Development of Habitable Climates

The Role of Plate Tectonic-Climate Coupling and Exposed Land Area in the Development of Habitable Climates on Rocky Planets

Bradford J. Foley 11affiliation: email: bradford.j.foley@gmail.com Department of Terrestrial Magnetism, Carnegie Institution for Science, Washington, DC 20015
Abstract

The long-term carbon cycle is vital for maintaining liquid water oceans on rocky planets due to the negative climate feedbacks involved in silicate weathering. Plate tectonics plays a crucial role in driving the long-term carbon cycle because it is responsible for CO degassing at ridges and arcs, the return of CO to the mantle through subduction, and supplying fresh, weatherable rock to the surface via uplift and orogeny. However, the presence of plate tectonics itself may depend on climate according to recent geodynamical studies showing that cool surface temperatures are important for maintaining vigorous plate tectonics. Using a simple carbon cycle model, I show that the negative climate feedbacks inherent in the long-term carbon cycle are uninhibited by climate’s effect on plate tectonics. Furthermore, initial atmospheric CO conditions do not impact the final climate state reached when the carbon cycle comes to equilibrium, as long as liquid water is present and silicate weathering can occur. Thus an initially hot, CO rich atmosphere does not prevent the development of a temperate climate and plate tectonics on a planet. However, globally supply-limited weathering does prevent the development of temperate climates on planets with small subaerial land areas and large total CO budgets because supply-limited weathering lacks stabilizing climate feedbacks. Planets in the supply-limited regime may become inhospitable for life and could experience significant water loss. Supply-limited weathering is less likely on plate tectonic planets, because plate tectonics promotes high erosion rates and thus a greater supply of bedrock to the surface.

astrobiology — planets and satellites: physical evolution — planets and satellites: terrestrial planets

1 Introduction

One of the biggest questions surrounding rocky extra-solar planets is whether they are potentially habitable for life. Water is thought to be necessary for the development of life, so a habitable planet is typically defined as one that is able to sustain liquid water oceans (e.g. Kasting & Catling, 2003). In order to be habitable a planet must lie within the habitable zone, the range of orbital distances where water can exist as a stable phase on a rocky planet’s surface (e.g. Hart, 1978, 1979; Kasting et al., 1993; Franck et al., 2000), and it must have accreted enough water to produce oceans. However, lying within the habitable zone does not guarantee that a planet, even one that accreted a sufficient supply of HO, will have liquid water oceans, or be able to maintain oceans for a significant portion of its history. The abundance of greenhouse gases in a planet’s atmosphere is also critical. A planet with a strong greenhouse effect could be hot enough to enter a moist greenhouse state, where dissociation of water in the upper atmosphere and hydrogen escape to space leads to rapid water loss (e.g. Kasting, 1988; Abbot et al., 2012). Furthermore, stellar evolution exerts a major influence on planetary climate that must be balanced by atmospheric greenhouse gas abundances (Kasting, 1989). Stars increase in luminosity as they age, so early in a planet’s history high greenhouse gas concentrations are needed to avoid a snowball climate (Sagan & Mullen, 1972; Newman & Rood, 1977; Gough, 1981), and late in a planet’s history low greenhouse gas concentrations are needed to keep surface temperatures below the moist greenhouse limit, where rapid water loss occurs (Kasting, 1988). The same effects apply to planets at different orbital distances. Planets closer to the inner edge of the habitable zone need a lower greenhouse effect to maintain oceans, while those closer to the outer edge need a strong greenhouse effect to prevent global glaciation.

A mechanism capable of regulating the strength of the atmospheric greenhouse effect, and thus counteracting the spatial and temporal variations in solar luminosity described above, exists via the long-term carbon cycle, which controls atmospheric CO concentrations (e.g. Walker et al., 1981; Kasting et al., 1993; Berner, 2004). The long-term carbon cycle refers to the cycling of CO between the atmosphere and ocean, carbonate rocks on the seafloor, and the mantle. Weathering of silicate minerals on continents and in the oceanic crust draws CO out of the atmosphere and ocean, depositing it on ocean plates in the form of carbonate rocks. CO resides on the seafloor until it is subducted, where some portion of the carbon reaches the deep mantle, and the rest is returned to the atmosphere through metamorphic degassing and arc volcanism. The carbon in the mantle eventually degasses back to the atmosphere and ocean at mid-ocean ridges, completing the cycle. The long-term carbon cycle stabilizes planetary climate due to the temperature sensitivity of silicate weathering. At high temperatures weathering rates increase, and more rapid weathering draws CO out of the atmosphere, cooling the climate. Meanwhile at low temperatures weathering rates decrease (or cease entirely in the case of an ice covered planet), and sluggish weathering allows volcanic degassing to build CO back up in the atmosphere, warming the climate (e.g. Berner, 2004). The same feedback buffers climate against changes in solar luminosity. When luminosity is low, slower weathering rates allow large quantities of CO to buildup in the atmosphere, warming the climate, and when luminosity is high rapid weathering draws down atmospheric CO, cooling the climate (e.g. Walker et al., 1981). As a result, an active global carbon cycle is thought to be essential for the long-term maintenance of liquid water oceans on rocky planets within the habitable zone (Kasting & Catling, 2003).

Plate tectonics plays a vital role in the operation of the long-term carbon cycle. Plate tectonics drives volcanism at ridges and arcs, the major sources of atmospheric CO, and facilitates silicate weathering, the primary sink of atmospheric CO, by providing a continuous supply of fresh, weatherable rock (both on continents and on the seafloor) through orogeny and volcanic resurfacing. Therefore plate tectonics is often thought to be necessary for the operation of the long-term carbon cycle, and thus crucial for allowing planets throughout the habitable zone to sustain liquid water oceans over geologic timescales (Gonzalez et al., 2001; Kasting & Catling, 2003) (though it is unknown whether some form of carbon cycling and climate stabilization can occur on non-plate-tectonic planets). Studies of carbon cycling and planetary habitability typically assume that plate tectonics and climate are independent; the presence of plate tectonics is imposed by the modeler, and the plate speed is often taken as a free parameter (Driscoll & Bercovici, 2013), or as solely a function of mantle temperature (e.g. Tajika & Matsui, 1992; Franck et al., 1999; Sleep & Zahnle, 2001). However, recent geodynamical studies have found that climate and plate tectonics might be linked. Cool surface temperatures promote plate tectonics by suppressing grain-growth, allowing weak plate boundaries to form through grainsize reduction (Landuyt & Bercovici, 2009; Foley et al., 2012; Bercovici & Ricard, 2014), and by increasing mantle convective stresses, such that convective forces can more easily exceed the lithosphere’s intrinsic strength (Lenardic et al., 2008). Plate speed may even depend on surface temperature, as higher surface temperatures lead to stronger lithospheric shear zones via rapid grain-growth, and these stronger shear zones provide a larger resistance to plate motions (Foley & Bercovici, 2014; Foley et al., 2014). Although there is uncertainty over the applicability of these geodynamical studies to the tectonics of terrestrial planets, as the mechanism responsible for plate tectonics on Earth is still not fully known (Tackley, 2000; Bercovici, 2003), the implications of a fully coupled climate-mantle system (where the presence and vigor of plate tectonics depends on surface temperature) for planetary habitability are worth exploring.

As stated in the previous paragraph, one of the key roles plate tectonics plays in the global carbon cycle is providing a supply of fresh rock at the surface. Without a sufficient supply of fresh rock continental weathering can enter a supply-limited regime. In the supply-limited regime the weathering reaction runs to completion in the regolith, and weathering can only continue by bringing unaltered bedrock into the weathering zone through physical erosion. As a result, supply-limited weathering depends on the physical erosion rate, rather than the kinetics of the weathering reaction (e.g. Stallard & Edmond, 1983; Edmond et al., 1995; Kump et al., 2000; Riebe et al., 2004; West et al., 2005; Hilley et al., 2010). If continental weathering on a planet becomes globally supply-limited, it would no longer depend on atmospheric CO and surface temperature, and the climate stabilization provided by the long-term carbon cycle could be lost (West, 2012). Given the important link between plate tectonics and the supply of weatherable rock, including a treatment for supply-limited weathering is crucial for studying the coupling between plate tectonics, climate, and planetary habitability. Furthermore, supply-limited weathering could be especially important on planets with small amounts of subaerial land (i.e. waterworlds), where the supply of fresh rock at the surface would naturally be low. Yet supply-limited weathering is often ignored in studies of exo-planet habitability, which, so far, have optimistically concluded that climate stabilization via the long-term carbon cycle can still be maintained on planets where up to 99 % of the surface is covered by oceans (e.g. Abbot et al., 2012).

Therefore in this study I test the ability of the long-term carbon cycle to regulate planetary climate, and thus allow hospitable surface conditions to persist over geologic time, for a fully coupled climate-mantle system that includes the supply limit to weathering. In particular, I assess whether the negative climate feedbacks inherent in the long-term carbon cycle are preserved when the plate speed depends on surface temperature as suggested by Foley & Bercovici (2014) and Foley et al. (2014), and the influence initial atmospheric CO concentrations have on the final steady-state climate reached on a planet. Finally, I determine whether planets with small land areas can maintain a carbon cycle with negative climate feedbacks when supply-limited weathering is considered. This study only considers Earth-like planets, that is planets whose size and bulk composition are similar to the Earth’s, as these type of planets are most easily modeled based on our current understanding of climate, tectonics, and carbon cycling on Earth (the possible influences of planet size are discussed in §4.1.1).

This study also focus solely on CO cycling, which is thought to exert the leading order control on the climate evolution of rocky planets (e.g. Kasting & Catling, 2003; Berner, 2004). One reason for the prominence of CO is that it is a major component of the gases released by terrestrial magmatism on Earth today, likely because the upper mantle became oxidized during, or soon after, core formation (e.g. Catling & Claire, 2005; Wade & Wood, 2005; Frost et al., 2008). Thus mantle degassing is a major source of atmospheric CO, and without weathering to act as a sink for this CO, Earth’s climate would be inhospitable to life. Carbon dioxide degassing has likely been active for the majority of Earths’ history, because core formation on the Earth was rapid, due to the high interior temperatures reached during accretion (Kleine et al., 2002). Geochemical data indicate that the upper mantle was oxidized by 3.9-3.5 Ga (Delano, 2001; Frost & McCammon, 2008), and possibly as early as Ga (Trail et al., 2011). Early oxidation of the mantle during core formation is thought to be a general feature of terrestrial planet formation, at least for planets as large or larger than the Earth (Halliday & Wood, 2007), so CO is likely to be a significant component of the atmospheres of rocky exo-planets as well. Although carbon dioxide is not the only greenhouse gas that can play a key role in planetary habitability, methane in particular may be important for solving the faint young sun problem (Pavlov et al., 2000; Haqq-Misra et al., 2008; Feulner, 2012; Charnay et al., 2013; Wolf & Toon, 2013), the ability of silicate weathering to regulate atmospheric CO is likely to be crucial for the habitability of rocky planets.

The paper is organized in the following manner: §2 describes the global carbon cycle model used in this study, with the weathering formulation that includes supply-limited weathering laid out in §2.2, and the dependence of plate speed on surface temperature described in §2.3; results are shown in §3, with the results of models testing the stability of the fully coupled climate-mantle system given in §3.1, and the influence of supply-limited weathering on climate stabilization for planets with small land areas shown in §3.2; finally, results are discussed in §4 and the main conclusions are summarized in §5.

2 Model Setup

Global carbon cycle models of varying complexity have been used to study the long-term evolution of the Earth and other planets (e.g. Walker et al., 1981; Tajika & Matsui, 1990, 1992; Franck et al., 1999; Sleep & Zahnle, 2001; Sleep et al., 2001; Abbot et al., 2012; Driscoll & Bercovici, 2013), as well as for more detailed studies of Neoproterozoic and Phanerozoic climate evolution (e.g. Berner et al., 1983; Volk, 1987; Berner, 1994, 2004; Tajika, 1998; Mills et al., 2011). In this study a simple model is used to focus on the first order effects of both climate-tectonic coupling and exposed land area on climate stabilization via the global carbon cycle. Carbon is assumed to cycle between four reservoirs: the ocean crust (or plate) (), the mantle (), the atmosphere (), and the ocean () (Figure 1). A significant fraction of oceanic carbonates reside on continental shelves at the present day (e.g. Ronov & Yaroshevsky, 1969), and thus a separate continental reservoir is often included in global carbon cycle models (e.g. Tajika & Matsui, 1990; Sleep & Zahnle, 2001). Here all carbonates are assumed to form on the seafloor, i.e. they are part of the ocean crust reservoir, as in Driscoll & Bercovici (2013). The separate continental reservoir is neglected because much of this study focuses on planets with small land fractions, where the large majority of carbonate formation would occur on the seafloor. Furthermore, neglecting the continental reservoir does not significantly alter the first order climate feedbacks or climate-tectonic coupling, because the functional form of the degassing function for continental carbon is similar to the mantle degassing function, in that both depend linearly on plate speed (Sleep & Zahnle, 2001).

Carbon cycles between the four reservoirs in this model as follows (Figure 1): Exposed rock at the surface reacts with atmospheric CO to form bicarbonate, magnesium, and calcium ions that travel to the oceans via rivers and groundwater. Once in the ocean, these ions form carbonate minerals on the seafloor, both in the form of sediments and via hydrothermal alteration of basalt (CO dissolved in the oceans can also react directly with basalt and form carbonate minerals, regardless of whether silicate weathering is active) (e.g. Berner, 2004). Thus weathering acts as a net flux of carbon from the atmosphere and ocean to the plate reservoir. Carbon leaves the plate reservoir when it is subducted at trenches. Here, a fraction of the carbonate minerals will devolatilize, returning carbon to the atmosphere via arc volcanism, and the remainder will be subducted into the deep mantle. Once in the mantle, carbon is mixed throughout by convection, and degasses to the atmosphere and ocean at mid-ocean ridges. Return of carbon to the atmosphere and ocean via mantle degassing closes the long-term carbon cycle.

Figure 1: Schematic diagram of the global carbon cycle model used in this study.

The carbon cycle is modeled using the following mass balance equations (e.g. Tajika & Matsui, 1992; Sleep & Zahnle, 2001; Driscoll & Bercovici, 2013):

(1)
(2)
(3)

where (the continental or terrestrial weathering flux) is the flux of carbon from the atmosphere to the plate resulting from silicate weathering on land, (the seafloor weathering flux) is the flux of carbon from the ocean to the plate resulting from hydrothermal alteration of basalt by CO dissolved in seawater, (the subduction flux) is the flux of carbon from the plate into the mantle, (the degassing flux) is the flux of carbon from the mantle to the atmosphere and oceans via mid-ocean ridge volcanism, and (the arc volcanic flux; , where is the fraction of subducted carbon that is degassed at arcs) is the flux of carbon to the atmosphere from devolatilization of the subducting slab (see Table 1 for a list of all model variables). The continental weathering flux is divided by two because half of the carbon initially drawdown by weathering on land is rereleased to the atmosphere when carbonates form on the seafloor (e.g. Berner et al., 1983). In other words, represents the net flux of CO from the atmosphere and ocean reservoirs to the plate reservoir via continental weathering. The seafloor weathering flux only represents the alteration of basalt by CO dissolved in the oceans; formation of carbonates in basalt by calcium and bicarbonate ions derived from silicate weathering on land are included in . Thus, equation (1) describes how carbon is added to the plate reservoir through continental and seafloor weathering, and leaves via subduction, equation (2) tracks the addition of carbon to the mantle reservoir via deep subduction of carbonates (i.e. ), and its removal by ridge degassing, and equation (3) describes how arc volcanism and mantle degassing add carbon to the atmosphere and ocean, while silicate weathering removes it. This model also assumes that carbon is instantaneously mixed throughout the mantle after being subducted. This is a simplification because mantle mixing is not instantaneous, and can take up to 1 Gyr for whole mantle mixing in the present day Earth (e.g. Olson et al., 1984; Hoffman & McKenzie, 1985; Christensen, 1989; Kellogg & Turcotte, 1990). However, as mantle mixing with realistic geometries and rheological effects is still a major topic of research (Tackley, 2007), incorporating the influence of a finite mixing time for mantle carbon is left for future studies.

As in Sleep & Zahnle (2001), the atmosphere and ocean reservoirs are grouped together. The partitioning of carbon between the atmosphere and ocean is controlled by the solubility of CO in the oceans. These two reservoirs equilibrate on a very short timescale compared to the timescale over which the other reservoirs evolve. Thus, equilibration between the atmosphere and ocean is assumed to take place instantaneously, and the combined atmosphere plus ocean carbon reservoir is partitioned into and using Henry’s Law at each timestep. Henry’s Law is given by , where is the partial pressure of atmospheric CO, is the solubility and is the mole fraction of CO in the oceans; where is the number of moles of HO in an ocean mass of water. The temperature dependence of the solubility, , is ignored in this study for simplicity; a representative of warm greenhouse conditions, where less CO can be dissolved in the oceans, is chosen and used in the majority of the models (see Table 2 for a list of all model parameters and their assumed baseline value, and Table 3 for a list of all physical constants used in this study). Variations in are considered in §3.2.1 and found to have no significant influence on the results. The atmospheric CO reservoir can be related to the partial pressure of atmospheric CO () through

(4)

where is the molar mass of CO, is acceleration due to gravity, and is the surface area of the Earth. This relation is only an approximation, because relating the moles of atmospheric CO to depends on the composition of the rest of the atmosphere. If CO is the dominant component, then equation (4) is exact; if N or HO are the dominant components in the atmosphere, then equation (4) will overestimate by approximately a factor of 2. However, the impact of this error on the overall results of the study is negligible; test cases assuming that CO is a minor atmospheric component were found to be nearly identical to the results presented in the remainder of this paper, all of which use equation (4).

Symbol Definition and units Equation
Ocean plate carbon reservoir (mol) (1)
Mantle carbon reservoir (mol) (2)
Atmosphere carbon reservoir (mol) (3)
Ocean carbon reservoir (mol) (3)
Subduction flux (mol Ma) (5)
Degassing flux (mol Ma) (6)
Continental weathering flux (mol Ma) (14)
Supply-limited weathering flux (mol Ma) (10)
Kinetically-limited weathering flux (mol Ma) (11)
Seafloor weathering flux (mol Ma) (7)
Partial pressure of atmospheric CO (Pa) (4)
Area of ocean plates (, km) (5)
Land fraction (9)
Saturation vapor pressure (Pa) (12)
Surface temperature (K) (15)
Effective temperature (K) (16)
Solar irradiance (W m) (16)
Plate speed (cm yr or m Ma) (17)
Total planetary CO budget (mol) (18)
Table 1: Table of variables.

2.1 Subduction, Arc, Degassing, and Seafloor Weathering Fluxes

The subduction flux, , is given by the product of the area density of carbon on the plate, (where is the area of oceanic plates), the plate speed, , and the length of trenches, ,

(5)

The arc flux is then , and the flux of carbon into the deep mantle is . The fraction of subducting carbon that degasses through arc volcanism on the present day Earth is not well constrained, with estimates typically ranging from 25 % to 70 % (e.g. Sleep & Zahnle, 2001; Dasgupta & Hirschmann, 2010; Ague & Nicolescu, 2014), though a vary small can not be ruled out (Kelemen & Manning, 2015). Thus is chosen as a baseline; variations in are considered in §3.2.1, and do not significantly influence the main results of this study (see also §4.2).

The degassing flux is given by the flux of upwelling mantle into the melting region beneath the ridge, multiplied by the concentration of carbon in the mantle and the fraction of this carbon that degasses (Tajika & Matsui, 1992). The flux of mantle into the melting zone must balance the flux of mantle leaving this region, and is thus given by , where is the depth where mid-ocean ridge melting begins and the length of ridges is assumed to be equal to the length of trenches. The degassing flux is then

(6)

where is the fraction of upwelling mantle that degasses and is the volume of the mantle. The current day length of mid-ocean ridges is km (e.g. Fowler, 2005); this value is assumed to be constant and used for all models. The degassing fraction, , is fixed to reproduce the present day Earth’s degassing flux and atmospheric CO content, which gives , similar to the estimate of Tajika & Matsui (1992).

The seafloor weathering flux, in particular whether it has a strong climate feedback or not, is poorly understood. Drill cores of oceanic crust show that low temperature (0-60 C) off-axis hydrothermal alteration of basalt is a significant CO sink (Staudigel et al., 1989; Alt & Teagle, 1999; Gillis & Coogan, 2011) and laboratory measurements indicate that the rate of CO uptake via basalt alteration increases with both increasing temperature and (Brady & Gíslason, 1997). However, it is not clear if the temperature of the seawater circulating through crustal basalt during hydrothermal alteration is dictated by the geothermal heat flow (Alt & Teagle, 1999) or by the ocean bottom water temperature (Gillis & Coogan, 2011; Coogan & Gillis, 2013; Coogan & Dosso, 2015), which is related to climate. Furthermore, seafloor weathering can also become “supply-limited,” if all of the basalt accessible to hydrothermal fluids is completely carbonated (Sleep et al., 2001); in this case only the creation of new seafloor at mid-ocean ridges allows further seafloor weathering. Thus even with a direct temperature feedback there are limits to the amount of CO that can be sequestered in the ocean crust. In this study both the direct temperature dependence of seafloor weathering and its supply-limit are neglected due to the large uncertainties associated with these effects; this also results in a simpler model that can be more completely understood and analyzed. In §3.2.3 I show that including a direct temperature feedback and a supply-limit to seafloor weathering does not significantly change the results of this study, therefore justifying the exclusion of these two effects, as long as complete basalt carbonation cannot penetrate to great depths within the crust. Physically, the simplified seafloor weathering flux used in this study can be interpreted as assuming that crustal heat flow sets the temperature of the water reacting with basaltic crust (Abbot et al., 2012). The seafloor weathering flux therefore follows after Sleep & Zahnle (2001) and Mills et al. (2014) as,

(7)

where stars denote present day quantities, describes the dependence of basalt carbonation on atmospheric CO, and represents the effect of spreading rate, with a constant ridge length, . Spreading rate sets the rate at which fresh, weatherable basalt is created, and the dependence is based on Brady & Gíslason (1997), who find ; this is rounded up to 0.25 for the baseline value of used in this study. The current day seafloor weathering flux, , is set to mol/yr (Mills et al., 2014).

2.2 Continental Weathering Flux

The continental weathering flux typically used in global carbon cycle models is based on the kinetics of the weathering reaction between CO and silicate rocks, with parameterizations for additional effects, such as increases in weathering rates brought about by increases in runoff or tectonic activity (e.g. Walker et al., 1981; Tajika & Matsui, 1992; Sleep & Zahnle, 2001; Berner, 2004; Driscoll & Bercovici, 2013). However, the weathering flux will only be determined by the reaction kinetics when the weathering reaction is the rate limiting step (i.e. when weathering is kinetically-limited). When weathering rates are high, or physical erosion rates are low, weathering can be limited by the supply of fresh rock to the surface (e.g. Stallard & Edmond, 1983; Edmond et al., 1995; Kump et al., 2000; Riebe et al., 2004; West et al., 2005). A more general weathering function, that incorporates supply-limited weathering, is given by (Gabet & Mudd, 2009; Hilley et al., 2010; West, 2012),

(8)

where is weathering rate (with dimensions of mass area time), is the total denudation rate (mass area time), is the fraction of reactable cations in the continental crust and is unitless, is the rate of the silicate weathering reaction (time), and is the effective depth of the weathering zone (mass area). The weathering rate, , can be converted into the global weathering flux, , by integrating over the area of exposed land, and dividing by the molar mass of reactable cations in the bedrock. Thus,

(9)

where is the fraction of exposed land (i.e. the area of exposed land divided by the surface area of the Earth), and is the average molar mass of reactable elements in the continental crust. Combining equations (8) and (9),

(10)

where is the physical erosion rate (length time), is the density of the regolith, and is the supply limit to weathering. Physical erosion rates vary by a wide range on Earth, depending on relief, climate, lithology, and other factors. In this study, a full model of how erosion varies in response to these factors is not attempted. Instead, a maximum erosion rate, , representing the upper bound on the globally averaged erosion rate on a planet, is chosen. This maximum erosion rate, along with the area of exposed land, then sets the global supply limit to weathering, . is poorly constrained, so a wide range of maximum erosion rates are tested in §3.2.1, and the likely dependence of on tectonic activity, and the implications this has for weathering and climate regulation on stagnant lid planets, is discussed in §4.4.

The reaction rate, , is determined from the typical weathering flux equations used in previous global carbon cycle models (e.g. Walker et al., 1981; Tajika & Matsui, 1992; Sleep & Zahnle, 2001; Berner, 2004; Driscoll & Bercovici, 2013); in this way the total weathering flux will mimic previous models when weathering is kinetically-limited, and then plateau when weathering rates are high enough, or erosion rates low enough, to reach the supply limit. Following Driscoll & Bercovici (2013), the weathering flux for kinetically-limited weathering is

(11)

where is the present day weathering flux ( mol/yr (Gaillardet et al., 1999)), is the saturation vapor pressure, and are constants, is the activation energy of the weathering reaction, is the universal gas constant, is the surface temperature, and starred quantities represent present day values. The weathering flux increases with increasing or because both factors increase the rate of the weathering reaction (Berner, 2004), and the saturation vapor pressure term models the variation in runoff as surface temperature changes (Driscoll & Bercovici, 2013). The saturation vapor pressure is (e.g. Kasting et al., 1984; Nakajima et al., 1992)

(12)

where is the reference saturation vapor pressure, at reference temperature , is the molar mass of water, and is the latent heat of water (see Tables 2 & 3).

Figure 2: Comparison between the total weathering flux, including supply-limited weathering, (solid line), and the kinetically-limited weathering flux, (dashed line). Surface temperature depends on based on equation (15) and saturation vapor pressure depends on surface temperature as given by equation (12). A maximum erosion rate of 10 mm/yr and Earth-like land fraction of 0.3 were assumed, and the other parameters are listed in Tables 2 & 3.

Assuming that the kinetically-limited global weathering flux is equal to the reaction rate, , multiplied by the total amount of reactable cations within the weatherable bedrock,

(13)

where is the thickness of the weathering zone, which can be replaced by (West, 2012). Combining equations (10), (11), and (13), the definition of , and using ,

(14)

Inverting for the reaction rate using a weathering function normalized to the present day Earth (equation (13)) implicitly assumes that weathering on the present day Earth is entirely kinetically controlled. While the breakdown of the present day global weathering flux between supply-limited and kinetically-limited weathering is not well constrained, many estimates find that kinetically-limited weathering is the dominant contributor (Kump et al., 2000; West et al., 2005). Thus assuming Earth’s present day weathering flux is kinetically controlled is a reasonable first order approximation.

The weathering flux, , is shown in Figure 2, plotted against , i.e. the typical weathering flux used in global carbon cycle models that do not treat supply-limited weathering. is identical to at low (corresponding to low surface temperatures and low weathering rates), and diverges from at high when weathering becomes supply-limited; here plateaus at , the global supply limit to weathering, determined by the maximum erosion rate and area of exposed land at the surface.

2.3 Climate and Plate Tectonic Models

To relate atmospheric CO to surface temperature, a simple parameterization from Walker et al. (1981) is used:

(15)

where K is the present day surface temperature, is the effective temperature, and K is the present day effective temperature (Figure 3a). Note that this parameterization also includes the contribution of water vapor to greenhouse warming, assuming that HO is always saturated in the atmosphere. The effective temperature is related to the absorbed solar radiation as

(16)

where is the solar irradiance, is the albedo, and is the Stefan-Boltzmann constant. In this study albedo will be held constant for simplicity, so ice-albedo feedbacks, or the currently poorly understood feedbacks involving clouds (e.g. Leconte et al., 2013; Wolf & Toon, 2014) are not included. This parameterization provides a good first order approximation to the results of more sophisticated radiative-convective models for an Earth-like planet (e.g. Kasting & Ackerman, 1986). As discussed in §4.2, using a more advanced climate model may change some of the details of the results, particularly the temperatures calculated at high CO levels where equation (15) deviates the most from the models of Kasting & Ackerman (1986), but would not change the overall conclusions of this study.

Although temperatures reach K at Pa, liquid water is still stable at the surface. A true runaway greenhouse, where liquid water cannot exist as a stable phase on the surface, is thought to only be possible through increases in insolation, rather than through the increase in atmospheric opacity brought about by elevated CO levels (Nakajima et al., 1992). However, above K the atmosphere would be in a moist greenhouse state, where, although liquid water is stable at the surface, high mixing ratios of HO in the stratosphere lead to rapid water loss to space (Abbot et al., 2012). Such water loss is not modeled in this study; rather I look at the expected climate and tectonic states for terrestrial planets with varying land areas, CO budgets, and incoming solar fluxes. However, the implications of the modeled climate states for water loss and habitability are discussed in §3.2.2 and §4.

Figure 3: Surface temperature as a function of for a present day solar luminosity and albedo (a), and plate speed as a function of surface temperature (b). Both the plate speed parameterization used in this study, equation (17), and the plate speed-surface temperature curve calculated from the scaling laws of Foley & Bercovici (2014) are shown in (b).

As discussed in §1, surface temperature can exert a major influence over whether plate tectonics takes place on a planet (Lenardic et al., 2008; Landuyt & Bercovici, 2009; Foley et al., 2012), and influences the speed of plate tectonics when it is present (Foley & Bercovici, 2014; Foley et al., 2014). To include the effect of climate on plate tectonics, plate speed is assumed to depend on surface temperature using a linear approximation to the scaling laws of Foley & Bercovici (2014)

(17)

where high surface temperatures lead to slower plates due to increased grain-growth rates in the lithosphere, which cause effectively stronger plate boundaries. Equation (17) gives in units of cm yr. Mantle temperature also exerts an influence on plate speed. However, the influence of mantle temperature on plate speed is neglected in this study because it is weaker than the influence of surface temperature (Foley et al., 2014). Furthermore, ignoring the weaker influence of mantle temperature on plate speed also allows this study to focus solely on the coupling between plate tectonics, carbon cycling, and climate. Figure 3b shows that the plate speed parameterization, equation (17), is an accurate approximation of the full scaling laws from Foley & Bercovici (2014) up to K, corresponding to the highest surface temperatures explored here.

3 Results

In this section, I first explore the influence the full coupling between plate tectonics and climate has on the development of habitable conditions (§3.1), by: testing whether the ability of the long-term carbon cycle to buffer climate against changes in solar insolation is maintained when plate speed depends on surface temperature (§3.1.1); and by assessing whether habitable climates can develop from arbitrary initial surface temperature and atmospheric CO conditions (§3.1.2), or whether different initial conditions prevent temperate climates from forming. Next, I examine the role land area and planetary CO budget play in the development and maintenance of habitable climates, and map out the conditions necessary for stable, temperate climates to exist on rocky planets (§3.2). Throughout this section, all parameters are held constant (see Tables 2 & 3), save for plate speed which is allowed to vary with surface temperature, so that the coupling between climate and plate tectonics, and the influence of land area and total CO budget, can each be studied in isolation. The effect of varying key model parameters is explored in §3.2.1, and the influence of including both the temperature dependence of seafloor weathering and its supply-limit is tested in §3.2.3.

3.1 Carbon Cycling and Climate Feedbacks with a Surface Temperature-Dependent Plate Speed

3.1.1 Stability of Climate in Response to Changes in Insolation

Symbol Definition Baseline Value Equation
Solubility of CO in seawater Pa (derived) above (4)
Moles of HO in one ocean mass mol (derived) above (4)
Length of trenches km (F05) (5)
Fraction of subducted carbon that degasses 0.5 (A14) (5)
Fraction of upwelling mantle that degasses 0.32 (T92) (6)
Depth of melting beneath ridges 70 km (K06) (6)
Present day seafloor weathering flux mol Ma (M14) (7)
exponent for seafloor weathering (B97) (7)
Present day plate speed 5 cm yr (K06) (7)
Fraction of Mg, Ca, K, and Na in continental crust (W12) (8)
Average molar mass of Mg, Ca, K, and Na 32 g mol (derived) (9)
Maximum erosion rate mm yr (W13) (10)
Regolith density kg m (W12) (10)
exponent for silicate weathering 0.55 (D13) (11)
exponent for silicate weathering 0.3 (D13) (11)
Activation energy for silicate weathering 42 kJ mol (B91) (11)
Present day weathering flux mol Ma (G99) (11)
Present day atmospheric CO 33 Pa (K84) (11)
Present day surface temperature 285 K (W81) (11)
Present day land fraction 0.3 (A12) (11)
Reference saturation vapor pressure 610 Pa (K84) (12)
Latent heat of water 2469 J g (K84) (12)
Reference temperature 273 K (K84) (12)
Present day effective temperature 254 K (W81) (15)
Albedo 0.31 (derived) (16)
  • Key for citations: A12=(Abbot et al., 2012), A14=(Ague & Nicolescu, 2014), B91=(Brady, 1991), B97=(Brady & Gíslason, 1997), D13=(Driscoll & Bercovici, 2013), F05=(Fowler, 2005), G99=(Gaillardet et al., 1999), K84=(Kasting et al., 1984), K06=(Korenaga, 2006), M14=(Mills et al., 2014), T92=(Tajika & Matsui, 1992), TS=(Turcotte & Schubert, 2002), W81=(Walker et al., 1981), W12=(West, 2012), W13=(Willenbring et al., 2013).

Table 2: Table of parameters.
Symbol Definition Value Equation
Molar mass of CO 44 g mol (derived) (4)
Acceleration due to gravity 9.8 m s (TS) (4)
Volume of the mantle km (TS) (6)
Surface area of Earth km (TS) (4)
Gas constant 8.314 J K mol (N92) (11)
Molar mass of water 18 g mol (derived) (12)
Stefan-Boltzmann constant W m K (N92) (16)
Solar constant 1360 W m (dP10) §3.1.1
  • Key for citations: dP10=(de Pater & Lissauer, 2010), N92=(Nakajima et al., 1992), TS=(Turcotte & Schubert, 2002).

Table 3: Table of physical constants.

As outlined in §1, one of the most important roles the long-term carbon cycle plays in planetary habitability is stabilizing climate against changes in solar luminosity. In this section, the efficacy of the carbon cycle climate buffer, when the influence of surface temperature on plate speed is included, is assessed. The steady-state solution to the global carbon cycle model (equations (1)-(3)) as is varied is shown in Figure 4. Solar luminosities ranging from , appropriate for the early Earth (Gough, 1981), to , where Wm is the solar constant, are considered. Solving the full model in steady-state assumes that the global carbon cycle reaches steady-state on a timescale shorter than changes in solar luminosity; in reality the two timescales are similar, both on the order of 1 Gyr. A different approach would be to assume that the plate and mantle reservoirs are fixed, and solve for the atmospheric reservoir in steady-state (i.e. setting ). Both approaches were found to give nearly identical results.

Figure 4: Surface temperature (a) and atmospheric CO (b) as functions of insolation normalized by the solar constant. Surface temperature and atmospheric CO are determined by solving equations (1)-(3) in steady-state for each . The black line assumes CO is constant and equal to the present day value (), the blue line shows the results from the carbon cycle model with a plate speed that depends on surface temperature, and the red line shows the results when plate speed is fixed at 5 cm/yr. Note that the blue and red lines lie almost on top of each other, and are therefore difficult to distinguish.

Similar to other studies on climate stabilization and the long-term carbon cycle (e.g. Walker et al., 1981), the temperature dependence of kinetically controlled weathering allows higher atmospheric CO levels to build up at low solar luminosity, and lowers atmospheric CO levels through higher weathering rates when solar luminosity increases (Figure 4B). As a result, the changes in surface temperature brought about by variations in solar luminosity are less pronounced than the case where atmospheric CO is fixed (Figure 4A). Also consistent with previous studies (e.g. Sleep & Zahnle, 2001), the carbon cycle is not able to produce atmospheric CO levels high enough to keep surface temperatures above 273 K at solar luminosities lower than . Other greenhouse gases, such as methane, may be necessary to prevent global glaciation. However, recent results from three-dimensional global climate models have shown that low latitudes can remain ice free at globally averaged surface temperatures of 260 K (Charnay et al., 2013; Wolf & Toon, 2013), and possibly even lower (Abbot et al., 2011), so the ability of the global carbon cycle to maintain liquid water oceans at low solar luminosities may be stronger than previously thought.

Figure 4 shows that the dependence of plate speed on surface temperature has almost no influence on the effectiveness of carbon cycle induced climate stabilization. The weathering feedback is by far the dominant factor, and thus the fully coupled climate-plate tectonic system is stable to changes in solar luminosity. In fact, the influence of surface temperature on plate speed has a very small additional stabilizing effect on long-term climate. Plate speed increases at cooler temperatures, so lower insolation causes degassing to increase (both at volcanic arcs and mid-ocean ridges), in addition to lowering the weathering rate. A complimentary effect occurs at higher insolation, where the warmer surface temperature slows plates and decreases degassing. However, as plate speed only changes by cm/yr per 20 degree change in surface temperature, while the weathering flux changes by an order of magnitude, the feedbacks inherent in silicate weathering are by far the dominant factor in long-term climate stabilization.

3.1.2 Influence of Initial Conditions

The results in §3.1.1 show that habitable climates can be maintained even when plate speed depends on surface temperature. However, they do not demonstrate whether habitable climates develop in the first place. Fully time-dependent evolution models are needed to determine whether different initial conditions, especially different initial atmospheric CO concentrations, influence the final surface temperatures reached when the global carbon cycle comes to steady-state. In this section, the full time-dependent form of the carbon cycle model, equations (1)-(3), is solved using three different initial conditions: an initially hot case, where the entire CO budget of the planet is initially in the atmosphere and ocean, an initially cold case where the entire CO budget initially resides in the mantle, and an intermediary case where half the CO starts in the atmosphere and ocean, and half in the mantle. An Earth-like planet is assumed, with a total CO budget of moles (Sleep & Zahnle, 2001), and a constant land fraction of 0.3. Both the thermal evolution of the mantle and the evolution of solar luminosity are ignored in this section so that the dynamics of the coupled climate-plate tectonic system can be studied in isolation.

Figure 5: Time evolution of the mantle reservoir (a), plate reservoir (b), ocean reservoir (c), and atmosphere reservoir (d) for an Earth-like model starting from three different initial conditions: all of the carbon in the atmosphere and ocean, i.e. (red line), half of the carbon in the mantle and half in the atmosphere and ocean, i.e. (blue line), and all of the carbon in the mantle, i.e. (green line). Model parameters are listed in Tables 2 & 3.
Figure 6: Time evolution of the terrestrial weathering flux (a), degassing flux (b), subduction flux (c), and seafloor weathering flux (d) for the models shown in Figure 5. The arc flux, , and deep subduction flux, , are not shown because they are both given by the subduction flux multiplied by a constant.
Figure 7: Time evolution of surface temperature (a), atmospheric CO (b), and plate speed (c) for the models shown in Figure 5.

Figures 5, 6, and 7 demonstrate that all three initial conditions reach the same steady state after Gyr. When atmospheric CO is initially high, rapid weathering draws CO out of the atmosphere on a 1-10 Myr timescale, dramatically cooling the climate (Figures 7A & B). This rapid CO drawdown also ensures that the early moist greenhouse state caused by the high initial condition is short lived, and any water loss during this time would be kept to a minimum (see also §3.2.2). Most of the initial CO drawdown takes place via supply-limited weathering (Figure 6A), as the extremely high initial surface temperatures push the weathering rate into the supply-limited regime. During this initial CO drawdown phase, carbon rapidly builds up in the plate reservoir and is gradually subducted into the mantle (seafloor weathering also plays a role, but is much smaller than the continental weathering flux (Figure 6D)). After the rapid initial draw down of atmospheric CO, the atmosphere and ocean reservoirs reach a quasi-steady-state as arc degassing is balanced by continental and seafloor weathering. The evolution of carbon in the atmosphere and hydrosphere is then controlled by the gradual loss of carbon from the plate reservoir to the mantle. As the mantle is regassed by subduction, the degassing flux becomes an important contributor to atmospheric CO, and the final steady-state is reached as the degassing and arc fluxes balance the terrestrial weathering and seafloor weathering fluxes. Models where half of the CO is initially in the mantle and half in the atmosphere and ocean evolve in a similar fashion.

A different evolution occurs when all of the planet’s CO initially resides in the mantle. In this case carbon builds up in the atmosphere and ocean through ridge degassing. A quasi-steady-state is quickly reached between mantle degassing and silicate weathering. Weathering also causes carbon to accumulate on the seafloor, and subduct into the mantle. The final steady-state is then reached when the mantle has degassed a sufficient supply of carbon for the deep subduction flux to balance the degassing flux. The timescale for reaching steady-state is thus dictated by the plate speed for all three initial conditions; the plate speed sets how quickly the mantle can be regassed for the hot initial atmosphere models, and how quickly the mantle degasses for the cold initial atmosphere case.

Despite the different initial conditions leading to different temperature and plate speed evolutions, the same final state is reached. This occurs for two main reasons: 1) the feedbacks inherent in silicate weathering dominate the effect of climate on plate speed; and 2) the influence of climate on plate speed has a stabilizing effect, acting to drive different initial conditions to the same final state. At warm temperatures plate speeds are low, suppressing both mantle degassing and arc volcanism and aiding in the initial drawdown of atmospheric CO. Likewise when surface temperature is low, a higher plate speed increases the mantle degassing rate, helping to build CO back up in the atmosphere. The effect of variable plate speed is small, however, and the primary reason that different initial conditions do not lead to different final states is the weathering feedback.

The preceding discussion can be shown more generally by analyzing the steady-state solutions to equations (1)-(3). In particular it can be shown that, for a given total CO budget (), land fraction, and solar luminosity, only one steady-state solution exists, and that this steady-state solution is stable. In steady-state, the sum of the four carbon reservoirs must equal the total CO budget, . Using equations (1) and (2) in steady-state and Henry’s Law, can be written solely as a function of ,

(18)

where plate speed, , and both weathering fluxes, and , can be written in terms of the ocean carbon reservoir, rather than the atmospheric carbon reservoir, using Henry’s Law. If the derivative of equation (18) with respect to is always positive, then there can only be one steady-state solution.

Taking this derivative term by term, the first term on the right hand side of equation (18) gives

(19)

which is always positive. Taking the derivative of the second term on the right hand side of equation (18) yields

(20)

which shows that only the weathering fluxes and plate speed determine the sign of this term; as , , , and are all independent of , changing any of these parameters does not lead to multiple steady-states. increases with , because a higher ocean carbon reservoir means higher atmospheric CO concentrations, warmer temperatures, and a larger weathering flux. In addition, plate speed decreases with increasing because higher surface temperatures lead to slower plates, so is positive due to both the temperature and CO dependence of the weathering flux, and the temperature dependence of the plate speed (see Appendix A.1 for the full derivation of ). Outside of the supply-limited weathering regime, is a much stronger function of than , so the positivity of is primarily due to the temperature and dependence of the weathering flux. In the supply-limited regime, , but the inverse relationship between and ensures that remains positive. Finally,

(21)

which is also always positive because . Note that has two different dependencies on : increasing increases and therefore directly, but also decreases the plate speed, which acts to slow the seafloor weathering flux. However, as shown by equation (20), it is the slope of that determines whether multiple steady-state solutions exist, so the plate-speed effect cancels out. Therefore is positive, as and are both increasing functions of , and only one steady-state solution exists. Even with a plate speed dependent on surface temperature, initial conditions do not influence the final state models evolve to (the final steady-state is also shown to be stable in §A.2). An important caveat to this finding is that the carbon cycle model used here implicitly assumes the presence of liquid water oceans and water vapor in the atmosphere. If initial surface temperature and pressure conditions are beyond the critical point for water, it is not clear that silicate weathering, and subsequent CO drawdown, can occur; thus, whether carbon cycling can produce habitable climates starting from such high temperature and pressure conditions is unknown. However, as CO-rock reactions above the liquid water critical point are poorly constrained, such extreme conditions are beyond the scope of this study.

The only way to produce multiple steady-state solutions with the present model would be for plate speed to have the opposite dependence on surface temperature, where higher temperatures lead to faster plate speeds. In this case both a high surface temperature, high plate speed with rapid degassing, and a low surface temperature, slow plate speed with low degassing steady-state could exist. However, plate speed would have to be a very strong function of surface temperature in order to counteract the temperature and dependence of the weathering fluxes and cause to change sign. Furthermore, the result that only one steady-state solution exists is robust to uncertainties in the parameters , , , and , because the sign of all these parameters is well known; in other words the uncertainties in these parameters are not sufficient to introduce hysteresis to the system because they would not change the sign of either or .

3.2 Transition to Supply-Limited Weathering at Small Land Areas and Large CO Inventories

The results in §3.1 show that as long as atmospheric temperature and pressure conditions are below the critical point for liquid water, such that silicate weathering can occur, initial atmospheric CO concentrations do not influence the final state reached by an Earth-sized planet with a given exposed land area, total planetary CO budget, and solar insolation. The dependence of plate speed on surface temperature does not lead to multiple steady-states or hysteresis in the coupled carbon cycle-plate tectonic system. Furthermore, the dependence of plate speed on surface temperature does not diminish the ability of the carbonate-silicate cycle to buffer surface temperatures against changes in solar luminosity; in fact, climate stabilization is weakly enhanced because degassing is decreased at high surface temperatures. However, the area of exposed land and the planetary CO budget can have a major impact on whether hospitable surface temperatures, and the climate buffering effects of the long-term carbon cycle, can be maintained on a planet.

In particular, if weathering becomes globally supply-limited a planet will not be able to regulate atmospheric CO levels through continental weathering; only the weaker seafloor weathering feedback would be active. In this case the planet may enter a hot, moist greenhouse climate where water is lost rapidly to space, either due to high atmospheric CO concentrations or the inability of the carbon cycle to drawdown atmospheric CO as solar luminosity increases. In this section, I show that at small land fractions and large planetary CO inventories, weathering does become supply-limited, leading to planets with hot climates and rapid water loss; with no mechanism for decreasing these planets may lose their water and become inhospitable to life.

Figure 8: Steady-state surface temperature (a) and partial pressure of atmospheric CO (b) as a function of land fraction. The boundary between the kinetically-limited and supply-limited weathering regimes is shown with a dashed line, and the critical , where weathering becomes supply-limited, is shown in panel (b). A total CO inventory of mol is assumed, and an insolation of , the solar constant, is used. All other model parameters are listed in Tables 2 & 3.

The transition to a globally supply-limited weathering regime at small land fractions is shown in Figure 8. While weathering is kinetically-limited (i.e. for ), smaller land fractions lead to a modest increase in both and surface temperature. However, once weathering becomes supply-limited, at land fractions lower than , and surface temperature increase sharply as decreases, before flattening out as the majority of the planet’s total CO inventory accumulates in the atmosphere and ocean. The surface temperature and atmospheric CO trends, and the transition to supply-limited weathering, can be explained as follows. In the kinetically-limited regime, the change in in response to a decrease in is governed by

(22)

where the ridge and arc degassing fluxes can be considered constant in this regime. Therefore when decreases, (and in turn ) must increase in order to balance the degassing fluxes. However, as the terrestrial weathering flux is a strong function of , only a modest increase in atmospheric CO is needed to balance the ridge and arc degassing fluxes. The seafloor weathering flux also increases with increasing , helping to balance the degassing fluxes with only a small increase in atmospheric CO. However, as seafloor weathering is a weak function of , it plays a minor role compared to changes in the terrestrial weathering flux.

The sharp increase in atmospheric CO at represents the transition to globally supply-limited weathering. This transition point occurs at a critical , where ; approximating this condition as ,

(23)

which can be solved numerically to give the critical where weathering is supply-limited. Note that the critical is not a function of the land fraction, which cancels out in equation (23). When terrestrial silicate weathering is supply-limited, it is fixed at and therefore no longer a function of atmospheric CO. As a result, the only way to balance the degassing fluxes is through a higher seafloor weathering flux and by depleting the mantle and plate reservoirs, and hence lowering the CO degassing fluxes. The weak dependence of seafloor weathering on means that depleting the mantle and plate reservoirs, thus producing a large increase in atmospheric CO, is the primary mechanism for bringing the system back into equilibrium. As a result, atmospheric CO concentration increases sharply as the land fraction drops below (Figure 8B). A stronger dependence to seafloor weathering, possible if it is directly temperature-dependent, can potentially balance degassing without significant atmospheric CO buildup when continental weathering is supply-limited. However, as shown in §3.2.3, once seafloor weathering also becomes supply-limited the CO drawdown rate will again be unable to balance the degassing rate until the mantle and plate reservoirs are depleted, and hot climates will form.

Figure 9: Steady-state partial pressure of atmospheric CO (a) and surface temperature (b) as a function of land fraction and total planetary CO budget. The boundary between the kinetically-limited and supply-limited weathering regimes is marked by a solid line, and the approximate position of the present day Earth is shown in panel (b). Insolation is assumed equal to the solar constant, and all other model parameters are listed in Tables 2 & 3.

The transition between globally supply- and kinetically-limited weathering also depends on the planetary CO inventory (Figure 9). When decreases, atmospheric CO concentrations also decrease in response to lower mantle and arc degassing rates (owing to a smaller amount of CO throughout the mantle and surface reservoirs). As a result, the boundary between the kinetically-limited and supply-limited weathering regimes drops to lower as decreases (Figure 9A). Below , the supply-limited regime disappears entirely because seafloor weathering does not allow enough CO to build up in the atmosphere for the critical to be reached, even for a planet with an infinitesimally small land fraction. Planets with large CO inventories and small land fractions therefore lie in the supply-limited weathering regime, resulting in high atmospheric CO concentrations and hot climates, while planets with either low CO inventories or large land areas, like the Earth, lie in the kinetically-limited weathering regime where the carbon cycle can maintain a temperate climate. Overall, the kinetically-limited regime dominates, at least for the baseline value of mm/yr chosen here, and thus the carbon cycle should be able to stabilize climate on most rocky planets.

3.2.1 Influence of Erosion Rate and Sensitivity to Model Parameters

The boundary between the supply-limited and kinetically-limited weathering regimes is strongly sensitive to the maximum erosion rate () used to define the supply limit to weathering, and less sensitive to other model parameters (Figure 10). A lower means a lower supply limit to the weathering flux; as a result, the supply-limited weathering regime expands significantly in space. Thus, the ability to sustain high erosion rates is important for effective climate stabilization through the global carbon cycle. Given that uplift and orogeny are crucial for producing high erosion rates, tectonics may play a key role in keeping planets in the kinetically-limited weathering regime, and therefore enabling the long-term carbon cycle to maintain habitable climates over geologic timescales (see §4.4).

Figure 10: Boundary between the kinetically-limited (KL) and supply-limited (SL) weathering regimes for different (a), (b), (c), (d), (e), (f), (g), and (h); for all plots the red curve indicates the standard parameter value used throughout the paper (e.g. in Figures 8 & 9). All model parameters are as specified in Tables 2 & 3, except when explicitly varied. The reference seafloor weathering rate, , is given in units of mol/Myr in panel (g).

The parameters and , which describe the direct dependence of atmospheric CO and surface temperature, respectively, on the rate of the weathering reaction, have a relatively weak influence on the boundary between the kinetically-limited and supply-limited regimes (Figures 10B & C). Lower values of , which are applicable to silicate weathering in the presence of land plants (e.g. Berner, 1994), slightly decrease the size of the supply-limited regime, because a lower increases the critical CO level where weathering becomes supply-limited (see equation (23)), therefore allowing kinetically-limited weathering to prevail at smaller land fractions and larger planetary CO inventories. Decreasing the activation energy of the weathering reaction, , also shrinks the supply-limited regime for the same reason; the critical decreases at lower . Both larger and larger expand the supply-limited weathering regime (Figures 10D & E). When , the fraction of subducted carbon that degasses at arc volcanoes, is larger, atmospheric CO levels are higher and the supply limit to weathering is reached at larger land fractions or smaller planetary CO inventories. Likewise, a greater , the depth of melting at mid-ocean ridges, increases the mantle degassing rate and therefore causes higher atmospheric CO concentrations. The influence of and highlight the role of mantle temperature in maintaining kinetically controlled weathering, as both and are in principle functions of mantle temperature (where higher mantle temperatures increase both and ).

The parameters governing seafloor weathering, , which describes the dependence on , and , the reference seafloor weathering rate, also have important influences on the weathering regime boundary (Figures 10F & G). Both smaller and smaller expand the supply-limited regime, because less efficient seafloor weathering means higher atmospheric CO levels, and the critical where weathering becomes supply-limited is reached at larger or smaller . The case for shows where the boundary between the weathering regimes would lie without any seafloor weathering. Finally, varying solar luminosity has only a very minor effect on the weathering regime boundary (Figure 10H); the supply-limited weathering regime is slightly expanded as solar luminosity increases, mainly due to a decrease in the critical at higher luminosity. As a result, stellar evolution is not a major factor in determining whether a planet will be able to sustain kinetically-limited weathering. Other parameters, such as , the solubility of CO in the oceans, and , the exponent describing how changes in temperature influence precipitation rates, were found to have almost no effect on the boundary between the kinetically-limited and supply-limited weathering regimes.

3.2.2 Climate Stabilization and Habitability with Supply-Limited Weathering

Figure 11: Steady-state partial pressure of atmospheric CO (a) and surface temperature (b) as a function of land fraction and total planetary CO budget for and (panels c & d). Model parameters are listed in Tables 2 & 3.

In addition to producing hot climates at the present day solar luminosity, supply-limited weathering also eliminates the carbon cycle’s ability to regulate atmospheric CO levels in response to changes in luminosity. This is important for planets that lie near the boundary with the kinetically-limited regime, where atmospheric CO levels can still be moderate; these planets may have temperate surface temperatures at low solar luminosities, but as luminosity increases, the inability to draw down atmospheric CO will cause temperatures to rise significantly. Likewise, planets within the supply-limited regime that sit near the inner edge of the habitable zone will experience much higher surface temperatures than those near the outer edge. Climate stabilization in the kinetically- and supply-limited weathering regimes is shown in Figure 11. Within the kinetically-limited weathering regime, increasing solar luminosity is counteracted by a decrease in , thanks to the weathering feedback (Figures 11A & C). As a result, surface temperatures remain moderate (Figures 11B & D). However in the supply-limited regime, atmospheric CO cannot adjust to changes in solar luminosity, and temperatures increase more sharply as luminosity increases.

Figure 12: Weathering timescale, equation (24), as a function of land fraction for three different surface temperatures with a present day solar luminosity: K (green line), K (blue line), and K (red line). These surface temperatures correspond to atmospheric CO levels of bar, bar, and bar, respectively. The 100 Myr timescale for water loss, , is shown as a dashed line.

The hot, CO rich atmospheres that result from supply-limited weathering have negative implications for planetary habitability. First, the typical surface temperatures estimated for the supply-limited regime, 400-600 K, are higher than any known life form on the modern Earth can tolerate (Takai et al., 2008), and thus life may not be possible at these conditions. Second, although these surface temperatures are within the liquid water stability field, they would place planets in a moist greenhouse climate state where photo dissociation of HO and hydrogen escape leads to rapid water loss (e.g. Kasting, 1988). As a result, planets in the supply-limited weathering regime could lose their oceans, rendering them uninhabitable. However, whether complete water loss will occur in practice is unclear because as sea level drops more land will be exposed, enhancing continental weathering and potentially drawing enough CO out of the atmosphere to stop water loss. If planets that originally lie in the supply-limited weathering regime can reestablish temperate, stable climates with higher land fractions before their oceans are completely lost to space, they will once again be habitable (i.e. the waterworld self-arrest mechanism proposed by Abbot et al. (2012)).

A detailed model of water loss is beyond the scope of this study, so the likelihood of waterworld self-arrest on supply-limited planets is assessed using a simple order of magnitude estimate. Specifically, the timescale for complete water loss is compared to the timescale for CO drawdown via silicate weathering. If Earth were completely covered in oceans, the timescale for water loss is estimated at Myr (Abbot et al., 2012). To determine the timescale for CO drawdown, I assume a planet that initially lies within the supply-limited regime, and therefore has a hot, CO rich atmosphere, undergoes water loss to a given land fraction. The weathering timescale, , is therefore

(24)

Carbon dioxide dissolved in the oceans is included because rapid equilibration between the atmosphere and ocean means that CO from both reservoirs must be drawn down to significantly cool climate. Furthermore, continental weathering is supply-limited for all land fractions, and thus the CO drawdown rate is primarily controlled by the land fraction, because the planets being considered here have CO rich atmospheres. As shown by equation (23), the transition to supply-limited weathering is determined solely by , and is independent of . Weathering can only decrease from the supply-limit as atmospheric CO is drawn down, so using the supply-limit to continental weathering in equation (24) gives the most optimistic estimate.

For a planet with K, does not reach 100 Myr until the land fraction is (Figure 12). At this point significant water loss will have occurred, and the remaining oceans may be lost before weathering can cool the climate (i.e. the water loss timescale at this point will be less than 100 Myrs, because most of the oceans have already been lost). The atmospheric and oceanic CO reservoirs will also have declined by the time reaches , but given the long weathering timescales this effect is likely negligible. However, with K, corresponding to a planet that originally sat near the boundary between supply- and kinetically-limited weathering, the weathering timescale drops below 100 Myr at land fractions of , and waterworld self-arrest appears to be quite likely. Thus the amount of CO that weathering must draw down, which correlates to how far into the supply-limited regime a planet lies, is crucial to determining whether water loss can stabilize climate before the oceans are lost. A planet’s bathymetry will also be important. If seafloor topography is dominated by a few tall islands, with little remaining topography, waterworld self-arrest will be less effective because nearly all of the planet’s water will be lost before a significant portion of land is exposed. However, if there are large submerged continents surrounded by deep basins, a significant supply of water will remain even after a large land fraction has been exposed. Waterworld self-arrest may be viable, especially for planets that originally sit near the boundary with kinetically-limited weathering, but more sophisticated models taking into account bathymetry are needed to place better constraints on this process.

3.2.3 Temperature-dependent Seafloor Weathering

In this section I show that including temperature-dependent seafloor weathering and an upper limit on the basalt CO reservoir does not significantly impact the transition to supply-limited weathering, or the predicted steady-state climates in - space, justifying the simpler model used in the rest of this study where seafloor weathering depends solely on . The upper bound on the basalt CO reservoir, , is given by calculating the number of moles of CO that are stored by complete carbonation of the seafloor to a certain depth:

(25)

where is the depth of complete basalt carbonation, is the density of basalt, is the fraction of reactable elements in basalt (CaO, MgO, and FeO), and is the average molar mass of CaO, MgO, and FeO (Sleep et al., 2001). From Sleep et al. (2001), , g mol, and a baseline value of m is set based on the depth that carbonation extends to on the modern day Earth (Alt & Teagle, 1999). With a negligible land fraction mol (see Table 4).

Symbol Definition Baseline Value Equation
Depth of complete basalt carbonation 500 m (S01) (25)
Basalt density kg m (TS) (25)
Fraction of CaO, MgO, and FeO in basalt 0.3 (S01) (25)
Molar mass of CaO, MgO, and FeO 55 g mol (derived) (25)
Maximum size of basalt reservoir mol (S01) (25)
Seafloor sedimentary carbon reservoir - (26)
Seafloor basalt carbon reservoir - (27)
Temperature-dependent seafloor weathering flux - (30)
Activation energy for seafloor weathering 50 kJ mol (B97) (30)
  • Key for citations: B97=(Brady & Gíslason, 1997), S01=(Sleep et al., 2001).

Table 4: Table of variables and parameters for temperature-dependent seafloor weathering.

In order to incorporate temperature-dependent seafloor weathering with a supply-limit, as determined by equation (25), the carbon cycle model (equations (1)-(3)) is modified as follows: the plate reservoir is separated into a basalt reservoir, , and a sedimentary reservoir, . Seafloor weathering is assumed to supply carbon solely to the basalt reservoir, and terrestrial weathering supplies carbon solely to the sedimentary reservoir. The modified carbon cycle equations are therefore

(26)
(27)
(28)
(29)

where