Modeling studies indicate a net ablation zone at lower latitudes on Snowball Earth, where annual mean sublimation exceeds precipitation . This would have exposed sublimating bare sea ice under extremely cold temperatures for thousands of years during the global glaciation events of the late Neoproterozoic era (715–635 Ma). At sufficiently low temperatures, the salt in seawater becomes saturated and precipitates in crystal form . In the net ablation zone on Snowball Earth, precipitated crystals at the surface could have accumulated a lag deposit as the ice sublimated . Field studies and laboratory experiments have shown that salt crystals have a high albedo, which can result in a salt-albedo feedback mechanism . The effects of this salt-albedo feedback have not previously been implemented in a climate model for Snowball Earth.

We implemented a simplified salt-albedo feedback parameterization in a diffusive one-dimensional energy balance model (EBM) and solved it using analytical methods. The EBM describes the zonal mean surface temperature of the Earth, and its key features are meridional heat distribution through diffusion, the Planck response and a nonlinear albedo feedback. This is a well-known model that has been extensively studied in the literature e.g.. Using spherical coordinates, the model may be formulated as 1C∂T∂t-D1sin⁡x∂∂xsin⁡x∂T∂x+BT=Qs(x)(1-a(x,T))-A where x is the co-latitude, ranging from x=0 at the North Pole to x=π/2 at the Equator. Here C is the heat capacity of the atmosphere–ocean column, Q is solar irradiance and the diffusion coefficient D scales the meridional heat transport. Solar radiation is distributed across latitudes according to the function s(x), and energy is radiated into outer space through the parametrization A+BT . Additionally, the solution is subject to no-flux boundary conditions .

We adapted the albedo function a(x,T(x)) to include both an ice-albedo and a salt-albedo feedback mechanism. Following the standard approach for EBMs, we let latitudes with a zonal mean temperature below a specific isotherm Ts have an ice/snow cover albedo value denoted α2. This will be separate from the albedo value of ice-free latitudes denoted α1. The extent of the net ablation zone on Snowball Earth is denoted by the latitude r1. We let the snow/ice albedo take on a lower value α3 equatorward of r1 to reflect that of bare sea ice. This dynamic, known as the Jormungand mechanism, has been shown to produce Waterbelt states, i.e., a Snowball Earth with a tropical water belt, in several climate models . However, it is uncertain whether the Snowball Earth events could have been a Waterbelt state . We let the surface take a salt albedo value α4 for temperatures below some eutectic temperature Te in the bare sea ice region. The eutectic temperature refers to the temperature at which the salt in seawater becomes solid and starts to precipitate, and is about -36 °C for the entire seawater system . A lag deposit of precipitated salt crystals is expected to form at the surface when the sea ice sublimates . This dynamic only operates in the net ablation zone. We initially set the eutectic temperature to Te=-36 °C. The albedo of hydrohalite salt crystals is extremely high at α4=0.93 , and we used this value throughout this study. Other model parameters are given in Table . Mathematically, the proposed albedo function takes the form: 2a(x,T)=α1,T>Ts(ice-free latitudes)α2,T≤Ts,x<r1(ice/snow cover; poleward ofr1)α3,Te<T≤Ts,x≥r1(bare sea ice; equatorward ofr1)α4,T≤Te,x≥r1(salt deposit) Here r1 refers to the latitude in spherical coordinates corresponding to the geographic latitude (in degrees north) listed in Table .

Equilibrium solutions to Eq. (), with the albedo in Eq. (), were found using boundary integrals. Here we briefly outline the main procedures of the method and how it is adapted to handle the salt-albedo feedback parametrization.

We non-dimensionalized Eq. () and introduced the dimensionless, equilibrium temperature field u(x)=1/TsT(x), where Ts is the ice line temperature. In , it was shown that u(x) satisfy the following relation on some interval (x1,x2) in the domain: 3u(x)=∫x1x2dssin⁡sg(s,x)f(s,u)+g(s,x)sin⁡s∂u∂x(s)-u(s)sin⁡s∂g∂x(s,x)|s=x1s=x2, where g(x,s) is a Green's function associated with the problem and 4f(x,u)=QTsDs(x)(1-a(x,u(x)))-ATsD. Here the albedo function a(x,u(x)) takes dimensionless temperatures and the input is scaled to give the same albedo response as the function in Eq. (). We used the Green's function from . The primary objective for any method for solving diffusive, one-dimensional EBMs with a step-function albedo response at the ice line, is to locate the ice line. In the model, the defining property of the ice line latitude, denoted xs, is that T(xs)=Ts, and it represents the transition between snow/ice and open ocean. We let the salt-albedo feedback take on a similar temperature response associated with the eutectic temperature Te. The salt-albedo feedback can therefore introduce a salt line, which is the transition latitude between salt cover and bare sea ice within the bare sea ice region equatorward of r1. We let xe denote the salt line latitude. Mathematically, the defining property of the salt line is that T(xe)=Te. We located the ice line and the salt line by developing a set of boundary integral equations. Details are left out here, but the approach was the same as in and .

To understand the effects of the salt-albedo feedback mechanism on the system, we drew bifurcation diagrams for the equilibrium solutions to Eq. (). Figure  shows a bifurcation diagram obtained using the model parameters from Table . Following , we let the bifurcation parameter be A, which controls the outgoing longwave radiation. The x axis in Fig.  show a deviation in the bifurcation parameter from the reference value A0 given in Table , such that radiative forcing ΔA=A0-A. This represents forcing in the model as a reduction in A results in lower outgoing energy, e.g. due to increased atmospheric greenhouse gas concentrations. We chose to display the equatorial temperature on the y axis of the bifurcation diagram. In the literature on EBMs, bifurcation diagrams typically show ice line latitude on the y axis. However, in this study, we focused on alternative Snowball Earth states, and all states below the point b2 in the bifurcation diagram in Fig.  are fully ice-covered. To differentiate these states, we represented them using their equatorial temperature. Note that this projection partially obscures the small ice cap instability, and the branch labeled “High-latitude ice line” in Fig.  is not directly connected to the “Hothouse”-branch. Stability was determined using the slope stability theorem , which applies to the classical two-ice-line solutions of Eq. (), and through simulations performed with the finite-difference code from . The transitions shown in Fig.  were verified using finite-difference simulations.

The salt-albedo feedback gave rise to new climatic states not previously seen in modeling studies (Fig. ). It is well established that the EBM for an idealized aquaplanet has three stable states: the ice-free hothouse state, the high-latitude ice line state and the Snowball Earth state . We prescribe a lower bare sea ice albedo in the net ablation zone, which gives the EBM Waterbelt solutions as well . The salt-albedo feedback most notably gives rise to two new, stable, coexisting Snowball Earth states: a Snowball Earth with a salt deposit (<b6 in Fig. ) and a Snowball Earth without a salt deposit (b2-b3). The latter is characterized by global ice cover and equatorial bare sea ice. The salt-deposit Snowball Earth state is likewise fully ice covered, but with a salt deposit on the bare sea ice. In addition, several unstable states appeared, with two smaller salt deposits on the bare sea ice (b3-b4 and b5-b6 in Fig. ) or with more than two deposits (b4-b5). We will not focus on these states here.

The presence of two new Snowball Earth states raises the question of which state is more likely to represent the climate of the Snowball Earth events in the Late Neoproterozoic. To assess this, we adopted an approach previously applied to assess Waterbelt states as an explanation to Neoproterozoic geology . Namely, that the state had to be accessible from warm climates (Fig. ) and had to exist over a large range of atmospheric CO2 concentrations. This is to determine which of the two new states is consistent with the well-established hysteresis loop associated with the Snowball Earth events . That is, from a warm climate to Snowball Earth and back again following a substantial increase in atmospheric CO2 concentrations. The peculiar aspect of the Neoproterozoic Snowball Earth events is the lack of evidence for ice sheets in 1.5 billion years leading up to these global glaciations, known as the Proterozoic glacial gap . Thus, the climate system must have rested in some warm, more or less ice-free equilibrium state prior to these events. In the EBM, such states exist in the upper branch of the bifurcation diagram in Fig.  and the bifurcation point b1 is associated with the onset of Snowball Earth. For the model parameters used in Fig. , the Snowball Earth without a salt deposit is not accessible because a parameter drift below b1 drives the system into the salt-deposit state. Therefore, the transition from a warm climate to a Snowball Earth without a salt deposit is not possible here.

The Snowball Earth without a salt deposit is inaccessible for plausible model parameters. The accessibility of the Snowball Earth state without a salt deposit depends most prominently on the eutectic temperature Te and the ice formation temperature Ts. Figure  shows a selection of bifurcation curves with different model parameters. A transition into the salt-deposit Snowball Earth from a warm, ice-free state is favored by higher diffusion rates (Fig. a) and larger albedo differences α2-α1 (Fig. c), whereas lower diffusions and albedo differences prevent a direct transition (pink curves in Fig. a and c). Assuming a direct transition to Snowball Earth, the parameters controlling whether such a transition sends the system into a Snowball Earth with or without a salt deposit are Te, Ts and r1 (see Fig. b and d). The curves in the (Te,Ts)-plane in Fig.  mark the boundary between the accessible and inaccessible regimes of the parameter space for r1=20°N, r1=15°N and r1=10°N. The extent of the net ablation zone, here denoted by the latitude r1, varies among general circulation models simulating Snowball Earth atmospheres , therefore we investigate the sensitivity to this parameter. The Snowball Earth without a salt deposit is only accessible for large bare sea ice regions and extremely low eutectic temperatures, suggesting it is unlikely that Snowball Earth existed without salt deposits.

Based on this study, it is not possible to determine how long a salt deposit lasted on Snowball Earth. We observe that the range of atmospheric CO2 concentrations supporting Snowball Earth without a salt deposit in the model is limited. The physical interpretation of this is that there is a limited window where the temperature on Snowball Earth is high enough to avoid activating the salt-albedo feedback. However, the model includes no ice dynamics, and it is widely recognized that sea glaciers must have formed and flowed equatorwards on Snowball Earth . Thus, the ice within the net ablation zone would have become progressively fresher over time as meteoric ice (ice formed by precipitation) moved in from higher latitudes. This makes it difficult to assess the duration and extent of the salt deposit for the Snowball Earth events. found that only a narrow zonal band of marine ice (ice formed by basal freezing) remained near the Equator over million-year timescales. Consequently, the climatic effects of a salt deposit probably became less important towards the later stages of these glaciations. Furthermore, the model has no explicit CO2 input. The EBM is therefore not the optimal modeling framework to determine conditions for deglaciation, and we omit discussion of the range of atmospheric CO2 concentrations associated with the alternative Snowball Earth states.

Figure  illustrates why assessing the duration of the surface salt deposit is important for Snowball Earth. In the model, terminating the salt-deposit Snowball Earth requires a much higher atmospheric CO2 concentration than the classical Snowball Earth without a salt deposit. The deglaciation threshold for the salt-deposit Snowball Earth is the bifurcation point b6 in Fig. , and the radiative forcing required to trigger the melting is substantially higher than for the state without a salt deposit, b2. Perhaps the greatest challenge to the Snowball Earth hypothesis is explaining how deglaciation could occur within a realistic range of atmospheric CO2 concentrations consistent with the duration of these events. Several efforts have been made to lower the CO2 deglaciation threshold in climate models to align with geochemical proxy data . If a salt deposit lasted for a significant time, the salt-albedo feedback may have provided an additional cooling element to Snowball Earth.

Our results indicate that the salt-albedo feedback could have played an important role in the initial phase of Snowball Earth by rapidly lowering temperatures due to the high albedo of salt crystals. Particularly since the full seawater system contains numerous species that precipitate, and initial precipitation of, e.g., mirabilite and hydrohalite occurs around -8 and -23°C, respectively , suggesting that some precipitation can be expected above Te=-36°C. We note that our analysis only includes stationary states, and widespread glaciation must occur before a lag deposit of salt crystals forms. However, it is possible that the salt-albedo feedback promoted favorable conditions for the initiation of Snowball Earth by keeping sea ice albedo high as the Earth cooled. All states with partial salt deposits are unstable in the model. Once the initial salt deposits form, the system collapses into the Snowball Earth state with fully salt-covered deep tropics, given a warm pre-glacial climate. Moreover, it does not matter at what temperature this process begins, the system ultimately ends up in the fully salt-covered state (Fig. , for Te>-36°C). Although the causes of the Snowball Earth initiation remain uncertain, our results show that it is difficult to avoid activating the salt-albedo feedback shortly after global glaciation sets in.

There are some important caveats to our results. First, it is uncertain whether a lag deposit of salt will form and persist on Snowball Earth. The simplified parametrization used here neglects ice dynamics and may therefore provide an overly crude representation of this phenomenon. Accumulation of surface salt crystals ultimately depends on the rates of sublimation and the downward migration of liquid brine in the ice , processes that are not explicitly included in this model. The spatial and temporal extent of saline marine ice, where salt precipitation could occur during Snowball Earth, is also uncertain . Furthermore, our salt parametrization relies on the assumption that water evaporated in the tropics is largely transported poleward before falling as precipitation, as consistently found in GCM simulations of Snowball Earth atmospheres . This leads to a tropical net ablation zone, exposing bare sea ice and subsequently the precipitated salt crystals. Second, the strength of the salt-albedo feedback is uncertain. This feedback may be weaker in a more comprehensive model where processes such as cloud and wind dynamics are included. found that clouds can mask the albedo effect of bare sea ice, and clouds may likewise mask the salt crystals. Surface dust may also reduce the albedo by partially covering salt crystals over time. Moreover, wind-driven redistribution of salt crystals into regions with net precipitation may ensure larger areas of salt-free bare sea ice in the net ablation zone. We used a high salt crystal albedo value in this study, however a lower value merely shifts the location of the bifurcation points b4-b6 in Fig.  (see Fig. e). More research is needed to assess the strength of the salt-albedo feedback on Snowball Earth, and this study should be seen as an initial investigation into the possible impacts of a salt-albedo feedback.

The initial phase of Snowball Earth was likely marked by rapid cooling associated with both a salt-albedo feedback mechanism and the well-known ice-albedo feedback. We have shown that a salt-albedo feedback introduces two coexisting Snowball Earth states in a simple climate model, one with a lag deposit of salt crystals and one without, where the former is significantly colder. We find that the warmer state is unlikely to participate in the initial stage of the Snowball Earth hysteresis, suggesting that the colder state is consistent with Neoproterozoic geology. Although the persistence of the colder Snowball Earth state remains uncertain, our results highlight salt precipitation as an important physical process that warrants further research in future modeling studies of Snowball Earth.