Carbonate saturation (zsat) and compensation (zcc) depths change with deep-ocean acidification and basification. We present simple, explicit, mechanistic formulas for the positions of these two critical depths. In particular zcc is expressed as a function of the mean dissolved carbonate ion concentration of the deep ocean, [CO3]D, the supply of dissolvable CaCO3, Fc, and the dissolution rate constant at the sediment-water interface, kc, which we show to be essentially mass-transfer controlled. Calculations reveal that zsat and zcc are today some ∼0.9 km apart and will rise and separate by as much as 1.7 km with acidification; conversely, if [CO3]D increases, zsat and zcc will deepen, but their separation will asymptote to ∼0.7 km.
 Carbonate compensation refers to self-induced adjustments in the deep-sea carbonic acid system via interaction with sedimentary carbonate, also termed the oceanic homeostat [Archer, 1996; Sarmiento and Gruber, 2006]. Three dynamic horizons in the ocean are crucial to compensation [Zeebe and Westbroek, 2003], i.e., the saturation horizon, the compensation depth, and the snow line, all defined later. The position and time evolution of these horizons is needed to predict the fate of CO2, both natural and anthropogenic, in the oceans and to interpret the geological record in oceanic sediments [Broecker, 2009]. Forecast/hindcast of these positions has suffered from unclear or multiple meanings of these horizons; this paper provides clear definitions and simple formulas for these horizons for use in predictive models and for paleoceanographic studies.
2. Definitions and Formulas
 The calcite saturation depth, zsat, occurs where the oceans first become undersaturated with respect to this mineral. (Aragonite is ignored, henceforth, as a minor complication.) zcc is depth-dependent because the solubility of calcite increases with pressure and decreases with temperature. A simplified depth/pressure-dependent fit (Figure 1) of the solubility data for calcite from Mucci , with pressure corrections from Millero , produces the following formula for the carbonate ion concentration at saturation, Csat,
where p(z) is the gauge pressure (atm) at depth z (km), [Ca] is the dissolved calcium (Ca2+) concentration, taken as 0.010282 M (moles dm−3), Ksp0 = 4.35 × 10−7 M2 is the average solubility product of calcite at the ocean surface, and pc = 511 atm, which is a characteristics pressure. If the deep-ocean carbonate ion concentration, [CO3]D, is assumed constant below the pycnocline, as currently occurs in the oceans, then equation (1) can be inverted explicitly for zsat, i.e.,
where ρ is the mean seawater density and g is the accelaration due to gravity (ρg 100 atm/km). In past models, zsat has been calculated implicitly [e.g., Sundquist, 1990; Merico et al., 2008; Tyrell, 2008]. Calculation of zsat is also possible if [CO3]D is a function of depth, but that dependence must be known and a more complex equation will result; furthermore, while our emphasis here is on a mean ocean, one could also generate a form of equation (2) for a specific basin by employing the local temperature profile and an appropriate [CO3]D depth-function.
 Previously zcc has been computed laboriously by modelling the calcite content of sediments with ocean depth using diagenetic models [e.g., Schink and Guinasso, 1977; Takahashi and Broecker, 1977; Sundquist, 1990; Archer and Maier-Reimer, 1994; Ridgwell and Hargreaves, 2007]. These models predict that the zone of dissolution in the sediment is on-the-order-of a few millimeters at most. This extreme attenuation indicates that dissolution could be well approximated as a reaction at the sediment-water interface alone. Consequently, to write a formula for zcc, one balances input and dissolution at this interface. As a first step, if Fc is the net amount of CaCO3 that falls into the deep sea and reaches the bottom, i.e., the CaCO3 input below ∼2000 m [Berelson et al., 2007], and AD is the area of the deep ocean below this same depth, then the average input per unit area of seafloor (the rain) is simply the quotient Fc/AD.
 Next, we need to specify the calcite dissolution rate, which has been modeled as a high-order function of the undersaturation [e.g., Keir, 1980]. Hales and Emerson  have argued instead that the order is linear, if the correct solubility is employed. We also employ linear kinetics, but for a different reason.
 Most experimental studies of CaCO3 dissolution, including Keir , deal with dissolution of suspended calcite, which is not directly applicable to the seafloor because of vastly different mass-transfer characteristics. Keir  dissolved beds of natural carbonate-bearing sediments in a reaction chamber with a rotating mixer, and he interpreted his data as consistent with high order of reaction. However, plots of the product Csat (1 − Ω) b, where Ω = [CO3]D/Csat and b = fraction of CaCO3 in sediment, versus his rates of CO3= release are decidedly linear (see Figure S1) except for two, which are simply scattered. This observed linearity applies to the seafloor.
 The balance of carbonate rain and dissolution at zcc is then given as
where kc is the heterogeneous rate constant for dissolution at the sediment-water interface, with units of length per unit time, and Csat(zcc) is the CO3= saturation concentration at zcc. Equation (1) can be substituted into equation (3) to obtain an explicit equation for zcc:
This expression shows that zcc is always larger than zsat because the first term within the logarithm is always positive. If [CO3]D is a known function of depth, an expression for zcc can still be obtained, but of a different form. Equations (2) and (4) hold for both steady state and transient conditions.
 One consequence of equations (3) and (4) is that there is no “tailing” of the predicted CaCO3 content of sediments with ocean depth [e.g., Broecker, 2009, Figure 6]; the tail results from either a small recalcitrant calcite component or the preserving effect of bioturbation [Schink and Guinasso, 1977]. However, the tail contains far less than 1% of the CaCO3 flux and, as such, can be ignored when considering global, rather than local, carbonate dynamics.
 The snow line [Zeebe and Westbroek, 2003], zsnow, a third critical depth occurs where no solid calcite can be found in the sediment due to dissolution. zcc and zsnow are identical at steady state. The position of a transient snow-line depends on the amount of calcite in the sediments and the history of [CO3]D. If deep-sea sediments are essentially well mixed with respect to CaCO3 [e.g., Schink and Guinasso, 1977; Takahashi and Broecker, 1977], then zsnow is obtained from calcite mass balance for the sediment, which is, at constant calcite rain,
where b(z, τ) is the concentration of dissolvable CaCO3 in the sediment at water depth z after elapsed time τ (which is a function of water depth), L(z) is the depth in the sediment over which PDC can ultimately be dissolved [Takahashi and Broecker, 1977]. The elapsed time τ is the time since zcc was at the chosen depth z. The product b(z, τ)L(z) is the standing stock of dissolvable calcite at ocean depth z. L(z) is a function of the amount of CaCO3 and clay in sediments [see Archer et al., 1998]. Equation (5) is contingent on knowing the chemical history, [CO3]D(t), to resolve the integral. An equation similar to equation (5) can be written to calculate the geological compensation depth, zccsed, again given an appropriate CO3= history.
 Finally, another common term, the lysocline, was originally the subjective depth where dissolution of CaCO3 became apparent in the sediment [Kennett, 1982]. Archer  attempted to quantify this definition by stating that the lysocline occurs where the CaCO3 content drops below 80%. Morse and Berner  argued from dissolution experiments that the lysocline instead reflected a change in the mechanism of dissolution at some critical degree of undersaturation, but Figure S1 suggests no such change in mechanism. Broecker and Peng  and Burdige  believe that the lysocline is equal to zsat, in which case equation (2) would be its predictor. However, it is possible, even likely, that the lysocline reflects (in part) the depth at which metabolically-driven dissolution of CaCO3 within sediments [Emerson and Bender, 1981; Jahnke et al., 1994] produces visually obvious effects on tests. Metabolically-driven dissolution occurs when CO2 from oxic organic matter decay causes the local state of saturation of porewaters within sediments to drop below the solubility of calcite. This process is largely independent of the saturation state of the overlying waters, and it, in no way, alters the formulas and dynamic depths discussed above.
3. Computation of Saturation and Compensation Depths
 We now examine zcc and zsat as functions of the controlling parameters, i.e., [CO3]D, Fc and kc. The current/pre-industrial [CO3]D can be taken as constant below ∼1500 m [Key et al., 2004] at a value of 0.09 mM [Broecker, 2003; B. P. Boudreau et al., Ongoing transients in carbonate compensation, submitted to Global Biogeochemical Cycles, 2009]. The total rain of calcite, Fc, is between 20–60 × 103 Gmol y−1 [Archer, 1996; Sarmiento and Gruber, 2006; Berelson et al., 2007], and we will examine solutions within that range.
 The unknown parameter in equation (4) is kc, and Broecker  has recently bemoaned this uncertainty. A kc from on Keir  is inappropriate because (1) the dependence of kc on circulation velocity is unknown for that experiment, (2) a cylindrical reactor with an impeller system does not simulate the seafloor because of secondary flows [Broström and Nilsson, 1999], and (3) Keir's experiment simulates transient dissolution of a bed of pre-existing CaCO3. In a separate contribution (i.e., Boudreau et al., submitted manuscript, 2009), we have coupled our equations to a box model of the oceans and extracted a value for kc of 700 cm y−1 for steady state (pre-industrial) conditions at the seafloor.
 To provide context for this kc value, dissolution cannot occur faster than the limit placed by transport across the benthic boundary layer. A theoretical value of the mass-transfer coefficient is given by kT = 0.0671 u* Sc−⅔ [Boudreau, 2001], where u* is the mean shear velocity at the ocean bottom and Sc is the Schmidt number, about 3900 for carbonate/calcium ions. Wimbush  states that u* has a mean value of about 0.1 cm s−1, which gives kT = 965 cm y−1. Consequently, kc from fitting the oceanic carbonate system and kT differ only by 38%, and this concurrence suggests that CaCO3 dissolution at the seafloor is largely controlled by mass transfer, rather than intrinsic kinetics. Schink and Guinasso  and Takahashi and Broecker  have previously advanced a role for mass transfer in controlling benthic CaCO3 dissolution, but we go further and argue that mass transfer at the interface is the dominant factor.
Figures 2a–2c plot zsat and zcc as functions of [CO3]D, Fc and kc. Clearly, zsat and zcc are not at the same depth. With a present-day/pre-industrial [CO3]D of 0.09 mM, Figure 2a indicates a separation of ∼0.9 km. This distance increases rapidly as [CO3]D decreases, potentially reaching ∼1.7 km if [CO3]D is approximately 0.045 mM, which is anticipated within the next 2000 years [Ridgwell and Hargreaves, 2007]. If, however, [CO3]D increases, as in the geological past, the distance between zsat and zcc will asymptote to ∼0.68 km, or a ratio of about 0.9. Thus, the assumption of a constant separation when the zcc is deepening from its present position is correct.
Figure 2b illustrates the dependence of zcc on kc. The position of zcc is crucially dependent on the value of this parameter, i.e., small drops (bottom-water stagnation) will result in large increases in zcc. However, if kc increases towards the pure mass-transfer value, zcc will shift to the saturation horizon. Over the range of Fc values currently thought to apply in the oceans, Figure 2c shows that zcc is a linear function of the rain rate, with a modest slope; thus, zcc changes only in a limited way if the calcite rain remains with that range; that said, it should be realized that these calculations are made with [CO3]D fixed, while in reality zcc can shallow, rather than deepen with increasing Fc because the compensation mechanism lowers [CO3]D.
 The results in Figure 2 have implications not only to the current acidification of the oceans, but also to the interpretation of the geological record in terms of paleoceanographic conditions. The depths zsat and zcc are often equated in modeling and in reconstructions of past excursions of the carbonate compensation horizon [e.g. Merico et al., 2008; Roberts and Tripati, 2009], while our formulas predict that they are at least ∼0.7 km apart. At steady state, obtained on a 105 y time scale [Archer et al., 1998], zcc and zccsed should be equal, and this then implies a separation between zsat and zccsed of between ∼0.7 to ∼2 km (Figure 2), much greater than the ∼250 m uncertainty in global reconstructions of the zccsed record [Roberts and Tripati, 2009]. Moreover, the difference between zsat and zcc is highest during periods of rapidly increasing atmospheric carbon dioxide, such as the Anthropocene, the Paleocene-Eocene Thermal Maximum [Zeebe et al., 2009] and glacial to interglacial transitions, consistent with Munhoven . During periods of ocean de-acidification, such as the Eocene-Oligocene transition [Merico et al., 2008], zsat and zcc deepen and their separation asymptotes.
 Two horizons important to the mathematical/theoretical description of carbonate dynamics in the oceans, i.e., the saturation horizon, zsat, and the geochemical carbonate compensation depth, zcc, can be calculated via equations (2) and (4), respectively. At steady state, zcc can be considered to be first-order predictor for the operational/data-defined geological compensation depth, zccsed.
 The distance between zsat and zcc is not fixed, but a function of the deep-water carbonate ion concentration, [CO3]D, the input of calcite, Fc, and the dissolution rate constant, kc. Acidification causes these horizons to rise and to separate, whereas, a higher carbonate ion concentration causes deepening and the difference between these depths to asymptote to about 0.68 km (or a ratio of 0.9). In transient situations, zccsed will lag the movement of zcc, but approach this latter depth as a steady state is established.
 Our analysis further indicates that CaCO3 dissolution at the seafloor is (largely) mass-transfer controlled, based on the dissolution rate constant derived by Boudreau et al. (submitted manuscript, 2009).
 This research was funded by the Natural Sciences and Engineering Research Council of Canada, the Netherlands Organization for Scientific Research, and the EU (European Project on Ocean Acidification). We thank Bob Berner, Ken Caldeira, Robin Keir, Fred Mackenzie, Robert Moore, Greg Raviza, Helmut Thomas, and Richard Zeebe for critical comments on earlier versions of our manuscript. We thank the two anonymous reviewers and Wallace Broecker for their insightful comments. The lead author conducted part of this research while on sabbatical at SOEST at the University of Hawai'i at Mānoa, a sojourn that would not have been possible without the aid of Kathleen Ruttenburg.