ate evolution. The arc degassing flux is typically written as $$\eqlbl{arc}
F_{arc} = \frac{f v_p L R_p}{A_p} ,$$ where $f$ is the fraction of subducted carbon that degasses, $v_p$ is the plate speed, $L$ is the length of subduction zone trenches, $R_p$ is the amount of carbon on the seafloor, and $A_p$ is the area of the seafloor. The ridge degassing flux is given by $$F_{ridge} = \frac{f_d v_p L d_{melt} R_{man}}{V_{man}} ,$$ where $f_d$ is the fraction of upwelling mantle that degasses, $L$ is the length of ridges (assumed equal to the length of trenches), $d_{melt}$ is the depth where mid-ocean ridge melting begins, $R_{man}$ is the amount of carbon in the mantle, and $V_{man}$ is the volume of the mantle [e.g. @TajikaMatsui1990; @TajikaMatsui1992; @Sleep2001b; @Driscoll2013; @Foley2015_cc].

Silicate weathering can exert a negative feedback on climate because mineral dissolution rates, and hence CO$_2$ drawdown rates, increase with increasing temperature and precipitation [e.g. @walker1981; @berner1983] (see [@Kump2000] and [@Brantley2014] for reviews on mineral dissolution kinetics and atmospheric CO$_2$). A similar link between climate and weathering rate can be seen in many field studies [e.g. @Velbel1993; @White_Blum1995; @Dessert2003; @Gislason2009; @White2014; @Viers2014]. However, many other locations show no link between weathering and climate; instead the weathering flux is linearly related to the physical erosion rate [@Stallard1983; @Edmond1995; @Gaillardet1999; @Oliva1999; @Millot2002; @Dupre2003; @Riebe2004; @West2005; @Viers2014]. These conflicting field observations are likely due to the distinction between “kinetically limited" (or “reaction limited") weathering and “supply limited" weathering. When weathering is kinetically limited, the kinetics of mineral dissolution controls the weathering rate. However, when weathering is supply limited, the supply of fresh rock brought to the weathering zone by erosion controls the weathering rate.

Supply limited weathering occurs when the reactio