Production of Neoproterozoic banded iron formations in a partially ice-covered ocean

Abstract

The meridional extent of marine ice during the Neoproterozoic snowball Earth events is debated. Banded iron formations associated with the Sturtian glaciation are considered evidence for a completely ice-covered, ferruginous ocean (hard snowball). Here, using an ocean general circulation model with thick sea glaciers and Neoproterozoic biogeochemistry, we find that circulation in a partially ice-covered ocean (soft snowball) yields iron deposition patterns similar to the observed distribution of Sturtian banded iron formations.

Main

Banded iron formations (BIFs) are iron- and silica-rich sedimentary rocks, the genesis of which is thought to require low oceanic O₂ concentrations, in agreement with their occurrence before ~1,800 million years ago (Ma), when Earth’s oceans are proposed to have been anoxic^1,2. An exception to this pattern of occurrence, Neoproterozoic (1,000 to 541 Ma) iron formations, were deposited when atmospheric O₂ levels were 1–10% of today’s³. Neoproterozoic BIFs occur in association with the Sturtian pan-glacial, which is the earlier and longer of two low-latitude glaciations (the ‘snowball Earth’ events^4,5,6,7). The ocean and continents are suggested to have been ice-covered during these pan-glacials that lasted for tens of millions of years^8,9. Widespread sedimentary and geochemical evidence supports the occurrence of the Neoproterozoic pan-glacials, though the meridional extent of marine ice is still debated. The Neoproterozoic BIFs have historically been suggested to support a ‘hard’ snowball^4,5, in which pole-to-pole ice cover renders the ocean anoxic, allowing the accumulation of ferrous iron (Fe²⁺). Although viable explanations for the survival of the photosynthesis-based marine biosphere through a hard snowball have been proposed¹⁰, such an extreme climate poses a serious challenge to life⁵. Alternatively, a ‘soft’ snowball⁷ with tropical swaths of open water does not pose a similar challenge to the biosphere, but O₂ input to the ocean by air–sea gas exchange may have prevented the mobility of Fe²⁺ required for widespread deposition of BIFs^10,11.

In this study, we tested the sensitivity of precipitation of BIFs to the extent of ice cover using an ocean general circulation model (GCM), which includes the representation of thick sea glaciers extending to a prescribed latitude and a new biogeochemical model tailored to the Neoproterozoic (Fig. 1; see Methods for details). In the model iron cycle, a hydrothermal source of Fe²⁺ at mid-oceanic ridges is balanced by oxidation and deposition of ferric iron (Fe³⁺) oxides, which occurs where Fe²⁺-bearing waters meet oxygenated waters. The transport of Fe²⁺ and O₂ from their respective hydrothermal and photosynthetic sources determines the location of such reaction and deposition fronts. Note that photosynthesis and O₂ production in the partially glaciated oceans that we simulate is confined to the ice-free regions and there is no primary production beneath the >100-m-thick sea glaciers. Patterns of phosphate concentration ([PO₄³⁻]) and [O₂] obtained in an idealized model configuration (Supplementary Section 1) with a 100% modern [PO₄³⁻] and a stoichiometry of organic matter oxidation (O₂ consumed to P released) representative of modern marine organic matter are consistent with today’s ocean (Supplementary Fig. 1), and atmospheric O₂ stabilizes at ~62% present atmospheric levels.

Fig. 1: Schematic representation of the biogeochemical model.

Cyanobacteria, limited by low seawater phosphate concentration ([PO₄³⁻]), produce organic matter (DOP and POP) and oxygen (O₂) in ice-free regions. Aerobic respiration of the organic matter consumes some of the O₂, as do the bacteria that oxidize ferrous iron (Fe²⁺) and produce ferric iron (Fe³⁺) oxides. The fraction of Fe³⁺ oxides that is not used by anaerobic bacteria to remineralize organic matter settles to the seabed, forming iron deposits.

Previous ocean–atmosphere–cryosphere modelling efforts have identified multiple stable climate states in which ice-free ocean exists equatorwards of latitudes between 60° and 5° (refs. ^11,12). However, not many climate models can sustain a narrow equatorial band of ice-free ocean. Accordingly, we examined a completely ice-free ocean and partially glaciated oceans in which the ice-free region extended from 30° S to 30° N and 12° S to 12° N. The exact size of the marine PO₄³⁻ pool during snowball Earth episodes is unknown, though several factors lead to expectations of lower-than-present seawater [PO₄³⁻]. These include low continental weathering rates, photosynthetic carbon fixation and nutrient uptake during the glacial advance and in regions of open water, limited organic matter remineralization in a partly ice-covered ocean, and PO₄³⁻ adsorption onto iron oxides in an ocean with relatively high Fe²⁺ concentrations. Hence, we examined iron deposition patterns at 100%, 10%, 3% and 1% the mean present-day seawater [PO₄³⁻] (3 μM), the lower three of which are within the proposed range required to sustain low Proterozoic atmospheric O₂ levels^13,14, and consistent with low Proterozoic primary productivity¹⁵. With these parameter combinations, cases in which 90% or more of the iron deposited within the model grid points corresponding to the hydrothermal Fe²⁺ sources were considered less viable for BIF deposition on multiple continental margins, as suggested by the sedimentary record¹⁰.

Our simulations show that the spatial pattern of Fe³⁺ precipitation is sensitive to both the extent of ice cover and the mean seawater [PO₄³⁻]. To elucidate Fe³⁺ precipitation in a partially glaciated ocean, we consider the 12° S to 12° N ice-free ocean snowball with a mean [PO₄³⁻] of 3% modern levels. The flux of O₂ produced (via photosynthesis) in this snowball is limited due to both a low ice-free surface area and low PO₄³⁻ availability. This small amount of O₂ is distributed within the ocean by the circulation. However, the circulation in the juvenile, narrow ocean basins that were formed by the break-up of the supercontinent Rodinia is much weaker than in other regions (Supplementary Fig. 2). Since a hydrothermal source is also located within these poorly ventilated basins, they are a suitable site for accumulation and mobility of Fe²⁺, which is oxidized at the ‘chemical reaction and deposition fronts’ in this region (Fig. 2a–f). The intersection of Fe²⁺-bearing water and O₂-bearing water at these fronts produces Fe³⁺, which is insoluble in seawater. The seawater insoluble Fe³⁺ sinks through the water column and a fraction of it is used by anaerobic bacteria to remineralize organic matter to replenish the marine PO₄³⁻ and Fe²⁺ pools, while the rest is deposited on the seafloor in the region. In contrast, we do not observe Fe²⁺ mobility around the hydrothermal vent located in the well-ventilated southeastern ocean and all the Fe²⁺ is oxidized (and precipitates) near the vent (Supplementary Fig. 3).

Fig. 2: Results of the snowball simulation with 12° S to 12° N ice-free ocean and mean seawater phosphate concentration ([PO₄³⁻]) of 0.1 μM (3% modern).

Latitude–longitude cross-section at a depth of −2.5 km (top) and latitude–depth cross-section at 166.5° E (bottom) of oxygen ([O₂]) (a,d), ferrous iron ([Fe²⁺]) (b,e) and ferric iron (Fe³⁺) precipitation rate (${{R}}_{{\mathrm{Fe}}^{3+}}$) (c,f). The solid black contours in a–c mark the sites of hydrothermal Fe²⁺ injection, based on estimated locations of mid-oceanic ridges. The dashed black lines enclose the region where [O₂] = 0 and the ‘chemical reaction front’ is adjacent to it. The dotted line in the top panels mark the 166.5° E longitude; the latitude–depth cross sections in d–f are along this longitude. The light-brown shading in a–c indicates the approximate continental configuration during the Sturtain pan-glacial¹⁸.

In partially ice-covered oceans with a mean [PO₄³⁻] that is 1–10% of today’s, the distribution of model iron deposition rates (Fig. 3a,b and Supplementary Figs. 4g,h and 5e,f) away from the hydrothermal sources are within the range of estimated Sturtian BIF deposition rates (between ~0.003 and ~0.4 mol m⁻² yr⁻¹; Supplementary Section 2). This model consistency with the observations is achieved at [Fe²⁺] approximately two orders of magnitude lower than previously thought to be required for widespread Neoproterozoic BIF deposition^16,17. Thus, we suggest that Fe²⁺ supply and mobility, not concentration, are the actual requirements for widespread BIF deposition, in the Neoproterozoic and over Earth history in general. At modern mean seawater [PO₄³⁻], most of the ocean interior is oxygenated, and >95% of the iron precipitates near the hydrothermal Fe²⁺ sources.

Fig. 3: Iron deposition rates in the snowball simulation with 12° S to 12° N and 30° S to 30° N ice-free ocean; the mean seawater phosphate concentration ([PO₄³⁻]) is 0.1 μM (3% modern).

a,b, Spatial distribution of iron deposition rates (${{R}}_{{\mathrm{Fe}}^{3+}}$) in snowballs with an ice-free ocean between 12° S to 12° N (a) and 30° S to 30° N (b), respectively. Iron deposition rates are calculated by vertical integration of the profiles corresponding to net ferrous iron (Fe²⁺) oxidation. The solid black contours mark the sites of hydrothermal Fe²⁺ injection, based on estimated locations of mid-oceanic ridges. The dotted black horizontal lines denote the extent of the ice-free region. The dashed black lines enclose the region where iron deposition rates are within the range estimated from Neoproterozoic BIFs. The stars indicate the approximate location of BIFs associated with the Sturtian pan-glacial. BIFs denoted by the grey stars have been excluded from the estimation of BIF deposition rate (Supplementary Section 2). The light-brown shading indicates the approximate continental configuration during the Sturtain pan-glacial.

Multiple BIF deposition sites emerge in the snowball simulations with 1–10% modern [PO₄³⁻] (Fig. 3a,b and Supplementary Figs. 4g,h and 5e,f), several of which are located in the deep ocean away from the continents. BIFs deposited at these deep sites are unlikely to be preserved, in contrast with the BIFs deposited on the continental margins facing the young ocean basins that formed during the break-up of Rodinia. This is consistent with the clustering of all but one of the BIFs dated to the Sturtian pan-glacial around these young ocean basins (Supplementary Fig. 6). The palaeogeographic reconstruction¹⁸ does not resolve inland seas, which host some of the Neoproterozoic BIFs, but we postulate that iron deposition is expected also in such seas that were connected to the basins represented in the model (for example, apparent inland BIF locations in Fig. 3a,b). We assessed the agreement between iron deposition patterns obtained in different simulations with the spatial distribution of Neoproterozoic BIFs (Supplementary Section 3 and Supplementary Fig. 7). By and large, in simulations with mean seawater [PO₄³⁻] between 1% and 10% modern levels, O₂-bearing and Fe²⁺-bearing water masses meet in the vicinity of Sturtian BIFs for the two partially ice-covered oceans considered. However, neither the ice-free ocean nor the partially ice-covered oceans show an affinity for precipitation of BIFs at modern [PO₄³⁻] levels, and all the Fe²⁺ is deposited close to the hydrothermal source. These results suggest that iron deposition patterns and rates consistent with the Sturtian BIFs can be obtained in partially ice-covered oceans with lower-than-present seawater [PO₄³⁻] (<~10% modern levels). Due to model uncertainties, we stop short of attempting to identify the precise [PO₄³⁻] levels and meridional ice extent that yield the most consistent iron deposition patterns.

To conclude, we contextualize our findings within the hard versus soft snowball debate. The hard snowball state^19,20,21,22 is suggested to have led to an anoxic ocean interior in which dissolved Fe²⁺ accumulated, to be oxidized and deposited only in spatially limited regions where O₂ was available, such as the interface between anoxic seawater and oxygenated meltwater plumes near the ice grounding line^10,23. Under such a scenario, production of BIFs is expected on all glaciated continental margins¹⁰, in contrast with the occurrence of Neoproterozoic BIFs only on some continental margins (Fig. 3a,b). However, fortuitous combinations of ice-sheet hydrology, meltwater residence times and subglacial bedrock geology may result in a distribution akin to the observed distribution of Sturtian BIFs even in a hard snowball. Our analysis shows that a phosphate-deficient ocean is conducive to production of BIFs for different extents of marine ice cover, wherever Fe²⁺-bearing and oxygenated water masses meet. We find several combinations of [PO₄³⁻] and marine ice cover that predict BIF deposition at multiple sites, including on the continental margins facing the young ocean basins that formed during the fragmentation of Rodinia, consistent with the observed distribution of Sturtian BIFs (Fig. 3a,b and Supplementary Figs. 4g,h and 5e,f). The coarse resolution of our model and other limitations (for example, absence of a dynamic atmosphere–ice–ocean coupling, uncertainties in seafloor topography) prevent us from predicting the meridional extent of marine ice cover during the Neoproterozoic pan-glacials. Nevertheless, our findings prompt investigations of the plausibility of a soft snowball state in more complete climate models, which include atmosphere–ocean–cryosphere coupling and a realistic continental configuration. Lastly, our findings highlight the role of Fe²⁺ fluxes into, and mobility within, Earth’s oceans as the necessary conditions for widespread BIF deposition. In contrast with previous suggestions, a high Fe²⁺ concentration is found unnecessary.

Methods

Ocean

We use the Massachusetts Institute of Technology General Circulation Model (MITgcm)²⁴ to perform the simulations discussed in this study. The MITgcm is a free, open-source GCM in which a finite volume method is implemented to solve the equations of fluid motion (that is, momentum equations, continuity equation and diffusion equations of temperature and salinity and equation of state). We employ the full equation of state provided in the MITgcm, MDJWF²⁵, and make the hydrostatic and Boussinesq approximations. The model uses a spherical grid spanning 72° S to 72° N with a resolution of 3° in latitude and has 37 vertical levels spanning a depth of 3,000 m (height of vertical levels from the surface to the bottom: 10 m, 12 m, 16 m, 20 m, 25 m, 32 m, 40 m, 50 m, 60 m, 75 m, 80 m, 2 levels × 90 m, and 24 levels × 100 m). The longitudinal resolution is 3° as well, with 20 grid points in the idealized box configuration simulations and 120 grid points in the Neoproterozoic simulations. Unresolved eddies in the ocean are parameterized using the Gent-McWilliams/Redi (GM-Redi) scheme^26,27, and the background diffusion coefficient is set to 1,000 m² s⁻¹. The lateral and vertical viscosities are set to 5 × 10⁵ m² s⁻¹ and 10⁻³ m² s⁻¹, and the vertical diffusivity and implicit vertical diffusivity for convection are set to 10⁻⁴ m² s⁻¹ and 10 m² s⁻¹. We also use a non-dimensional grid-dependent biharmonic viscosity of 0.1 to suppress the grid-scale noise. The vertical diffusivity for biogeochemical variables is set to 3 × 10⁻⁵ m² s⁻¹. The third-order direct space-time flux limiter advection scheme (no. 33) is used for both momentum tracers and biogeochemical variables. The flat bathymetry ocean is subjected to a constant geothermal heat flux of 0.1 W m⁻² (ref. ²⁸). At the surface, the ocean is forced by a zonal wind stress that changes for the different meridional extent of the sea glaciers. The wind stress field for the idealized box configuration simulations is similar to the present-day, and for ice-free Neoproterozoic simulation is the zonally averaged wind stress obtained from fully coupled ocean–atmosphere simulation of Cambrian–Precambrian boundary²⁹. The wind stress fields for our snowball simulation conditions (spatial grid and ice opening) are generated by setting the wind stress over the ice-covered region to zero and using zonally averaged wind stress profiles over the ice-free region that are based on results from simulations of partially glaciated climate states carried out on fully coupled GCMs^30,31. The temperature at the surface (z = 0 m) is restored to temperature fields that are also generated on the basis of coupled model results^11,12. Supplementary Fig. 8a,b shows the zonally averaged restoration temperature and wind stress over the ocean for different meridional extents of sea glaciers. Supplementary Fig. 8c shows the evaporation minus precipitation (E − P) fields over the ocean in the box simulations and in the Neoproterozoic ice-free simulations, these idealized E − P fields are generated by introducing a small asymmetry to the E − P field described in ref. ³². In both the snowballs, melting under the sea glaciers is balanced by evaporation in the ice-free region, which conserves the mean salinity of the ocean.

Sea glaciers

Given the uncertainties regarding the thickness of ice during the Neoproterozoic glaciations, we estimate ice thickness using a simplified energy balance. A similar calculation has been used to estimate the thickness of ice on Jupiter’s moon Europa³³. We begin by assuming that a constant geothermal flux of 0.1 W m⁻² is applied to the ocean bottom and that the component of tidal heating in both the ice and ocean is negligible. Moreover, the temperature within the ice varies linearly with depth. In this case, the thickness of ice is given by

$$h={\rho }_{{{\mathrm{i}}}}{c}_{{\mathrm{p}},{{\mathrm{i}}}}\kappa \frac{{T}_{{{\mathrm{f}}}}-{T}_{{{\mathrm{i}}}}}{Q},$$

(1)

where ρ_i is the density, ${c}_{{\mathrm{p}},{\mathrm{i}}}$ is the heat capacity and κ is the temperature diffusion constant of ice. T_f and T_i are the freezing and surface temperature of the ice, and Q is the internal heat flux (the geothermal heat flux). T_f is a function of salinity and pressure:

$${T}_{{\mathrm{f}}}=273.16+0.0901-0.0575\times S-7.61\times {10}^{-8}\times {P}_{{\mathrm{b}}},$$

(2)

where S is the salinity and P_b = ρ_igh is the pressure at the bottom of the ice. The gradient in solar insolation yields a gradient in the surface temperature of the ice, and in turn, a gradient in ice thickness. This gradient in ice thickness causes it to flow smoothly, thereby partially relaxing the gradient³⁴. Hence, we approximate the thickness of the ice as the meridional average of h calculated using the prescribed temperature field. Under these assumptions, the ice thickness can iteratively be estimated as:

$${\langle h\rangle }_{{\mathrm{j}}+1}={\rho }_{{\mathrm{i}}}{c}_{{\mathrm{p}},{\mathrm{i}}}\kappa \frac{{\langle {T}_{{\mathrm{f}}}\rangle }_{{\mathrm{j}}}-\langle {T}_{{\mathrm{i}}}\rangle }{Q}.$$

(3)

The algorithm³³, when initialized from the typical seawater freezing temperature of 271 K and the surface temperature fields such as the ones given in refs. ^11,12, predicts a certain ice thickness, which implies a certain P_b value and a modified freezing temperature. For the temperature fields shown in Supplementary Fig. 8a, the ice thickness converges to 490 m and 253 m below the sea surface within a few iterations for snowballs with 12° S to 12° N ice-free ocean and 30° S to 30° N ice-free ocean, respectively. To avoid any numerical artefacts that may arise from partially filled grid cells, we use a thickness of 510 m and 265 m below the sea surface for the snowballs with 12° S to 12° N ocean and 30° S to 30° N ocean, respectively. The effect of this sea glacier is simulated using the SHELFICE package³⁵. Unlike the SEAICE package, the SHELFICE package in the MITgcm can handle a thick sea glacier spanning over multiple vertical levels, which makes it better suited to snowball Earth modelling.

Biogeochemical cycle

The biogeochemical model presented here is based on refs. ^36,37, implemented in the MITgcm using a combination of the GCHEM and DIC packages³⁸. MITgcm’s biogeochemical setup (GCHEM + DIC), in its default configuration, considers coupled cycles of carbon, oxygen, phosphorus and alkalinity. These cycles are expressed using five variables that do not affect the physical circulation: dissolved inorganic carbon (DIC), alkalinity (Alk), phosphate (${{\rm{PO}}}_{4}^{3-}$), dissolved organic phosphorus (DOP) and oxygen (O₂). The velocity fields and eddy diffusivities calculated by the physical model are used to transport these compounds, which are additionally produced and consumed locally by biogeochemical reactions, as described below.

For this study, we focus on the phosphorus and oxygen cycles. Additionally, we incorporate an iron (Fe²⁺) cycle and a well-mixed atmospheric box to the pre-existing setup. The atmospheric box only keeps track of the total O₂ in the atmosphere. The values of all parameters described below are provided in Supplementary Table 1. The sources and sinks for each tracer are summarized below:

$$\frac{\partial }{\partial t}\left[{{{\mathrm{PO}}}}_{4}^{3-}\right]=-{J}_{{{{\mathrm{production}}}}}-\frac{\partial }{\partial {\mathrm{z}}}{F}_{{\mathrm{P}}}+{\kappa }_{{{{\mathrm{remin}}}}}\left[{{\mathrm{DOP}}}\right],$$

(4)

$$\frac{\partial }{\partial t}\left[{{\mathrm{DOP}}}\right]={f}_{{{\mathrm{DOP}}}}\,{J}_{{{{\mathrm{production}}}}}-{\kappa }_{{{{\mathrm{remin}}}}}\left[{{\mathrm{DOP}}}\right],$$

(5)

$$\begin{array}{l}\frac{\partial }{\partial t}\left[{\mathrm{O}}_{2}\right]=\frac{{k}_{{\mathrm{w}}}}{\Delta {z}_{{{{\mathrm{surf}}}}}}\left(\frac{{n}_{{{\mathrm{O}}}_{2}}^{{{{\mathrm{model}} \; {\mathrm{atmosphere}}}}}}{{n}_{{{\mathrm{O}}}_{2}}^{{{{\mathrm{present}} \; {\mathrm{day}}}}}}{\left[{{\mathrm{O}}}_{2}\right]}_{{{sat}}}-{\left[{{\mathrm{O}}}_{2}\right]}_{{{{\mathrm{surf}}}}}\right)\\\qquad\qquad+f\left(\left[{{\mathrm{O}}}_{2}\right]\right)\times \left({r}_{{{\mathrm{O}}}_{2}:\mathrm{P}}\frac{\partial }{\partial t}\left[{{{\mathrm{PO}}}}_{4}^{3-}\right]\right)-{k}_{{{\mathrm{Fe}}}}\left[{{{\mathrm{Fe}}}}^{2+}\right],\end{array}$$

(6)

$$\frac{\partial }{\partial t}\left[{{{\mathrm{Fe}}}}^{2+}\right]={F}_{{{\mathrm{Fe}}}}^{{{{\mathrm{hydrothermal}}}}}-{k}_{{{\mathrm{Fe}}}}\left[{{{\mathrm{Fe}}}}^{2+}\right]-\frac{\partial }{\partial z}{F}_{{{\mathrm{Fe}}}}^{{{{{\mathrm{recycle}}}}}},$$

(7)

where square brackets denote concentration, J_production is primary production. During production, a fraction of ${{\rm{PO}}}_{4}^{3-}$ utilized by phytoplankton is converted to DOP (f_DOP) and the rest (1 − f_DOP) is converted to particulate organic phosphorus (POP); κ_remin is the rate constant of remineralization of DOP to ${{\rm{PO}}}_{4}^{3-}$ and $\frac{\partial }{\partial z}{F}_{{\mathrm{P}}}$ is the partial derivative with depth of the POP sinking flux that is remineralized at different rates during aerobic and anaerobic respiration (see below). The gas transfer velocity for O₂ is given by k_w, and Δz_surf is the thickness of the surface layer. The moles of O₂ in the model atmosphere and the present-day atmosphere are given by ${n}_{{{\mathrm{O}}}_{2}}^{{{{\mathrm{model} \; {\mathrm{atmosphere}}}}}}$ and ${n}_{{{\mathrm{O}}}_{2}}^{{{{\mathrm{present}}}}\;{{{\mathrm{day}}}}}$ respectively, and ${\left[{{\mathrm{O}}}_{2}\right]}_{{{{\mathrm{sat}}}}}$ and ${\left[{{\mathrm{O}}}_{2}\right]}_{{{{\mathrm{surf}}}}}$ are the saturation and surface concentration of O₂, respectively. The function $f\left(\left[{{\mathrm{O}}}_{2}\right]\right)=1$ for $\left[{{\rm{O}}}_{2}\right] > 0$ and 0 otherwise. The number of moles of O₂ consumed to remineralize 1 mole of the typical Neoproterozoic marine biomass is given by $-{r}_{{{\mathrm{O}}}_{2}:{\mathrm{P}}}$ (see below).

${F}_{\mathrm{Fe}}^{{{{\mathrm{hydrothermal}}}}}$ is the hydrothermal influx of Fe²⁺ to the ocean, k_Fe is the oxidation rate of Fe²⁺ to Fe³⁺, which accounts for both abiotic and microbial pathways (described below), and $\frac{\partial }{\partial z}{F}_{{{\mathrm{Fe}}}}^{{{{\mathrm{recycle}}}}}$ is the partial derivative with depth of the particulate Fe³⁺ sinking flux, which gives the rate at which particulate Fe³⁺ is recycled back to Fe²⁺ by anaerobic remineralization of organic matter. The tendency term for the number of moles of O₂ in the model atmosphere is

$$\begin{array}{l}\frac{\mathrm{d}}{{\mathrm{d}t}}{n}_{{\mathrm{O}}_{2}}^{{{{\mathrm{model}}}\; {{\mathrm{atmosphere}}}}}\\=-\xi \left({k}_{{\mathrm{w}}}{A}_{{{{\mathrm{surf}}}}}\left(\frac{{n}_{{{\mathrm{O}}}_{2}}^{{{{\mathrm{model}}}\; {{\mathrm{atmosphere}}}}}}{{n}_{{{\mathrm{O}}}_{2}}^{{{{\mathrm{present}}}}\;{{{\mathrm{day}}}}}}{\left[{{\mathrm{O}}}_{2}\right]}_{{{{\mathrm{sat}}}}}-{\left[{{\mathrm{O}}}_{2}\right]}_{{{{\mathrm{surf}}}}}\right)-{\varPhi }_{{{{\mathrm{red}}}}}\right),\end{array}$$

(8)

where $\xi ={10}^{4}$ is an arbitrary constant introduced to accelerate the convergence of the atmospheric box to a steady state, A_surf is the area of the ocean surface, Φ_red is the (mole equivalent) rate of consumption of O₂ required to balance the volcanic flux of reduced gases into the atmosphere (for example, SO₂, H₂S, CO, CH₄ and H₂). Other oxygen sinks relevant to the present-day Earth system, like oxidative weathering of the continents and respiration by the terrestrial biosphere, are considered negligible during a global glaciation.

J_production, which is a function of the light (L) and $\left[{{\rm{PO}}}_{4}^{3-}\right]$, is given by

$${J}_{{{{\mathrm{production}}}}}=\alpha \frac{L}{L+{\kappa }_{{\mathrm{L}}}}\frac{\left[{{{\mathrm{PO}}}}_{4}^{3-}\right]}{\left[{{{\mathrm{PO}}}}_{4}^{3-}\right]+{\kappa }_{{{{\mathrm{PO}}}}_{4}^{3-}}},$$

(9)

where κ_L and ${\kappa }_{{{{\mathrm{PO}}}}_{4}^{3-}}$ are Michaelis–Menten-type half-saturation constants typical of ocean biology, and α is the maximum community production. The expression for L is

$$L={f}_{{{{\mathrm{PAR}}}}}{Q}_{{{{\mathrm{SW}}}}}\times \exp \left(-{kz}\right),$$

(10)

where f_PAR is the fraction of shortwave radiation (SW) available for photosynthesis and k is the light attenuation constant. Since the light available for photosynthesis declines exponentially with depth, most of the biological activity in the ocean is close to the surface. Recall that primary production converts a fraction of ${{\rm{PO}}}_{4}^{3-}$ to DOP (f_DOP), which can be advected by the physical circulation in the ocean, while the rest is POP, which sinks³⁹. The downwards sinking flux of POP is given by

$${F}_{{{\mathrm{POP}}}}=\left\{\begin{array}{c}\left(1-{f}_{{{\mathrm{DOP}}}}\right){\int }_{0}^{{z}_{{{\mathrm{i}}}}}{J}_{{{{\mathrm{production}}}}}{\mathrm{d}z\; {\mathrm{if}}\; z} < {z}_{{{\mathrm{c}}}}\\ \left(1-{f}_{{{\mathrm{DOP}}}}\right){\int }_{0}^{{z}_{{{\mathrm{c}}}}}{J}_{{{{\mathrm{production}}}}}{\mathrm{d}z\; {\mathrm{if}}\; z}\ge {z}_{{\mathrm{c}}}\end{array},\right.$$

(11)

where z_i is a reference depth below which the downwards flux of POP is calculated and z_c is the compensation depth. In present-day ocean models, POP is remineralized according to a power law relation related almost entirely to aerobic respiration⁴⁰. Additionally, our model accounts for remineralization of POP by anaerobic microbes that reduce Fe³⁺ particles (see below).

As was previously mentioned, Fe²⁺ in the ocean can be oxidized by abiotic and microbial pathways, both O₂-related and light driven. The Fe²⁺ oxidation rate accounting for both abiotic and microbial processes is

$${k}_{{{\mathrm{Fe}}}}=\frac{{k}_{{{\mathrm{Fe}}}}^{{{{\mathrm{ab}}}}}}{1-{f}_{{{{\mathrm{bio}}}}}},$$

(12)

where ${k}_{{{\mathrm{Fe}}}}^{{{{\mathrm{ab}}}}}$ is the abiotic rate constant of oxidation and f_bio is the fraction of microbial Fe²⁺ oxidation out of the total oxidation as a function of available [O₂] (ref. ⁴¹); ${{k}}_{{\rm{Fe}}}^{{{\mathrm{ab}}}}$ is given by

$${k}_{{{\mathrm{Fe}}}}^{{{{\mathrm{ab}}}}}={k}_{1}\left[{{\mathrm{O}}}_{2}\right]{\left[{\mathrm{OH}}^{-}\right]}^{2}$$

$$\log {k}_{1}=21.56-\frac{\text{1,545}}{T}-3.29\sqrt{I}+1.52I,$$

(13)

where T is the temperature (in kelvin), $I=\frac{19.992{S}}{{10}^{3}-1.005{S}}$ and S is the salinity of seawater⁴². The fraction of microbial Fe²⁺ oxidation is given by

$${f}_{{{{\mathrm{bio}}}}}=1.074\times {10}^{-3}\times \exp \left(-3.446\log \left(\frac{\left[{{\mathrm{O}}}_{2}\right]}{{K}_{{{\mathrm{H}}}}}\right)\right),$$

(14)

where K_H is Henry’s law constant for O₂:

$${K}_{{{\mathrm{H}}}}=1.3\times {10}^{-3}\times \exp \left(\text{1,500}\left(\frac{1}{T}-\frac{1}{298.15}\right)\right).$$

(15)

f_bio is capped at 0.95, an upper bound based on experimental results⁴¹. To allow model timesteps that are long enough so that convergence is reached over reasonable runtimes, chemical rate constants could not be too large. Thus, we prescribed an upper limit on k_Fe of 10⁻⁵ s⁻¹. This upper limit was applied wherever O₂ concentrations were high enough to result in k_Fe that was too large to allow efficient model timesteps, typically in the well-oxygenated regions in the ocean (for example, near the surface in the snowball with 12° S to 12° N ocean). The upper limit of k_Fe was prescribed in ~68% of the ocean volume in the snowball simulation with 12° S to 12° N ocean with 1% modern [${{\rm{PO}}}_{4}^{3-}$], in >90% of the ocean volume for a lower ice cover or higher [${{\rm{PO}}}_{4}^{3-}$]. The model results were relatively insensitive to this choice of an upper limit on k_Fe (Supplementary Fig. 9).

The downwards flux of Fe³⁺ produced from oxidation of Fe²⁺ is given by

$${F}_{{{\mathrm{Fe}}}}^{{\rm{down}}}={\int }_{0}^{{z}_{{u}}}{k}_{{{\mathrm{Fe}}}}[{{{\mathrm{Fe}}}}^{2+}]{\mathrm{d}z},$$

(16)

where z_u is the reference up to which the downwards flux of Fe³⁺ is calculated. We assume that a fraction of sinking Fe³⁺ is preserved (f_preserved) while the rest is remineralized by anaerobic microbes. This process restores some Fe²⁺ and is summarized by the following reaction:

$$4{{\rm{Fe}}}^{3+}+{{\rm{CH}}}_{2}{\rm{O}}+5{{\rm{OH}}}^{-}\to 4{{\rm{Fe}}}^{2+}+{{\rm{HCO}}}_{3}^{-}+3{{\rm{H}}}_{2}{\rm{O}}.$$

The anaerobic reduction of Fe³⁺ by bacteria follows Michaelis–Menten-type kinetics with respect to the concentrations of both organic matter and Fe³⁺:

$$\frac{d}{{dt}}\left[{\mathrm{Fe}}^{3+}\right]=\mu \min \left(\frac{[{\mathrm{Fe}}^{3+}]}{{\kappa }_{{{\mathrm{Fe}}}}+[{\mathrm{Fe}}^{3+}]},\frac{[{\mathrm{C}}]}{{\kappa }_{{\mathrm{C}}}+[{\mathrm{C}}]}\right),$$

(17)

where μ is the typical respiration rate, and [Fe³⁺] and [C] are the concentrations of Fe³⁺ and organic matter, respectively. The relevant half-saturation constants for Fe³⁺ and C are κ_Fe and κ_C, respectively, and the smaller of the saturation terms is considered to limit the Fe³⁺ reduction rate. Assuming a typical C:P ratio of 100:1 yields

$$\frac{\mathrm{d}}{{\mathrm{d}t}}\left[{\mathrm{Fe}}^{3+}\right]=\mu \min \left(\frac{\left[{\mathrm{Fe}}^{3+}\right]}{{\kappa }_{{{\mathrm{Fe}}}}+\left[{\mathrm{Fe}}^{3+}\right]},\frac{\left[{{\mathrm{POP}}}\right]}{\frac{{\kappa }_{{\mathrm{C}}}}{100}+\left[{{\mathrm{POP}}}\right]}\right),$$

(18)

and

$$\frac{\mathrm{d}}{{\mathrm{d}t}}\left[{{\mathrm{POP}}}\right]=\frac{\mu }{400}\min \left(\frac{\left[{\mathrm{Fe}}^{3+}\right]}{{\kappa }_{{{\mathrm{Fe}}}}+\left[{\mathrm{Fe}}^{3+}\right]},\frac{\left[{{\mathrm{POP}}}\right]}{\frac{{\kappa }_{{\mathrm{C}}}}{100}+\left[{{\mathrm{POP}}}\right]}\right).$$

(19)

The factor of 400 comes from the Fe:C:P stoichiometry of the reaction. To calculate the flux of recycled Fe³⁺ (${F}_{{{\mathrm{Fe}}}}^{{{{\mathrm{recycle}}}}}$) and the effect of Fe³⁺ reduction on the vertical POP remineralization profile, we further assume that [POP] and $[{\rm{F}}{{\rm{e}}}^{3+}]$ are proportional to F_P and ${F}_{{{\mathrm{Fe}}}}^{{{{\mathrm{down}}}}}$ respectively. If ${F}_{{{\mathrm{Fe}}}}^{{{\mathrm{down}}}} >\left(\right.{F}_{{\mathrm{P}}}$/400), the fluxes of recycled Fe³⁺ and POP are given by:

$${F}_{{\mathrm{P}}}(z)=\left\{\begin{array}{l}{F}_{{\mathrm{POP}}}{\left(\frac{z}{{z}_{{{\mathrm{i}}}}}\right)}^{-\left({a}_{{{\mathrm{remin}}}}+\frac{\mu }{4{\kappa }_{{\mathrm{C}}}\times a}\right)}\,\mathrm{if}\,{F}_{{\mathrm{P}}} < w(z)({\kappa }_{{\mathrm{C}}}/100)\\ {F}_{{\mathrm{POP}}}{\left(\frac{z}{{z}_{{\mathrm{i}}}}\right)}^{-{a}_{{\mathrm{remin}}}}\,{\mathrm{otherwise}} \end{array}\right. ,$$

(20)

and

$${F}_{{{\mathrm{Fe}}}}^{{{{\mathrm{recycle}}}}}\left(z\right)=\left(1-{f}_{{{{\mathrm{preserved}}}}}\right){F}_{{{\mathrm{Fe}}}}^{{{{\mathrm{down}}}}}-\mu \frac{{F}_{{\mathrm{P}}}\left(z\right)}{{w(z)({\kappa }_{{\mathrm{C}}}/100)+F}_{{\mathrm{P}}}\left(z\right)}.$$

(21)

Similarly, when ${F}_{\mathrm{Fe}}^{{\mathrm{down}}} < ({F}_{\mathrm{P}}/400)$, the fluxes of recycled Fe³⁺ and POP are given by

$${{F}_{{\mathrm{P}}}\left(z\right)=F}_{{{\mathrm{POP}}}}{\left(\frac{z}{{z}_{{{\mathrm{i}}}}}\right)}^{-{a}_{{{{\mathrm{remin}}}}}},$$

(22)

and

$${F}_{{\mathrm{Fe}}}^{{{\mathrm{recycle}}}}(z)=\left\{\begin{array}{l}(1-{f}_{{{\mathrm{preserved}}}}){F}_{\mathrm{Fe}}^{{{\mathrm{down}}}}{\left(\frac{z}{{z}_{{{\mathrm{i}}}}}\right)}^{-\frac{\mu }{a{\kappa }_{{\mathrm{Fe}}}}}\,\mathrm{if}\,{F}_{{\mathrm{Fe}}}^{{{\mathrm{down}}}} < w(z)\,{\kappa }_{{\mathrm{Fe}}}\\ (1-{f}_{{{\mathrm{preserved}}}}){F}_{{\mathrm{Fe}}}^{{{\mathrm{down}}}}\,{\mathrm{otherwise}}\end{array}.\right.$$

(23)

Here z_i is a reference depth below which the downwards flux of POP is calculated (note that z_i = z_c when z > z_c), a is the vertical gradient of particle sinking velocity such that w(z) = az yields the speed of the sinking particles at depth z, and a_remin is the exponent in Martin’s curve for POP remineralization in the modern ocean. Note that even though F_P and F_POP have the same units, the quantities are not the same. F_POP is the depth-dependent downwards sinking flux of POP. On the other hand, F_P is the depth-dependent POP remineralization flux, which can change based on whether remineralization of organic matter is anaerobic or aerobic.

The Fe²⁺ influx to the ocean from axial and off-axis hydrothermal circulation depends on the Fe²⁺ concentration in the fluid and the total water fluxes. The concentration of Fe²⁺ in axial and off-axis hydrothermal fluids is estimated to be in the range (0.75–6.5) × 10⁻³ mol kg⁻¹ and (0.4–6.0) × 10⁻⁶) mol kg⁻¹, respectively^43,44,45. Assuming that the Fe²⁺ concentration is uniformly distributed in these ranges and the axial and off-axis water fluxes follow the distributions shown in ref. ⁴⁶, we generated probability distributions corresponding to the axial and off-axis influx of hydrothermal Fe²⁺ to the ocean (Supplementary Fig. 10a,b). Separate from hydrothermal Fe²⁺ fluxes, weathering of seafloor basalt can deliver Fe²⁺ to the ocean^47,48. The Fe²⁺ flux from seafloor weathering is rather difficult to constrain in the well-oxygenated modern ocean but it is estimated to be ~0.1–0.3 Tmol yr⁻¹ and as high as 1.0 Tmol yr⁻¹ (ref. ⁴⁹), though it is unclear that all the Fe²⁺ lost from the basalt actually reaches the ocean. Given the existing estimates, we adopt a weathering Fe²⁺ flux uniformly distributed between 0.1 and 0.3 Tmol yr⁻¹ (Supplementary Fig. 10c).

Convolving the distributions of axial and off-axis hydrothermal fluxes and seafloor weathering flux yields a range of Fe²⁺ influxes between 0.2 and 0.67 Tmol yr⁻¹ (95% confidence interval) with a mode of 0.36 Tmol yr⁻¹ (Supplementary Fig. 10d). Though some have suggested higher hydrothermal fluxes in Earth’s deep past due to higher radiogenic heat production in the mantle⁵⁰, it remains unclear whether the higher heat production led instead to more sluggish seafloor spreading⁵¹ and a hydrothermal flux like the Phanerozoic. We do not upscale the hydrothermal water flux, leading to conservatively low Fe²⁺ influx estimates. Following the above analysis, we distribute the mode Fe²⁺ influx of 0.36 Tmol yr⁻¹ on the model grid points corresponding to the suggested locations of seafloor spreading centres during the Sturtian glaciation. Moreover, we do not taper the Fe²⁺ influx with increasing distance from the spreading centres, since both near-axis and off-axis hydrothermal inputs occur close to the spreading centres (considering the spatial resolution of the ocean model), and the dissolved products of seafloor weathering are expected to reach the ocean mostly where the sedimentary cover on the basaltic crust is thin to absent (that is, near the spreading centres).

The marine phytoplankton community during the Neoproterozoic primarily composed of cyanobacteria. Previous studies^52,53,54,55 show that the metabolite content of typical cyanobacterial biomass is 51% protein, 16% carbohydrates, 23% lipid and 10% nucleic acid, which is slightly different from the average composition of modern phytoplankton (which includes diatoms, coccolithophores, dinoflagellates and cyanobacteria): 54% protein, 26% carbohydrates, 16% lipid and 4% nucleic acid⁵⁶. Despite these differences in the metabolite content, the oxidation state of cyanobacterial biomass is not substantially different from the gross modern assemblage (we estimated <1% variation in H:C_org and <10% variation in O:C_org ratios), which suggests that the difference between the Neoproterozoic and present-day O₂:P ratio is expected to arise from a difference between the C:P ratios of cyanobacteria and the modern assemblage and not from the difference in the carbon oxidation state. Using the mean C:N:P ratio of 152:25:1 for cyanobacteria⁵⁷, we calculate the following elemental composition of typical Neoproterozoic plankton: C₁₅₂H₂₄₇O₆₁N₂₅P. As per this mean composition, the ${{r}}_{{{\rm{O}}}_{2}:{\rm{P}}}$ ratio is estimated to be −216, which is ~45% more negative than its present-day value of −150 (ref. ⁵⁶). In other words, 45% more O₂ is consumed per mole of organic phosphorus liberated during organic matter remineralization.

As mentioned above, the majority of O₂ sinks on the present-day Earth were negligible during the Neoproterozoic snowball Earth events. However, reducing volcanic emissions (SO₂, H₂S, CO, CH₄ and H₂) can be an important sink for O₂ in a globally glaciated climate (Supplementary Table 2). All listed fluxes are approximate but can be used to reasonably estimate the equivalent moles of O₂ consumed annually by oxidation of the reduced gases.