1. Introduction
Glaciogenic deposits near the paleoequator indicate that Earth might have entered global glaciation a few times during the Neoproterozoic (Hoffman et al. 1998, 2017; Kirschvink 1992). These events have been commonly referred to as snowball Earth events. Due to the high reflectivity of ice and snow, once Earth enters a snowball state, a very high level of atmospheric CO2 (pCO2) is required to get out, likely on the order of 0.1 bar (Abbot et al. 2012; Hu et al. 2011; Le Hir et al. 2007; Pierrehumbert 2004, 2005; Wu et al. 2021). It will take multiple millions of years for Earth’s atmosphere to accumulate that much CO2 (Hoffman et al. 1998). A snowball Earth thus experienced many orbital cycles before it was melted. Indeed, all the Milankovitch cycles similar to today have been identified in the banded iron formation (BIF) at Holowilena, South Australia, in a recent study (Mitchell et al. 2021).
Very few studies on the climatic influence of orbital forcing during a snowball Earth have been carried out, the only one known to the authors being that of Benn et al. (2015). They used an asynchronously coupled climate (LMDZ; Hourdin et al. 2006)–ice sheet model [Grenoble Ice Shelf and Land Ice model (GRISLI); Ritz et al. 2001] with 20-mbar CO2 to study the fluctuations of continental ice sheets due to orbital forcing during the Marinoan snowball Earth event. They found that the spatial distribution of ice sheets was sensitive to orbital forcing, especially when pCO2 was relatively high. In some local regions, the ice sheet could have advanced and retreated multiple times even if the oceans were fully covered by sea ice, helping explain the cyclic sedimentological features that are often used to argue against the snowball Earth hypothesis. However, the influence of orbital forcing on the climate and hydrological cycle in a snowball Earth is still poorly understood and deserves more work.
Here, we use a coupled atmosphere–land model (see section 2) to investigate the climate variability induced by orbital forcing. Specifically, we focus on how the hydrological cycle and surface temperature over the thick sea ice (also known as sea glacier; Goodman and Pierrehumbert 2003) vary with the solar insolation, and thus remove the land and the ice sheet on it altogether to simplify the problem. The results will show what the global-mean surface temperature is and how the location of the maximum monthly temperature changes with the orbital configuration. The results will also provide us with some clue as to whether the orbital configuration has an influence on the deglaciation of a snowball Earth. It has been shown that the initial melting may appear in the midlatitude region due to large seasonal insolation there (de Vrese et al. 2021; Wu et al. 2021), which is sensitive to orbital configuration.
The rest of the paper is organized as follows. A brief description of the model used, the orbital setting, and our experimental design is given in section 2. The influence of the orbital forcing on the hydrological cycle and surface temperature in a snowball Earth as well as discussions are presented in section 3. Finally, a summary is given in section 4.
2. Model and experimental design
a. Climate model
The Community Atmosphere Model, version 3.0 (CAM3.0), and Community Land Model, version 3 (CLM3), are used to simulate the snowball Earth climate and surface processes, respectively. Both are components of the Community Climate System Model, version 3 (CCSM3), developed and maintained by the National Center for Atmospheric Research. CAM3.0 solves the primitive equations in a generalized terrain-following vertical coordinate. The equations are solved with a spectral dynamic core with triangular truncation at wavenumber 31 (T31), which is equivalent to a horizontal spatial resolution of ∼3.75° × 3.75°. The model has 26 vertical levels, and the top of the atmosphere (TOA) is approximately 2 hPa (Collins et al. 2004). The module CLM3 can simulate the hydrological processes on land and the thermodynamics of snow and soil (Oleson et al. 2004). It shares the same horizontal grid with CAM3.0.
b. Experimental design
The atmospheric concentrations of CH4 and N2O are all at preindustrial levels. Chlorofluorocarbons (CFCs) are set to zero. The solar constant is 94% of the present-day value, and pCO2 is 0.1 bar. A high pCO2 is used because this is when a snowball Earth is near its melting point (Abbot et al. 2012), at least for the model used here (Wu et al. 2021). The climate sensitivity to orbital forcing at this point is the most interesting since 1) the sensitivity may be larger than when pCO2 is high (Benn et al. 2015) and 2) it can help us get a sense of how important orbital configuration might be for the snowball deglaciation. It is found very challenging to extract the mechanisms for the climate response to orbital forcing even in a snowball Earth with no continent. To further simplify the analysis, the effect of melt pond proposed by Wu et al. (2021) is not considered in this study.
The orbital configuration is controlled by three parameters: eccentricity, precession, and obliquity (Berger 1978). Eccentricity varies from 0.000 055 (nearly circular orbit) to 0.0679 mainly at periods of ∼100 and ∼400 kyr (Laskar et al. 2011). Eccentricity is the only orbital parameter that can alter the globally averaged annual-mean solar insolation, which increases by about 0.7 W m−2 when the eccentricity is increased from 0.017 to 0.067. These are also the two eccentricities that we chose to test because 0.017 is similar to the present-day eccentricity. Low eccentricity is denoted “El” and high eccentricity is denoted “Eh.” Under snowball Earth conditions, the high eccentricity might have induced significant seasonality directly near the equator due to the low effective heat capacity of the surface ice or frozen soil (Liu et al. 2020). However, the major impact of eccentricity on climate is probably through its modulation on the precession (Milankovitch 1941), which is defined here as the longitude of the vernal equinox (of the Northern Hemisphere) measured counterclockwise from the perihelion when looking down on the North Pole. In the extreme case when eccentricity is zero, the precession will have no influence on climate except that the orbital location at which a certain season occurs will change; the amplitude of the seasonal cycle at any latitude will be determined by the obliquity alone. The seasonal variation of solar insolation at the TOA at different latitudes for two different eccentricities and two different precessions can be seen (Figs. 2a, 2c, 2e, and 2g). Note that the annual-mean solar insolation at any location does not change with precession.
For a glaciated aquaplanet, the northern and southern hemispheres are symmetrical. This makes it unnecessary to test all the precessions. For example, the summer and winter solstices of the Northern Hemisphere occur at perihelion when the precession is 270° and 90°, respectively (Fig. 1), but the global climate is the same in these two situations except that the climates in the two hemispheres are reversed. Therefore, the greatest contrast in climate is between precessions of 270° (the northern summer solstice occurs at perihelion; denoted “Ps”) and 180° (the northern autumn equinox occurs at perihelion; denoted “Pe”), so only these two precessions are tested. The difference in TOA insolation between these two precessions (Figs. 3a and 3b) shows that the maximum monthly difference can reach >70 W m−2 at high eccentricity. For obliquity, only the minimum (22°; denoted “Ol”) and the maximum (24.5°; denoted “Oh”) are tested. The annual-mean insolation decreases/increases with obliquity at low/high latitudes, but the global-mean remains the same. Thus, there are eight combinations of orbital configurations that we test (see a summary in Table 1 and their solar forcing in Fig. 2). All simulations are run for 600 years, which is sufficient for the surface snow and temperature to reach equilibrium (Fig. S1 in the online supplemental material), and the results are averaged over the last 100 years for analysis.
The precessions tested in this study.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
Summary of all simulations. Here, TGlo and TTp are the globally and tropically (20°S–20°N) averaged annual-mean surface temperature, respectively; Tmax is the maximum monthly mean surface temperature.
Seasonal variation of solar insolation at TOA for various orbital parameters. Eccentricities are (a)–(d) 0.067 (Eh) and (e)–(h) 0.017 (El); precession is 270° (Ps) in the first two columns and 180° (Pe) in the last two columns; obliquity is 24.5° (Oh) for the first and third columns, and 22° (Ol) for the second and fourth columns.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
The difference in the TOA solar insolation between different precessions at (a) high eccentricity and (b) low eccentricity, and (c) between different obliquities.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
3. Results and discussion
a. Surface temperature
As described in section 2, the change of precession has a smaller influence on solar insolation and climate when the eccentricity is small. This is verified by the results of numerical experiments. The globally averaged annual-mean surface temperature (TGlo) changes by less than 0.8°C among the four experiments with lower eccentricity (the El cases in Table 1), while the difference can be as large as 2°C among the four Eh cases. For the tropical-mean surface temperature (TTp), the difference can be as large as 3.8°C among different Eh cases. These indicate that the orbital configuration does have a significant influence on the global climate even though the global-mean insolation is the same. For the same eccentricity, there is no clear relationship to explain how the TGlo or TTp changes with precession (Table 1).
When the eccentricity and precession are fixed, higher obliquity produces colder TTp (Table 1). This is understandable since the higher the obliquity, the less insolation is received in the tropical region (Fig. 3c). This behavior contrasts with the ice ages of the Pleistocene, where higher obliquity favors deglaciation, because for the Pleistocene ice ages, the critical latitudes are ∼60°N rather than the tropics. The difference in TTp caused by obliquity is smaller than 1°C among most cases, but can be as large as 3.8°C between the cases “Eh_Pe_Oh” and “Eh_Pe_Ol.” Therefore, the influence of different orbital parameters on climate has complex interdependence. The increase of eccentricity may not only amplify the influence of precession, but also the influence of obliquity through nonlinear processes such as snow accumulation. Moreover, the influence of obliquity on TTp is dependent on the phase of precession; its influence is larger when the subsolar point is at the equator when Earth is closest to the sun (i.e., the Pe cases). These are clearly demonstrated by the difference in TTp between different pairs of Oh and Ol simulations (Table 1).
The large difference between Eh_Pe_Oh and Eh_Pe_Ol is due to the difference in surface albedo in the tropical region, especially the southern tropics (cf. the black curves in Fig. 4c), which determines the difference in the planetary albedo in these cases (Fig. 4e). The difference in surface albedo itself is due to the different snow thickness in the tropical region between the two cases, as will be described in section 3b. Note that the TGlo may not change in the same direction as TTp. For example, “El_Ps_Oh” has lower TTp than the case “El_Ps_Ol” but has higher TGlo, indicating that the temperature change in the mid–high-latitude region (Figs. 4a,b) may dominate the change of the global mean in some cases.
Annual- and zonal-mean (a),(b) surface temperature, (c),(d) planetary albedo, and (e),(f) surface albedo obtained in each experiment (Table 1). Experiments at (left) high and (right) low eccentricity.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
Despite the finding that the globally averaged annual-mean solar insolation at TOA is 0.7 W m−2 higher for the Eh cases than the El cases, both TGlo and TTp of the former are, on average, lower than those of the latter (Table 1). This is likely because the seasonality of the insolation strengthens almost everywhere, including the tropical region, as the eccentricity increases (Fig. 3). The seasonality of the surface temperature is highly correlated with the insolation (cf. Figs. 5 and 2), and thus the seasonality of TTp increases with eccentricity. Such an increase in seasonality ends up increasing both the annual-mean surface and planetary albedos within the tropical region (Fig. 4). The exact mechanism is unclear but is likely related to how snow accumulation and aging change, which will be described in more detail in sections 3b and 3c.
Seasonal variation of surface temperature. The black solid lines are the 0°C contour lines of surface temperature.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
The maximum monthly mean surface temperature (Tmax) is found in the subtropical region, where the low-albedo glacier is exposed and the summer insolation is strong. The Tmax is above 0°C and is reached in July in all cases, but the values and the latitudinal locations are different (Fig. 6). Higher Tmax is obtained for larger eccentricity (red markers in Fig. 6) since Earth can always get to a closer position to the sun when the orbit is more elliptic. The highest Tmax is 6.4°C, obtained in the case “Eh_Ps_Oh,” for which the eccentricity is larger and the summer solstice occurs at the perihelion. The lowest Tmax is 2.7°C, obtained in the case “El_Pe_Ol,” for which the eccentricity is smaller and the vernal equinox occurs at the perihelion. Note that although the Tmax is high in the case Eh_Ps_Oh, its TGlo is relatively low.
The maximum monthly mean surface temperature Tmax and its position for each experiment (Table 1). The red color represents Eh and the blue represents El in Table 1. Crosses denote Pe and dots denote Ps in Table 1. The marker without a circle denotes Oh and the marker with a circle denotes Ol in Table 1. The Tmax of all experiments occurs in July.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
Interestingly, both the value and the latitudinal location of Tmax increase with obliquity (cf. the markers with circles and the corresponding ones without circles in Fig. 6). For example, the Tmax is 1.4°C higher in the case Eh_Ps_Oh than in the case “Eh_Ps_Ol,” and the latitude is 42.7° and 38.9°N for the two cases, respectively. The location change is easily understandable since the increase of obliquity shifts the solar insolation, especially the summer insolation, from the lower latitude to the higher latitude (Fig. 3c). The higher value of Tmax is because at the locations where Tmax is obtained (i.e., between 35° and 43°N), solar insolation always increases when the obliquity is increased (Fig. 3c).
It is worth pointing out that the tropical temperature TTp is not consistent with Tmax. For example, the case Eh_Ps_Oh has the highest Tmax (6.4°C) but has a medium TTp (−11.4°C), while the case El_Ps_Ol has the second-lowest Tmax (3.2°C) but has the maximum TTp (−9.4°C). It has been proposed recently that the snowball deglaciation might be initiated from the midlatitude region due to formation of seasonal melt ponds (Wu et al. 2021) or accumulation of dust (de Vrese et al. 2021) there. However, Wu et al. (2021) also demonstrated that only the equatorial region would accumulate perennial meltwater to a thickness sufficient to break the sea ice lid, meaning that the tropical temperature was also important for snowball deglaciation. This makes it uncertain which orbital configuration is optimal for triggering snowball deglaciation and warrants further study, for example, by considering explicitly the formation of melt ponds as in Wu et al. (2021).
b. The response of the hydrological cycle
When the precession is 270°, the Northern Hemispheric summer receives more solar insolation than the Southern Hemispheric summer (Figs. 3a,b), especially when the eccentricity is high; the hemispheric asymmetry is much weaker when the precession is 180° and when the eccentricity is low (Fig. 2). This results in stronger and higher Hadley circulation in the northern summer (Fig. 7g) than in the southern summer (Fig. 7d) in the case Eh_Ps_Oh. Similarly, the Hadley circulation of the Northern Hemispheric summer is stronger and higher in the case Eh_Ps_Oh than in the case Eh_Pe_Oh (Figs. 7g–i). For the Northern Hemispheric winter, the results are the opposite (Figs. 7d,e). The annual-mean Hadley circulation is much weaker and shallower than the seasonal circulation (Fig. 7), as has been pointed out in previous studies (Voigt 2013; Yang et al. 2012).
Mass streamfunction of the atmosphere for (left) Eh_Ps_Oh, (center) Eh_Pe_Oh, and (right) their difference (left − center). The (a)–(c) annual, (d)–(f) January, and (g)–(i) July means. Red solid curves indicate clockwise circulation, and blue dashed curves indicate counterclockwise circulation. The zero contour is indicated by the black solid curves. Note that the contour spacing is different for different panels; it is 30 × 1010 kg s−1 in (d), (e), (g), and (h), and 3 × 1010 kg s−1 for the rest.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
The precipitation is highly influenced by the Hadley circulation in these simplified experiments, where the zonal variabilities due to continents are removed. The peak of snow precipitation (Figs. 8a–c) is coincident with the upwelling branch of the Hadley circulation (Fig. 7). Within this region (averaged over ∼10°–20° latitude), the annual snow precipitation is dominated by precipitation during the extended summer [May to September, as often used in the study of summer monsoons (Wang et al. 2017); Fig. 9]. Since the northern summer has a stronger Hadley circulation (Fig. 7g) than the southern summer (Fig. 7d) in the case Eh_Ps_Oh, the annual-mean precipitation exhibits a much higher peak precipitation in the Northern Hemisphere than in the Southern Hemisphere (Fig. 8a). This is true for all the cases with precession equaling to 270° (the orange curves in Figs. 8a–d), but the hemispheric asymmetry is again much weaker when the eccentricity is low (cf. the orange curves in the right column with those in the left column).
Annual- and zonal-mean (a),(b) snow rate, (c),(d) rain, (e),(f) evaporation, and (g),(h) net precipitation obtained in each experiment (Table 1). Experiments with (left) high and (right) low eccentricity. The gray line in (g) and (h) denotes the zero line. Note that the total precipitation is the sum of the snow rate and rain, and the net precipitation is the total precipitation minus the evaporation.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
Seasonal variation of the snow precipitation rate averaged over (a),(b) 10°–20°N and (c),(d) 10°–20°S. Experiments with (left) high and (right) low eccentricity.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
The precipitation is also affected by the surface temperature and is most clearly seen in the precipitation of rain (Figs. 8c,d). The monthly mean temperature of the southern summer does not reach 0°C in the case Eh_Ps_Oh (Fig. 5a); there is thus almost no rain there (orange dashed curve in Fig. 8c; it is nonzero because daily temperature can still exceed 0°C). For all other cases, a significant amount of rain (∼0.4–0.5 mm day−1) is precipitated over the Southern Hemisphere (Figs. 8c,d). Note that the peak of the rain falls at a higher latitude than the snow in all cases (Fig. 8). This is because the regions where surface temperature is higher than 0°C are generally between 20° and 60° latitude (Fig. 5), while snow can occur anywhere and is the strongest where solar insolation is high and upwelling is strong (cf. Figs. 8a–d with Figs. 2 and 7).
Unlike the double-peak structure of the annual-mean precipitation (Figs. 8a–d), evaporation has three peaks (Figs. 8e,f). One peak is at the equator, and the other two peaks are between 20° and 30° latitude. There are two minima located between 10° and 20° latitude where the upwelling (rising branch of the Hadley cell) is overall the strongest (Figs. 7a–c) and thus cloudy (not shown). The Hadley cell structure here is different from the present day probably due to the lack of oceans and the meridional overturning ocean circulation on a snowball Earth. The most significant difference may be in the annual-mean circulation; the upwelling in the Northern Hemisphere dominates that in the Southern Hemisphere in the present day, while the difference is small in a snowball Earth such that they largely cancel each other and the remainder is shallow and weak (Figs. 7a,b). The net precipitation is then strongly negative at the equator with values between −0.5 and −0.3 mm day−1, and positive between 8° and 25° latitude with values ∼0.3 mm day−1, but is hemispherically asymmetric when the eccentricity is high (Figs. 8g,h). Further away from the equator (25°–35° latitude), the net precipitation becomes negative (around −0.1 mm day−1) again, corresponding to the strong evaporation in these two zones (Figs. 8e,f). These have been partially shown before (Pierrehumbert 2005; Yang et al. 2012) for the present-day orbital configuration, which has a small eccentricity. In the wide regions beyond 35° latitude, the net precipitation is small (<0.1 mm day−1) but mostly positive. This positive value does not mean that there is a net accumulation of snow, as much of the precipitation is in the form of rain (Figs. 8c,d). The change of orbital configuration has a significant influence on the structure of net precipitation and thus the snow depth, which then has a strong influence on the surface albedo and temperature, as will be discussed below.
The snow thickness, which is important for the growth of continental ice sheets (not considered in this study) and maybe also for the initiation of deglaciation, has significant variation in the meridional direction and strong dependence on orbital configuration. Note that the snow thickness is not only determined by the net precipitation in Figs. 8g and 8h, but also by the surface melting, which is not shown but can be inferred from the surface temperature in Fig. 5. Because of this factor, the annual-mean SWEs within the tropical region are all smaller than the model limit of 1 m, even within the net accumulation zone (Fig. 10). Between 25° and 50° latitude, the annual-mean SWEs are almost zero. Beyond 50° latitude, the SWE is large, either 0.45 or 1 m, depending on the orbital configuration.
Annual- and zonal-mean SWE obtained in each experiment (Table 1). Experiments at (a) high eccentricity and (b) low eccentricity.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
Overall, the annual-mean SWE is thicker within the tropical region and thinner over the polar regions when the obliquity is high, and vice versa (cf. the solid and dashed curves in Fig. 10), as expected from the influence of obliquity on solar insolation (Figs. 2 and 3). A strong influence of precession on SWE is seen only when the eccentricity is high; tropical SWE is thicker when precession is 270° than when it is 180° (cf. the orange curves with the black curves in Fig. 10a). This difference is not because of the difference in the net precipitation (Figs. 8g,h), but because of the stronger melting during the warmer months when precession is 180° (cf. the highest surface temperature in Figs. 5c,d with Figs. 4a,b). This is better seen from the seasonal snow depth shown in Fig. 11.
SWE under different orbital configurations. The red solid contour lines represent that snow depth is 0.6 m, and the dashed contour lines represent that snow depth is 0.1 m.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
The convective precipitation responds to the radiative heating almost immediately, whereas the surface temperature takes more time (less than a month; not shown) to be high enough to melt the snow. Consequently, the snow depth grows first, and then melting follows. For example, the snow depth in the northern polar region (greater than 60°N) increases in May and decreases in June for all cases (Fig. 11). This melting occurs earlier (in spring) in the northern tropical region because of the different seasonal cycle of solar insolation and temperature there (Figs. 2 and 5). For this region, another melting season occurs in the autumn (Fig. 11).
c. Surface albedo and the snow aging
Due to the aging of snow, the surface albedo is not a simple function of the snow thickness or SWE. For example, the case Eh_Pe_Oh has a small SWE in the tropical region (black solid curve in Fig. 10a), but the surface albedo is the highest among all cases (Fig. 4c). This is because the snow age is near zero (i.e., new snow) during the whole year (Fig. 12c). The generally smaller tropical snow age in the Eh cases than in the El cases (Fig. 12) is the major reason why the tropical temperature in the former is lower than in the latter (Table 1). The case Eh_Pe_Oh also has the highest surface albedo in the northern polar region (Figs. 4c,d) because of its small snow age there (Fig. 12).
As in Fig. 11, except nondimensional snow age is shown. The red solid contour lines represent that nondimensional snow age is 0.5.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
The snow thickness also plays a role in determining the surface albedo. For example, the snow age in the tropical region in the case Eh_Pe_Ol is even smaller than in the case Eh_Pe_Oh (cf. Figs. 12c,d), but the surface albedo is lower (cf. the black dashed curve to the solid curve in Fig. 4c). This is due to the smaller snow thickness in the case Eh_Pe_Ol than in Eh_Pe_Oh (cf. Figs. 11c,d or the black curves in Fig. 10a), which reduces the surface albedo by exposing more low-albedo ice [Eq. (1)]. The high tropical surface albedo of the case Eh_Pe_Oh makes its tropical region the coldest among all cases (Table 1).
Therefore, orbital configuration affects the distribution of snow thickness by affecting the hydrological cycle, but thin snow does not necessarily mean small surface albedo or high surface temperature. This makes a straightforward inference of the response of snowball Earth temperature to orbital forcing almost impossible except in the midlatitude region. In the wide midlatitude region, there is little or no snow during most of the year (Fig. 12). The surface temperature is thus highly dependent on the solar insolation received. We can expect that the highest monthly temperature should be generated in the case Eh_Ps_Oh since it provides the highest monthly solar insolation in the northern midlatitude among all cases (Fig. 2). This is indeed the case, as shown in Fig. 6.
Note that the exact albedo of ice in a late-stage snowball is uncertain due to the presence of surface dust and volcanic ashes (Abbot and Halevy 2010), as well as the exposure of low-albedo marine ice (Hoffman et al. 2017). This uncertainty in ice albedo will likely induce an uncertainty in the dependence of climate on orbital configuration as the snowball Earth climate is very sensitive to surface albedo. Such sensitivity has already been pointed out by Pierrehumbert et al. (2011), who showed that 10 times more CO2 is required to deglaciate the snowball Earth when the bare-ice albedo increases from 0.55 to 0.65. Moreover, the positive feedback strengthens when the contrast of bare-ice albedo and snow albedo increases; once the snow cover disappears (either seasonally or perennially; e.g., Fig. 11), the exposed bare ice will absorb more solar insolation and induce more warming. If this feedback is strong enough that the tropical region becomes snow free, a much warmer climate will be obtained for the high-eccentricity cases than for the low-eccentricity cases because the former have wider snow-free regions in the subtropics (Figs. 10 and 11). That is, the opposite results presented above may be obtained. The strengthening of positive feedback described above is quite similar to that shown in Wu et al. (2021), where the bare-ice albedo was lowered by forming melt ponds on the ice surface. Therefore, it will be useful to investigate how the deglaciation of snowball Earth is affected by orbital configuration in the future.
Similarly, the snow albedo and the aging parameterization could also have a significant influence on the results shown here. However, we decided not to explore such complications because the results are already quite complex to understand. More importantly, the results here are sufficient to demonstrate that orbital configuration could have a significant impact on the snowball Earth climate and likely on the deglaciation as well, which is our main purpose herein.
d. The Milankovitch hypothesis
Milankovitch hypothesized that the growth of polar continental ice sheets was promoted by cooler summer, not colder winter (Milankovitch 1941), and the Quaternary glacial–interglacial cycles were thus modulated by the orbital cycles of Earth which change the summer insolation. His hypothesis was later largely confirmed by observation (Hays et al. 1976). Here, we want to test whether this hypothesis would hold in a snowball Earth, especially in the tropical region, where glaciations did not occur during the Quaternary. We use SWE to mimic the growth of a continental ice sheet and find that the regional SWE decreases with the maximum monthly insolation (MMI) in all regions including the tropics (Fig. 13). This linear relation is almost perfect in the southern extratropical region for the simulations carried out herein (Fig. 13a). This means that the Milankovitch hypothesis is applicable to the snowball Earth situation, even in the tropical region. This will allow us to predict that the low-latitude continental ice sheet would most likely develop when the eccentricity is low, obliquity is high, and precession is 270° or 90° (Fig. 9c), all of which prevent the equatorial region from having warm summers. To achieve this, we would want the summer or winter solstice to occur at the perihelion point so that the equator is farthest from the sun at this point. This means that the precession has to be 270° or 90° (Fig. 1).
Scatterplot of the regional- and annual-mean SWE against MMI for various orbital configurations. The definition of the markers is similar to those in Table 1 and Fig. 7. Note that the tropic MMI is defined at 0° to mimic the 65°N/S insolation in Milankovitch theory.
Citation: Journal of Climate 36, 18; 10.1175/JCLI-D-23-0041.1
Roe (2006) showed that it was the rate of change rather than the value of global ice volume that had responded to the summer insolation of the northern high latitudes at zero lag during the Quaternary. This is because the full development of the continental ice sheet takes thousands of years, rendering the appearance of an apparent lag to the orbital forcings at times of fast change of solar insolation. The SWE in the model used herein is limited to only 1 m, which enables its rapid response (a few hundred years; Fig. S1) to the local solar forcing induced by orbital changes.
4. Conclusions
The response of the snowball Earth climate to orbital forcings is studied using CAM3.0 coupled with CLM3. It is found that both the local and global-mean surface temperatures are sensitive to orbital perturbations. The TGlo, TTp, and Tmax can change by as much as 2.4°, 4.3°, and 3.7°C, respectively, when the orbital configuration is changed (Table 1). Such a magnitude of temperature change is comparable to that induced by doubling CO2 in the same model in a snowball Earth (Table S2 of Liu et al. 2020). This means that the snowball deglaciation may be triggered much more easily at some orbital configurations than others, and the duration of the snowball Earth events is thus influenced by the orbital configuration.
In certain cases, the variations of surface temperature correspond to those of local solar insolation in a straightforward way. The TTp is higher with lower obliquity when the other two orbital parameters are fixed. The Tmax is obtained in July between 35° and 43°N; the higher the obliquity and the higher the eccentricity, the higher the Tmax (Fig. 6). The Tmax is larger when the northern summer occurs at the perihelion than when the autumn equinox occurs at the perihelion.
However, the response of the surface temperature to the change of certain orbital parameters is not always intuitive due to the complex interaction between solar insolation, the hydrological cycle, and snow aging. Such interactions might result in higher surface albedo for thinner snow and thus lower surface temperature. For example, the tropical snow depth in the case Eh_Pe_Oh is the second thinnest (Fig. 11), but its tropical temperature TTp is the lowest (Table 1). Due to such nonlinearities, the TGlo and TTp for the high-eccentricity cases are generally lower than those for the low-eccentricity cases, despite the fact that the annually averaged global-mean solar insolation received for the former is higher than that for the latter; the increase of eccentricity may not only amplify the influence of precession but also the influence of obliquity so that TTp can differ by 3.8°C between Eh_Pe_Oh and Eh_Pe_Ol; the influence of obliquity on TTp is larger when the subsolar point is at the equator when Earth is at the perihelion (i.e., the Pe cases; Table 1).
It is uncertain which orbital configuration would favor the deglaciation of a snowball Earth because important feedback mechanisms such as the formation of melt ponds are not included in this preliminary study. If the deglaciation is triggered more easily at higher TTp, then the deglaciation would more likely occur for orbit El_Ps_Ol, which gives a TTp of −9.4°C. If the deglaciation is triggered more easily at higher Tmax, then the deglaciation would more likely occur for orbit El_Ps_Oh, which gives a Tmax of 6.4°C. The actual deglaciation such as that done by Wu et al. (2021) should be investigated, and that is currently underway.
Moreover, we found that the Milankovitch hypothesis was likely valid within the tropical region too. That is, the snow depth in the tropical region during a snowball Earth is inversely proportional to the maximum monthly solar insolation at the equator. The largest monthly solar insolation at the equator is found when the eccentricity is high so that the perihelion is closer to the sun and when the obliquity is low. Therefore, the equatorial glaciation should occur most easily when the eccentricity is low and the obliquity is high.
Acknowledgments.
We thank Dorian Abbot and two other anonymous reviewers for their constructive review. This work is supported by the National Natural Science Foundation of China (42225606 and 41875090).
Data availability statement.
The model results that support the findings of this study are available on Zenodo with the identifier https://doi.org/10.5281/zenodo.7554223.
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