Lithologic Controls on Silicate Weathering Regimes of Temperate Planets

Draft version March 10, 2021 Typeset using LATEX twocolumn style in AASTeX63

Kaustubh Hakim ,1 Dan J. Bower ,1 Meng Tian ,1 Russell Deitrick ,1 Pierre Auclair-Desrotour ,2, 1 Daniel Kitzmann ,1 Caroline Dorn ,3 Klaus Mezger ,4 and Kevin Heng 1, 5

1University of Bern, Center for Space and Habitability, Gesellschaftsstrasse 6, 3012 Bern, Switzerland 2ASD IMCCE, CNRS-UMR 8028, Observatoire de Paris, PSL, UPMC, 77 avenue Denfert-Rochereau, 75014 Paris, France 3University of Z¨urich,Institute for Computational Science, Winterthurerstrasse 190, 8057 Z¨urich,Switzerland 4University of Bern, Institute of Geology, Baltzerstrasse 1+3, 3012 Bern, Switzerland 5University of Warwick, Department of Physics, Astronomy and Astrophysics Group, Coventry CV4 7AL, UK

(Received August 26, 2020; Accepted January 29, 2021; The Planetary Science Journal; https://doi.org/10.3847/PSJ/abe1b8)

ABSTRACT

Weathering of silicate rocks at a planetary surface can draw down CO2 from the atmosphere for eventual burial and long-term storage in the planetary interior. This process is thought to provide an essential negative feedback to the carbonate-silicate cycle (carbon cycle) to maintain clement climates on Earth and potentially similar temperate exoplanets. We implement thermodynamics to determine weathering rates as a function of surface lithology (rock type). These rates provide upper limits that allow estimating the maximum rate of weathering in regulating climate. This modeling shows that the weathering of mineral assemblages in a given rock, rather than individual minerals, is crucial to determine weathering rates at planetary surfaces. By implementing a ﬂuid-transport controlled approach, we further mimic chemical kinetics and thermodynamics to determine weathering rates for three types of rocks inspired by the lithologies of Earth’s continental and oceanic crust, and its upper mantle. We ﬁnd that thermodynamic weathering rates of a continental crust-like lithology are about one to two orders of magnitude lower than those of a lithology characteristic of the oceanic crust. We show that when the CO2 partial pressure decreases or surface temperature increases, thermodynamics rather than kinetics exerts a strong control on weathering. The kinetically- and thermodynamically-limited regimes of weathering depend on lithology, whereas, the supply-limited weathering is independent of lithology. Our results imply that the temperature-sensitivity of thermodynamically-limited silicate weathering may instigate a positive feedback to the carbon cycle, in which the weathering rate decreases as the surface temperature increases.

Keywords: carbon cycle — weathering — exoplanets — lithology — habitable zone

1. INTRODUCTION climate (Kasting et al. 1993). However, it remains un- clear how this cycle may operate on rocky exoplanets Greenhouse gases such as CO2 are essential in raising Earth’s surface temperature (Kasting et al. 1993; Kop- where surface conditions could depart from those on modern Earth. The essence of the inorganic carbon cy- parapu et al. 2013). To regulate the amount of CO2 in the atmosphere, processes such as the weathering of cle is captured by the Ebelman-Urey reaction involving rocks, degassing and regassing are necessary (Ebelmen the conversion of a silicate mineral (e.g., wollastonite 1845; Urey 1952); but see Kite & Ford(2018). One of CaSiO3) to a carbonate mineral (e.g., calcite CaCO3) in the basic assumptions for the definition of the classical the presence of atmospheric CO2, habitable zone is therefore the presence of the carbonate- CaSiO + CO CaCO + SiO . (1) silicate cycle (carbon cycle) that regulates the long-term 3 2 ←→ 3 2 The reverse of reaction (1), metamorphism, converts Corresponding author: Kaustubh Hakim carbonates back to silicates and releases CO2 to the at- [email protected] mosphere. In addition to metamorphic decarbonation, the degassing of the mantle at mid-ocean ridges and at 2 Hakim et al. volcanic arcs also contributes to the CO2 supply to the and thus neglect it (e.g., Walker et al. 1981; Berner et al. atmosphere. Although wollastonite is used to exemplify 1983; Caldeira 1995; Berner & Kothavala 2001). How- the carbon cycle, in reality its contribution is insignifi- ever, recent studies have claimed that seafloor weather- cant. In this study, we evaluate the contribution of sil- ing is likely a significant component of global weathering icate minerals and rocks on the weathering component during Earth’s history (e.g., Coogan & Gillis 2013; Char- of the carbon cycle. nay et al. 2017; Krissansen-Totton et al. 2018); but see An important feature of the carbon cycle on Earth is Isson & Planavsky(2018). Seafloor weathering may be the negative feedback of silicate weathering (e.g., Walker of critical importance on oceanworlds (e.g., Abbot et al. et al. 1981; Berner et al. 1983; Kump et al. 2000; Sleep & 2012; Foley 2015; Höninget al. 2019); but see Kite & Zahnle 2001; Abbot et al. 2012; Foley 2015; Krissansen- Ford(2018). Totton & Catling 2017; Graham & Pierrehumbert 2020). A prevalent assumption among studies modeling sil- This feedback buffers the climate against changes in stel- icate weathering is that kinetics of mineral dissolution lar luminosity and impacts the extent of the habitable reactions determines the weathering flux zone (e.g., Kasting et al. 1993; Kopparapu et al. 2013). This feedback is sensitive to partial pressure of CO in P β E 1 1 2 w = w CO2 exp (2) the atmosphere, surface temperature, and runoff which 0 PCO2,0 − R T − T0 facilitates weathering reactions and transport of aqueous where the subscript ‘0’ denotes reference values, P chemical species from continents to oceans (e.g., Walker CO2 is the partial pressure of CO in the gaseous state, and et al. 1981; Berner et al. 1983). In oceans, cations and 2 β is the kinetic power-law constant constrained empir- bicarbonate or carbonate ions react to form carbonates ically from laboratory or field data (e.g., Walker et al. that are deposited on the seafloor, thereby removing 1981; Berner et al. 1983; Kasting et al. 1993; Sleep & CO2 from the atmosphere-ocean system (e.g., calcite, Zahnle 2001; Foley 2015; Krissansen-Totton & Catling Ridgwell & Zeebe 2005). High concentrations of CO2 in 2017). The activation energy E is also determined em- the atmosphere elevate the surface temperature due to pirically from the dependence of kinetic rate coefficients the greenhouse effect of CO2. High temperatures give on the Arrhenius law with T as the surface temper- rise to high evaporation and high precipitation (rainfall) ature and R as the universal gas constant (Palandri rates. As precipitation intensifies, runoff also intensifies. & Kharaka 2004; Brantley et al. 2008). Walker et al. As a result, silicate weathering intensifies, reducing the (1981) adopt β = 0.3 based on kinetic rate measure- abundance of atmospheric CO2 which in turn decreases ments of feldspar weathering in a laboratory (Lagache the temperature. When temperatures become low, evap- 1965). Later studies adjusted the value of β using more oration and precipitation rates become low thereby de- laboratory and field measurements or balancing car- creasing runoff and weathering. In the absence of intense bon fluxes (Brantley et al. 2008). A Bayesian inver- weathering, volcanic degassing increases the abundance sion study performed by Krissansen-Totton & Catling of CO2 in the atmosphere sufficiently to increase the (2017) for the carbon cycle on Earth based on data from temperature again. In the absence of a self-regulating the past 100 Myr shows that the value of β for con- mechanism, the fate of Earth’s atmosphere may have tinental weathering is largely unconstrained: between been like that of Venus (Gaillard & Scaillet 2014). 0.21 0.48 (prior: 0.2 0.5) for their nominal model and In addition to silicate weathering on continents, sil- − − between 0.05 0.95 (prior: 0 1) for their Michaelis- icate weathering on the seafloor also has the potential − − Menten model (Volk 1987; Berner 2004). Moreover, to provide an equivalent negative feedback (e.g., Fran- the power-law exponent of seafloor weathering in Equa- cois & Walker 1992; Brady & G´ıslason 1997; Sleep & tion (2) is either assumed to be equal to 0.23 (Brady Zahnle 2001; Coogan & Gillis 2013; Krissansen-Totton & G´ıslason 1997) or varied between 0 1 (e.g., Sleep & & Catling 2017; Charnay et al. 2017). Seafloor weath- − Zahnle 2001; Coogan & Dosso 2015). More sophisti- ering is the low-temperature (<313 K) carbonation of cated formulations of seafloor weathering assume a de- the basaltic oceanic crust facilitated by the circulation pendence on the oceanic crustal production rate with yet of seawater through hydrothermal systems (Coogan & another power-law exponent between 0 2 (Krissansen- Gillis 2018). Seafloor weathering reactions, like conti- − Totton et al. 2018). nental weathering reactions, dissolve silicate minerals The flow of fluids such as rainwater through the pore- constituting rocks in the presence of water and CO2. space of soils facilitates silicate weathering reactions on Most studies modeling the carbon cycle expect the con- continents. If the fluid residence time is shorter than tribution of seafloor weathering to be smaller than con- the time needed for a reaction to attain chemical equi- tinental weathering by up to a few orders of magnitude librium, the reaction becomes rate limiting and reaction Lithologic Controls on Silicate Weathering 3 kinetics govern the amount of solute released (e.g., Aa- (CHemical weatherIng model based on LIthology). This gaard & Helgeson 1982; Helgeson et al. 1984). Then the model tracks the aqueous carbon reservoir assimilating weathering is kinetically-limited. In contrast, at long three weathering regimes as a function of lithology. In fluid residence times, silicate weathering reactions reach addition to continental weathering, this model is applied chemical equilibrium resulting in a thermodynamic limit to seafloor weathering. The primary goal is to determine (also known as the runoff limit, Kump et al. 2000; Ma- lithology-based weathering fluxes on the surface of tem- her 2011). The thermodynamic limit of weathering is re- perate exoplanets by mitigating the impact of present- ferred to as the transport limit in recent literature (e.g., day Earth calibrations. The key philosophy behind this Winnick & Maher 2018); however, the transport limit study is to investigate the extent to which the silicate is traditionally used to describe weathering limited by weathering model may be generalized, beyond its Earth- the supply of fresh minerals, i.e., supply-limited weath- centric origins, in order to apply it to rocky exoplanets ering (Thompson 1959; Stallard & Edmond 1983; Maher with secondary atmospheres. 2011; Kump et al. 2000). In this study, the usage of the term transport limited is avoided. Observations that 2. WEATHERING MODEL regional weathering fluxes on Earth depend on runoff (e.g., Riebe et al. 2004) in addition to reaction kinet- 2.1. Proxies for Weathering ics suggest that global weathering fluxes are a mixture Continental weathering occurs in continental soils and of kinetically- and thermodynamically-limited regimes. saprolite where runoff (water discharge per unit surface The fluid-transport approach (Steefel et al. 2005; Maher area) facilitates fluid-rock weathering reactions (Fig- 2010; Maher & Chamberlain 2014; Li et al. 2017) allows ure1). Seafloor weathering occurs in pores and veins the modeling of both thermodynamic and kinetic limits of the oceanic crust (hereafter, pore space) where sea- of weathering on temperate planets (Winnick & Maher water reacts with basalts. Analogous to runoff, the fluid 2018; Graham & Pierrehumbert 2020). flow in off-axis low-temperature hydrothermal systems The lithology (rock type) is anticipated to play a role facilitates seafloor weathering reactions (Stein & Stein in the intensity of the silicate weathering feedback (e.g., 1994; Coogan & Gillis 2018). The fluid flow rate q (ei- Walker et al. 1981; Stallard & Edmond 1983; Kump ther runoff or hydrothermal fluid flow rate) is a free pa- et al. 2000). At the thermodynamic limit of weather- rameter in CHILI (Table1). On Earth, the present- 1 ing, Winnick & Maher(2018) demonstrate that miner- day continental runoff varies between 0.01 3 m yr− 1 − als in the feldspar mineral group exhibit feedback that is with a mean of approximately 0.3 m yr− (Gaillardet weaker to stronger than the kinetic weathering feedback. et al. 1999; Fekete et al. 2002). On the seafloor, the For example, their calculations show that the thermo- non-porosity corrected hydrothermal fluid flow rates 1 dynamic β (or βth) is 0.25 for albite and K-feldspar are between 0.001 0.7 m yr− with a mean of about 1 − and 0.67 for anorthite, compared to the classic kinetic 0.05 m yr− (Stein & Stein 1994; Johnson & Pruis 2003; β = 0.3 from Walker et al.(1981). The kinetic weather- Hasterok 2013). ing models are based on reaction rates of feldspars, com- In the fluid transport-controlled model of continental 2 1 mon silicate minerals comprising granitic rocks on mod- weathering, the weathering flux w (mol m− yr− ) is ern continents (e.g., Walker et al. 1981). However, rocks the product of the concentration of a solute of interest on the Hadean and early Archean Earth were more mafic C and runoff q (e.g., Maher & Chamberlain 2014), than present-day Earth and therefore poorer in SiO2 and feldspar content (e.g., Chen et al. 2020); but see Keller & Harrison(2020). The modern seafloor mostly contains w = C q. (3) basaltic rocks. Observations of clay minerals on Mars suggest past weathering processes (Bristow et al. 2018). We also apply the same approach to seafloor weathering. 1 1 Clues from stellar elemental abundances point to diverse The total silicate weathering rate W (mol yr− ) is the planetary compositions (e.g., Wang et al. 2019; Spaar- sum of the continental (Wcont) and seafloor weathering garen et al. 2020). Consequently, the surface lithologies rates (Wseaf ). These rates are given by product of the (and therefore weathering fluxes) on rocky exoplanets continental (wcont) or seafloor weathering flux (wseaf ) are not necessarily similar to modern Earth. and the continental (f As) or seafloor surface area ((1 In this study, we develop a silicate weathering model − of the inorganic carbon cycle with key inclusion of lithol- 1 ogy by applying the fluid-transport model of Maher The weathering rate has dimensions of moles per unit time, whereas the weathering flux has dimensions of moles per unit & Chamberlain(2014) to fluid-rock reactions: CHILI area per unit time (Table2). 4 Hakim et al.

Absorbed Outgoing Precipitation Stellar Longwave Flux Radiation

CO% Partial Pressure (&'() ) Evaporation Surface Temperature (") Continental Runoff (!) SaproliteSoils

Hydrothermal Fluid Flow Rate (!)

Seafloor Pore-space Hydrothermal Heat Flux

Continental Area Fraction (*) Sea Area Fraction (1 − *)

Figure 1. Silicate weathering model based on weathering reactions and fluid flow rates. The continental runoff and the seafloor fluid flow rate enable weathering reactions in continental soils and saprolite, and seafloor pore-space, respectively. Key parameters and processes are labeled (see Tables1,2 and3 for all parameters and computed quantities). 2 f)As), where As is the planet surface area and f is the ceeding [HCO3−]. Both CO3− and HCO3− have the po- continental area fraction: tential to contribute to Ca-Mg-Fe carbonate precipitation on the seafloor (e.g., calcite precipitation, Plummer W = w f A + w (1 f) A . (4) et al. 1978), cont s seaf − s To model silicate weathering, the aqueous bicarbonate 2+ ion concentration [HCO−] is normally used as a proxy 2HCO− + Ca CaCO + CO + H O, 3 3 −→ 3 2 2 for weathering because it is a common byproduct of sil- 2 2+ (6) CO − + Ca CaCO3. icate weathering (e.g., anorthite weathering, Winnick & 3 −→ Maher 2018), Equation (6) exemplifies that, in addition to bicarbonate and carbonate ions, divalent cations are required to 2+ CaAl Si O + 2CO + 3H O 2HCO− + Ca drive the flux of CO2 out of the atmosphere-ocean sys- 2 2 8 2 2 −→ 3 (5) tem. Nonetheless, since twice as many HCO3− ions as +Al2Si2O5(OH)4. 2 CO3− ions are needed to precipitate one mole of car- This is a good assumption under modern Earth condi- bonate (Krissansen-Totton & Catling 2017), in addition to HCO3− (C = [HCO3−] in Equation3), we consider car- tions since HCO3− is the primary CO2-rich product of continental silicate weathering that is carried to oceans bonate alkalinity A resulting from reactions between sil- by rivers and reacts with Ca2+ to precipitate Ca-rich icate rocks and fluids as a proxy for weathering (C = A carbonates on the seafloor. We are interested in model- in Equation3) where ing weathering under conditions more diverse than those 2 on modern Earth. In highly alkaline conditions, the car- A = [HCO3−] + 2[CO3−]. (7) 2 bonate ion CO3− is produced in amounts similar or ex- Lithologic Controls on Silicate Weathering 5

(a) Pyroxene Quartz

Peridotite Basalt Granite (Ultramafic) (Mafic) (Felsic) Pyroxene Olivine Feldspar Olivine Feldspar Mica

(b) Wollastonite Anorthite Muscovite CaSiO3 CaAl2Si2O8 KAl3Si3O10(OH)2

Pyroxene Feldspar Mica Enstatite Ferrosilite Albite K-feldspar Phlogopite Annite

MgSiO3 FeSiO3 NaAlSi3O8 KAlSi3O8 KMg3AlSi3O10(OH)2 KFe3AlSi3O10(OH)2

Forsterite Olivine Fayalite Quartz Anthophyllite Amphibole Grunerite

Mg2SiO4 Fe2SiO4 SiO2 Mg7Si8O22(OH)2 Fe7Si8O22(OH)2

Figure 2. (a) The composition of peridotite, basalt and granite considered in this study in terms of major endmember mineral groups. All mineral groups for a given rock are assumed to be present in significant proportions (shaded regions). (b) The major weatherable silicate minerals constituting silicate rocks: ternary endmembers of pyroxene, feldspar and mica, binary endmembers of olivines and amphibole, and quartz. Our choice of rock compositions is limited to certain endmember minerals in a mineral group. Mica and amphibole exhibit a large number of endmembers (Holland & Powell 2011). The endmembers of mica and amphibole chosen here are merely illustrative. Table 1. Control parameters and their reference values for Table 2. Computed quantities. modern Earth. Symbol Description Units Symbol Description Reference w Weathering flux mol m−2 yr−1 − PCO2 CO2 partial pressure 280 µbar W Weathering rate mol yr 1 T Surface temperature 288 K T 0 Seafloor pore-space temperature K P Surface pressure 1 bar A Carbonate alkalinity mol dm−3 −1 q Runoff or fluid flow rate 0.3 m yr pH Negative logarithm of H+ activity 5 − ts Soil or pore-space age 10 yr C Concentration mol dm−3 ψ Dimensionless pore-space parameter 222 750 [C] Concentration of a species C mol dm−3

Note— T is not a control parameter for some calculations (T = aC Activity of a species C − −1 f(PCO2 ), Section 2.5). Dw Damköhlercoefficient m yr K Equilibrium constant −2 −1 keff Kinetic rate coefficient mol m yr E Activation energy kJ mol−1 −1 2.2. Major Silicate Lithologies Eth Thermodynamic activation energy kJ mol β Power-law exponent Previous studies applying the fluid transport- − βth Thermodynamic power-law exponent controlled model to continental weathering limit the − lithology to monomineralic (e.g., oligoclase feldspar, Graham & Pierrehumbert 2020) or one rock type (e.g., granite, Maher & Chamberlain 2014; Winnick & Maher 2018). However, the chemistry of common silicate rocks exoplanets, we approximate an ultramafic, a mafic and a ranges from ultramafic to felsic, with increasing SiO2 felsic lithology with major mineralogical compositions of content. To model the surface mineralogy of terrestrial peridotite, basalt and granite, respectively. Peridotites 6 Hakim et al. are upper mantle rocks rich in MgO and FeO and poor actions and thermodynamic activities of reactants and in SiO2 relative to basalts and granites. Basalts are ig- products are given in AppendixB. Thermodynamic ac- neous rocks that are common examples of rocks present tivities quantify the energetics of mixing of constituent in the oceanic crust of Earth, lunar mare on the Moon, components in solid or aqueous solutions (DeVoe 2001). and the crust of Mars. Granites are present on the mod- Since no non-ideal solid-solution behaviors are consid- ern continental crust of Earth. These are highly differ- ered, the endmember compositions of the minerals are entiated rocks that are rich in SiO2, Na2O and K2O due used for calculations and thus the activities of minerals to partial melting and crystallization processes. are set at unity. Furthermore, in a dilute solution, the The three rock types (peridotite, basalt and granite activity of liquid water is approximately unity. considered in this study are assumed to be composed of The activity of an aqueous species (e.g., HCO3−(aq)) 2 3 major mineral groups, and can thus be projected is given by the product of its concentration [HCO−(aq)] − 3 onto binary or ternary diagrams spanned by the major and the activity coefficient γ normalized to the standard 3 mineral groups considered (Figure2a). Each mineral state of concentration (C0 = 1 mol dm− ), a − = HCO3 (aq) group is similarly defined by 1 3 endmember miner- γ [HCO−(aq)] − 3 (DeVoe 2001). In an ideal, dilute solution, als (Figure2b). For instance, olivine is a solid solution C0 γ 1 and [HCO3−(aq)] = aHCO−(aq) C0, an assumption of two endmember minerals, forsterite and fayalite. In → 3 made throughout the study for all aqueous species. The the reduced set of mineral groups with only endmember activity of a gaseous species such as CO (g) is given minerals, we represent peridotite using pyroxenes (wol- 2 by the ratio of its fugacity in the gas mixture to the lastonite and enstatite) and olivines (forsterite and fay- fugacity of pure CO (g) at the total pressure P (i.e., alite), basalt using plagioclase feldspars (anorthite and 2 fCO2 surface pressure in this study). Thus, aCO2(g) = f tot = albite) and pyroxenes (wollastonite, enstatite and fer- CO2 ΓP rosilite), and granite using alkali feldspars (K-feldspar CO2 ΓtotP , where PCO2 is the CO2 partial pressure, P is the and albite), quartz and biotite micas (phlogopite and total surface pressure, Γ and Γtot are fugacity coefficients annite). Halloysite, a phyllosilicate mineral, is a byprod- for the CO2 component and pure CO2, respectively. The uct (secondary mineral) of the weathering of basalt and fugacity coefficients give a correction factor for the non- granite but not of peridotite (see AppendixB). Kaolin- ideal behavior due to mixing and/or pressure effects. For ite, another phyllosilicate secondary mineral is also con- CO2(g), the fugacity coefficient varies between 0.5 and 1 sidered but no secondary minerals containing divalent up to pressures of 200 bar (Spycher & Reed 1988). Our and monovalent cations are modeled. The choice of end- assumptions of unity fugacity coefficients and P = 1 bar member primary minerals to define these rocks makes lead to aCO2(g) = PCO2 . them idealized compared to natural rocks. Moreover, Equilibrium constants depend on temperature and for a rock, not all endmember minerals of a mineral pressure (see AppendixA). Chemical reactions on con- group can be modeled due to the unity activity assump- tinents are characterized by the surface temperature T tion for endmember minerals (Section 2.3). Only major and surface pressure P (Table1). Seafloor weathering minerals in typical rock types are considered and mi- reactions are characterized by the seafloor pore-space nor minerals such as magnetite, hematite and pyrite are temperature T 0 and pressure P 0. In the temperature not discussed. Since the net contribution of carbonate range 273 373 K, data suggest that T 0 is within 1% − weathering on the carbon cycle is small on timescales of T (Krissansen-Totton & Catling 2017, and references of the order of 100 kyr for Earth (e.g., Sleep & Zahnle therein) and hence T 0 = T is assumed. Moreover, pres- 2001), the weathering of carbonate minerals is neglected. sure in the range of 0.01 1000 bar has a negligible effect − on equilibrium constants and resulting concentrations 2.3. Maximum Weathering Model (see Figure A.1 and Figure B.3). However, seafloor pres- Concentrations of products of weathering reactions at sure affects carbonate stability. Nonetheless, P 0 is fixed chemical equilibrium allow one to calculate a thermo- to 200 bar which is the pressure at approximately 2 km dynamic upper limit to weathering (maximum weather- depth in Earth’s present-day oceans. When extending ing). To calculate concentrations of silicate weathering the continental weathering model to seafloor weather- products, a number of chemical reactions need to be ing, we consider a fresh water ocean, in which ocean chemistry does not limit the production of cations and considered. Reactions between water, CO2 and silicate carbonate alkalinity. minerals produce the bicarbonate ion HCO3−. The bicarbonate ion further dissociates into the carbonate ion To demonstrate the computation of thermodynamic 2 + + solute concentrations resulting from fluid-rock reactions, CO3− and H . Water dissociates into H and OH−. The relations between equilibrium constants of these re- weathering of peridotite is considered. Since two pyrox- Lithologic Controls on Silicate Weathering 7 ene endmembers (wollastonite and enstatite) and two (a) olivine endmembers (forsterite and fayalite) are consid- Maximum Concentrations ered to constitute peridotite, dissolution reactions of 12 1 these four minerals are of interest (AppendixB, Ta- 10 ble B.1, rows: (a), (b), (d), (e)). Besides, reactions in 1 10− 10 the water-bicarbonate system are needed (AppendixB, ] 3

Table B.1, rows: (o), (p), (q), (r)). These eight chemi- − 3 cal reactions result in eight equations and ten unknowns. 10− eq The eight equations are given by the relations between 8 pH

activities and equilibrium constants (AppendixB). The [mol dm 5 10− Aeq ten unknowns are the activities of CO2(g), CO2(aq), C 2+ 2+ 2+ + [HCO3−]eq 6 SiO2(aq), Ca , Mg , Fe ,H , OH−, HCO3− and 7 10− 2 2 [CO3−]eq CO3−. An additional equation is given by balancing the charges of all cations and anions present in the solution. pHeq 10 9 4 These nine equations are further reduced to one polyno- − 7 5 3 1 10− 10− 10− 10− mial equation with two unknowns, activities of HCO− 3 PCO2 [bar] and CO2(g) (see AppendixB, Table B.2). If ferrosilite, the third endmember of pyroxene, is also considered, an (b) additional equation is introduced but the unknowns re- Maximum Weathering 104 main the same and the system is over-determined. This scenario stems from the assumption of endmember min- 102 erals with unity activities instead of solid solutions. In ] 1 the case of solid solutions, the activities of endmem- − ber minerals become unknowns rather than being fixed yr 100 2 at unity. This increases the number of unknowns for − a given set of equations, and requires more equations 2 10− that can be derived from additional reactions (e.g., see [mol m Galvez et al. 2015, for a treatment of solid solutions). w 4 Aeq Although mineral solid solutions are commonplace in 10− [HCO3−]eq 2 rocks, endmember considerations allow us to establish a [CO3−]eq 6 simple framework in which the effects of various litholo- 10− 7 5 3 1 gies can be explored on the weathering of exoplanets. 10− 10− 10− 10− Therefore, the presence of ferrosilite in peridotite is ig- PCO2 [bar] nored. Moreover, the presence of K-feldspar and quartz in basalt and anorthite in granite is not considered. For Figure 3. (a) Maximum concentrations of the components of carbonate alkalinity A and the solution pH of the network − a given PCO2 , aHCO is calculated by finding the sole 3 of reactions defining peridotite weathering assuming chem- physical root of such a polynomial equation. The bi- ical equilibrium as a function of PCO2 at T = 288 K. (b) carbonate ion concentration at chemical equilibrium is − Maximum weathering flux corresponding to A, [HCO3 ] and − 2− −1 obtained from the standard concentration, [HCO3 ]eq = [CO3 ] at modern mean runoff of q = 0.3 m yr (other 3 aHCO− 1 mol dm− . Similarly, activities of all aqueous parameters take reference values). Yellow background de- 3 × species are converted to concentrations using the stan- notes the thermodynamic regime of weathering. The scales dard concentration. Subsequently, all unknowns are cal- on vertical and horizontal axes are equal to those in Figure4. culated from the relations between activities and equilibrium constants (Table B.1). centrations gives the maximum weathering flux. Since At chemical equilibrium, the carbonate alkalinity in- no secondary minerals are modeled for peridotite weath- creases and the pH of solution decreases with an in- ering, the weathering flux for PCO2 > 0.1 bar is likely crease in PCO2 at T = 288 K (Figure3a). As PCO2 in- an overestimate (e.g., aqueous alteration of olivine pro- 2 creases, [HCO3−]eq and [CO3−]eq increase monotonically, duces secondary minerals, Kite & Melwani Daswani thereby increasing Aeq, whereas pHeq decreases mono- 2019). Figure3b shows that the thermodynamic weath- tonically. For all except PCO2 < 20 µbar, [HCO3−]eq ering flux calculations using [HCO3−]eq and Aeq increase contributes significantly to Aeq. The weathering flux monotonically with PCO2 . We follow the same proce- (Equation3) resulting from thermodynamic solute con- dure as for peridotite to compute the maximum solute 8 Hakim et al. concentrations and the maximum weathering flux for exceeds the fluid residence time, or the fluid flow rate basalt, granite or individual endmember silicate miner- exceeds the net reaction rate (q > Dw), [HCO3−] Dw → als by solving polynomial equations given in AppendixB [HCO3−]eq q and w = [HCO3−]eq Dw, making w in- and corresponding charge balance equations. dependent of [HCO3−]eq because Dw is modeled to be inversely proportional to [HCO3−]eq (see below). This 2.4. Generalized Weathering Model regime is known as kinetically-limited weathering (here- In natural environments, fluid-rock reactions may not after, kinetic regime). The transition between these two have the time to attain chemical equilibrium due to regimes occurs at q = Dw. The comparison of timescales high fluid flow rates. The maximum weathering model described here is conceptually identical to the ‘quench- (Section 2.3) does not accurately represent the weathering approximation’ employed in atmospheric chemistry ing flux when chemical equilibrium for mineral dissolu- (Prinn & Barshay 1977; Tsai et al. 2017). tion reactions is not attained. We present the general- Rewriting the formulation of Dw from Maher & ized weathering model by extending the fluid transport- Chamberlain(2014), controlled approach of Maher & Chamberlain(2014). Mineral dissolution reactions, being rate limiting, es- ψ Dw = 1 (9) sentially regulate the concentration of reaction products [HCO3−]eq(keff− + mAspts) including [HCO−]. In this limit, kinetics plays a more 3 where [HCO3−]eq is the equilibrium solute concentration, dominant role than thermodynamics. Maher & Cham- keff is the effective kinetic rate coefficient given by ki- berlain(2014) provide a solute transport equation to netics data (AppendixA), ts is the age of soils or pore- calculate such a transport-buffered (dilute) solute con- space, Asp is the specific reactive surface area per unit centration from its value at chemical equilibrium, fluid mass of the rock, m is the mean molar mass of the rock, flow rate (runoff) q and the Damköhlercoefficient D w ψ = L(1 φ)ρXrA is a dimensionless pore-space pa- − sp which gives the ‘net reaction rate’ (see below). The so- rameter that combines five parameters including flow- lute transport equation with HCO3− is path length L, porosity φ, rock density ρ and fraction of reactive minerals in fresh rock Xr. Although Ma- [HCO3−]eq [HCO−] = . (8) her & Chamberlain(2014) parameterize Dw using nine 3 1 + q Dw quantities, in AppendixD, we show that Dw is mostly In their formulation, Maher & Chamberlain(2014) mul- sensitive to four of these quantities given their plausi- 2 tiply Dw by an arbitrary scaling constant τ = e 7.389 ble ranges: [HCO3−]eq, keff , ts and L (L is absorbed in ≈ in order to scale up the solute concentration to 88% of ψ). The remaining five parameters are fixed to reference its equilibrium value at q = Dw. Ibarra et al.(2016) values (Table3). The parameter ψ scales keff with di- when applying the solute transport equation of Maher mensions of moles per unit reactive surface area of rocks & Chamberlain(2014), use τ = 1 without any scal- per unit time to w with dimensions of moles per unit exposed continental or seafloor area for a given lithology ing. For simplicity, we use τ = 1, implying [HCO3−] = per unit time. In Equation (9), the solid mass to fluid 0.5 [HCO3−]eq at q = Dw. The resulting [HCO3−] of our model at q = Dw is about 43% lower than that of the volume ratio ρsf given in the Dw formulation of Maher Maher & Chamberlain(2014) model, which is a much & Chamberlain(2014) is rewritten in terms of solid den- smaller difference than the 5 10 orders of magnitude sity and porosity using ρsf = ρ(1 φ)/φ. The kinetic − − range of solute concentrations explored in this study. rate coefficients of mineral dissolution reactions are ob- Dw tained from Palandri & Kharaka(2004) as a function of The quantity q in Equation (8) is the Damköhler number, the ratio of fluid residence time and chemi- T and pH (see AppendixA). The solution pH at chem- cal equilibrium timescale (Steefel & Maher 2009). The ical equilibrium is used to calculate keff . For minerals D w with no keff data, keff of a corresponding endmember q ratio is essentially the ratio of the ‘net reaction rate’ and the fluid flow rate. When the fluid resi- mineral from the same mineral group is adopted. For dence time exceeds the chemical equilibrium timescale, rocks, we adopt keff given by the minimum keff among or the net reaction rate exceeds the fluid flow rate constituent minerals since the slowest reaction is nor- (q < Dw), [HCO−] [HCO−] and the weathering mally rate limiting; but see, e.g., Milliken et al.(2009) 3 → 3 eq flux reaches its maximum value for a given q, w = where the fastest dissolving mineral sets the pace of rock dissolution. [HCO3−]eq q (Figure4). This weathering regime is called the thermodynamically-limited weathering (hereafter, In Equation (9), when ts = 0 (young soils), Dw = keff ψ thermodynamic regime), also known as runoff-limited − and w = keff ψ. In this ‘fast kinetic’ regime, [HCO3 ]eq weathering. When the chemical equilibrium timescale the weathering flux is directly proportional to the ki- Lithologic Controls on Silicate Weathering 9

Table 3. Fixed Parameters. (a) Generalized Concentrations Symbol Description Value 12 φ Porosity 0.175 † 101 † L Flowpath length 1 m Kinetic ∗ 1 Thermo- Xr Fraction of reactive minerals 1 10− 10 ] dynamic in fresh rock 3 2 −1 † − Asp Speciﬁc surface area 100 m kg 3 10− of mineral or rock 8 pH ρ Density of mineral or rock 2700 kg m−3 †

[mol dm 5 m Mean molar mass of rock 0.27 kg mol−1 † 10−

2 C A As Planet surface area 510.1 Mm [HCO−] 6 ‡ 7 3 f Continental area fraction 0.3 10− 2 ‡ [CO3−] S Stellar flux 1360 W m−2 ‡ pH α Planetary albedo 0.3 ‡ 9 10− 7 5 3 1 4 P 0 Seafloor pore-space pressure 200 bar 10− 10− 10− 10− ‡ PCO [bar] Note—†Maher & Chamberlain(2014), ‡Present-day Earth, 2 ∗ All considered minerals are reactive. (b) Generalized Weathering 104 netic rate coefficient, as assumed in traditional kinetic 102 Kinetic ] Thermo- weathering models (e.g., Walker et al. 1981). When 1

− dynamic 1 ψ t (old soils), D = − and s w yr keff mAsp [HCO ] mA t 0 3 eq sp s 10 ψ 2 w = . This regime may be termed as the − mAspts ‘slow kinetic’ regime, however an accepted terminology 2 is supply-limited weathering (hereafter, supply regime) 10− which is limited by the supply of fresh rocks in the [mol m

w 4 weathering zone (Riebe et al. 2003; West et al. 2005). 10− A [HCO−] An increase in ts decreases the influence of chemical ki- 3 2 [CO3−] netics on weathering. Although ts depends on the soil 6 10− production and physical erosion rates which in turn are 7 5 3 1 10− 10− 10− 10− sensitive to climate, topography and fluid flow proper- P [bar] ties, there is no consensus on the formulation of soil CO2 production and physical erosion rates (Riebe et al. 2003; Figure 4. (a) Generalized concentrations of carbonate West et al. 2005; Gabet & Mudd 2009; West 2012; Ma- alkalinity and the solution pH of peridotite weathering as a her & Chamberlain 2014; Foley 2015). Treating ts as a function of PCO2 at T = 288 K and modern mean runoff free parameter makes it possible to model both the age of q = 0.3 m yr−1 (other parameters take reference values). of continental soils and seafloor pore-space. The tran- From left to right, colored disks mark the transition between sition between the kinetic and supply regimes can be thermodynamic and kinetic regimes. (b) The corresponding 1 weathering flux of carbonate alkalinity, [CO2−] and [HCO−]. defined at ts = k mA where the kinetic reaction rate 3 3 eff sp The scales on vertical and horizontal axes are equal to those equals the supply rate of fresh rocks (see Equation9). in Figure3. In the fluid transport-controlled model, the mineral dissolution reactions (AppendixB, rows (a n) in Ta- − ble B.1) do not attain chemical equilibrium and their bicarbonate system is of the order of milliseconds to reaction products are given by the solute transport equa- days, unlike mineral dissolution reactions that may take tion (Equation8). On the other hand, reactions in the months to thousands of years (e.g., Palandri & Kharaka water-bicarbonate system (AppendixB, rows (o r) in 2004; Schulz et al. 2006). Thus, the generalized con- − centrations of aqueous species in the water-bicarbonate Table B.1), are considered to reach chemical equilib- 2 + rium. This assumption is justified because the equi- system such as [CO3−], [H ] and [OH−] are calculated librium timescale of chemical reactions in the water- from the transport-buffered [HCO3−] and equilibrium 10 Hakim et al. constants (see AppendixB, Figure B.1 for the flow chart formed 1D radiative-convective calculations to obtain T 2 of methodology). Rewriting [CO3−] in terms of [HCO3−], as a function of PCO2 , α and S. Studies such as Haqq-

PCO2 and equilibrium constants in Equation (7), Misra et al.(2016) and Kadoya & Tajika(2019) provide fitting functions to the models of Kopparapu et al.(2013, 2 KCar[HCO3−] 2014). We use the fitting function provided by Kadoya A = [HCO3−] + 2 . (10) KBicPCO2 & Tajika(2019) to couple T in the range 150 350 K 5 − with PCO2 in the range 10− 10 bar. For α = 0.3 Figure4(a) shows the carbonate alkalinity and the so- − 2 lution pH for the generalized model of peridotite weath- (present-day albedo of Earth) and S = 1360 W m− (present-day solar flux), the Kadoya & Tajika(2019) fit- ering as a function of PCO2 at a constant surface temper- 2 ting function results in T between 280 350 K for PCO2 ature (T = 288 K). For PCO2 < 1 µbar, [HCO3−], [CO3−] − and pH in Figure4(a) exhibit the same behavior as in between 10 µbar and 0.5 bar (see AppendixE for de- tails). Figure3(a) because q < Dw, reiterating that chemical equilibrium calculations are valid in the thermodynamic 3. WEATHERING ON TEMPERATE PLANETS regime resulting in the maximum weathering flux. The Maximum Weathering for Various Lithologies maximum ([HCO3−]eq and Aeq, Figure3b) and general- 3.1. ized ([HCO3−] and A, Figure4b) weathering fluxes di- The thermodynamic solute concentrations provide an verge from each other beyond the thermodynamic to upper limit to weathering (e.g., peridotite weathering, kinetic regime transition that occurs at P 1 µbar. CO2 ∼ Figure3). To evaluate the case of maximum weath- The choice of q determines this transition. At a higher ering on temperate planets, in Figure5, we provide q, the regime transition would shift to a lower PCO 2 the [HCO3−]eq weathering flux of three common rocks and vice versa. For PCO2 > 1 µbar, [HCO3−] becomes (basalt, peridotite, granite) as a function of climate independent of PCO in the kinetic weathering regime. 2 properties, PCO2 and T , at present-day mean runoff 1 This is because the kinetic rate coefficient of the slowest of 0.3 m yr− . The dependence of weathering on to- reaction (fayalite dissolution) is constant at a fixed T tal surface pressure is negligible (see AppendixB). The and does not vary when pH is basic (see AppendixA). [HCO3−]eq weathering flux for all rocks increases mono- However, the results shown in Figure4 are expected to tonically with PCO2 at a constant temperature because change when temperature is not held constant and de- weathering intensifies as PCO2 increases (Figure5a). pends on PCO2 where the relation between the two is This is a direct consequence of the calculations of solute given by a climate model (Section 2.5). concentrations at chemical equilibrium (Section 2.3). 2 In the kinetic weathering regime, [CO3−] decreases Table4 gives fitting parameters of the thermodynamic 2 with PCO (Figure4a) because [CO −] is directly pro- 2 3 weathering flux of [HCO3−]eq to the kinetic weathering portional to [HCO3−] and inversely proportional to PCO2 expression (Equation2) for silicate rocks and minerals (Equation 10). Since [HCO3−] is constant in the kinetic considered in this study. Such a fit, although not the 2 regime, [CO3−] shows a strong decrease because of its best approximation of the calculated values, provides a sole dependence on PCO2 . We follow the same procedure way to compare the sensitivity of thermodynamic weath- as for peridotite to compute the generalized concentra- ering to climate properties with studies assuming kinetic tions for the weathering of basalt, granite or individual weathering (e.g., Walker et al. 1981; Sleep & Zahnle minerals and find that lithology strongly impacts the oc- 2001; Foley 2015). currence of weathering regimes (AppendixB, Fig. B.2). Fluid-rock reactions produce both monovalent (e.g., K+, Na+) and divalent cations (e.g., Ca2+, Mg2+). The 2.5. Climate Model sensitivity of [HCO3−]eq flux to PCO2 depends on the The greenhouse effect of CO2 exerts a strong control capacity of a rock to produce divalent cations. More on the planetary surface temperature. A climate model divalent cations in the solution require more HCO3− 2 enables one to express the surface temperature T as a and CO3− ions to balance the charges. Consequently, function of the CO2 partial pressure PCO2 for a given the higher the fraction of divalent cations in a rock, planetary albedo α and top-of-atmosphere stellar flux the higher the thermodynamic PCO2 sensitivity (βth). S. Such a climate model is essential to assess the role of For instance, peridotite, which produces only divalent climate in weathering on temperate planets. CHILI pro- cations, exhibits the highest βth among the three rocks vides the functionality to couple T and PCO2 using any (Figure3). Granite produces more monovalent cations climate model. Previous studies provide formulations of than peridotite and basalt, and therefore has the lowest climate models (e.g., Walker et al. 1981; Kasting et al. βth. This effect has been discussed for feldspar minerals 1993). Recently, Kopparapu et al.(2013, 2014) per- by Ibarra et al.(2016); Winnick & Maher(2018). Lithologic Controls on Silicate Weathering 11

Table 4. Parameters for ﬁtting thermodynamic weathering ﬂux to kinetic weathering expression (Equation2).

Rock/Mineral Halloysite or no secondary mineral Kaolinite as a secondary mineral

Rock/Mineral βth Eth w0 βth Eth w0 (kJ mol−1) (mol m−2 yr−1) (kJ mol−1) (mol m−2 yr−1) Rocks Peridotite 0.71 45 20.5 − Basalt 0.69 52 5.0 0.65 57 125 − − Granite 0.65 31 0.5 0.54 26 1.5 − − P yroxenes Wollastonite 0.53 27 2.74 − Enstatite 0.52 29 0.99 − Ferrosilite 0.51 24 0.10 − Olivines Forsterite 0.59 19 2.99 − Fayalite 0.58 16 0.70 − F eldspars Anorthite 0.69 52.0 5.0 0.72 55.0 50.1 − − Albite 0.26 +2.0 0.04 0.26 +0.04 0.12 K-feldspar 0.26 +8.0 0.01 0.26 +5.9 0.03 Micas Muscovite 0.50 7 0.001 0.50 18 0.076 − − Phlogopite 0.54 27 0.21 0.55 27 0.28 − − Annite 0.54 23 0.04 0.54 23 0.057 − − Amphiboles Anthophyllite 0.50 8 0.001 − Grunerite 0.50 8 0.001 −

Note— w0 is not a ﬁtting parameter but the value of w at PCO2 = 280 µbar, T = 288 K and q = 0.3 m yr−1. No secondary minerals are assumed to be produced during the weathering of peridotite, pyroxenes, olivines and amphiboles.

The βth values for endmember minerals within the librium constants of all reactions considered in addition same mineral group (pyroxene, olivine, mica and amphi- to stoichiometry and hence βth = 0.26 for albite and K- bole), with the exception of feldspars, are similar to each feldspar (Table4). The βth value of the divalent cation- other (Table4). This result is again attributed to the producing anorthite is 0.69, considerably higher than presence of monovalent or divalent cations. The divalent other divalent cation-producing minerals. This is again cation-producing pyroxene, olivine, mica and amphibole attributed to reaction stoichiometry as demonstrated by endmembers exhibit β 0.5 and monovalent cation- Winnick & Maher(2018), although their study results th ∼ producing albite and K-feldspar exhibit β 0.25. in a slightly smaller β = 0.67 because of neglecting th ∼ th The deviation from these ideal values of 0.5 and 0.25 reactions in the water-bicarbonate system. is a result of the simultaneous consideration of the min- The weathering flux produced by the rock with a eral dissolution reaction and the water-bicarbonate reac- higher βth is higher (Figure3a). However, the choice tions. For instance, Winnick & Maher(2018) show that of secondary minerals produced during weathering in- reaction stoichiometry controls βth values by consider- fluences this result. For instance, if kaolinite is consid- ing dissolution reactions of individual feldspar minerals ered instead of halloysite, the weathering flux of basalt and find βth = 0.25 for albite and K-feldspar. How- is higher by more than an order of magnitude, which ever, we find that the presence of ions produced by the is even higher than that of peridotite (see Table4). water-bicarbonate system makes βth dependent on equi- The effect of a secondary mineral on the weathering 12 Hakim et al.

(a) ering ﬂux when kaolinite is used instead of halloysite Thermodynamic PCO sensitivity (Table4). Whereas, the weathering ﬂux of albite and K- 4 2 10 feldspar, constituents of granite in our model, increases βth w = [HCO−]eq q P βth = 0.71 3 ∝ CO2 by a factor of three for the same change. Moreover, the 103 non-consideration of secondary minerals for peridotite

] weathering results in high weathering ﬂux at high PCO2 , 1

− which in reality may be lower (e.g., Kite & Melwani 102 yr βth = 0.69 Daswani 2019). Thus, the choice of secondary mineral

2 strongly aﬀects w and βth of rocks. However, this ef- − 101 fect is small for the weathering of individual minerals βth = 0.65 (Table4). 0 Contrary to observations of the increase in kinetic

[mol m 10 weathering ﬂux with temperature (e.g., Walker et al.

w Peridotite 1981), the thermodynamic weathering flux for rocks de- 10 1 − Basalt creases with temperature at a fixed PCO2 (Figure5b). Granite The fitting parameter, the activation energy of thermo- 2 dynamic weathering provides a scaling relation between 10− 6 5 4 3 2 1 0 10− 10− 10− 10− 10− 10− 10 weathering and temperature. Unlike kinetic weathering,

PCO2 [bar] Eth is negative for weathering of all rocks and minerals except albite and K-feldspar (Table4). This result is (b) a consequence of the decrease in equilibrium constants Thermodynamic T sensitivity as a function of temperature for all mineral dissolution 104 reactions except those of albite and K-feldspar (see Ap- w = [HCO−] q exp ( E /RT ) 3 eq ∝ − th pendixA). The choice of secondary mineral inﬂuences 103 the magnitude of Eth but not its sign. Winnick & Maher

] (2018) observe this eﬀect for plagioclase feldspars which 1

− contain anorthite in addition to albite. This eﬀect is 2 1 10 Eth = -45 kJ mol− yr discussed further in Section 4.2.

2 The ﬁtting parameters provided in Table4 should be − 1 10 used with caution as βth depends on T and Eth depends

on PCO2 . For example, βth for peridotite at PCO2 = Eth = -52 1 0 kJ mol− 280 µbar varies between 0.71 at T = 273 K and 0.77 at

[mol m 10 T = 373 K. Whereas, Eth for peridotite at T = 288 K

w 1 varies between 55 kJ mol− at PCO2 = 1 µbar and 1 1 1 − 10− Eth = -31 kJ mol− 46 kJ mol− at P = 1 bar. This provides fur- − CO2 ther reason to calculate thermodynamic concentrations 2 consistently by considering all necessary reactions simul- 10− 275 300 325 350 375 taneously, as formulated in this study. T [K] 3.2. Climate Sensitivity of Peridotite Weathering Figure 5. Sensitivity of thermodynamic [HCO−] flux 3 eq Figure6 shows the sensitivity of both maximum and (maximum weathering) of rocks to (a) PCO2 at T = 288 K generalized weathering fluxes of peridotite to CO2 par- and (b) T at PCO2 = 280 µbar, at present-day mean runoff q = 0.3 m yr−1. The labeled fitting parameters are obtained tial pressure and surface temperature. The maximum by fitting the thermodynamic weathering flux to the kinetic weathering model gives an upper limit to the weather- weathering expression (Equation2). ing flux. The generalized weathering flux is either equal to or smaller than the maximum weathering flux de- flux of granite is smaller than on basalt. This differ- pending on if the generalized model encounters the ther- ence is attributed to the impact of the secondary mineral modynamic regime or not. The generalized weathering on weathering reactions of feldspar endmember miner- flux in the kinetic regime is lower than that in the ther- als. Anorthite, a constituent of basalt in our model, modynamic regime and higher than that in the supply exhibits an order of magnitude increase in the weath- regime. Since the weathering flux in the thermodynamic regime is an upper limit to weathering, the kinetic flux Lithologic Controls on Silicate Weathering 13

(a) (b)

Without climate model (T = 288 K) Without climate model (PCO2 = 280 µbar)

Peridotite Weathering 3 3 10 A 10

[HCO3−] ] ] 1 2 1 2 − 10 − 10

yr yr Thermodynamic 2 2

− 101 Thermodynamic Maximum − 101 Generalized (Young soils)

0 Generalized (Old soils) 0 10 10 Kinetic [mol m Kinetic [mol m w w 1 1 10− 10− Supply Supply 2 2 10− 5 4 3 2 1 10− 10− 10− 10− 10− 10− 280 300 320 340 PCO2 [bar] T [K]

With climate model (T = f (PCO2 )) Without climate model (PCO2 = 0.1 bar)

103 103 Thermodynamic ] ] 1 2 1 2 − 10 Thermodynamic − 10 yr yr 2 2

− 101 − 101 Kinetic Kinetic

100 100 [mol m [mol m w w 1 1 10− 10− Supply Supply 2 2 10− 5 4 3 2 1 10− 10− 10− 10− 10− 10− 280 300 320 340 PCO2 [bar] T [K]

Figure 6. Sensitivity of peridotite weathering ﬂux of the maximum, generalized young soils (ts = 0) and generalized old soils −1 (ts = 100 kyr) models to climate properties at present-day mean runoﬀ q = 0.3 m yr (other parameters take reference values).

(a) Sensitivity to PCO2 at T = 288 K. (b) Sensitivity to T at PCO2 = 280 µbar. (c) Sensitivity to PCO2 at T = f (PCO2 ) given 5 by the climate model (Section 2.5). (d) Sensitivity to T at PCO2 = 10 µbar. Colored disks denote the transition between kinetic and thermodynamic regimes. cannot exceed the thermodynamic flux. To isolate the (Figure6b,d). In reality, T depends on PCO2 due to the effect of PCO2 on weathering, we present the peridotite greenhouse effect of CO2 which is normally modeled us- weathering flux as a function of PCO2 at a fixed surface ing a climate model (Section 2.5). The strength of the temperature (Figure6a). The sensitivity of weather- coupling between T and PCO2 is sensitive to numerous ing to temperature is demonstrated at two PCO2 values parameters including incident solar flux and planetary 14 Hakim et al. albedo that are fixed to present-day values (Table3). kinetic to thermodynamic regime occurs at high T and

Since the solar ﬂux is held constant, the coupling below PCO2 . The sequence of this transition is in contrast tween T and PCO2 becomes stronger than that during to the switch from the thermodynamic to kinetic regime

Earth’s early history where the solar ﬂux dropped to that occurs at PCO2 = 1 µbar as a function of PCO2 about 70% of its present-day value. Figure6(c) demon- (see Figure4). If the climate model is invoked, the strates peridotite weathering under the limit of strong generalized (ts = 0) weathering ﬂux increases steeply coupling between T and PCO2 . with PCO2 and encounters the thermodynamic regime

The maximum (thermodynamic) carbonate alkalinity at about PCO2 = 0.1 bar, a result of the kinetic temper- and bicarbonate ion ﬂuxes increase monotonically with ature dependence (Figure6c). When T is strongly cou-

PCO2 at T = 288 K (Figure6a). This monotonic behav- pled to PCO2 , the generalized model may switch from the ior is a direct result of chemical equilibrium calculations thermodynamic to kinetic regime at low PCO2 and from with PCO2 as a free parameter at a ﬁxed temperature. the kinetic regime back to the thermodynamic regime

The diﬀerence between the maximum A and [HCO3−] at high PCO2 .

fluxes at low PCO2 is due to the excess contribution of When soils are old (present-day characteristic soil age, 2 [CO3−] to A when the solution pH is basic (see Fig- ts = 100 kyr), the [HCO3−] flux at T = 288 K as a func- ure3). The thermodynamic weathering flux decreases tion of PCO2 is constant and in the supply regime (Fig- with T at a fixed PCO2 (Figure6b,d). As explained in ure6a). This regime is limited by the supply of fresh Section 3.1 and later discussed in Section 4.2, this de- rocks and the influence of chemical kinetics on weath- crease is a result of the decrease in equilibrium constants ering is small compared to when soils are young (ts is of mineral dissolution reactions as a function of tem- small). This regime is almost independent of T (Fig- perature (see AppendixA). The thermodynamic flux at ure6b,d). Consequently, unlike the kinetic weathering

PCO2 = 0.1 bar is higher than at PCO2 = 280 µbar by ﬂux, the supply-limited weathering ﬂux is independent about a factor of 30. Unlike at PCO2 = 280 µbar, at of PCO2 when T -dependence via the climate model is in-

PCO2 = 0.1 bar there is a negligible difference between voked (Figure6c). In the supply regime, the weathering the thermodynamic A and [HCO3−] fluxes. When T de- flux depends on the age of soils which is held constant. pends on PCO2 via a climate model, the behavior of ther- There is almost no difference between the generalized modynamic weathering as a function of PCO2 depends (ts = 100 kyr) A and [HCO3−] fluxes between Figure6(a) on the trade-off between the individual effects of T and and (c) because the inclusion of the T effect via the cli-

PCO2 (Figure6c). Up to PCO2 = 0.01 bar, the ther- mate model is not strong enough to escape from the modynamic fluxes increase with PCO2 because the PCO2 supply regime. In contrast, for the generalized (ts = 0) effect dominates. Whereas, beyond PCO2 = 0.05 bar, model, invoking the climate model increases the weath- there is a small decrease in the thermodynamic fluxes ering fluxes in the kinetic regime enough to enter the because the effect of T takes over. thermodynamic regime at about PCO2 = 0.1 bar. When soils are young (soil age is zero, ts = 0), the generalized [HCO3−] flux at T = 288 K, which is in the 3.3. Endmember Cases of Continental and Seafloor kinetic regime, is almost constant for the given PCO2 Weathering range (Figure6a). The kinetic rate coefficient depends We apply the generalized weathering model to both on T as well as the solution pH which in turn depends continental (Figure7a,b) and seafloor silicate weather- on PCO (Section 2.4). However, the kinetic weathering 2 ing (Figure7c,d) for diverse cases that may represent flux is independent of PCO at a fixed temperature be- 2 weathering scenarios on temperate rocky exoplanets. cause keff of peridotite is independent of PCO2 when the The climate model is used to couple T with PCO2 by fix- solution pH is basic (Figure6a). This keff is determined ing the stellar flux and planetary albedo to present-day by the fayalite dissolution reaction as it is rate limiting Earth values (Section 2.5). This implies a strong cou- among the considered mineral dissolution reactions for pling between T and P such that T varies from 280 K peridotite (see AppendixA). As a function of T at con- CO2 to 350 K when PCO2 varies from 10 µbar to 0.5 bar. The stant PCO , kinetic weathering exhibits a strong depen- 2 lithology and pore-space properties of continents and dence on temperature as seen in Figure6(b,d). For the seafloor on present-day Earth are different from each PCO = 280 µbar case, the generalized (ts = 0) model 2 other. This presents an opportunity to test the general- switches from the kinetic to thermodynamic regime at ized weathering model in an extended parameter-space 310 K where the limit of maximum weathering is en- ∼ beyond applications to continental weathering. We show countered. For the PCO = 0.1 bar case, this transition 2 the results for basalt and granite in Figure7. The sen- temperature increases to 340 K. The transition from ∼ sitivity of the carbonate alkalinity weathering flux to Lithologic Controls on Silicate Weathering 15

(a) (b) Humid vs. Arid Climate Young vs. Old Soils 1 1 Humid (q = 10 m yr− ) 1 10 10 1 1 q = 0.3 m yr− Thermo. Modern-mean (q = 0.3 m yr− ) ψ = ψ0 Kinetic

] 1 ] 1 0 Arid (q = 0.01 m yr− ) 1 0

− 10 − 10 ts = 100 kyr

yr yr Thermodynamic ψ = ψ0 2 1 Supply 2 1 − 10− − 10−

Supply Supply 2 Thermo. 2 10− 10− [mol m Thermodynamic [mol m

cont 3 cont 3 10− 10− Young soils (ts = 0) w w Basalt Modern-mean (ts = 100 kyr) 4 Granite 4 Old soils (ts = 1 Myr) 10− 10− 5 4 3 2 1 5 4 3 2 1 10− 10− 10− 10− 10− 10− 10− 10− 10− 10−

PCO2 [bar] PCO2 [bar]

(c) (d) High vs. Low Hydrothermal Heat Young vs. Old Pore-space 1 1 1 10 High heat ﬂux (q = 1 m yr− ) 10 1 1 Modern-mean (q = 0.05 m yr− ) q = 0.05 m yr−

] 1 ] 1 0 Low heat ﬂux (q = 0.001 m yr− ) 1 0

− 10 − 10 t = 50 Myr s Thermodynamic yr ψ = 100ψ0 yr 2 1 2 1 − 10− − 10− Supply Supply 2 2 10− 10− [mol m [mol m Thermo. Young pore-space (ts = 0, ψ = 100ψ0)

seaf 3 seaf 3 Modern-mean (t = 50 Myr, ψ = 100ψ ) 10− 10− s 0 w w Thermodynamic Small ﬂowpath (ts = 50 Myr, ψ = ψ0)

4 4 Supply 10− 10− 5 4 3 2 1 5 4 3 2 1 10− 10− 10− 10− 10− 10− 10− 10− 10− 10−

PCO2 [bar] PCO2 [bar]

Figure 7. Carbonate alkalinity ﬂux on continents (a,b) and seaﬂoor (c,d) for basalt and granite as a function of PCO2 , where T = f(PCO2 ) is given by the climate model (Section 2.5). Colored disks denote the transition between kinetic and thermodynamic regimes, and inverted triangles denote the transition between thermodynamic and supply regimes.

PCO2 is a complex function of climate, fluid flow rate, In Figure7(a), the continental A flux is calculated rock and pore-space properties. This result implies that for three values of runoff that are representative of the weathering flux cannot be simply approximated by arid, modern-mean and humid climates. Gaillardet Equation (2) assuming either kinetic weathering (e.g., et al.(1999) report regional variations in the present- 1 Walker et al. 1981; Berner et al. 1983; Sleep & Zahnle day runoff from 0.01 3 m yr− . Since the choice of hu- − 2001; Foley 2015) or thermodynamic weathering (Win- mid runoff in our model is higher than the arid runoff by nick & Maher 2018, and Section 3.1, this study). three orders of magnitude, the corresponding weathering 16 Hakim et al.

fluxes should differ by the same amount if the weather- decreases by two orders of magnitude at a much lower ing fluxes are in the thermodynamic regime. However, PCO2 . For the young seafloor pore-space case, the A the differences are smaller in the given PCO2 range. This fluxes of basalt and granite are in the thermodynamic is because non-thermodynamic regimes exhibit smaller regime, and consequently the impact of lithology is pro- weathering fluxes than the thermodynamic regime. For nounced. Compared to the ts = 0 basalt model in Fig- example, up to PCO2 = 0.2 bar, the arid model of gran- ure7(b) that is in the kinetic regime, the ts = 0 basalt ite is in the thermodynamic regime (upper limit for this model in Figure7(d) is in the thermodynamic regime. model), whereas the humid model of granite is in the This difference arises due to the choice of ψ that makes supply regime (lower limit for this model). Thus, the the net reaction rate (Dw) higher than the fluid flow differences between the two models are smaller than rate, pushing basalt into the thermodynamic regime. the maximum possible differences. For more arid cli- Being in the thermodynamic regime, chemical equilib- mates, lithology plays an even more important role as rium controls the weathering flux of basalt. At high the weathering becomes thermodynamically-limited for PCO2 , the thermodynamic weathering flux of basalt de- the whole PCO2 range. In contrast, the supply regime is creases slightly as the decreasing effect of T takes over largely independent of lithology. For this reason, lithol- the increasing effect of PCO2 . This decreasing effect ogy has negligible impact on the modern-mean and hu- of T is due to the decrease in equilibrium constants of mid cases. weathering reactions as a function of temperature (Ap- The age of soils is a key parameter that determines pendixA). The effect of T on the weathering flux of if the weathering is limited by reaction kinetics or lim- granite is small and hence there is no net decrease in ited by the supply of fresh rocks. Figure7(b) shows the weathering flux at high PCO2 . that lithology has no influence on the weathering flux In Figure7(c), we calculate the seafloor A flux for of old soils (ts = 100 kyr and ts = 1 Myr) as opposed three hydrothermal fluid flow rates, where the two ex- to young soils (ts = 0). The old soil models are in the treme q values differ by three orders of magnitude, sim- supply regimes. Granite and basalt in young soils are ilar to the strategy in Figure7(a). Since the fluid in different weathering regimes. Granite is in the ther- flow rates are directly proportional to the hydrother- modynamic regime for the whole PCO2 range because mal heat flux (Stein & Stein 1994; Coogan & Gillis 1 the net reaction rate (Dw = 1 3 m yr− ) is higher than 2013), a variation in the hydrothermal heat flux implies − 1 the fluid flow rate (q = 0.3 m yr− ). In contrast, Dw a variation in the fluid flow rate. Depending on the 1 1 (0.05 4.5 m yr− ) of basalt is lower than q = 0.3 m yr− age of oceanic crust, present-day non-porosity-corrected − 1 up to P = 0.5 bar, implying a transition from kinetic fluid flow rates are observed between 0.001 0.7 m yr− CO2 − to thermodynamic regime above this PCO2 . (Johnson & Pruis 2003). The three cases of seafloor On Earth, the continental and seafloor pore-space dif- weathering shown in Figure7(c) are broadly similar fer in terms of pore-space properties besides the differ- to their continental counterparts. The modern-mean ences in lithology. The dimensionless pore-space param- and high fluid flow rates result in lithology-independent eter ψ depends on the flowpath length L that is nor- weathering fluxes that are in the supply-limited regimes. mally assumed to be of the order of the regolith thick- The low hydrothermal fluid flow rate causes the gran- ness (Section 2.4). Since the thickness of the oceanic ite model to be in the thermodynamic regime for the crust where seafloor weathering occurs is of the order full PCO2 range, although the basalt model transition of 100 m (e.g., Alt et al. 1986; Coogan & Gillis 2018), from the thermodynamic to supply regime at PCO2 = we assume L = 100 m and ψ = 100ψ0. Moreover, the 0.6 mbar. average age of the oceanic crust on Earth at present day is approximately equal to 50 Myr, about 500 times the 4. DISCUSSION AND IMPLICATIONS characteristic age of continental soils. 4.1. Weathering Regimes and the Role of Lithology In Figure7(d), we compare the seafloor A flux for In this study, silicate weathering rates are computed characteristic seafloor values of ts = 50 Myr and ψ = by the simultaneous consideration of dissolution reac- 100ψ0 with two models, one with ts = 0 and another tions of all minerals present in a rock as well as re- with ψ = ψ0. The present-day seafloor A flux is in the actions in the water-bicarbonate system. Three com- supply regime making it independent of lithology. The mon silicate rocks (peridotite, basalt and granite) are supply-limited fluxes of seafloor weathering are smaller examined. We develop the maximum weathering model than those of continental weathering by a factor of 5 ψ (Section 2.3) presuming chemical equilibrium and the because w in this regime. When ψ is lowered ∝ ts generalized weathering model (Section 2.4) that ap- from 100ψ0 to ψ0, the supply-limited weathering flux plies to both equilibrium and non-equilibrium condi- Lithologic Controls on Silicate Weathering 17

(a) (b) Transport vs. Regimes Climate vs. Regimes

1 1 Thermo- 10 10 dynamic Kinetic ] ] 1 1

− Kinetic −

yr yr Thermodynamic

2 1 2 1 10− Thermo- 10− − dynamic Supply − Supply

[mol m 3 1 [mol m 3 10− q = 0.3 m yr− 10− PCO2 = 280 µbar w w Peridotite Basalt Young soils (ts = 0) Granite Old soils (ts = 100 kyr) 5 5 10− 5 3 1 1 3 10− 5 4 3 2 1 10− 10− 10− 10 10 10− 10− 10− 10− 10− 1 q [m yr− ] PCO2 [bar]

Figure 8. Weathering regimes with respect to carbonate alkalinity flux of rocks for two generalized weathering models (ts = 0 and ts = 100 kyr). (a) Weathering flux as a function of fluid flow rate at PCO2 = 280 µbar and T = 288 K. The vertical −1 black line is the modern-mean runoff (q = 0.3 m yr ). (b) Weathering flux as a function of PCO2 at modern-mean runoff of −1 q = 0.3 m yr where T = f(PCO2 ) is given by the climate model (Section 2.5). The vertical black line is the pre-industrial CO2 partial pressure (PCO2 = 280 µbar). Colored disks mark the transition between thermodynamic and kinetic/supply regimes. tions. The generalized weathering model allows us to ite. The thermodynamic weathering flux is proportional explore weathering in three different regimes (Figure8). to the equilibrium carbonate alkalinity which is strongly To simulate the transport-based dilution of equilibrium sensitive to the mineralogy considered for a given rock. concentrations of weathering products, the solute trans- In the kinetic regime, the differences are smaller but sig- port equation of Maher & Chamberlain(2014) is im- nificant. The kinetic weathering flux is proportional to plemented. This equation is based on the interplay be- the effective rate coefficient of a given rock. In contrast, tween the fluid flow rate (q) and the net reaction rate in the supply-limited regime, the A fluxes almost overlap (Dw). When q < Dw, reactions are limited by ther- with each other because there is no impact of lithology. modynamics or transport (runoff) and the weathering The supply regime is strongly sensitive to pore-space is thermodynamically-limited (Figure8a). In contrast, space properties that are held constant. when q > Dw, reactions are limited by kinetics (and in- The climate sensitivity of the weathering flux show- dependent of runoff) and the weathering is kinetically- cases that the weathering regime may depend on lithol- limited. Another parameter, the age of soils (ts), is in- ogy. Figure8(b) shows that the generalized ts = 0 model troduced by Maher & Chamberlain(2014) to model the of the weathering fluxes of peridotite and basalt increase effect of limited supply of fresh rocks on the net reac- steeply with P up to 0.1 bar where T = f(P ) CO2 ∼ CO2 tion rate. The higher the ts, the lower is the Dw. This is obtained from the climate model. Beyond this PCO2 gives rise to another regime, supply-limited weathering value, peridotite and basalt enter the thermodynamic (Figure8a). regime. This is because the weathering flux in the ki- As a function of transport and climate properties, the netic regime cannot keep increasing indefinitely. As soon carbonate alkalinity weathering flux shows a strong de- as the model hits the thermodynamic upper limit, the pendence on lithology in the thermodynamic and kinetic model follows the thermodynamic sensitivities of weath- regimes and a weak dependence on lithology in the sup- ering that are independent of pore-space properties and ply regime (Figure8). This impact of lithology for the depend on chemical equilibrium and lithology. It is im- three weathering regimes as a function of q is seen in portant to note that extrapolations of kinetic weathering Figure8(a). There is approximately an order of mag- expressions (Equation2) used in previous studies (e.g., nitude difference between the thermodynamic fluxes of Walker et al. 1981; Berner et al. 1983; Sleep & Zahnle peridotite and basalt as well as those of basalt and gran- 2001; Foley 2015; Krissansen-Totton & Catling 2017) 18 Hakim et al. may incorrectly predict the weathering flux to be higher for alternate viewpoints. This T -dependence of weath- than the upper limit provided by the thermodynamic ering is due to the increase in kinetic rate coefficients flux. of mineral dissolution reactions as a function of T (see Unlike basalt and peridotite, the generalized young- AppendixA). Laboratory and field measurements of ki- soils (ts = 0) model of granite is in the thermodynamic netic rate coefficients are fitted to the Arrhenius law, regime for the given P range (Figure8b). This is w exp ( E/RT ), where E is the activation energy CO2 ∝ − also evident from Figure8(a) where the vertical line and R is the universal gas constant (Palandri & Kharaka 1 (q = 0.3 m yr− ) falls right before the thermodynamic 2004). The generalized weathering model based on the to kinetic regime transition for granite. Thus, the cli- fluid transport-controlled approach (Maher & Chamber- mate sensitivity of the weathering of fresh granite at lain 2014) captures the diversity of weathering regimes 1 q = 0.3 m yr− is determined largely by thermodynam- in a generic formulation, which is particularly useful for ics instead of kinetics. On the other hand, for the gener- applications to exoplanets with potentially diverse sur- alized models at the present-day characteristic soil age face environments. Studies applying the fluid transport- of ts = 100 kyr, the weathering fluxes of the three rocks controlled model find that T has a small effect on weath- overlap with each other. This is because this ts value is ering (Winnick & Maher 2018; Graham & Pierrehum- so high that the effect of keff on the ‘net reaction rate’ bert 2020). This statement holds in the thermodynamic Dw is negligible in pushing the model out of the supply regime of weathering for certain plagioclase feldspars in- regime. In this regime, the weathering flux is indepen- cluding oligoclase which coincidentally exhibit a small dent of the climate and transport properties. T sensitivity, although Winnick & Maher(2018) find

Since most laboratory measurements of kinetic rate that the equilibrium [HCO3−] resulting from plagioclase coefficients are available for individual minerals, previ- feldspars decreases with T . ous studies discuss weathering of individual minerals in- To isolate the effect of temperature on weathering stead of rocks (e.g., Walker et al. 1981; Berner et al. from that of the PCO2 effect, we first present weathering

1983). In reality, all minerals in rocks undergo weath- as a function of T at a fixed PCO2 in Figure9(a). The 1 ering contemporaneously, rendering consideration of in- generalized weathering model at ts = 0, q = 0.3 m yr− dividual minerals in isolated systems as less informa- and PCO2 = 280 µbar (no climate model) shows that tive. Since the minerals in a rock are in contact with the carbonate alkalinity flux of rocks is in the kinetic the aqueous solution, solute concentrations are buffered regime at low temperatures and in the thermodynamic by the dissolution reactions of these minerals. It is regime at high temperatures. In the kinetic regime, essential to consider these reactions simultaneously to there is a steep increase in the weathering flux of gran- solve for solute concentrations. The generalized weath- ite up to 285 K, basalt up to 310 K, and peridotite ∼ ∼ ering model shows that the choice of individual min- up to 320 K because the effective kinetic rate co- ∼ erals or rocks determines the weathering regime. For efficients show an exponential increase with tempera- example, the feldspar endmember minerals (anorthite, ture. Beyond these transition temperatures (where the albite and K-feldpsar) are in the thermodynamic regime net reaction rate equals the fluid flow rate), the mod- for the PCO2 range considered, whereas rocks are ex- els enter their respective thermodynamic regimes. This hibit both thermodynamic and kinetic regimes (see Fig- transition implies a switch from the negative feedback ure B.2). In their implementation of the fluid transport- of silicate weathering to the carbon cycle to a poten- controlled model, Graham & Pierrehumbert(2020) find tial positive feedback. Since the thermodynamic regime that weathering is largely independent of kinetics be- gives the maximum possible weathering flux, the ki- cause of their choice of oligoclase (a type of plagioclase netic weathering flux cannot exceed the thermodynamic feldspar mineral) to model weathering, which is in the weathering flux in the generalized model. Importantly, thermodynamic regime of weathering for a wide range the kinetic weathering flux of granite is higher than that of CO2 partial pressures similar to feldspar endmembers of basalt and peridotite because the slowest kinetic re- shown in Figure B.2. action among the constituent endmember minerals of granite has a higher kinetic rate coefficient than those 4.2. Positive Feedback of Weathering at High of basalt and peridotite. This result is based on the Temperature choice of endmember minerals to define rocks and is expected to change with the choice of other endmember It is widely accepted that weathering intensifies with minerals or mineral solid solutions. surface temperature (e.g., Walker et al. 1981; Berner The kinetic weathering flux increases with T as ex- et al. 1983; Kump et al. 2000; Brantley et al. 2008); pected, however the thermodynamic weathering flux de- but see Gaillardet et al.(1999) and Kite et al.(2011) Lithologic Controls on Silicate Weathering 19

(a) a function of T . This negative exponential decrease in Without climate model (PCO = 280 µbar) weathering is a result of the negative slope exhibited by 2 2 10 equilibrium constants of mineral dissolution reactions as a function of T except for albite and K-feldspar (see AppendixA). This T -dependence is a thermodynamic

] property of mineral dissolution reactions in the aqueous 1 1 − 10 system. For rocks, constituent minerals determine the

yr overall dependence of thermodynamic weathering on T .

2 For example, for a rock consisting of only albite and K- − feldpsar, the thermodynamic weathering ﬂux is expected Kinetic Thermodynamic 100 to increase with T , similar to the prevalent understand- ing of the T -dependence of kinetic weathering. How- [mol m ever, apart from these two minerals, all other minerals

w considered in this study show a negative slope. Since granite consists of both albite and K-feldspar in addi- 10 1 − tion to quartz, phlogopite and annite, granite shows the least negative slope among the three rocks considered 280 300 320 340 (Figure9a). Moreover, weathering of basalt in the ther- T [K] modynamic regime decreases with T more steeply than peridotite because of the presence of anorthite in basalt (b) that exhibits the most negative slope in the equilibrium

With climate model (PCO2 = f (T )) constant temperature parameter-space. 102 − 1 When the climate model is invoked assuming present- q = 0.3(1 + (T T0)) m yr− − day solar ﬂux and present-day planetary albedo, PCO 1 2 q = 0.3 m yr− increases from 10 µbar to 0.5 bar as a function of the T

] varying from 280 K to 350 K and the decreasing eﬀect 1 1 − 10 of T on weathering disappears (Figure9b). This is be-

yr Thermodynamic cause in the thermodynamic regime the weathering ﬂux 2 shows a positive power-law dependence on PCO2 , which − is stronger than the exponential decrease in weathering 100 with temperature. And in the kinetic regime, the exponential increase in weathering with temperature domi- [mol m nates. It is interesting to note that basalt and peridotite

w Kinetic Peridotite enter the thermodynamic regime only at 325 K and ∼ 1 Basalt 345 K respectively, whereas granite is in the thermody- 10− ∼ Granite namic regime for the given temperature range. For this climate model, assuming a constant stellar ﬂux results

280 300 320 340 in a strong coupling between T and PCO2 , in contrast to the behavior when P is held constant. Depending T [K] CO2 on the stellar ﬂux and planetary albedo, the strength of

Figure 9. Generalized carbonate alkalinity flux of rocks coupling between T and PCO2 is probably in between as a function of temperature at ts = 0, PCO2 = 280 µbar the two panels shown in Figure9, pushing the ther- for a constant q = 0.3 m yr−1 and a T -dependent q with modynamic to kinetic transition temperature given in −1 = 0.03 K and T0 = 288 K. Colored disks mark the Figure9(a) to higher values. The supply-limited weath- transition from kinetic to thermodynamic regimes. ering flux, on the other hand, is independent of T (e.g., Figure6c,d). This is because the supply-limited weath- creases with T (Figure9a). As described in Section 3.1, ering flux depends on pore-space parameters including if fitted to a kinetic weathering expression (Equation2), ts and ψ that are assumed to be constant for a given the thermodynamic model gives a negative value for the model. In reality, the age of soils may indirectly depend activation energy (Eth) for the weathering of rocks and on temperature through the effect of precipitation and most minerals. Such a negative value highlights that runoff on physical erosion rates (e.g., West et al. 2005; thermodynamic weathering exhibits a negative slope as West 2012; Foley 2015). 20 Hakim et al.

Previous studies have argued that since runoff de- time-dependent median models of PCO2 and T from the pends on precipitation which in turn depends on T , the same KT18 model are used as inputs to the generalized weathering flux should increase even more strongly with model. The same inputs are used to obtain the con- T when a T -dependent runoff is assumed (e.g., Berner tinental weathering rate from Walker et al.(1981) and & Kothavala 2001). Figure9 shows that this statement the seafloor weathering from Brady & G´ıslason(1997), does not hold in the kinetic regime of weathering. This where the present-day weathering rates of both models is because the kinetic regime is independent of runoff in are normalized to those of KT18 models. the fluid transport-controlled model (Maher & Cham- To first order, one may expect that continental and berlain 2014). Considering a linear dependence of runoff seafloor weathering rates depend on the continental sur- on temperature (e.g., Equations 41, 42 in Graham & face area (f A ) and the seafloor surface area ((1 f) A ), s − s Pierrehumbert 2020) instead of a constant runoff, there respectively (Equation4). However, not all continen- is no impact on weathering because the kinetic regime tal and seafloor surface area undergoes weathering on is independent of runoff in the generalized model. Earth. Fekete et al.(2002) report the value of continental weatherable area to be equal to 93 Mm2, about 4.3. Global Silicate Weathering Rates 60% of modern continental area (= 0.6 f As). About 2 147 Mm of the seafloor area (= 0.41 (1 f) As) is ex- During the Archean (2.5 4 Ga), the incident solar − − pected to contribute to seafloor weathering (given by the radiation was about 70 80% of its present-day value, − exposed area of low-temperature hydrothermal systems, not high enough to maintain a temperate climate with Johnson & Pruis 2003). Therefore, in Figure 10, the present-day atmospheric CO levels (Sagan & Mullen 2 continental and seafloor weathering rates are given by 1972; Charnay et al. 2020). Although there are no di- 2 2 Wcont = wcont 93 Mm and Wseaf = wseaf 147 Mm , rect measurements of historical weathering rates, the × × respectively. A more precise definition of continental Archean geological record suggests a steady decrease weatherable area is the area susceptible to precipita- in P from the order of 0.1 bar to modern values CO2 tion and runoff, and that of seafloor weatherable area while maintaining surface temperatures between 280 K is the area covered by low-temperature hydrothermal and 315 K (Krissansen-Totton et al. 2018, and refer- systems that are younger than approximately 60 Myr ences therein). The lower insolation during the Archean (Stein & Stein 1994; Johnson & Pruis 2003; Coogan & was compensated by the greenhouse effect of CO . As 2 Gillis 2018). Nonetheless, the definition given in Equa- the insolation increased, climates should have become tion (4) assumes all surfaces are weatherable, thereby warmer than the observations suggest. Without a neg- giving upper estimates of the global weathering rates on ative feedback of silicate weathering that allowed a de- exoplanets. crease in CO levels as insolation increased, modern cli- 2 The present-day continental weathering rate for young mates would not have been temperate (Walker et al. soils (ts = 0) in Figure 10(c) is in the thermodynamic 1981; Berner et al. 1983; Kasting et al. 1993). The ex- regime (see also Figure8a). Similarly, the present-day tent of silicate weathering during the history of Earth seafloor weathering rate for young pore-space is in the and the contribution of seafloor weathering are debated thermodynamic regime (Figure 10d). As the CO2 lev- (e.g., Sleep & Zahnle 2001; Berner & Kothavala 2001; els increase (prior to 0.5 Ga), continental weathering for Foley 2015; Krissansen-Totton et al. 2018; Coogan & young soils is driven by kinetics (Figure 10c). Whereas, Gillis 2018). By applying the generalized weathering seafloor weathering for young pore-space is in the ther- model to Earth, we lay the foundation for understand- modynamic regime even at high PCO . The weathering ing climate regulation on temperate planets. 2 rate of the young-soils granite model decreases between Since present-day continents on Earth are largely fel- 1 2 Ga. This is because the effective kinetic rate coeffi- sic and the seafloor is mafic, we approximate the lithol- − cient of granite is almost constant as a function of pH in ogy of continents by granite and the seafloor by basalt. basic solutions resulting from our calculations and de- Continental and seafloor weathering rates on Earth are pends mostly on T , which decreases between 1 2 Ga. calculated up to 4 Ga (Figure 10c,d). This is the first − Seafloor weathering in the thermodynamic regime in- application of the fluid transport-controlled weather- creases monotonically because the effect of PCO on ing model of Maher & Chamberlain(2014) to seafloor 2 weathering wins over the T effect. In the thermody- weathering on Earth. Since there are no direct measure- namic regime, the choice of secondary minerals also af- ments of historical weathering rates on Earth, we com- fects the weathering rates. For example, on one hand, pare the generalized model developed in this study with the granite and basalt weathering rates for ts = 0 mod- a model from Krissansen-Totton et al.(2018, Figure 3, els would be higher by a factor of 20 and 3 when kaolin- hereafter, KT18). Instead of using a climate model, Lithologic Controls on Silicate Weathering 21

(a) (b) 300 Krissansen-Totton et al. (2018) 1 10− 295

2 10− [bar] [K]

290 2 T CO P 10 3 285 −

Krissansen-Totton et al. (2018) 4 280 10− 0 1 2 3 4 0 1 2 3 4 t [Ga] t [Ga]

This work (Granite, Young soils) Thermo. This work (Granite, Old soils) 103 103 Walker et al. (1981) ] ] 1 Krissansen-Totton et al. (2018) 1 − −

102 Kinetic 102 Thermo. This work (Basalt, Young pore-space) This work (Basalt, Old pore-space) [Tmol yr [Tmol yr Brady & Gislason (1997) 1 1 10 10 Krissansen-Totton et al. (2018) seaf

cont Supply W W 100 100 A [X2+] Supply 0 P 1 2 3 4 0 1 2 3 4 t [Ga] t [Ga]

Figure 10. Continental granite and seaﬂoor basalt weathering rates derived from carbonate alkalinity A and the sum of concentrations of divalent cations P[X2+] compared with a model from Krissansen-Totton et al.(2018, Figure 3E,F, hereafter,

KT18) for the past 4 Ga on Earth. (a) T and (b) PCO2 are the median values of the same model from KT18 that are used as inputs to the weathering models in (c) and (d). (c) Generalized young soils (ts = 0) and old soils (ts = 100 kyr) granite models −1 2 take the same values for ψ = ψ0, q = 0.3 m yr (modern mean runoff) and continental weatherable area of 93 Mm (Fekete et al. 2002). (d) Generalized young pore-space (ts = 0) and old pore-space (ts = 50 Myr) basalt models take different values −1 for ψ (ψ = 100ψ0 and ψ = 18ψ0, respectively) but the same value for q = 0.05 m yr (modern mean hydrothermal fluid flow rate) and seafloor weatherable area of 147 Mm2 (Johnson & Pruis 2003). Gray shaded regions are 95% confidence intervals of KT18 models. Vertical gray lines are uncertainties in the estimates of present-day silicate weathering rates given by geological measurements and models. Colored disks denote the transition between thermodynamic and kinetic regimes. ite is used as a secondary mineral instead of halloysite. On the other hand, if secondary minerals incorporat- 22 Hakim et al. ing divalent cations are modeled, the weathering rates the soil age is set to the characteristic soil age (100 kyr). would be lower because bicarbonate and carbonate ions The soil age encapsulates several processes including tec- equivalent to cations locked up in secondary minerals tonics, erosion and soil production (Heimsath et al. 1997; would be removed from the solution (e.g., Kite & Mel- Riebe et al. 2003; West et al. 2005; Maher & Chamber- wani Daswani 2019). lain 2014). For example, the soil age decreases with in- The continental weathering rate derived from carbon- creasing erosion or increasing mineral supply rates (Ma- ate alkalinity for young soils and the seafloor weath- her & Chamberlain 2014). On continents, the dimen- ering rate derived from carbonate alkalinity for young sionless pore-space parameter varies with topography pore-space are respectively about an order of magnitude and its mean value for present-day continents is not and up to three orders of magnitude higher than that of well-constrained as previous studies adopt values that KT18 models (Figure 10c,d). The Walker et al.(1981) span an order of magnitude either side of the reference continental weathering rate and the Brady & G´ıslason value used in this study (Maher 2010, 2011; Maher & (1997) seafloor weathering rate exhibit a monotonic rise Chamberlain 2014; Winnick & Maher 2018). Given the (β = 0.3 and β = 0.23, respectively). There are mul- validity range in parameters and weathering proxies, our tiple reasons for the discrepancy between our ts = 0 model allows the present-day global continental weath- models and other models in Figure 10. Our models are ering on Earth to either be in thermodynamic, kinetic designed to provide absolute weathering rates for gen- or supply-limited regimes of weathering. erality, and are therefore not intrinsically calibrated to To match the present-day seafloor weathering rate de- present-day weathering rates unlike other studies. The rived from carbonate alkalinity with that of KT18, the entire exposed planetary surface area may not contain pore-space age is increased to 50 Myr (characteristic fresh rocks for weathering and continents do not nec- seafloor age) and the dimensionless pore-space param- essarily have a uniform lithology and topography. On eter is decreased from 100ψ0 to 18ψ0. Since these two Earth’s continents, orogeny (mountain building) exposes pore-space parameters have never been discussed in the fresh rocks to the surface that are highly susceptible to context of seafloor weathering, future studies must eval- weathering (low ts), whereas the contribution of cra- uate their magnitude and extent of validity. For the tons (ancient continental crust, high ts) to weathering is same parameters, the seafloor weathering rate derived smaller than mountains (Maher & Chamberlain 2014). from divalent cations is smaller by less than a factor Similarly, ridge volcanism exposes fresh basalt at mid- of two. Both these models are in the supply-limited ocean ridges but the majority of seafloor area is older regime of weathering and therefore exhibit almost con- than a million years. stant weathering rates. For a planet with surface conditions where all diva- The present-day regional variation in continen- 1 lent and monovalent cations react with carbonate and tal runoff (0.01 3 m yr− , Gaillardet et al. 1999; − bicarbonate ions to produce carbonates, carbonate al- Fekete et al. 2002) and seafloor fluid flow rate 1 kalinity is a proxy for the flux of CO out of the (0.001 0.7 m yr− , Stein & Stein 1994; Johnson & Pruis 2 − atmosphere-ocean system. However, carbonate alkalin- 2003; Hasterok 2013) is more than two orders of mag- ity overestimates the global weathering rate on modern nitude. At fluid flow rates lower than the mean values Earth, which is limited by the flux of divalent cations (arid climates or low hydrothermal heat flux, see Fig- instead of carbonate alkalinity (France-Lanord & Derry ure7), the models might escape kinetic/supply regimes 1997). This is because monovalent cations do not con- and enter the thermodynamic regime. These models tribute to the carbonate precipitation in the present- rely on characteristic values of ts, ψ and q to be suitable day ocean. Figure 10c shows that the weathering rate proxies for calculating global weathering rates, that are derived from divalent cations in our granite weathering not necessarily appropriate for present-day Earth. For model is smaller than the carbonate alkalinity weather- example, mountains contribute an order of magnitude ing rate by up to a factor of five. Alternatively, a planet more weathering flux than cratons (Maher & Chamber- with no rivers but only coastal springs that are isolated lain 2014). Additionally, basaltic regions on continents from the atmosphere would produce divalent cations but (omitted in our calculations) may contribute a weather- no carbonate alkalinity and effectively no CO2 flux out ing flux (weathering rate per unit area) higher than that of the atmosphere-ocean system. Therefore, care should of granitic regions by a factor of five (Ibarra et al. 2016). be taken when extrapolating a single weathering proxy On Earth, local weathering rates can be estimated from to diverse planetary conditions. data and integrated to compute the global weathering The present-day continental weathering rate derived rate for the planetary surface. However, exoplanets lack from carbonate alkalinity matches that of KT18 when (Earth independent) data constraints on global proper- Lithologic Controls on Silicate Weathering 23 ties such as fluid flow rates and pore-space parameters, evident from Figure8, where the supply-limited weath- let alone their potential regional variations. Hence, we ering flux is seen to be insensitive to changes in T and propose to study exoplanets by implementing the gen- PCO2 . However, if the soil age indirectly depends on T eralized weathering model to various lithologies using and PCO2 on geological timescales, the negative feedback characteristic values of fluid flow rates and pore-space may reinstate. Our models of thermodynamic weather- parameters. ing flux show that the weathering flux decreases with T

Holding modeling parameters constant throughout and increases with PCO2 (Figure9). In the parameter- Earth’s dynamic history is another critical approxima- space where the effect of T dominates, it is possible to tion. The uplift of the Himalayan-Tibetan Plateau in have a positive climate feedback associated with weath- the past 40 million years has decreased the characteris- ering for lithologies tested in this study. Such a scenario tic soil age which has likely increased the global weath- may have implications on the boundaries of the habit- ering rates (Kump et al. 2000). The lithology of conti- able zone. Future studies should combine the climate nents has also evolved over time and therefore the ap- and weathering models to investigate these possibilities. plication of one type of lithology may result in inaccu- rate weathering rates for certain periods of Earth’s history. The continental area fraction (and consequently 5. SUMMARY AND CONCLUSIONS the weatherable area) was smaller in the Archean than Silicate weathering is a key process in the carbon cycle today (Dhuime et al. 2017). Even if the strength of con- that transfers CO from the atmosphere to the surface tinental weathering flux was similar at the beginning of 2 of a planet. The intensity of silicate weathering has the Archean compared to today but the continental area previously been attributed to the kinetics of fluid-rock was significantly smaller, then the continental weather- reactions (e.g., Walker et al. 1981; Berner et al. 1983; ing rate must have been significantly smaller. The true Kump et al. 2000). Maher(2011); Maher & Chamber- contribution of the silicate weathering flux to the car- lain(2014) show that if the reaction rate exceeds the bon cycle also depends on the carbonate compensation fluid flow rate, thermodynamics of fluid-rock reactions depth in the oceans (Pytkowicz 1970; Ridgwell & Zeebe at chemical equilibrium drives weathering instead of ki- 2005) as well as reverse weathering (Mackenzie & Gar- netics. This fluid transport-controlled approach models rels 1966; Isson & Planavsky 2018; Krissansen-Totton & both thermodynamic and kinetic regimes of weather- Catling 2020) that requires knowledge of ocean salinity ing with a single formulation. Moreover, if there is a and ocean pH, which suggests that an ocean chemistry limited supply of fresh rocks, the weathering is supply- model should form the basis of future research. limited. The applications of this approach to continental weathering on Earth (Winnick & Maher 2018) and 4.4. Weathering Regimes and the Habitable Zone temperate exoplanets (Graham & Pierrehumbert 2020) It has been proposed that measuring the gaseous consider weathering reactions of individual minerals. abundance of CO2 in the atmospheres of Earth-sized In this study, we extend this approach to the weather- exoplanets will allow one to statistically distinguish being of any rock type (lithology) and apply it to seafloor tween Venus-like and Earth-like climates (Bean et al. weathering in addition to continental weathering. We 2017; Checlair et al. 2019; Graham & Pierrehumbert find that the simultaneous consideration of weathering 2020). A negative feedback associated with weathering reactions of the major minerals present in a rock as is necessary to control the climatic impact of CO2 over well as the reactions in the water-bicarbonate system geological timescales on an Earth-like planet (Walker instead of weathering reactions of individual minerals et al. 1981). One-dimensional, radiative-convective at- impacts weathering rates as well as weathering regimes. mospheric models used to study the boundaries of the Moreover, this model allows the calculation of absolute classical habitable zone usually assume the presence of weathering rates instead of weathering rates normalized the negative weathering feedback to justify the assump- to present-day values as most previous weathering stud- tion of CO2 being present as a greenhouse gas at the ies. In addition to climate properties (T and PCO2 ) and outer boundary of the habitable zone and having only a runoff or fluid flow rate (q), this model is mainly sen- minor abundance at its inner edge (Kasting et al. 1993; sitive to age of soils (ts) and a dimensionless scaling Kopparapu et al. 2013). The effects of CO2 on the run- parameter (ψ) based on pore-space and rock properties. away greenhouse effect associated with water is debated The equilibrium constants and kinetic rate coefficients

(Nakajima et al. 1992; Abe 1993). Moreover, if weath- are eﬀectively a function of T and PCO2 . Depending on ering is limited by the supply of fresh rocks, the nega- these ﬁve parameters, the weathering for a given lithol- tive feedback is lost (West 2012; Foley 2015). This is ogy is in the thermodynamic, kinetic or supply regimes. 24 Hakim et al.

Close to the regime transition points, the contribution with T . This is in contrast with the prevalent under- of both regimes to weathering is similar. standing that weathering intensifies with an increase in Weathering reactions at chemical equilibrium give the T which is attributed to an increase in kinetic rate coef- maximum concentrations of weathering products. We ficients of mineral dissolution reactions with T (Lagache use this approach to calculate the maximum weather- 1965; Palandri & Kharaka 2004; Brantley et al. 2008). ing flux for a given lithology. The larger the fraction An important implication of this finding is that when of divalent cations in rocks, the higher the sensitivity T increases without a strong variation in PCO2 , silicate of maximum weathering flux to CO2 partial pressure. weathering has the potential to instigate a positive feed-

This thermodynamic PCO2 sensitivity (power-law expo- back to the carbon cycle. The focus of future studies nent βth) is 0.71, 0.69 and 0.65 for peridotite, basalt and should be on applying a generalized weathering model granite, respectively. These values are subject to change encompassing multiple weathering regimes to model the depending on the choice of minerals to define a rock type carbon cycle. as well as secondary minerals produced during weather- ACKNOWLEDGMENTS ing. For example, the consideration of kaolinite as a sec- We thank Edwin Kite and an anonymous reviewer ondary mineral instead of halloysite changes βth to 0.65 and 0.54 for basalt and granite, respectively. The ther- for their constructive comments that helped to improve this manuscript. We thank Eric Gaidos for stimu- modynamic PCO sensitivity of these rocks is stronger 2 lating discussions. We acknowledge financial support than the kinetic PCO2 sensitivity implemented in previous studies (0.22 0.55, Walker et al. 1981; Berner 1991; from the European Research Council via Consolidator − Driscoll & Bercovici 2013). However, the combined ef- Grant ERC-2017-CoG-771620-EXOKLEIN (awarded to KeHe) and Center for Space and Habitability, Univer- fect of PCO and T results in a weaker thermodynamic 2 sity of Bern. DJB acknowledges the Swiss National PCO2 sensitivity. The fluid transport-controlled model demonstrates Science Foundation (SNSF) Ambizione Grant 173992. that the weathering flux cannot necessarily be approx- CD acknowledges the SNSF Ambizione Grant 174028. imated by the kinetic weathering expression (Equa- We thank the financial support of the National Centre tion2). In our model, planets with arid climates (low of Competence in Research PlanetS supported by the runoff) and elevated topography (young soils) are likely SNSF. in the thermodynamic regime of weathering, exhibiting Software: astropy (Astropy Collaboration et al. weathering rates higher than that of modern Earth by 2013, 2018), CHNOSZ (Dick 2019), ipython (Perez & three to four orders of magnitude. Moreover, limited Granger 2007), matplotlib (Hunter 2007), numpy (van supply of fresh rocks mitigates the role of kinetics. The der Walt et al. 2011), scipy (Virtanen et al. 2020) thermodynamic T sensitivity of weathering of rocks is negative, implying that the weathering flux decreases

APPENDIX

A. THERMODYNAMICS AND KINETICS DATA ergy of formation at reference pressure P0 and reference The equilibrium constant K of a reaction is given by temperature T0 by the diﬀerence of the Gibbs energy of formation of products and reactants as follows ∆f GP,T = ∆f GP ,T SP ,T (T T ) 0 0 − 0 0 − 0 T T P (A2) + CP dT T CP d ln T + V dP, − products reactants ZT0 ZT0 ZP0 RT ln K = νi∆f GP,T,i νj∆f GP,T,j, − − i j where SP ,T is the entropy at the reference pressure X X 0 0 (A1) and temperature, CP is the heat capacity at constant th where ∆f GP,T,i is the Gibbs energy of formation of i pressure as a function of temperature, and V is volume species at pressure P and temperature T , νi is the sto- as a function of pressure. In this study, ∆f G values are ichiometric coeﬃcient and R is the universal gas con- obtained from the CHNOSZ database (Dick 2019). The stant. The Gibbs energy of formation of each species equilibrium constants of reactions given in Table B.1 is computed at any P and T in terms of the Gibbs en- are shown as a function of P and T in Figure A.1. Lithologic Controls on Silicate Weathering 25

(a) (b)

1 1 10 P = 1 bar 10 T = 278 K 1 1 10− 10−

3 3 10− 10−

5 5 10− 10−

K 7 K 7 10− 10−

9 9 10− 10−

11 11 10− 10−

13 13 10− 10−

15 15 10− 10− 2 1 0 1 2 3 280 300 320 340 360 10− 10− 10 10 10 10 T [K] P [bar]

Reaction (f) (Anorthite weathering) Reaction (j) (Phlogopite weathering) Reaction (d) (Forsterite weathering) Reaction (c) (Ferrosilite weathering)

Reaction (a) (Wollastonite weathering) Reaction (p) (CO2 dissociation)

Reaction (o) (CO2 dissolution) Reaction (k) (Annite weathering) Reaction (b) (Enstatite weathering) Reaction (m) (Grunerite weathering) Reaction (i) (Muscovite weathering) Reaction (g) (Albite weathering) Reaction (n) (Quartz dissolution) Reaction (q) (Bicarbonate dissociation) Reaction (l) (Anthophyllite weathering) Reaction (h) (K-feldspar weathering) Reaction (e) (Fayalite weathering) Reaction (r) (Water dissociation)

Figure A.1. Equilibrium constants for the reactions listed in Table B.1 obtained from CHNOSZ (Dick 2019). (a) K as a function of T . (b) K as a function of P . Please note that some reactions have fractional stoichiometry in order to ensure high numerical precision.

The kinetic rate coefficient keff of mineral dissolution dissolution reactions. For minerals not present in this reactions are obtained from the compilation of Palan- compilation (ferrosilite, annite, grunerite), the kinetic dri & Kharaka(2004). This compilation (Table A.1) is rate coefficients are obtained from respective endmem- based on the fitting of the following equation to experi- ber minerals of the same mineral group (enstatite, phl- mental kinetics data, ogopite, anthophyllite).

B. Eacid pH n MAXIMUM AND GENERALIZED k = A exp − 10− acid eﬀ acid RT CONCENTRATIONS OF ROCKS AND E MINERALS +A exp − neut (A3) neut RT Sects. 2.3 and 2.4 introduce the methods to compute maximum (thermodynamic) and generalized solute con- Ebase pH n +A exp − 10− base base RT centrations for peridotite weathering. Table B.1 lists the mineral dissolution (reactions (a n)) and water- − where E , E and E are the activation ener- bicarbonate reactions (reactions (o r)), and the rela- acid neut base − gies at acidic, neutral and basic pH, Aacid, Aneut and tion between equilibrium constants of these reactions Abase are the preexponential factors at acidic, neutral and thermodynamic activities. Table B.2 gives the poly- and basic pH, and nacid and nbase are the neutral and nomial equations to calculate the activity of HCO3− basic power-law exponents. Figure A.2 shows the vari- at chemical equilibrium as a function of PCO2 for the ation of keﬀ with T and pH for a number of mineral weathering of all rocks and minerals considered. As an 26 Hakim et al.

(a) (b)

104 pH = 7 104 T = 288 K

] 2 ] 2

1 10 1 10 − −

yr 0 yr 0

2 10 2 10 − −

2 2 10− 10−

[mol m 4 [mol m 4 10− 10− eﬀ eﬀ k k 6 6 10− 10−

8 8 10− 10− 280 300 320 340 360 0 2 4 6 8 10 12 14 T [K] pH (a) (b)

104 pH = 10 104 T = 348 K

] 2 ] 2

1 10 1 10 − −

yr 0 yr 0

2 10 2 10 − −

2 2 10− 10−

[mol m 4 [mol m 4 10− 10− eﬀ eﬀ k k 6 6 10− 10−

8 8 10− 10− 280 300 320 340 360 0 2 4 6 8 10 12 14 T [K] pH

Wollastonite (CaSiO3) Enstatite (MgSiO3)

Anorthite (CaAl2Si2O8) Phlogopite (KMg3AlSi3O10(OH)2)

Forsterite (Mg2SiO4) K-Feldspar (KAlSi3O8)

Fayalite (Fe2SiO4) Muscovite (KAl3Si3O10(OH)2)

Albite (NaAlSi3O8) Anthophyllite (Mg7Si8O22(OH)2)

Quartz (SiO2)

Figure A.2. Kinetic rate coefficients for the dissolution of minerals obtained from Palandri & Kharaka(2004). (a) keff as a function of T at pH = 7. (b) keff as a function of pH at T = 288 K. (c) keff as a function of T at pH = 10. (d) keff as a function of pH at T = 348 K. example, the maximum [HCO3−] for peridotite weathering is obtained as a function of CO2 partial pressure, Lithologic Controls on Silicate Weathering 27

Table A.1. Kinetics data from Palandri & Kharaka(2004).

Mineral Eacid nacid Eneut Ebase nbase Thermodynamics Data kJ mol−1 kJ mol−1 kJ mol−1

Wollastonite 54.7 0.400 54.7 − − Enstatite 80.0 0.600 80.0 − − Forsterite 67.2 0.470 79.0 Network of Mineral − − Fayalite 94.4 94.4 Climate Dissolution and Water- − − − Anorthite 16.6 1.411 17.8 Properties − − Bicarbonate Reactions at Albite 65.0 0.457 69.8 71.0 0.572 − Equilibrium K-feldspar 51.7 0.500 38.0 94.1 0.823 − Muscovite 22.0 0.370 22.0 22.0 0.220 − Phlogopite 29.0 − − − − Anthophyllite 51.0 0.440 51.0 Maximum Solute − − Kinetics Data Quartz 90.1 108.4 0.5 Concentrations − − − Note— For minerals that are not listed, data from a corresponding endmember mineral from the same mineral group is adopted.

Rock and Transport-Controlled Pore-space Dilution of Maximum Properties Solute Concentrations surface temperature and total pressure in Figure B.3. As

described in Section 2.3, [HCO3−]eq is strongly sensitive to P and T . However, P has a negligible effect on Fluid Flow CO2 Transport-Controlled Rate [HCO3−]eq because the equilibrium constants of reactions Solute Concentrations are largely unchanged up to 1000 bar (AppendixA). (Runoff) This figure demonstrates that the total pressure plays a negligible role in determining the solute concentrations of aqueous species. The effect of precipitation of Network of Water- amorphous silica (Winnick & Maher 2018) is not mod- Bicarbonate Reactions at eled since this effect changes the weathering flux by less Equilibrium

than an order of magnitude (only at high PCO2 ) which is smaller than the 5 10 orders of magnitude spread in − the weathering ﬂuxes discussed in this study. Generalized Solute

Once the maximum [HCO3−] is determined for a given Concentrations rock or mineral, the solute transport equation of Maher & Chamberlain(2014) is implemented to dilute the equi-

librium value of [HCO3−] as a function of runoff or fluid flow rate q (Figure B.1). This equation allows to cal- Generalized Weathering culate the non-equilibrium concentrations using equilibrium concentrations. Higher the fluid flow rate, more di-

luted is the resulting [HCO3−] (Equation8). This diluted [HCO3−] is then used as an input to solve for concentra- 2 + tions of other aqueous species such as CO3−,H and Weathering Flux OH− by assuming that the water-bicarbonate reactions obey chemical equilibrium. Figure B.2 demonstrates that generalized solute concentrations (Section 2.4) are Figure B.1. Schematic describing the methodology of strongly sensitive to lithology. For example, the tran- the weathering model CHILI. Square boxes represent soft- sition between thermodynamic and kinetic weathering ware modules, ovals denote parameters (Tables1 and3) and rounded squares represent computed quantities (Table2). regimes of peridotite occurs at PCO2 = 1 µbar for 1 The solute transport equation of Maher & Chamberlain q = 0.3 m yr− . Whereas, this transition occurs at P = 1 mbar for granite. Once these generalized con- (2014) is implemented to calculate diluted solute concentra- CO2 tions. Thermodynamics and kinetics data are obtained from centrations are obtained, the generalized weathering ﬂux Dick(2019) and Palandri & Kharaka(2004), respectively is calculated using Equation (3). (see AppendixA). 28 Hakim et al.

(a) Peridotite (b) Basalt (c) Granite 100 12 100 12 100 12 ] ] ] 3 3 3 3 3 3 − 10− 10 − 10− 10 − 10− 10 6 8 6 8 6 8 10− pH 10− pH 10− pH [mol dm [mol dm [mol dm 10 9 6 10 9 6 10 9 6 C − C − C −

12 12 12 10− 4 10− 4 10− 4 8 6 4 2 0 8 6 4 2 0 8 6 4 2 0 10− 10− 10− 10− 10 10− 10− 10− 10− 10 10− 10− 10− 10− 10