Increased Sea Level Sensitivity to CO2 Forcing across the Middle Pleistocene Transition from Ice-Albedo and Ice-Volume Nonlinearities

Parker Liautaud aDepartment of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts

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Peter Huybers aDepartment of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts

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Abstract

Proxy reconstructions indicate that sea level responded more sensitively to CO2 radiative forcing in the late Pleistocene than in the early Pleistocene, a transition that was proposed to arise from changes in ice-sheet dynamics. In this study we analyze the links between sea level, orbital variations, and CO2 using an energy-balance model having a simple ice sheet. Model parameters, including for age models, are inferred over the late Pleistocene using a Bayesian method, and the inferred relationships are used to evaluate CO2 levels over the past 2 million years in relation to sea level. Early Pleistocene model CO2 averages 244 ppm (241–246 ppm 95% confidence interval) across 2 to 1 million years ago and indicates that sea level was less sensitive to radiative forcing than in the late Pleistocene, consistent with foregoing δ11B-derived estimates. Weaker early Pleistocene sea level sensitivity originates from a weaker ice-albedo feedback and the fact that smaller ice sheets are thinner, absent changes over time in model equations or parameters. An alternative scenario involving thin and expansive early Pleistocene ice sheets, in accord with some lines of geologic evidence, implies 15-ppm-lower average CO2 or ~10–15-m-higher average sea level during the early Pleistocene relative to the original scenario. Our results do not rule out dynamical transitions during the middle Pleistocene, but indicate that variations in the sea level response to CO2 forcing over the past 2 million years can be explained on the basis of nonlinearities associated with ice-albedo feedbacks and ice-sheet geometry that are consistently present across this interval.

Significance Statement

Insight into the consequences of atmospheric CO2 variations for sea level is possible from studying past changes in these two variables. Previous studies showed, in particular, that sea level was more sensitive to CO2 forcing in the late Pleistocene [~0–1 million years ago (Ma)] than in the early Pleistocene (~2–1 Ma). Using a Bayesian methodology, we condition a physical model on late Pleistocene sea level and CO2 observations and show that the inferred relationships also predict the lower early Pleistocene sensitivity. Temporal changes in sensitivity come from nonlinearities involving the ice-albedo feedback and ice-sheet geometry that intensified when ice sheets grew larger after 1 Ma. An alternative scenario involving changes in conditions at the base of the ice sheet also satisfies observations but such changes are not required for explaining sensitivity changes.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Parker Liautaud, parker_liautaud@g.harvard.edu

Abstract

Proxy reconstructions indicate that sea level responded more sensitively to CO2 radiative forcing in the late Pleistocene than in the early Pleistocene, a transition that was proposed to arise from changes in ice-sheet dynamics. In this study we analyze the links between sea level, orbital variations, and CO2 using an energy-balance model having a simple ice sheet. Model parameters, including for age models, are inferred over the late Pleistocene using a Bayesian method, and the inferred relationships are used to evaluate CO2 levels over the past 2 million years in relation to sea level. Early Pleistocene model CO2 averages 244 ppm (241–246 ppm 95% confidence interval) across 2 to 1 million years ago and indicates that sea level was less sensitive to radiative forcing than in the late Pleistocene, consistent with foregoing δ11B-derived estimates. Weaker early Pleistocene sea level sensitivity originates from a weaker ice-albedo feedback and the fact that smaller ice sheets are thinner, absent changes over time in model equations or parameters. An alternative scenario involving thin and expansive early Pleistocene ice sheets, in accord with some lines of geologic evidence, implies 15-ppm-lower average CO2 or ~10–15-m-higher average sea level during the early Pleistocene relative to the original scenario. Our results do not rule out dynamical transitions during the middle Pleistocene, but indicate that variations in the sea level response to CO2 forcing over the past 2 million years can be explained on the basis of nonlinearities associated with ice-albedo feedbacks and ice-sheet geometry that are consistently present across this interval.

Significance Statement

Insight into the consequences of atmospheric CO2 variations for sea level is possible from studying past changes in these two variables. Previous studies showed, in particular, that sea level was more sensitive to CO2 forcing in the late Pleistocene [~0–1 million years ago (Ma)] than in the early Pleistocene (~2–1 Ma). Using a Bayesian methodology, we condition a physical model on late Pleistocene sea level and CO2 observations and show that the inferred relationships also predict the lower early Pleistocene sensitivity. Temporal changes in sensitivity come from nonlinearities involving the ice-albedo feedback and ice-sheet geometry that intensified when ice sheets grew larger after 1 Ma. An alternative scenario involving changes in conditions at the base of the ice sheet also satisfies observations but such changes are not required for explaining sensitivity changes.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Parker Liautaud, parker_liautaud@g.harvard.edu

1. Introduction

Over the past million years, kilometers-thick ice sheets periodically grew to cover large portions of North America and Eurasia, causing intermittent reductions in global sea level as large as 140 m (Siddall et al. 2010; Bintanja and van de Wal 2008; Willeit et al. 2019) over intervals of 80 000–120 000 years. These large glacial build-ups contrast with the thinner (Clark and Pollard 1998) or less expansive (Willeit et al. 2019) ice sheets of the early Pleistocene that effected more modest sea level reductions of ~50–70 m (Bintanja and van de Wal 2008; Siddall et al. 2010; Rohling et al. 2014) on shorter periods of ~40 000 years. Proxy reconstructions of atmospheric CO2 derived from foraminiferal boron isotope ratios offer potentially important insight into the nature of this change in glacial variability. Specifically, regressions of δ11B-derived CO2 radiative forcing estimates against sea level reconstructed using δ18O records imply that sea level is twice as sensitive to CO2 forcing in the late Pleistocene than in the early and middle Pleistocene (Chalk et al. 2017; Dyez et al. 2018).

One line of reasoning for increases in sea level sensitivity involves changes in the processes controlling ice-sheet responses to forcing. A prominent hypothesis is that a layer of deformable till allowed a thin early Pleistocene Laurentide ice sheet to spread expansively, and that scouring of the sediment layer over time exposed the crystalline bedrock, increased basal shear stresses, and permitted the growth of a thicker ice sheet (Clark and Pollard 1998). A consequence of such thickening would be greater ice-volume variations for a given change in ice-sheet extent and, therefore, a plausible explanation for the heightened late Pleistocene sea level sensitivity. Chalk et al. (2017) suggested that phase-locking of glacial cycles to orbital variations (e.g., Tziperman et al. 2006) might also contribute to changes in sensitivity, and additional factors such as the presence of sea ice (Gildor and Tziperman 2000) could play a role.

A second line of reasoning is for the change in sensitivity to reflect stronger feedbacks inherent to the growth of larger ice sheets. A prime candidate is for the ice-albedo feedback to strengthen as ice sheets expand (Milankovitch 1941; Budyko 1969) both because Northern Hemisphere ice sheets that extend to low latitudes can have greater zonal extent and because surface radiation increases from high to middle latitudes. The albedo effect was proposed earlier as an explanation for increased climate sensitivity after the first appearance of Northern Hemisphere ice in the Pliocene (Martínez-Botí et al. 2015). A greater sea level response to radiative perturbations also follows inherently from greater thickness and zonal extent of large ice sheets (Weertman 1976; Källén et al. 1979), implying greater sea level consequences for a change in length. Increased frictional coupling of ice sheets at their base would, of course, heighten this volume effect (Clark and Pollard 1998).

In the following we explore the origins of changes in sea level sensitivity using a Bayesian methodology that combines observations of CO2 and sea level with a physical model relating the two. Bayesian methods are well suited for inferring past climate conditions from uncertain proxy data sources (Tingley and Huybers 2010) and have also been successfully applied to selecting between competing models for the origin of glacial cycles (Carson et al. 2017; Feng and Bailer-Jones 2015; Crucifix et al. 2016), constructing probabilistic age models (Werner and Tingley 2015; Lin et al. 2014), and combinations thereof (Parnell et al. 2015; Carson et al. 2019). Kalman filters have, alternatively, been used to infer paleoclimate phenomena (e.g., Annan et al. 2005a,b; Okazaki and Yoshimura 2017), but the fact that the processes of interest feature nonlinearities, nonnormal distributions, and age model uncertainty makes a more general probabilistic approach preferable.

Many heuristic models of Pleistocene glaciation readily permit for Bayesian sampling (Imbrie and Imbrie 1980; Huybers 2007; Paillard 1998; Parrenin and Paillard 2003; Imbrie et al. 2011) but their relationship with observational constraints is obscured by highly simplified representations of relevant processes. More comprehensive representations are, of course, available in global Earth system models (Abe-Ouchi et al. 2013), but such sophistication makes it difficult to explore the full range of parameters. When using an intermediate complexity model, for example, Willeit et al. (2019) were constrained to using 16 different combinations of regolith removal and volcanic CO2 outgassing scenarios in exploring the causes of the transition between early and late Pleistocene glacial cycles.

In this study we first construct a model that represents key components of the relationship between CO2 and sea level in a manner that permits for rapid sampling from a joint posterior distribution (section 2). We then fit the model to late Pleistocene sea level and CO2 observations and use it to predict CO2 from sea level during the early Pleistocene (section 3). Specifically, we first estimate early Pleistocene CO2 based exclusively on its late Pleistocene relationship with sea level and, second, explore a scenario with a thinner and more expansive early Pleistocene ice sheet. We conclude with some suggestions for further lines of investigation (section 5).

2. A Bayesian sea level model

We build a simple model to investigate the factors affecting the relationship between sea level and CO2. Among the processes that we include, we highlight two. First, orbital forcing of ice sheets should be accounted for because changes in insolation occur at similar time scales as those of CO2 and contribute independently to variations in sea level (Weaver et al. 1998). Furthermore, orbital amplitude changes over the Pleistocene (Laskar et al. 2004; Hinnov 2000) could affect inferences of changes in sensitivity to radiative forcing. Second, the ages of sea level and CO2 are only known approximately and were partially inferred through matching against an orbital template. We are interested in explicitly including age as part of the model because relative timing can be a first-order control on any inferred behavior (Carson et al. 2019).

a. Planetary energy balance model

Budyko (1969) represented the equilibrium position of the edge of a zonally averaged ice sheet as a function of the incoming shortwave and outgoing longwave radiation. The zonally averaged energy balance is
F4I(x)[1α(x)]=A+BT(x)+H,
where x is the latitude in degrees, F is the globally averaged incoming shortwave radiation, α (x) is the surface albedo, A + BT(x) is the linearized outgoing longwave radiation (OLR), and I(x) is the normalized pattern of zonally and annually averaged insolation as a function of latitude. The term ∇ ⋅ H is the divergence of the meridional heat flux and, as in Budyko (1969), is approximated as M(TT¯), where T¯ is the global mean surface temperature. This approach affords rapid numerical integration of our overall model. Figure 1 gives a schematic of the model.
Fig. 1.
Fig. 1.

Schematic of a coupled energy-balance and ice-sheet model. Incoming shortwave radiation (blue) varies with the orbital configuration and latitude, and is absorbed according to the surface albedo of either ice-free or ice-covered surfaces. Outgoing longwave radiation (red) is a function of CO2 concentration and surface temperature, and heat is fluxed across latitudes (green) according to the local deviation from the global mean surface temperature. An ice sheet is represented between a fixed northern terminus xn and a southern terminus xs that responds to the temperature determined by the energy-balance model. Rates of change in xs are also influenced by ice-sheet thickness that is controlled by a basal shear stress τ and the length of the ice sheet.

Citation: Journal of Climate 34, 24; 10.1175/JCLI-D-21-0192.1

Equation (1) is extended to include effects of atmospheric CO2 on outgoing longwave radiation and of orbital variations on incoming shortwave radiation. CO2 variations are represented by modifying the OLR parameter, A, to equal A0 + A1φ, where φ=log([C]/[Co]), [C] is the atmospheric CO2 concentration in parts per million, and [Co] is a reference concentration. Spatial and seasonal variation in CO2 radiative forcing are neglected.

We follow adaptations of the Budyko model in using polynomial approximations for the latitude dependence of mean annual insolation to obtain tractable solutions for the position of the ice line (e.g., Chylek and Coakley 1975; North 1975b; Lindzen and Farrell 1977; North 1975a; Held and Suarez 1974). Here I(x) is approximated as I(x) = 1 + sP2(x), where P2 = [3sin(x)2 − 1]/2 is the second Legendre polynomial and s = −0.482. Orbital variations are represented by way of modifications to F:
F=S0+ηpesin(ϖ+ϕ)+ηoε,
where S0 is the solar constant, e is the eccentricity, ϖ is the longitude of the perihelion relative to the vernal equinox, ϕ is the phase of the climatic response to precession, ε is obliquity, and ηp and ηo are amplitudes respectively for the climatic precession and obliquity contributions. The astronomical solution of Berger and Loutre (1991) is used for orbital values and, for convenience in interpreting ηp and ηo, ε and esin(ϖ + ϕ) are normalized to zero mean and unit variance over 0–805 ka.

Although represented as changing F, climatic precession does not influence annual average insolation at any latitude and obliquity does not influence the annual energy summed across all latitudes (Milankovitch 1941; Rubincam 1994). Rather, Eq. (2) aggregates seasonal and spatial patterns that influence the position of the ice line, such as changes in the seasonal distribution of insolation above a melt threshold (Huybers 2006). Modifying the model such that obliquity modulates s, instead of influencing F, leads to an inference of smaller obliquity contributions but does not qualitatively change other results.

The globally integrated energy balance is
1αp4F=A0+A1φ+BT¯,
and following Roe and Baker (2010), the mean planetary albedo is
αp(xs)=α2+(α1α2)[sin(xs)+s5(P3P1)].
Terms α2 and α1 represent, respectively, average albedo at latitudes with and without ice, and xs is the latitude of the southern ice-sheet margin, or the ice line. The term Pn is the nth Legendre polynomial evaluated at sin(xs).
Linearization of the outgoing longwave radiation permits cancellation of the global-mean surface temperature, T¯. Equation (3) is rearranged and substituted into Eq. (1) at xs, following Roe and Baker (2010), to obtain an expression for surface temperature at the ice line:
Ts=1B+M{F4(1αs)I(xs)MB[F4(αp1)+A0+A1φ]A0A1φ},
where αs is a transitional albedo equal to (α1+α2)/2.

b. Ice sheet

In Eq. (5) surface temperature is assumed to be in equilibrium with radiative forcing. Orbital forcing and changes in atmospheric CO2, however, vary at time scales comparable with that of ice sheet adjustment (Weertman 1964; Cuffey and Paterson 2010), suggesting the need to represent ice sheet time dependence. An ice sheet is assumed to exist between xs and a fixed northern terminus xn (Fig. 1). Following Weertman (1976), the ice sheet is zonally averaged and assumed to follow a plastic rheology:
h(x)=[λ(|xmxs||xxm|)]1/2.
Here h is the height of the ice sheet above the surrounding land surface, λ = 4τ/(3ρigd), with τ being the basal shear stress, ρi the ice density, g gravitational acceleration, xm the midpoint of the ice sheet, and d = 111.319 × 103 m per degree of latitude. To keep both the height and length of the ice sheet in the same units and because our analysis is focused on the position of the ice line, values of h, xm, and λ are in units of degrees of latitude.
Total ablation over an ice sheet increases with both temperature and area of the ice sheet, but because the interiors of large ice sheets tend to be cold, most melting occurs at the margins (Van den Broeke et al. 2011). We therefore focus on conditions at the ice line for representing changes in ablation rates across the whole ice sheet, for which purposes it is simpler to consider the cross-sectional area of the ice sheet above the equilibrium land surface:
Y=xsxnh(x)dx=4λ3(xnxs2)3/2.
Assuming that net ablation along the southern terminus scales linearly with zonal length, we represent the change in cross-sectional area as linearly proportional to temperature at the ice line,
dYdt=p(zTs),
where z controls the ice-line temperature at which ablation overtakes accumulation, and accumulation is assumed constant. The ablation sensitivity, p, represents the amount of melting over the whole ice sheet for each degree of temperature increase at the ice line multiplied by a scaling factor of 2/3. The 2/3 adjustment reflects isostatic adjustment of bedrock in response to the removal or addition of ice, where bedrock density is assumed to be 3 times that of ice (Weertman 1976). Bedrock rebound is taken as instantaneous. For simplicity we aggregate the isostatic and melting components into the single parameter p, which has units of degrees latitude squared per 1000 years (ky) per 1°C. Combining Eqs. (7) and (8), we obtain the rate of change in xs as a function of ice-line temperature:
dxsdt=p(Tsz)[λ(xnxs)2]1/2,
where we applied that dxs/dt = (dY/dt)(dxs/dY) and that xn is fixed.

To account for the fact that the ice-sheet only spans a fraction of the Earth circumference, we represent α2 = flαi + (1 − fl)α1, where αi is the average ice albedo. Regions above xn are assumed to be covered with snow or sea ice with the same average fraction fl as in regions with an ice sheet.

c. Coupled model inversion

Most Budyko-type energy balance models achieve closure by specifying a value for Ts, typically −10°C (Roe and Baker 2010; North 1975a). We instead couple the energy-balance and ice-sheet ablation expressions by solving Eq. (9) for Ts and combining it with Eq. (5). The coupled model can be rearranged to give the log of the CO2 fractional anomaly φ as a function of the ice line:
φ=β0β1dxsdt.
The β0 term represents the equilibrium relationship between CO2 and the ice line:
β0=BA1(B+M)[F4(1αs)I(xs)A0MB(A0F4+Fαp4)]BzA1.
Implied CO2 concentration increases with temperature at the ice line, which itself is influenced by insolation, the efficiency with which heat is transported through the atmosphere, and the planetary albedo. The term β1 scales the effect of ice-sheet disequilibrium:
β1=BA1p[λ(xnxs)2]1/2.
Larger rates of change in the ice line are induced when ablation sensitivity is large or the ice sheet is thin either because of a small latitudinal extent or low basal shear stress.
To relate xs to sea level we approximate the ice sheet as a partial torus, neglecting meridional curvature. Ice volume is obtained by multiplying Y by the zonal extent of the ice sheet at its midpoint and converted to sea level:
S=ρiaoρwflEccos(xm)Y,
where Ec is Earth’s circumference at the equator, fl is the zonal fraction of Earth covered by the ice sheet, ρi and ρw are the densities of ice and water, and ao is the area of the ocean.

d. Observations

Atmospheric CO2 measurements are from the revised EPICA Dome C composite record (Bereiter et al. 2015) that extends to 806 ka (i.e., 806 000 years ago; ka indicates “thousand years ago”) has an expected measurement error averaging 1.34 ppm. Ages are assigned by the chronology outlined in Bereiter et al. (2015) and are based on ice and gas stratigraphic age markers, orbital variations, and ice flow modeling across multiple ice cores (Bazin et al. 2013). Age uncertainties associated with CO2 are estimated to have a 6-ky standard deviation on account of uncertainty in accumulation rates, rates of ice-layer thinning, phasing of CO2 with respect to orbital variations, and ages of stratigraphic markers. Additional proxy-based CO2 reconstructions from foraminiferal δ11B (Chalk et al. 2017; Hönisch et al. 2009; Dyez et al. 2018) are used for evaluating how well the model reproduces a temporal shift in sea level sensitivity. Other published δ11B-based estimates are not used because of large discrepancies with ice-core CO2 (Sosdian et al. 2018), few samples across the 0–2-Ma (million years ago) interval that is the focus of our study (Seki et al. 2010), or both (Stap et al. 2016).

For sea level, we use a reconstruction that is derived from a nonlinear transfer function relating benthic δ18O to sea level (Siddall et al. 2010). In this record, a series of independent sea level constraints spanning the past ~240 ky as well as from the Pliocene are compared with coeval δ18O values to establish a nonlinear relation between the two. This relation was used to generate a 5-My (i.e., 5-million-yr) sea level estimate based on δ18O under an assumption that their relationship has remained stationary. We use this record because it broadly agrees with other sea level reconstructions over the late Pleistocene (Spratt and Lisiecki 2016; Shakun et al. 2015; Waelbroeck et al. 2002; Rohling et al. 2009) and extends into the early Pleistocene, where the latter is a requirement for our purposes. We adjusted sea level to have the model reference level of no Northern Hemisphere ice by subtracting a constant from all values such that present-day sea level is at −8 m. The choice of offset reflects the present-day ice volume in Greenland, Iceland, and ice caps (Farinotti et al. 2019; Morlighem et al. 2017, 2020), but inferred CO2 values are not qualitatively sensitive to using a smaller offset.

Whereas orbital variations can be accurately computed over the last several million years (Laskar et al. 2011), sea level and CO2 ages are uncertain. Baseline chronologies are adopted for sea level and CO2, respectively, from Shackleton et al. (1990) and Bazin et al. (2013). Adjustments to these age scales are permitted by way of eight age control points (ACPs) placed every 100 ky from 100 to 800 ka in the sea level and CO2 records, respectively denoted k1 through k8 and q1 through q8. Adjustments to ACPs are propagated to other data points using linear interpolation. The age of the most recent sample in each record is treated as fixed.

e. Priors and sampling

The parameters that we seek to constrain and the priors that we impose are listed in Table 1. Uniform priors are assigned either where bounds are considered physically strict or are used across wide intervals where uninformative priors are desired. Normal priors are used where an informed guess can be made as to a parameter value.

Table 1.

Prior and posterior distributions for the parameters in the Bayesian sea level and CO2 model, as well as values of the constants used.

Table 1.

We expect a net-zero mass balance where mean annual temperatures are somewhat below 0°C (Cuffey and Paterson 2010), and, in accord with foregoing energy balance models (North 1975b), assign a normal prior to z with a mean of −10°C and a standard deviation of 5°C. Orbital amplitude parameters ηp and ηo are uncertain but should be nonnegative, and are therefore assigned diffuse uniform priors spanning 0–10 W m−2. Priors for the ice-covered and ice-free surface albedos are also uniform and allow variations from 0.4 to 0.95 and from 0.15 to 0.5. The ablation sensitivity parameter p has a uniform prior allowing values between 8 × 10−3 and 8 × 10−2 degrees2 ky−1 °C−1, which corresponds to lags of sea level behind radiative forcing of roughly 5–20 ky when sea level varies with an amplitude of 100 m.

Prior analyses of Budyko-type models prescribed values for the linearized outgoing longwave radiation parameters A and B of approximately 200–215 W m−2 and 1.4–1.6 W m−2 °C−1, respectively (Lindzen and Farrell 1977; Roe and Baker 2010; North 1975a; Chylek and Coakley 1975). The appropriate values are uncertain because the correct linearization depends on a number of unrepresented factors, including cloudiness (North 1975a). We choose a prior mean of 211 W m−2, a commonly used value for A, and a standard deviation of 10 W m−2. Note that B and M are excluded from the Bayesian inference on account of strong covariation with other parameters, which otherwise leads to slow convergence of our sampling approach. Following Lindzen and Farrell (1977), B is specified as 1.55 W m−2 °C−1. Typically M is considered in relation to the value of B because the nondimensional quantity M/B represents the efficiency of poleward energy transport (Held and Suarez 1974), with commonly used values of around 2.5. We follow Chylek and Coakley (1975) in prescribing a ratio of 2.61 and, therefore, specify M equal to 4.05.

Well-constrained physical parameters are specified as constants, including the solar constant S0 (Kopp and Lean 2011), CO2 radiative forcing parameter A1 (Myhre et al. 1998), ice and water densities ρi and ρw, and the ocean area ao (Sima et al. 2006). Fixed values are also used for the Earth-circumference fraction covered by the ice sheet fl and upper ice-sheet terminus position xn (see Table 1). In the initial set of simulations the basal shear stress τ is also held constant.

Finally, age-control parameters are assigned normal priors with means at their respective ages in the time scale of Shackleton et al. (1990) and Bazin et al. (2013) and standard deviations of 6 ky. Evidence that the Brunhes-Matuyama magnetic reversal is identifiable in ice-core records (e.g., Dreyfus et al. 2008) suggests, however, that ages for the oldest ice-core samples should not be younger than the reversal, and the timing of the reversal in the age model of Shackleton et al. (1990) aligns with radiometric estimates of ~780 ka (Cande and Kent 1995). Priors for k8 and q8, centered at 800 ka, are therefore additionally truncated so as to disallow values younger than 780 ka.

Likelihoods for combinations of parameters are computed from the residuals between modeled and observed CO2 values. The magnitude and structure of the residual have several contributions, including uncertainty in the sea level reconstruction, omission of relevant processes in our model, and age error. These factors, when combined with our having aggregated climate lags into a single ablation sensitivity parameter, lead errors to be autocorrelated. We therefore parameterize errors using an autoregressive order-one process and infer the variance and autocorrelation coefficient as part of the Bayesian inversion.

To build up the joint posterior distribution, we use an affine-invariant ensemble sampling algorithm in which each of a collection of sampling chains explores the parameter space (Goodman and Weare 2010). Affine invariance assures that sampling is insensitive to the aspect ratio of the state space, avoiding the need to tune step sizes for each parameter. 30 chains are used for inference, with parameters in each chain initialized at a random perturbation about an initial guess. Further details on our sampling approach are in the appendix.

The inference is conducted using sea level and CO2 data spanning 780 ka to 2 ka after age adjustments. Data across 0–2 ka are excluded to omit anthropogenic influences in the ice-core composite, and data older than 780 ka are omitted so that likelihoods are calculated over a consistent time interval that would otherwise vary with the inferred age of the end of the ice-core record. Alternatively allowing k8 and q8 to reach ages as young as 770 and using 770–0 ka as the inference interval does not qualitatively change our results.

3. Results

Realizations of late Pleistocene CO2 from our model have a residual standard deviation of 16.1 ppm [14.6–17.6 ppm 95% confidence interval (c.i.)] relative to late Pleistocene CO2 observations (Fig. 2), and inferred error properties appear appropriate in that 93% of ice-core CO2 values lie within the inferred 95% c.i. given by the combined uncertainty in model fit and model parameters. Below we interpret the posterior values for the physical and age-control parameters and discuss their implications for the relationship between sea level and CO2.

Fig. 2.
Fig. 2.

Variations in sea level and CO2 values over the past 2 Ma. (a) Maximum likelihood model CO2 levels (black) agree closely with ice-core CO2 data (orange). 95% of CO2 predictions fall within the dark gray interval, or when summing values with model-data error, the lighter gray interval. Model CO2 variations also broadly reproduce patterns in δ11B-derived CO2 estimates, shown in blue for Chalk et al. (2017), green for Dyez et al. (2018), and purple for Hönisch et al. (2009). (b) Variations in sea level and ice line implied by the model. Sea level (gray) is from Siddall et al. (2010) and is converted to a zonally averaged ice-sheet length (blue) that is controlled by the basal shear stress τ and the fraction of Earth circumference covered by the ice sheet fl. Adjustments to CO2 and sea level ages are achieved by way of eight age-control parameters whose prior means are displayed in empty gray circles and posterior 95% c.i. displayed as orange and blue horizontal bars. Uncertainty in the ages for sea level, and therefore for ice lines, is reflected in their shaded 95% c.i.

Citation: Journal of Climate 34, 24; 10.1175/JCLI-D-21-0192.1

a. Joint posterior distribution of physical parameters

A first issue relates to distinguishing the orbital and CO2 forcing contributions to sea level changes. Orbital amplitude parameters ηp and ηo respectively accumulate density into modes at 0.96 W m−2 (0.72–1.22 W m−2 95% c.i.) and 0.94 W m−2 (0.67–1.25 W m−2), values that are well constrained with respect to their uniform priors. The climatic precession phase is also well constrained with a posterior median at 18.9° (−12.7° to 39.8°) such that forcing is strongest when perihelion occurs on 10 July (8 June–31 July). These results are consistent with previous inferences of the orbital configuration most associated with net ablation (Milankovitch 1941; Imbrie et al. 1984; Carson et al. 2019). If orbital forcing is omitted from the model, a greater CO2 contribution is inferred because, in that case, CO2 assumes all responsibility for sea level changes. Keeping all other parameters at their inferred values, the differences between inferred CO2 values in the models with and without orbital forcing have an average variance of 75 ppm2, and variations of up to 47 ppm.

Ice-free surface albedo α1 is inferred to be centered at 0.30 (0.21–0.38) and ice-albedo αi at 0.80 (0.79–0.82), both occupying narrow distributions relative to their uniform priors. The substantial inferred difference between α1 and αi implies a strong ice-albedo feedback, and their relatively narrow posterior distributions reflect that the model is quite sensitive to even small albedo changes. Increasing the ice albedo by 0.1 increases the variance of model CO2 by an average of 54%, and increasing the ice-free albedo by just 0.01 increases the variance of the prediction by 180%, although changes in other parameters, such as the outgoing longwave radiation parameter A0 or the equilibrium ice-line temperature, z, can offset those effects. It should also be noted that although α1 and αi are expected to capture an ice-albedo feedback present in the observations, their values are approximate on account of our simple model lacking multiple land-cover types, representative geography, or oceans.

The terms A0 and z have posterior medians of 210.7 W m−2 (190.8–229.6 W m−2) and −9.7°C (−19.3° to −0.6°C). Inferred values are within the expected range, but their standard deviations of 10.0 W m−2 and 4.8°C are comparable to their priors. The influence of the priors here mainly reflects that these parameters covary with the albedo parameters (correlation between −0.49 and −0.79 for pairs of A0 and z with α1 and αi), as dictated by Eq. (10). Because opposing changes among these parameters can preserve similar predictions of CO2 values, various combinations of A0, z, α1, and α2, are difficult to distinguish. In this case basic physics help us constrain values for A0 and z. Repeating the analysis with doubled prior standard deviations for A0 and z, however, negligibly changes the inferred CO2 values.

Ablation sensitivity p is strongly constrained by its prior, with posterior values accumulated against the upper limit (Fig. 3). The need for a prior to constrain p reflects uncertainties in sea level ages and magnitudes relative to CO2, and the prior accords with estimates of ice-sheet response time scales (Weertman 1964; Cuffey and Paterson 2010), but our conclusions do not depend on our choice of prior. If we repeat the inference after increasing the prior upper bound by a factor of 10, inferred values of p imply ice-sheet equilibration times of approximately 0.5–5 ky that appear unrealistically small, but early Pleistocene CO2 values are negligibly altered relative to the main model.

Fig. 3.
Fig. 3.

Prior (gray) and posterior values for each of the adjustable model parameters when ages are allowing age adjustments (blue) or disallowing them (orange). Panels with age-control parameters represent sea level ACPs (darker blue) and corresponding ACPs in the CO2 record (lighter blue).Vertical lines at k4, q4 indicate the prior age for termination 5 in the sea level record we use here (Siddall et al. 2010, thick gray line), and in the records of Rohling et al. (2009) (yellow), Rohling et al. (2014) (purple), Bintanja and van de Wal (2008) (red), De Boer et al. (2010) (cyan), Shakun et al. (2015) (dark green), Spratt and Lisiecki (2016) (black), and Elderfield et al. (2012) (orange). Axis labels correspond to the parameters listed in Table 1 and, for visual clarity, the x axis is concentrated to posterior values even if the prior is wider.

Citation: Journal of Climate 34, 24; 10.1175/JCLI-D-21-0192.1

b. Age inferences

Estimated ages for sea level and CO2 are mostly within the expected error of orbitally tuned age models and have small variance relative to the priors (Table 1 and Fig. 3), with the exception of larger anomalies during stages 14–11 (550–400 ka). In particular, CO2 age parameter q5 is adjusted from 500 ka to a posterior median value of 481 ka, and q4 and sea level age parameter k4 are adjusted from 400 ka to posterior median values at 418 and 422 ka, respectively. Sea level age revisions coincide with differences between the Shackleton et al. (1990) age-scale adopted by Siddall et al. (2010) and other sea level reconstructions (Rohling et al. 2009, 2014; Elderfield et al. 2012; Bintanja and van de Wal 2008; Shakun et al. 2015; De Boer et al. 2010; Spratt and Lisiecki 2016) that assign ages that are 10–30 ky older than Siddall et al. (2010) at termination 5 and are independently consistent with our revised age estimates (see vertical lines in Fig. 3).

In general, relative age is better constrained than absolute age. Several pairs of ACPs move in tandem on account of astronomical variability not fully constraining absolute ages (Fig. 3). The median correlation between pairs of analogously placed ACPs in sea level and CO2 is 0.58, with values as high as 0.87 for (k3,q3). The median variance of the difference between pairs of sea-level and CO2 ACPs, therefore, is 29% less than the variance of individual CO2 ACPs.

It might be expected that explicit inclusion of age model uncertainty would lead to greater uncertainty in other parameters. We find, however, little systematic difference in parameter uncertainties if repeating the Bayesian analysis without permitting age refinements (orange vs blue distributions in Fig. 3). In this case the standard deviations for the eight non-age-related physical parameters are between 30% less and 20% more than in the flexible-age case. The average ratio of parameter standard deviations between fixed- and flexible-age cases is 1.0, indicating that parameters are equally likely to be slightly more versus slightly less uncertain between the two cases.

Disallowing age refinements lead to qualitatively insignificant changes in all parameters except for ηo, the obliquity forcing amplitude, which is reduced by 43%, compared with only a 7% decrease in the precession amplitude ηp. Although the precession and obliquity forcing are both statistically significant in each case, we note that their more-equal contributions in the case of flexible ages accord with the predictions of Milankovitch (1941) and the subsequent studies of orbital forcing of late-Pleistocene glacial cycles that confirmed those predictions. We suspect that allowing for age refinements leads to inference of a more consistent physical relationship between sea level and CO2 as compared with inferences based upon ages that are fixed, albeit uncertain.

4. Discussion

Although sea level and CO2 follow similar broad patterns across glacial–interglacial cycles, one is not merely a linear rescaling of the other. Changes in CO2 radiative forcing, denoted Rf and equal to A1φ, cause changes in sea level after a delay. The adjustment time scale leads to a looping structure about an equilibrium curve when plotting sea level against Rf (blue and orange traces in Fig. 4) about a central equilibrium relationship between Rf and sea level. A curve representing the equilibrium relationship (black line in Fig. 4) indicates that ice sheets are more sensitive when they are large. It follows that sea level sensitivity to radiative forcing is smaller when sea level is higher. Below we investigate whether the model-inferred structure of sea level sensitivity is sufficient for explaining empirical evidence for changes in sea level sensitivity over time. Unless otherwise noted, references to CO2 in this section refer to the model inference of CO2 derived from sea level, not the late Pleistocene ice-core CO2 record on which the model is calibrated.

Fig. 4.
Fig. 4.

A nonlinear relationship is present between sea level and CO2 radiative forcing Rf that, on average, indicates greater sensitivity in the late Pleistocene (1–0 Ma; blue trace and markers) than in the early Pleistocene (2–1 Ma; orange trace and markers). Owing to the time taken to melt or grow an ice sheet, the time-dependent model exhibits width about a maximum likelihood equilibrium relationship (dark gray curve, with 95% of equilibrium curves in lighter gray).

Citation: Journal of Climate 34, 24; 10.1175/JCLI-D-21-0192.1

a. Changes in sea level sensitivity

We use the derivative of the equilibrium relationship between sea level and CO2 radiative forcing, dS/dRf, to describe sea level sensitivity to CO2 forcing (Fig. 5). Averaged across the early Pleistocene (2–1 Ma), model equilibrium sensitivity is 47 m (W m−2)−1 [44–50 m (W m−2)−1 95% c.i.]. This increases to 106 m (W m−2)−1 [89–141 m (W m−2)−1] during the late Pleistocene (0–1 Ma), or a 126% (103%–182%) increase relative to 2–1 Ma. Chalk et al. (2017) gave a specific estimate of an 80% (40%–130%) increase, and if we only average over the intervals covered in their study—0 to 260 ka and 1090 to 1250 ka—the model fractional sensitivity increase is 90% (63%–117%).

Fig. 5.
Fig. 5.

Changes in equilibrium sea level sensitivity to radiative forcing through the Pleistocene. (a) Sensitivity increases with ice-sheet length (gray curve, maximum likelihood estimate) and therefore varies across glacial–interglacial cycles. Long-term patterns, obtained by smoothing sensitivity with a 300-ky Gaussian filter, are also displayed with 95% of curves shaded. Denial of mechanism experiments show that sensitivity increases over time are muted when suppressing ice-albedo feedbacks (blue), the nonlinear relationship between ice-sheet volume and length (yellow), or both (orange). (b) As in (a), but only smoothed sensitivities are shown and they are taken relative to their values at 2 Ma.

Citation: Journal of Climate 34, 24; 10.1175/JCLI-D-21-0192.1

The sensitivity magnitudes estimated here differ from the values of ~20–30 and ~40–50 m (W m−2)−1 for the early and late Pleistocene that were estimated by Chalk et al. (2017). The difference arises mainly because we define sensitivity differently: we use the slope of a nonlinear equilibrium relationship that was inferred accounting for age adjustments and orbital variations, whereas Chalk et al. (2017) used the slope of a York linear regression between Rf and S. Two further differences are 1) that we use the sea level record of Siddall et al. (2010), whereas Chalk et al. (2017) used reconstructions from Rohling et al. (2014), De Boer et al. (2010), and Elderfield et al. (2012), and 2) that our model is conditioned on ice-core CO2 over the entire late Pleistocene and not δ11B-based estimates. Despite the differences in sea level and CO2 data used, model sensitivity magnitudes are similar to those reported in Chalk et al. (2017) if we change our method to instead use a York regression between model Rf and S. In this case, average model sea level sensitivity is 35.7 m (W m−2)−1 [33.9–37.2 m (W m−2)−1] for the early Pleistocene and 55.2 m (W m−2)−1 [53.7–56.6 m (W m−2)−1] for the late Pleistocene. Both the percentage sensitivity increase across the MPT implied by the regression, 55%, and the magnitude of the increase, 20 m (W m−2)−1, are within the uncertainty range reported in Chalk et al. (2017).

We investigate the origins of sensitivity changes by suppressing physical processes in the model. To omit the ice-albedo feedback, we set α1 = α2 with a value of 0.35, chosen to give an maximum-likelihood CO2 with a similar average to that in the full model. Elimination of the feedback dramatically reduces equilibrium sensitivity to an average of just 10.5 m (W m−2)−1 across the early Pleistocene, increasing only to 13.4 m (W m−2)−1 in the late Pleistocene (blue lines in Fig. 5). Unsurprisingly, much larger glacial–interglacial variations in CO2 would be required to generate the glacial cycles if the albedo feedback were absent, and the increase in sensitivity between early and late Pleistocene is reduced by 78%. Similarly, the effect of nonlinear scaling of ice-sheet volume with its length on equilibrium sensitivity may be estimated by replacing the model ice sheet with a simple linear relationship between xs and sea level (Fig. 6c). If the width of the ice sheet were constant, the volume would scale with length to the ~3/2 power (Weertman 1976), but because zonal extent also increases with length, the power law is ~5/2. In this experiment, equilibrium sensitivity averages 55.6 m (W m−2)−1 for the early Pleistocene and 96.9 m (W m−2)−1 for the late Pleistocene (yellow lines in Fig. 5), yielding a 41% weaker sensitivity increase across the MPT than the full model. Suppressing both the ice-albedo and ice-sheet-geometry contributions effectively eliminates model nonlinearity (Fig. 6d), and gives nearly equivalent early and late Pleistocene equilibrium sensitivities of 12.8 and 13.3 m (W m−2)−1 (orange lines in Fig. 5).

Fig. 6.
Fig. 6.

Consequences of suppressing model nonlinearities. (a) Relationship between sea level and Rf in the full model, including both the maximum-likelihood time-dependent response in the early (orange) and late (blue) Pleistocene and the maximum-likelihood equilibrium relationship (gray). (b) As in (a), but ice-albedo feedback is omitted from the model. (c) As in (a), but the nonlinear relationship between ice-sheet volume and length is omitted from the model. (d) As in (a), but both the ice-albedo feedback and ice-sheet geometry are omitted. These denial-of-mechanism experiments show that the ice-albedo feedback and ice-sheet geometry explain more than 95% of the change in sea level sensitivity across the MPT.

Citation: Journal of Climate 34, 24; 10.1175/JCLI-D-21-0192.1

Finally, our inference of the nonlinear relationship between sea level and CO2 radiative forcing is conditional on the choice of sea level record, and it is useful to verify that similar patterns are observed if alternative choices are made. Repeating the inference procedure using the reconstructions of Willeit et al. (2019), Elderfield et al. (2012), and Rohling et al. (2014) gives an average equilibrium sensitivity for sea levels between −100 and −50 m that is double or more than for sea levels between −50 and 0 m, consistent with the results obtained if using the record of Siddall et al. (2010).

b. Mean early Pleistocene CO2 levels

The δ11B-based CO2 estimates are an order of magnitude more uncertain than ice-core CO2 measurements (Chalk et al. 2017; Dyez et al. 2018; Hönisch et al. 2009). The proxy requires specifying at least one parameter for the marine carbonate system, such as total alkalinity or dissolved inorganic carbon, whose values are uncertain. Alkalinity may have increased through the Pleistocene, for example, due to weathering of the Canadian Shield (Clark et al. 2006). Different species of foraminifera require different calibrations and, typically, an assumption that the conditions in which they proliferated did not change over time (Dyez et al. 2018). As such, although we use proxy estimates as they were originally published, different realizations are possible. For example, Dyez et al. (2018) used revised salinity and alkalinity estimates to obtain ~12-ppm-higher average CO2 than Chalk et al. (2017) and ~26-ppm-higher average CO2 than Seki et al. (2010) using the originally published δ11B values.

We find mismatches between our model predictions and δ11B-based proxy reconstructions where they overlap (Hönisch et al. 2009; Chalk et al. 2017; Dyez et al. 2018) that are qualitatively similar in the early and late Pleistocene. Combining uncertainties in both proxy and model CO2 values using a Monte Carlo approach with 104 samples, and assuming reported uncertainties follow a normal distribution, indicates that δ11B-derived estimates are, on average, 11 ppm (6–16 ppm) higher than our model predictions between 2–1 Ma and 8 ppm (2–14 ppm) higher across 0–1 Ma. A similar offset arises between δ11B-derived and ice-core CO2 in the late Pleistocene of 9 ppm (5–13 ppm). Subtracting this difference to align δ11B-derived and ice-core CO2 values also brings the δ11B-derived values into closer agreement with the model, reducing the mean offset to 1 ppm in the late Pleistocene and 2 ppm in the early Pleistocene. Our results thus support the accuracy of the reconstructions of Hönisch et al. (2009), Chalk et al. (2017), and Dyez et al. (2018) and may also support a modest downward revision.

There are also qualitatively similar mismatches in the range of CO2 variability between the proxy and ice-core values and between the proxy and model values. In the late Pleistocene, the point estimates for the δ11B-derived estimates have 57% greater standard deviation than ice-core values. This aligns with the δ11B-derived values having 52% (43%–61%) greater standard deviation than the model CO2 over the late Pleistocene, and 70% (59%–82%) greater standard deviation in the early Pleistocene. These differences partly explain why we obtained somewhat larger sensitivity magnitudes than Chalk et al. (2017) when using their York regression method with model CO2 (see section 4a). The fact that differences in the range of CO2 values in the δ11B-based values relative to either the ice-core or model values are similar over the late Pleistocene is expected on account of our model being conditioned on ice-core CO2. Furthermore, that the differences in CO2 ranges between model and δ11B-derived values are similar in both the early and late Pleistocene is consistent with much of the proxy–model disagreement arising from systematic proxy bias. In particular, Chalk et al. (2017) note that a more complete treatment of the relationship between alkalinity and pH would somewhat reduce the glacial–interglacial range in the δ11B-based estimates.

c. Interpreting the regolith hypothesis in the context of proxy CO2 reconstructions

Geologic evidence suggests that the maximal extent of early Pleistocene ice sheets was similar to that of late Pleistocene ice sheets despite their containing smaller ice volume. Glacial tills as far south as Iowa (Boellstorff and Te Punga 1977; Boellstorff 1978) and meltwater events in Gulf of Mexico δ18O records (Joyce et al. 1993; Shakun et al. 2016) support a low-lying early Pleistocene Laurentide that flowed rapidly and spread across a large area (Clark and Pollard 1998). Repeated glacial advance and retreat could have stripped upper North America of its regolith, leading to greater friction at the base of the ice sheet and causing it to grow thicker relative to its length (Clark and Pollard 1998).

We use our model to investigate the implications of regolith removal for inferences of early Pleistocene CO2 levels. We emphasize, however, that by adopting sea level as an input we do not address the root causes of glacial intensification through the Pleistocene. It is possible to simulate thin, expansive early Pleistocene ice sheets by specifying a basal shear stress that increases from 25 to 100 kPa across a transition centered at 1 Ma (Figs. 7a,b). In this case, the sea level sensitivity increase across the mid-Pleistocene transition primarily arises not from an ice-albedo feedback, but because a given CO2-induced change in the ice line has greater consequences for sea level when the ice sheet becomes thicker.

Fig. 7.
Fig. 7.

Consequences of changing the ice-sheet basal conditions for early Pleistocene ice line and CO2. (a) Two scenarios for the mid-Pleistocene transition. A first case is the assumption of a constant basal shear stress of 100 kPa (blue). A second case is that, in accord with Clark and Pollard (1998), basal shear stress increased from 25 to 100 kPa across a transition centered at ~1 Ma (orange). (b) Ice lines implied by the two basal shear stress scenarios. Early Pleistocene ice sheets are thinner and more extensive if assuming a lower basal shear stress (blue), with a value of 25 kPa giving similar early and late Pleistocene glacial-stage ice lines. (c) Model CO2 levels in the two scenarios (maximum likelihood parameters in bold, with 95% of model estimates in the shaded region). Lower CO2 values are required to explain the lower ice lines implied by thin early Pleistocene ice sheets, leading to greater disagreements with δ11B-derived CO2 reconstructions in comparison to an assumption of a constant τ [yellow, Chalk et al. (2017); green, Dyez et al. (2018); purple, Hönisch et al. (2009)].

Citation: Journal of Climate 34, 24; 10.1175/JCLI-D-21-0192.1

The regolith hypothesis has implications for early Pleistocene CO2 values inferred by our model because lower CO2 levels are required to generate low-latitude ice lines (Fig. 7c). Specifying τ = 25 kPa gives an average early Pleistocene CO2 of 229 ppm (227–230 ppm), compared to a value of 244 ppm (241–246 ppm) when retaining τ = 100 kPa. These lower values imply similar early and late Pleistocene mean CO2 levels under the regolith hypothesis and induce a modest average difference of 24 ppm (20–28 ppm) relative to δ11B-derived reconstructions in the early Pleistocene (Fig. 7c). This difference does not necessarily imply that regolith removal is unlikely to have occurred and can be explained in several ways. First, subtracting the 9 ppm offset between the δ11B-derived and ice-core CO2 values would already reduce the discrepancy to 15 ppm. For the remaining CO2 difference, there are several plausible explanations.

One factor is that early Pleistocene sea level records are uncertain, and we have used just one reconstruction. Our model can produce both ice lines and average CO2 values that are in keeping with the regolith hypothesis if, for example, sea level averages 12 m higher than in Siddall et al. (2010) and basal shear stress averages 15 kPa in the early Pleistocene. Such a revision implies closer agreement of sea level with estimates by Rohling et al. (2014), whose reconstruction averages 21 m higher than Siddall et al. (2010) across 2–1 Ma.

A second obvious possibility is that shortcomings in our model play a role, particularly because a number of simplifications were made to permit solving it efficiently for purposes of Bayesian sampling. We reduced the process of ice-sheet growth and decline to a linear scaling with temperature at the ice-margin and assumed that the ice sheet retains the aspect ratio set by the basal shear stress parameter. We excluded the effects of delayed isostatic rebound on ablation rates (Pollard 1982), as well as calving (Pollard 1983) or any thermodynamic considerations such as melting at the base of the ice sheet (Jenson et al. 1996), all of which could lead to greater ice-sheet disequilibrium than we have considered here. Our model may also overstate the importance of a low-latitude Laurentide for the planetary albedo—and therefore for CO2 levels—because the position of the Eurasian ice sheet, sea ice, and variations in land cover would all have played a role.

A third possibility is that the Laurentide did not actually attain similar glacial ice lines in the early and late Pleistocene. Balco and Rovey (2010) propose that absolute dating of tills supports an expansive Laurentide only immediately after Northern Hemisphere glacial intensification and that ice-sheet advances between 2 and 0.75 Ma are not, on average, consistent with lower-latitude early Pleistocene ice margins. Greater consistency between a version of the regolith hypothesis and δ11B-based CO2 reconstructions could thus be achieved if a milder or more gradual change in basal conditions took place than those required for consistent ice lines through the entire Pleistocene. An intermediate scenario is that the early Pleistocene Laurentide might have episodically expanded to large areas, consistent with intermittent early Pleistocene meltwater events identified in Gulf of Mexico δ18O records (Joyce et al. 1993; Shakun et al. 2016).

A combination of factors is also plausible. Notably, Willeit et al. (2019) simulated ice sheets that scour regolith while also producing sea level that averages ~20% higher than in Siddall et al. (2010) across 2–1 Ma. The model retained early Pleistocene ice margins that, when averaged across longitudes, were at substantially higher latitudes than in the late Pleistocene, implying a combination of ice-sheet thickening and geographic expansion occurred across the middle Pleistocene. Higher early Pleistocene sea level was, however, achieved by way of an imposed CO2 trend and required early Pleistocene CO2 levels averaging 266 ppm that are at odds with δ11B reconstructions, especially in light of evidence that δ11B CO2 proxy values are biased modestly high.

5. Conclusions

Several climate feedbacks and nonlinearities that govern the sea level response to CO2 radiative forcing on glacial–interglacial time scales, such as ice-albedo feedbacks (Budyko 1969; Milankovitch 1941) and power-law scaling of ice-sheet volume with length (Weertman 1976), are well established in theory. Their actual manifestation in early Pleistocene climate is not obvious, however, because neither the ages nor magnitudes of sea level or CO2 are well constrained during that interval. Indeed, several foregoing attempts to reconstruct early Pleistocene CO2 use scaling constants or aggregate feedback factors (van de Wal et al. 2011; Stap et al. 2016; Berends et al. 2020) for lack of means to identify more precise relationships. Given the uncertainties, it is not surprising that explanations for changes in sea level sensitivity to CO2 forcing across the mid-Pleistocene transition (Chalk et al. 2017; Dyez et al. 2018) have not been statistically tested.

A Bayesian analysis of late Pleistocene sea level and CO2 suggests that increases in sea level sensitivity through the Pleistocene are a natural consequence of ice sheets growing more expansive. The primary implication of our result is that there need not have been a change in the processes governing the ice-sheet sensitivity to radiative forcing over the last 2 million years, but rather that feedbacks inherent to the climate amplified when ice sheets began to vary with greater amplitude after ~1 Ma. The processes involved are clarified by our having jointly inferred sea level and CO2 ages with physical parameters in our model. Inferred ages are necessarily sensitive to model restrictions and simplifications, and future work could more explicitly consider depth–age relationships in ice and sediments, but the present approach is arguably more complete than orbital-tuning methods that assume a constant phase relative to orbital forcing and do not account for CO2.

The record from Siddall et al. (2010) that we use for conditioning the model agrees broadly with multiple other Pleistocene sea level reconstructions (Bintanja and van de Wal 2008; Willeit et al. 2019; Berends et al. 2020; De Boer et al. 2010; van de Wal et al. 2011), but there are two potentially important further considerations. First, in assuming a fixed nonlinear relationship between δ18O and sea level over time, the record of Siddall et al. (2010) does not account for possible changes in this relationship that could, for example, arise if ice sheets were thinner in the early Pleistocene. For example, a shift toward lower fractionation of δ18O during ice ages would imply larger sea level changes and larger-amplitude changes in CO2.

Second, two other sea level reconstructions suggest early Pleistocene sea level could have reached 20–50 m higher than the present (Elderfield et al. 2012; Rohling et al. 2014). Given that Greenland and ice caps collectively contain approximately 8 m of sea level-equivalent ice, the records of Elderfield et al. (2012) and Rohling et al. (2014) imply substantial Antarctic melting in the early Pleistocene. Foregoing studies suggest that even in the late Pleistocene, Antarctica might have been responsible for 10%–15% of glacial–interglacial ice-volume change (Simms et al. 2019; Dutton et al. 2015). To represent these possibilities, our model could be extended to include multiple ice sheets, Southern Hemisphere insolation forcing, and a parameterization for the grounding line of marine ice sheets. Although we cannot rule out the relevance of such factors, we suggest that the excellent fit between modeled and observed CO2 during the early and late Pleistocene suggests that major additions to our model are not required.

That changes in the dynamics controlling glacial cycles, including changes in the frictional coupling at the base of the ice sheet, are unnecessary for explaining early Pleistocene CO2 does not preclude their having occurred, of course. They may have been involved in causing glacial intensification or ~100-ky cycles in the late Pleistocene, the origins of which we have not investigated in this study. A role for regolith removal in sea level sensitivity changes would, however, likely imply that early Pleistocene sea levels were higher than those reported in Siddall et al. (2010), possibly closer to those reported in Rohling et al. (2014).

The discrepant climate states implied by different sets of observations appear to present an opportunity for better constraining the Pleistocene progression of sea level and CO2. It may be possible to identify the most likely combinations of ice-sheet geographic extent, sea level, and CO2 in the early Pleistocene by extending our Bayesian framework to evaluate a more comprehensive collection of constraints. These would include, in particular, CO2 measurements from Allan Hills blue ice (Yan et al. 2019; Higgins et al. 2015), the ages and locations of dated tills (Balco and Rovey 2010), and a more comprehensive set of sea level records.

Acknowledgments

This work was supported by NSF Award 1338832. Peter Clark (Oregon State University), David Hodell (University of Cambridge), and Richard Katz (Oxford University) provided comments that helped to improve this manuscript. Three anonymous reviewers provided helpful feedback.

Data availability statement

Supplementary materials contain the sea-level-based estimate of CO2 over past 2 million years and its uncertainties, the sea-level record of Siddall et al. (2010) on the original and adjusted age-models, posterior values for all parameters, and code to produce realizations of CO2 from sea-level and model parameters. Materials are available for download on the Harvard University Dataverse repository or by contacting the corresponding author.

APPENDIX

Bayesian Algorithm

a. Sampling from the joint posterior distribution

The Bayesian methodology is written in Matlab expressly, and no external packages or software are used for inference. The approach we use to sample the physical and age parameters follows Goodman and Weare (2010) and employs an ensemble of K chains that explore the parameter space. A vector of parameter values is initiated for each of the K samplers, θk for k = 1, 2, …, K. At each iteration of the algorithm and for each k, a new proposed parameter vector, denoted θ^k, is generated and either accepted or rejected on the basis of its posterior probability. The proposal is computed by multiplying a step size, randomly drawn from a proposal distribution, with the distance between the most recent value of θk, denoted θ˜k, and the most recent value of another sampler θ˜j where j is randomly selected from the complementary set of samplers (i.e., all samplers where jk). The move is taken from the complementary sampler θ˜j:
θ˜kθ^k=θ˜j+Z(θ˜kθ˜j).
The term θ^k indicates the proposed new position of θj, and Z is a random draw from the proposal distribution g(Z = ζ), where g(ζ) must be symmetric, that is, Pr(θ˜jθ^k)=Pr(θ^kθ˜j). A commonly used proposal distribution (Goodman and Weare 2010), and the one we use here, is
g(ζ){ζ1/2ifζ[1a,a],0otherwise.
Here a is a scaling parameter that influences convergence rate but not the result and must be greater than 1. We specify a = 2.5, chosen so that jumps are sufficiently large to escape local minima but not so large as to consistently preclude acceptance. The candidate vector θ^k is accepted with probability
min{1,Zm1Pr(θ^k|[C])Pr(θ˜k|[C])},
where m is the dimensionality of the parameter space. If accepted, we set θ˜k=θ^k, otherwise we leave θ˜k unchanged.

b. Inference of the error properties

The error between observed and modeled CO2, denoted ε, is represented as a first-order autoregressive process,
εt=ρεt1+ψt,
where ψ ~ N(0, σ2). The likelihood is the multivariate normal probability density with zero mean and AR(1) covariance matrix whose entries i, j are σ2ρ|titj|/(1ρ2).
The autoregressive parameter ρ and error variance σ2 are inferred using a Gibbs sampling approach. When a candidate set of physical and age parameters is accepted in the random-step stage of the algorithm, a new value for each of the hyperparameters within the same chain is drawn from their conditional posterior distributions before proceeding to new proposals for θk. The full conditional posterior distributions for ρ and σ2 are (Tsay 2005; Gelman et al. 2013)
ρ|ε,σ2~N[1,1]([t=2nεt2t=2nεt1εt+γ1]1,[t=2nεt2σ2+γ2]1)
and
σ2|ε,ρ~Γ1(n2+γ3,2n(εtρcεt1)2/2+γ4).
The term N[−1,1] indicates truncation of the normal distribution at [−1, 1], and γ1 and γ2 are respectively the prior hyperparameters for the mean and variance of ρ; Γ−1 is the inverse-gamma distribution and β3 and β4 are prior hyperparameters.

c. Data interpolation, time steps, and processing of posterior samples

For purposes of model–data comparison, sea level and CO2 data are placed on a common time vector with 2-ky time steps. The chosen time step reflects a balance between computational efficiency, resolution of the data series, and independence of sequential points. To avoid aliasing the CO2 record, the ice-core measurements are first interpolated to a fine grid with 0.1-ky time steps and smoothed with a 2-ky low-pass filter before downsampling to the 2-ky time steps. To reduce numerical error, the step involving estimation of CO2 from sea level is solved with time steps of 0.2 ky, or 10 times the resolution of that used for calculating likelihoods from the model output and observations. If a sea level input value is outside the model domain—for example, if sea level is too high to be explained by Northern Hemisphere ice-sheet behavior alone—the model returns no solution.

The sampling algorithm is run until 3 × 105 parameter combinations are accepted. Chains are combined into a single posterior distribution, and the first 2 × 105 samples are discarded as a “burn-in” interval. Nine in every 10 samples thereafter are discarded to reduce autocorrelation between successive samples. The final joint posterior distribution retains 104 samples after removal of the burn-in interval and thinning.

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