (Figure 11.9b),Reference Workman and Hart⁴⁶ C_BSE < ~192 ppm. In any case, C_BSE can be no higher than <350 ppm (Table 11.1), consistent with a recent estimate of C_BSE of ~140 ppm.Reference Hirschmann⁵² These values of C_BSE would imply that the DM reservoir has lost no more than 40–90% of its original C during silicate differentiation; however, the DM reservoir is unlikely to represent the entire convecting mantle, so the fractional loss of C from the whole mantle must be far smaller. We note that the C concentration of the mantle source of ocean island basalts (OIBs) ranges from 33 to 500 ppm C.Reference Dasgupta and Hirschmann¹⁸ The source region of OIBs is thought to represent primitive undegassed mantle and/or enriched domains associated with recycled crust.Reference Hofmann and White^53, Reference Garapić, Mallik, Dasgupta and Jackson⁵⁴

11.3.2 Continental Crust and Continental Lithospheric Mantle

Carbon in the continental crust is represented by ancient sedimentary rocks as well as carbonate and graphite stored in metamorphic and igneous basement rocks. The sedimentary rocks are dominated by carbonates (limestones and dolostones) with organic C-rich sediments making up a smaller fraction.Reference Wilkinson and Walker^55, Reference Wilkinson and Algeo⁵⁶ The carbonate content of the continental crust is calculated by estimating the volume of carbonate sediments in the continents.Reference Ronov^57, Reference Wedepohl⁵⁸ Due to high variability in the organic C content of sediments, the average organic C content of continents is usually estimated by assuming that organic C makes up ~20% of the total C budget. This approach is based on the assumption that the C isotopic composition of seawater, often assumed to represent the fraction of organic C (f_org), has remained relatively constant over most of Earth’s history. The C content of the rest of the continental crust is estimated from measurements of basement rocks and extrapolated to the upper, middle, and lower crusts based on a model of the compositional structure of the crust. Because of the high degree of variability in carbonate content of igneous and metamorphic rocks, any estimate of the C content of the nonsedimentary part of the continental crust is fraught with large errors. For example, WedepohlReference Wedepohl⁵⁸ estimated that the continental basement amounted to one-fifth of the C in sedimentary rocks, yielding a total continental crust budget of 4.2 × 10⁷ Gton C. A more comprehensive study of basement rocks in China found higher concentrations of C in igneous and metamorphic rocks, leading to a much higher estimate of the total continental crust budget of 2.6 × 10⁸ Gton C.Reference Gao⁵⁹

We now turn to the continental lithospheric mantle (CLM), the cold and hence rheologically strong part of the mantle underlying the continental crust, which extends down to depths of ~100 km beneath Phanerozoic terranes and up to ~200–250 km beneath many stable cratons.Reference Jordan^60–Reference Gung, Panning and Romanowicz⁶³ Because of their Gy stability,Reference Pearson^64–Reference Walker, Carlson, Shirey and Boyd⁶⁷ CLM is subjected to numerous metasomatic events, which could lead to hydration or carbonation.Reference Foley^68–Reference Dasgupta⁷⁵ We can place an upper bound on how much C could be in the CLM. If we assume that the volume of DM is equivalent to the entire upper mantle down to 670 km depth (1.06 × 10²⁴ kg), the total amount of C degassed is <3.4 × 10⁸ Gton C as inferred above from C/Ba systematics and >1.7 × 10⁸ Gton C from C/U systematics. From these quantities we subtract the amount of C in the continental crust. If we use the higher crustal abundances of Gao et al.Reference Gao⁵⁹ and the DM C content inferred from C/Ba, we are left with 20% of unaccounted C that must be housed in the CLM or as recycled components deep in the mantle (Figure 11.9c). If we use the crustal abundances of Wedepohl,⁵⁸ 80% is unaccounted for. If we use the lower estimates of C in DM as inferred from C/U systematics, we find that the C budget of the continental crust accounts for or even exceeds the amount of C degassed from the upper mantle, which seems unreasonable; reconciling this discrepancy would require a volume of mantle greater than the upper mantle to have degassed to the extent of the DM reservoir. In any case, if we restrict the DM to the upper mantle and use the C/Ba-constrained DM composition and Gao et al.’s crustal estimate, we find a maximum missing C of 4.8 × 10⁷ Gton C. All of this C in the CLM would translate to a hard upper bound of ~740 ppm C in the CLM, assuming an average CLM thickness of ~150 km (Figure 11.9d and Table 11.1). No doubt, some of the missing C is in the form of recycled crust deep in the mantle, as balancing present-day arc flux requires a small fraction of subducted C to be supplied to sub-arc mantle source regions via fluids or melts (e.g. Ref. Reference Duncan and Dasgupta76). Whether the missing C is in the CLMReference Kelemen and Manning⁴⁹ or recycled into the deep mantleReference Hirschmann⁵² is currently debated.

11.3.3 Exogenic Reservoirs

For completeness, we briefly discuss the C budget of the exogenic system (Table 11.1). In total, the exogenic system accounts for <0.01% of Earth’s C budget (excluding the core), with reservoir sizes increasing as follows (Figure 11.3): atmosphere < terrestrial biosphere and soil < reactive marine sedimentary carbon < ocean.Reference Sundquist and Visser¹¹ For a modern preindustrial atmospheric CO₂ content of 280 ppmv, the total amount of pre-anthropogenic C in the atmosphere today is 590 Gton or ~1.4% of the total C in the exogenic system.Reference Sundquist and Visser¹¹ The oceans account for ~87% of the total exogenic C, with most of the C in the oceans in the form of HCO₃^–. The terrestrial biosphere and soils account for ~4% and the reactive marine sedimentary reservoir accounts for ~8% of the total exogenic C. It is possible that the reactive marine sedimentary reservoir is substantially larger than noted here if the size of gas-hydrate reservoirs has been underestimated.Reference Dickens⁷⁷

In summary, although the amount of C in the exogenic system is nearly negligible compared to that of the bulk Earth, all C degassed from the interior of Earth passes through the exogenic system before being buried as carbonates or organic C, and it is through the exogenic system that C influences climate.

11.4.1 Inputs

11.4.1.1 Volcanic Inputs

On long timescales, the exogenic system is supported by C inputs from the endogenic system through volcanism, metamorphism, and weathering of ancient carbonates and organic C. Outputs from the exogenic system occur through carbonate and organic C burial via the intermediate steps of silicate weathering and photosynthesis. Ultimately, some fraction of carbonate and organic C is removed from the exogenic system by deep burial in the crust or subduction back into the mantle (Figure 11.3).

Fluxing of C into and out of the exosphere can occur through many different species of C (CO₂, CH₄, CO, etc.). We denote all C fluxing, regardless of species, in terms of Gton of elemental C per year. In the case of volcanism and metamorphism, the dominant C species today is CO₂. The most widely used estimate of global volcanic degassing is that of Marty and Tolstikhin.Reference Marty and Tolstikhin⁷⁸ Based on various approaches to estimating CO₂ fluxes through mid-ocean ridges (MORs), arcs, and intraplate magmas (Figure 11.10), they estimated a global volcanic C flux of 0.03–0.13 Gton C/y, with MORs > arcs > intraplate magmatism. These estimates are highly uncertain. In both cases, C fluxes were estimated by measuring C/³He ratios in gases or hydrothermal plumes and then multiplying by the ³He flux, which is either measured or calculated. Regardless of these uncertainties, the Marty and Tolstikhin estimates are lower bounds because they did not consider diffuse fluxes, which Burton et al.Reference Burton, Sawyer and Granieri²¹ showed to be significant. Burton et al. estimate that the total subaerial flux, dominated by arc volcanoes, could be as high as 0.15 Gton C/y, significantly greater than the ~0.030 Gton C/y arc flux estimated by Marty and Tolstikhin (Figure 11.8). Diffuse degassing through faults in continental rifts can also be important (Figure 11.10b); diffuse degassing in the East African Rift alone currently emits 0.05–0.10 Gton C/y, comparable to the flux emitted through the entire length of MORs (Figure 11.8).Reference Lee⁷⁹ It is also important to consider that some anomalous volcanoes like Mount Etna, whose magmas have interacted with crustal carbonates, have unusually high emissions (~0.006 Gton/y for Mount EtnaReference Allard²⁴), with one volcano accounting for a substantial proportion (>10%) of the modern arc flux (Figure 11.8).

Figure 11.10

The four most important regions of magmatically related degassing (not to scale): (a) MORs, (b) continental rifts, (c) island arcs, and (d) continental arcs. In continental rifts, metasomatized domains within the CLM may contain excess C in the form of carbonate or graphite. Destabilization of such C can enhance magmatic flux during rifting. In continental arcs, magmatic degassing can be enhanced by decarbonation of crustal carbonates.

Global volcanic inputs of C are thus at least 0.54 Gton C/y if diffuse degassing estimates from Burton et al.Reference Burton, Sawyer and Granieri²¹ are considered. Global volcanic flux of C through geologic time, however, is likely to vary significantly due to variations in plume activity or global oceanic crust production rates.Reference Larson^80, Reference Larson⁸¹ It has also been shown that continental arcs, because of magmatic interactions (via assimilation, melt-rock reaction, contact, and regional metamorphismReference Carter and Dasgupta^82–Reference Carter and Dasgupta⁸⁵) with ancient crustal carbonates stored in the continents emit significantly more CO₂ than island arcs (Figure 11.10c and d); hence, the waxing and waning of continental arcs through time will also lead to increases in volcanic CO₂ fluxing.Reference Lee^22, Reference Lee and Lackey^23, Reference Mason, Edmonds and Turchyn^86–Reference McKenzie⁸⁸ Enhanced continental rifting could also lead to enhanced CO₂ production.Reference Lee^79, Reference Johansson, Zahirovic and Muller^89, Reference Brune, Williams and Muller⁹⁰

11.4.2 Carbon Outputs

11.5.1 Seafloor Weathering Feedback

In Section 11.4.2, we showed that carbonate sequestration via seafloor alteration was comparable to the CO₂ flux out of MORs, begging the question of how important seafloor alteration is as a negative feedback. Coogan and DossoReference Coogan and Dosso¹⁰⁸ inferred that the activation energy for rock dissolution during seafloor alteration is ~92 kJ/mol, similar to that for the dissolution of various silicate minerals under conditions relevant for terrestrial weathering.Reference Kump, Brantley and Arthur¹¹⁵ If direct temperature effects alone were the dominant control on the negative weathering feedback, one would expect the strength of the seafloor alteration feedback to be equivalent to that of the terrestrial feedback, all other variables being equal. However, as discussed above, many other processes operate. In the case of terrestrial weathering, the effects of runoff on erosion will amplify the strength of the negative feedback. In the case of seafloor alteration, there are no erosional mechanisms that replenish fresh rock for dissolution, so the weathering system would be closer to equilibrium and the kinetics of weathering would be slow. These factors could reduce the net sensitivity of seafloor weathering to global temperature. A recent inverse study of the paleoclimate record suggested that seafloor weathering was not as important as continental weathering over the last 100 My.Reference Krissansen-Totton and Catling¹³³ However, during periods of low orogenic activity, the proportional contribution of the seafloor to the global feedback indeed increases (Figure 11.11). For example, silicate weathering could have been dominated by seafloor weathering in the Archean if most of the continents were below sea level.Reference Lee¹³⁴ In such a case, the strength of the seafloor weathering feedback becomes critically important to understand.

11.6.1 Framework for Modeling Whole-Earth C Cycling

Earth systems modeling is traditionally rendered in terms of boxes with estimates of reservoir sizes and fluxes (Figure 11.3 and Sections 11.3 and 11.4). However, in such renditions, fluxes and reservoir sizes are snapshots in time and cannot be applied back in time if reservoir sizes change. Instead of using fluxes, it is better to use the kinetics of C transfer between reservoirs, as kinetic rate “constants” do not depend on reservoir size even if the system is not at steady state. Rate constants can of course change if the fundamental physics and chemistry change. If the rate constants are known, (11.1) can be expressed as follows for a linear system:

$\frac{{\mathit{dM}}_i}{\mathit{dt}}=\sum k_{\mathit{ji}}{M}_j-\sum k_{\mathit{ij}}{M}_i,$(11.29)

where k_ij represents the rate constant describing the transfer of C from reservoir i to j. The first term represents inputs into reservoir j from all reservoirs i. The second term represents outputs from reservoir i into different reservoirs j. Solving (11.29) for a system of interacting reservoirs involves solving a system of equations using a multidimensional (i × j) matrix k_ij, which can be estimated from modern fluxes if the system is near steady state. We can divide the flux associated with any given process by the size of the reservoir from which C is removed, resulting in a fractional rate constant. The sum of all rate constants describing removal of C from a reservoir i is the total rate constant, k_i; that is:

$k_i=\sum _j{k}_{\mathit{ij}}.$(11.30)

The inverse of the total rate constant represents the residence time τ_i of C in reservoir i from which C is extractedReference Lee, Lee and Wu¹³⁵:

${\tau }_i=\frac{1}{k_i}=\frac{1}{\sum _j{k}_{\mathit{ij}}}={\left(\frac{1}{{\tau }_{\mathit{ij}}}\right)}^{-1}.$(11.31)

In Figure 11.12, we present reservoir residence times (leftmost column) and the rate constant matrix, the latter represented in the form of fractional response times 1/k_ij, which are more intuitive than fractional rate constants. In Figure 11.12, the row denotes the reservoir (i) from which C is being removed and the column denotes the reservoir (j) receiving C from i. Estimates of exogenic rate constants were guided by modern fluxes and reservoir sizes from Table 11.2 and an assumption of near steady state. For endogenic reservoirs, which are larger and may not be at steady state, our estimated rate constants are speculative. The modeler is encouraged to modify the endogenic rate constants to explore the behavior of the system. Empty boxes indicate the absence of flux. Boxes with a question mark indicate that there is a flux, but the magnitude of the flux and the rate constant are poorly known or change considerably with time. For example, degassing of the deep ocean to the atmosphere varies considerably between glacial–interglacial cycles. Transfer of C between the upper mantle and lower mantle is uncertain because it depends on the flux of mantle material between the two reservoirs, which is not known.

Figure 11.12

Matrix of kinetic rate constants between reservoirs, expressed as fractional response times or the inverse of the kinetic rate constant (τ_ij = k_ij^–1). Each cell corresponds to the transfer of C from reservoir i to j, where i correspond to the row and j corresponds to the column. Empty or gray cells indicate a lack of C transfer from i to j. Cells with a question mark indicate that C transfer occurs, but the magnitude and rate constant are not known or vary significantly in time; these are left for the reader to vary in their own studies. Blue-shaded cells represent exogenic reservoirs. The blue-outlined group of cells represents ocean reservoirs, including reactive marine sediments. The leftmost column shows residence time τ_i of reservoir i. Reservoir symbols as follows: Atm = atmosphere; Bio = terrestrial biosphere; Oc-S = surface ocean; Oc-D = intermediate and deep ocean; Oc-rs = reactive marine sediment; Oc Lith = oceanic lithosphere, which includes oceanic crust and oceanic lithospheric mantle; Cont Lith = continental lithosphere, which includes continental crust (sediments + basement) and CLM; UM = upper mantle (<670 km); LM = lower mantle (>670 km). Earth’s core is excluded. “f” corresponds to the fraction of C from the subducting oceanic lithosphere that is lost to a particular reservoir.

Table 11.2Fluxes of carbon

11.6.2 Is Earth’s Mantle Degassing or “Ingassing”?

A key question is whether mantle degassing rates are balanced by recycling of C back into the mantle through subduction. The only direct way to answer this question is to compare mantle degassing fluxes to subduction fluxes, but these quantities are uncertain. A better approach is to explore possibilities through a forward box modeling approach as discussed in Section 11.6.1 and depicted in Figure 11.12. Rate constants can be allowed to vary with time as constrained by the physical evolution of Earth following simple, parameterized convectionReference McGovern and Schubert¹³⁶ rather than full-scale coupling of thermodynamic and geodynamic models. For example, it is relatively straightforward to develop petrologically constrained scalings for mantle and crustal degassing as a function of mantle thermal state and convective vigor, with the goal of quantifying degassing at MORs, subducting slabs, arcs, and intraplate environments. Important quantities would be the kinetics of chemical weathering, but also where carbonates are deposited. In the extreme scenario in which all carbonates are deposited on continental shelves, which do not subduct, then there is little recycling of C back into the mantle. As far as we know, there is not yet any whole-Earth box model that has incorporated the nature of how C is deposited (oceans versus continents). The important controls of where carbonates are deposited will be total area of shallow continental margins, the evolution of life, and the chemistry of the oceans.Reference Ridgwell and Zeebe^137, Reference Ridgwell¹³⁸ Variations of continental shelf area with time could be one of the most important factors controlling the rate at which C accumulates at the surface, as there are times in Earth’s history when continents may have been wholly submerged,Reference Lee^134, Reference Korenaga, Planavsky and Evans^139, Reference Flament, Coltice and Rey¹⁴⁰ with most C being deposited in epicontinental seas, leaving little to be deposited in the deep sea for subduction.

In any case, we know that sedimentary rocks have accumulated on the continents over Earth’s history. We also know that other incompatible elements, such as U, Th, and K, among others, have progressively migrated into the continental crust, leaving behind a depleted upper mantle. If C behaves incompatibly, then the continental crust and CLM C budget have likely grown with time as well. This growing continental reservoir of C serves as a capacitor, which can enhance outgassing rates by metamorphic or magmatically driven decarbonation of the lithosphere in continental arcsReference Lee²² or continental rifts.Reference Lee⁷⁹ If direct mantle degassing rates have not changed with time, total outgassing rates from the endogenic system should have increased with time. Alternatively, if mantle degassing rates are thought to have decreased significantly through time, lithospheric amplification could have partly compensated for this decline. Understanding how total outgassing rates from Earth’s interior have changed through time will have profound implications for the evolution of other volatiles, whose chemistries may be linked to C. For example, an increasing outgassing flux of C would result in increased photosynthetic production of atmospheric oxygen.Reference Lee⁹⁵

11.6.3 What Drives Greenhouse and Icehouse Conditions?

On 10–100‑My timescales, Earth’s climate fluctuates between greenhouse and icehouse states, the latter being characterized by permanent polar ice sheets and the former without permanent polar ice. For example, the late Jurassic to the early Cenozoic was characterized by greenhouse conditions, where temperatures were thought to be >10°C higher than today and any ice sheets were thought to be ephemeral. This greenhouse interval was punctuated by a number of short-term (<100‑ky) excursions to even warmer conditions, often referred to as “hothouse” events.Reference Jenkyns, Forster, Schouten and Sinninghe Damste¹⁴¹

Earth began to cool around 50 Ma in the mid-Cenozoic based on the seawater oxygen isotope record, with southern hemisphere glaciation (Antarctica) beginning ~35 Ma, followed by the development of the northern hemisphere ice sheet ~5 Ma.Reference Zachos, Dickens and Zeebe^4, Reference Zachos, Pagani, Sloan, Thomas and Billups³⁶ Earth in the last several million years has been characterized by bipolar glaciation, and because of the albedo effect of ice, subtle variations in solar insolation driven by orbital dynamics can drive short-term cycles (10–100 ky) of ice sheet growth and retreat. Thus, superimposed on the late Cenozoic icehouse baseline are geologically rapid periodic fluctuations of temperature, ice volume, and eustatic sea level tuned to orbital dynamics. During greenhouse intervals, apparent changes in relative sea level (e.g. sequence boundariesReference Haq, Hardenbol and Vail^142, Reference Vail, Mitchum and Thompson¹⁴³) recorded during greenhouse conditions most likely have origins that are different from glacioeustasy, such as external changes in sediment supply. While rapid fluctuations in sea level are a defining characteristic of an icehouse world, short-lived hothouse excursions – a characteristic feature of greenhouse intervals – appear to be absent from icehouse worlds.

Two questions follow. First, what controls baseline climate? Second, what conditions allow for and trigger short-term excursions to extreme warm and cold climates? In general, baseline climate on >10‑My timescales is ultimately controlled by the amount of greenhouse gases in the atmosphere, namely CO₂ for most of Earth’s history. Orbital dynamics are too rapid, while changes in the sun’s luminosity only explain changes over Gy. Understanding long-term baselines comes down to understanding what controls the steady state. The combination of volcanic/metamorphic forcings (input) and the efficiency of the silicate weathering feedback defines a given steady state, and it is the variations in these two quantities that we seek to understand.Reference Caves, Jost, Lau and Maher³⁰

11.6.3.2 Icehouse Drivers

Are long-lived icehouses driven by enhanced silicate weathering feedbacks or decreased outgassing from Earth’s interior? The Cretaceous greenhouse and mid-Cenozoic to present icehouse represent an ideal case study. It is widely thought that the Himalayan orogeny played a strong role in Cenozoic cooling beginning ~50 Ma by enhancing the silicate weathering feedback through enhanced erosion rates.Reference Raymo and Ruddiman¹²⁵ However, the rise in ⁸⁷Sr/⁸⁶Sr often touted as an indicator of terrestrial weathering occurs ~38 Ma, well after cooling initiated.Reference Kent and Muttoni^150, Reference Hodell¹⁵¹ Furthermore, radiogenic isotopes do not necessarily correlate with silicate weathering fluxes because source age and composition can be just as important.Reference Bataille, Willis, Yang and Liu¹⁵² For example, local weathering of ultramafics or black shales has an outsized influence on seawater ¹⁸⁷Os/¹⁸⁸Os.Reference Peucker-Ehrenbrink, Ravizza and Hofmann^153, Reference Peucker-Ehrenbrink and Ravizza¹⁵⁴ Similarly, weathering of carbonates or the very ancient basement influences ⁸⁷Sr/⁸⁶Sr.Reference Bataille, Willis, Yang and Liu¹⁵²

These complications in interpreting seawater Sr isotopes raise a number of important questions. Is the Himalayan orogeny, in a regional sense, a net producer or consumer of CO₂?Reference Evans, Derry and France-Lanord^94, Reference Evans, Derry and France-Lanord^96, Reference Jacobson, Blum and Walter⁹⁷ This, combined with the possibility that regional metamorphism causes metamorphic decarbonation of crustal carbonates, should encourage vigorous discussion on the role of the Himalayan orogeny on global cooling. Recent studies have suggested that the closing of the Tethys ocean, which culminated in a transcontinental orogenic belt, of which the Himalayas are but one segment, involved the obduction of numerous island arc terranes and ophiolites, increasing global silicate weathering efficiency.Reference Jagoutz, Macdonald and Royden¹⁵⁵ In an alternative scenario, Kent et al.Reference Kent¹⁵⁶ suggested that the ~65‑Ma Deccan flood basalts in India may have increased global weathering efficiency as India migrated north into wet equatorial latitudes in the Cenozoic.

There is also the possibility that cool climates simply represent a decline in magmatic outgassing.Reference Berner, Lasaga and Garrels¹⁵⁷ Evidence from detrital zircons and arc reconstructions have been used to suggest that long icehouse intervals coincide with periods of reduced continental arc activity, just as greenhouses have been suggested to correlate with enhanced continental arc activity.Reference Lee^22, Reference McKenzie, Hughes, Gill and Myrow^87, Reference McKenzie^88, Reference Cao, Lee and Lackey¹⁴⁵ The rise and fall of continental arcs offer an intriguing process for modulating the exogenic C system. When continental arcs are magmatically active, they are likely to be net producers of CO₂, but after magmatism wanes, the remnant topography transforms continental arcs into regional sinks of CO₂, resulting in a global increase in the efficiency of silicate weathering.Reference Jiang and Lee^147, Reference Lee, Thurner, Paterson and Cao¹⁴⁸ In such a scenario, magmatic outgassing drives greenhouse intervals, which are followed by icehouse intervals when the same orogens die magmatically but continue to weather (Figure 11.13).

Figure 11.13

(a) Case study of a magmatic orogen: Cretaceous Peninsular Ranges batholith in southern California, USA.Reference Jiang and Lee¹⁴⁷ Red line shows the observed magmatic production rate and symbols with error bars represent inferred erosion rates. The orogen is characterized by both magmatism (and CO₂ degassing) and erosion, but they are not in phase. After magmatism ends, remnant topography allows for continued erosion and regional drawdown of CO₂ unsupported by magmatic degassing. (b) Global effect of magmatic orogen on steady-state pCO₂. Details of the plot are identical to those of Figure 11.5. Magmatic orogeny increases CO₂ outgassing, so pCO₂ increases from states 1 to 2. Uplift during magmatism enhances erosion, increasing the sensitivity of the global weathering feedback (states a to b), which buffers the rise of pCO₂. After magmatism ends, physical and chemical weathering persist, driving pCO₂ to low levels. Magmatic orogens can potentially drive greenhouses, but are followed by global cooling due to protracted weathering.

Ice sheets may also influence the long-term climate. Growth of ice sheets increases albedo, which cools the climate and encourages additional ice sheet growth. Could ice sheet expansion have further facilitated Cenozoic cooling by enhancing the silicate weathering associated with glacial erosion? This is a tantalizing thought as the Antarctic ice sheet is situated on the ancient Archean basement, raising the question of whether the onset of Antarctic glaciation at ~35 Ma may have played a role in the rise of seawater Sr isotopes at about the same time. On the other hand, glacial weathering might even result in the net production of CO₂ through the oxidation of sulfides, which in turn drives carbonate dissolution; because the resulting acidity is driven by sulfide oxidation rather than carbonation itself, this results in a net release of CO₂, counteracting the effects of glacial weathering and the ice–albedo positive feedback.Reference Tranter^158, Reference Torres, Moosdorf, Hartmann, Adkins and West¹⁵⁹

11.6.4 Climatic Excursions and Runaways