Generated using version 3.2 of the official AMS LATEX template 1 Ocean circulation under globally glaciated Snowball Earth 2 conditions: steady state solutions Yosef Ashkenazy 3 ∗ Ben-Gurion University, Midreshet Ben-Gurion, Israel Hezi Gildor 4 The Fredy and Nadine Herrmann Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel Martin Losch 5 Alfred-Wegener-Institut, Helmholtz-Zentrum für Polar- und Meeresforschung, Bremerhaven, Germany Eli Tziperman 6 Dept. of Earth and Planetary Sciences and School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA ∗ Corresponding authors’ address: Yosef Ashkenazy, Department of Solar Energy and Environmental Physics, BIDR, Ben-Gurion University, Midreshet Ben-Gurion, 84990, Israel. E-mail:[email protected]; Eli Tziperman, Dept. of Earth and Planetary Sciences and School of Engineering and Applied Sciences, Harvard University; 20 Oxford St, Cambridge, MA, 02138, USA. E-mail:[email protected] 1 7 ABSTRACT 8 Between ∼750 to 635 million years ago, during the Neoproterozoic era, the Earth experienced 9 at least two significant, possibly global, glaciations, termed “Snowball Earth”. While many 10 studies have focused on the dynamics and the role of the atmosphere and ice flow over the 11 ocean in these events, only a few have investigated the related associated ocean circulation, 12 and no study has examined the ocean circulation under a thick (∼1 km deep) sea-ice cover, 13 driven by geothermal heat flux. Here, we use a thick sea-ice flow model coupled to an ocean 14 general circulation model to study the ocean circulation under Snowball Earth conditions. 15 We first investigate the ocean circulation under simplified zonal symmetry assumption and 16 find (i) strong equatorial zonal jets, and (ii) a strong meridional overturning cell, limited 17 to an area very close to the equator. We derive an analytic approximation for the latitude- 18 depth ocean dynamics and find that the extent of the meridional overturning circulation cell 19 only depends on the horizontal eddy viscosity and β (the change of the Coriolis parameter 20 with latitude). The analytic approximation closely reproduces the numerical results. Three- 21 dimensional ocean simulations, with reconstructed Neoproterozoic continents configuration, 22 confirm the zonally symmetric dynamics, and show additional boundary currents and strong 23 upwelling and downwelling near the continents. 1 24 1. Introduction 25 The Neoproterozoic Snowball events are perhaps the most drastic climate events in 26 Earth’s history. Between 750 and 580 million years ago (Ma), the Earth experienced at 27 least two major, possibly global, glaciations (e.g., Harland 1964; Kirschvink 1992; Hoffman 28 and Schrag 2002; Macdonald et al. 2010; Evans and Raub 2011). During these events (the 29 Sturian and Marinoan ice ages), ice extended to low latitudes over both ocean and land. It 30 is still debated whether the ocean was entirely covered by thick ice (“hard” Snowball) (e.g., 31 Allen and Etienne 2008; Pierrehumbert et al. 2011), perhaps expect very limited regions of 32 sea-ice free ocean, e.g., around volcanic islands (Schrag et al. 2001) (that could have pro- 33 vided a refuge for photosynthetic life during these periods), or whether the tropical ocean 34 was partially ice free or perhaps covered by thin ice (“soft” Snowball) (e.g., Yang et al. 35 2012c). 36 The initiation, maintenance, and termination of such a climatic condition pose a first- 37 order problem in ocean and climate dynamics. One may argue that the Snowball state was 38 predicted by simple energy balance models (EBMs) (Budyko 1969; Sellers 1969). Snowball 39 dynamics also provide a test-case for our understanding of the climate system as manifested 40 in climate models. Therefore, in recent years, these questions have been the focus of nu- 41 merous studies and attempts to simulate these climate states using models with different 42 levels of complexity. The role and dynamics of atmospheric circulation and heat trans- 43 port, CO2 concentration, cloud feedbacks, and continental configuration have been studied 44 (Pierrehumbert 2005; Le-Hir et al. 2010; Donnadieu et al. 2004a; Pierrehumbert 2002, 2004; 45 Le-Hir et al. 2007). Recently, the effect of clouds, as well as the role of atmospheric and 46 oceanic heat transports in the initiation of Snowball Earth events was studied; these studies 47 were based on atmospheric GCMs and used different setups and configurations including 48 different CO2 concentrations, different continental configurations, and different sea-ice dy- 49 namics (Yang et al. 2012c,b,a; Voigt and Abbot 2012; Abbot et al. 2012). It was concluded, 50 e.g., that sea-ice dynamics has important role in the initiation of Snowball events (Voigt 2 51 and Abbot 2012). Additionally, perceived difficulties in exiting a Snowball state by a CO2 52 increase alone motivated the study of the role of dust over the Snowball ice cover (Abbot 53 and Pierrehumbert 2010; Le-Hir et al. 2010; Li and Pierrehumbert 2011; Abbot and Halevy 54 2010). 55 A simple scaling calculation of balancing geothermal heat input into the ocean with heat 56 escaping through the ice by diffusion leads to an estimated ice thickness of 1 km. The ice 57 cover is expected to slowly deform and flow toward the equator to balance for sublimation 58 (and melting at the bottom of the ice) at low latitudes and snow accumulation (and ice 59 freezing at the bottom of the ice) at high latitude. The flow and other properties of such 60 thick ice over a Snowball ocean (“sea glaciers”, Warren et al. 2002) were examined in quite 61 a few recent studies (Goodman and Pierrehumbert 2003; McKay 2000; Warren et al. 2002; 62 Pollard and Kasting 2005; Campbell et al. 2011; Tziperman et al. 2012; Pollard and Kasting 63 2006; Warren and Brandt 2006; Goodman 2006; Lewis et al. 2007). Snowball Earth global 64 ice cover is an extreme example within a range of multiple ice cover equilibrium states, 65 which have been studied in a range of simple and complex models (e.g., Langen and Alexeev 66 2004; Rose and Marshall 2009; Ferreira et al. 2011). In contrast to these many studies of 67 different climate components during Snowball events, the ocean circulation during Snowball 68 events has received little attention. Most model studies of a Snowball climate used an ocean 69 mixed layer model only (Baum and Crowley 2001; Crowley and Baum 1993; Baum and 70 Crowley 2003; Hyde et al. 2000; Jenkins and Smith 1999; Chandler and Sohl 2000; Poulsen 71 et al. 2001b; Romanova et al. 2006; Donnadieu et al. 2004b; Micheels and Montenari 2008). 72 The studies that used full ocean General Circulation Models (GCMs) concentrated on the 73 ocean’s role in Snowball initiation and aftermath (Poulsen et al. 2001a; Poulsen and Jacob 74 2004; Poulsen et al. 2002; Sohl and Chandler 2007), or other aspects of Snowball dynamics 75 in the presence of oceanic feedback (Voigt et al. 2011; Le-Hir et al. 2007; Yang et al. 2012c; 76 Ferreira et al. 2011; Marotzke and Botzet 2007; Lewis et al. 2007; Voigt and Marotzke 2010; 77 Abbot et al. 2011; Lewis et al. 2004, 2003). Yet none of these studies employing ocean 3 78 GCMs accounted for the combined effects of thick ice cover flow and driving by geothermal 79 heating. Ferreira et al. (2011) simulated an ocean under a moderately thick (200 m) ice cover 80 with no geothermal heat flux, and calculated a non steady-state solution with near-uniform 81 temperature and salinity. They described a vanishing Eulerian circulation together with 82 strongly parameterized eddy-induced high latitude circulation cells. 83 With both the initiation (Kirschvink 1992; Schrag et al. 2002; Tziperman et al. 2011) 84 and termination (Pierrehumbert 2004) of Snowball events still not well understood, and the 85 question of hard vs. soft Snowball still unresolved (Pierrehumbert et al. 2011), our focus here 86 is the steady state ocean circulation under a thick ice cover (hard Snowball). By examining 87 ocean dynamics under such an extreme climatic state, we aim to better understand the 88 relevant climate dynamics, and perhaps even provide constraints on the issues regarding soft 89 vs. hard Snowball states. 90 To study the 3D ocean dynamics under a thick ice cover, it is necessary to have a two- 91 dimensional (longitude and latitude) ice-flow model, and this was recently developed by 92 Tziperman et al. (2012), based on the ice-shelf equations of Morland (1987) and MacAyeal 93 (1997), extending the 1D model of Goodman and Pierrehumbert (2003). This model is 94 coupled here to the MITgcm (Marshall et al. 1997). Another challenge in studying the 3D 95 ocean dynamics under a thick ice cover is that thick ice with lateral variations of hundreds of 96 meters (as under Snowball conditions) poses a numerical challenge as standard ocean models 97 cannot handle ice that extends through several vertical layers; we use the ice-shelf model of 98 Losch (2008), which allows for this. An alternative, vertically scaled coordinates, was used 99 by Ferreira et al. (2011). 100 This paper expands on results briefly reported in Ashkenazy et al. (2013) (hereafter 101 AGLMST), and we report the details of the steady state ocean dynamics under a thick ice 102 (Snowball) cover, analytically and numerically, when both geothermal heating and a thick ice 103 flow are taken into account. We find the ocean circulation to be quite far from the stagnant 104 pool envisioned in some early studies, and very different from that in any other period in 4 105 Earth’s history. In particular, the stratification is very weak as might be expected (Ferreira 106 et al. 2011), and is dominated by salinity gradients due to melting and freezing of ice; we 107 find a meridional overturning circulation that is confined to the equatorial region, significant 108 zonal equatorial jets, and strong equatorial meridional overturning circulation (MOC). 109 The paper is organized as follows. We first describe the models and configurations used 110 in this study (section 2). We then present the results of the latitude-depth ocean model 111 coupled to a 1D (latitude) ice-flow model when geothermal heating is taken into account 112 (section 3). Analytically approximated solutions of the 2D, latitude-depth ocean model are 113 then presented (section 4). Section 5 presents sensitivity runs to study the robustness of 114 both the numerical results and the analytical approximations, followed by the steady state 115 results of a 3D ocean model coupled to a longitude-latitude 2D ice-flow model in section 6. 116 The results are discussed and summarized in section 7. 117 2. Model description 118 a. Ice-flow model 119 The ice-flow model solves for the ice depth and velocity over an ocean as a function of 120 longitude and latitude, in the presence of continents (Tziperman et al. 2012). The model 121 extends the 1D model of Goodman and Pierrehumbert (2003), which was based on the Weert- 122 man (1957) formula for ice shelf deformation. Because this specific formulation cannot be 123 extended to ice flow in two horizontal dimensions, we instead used the ice-shelf approxima- 124 tion (Morland 1987; MacAyeal 1997) that can be extended to two dimensions. The ice-shelf 125 approximation implies a depth-independent ice velocity, and in addition, the vertical tem- 126 perature profile within the ice is assumed to be linear (Goodman and Pierrehumbert 2003). 127 The temperature at the upper ice surface and surface ice sublimation and snow accumulation 128 are prescribed from the energy balance of Pollard and Kasting (2005) and are assumed to be 129 constant in time. The temperature and melting/freezing rates at the bottom of the ice are 5 130 calculated by the ocean model. The model’s spatial resolution is set to that of the ocean, 131 and the model is run in either 1D (latitude only) or 2D configurations, depending on the 132 ocean model used; it is typically 1-2◦ . 133 b. The ocean model—MITgcm 134 We used the Massachusetts Institute of Technology general circulation model (MITgcm, 135 Marshall et al. 1997), a free-surface, primitive equation ocean model that uses z coordinates 136 with partial cells in the vertical axis; we use a longitude-latitude grid. To account for the thick 137 ice, we used the ice-shelf package of the MITgcm (Losch 2008) that allows ice thicknesses 138 that span many vertical layers. Parameter values followed Losch (2008). The ocean was 139 forced at the bottom with a spatially variable (but constant in time) geothermal heat flux. 140 The equation of state used here (Jackett and McDougall 1995) was tuned for the present 141 day ocean, while the temperature and salinity we used to simulate Snowball conditions were 142 somewhat outside this range. Sensitivity tests, using mean present day salinity and mean 143 salinity that is two times larger than the present day value, showed no sensitivity of the results 144 for the circulation. The ocean model was run at two different configurations, including a 145 zonally symmetric 2D configuration and a near-global 3D configuration, described as follows. 146 1) Latitude-depth configuration 147 In the 2D runs, the spatial resolution was 1◦ with 32 vertical levels spanning a depth 148 of 3000 m, with vertical level thicknesses (from top to bottom) of 920, 15×10, 12, 17, 23, 149 32, 45, 61, 82, 110, 148, 7×200 m; the uppermost level was entirely within the ice. The 150 steady state ice thickness was calculated by the ice model to be approximately 1 km with 151 lateral variations of less than 100 m. The latitudinal extent of the 2D configuration was from 152 84◦ S to 84◦ N with walls specified at these boundaries to avoid having to deal with the polar 153 singularity of the spherical coordinates. The bathymetry was either flat or had a Gaussian 6 154 ridge centered at φ0 with a height of h0 = 1500 m and width 2 /(2σ 2 ) h(φ) = h0 e−(φ−φ0 ) √ 2σ=7◦ : . (1) 155 In the standard configuration, the ridge was located at φ0 = 20◦ N, to schematically represent 156 paleoclimatic estimates of more tectonic divergence zones in the Northern Hemisphere (NH). 157 We choose the bottom geothermal heat flux to have the same form of Eq. (1) such that it 158 is proportional to the height of the ridge (Stein and Stein 1992). The maximal geothermal 159 heating was four times larger than the background, with a spatial mean value of 0.1 W/m2 , 160 as for present day; in the standard 2D run presented below, the maximal geothermal heat 161 was ∼0.3 W/m2 while the background geothermal heat, far from the ridge, was ∼0.08 W/m2 . 162 The mean value of 0.1 W/m2 was based on the mean present day oceanic geothermal heat 163 fluxes, given in Table 4 of Pollack et al. (1993). 164 The lateral and vertical viscosity coefficients were 2×104 m2 s−1 and 2×10−3 m2 s−1 . The 165 lateral and vertical tracer diffusion coefficients were 200 m2 s−1 and 10−4 m2 s−1 . To be 166 conservative, the horizontal viscosity and diffusion coefficients were chosen to be larger than 167 those estimated based on eddy resolving runs presented in AGLMST. Static instabilities 168 in the water column were removed by increasing the vertical diffusion to 10 m2 s−1 . Their 169 large values required an implicit scheme for solving the diffusion equations. We note that 170 our simulations do not incorporate the effect of vertical diffusion of momentum which was 171 shown to be important in atmospheric dynamics under Snowball Earth conditions (Voigt 172 et al. 2012). 173 For efficiency, we used the tracer acceleration method of Bryan (1984), with a tracer 174 time step of 90 minutes and a momentum time step of 18 minutes. We did not expect major 175 biases due to the use of this approach as time-independent forcing was used here. 7 176 2) 3D configuration 177 The domain of the 3D configuration was 84◦ S to 84◦ N, again with walls specified at these 178 boundaries, with a horizontal resolution of 2◦ . The ocean depth was 3000 m, and there were 179 73 levels in the vertical direction with thicknesses (from top to bottom) of: 550 m, 57 layers 180 of 10 m each, 14, 20, 27, 38, 54, 75, 105, 147, and then 7 layers of 200 m each. In a steady 181 state, the upper 33 levels were inside the ice — the high 10 m depth resolution was needed 182 to resolve the relatively small variations in ice thickness. We used a reconstruction of the 183 land configuration at 720 Ma of Li et al. (2008). The standard run used a flat ocean bottom, 184 reflecting the uncertainty regarding Neoproterozoic bathymetry. To address this uncertainty, 185 we showed sensitivity experiments to bathymetry using prescribed Gaussian sills and ridges 186 of 1 km height. 187 The average geothermal heat flux was 0.1 W m−2 , as in the 2D case. The 720 Ma config- 188 uration of Li et al. (2008) also included estimates of the location of divergence zones (ocean 189 ridges). In these locations, the geothermal heat flux was up to four times the background; 190 we also presented sensitivity runs with uniform geothermal heat flux and with additional 191 geothermal heat flux at the ocean ridges. 192 The horizontal and vertical viscosity coefficients were 5×104 m2 s−1 and 2×10−3 m2 s−1 , 193 respectively. The lateral and vertical diffusion coefficients for both temperature and salinity 194 were 500 m2 s−1 and 10−4 m2 s−1 . As in the 2D configuration, the implicit vertical diffusion 195 scheme was used with an increased diffusion coefficient of 10 m2 s−1 in the case of statically 196 unstable stratification. The tracer acceleration method (Bryan 1984) was used in these runs 197 with a tracer time step of three hours and a momentum time step of 20 minutes. 198 c. Initial conditions 199 The initial ice thickness, both for the 2D and 3D ocean runs, was chosen with a balance 200 between the geothermal heat flux of 0.1 W m−2 and the mean atmospheric temperature of 8 201 -44◦ C in mind. As the 3D ocean model runs were highly time consuming, we choose an 202 initial ice-depth that is closer to the final steady state, instead of initiating the ocean model 203 with an uniform ice-depth. The initial ice depth was calculated by running the much faster 204 ice-flow model for thousands of years to a steady state when assuming zero melting at its 205 base. For the zonally symmetric 2D ocean runs, the initial ice depth for the ocean model 206 was chosen to be uniform in space. 207 Recent estimates of the mean ocean salinity in Snowball states lie somewhere between 208 the present day value of ∼35 and two times this value (∼70) (although see Knauth 2005), 209 based on the assumption that the ocean’s Neoproterozoic salt content prior to the Snowball 210 events was similar to present day values and that the mean ocean water depth was about 211 two kilometers, about half of present day values. This is based on an assumed 1 km sea level 212 equivalent land ice cover (Donnadieu et al. 2003; Pollard and Kasting 2004) and 1 km ice 213 cover over the ocean. We chose (somewhat arbitrarily) an initial salinity of 50. The initial 214 temperature was set to be uniform and equal to the freezing temperature based on an ice 215 depth of 1 km and the initial salinity described above, following Losch (2008), Tf = (0.0901 − 0.0575Sf )o − 7.61 × 10−4 pb , (2) 216 where Sf is the freezing salinity (in our case, the initial salinity), and pb is the pressure at 217 the bottom of the ice and is given in dBar. For an ice depth of 1 km and a salinity of 50, we 218 obtained an initial temperature of about −3.55◦ C. For salinities of 35 and 70, we obtained 219 freezing temperatures of ≈ −2.7◦ C and ≈ −4.7◦ C, respectively. 220 d. Coupling the models 221 The ice and ocean models were asynchronously coupled, each run for 300 years at a time. 222 The ice thickness was fixed during the ocean run, at the end of which the melting rate at 223 the base of the ice and the freezing temperature, calculated at each horizontal location by 224 the ocean model, were passed to the ice-flow model. The ice model was then run to update 9 225 the ice-thickness. The simulation ended after both models reached a steady state. Typically, 226 more than 30 ice-flow-ocean coupling steps (9,000 years) were required. 227 3. Zonally-averaged fields and MOC using a latitude- 228 depth ocean model 229 The ice thickness, the bottom freezing rate of the ice together with the atmospheric snow 230 accumulation minus sublimation, and the ice velocity of the 2D configuration at steady 231 state were already presented in AGLMST. The ice surface temperature and the net surface 232 accumulation rate are symmetric about the equator (following Pollard and Kasting 2005), 233 but the ice depth, the freezing rate at the bottom of the ice (calculated by the ocean model), 234 and the ice velocity are not, because the enhanced geothermal heat flux over the ridge at 235 20◦ N leads to thinner ice, larger melting, and a smaller ice velocity in the NH. The bottom 236 ice melting rate is maximal in two locations: (i) 20◦ N due to the maximum geothermal 237 heating, and (ii) at the equator due to the strong ocean dynamics (as will be shown below). 238 The ice thickness is around 1150 m on average, and varies over a range of only about 80 239 m. This small variation is due to the efficiency of the ice flow in homogenizing ice thickness 240 (Goodman and Pierrehumbert 2003). The small variations in ice-thickness are consistent 241 with previous studies (Tziperman et al. 2012; Pollard and Kasting 2005). 242 The density, and the vertical derivative of the density are plotted in Fig. 1a,b while the 243 oceanic potential temperature and salinity of AGLMST are presented in the top panels of 244 Fig. 2. Variations in temperature, salinity, and density are ∼0.3◦ C, ∼0.5, and ∼0.3 kg/m3 , 245 respectively. The ocean temperature is low because the high pressure at the bottom of the 246 (∼1 km) thick ice and the high salinity (∼49.5) reduce the freezing temperature. The small 247 variations in temperature at the top of the ocean (bottom of the ice), the large variations 248 in surface salinity, the similarity between the density and salinity fields, and an analysis 249 based on a linearized equation of state all indicate that changes in density are dominated by 10 250 salinity variations. The changes in salinity are brought about by melting over the enhanced 251 geothermal heat flux in the NH: the warmest water is close to the warm ridge, and the 252 freshest water is located above the top of the ridge. 253 A notable feature of the solution is the vertically well-mixed water column, except in the 254 vicinity of the geothermally heated ridge and the equator, where a very weak stratification 255 exists. This weak stratification is associated with melt water at the base of the ice as a 256 result of the enhanced heating there. This is also related to the zonal jets that are discussed 257 below and in the next section. The nearly vertically homogeneous potential density is used 258 to simplify the analytic analysis in the next section. 259 The zonal, meridional, vertical velocities, and the MOC, are shown in Fig. 1c,d and in 260 the top panel of Fig. 2. Surprisingly, the counterclockwise circulation is concentrated around 261 the equator, while velocities away from the equator, including over the ridge and enhanced 262 heating, are very weak. This result is explained in the next section. The simulated currents 263 are not small, as one would naively expect from a “stagnant” ocean under Snowball Earth 264 conditions (Kirschvink 1992), and the intensity of the circulation is close to that of the 265 present day. 266 Several additional features of the solution are worth noting: (i) there are two relatively 267 strong and opposite (anti-symmetric) jets (of a few cm s−1 ) in the zonal velocity, u (top 268 panel of Fig. 2). At the surface, we observe a westward current north of the equator and 269 an eastward current south of the equator. The meridional velocity (Fig. 1c) is symmetric 270 around the equator, with negative (southward) direction at the top of the ocean and positive 271 (northward) direction at the bottom of the ocean. (ii) The zonal and meridional velocities 272 are maximal (minimal) at the top and the bottom of the ocean, change sign with depth, 273 and vanish at the middle of the ocean. (iii) Both the zonal and meridional velocities decay 274 away from the equator where the zonal velocity decays much slower than the meridional 275 and vertical velocities. (iv) The MOC (top panel of Fig. 2) stream function, implied by the 276 vertical and meridional velocities, is largest at the equator and concentrated close to the 11 277 equator. (v) The vertical velocity w (Fig. 1d) is upward (positive) north of the equator, 278 downward (negative) south of the equator, vanishes at the equator and maximal at mid 279 ocean depth. 280 4. The dynamics of the equatorial MOC and zonal jets 281 Our goal in this section is to explain the dynamical features listed in the previous section. 282 We consider the steady state, zonally symmetric (x-independent) hydrostatic equations. For 283 simplicity, we use a Cartesian coordinate system centered at the equator with an equatorial β- 284 plane approximation. Then, following the numerical simulations, the advection and vertical 285 viscosity terms can be neglected from the momentum equations (not shown). Apart from the 286 fact that they are found to be small in the numerical simulation, the momentum advection 287 terms and the vertical viscosity may be shown to be small based on scaling arguments (see 288 Appendix). Based on the numerical results presented in section 3 and Fig. 1a, the density 289 is assumed to be independent of depth and the meridional density (pressure) gradient is 290 assumed to be approximately constant near the equator. 291 The dominant momentum balances are found to be −βyv = νh uyy , βyu = −py /ρ0 + νh vyy , pz = −gρ, (3) (4) (5) 292 where y and z are the meridional and depth coordinates, u and v are the zonal and meridional 293 velocities, β = df /dy (where f is the Coriolis parameter), νv and νh are the vertical and 294 horizontal eddy-parameterized viscosity coefficients, ρ is the density, ρ0 is the mean ocean 295 density, and g is the gravity constant. Vertically integrating the hydrostatic equation and 296 differentiating with respect to y we find that py = −ρy g(z + F (y)), where z = 0 is defined 12 297 to be at the ocean-ice interface and F (y) is an arbitrary function of y so that, βyu = 1 g(z + F (y))ρy + νh vyy . ρ0 (6) 298 It is possible to show that F (y) = H/2, by depth-integrating Eqs. (3),(6), using the fact that 299 the integrated meridional velocity should be zero due to the mass (or volume) conservation, 300 and by assuming that the depth-integrated zonal velocity vanishes at y → ±∞1 . 301 Eqs. (3) and (4) may be solved in terms of Airy functions, but we instead solve them 302 separately for the off-equatorial and equatorial regions and then match the two solutions, 303 leading to a more informative solution. As shown in AGLMST, for the off-equatorial region, 304 the viscosity term in Eq. (4) is negligible compared to the Coriolis term, leading to uoe = 305 g(z + H/2)ρy 1 . βρ0 y (7) This leads, based on Eq. (3), to the following meridional velocity away from the equator, voe = − 2g(z + H/2)νh ρy 1 , β 2 ρ0 y4 (8) 306 where the subscript “oe” stands for “off-equatorial”. Based on Eqs. (7), (8), it is clear that: 307 (i) both the zonal (u) and meridional (v) velocities decay away from the equator, where v 308 decays much faster than u; (ii) u is anti-symmetric about the equator, while v is symmetric; 309 and (iii) both u and v change signs at the mid-ocean depth, z = −H/2. 310 In the equatorial region, the Coriolis term is negligible in the meridional momentum 311 balance, while it still balances eddy viscosity in the zonal momentum equation, so that 312 Eqs. (3, 4) become 1 νh ue,yy + βyve = 0, (9) 1 g(z + H/2)ρy + νh ve,yy = 0, ρ0 (10) The integration of Eqs. (3),(6) leads to −βyV = νh Uyy = 0 and hence U = ρy gH(F (y) − H/2)/(ρ0 βy) where U ,V are the vertically integrated velocities. Thus V = 0 and U must be a linear function of y. Since U must vanish when y → ±∞, F (y) = H/2 and hence U = 0 for every y. 13 313 where the subscript “e” denotes the equatorial solution. These balances were verified from 314 the numerical solution, and it was found that the eddy viscosity term indeed varies linearly in 315 latitude around the equator. Continuing to assume, for simplicity, that the pressure gradient 316 term is approximately constant in latitude near the equator, the solution is a second-order 317 polynomial for v and a fifth-order polynomial for u. Requiring that the equatorial and 318 off-equatorial solutions match continuously at some latitude y0 one finds, 40νh2 10 y 3 80νh2 7 y gβρy (z + H/2) 5 y 5 y0 5 + − + + , ue = 40ρ0 νh2 y0 3β 2 y06 3 y03 3β 2 y06 3 y0 gρy (z + H/2) 2 y 2 4νh2 1 ve = − y0 + 2 6 −1 . 2ρ0 νh y02 β y0 (11) (12) 319 It is clear that ue is anti-symmetric in latitude, while ve is symmetric, as in the off-equatorial 320 region. The matching point between the off-equatorial and the equatorial velocities, y0 , can 321 be found by requiring that the derivative of the zonal velocity is continuous at y0 as well, 322 giving, 1/6 y0 = 40 323 324 325 326 327 328 νh β . Using y0 , the overall solution is y3 y 5 y5 gβρy (z+H/2) |y| < y0 y 0 y 5 − 3 y 3 + 3 y0 , 40ρ0 νh2 0 0 u(y) = g(z+H/2)ρy 1 , |y| ≥ y0 βρ0 y gρy (z+H/2) y 2 9 − y22 , |y| < y0 0 10 2ρ0 νh y0 v(y) = h ρy 1 − 2g(z+H/2)ν , |y| ≥ y0 β 2 ρ0 y4 The vertical velocity can be found from the continuity equation gρy (z + H/2)2 − H 2 y, |y| < y0 2ρ0 νh 4 w(y) = − 4gν2h ρy (z + H/2)2 − H 2 15 , |y| ≥ y0 β ρ0 4 y (13) (14) (15) (16) Note that w is not continuous at y0 . The half-width of the MOC cell, y1 , can be estimated by finding the location at which the meridional velocity vanishes and is 3 y1 = √ y0 . 10 14 (17) 329 The maximum meridional velocity vmax is found at the equator, either at the top or the 330 bottom of the ocean as vmax = 331 9gρy H 2 y . 40ρ0 νh 0 (18) The mean meridional velocity within the MOC cell boundaries is 2 hvi = vmax . 3 (19) 332 The maximal zonal velocity umax can be shown to be either at the surface or bottom of the 333 ocean with a value of umax ≈ 0.44vmax , 334 335 (20) q √ at y = ±y0 (9 − 21)/10 ≈ ±0.66y0 . ∗ The MOC stream function ψ(y, z) can be found by integrating v(y, z) = −ψz as gρy 2 ψ(y, z) = y 4ρ0 νh 0 y2 9 − 2 y0 10 H2 2 (z + H/2) − , 4 (21) 336 such that the stream function vanishes at the top (z = 0) and bottom (z = −H) of the 337 ocean. The maximum of the stream function is at mid-ocean depth at the equator (i.e., 338 y = 0 and z = −H/2) and is found to be ψmax = H vmax . 4 (22) 339 The stream function MOC, in Sv, is obtained by multiplying the above stream function by 340 the Earth’s perimeter. 341 The solution presented above accounts for nearly all the characteristics of the numerical 342 properties listed at the end of section 3. Namely: (i) the zonal velocity is anti-symmetric in 343 latitude (vanishing at the equator), and the meridional velocity is symmetric (maximal at 344 the equator); (ii) horizontal velocities obtain their maximum absolute value at the bottom 345 and the top of the ocean and change signs with depth; (iii) velocities decay away from the 346 equator, and the decay is faster for the meridional velocity; (iv) the meridional extent of the 347 MOC cell and its maximal value at the mid-depth at the equator are well predicted; and 15 348 (v) vertical velocity shows upwelling north of the equator, downwelling south of the equator, 349 zero at the equator, and the maximal vertical velocity at the mid-depth of the ocean. The 350 length scale associated with the dynamics depends on the horizontal viscosity and the β 351 Coriolis parameter. While β is well defined, the horizontal viscosity is unknown for Snowball 352 conditions. In our simulations, we used a value that is comparable to present day values for 353 1◦ resolution models; for larger horizontal viscosity, the approximations above (neglecting 354 the advection terms and vertical viscosity) become even more accurate. Horizontal viscosity 355 that is consistent with mixing length estimates, based on a high resolution, eddy resolving 356 1/8 of a degree calculations for the Snowball ocean AGLMST, yielded a higher value. 357 While the extent of the MOC cell is well constrained (by νh and β), its magnitude and the 358 magnitude of the velocities depend on the meridional density gradient, ρy , which we assumed 359 to be roughly constant and specified (from the numerical solution) near the equator. We now 360 attempt to develop a rough approximation for this density gradient, completing the above 361 discussion. 362 We integrate the time independent, zonally symmetric, salinity equation (vS)y +(wS)z = 363 κv Szz + κh Syy from bottom to top and from the southern boundary of the MOC cell (i.e., 364 from y = −y1 given in Eq. (17)) to the equator (y = 0), where we assume vS ≈ 0 and 365 κh Sy ≈ 0 at the southern edge of the MOC cell. We then use the surface boundary conditions 367 −κv Sz = S0 q/ρ0 where q is the freshwater flux due to ice melting/ freezing (in kg m−2 s−1 ), R R finding H −1 dz(vS − κh Sy ) = H −1 dy qS0 /ρ0 = (y1 /H)qS0 /ρ0 ; here we assume a constant 368 melting rate difference, q, over the MOC cell. Since the salinity contribution to density 369 variations dominates that of the temperature, we can multiply the equation by βS ρ0 (where 366 370 βS = 7.73 × 10−4 is the haline coefficient) to find an equation for the potential density, Z βS S0 M δ 1 0 dz(κh ρy − vρ) ≈ κh ρy − vmax (ρy y1 ) = βS S0 qy1 /H ≈ , (23) H −H λH 371 where vmax is the maximal meridional velocity (18). The freshwater flux over the MOC cell 372 may be related to the difference between the maximal geothermal heating and that of the 373 equator, δ (in W m−2 ) as follows: q ≈ M δ/(y1 λ), where λ = 334000 J kg−1 is the latent heat 16 374 of fusion and M is the distance between the central heating and the equator. The above is 375 based on the ice-shelf equations of the MITgcm (Losch 2008). Since vmax depends on ρy , it 376 is necessary to solve a quadratic equation to find ρy 2 . Following the above, we obtain the 377 following expression for ρy , 10ρ0 κh β ρy = 27gH 378 s 27δgM βS S0 1+ 5λκ2h ρ0 β 1− ! . (24) 5. Sensitivity tests of the 2D solution 379 We now present the results of sensitivity experiments for the latitude-depth 2D ocean 380 configuration, having two objectives in mind: (1) to examine the robustness of the results 381 discussed above, and (2) to examine the predictive power and accuracy of the analytic 382 approximations presented in section 4. 383 a. Sensitivity of the 2D numerical solution 384 The latitude-depth profiles of the temperature, salinity, meridional velocity, and the MOC 385 of the standard run and of the following sensitivity experiments are shown in Fig. 2 (from 386 the top row downward). All experiments started from the standard case described in section 387 3, with modifications from that configuration as follows, 388 389 390 i. Without a ridge. The geothermal heating is as in the standard case. ii. With the ridge and the geothermal heating centered at the equator. iii. Same as ii, including enhanced equatorial heating, but without the ridge. 2 It is possible to find the velocities when the density gradient is parabolic (ρ = γρ y 2 ) rather than linear. In this case, in off-equatorial regions, the meridional velocity is zero, while the zonal velocity is constant and equals to gz/γρ /βρ0 . Such an approximation is useful when geothermal heating is concentrated at the equator, a situation that, most probably, does not resemble Snowball conditions. 17 391 392 iv. With the ridge and geothermal heating located at 40◦ N instead of 20◦ N. v. With mean geothermal heating of 0.075 W/m2 instead of 0.1 W/m2 . 393 There are several common characteristics to the steady state solutions in all experiments. 394 First, the spatial variations in ice thickness do not exceed 100 m. Second, the temperature 395 and salinity are nearly independent of depth. Third, the ocean circulation is centered around 396 the equator, where the MOC cell is only a few degrees of latitude wide. Fourth, the zonal 397 velocity close to the bottom has an opposite sign from the zonal velocity at the top of the 398 ocean. All the above features are similar to those of the standard run and in agreement with 399 the analytic approximations presented in section 4. This indicates that the solutions shown 400 and analyzed above are indeed robust and represent a wide range of geometries and forcing 401 fields. 402 As expected, the warmest and freshest waters are located close to the location of en- 403 hanced geothermal heating. Still, the equatorial ocean response (velocities and MOC) is 404 not sensitive to the location of the ridge or geothermal heating once the heating is located 405 outside the tropics (top, fourth, and bottom rows of Fig. 2). This is expected from the ana- 406 lytic approximation, presented above, that basically depends on the density gradient across 407 the equator, which does not change dramatically when the ridge and heating are located at 408 different latitudes outside the equatorial region. 409 However, when the ridge and/or geothermal heating are located exactly at the equator 410 (second and third rows of Fig. 2), the density gradient exactly at the equator is almost 411 zero, and the equatorial water depth is affected by the ridge. In these cases, the zonal 412 velocity does not change signs across the equator, as in all the other, off-equatorial heating 413 experiments. This is consistent with a parabolic density profile, which may be analyzed 414 similarly to the linear profile discussed in section 4. The zonal and meridional velocities still 415 change signs with depth in this case, and are still limited to near the equator. Moreover, the 416 MOC in the absence of an equatorial ridge (third row of Fig. 2) is about four times larger 417 compared to the case with the equatorial ridge (second row of Fig. 2), consistent with the 18 418 analytic approximation [Eqs. (21),(22)] that predicts that the MOC intensity will increase as 419 a function of the water depth at the equator. In the case of equatorial heating, the system is 420 symmetric, and the MOC can be either clockwise (second row of Fig. 2) or counterclockwise 421 (third row of Fig. 2). We did not observe a solution with two equatorial MOC cells in these 422 2D latitude-depth experiments, although in principle such a situation may be possible. 423 When the mean geothermal heating is reduced from 0.1 to 0.075 W/m2 (bottom row of 424 Fig. 2), the ice becomes thicker by about 25% and the circulation is weaker compared to the 425 standard case, due to the weaker meridional density gradient that results from the weaker 426 geothermal heating gradients. 427 In addition to the above experiments, we also performed an experiment without a ridge 428 and with uniform geothermal heating; these changes led to an MOC cell of ∼8 Sv, sig- 429 nificantly weaker than the standard case. This experiment suggests that the atmospheric 430 temperature, which is now the only source of meridional gradients in melting and freezing, 431 is responsible for about one quarter of the MOC intensity, as the circulation with local- 432 ized geothermal heating is about 35 Sv. When using uniform atmospheric temperature and 433 uniform geothermal heating, the circulation vanishes. We also initialized the model with 434 present day salinity (35 ppt) and two times the present day salinity (70 ppt), and obtained a 435 circulation that is similar to the standard run; these salinity sensitivity experiments suggest 436 that the dynamics of Snowball ocean do not strongly depend on the mean salinity. 437 b. A broader exploration of parameter space 438 To examine the range of applicability of the analytic approximations presented in section 439 4, we used an idealized configuration and large parameter variations, covering and exploring 440 a large regime in the parameter space. 441 In the reference experiment of this set, the ice thickness was kept constant in time and 442 space (i.e., the ocean was not coupled to the ice-flow model); the ice thickness was set to 443 1124 m so that the base of the ice was 1124 × ρi /ρw = 1011 m, as heat diffusion through this 19 444 ice thickness exactly balances a mean geothermal heat flux of 0.1 W/m2 , based on a globally 445 averaged ice-surface temperature; we used a flat ocean bottom (no ridge), a geothermal 446 heat flux as for the standard case discussed above with the difference between the maximal 447 heating and background heating of ∆Q = 0.225 W/m2 (i.e., mean geothermal heating of 448 0.1 W/m2 with enhanced heating concentrated around 20◦ N, at which the maximal heating 449 is four times larger than the background), a horizontal viscosity of νh = 2 × 105 m2 s−1 , a 450 vertical viscosity of νv = 2 × 10−3 m2 s−1 , horizontal and vertical diffusion coefficients of 451 temperature and salinity of κh = 2000 m2 s−1 and κv = 2 × 10−4 m2 s−1 , and an ocean depth 452 of H =2000 m. We used a latitude-depth configuration with a meridional extent from 84◦ S 453 to 84◦ N and 2◦ resolution (the edge grid points are assumed to be land points); 21 vertical 454 levels were used, with an upper level, completely embedded within the ice, having thickness 455 of 1 km and additional 20 levels, each of them 100 m thick. The different experiments were 456 run until a steady state was reached. 457 We performed the following experiments, all starting from the reference experiment de- 458 scribed above with the following modifications, 459 1. Reference experiment as described above. 460 2. Ten times deeper ocean, 10H. 461 3. Ten times shallower ocean, H/10. 462 4. Uniform geothermal heat flux, ∆Q = 0. 463 5. Difference between the maximal geothermal heat flux and the background of 3∆Q ≈ 0.608 464 W/m2 ; the maximum heat flux is 18 times larger than the background. 465 6. Rotation that is 1/4 of the Earth’s rotation; i.e., the β-plane coefficient becomes β/4. 466 7. Rotation that is 1/9 of Earth’s rotation; i.e., the β-plane coefficient becomes β/9. 467 8. Sixteen times larger horizontal viscosity coefficient, 16νh . 20 468 9. Four times smaller horizontal viscosity coefficient, νh /4. 469 10. Four times larger horizontal diffusion coefficient, 4κh . 470 11. Four times smaller horizontal diffusion coefficient, κh /4. 471 12. Sixteen times larger horizontal viscosity coefficient, 16νh , and four times larger horizontal 472 473 474 diffusion coefficient, 4κh . 13. Four times larger horizontal viscosity coefficient, 4νh , and four times larger horizontal diffusion coefficient, 4κh . 475 14. Ten times smaller vertical diffusion coefficient, κv /10. 476 15. Four times smaller horizontal viscosity coefficient, νh /4, and a four times smaller horizon- 477 tal diffusion coefficient, κh /4. 478 The results of these numerical experiments are compared with the analytical scaling solu- 479 tions in Fig. 3. As the horizontal eddy viscosity becomes larger, the analytic approximations 480 become more accurate, as the neglected momentum advection terms become even smaller 481 than the horizontal eddy viscosity term. Four measures were considered: maximum zonal 482 velocity, maximum meridional velocity, maximum MOC, and half-width of the MOC cell. 483 All four measures yielded a good correlation between numerical experiments and analytic 484 expressions with a correlation coefficient higher than or equal to 0.87, pointing to a good 485 correspondence between the analytic approximations and the numerical results. Yet, there 486 are systematic quantitative biases in the analytic results relative to the numerical solutions. 487 The predicted maximal zonal velocity is more than two times smaller than the numerical one, 488 while the predicted maximal meridional velocity is about 30% larger than the numerical one. 489 In the analytic approximation, the maximal zonal velocity is 44% of the maximal meridional 490 velocity, while in the numerical simulations, the maximal zonal velocity is larger than 67% 491 of the maximal meridional velocity. Similarly, the predicted maximal MOC is 30% larger 492 than the numerical one. The difference between the numerical and analytic approximations 21 493 may be attributed to the terms neglected in the analytic approximation, to the piece-wise 494 analytic solution (solving for the equatorial and off-equatorial regions instead of solving for 495 both simultaneously using Airy and hypergeometrical functions), and to the assumption of 496 a linear latitudinal density gradient. 497 We found a relatively high correlation coefficient of 0.95 for the comparison between 498 the half-width of the MOC cell of the numerical results and the numerical approximation. 499 Still, the MOC cell width is larger in the numerical results by about 50%. According to 500 the analytic approximation, the half-width in the MOC cell only depends on the horizontal 501 viscosity and the β parameter (i.e., it is proportional to (νh /β)1/3 )–other parameters, such 502 as the density gradient, ρy , which may be associated with larger uncertainties, do not appear 503 in the expression for the width of the MOC cell. This high correlation coefficient strengthens 504 the first part of the analytic approximation, which can be obtained once a specific density 505 gradient ρy is given. 506 Our scaling estimate of the density gradient ρy (Eq. 24) leaves room for improvement. 507 Yet, overall, the analytic approximations provide a reasonable estimate, within factor 2, of 508 the numerical solutions. 509 6. 3D ocean model solution with a reconstructed Neo- 510 proterozoic continental configuration 511 We proceed to describe steady solutions of the 3D near-global ocean model coupled to 512 the 2D ice flow model. Our objective is to examine if and how the insights obtained above, 513 using the 2D ocean model, change due to the added dimension and presence of continents. 514 We can also examine a more realistic geothermal forcing, and study the sensitivity to the 515 geothermal heating and bathymetry that are not well constrained by observations. 22 516 a. Reference state 517 For the simulation using the 3D ocean model coupled to the 2D ice flow model, we 518 followed the configuration described in section 2. Our standard 3D run included enhanced 519 localized geothermal heating along spreading centers following Li et al. (2008), as indicated 520 by the solid black contour line in Fig. 4. 521 The ice thickness and velocity field are shown in Fig. 4a. The ice is generally thicker than 522 1 km. As in Tziperman et al. (2012), the ice is thinner in the constricted sea area between 523 the land masses, both due to the ice sublimation and melting there (see below) and due to 524 the reduced ice flow into this region due to the friction with the land masses. The differences 525 in ice thickness can reach 240 m, significantly more than in the 1D case without continents 526 (Campbell et al. 2011; Tziperman et al. 2012). As expected, the general ice flow is directed 527 from the high latitudes towards the equator (i.e., from snow/ice accumulation areas to ice 528 sublimation/melting areas) with a velocity of up to 35 m y−1 in the region of the constricted 529 sea. 530 The temperature, salinity, and density fields close to the base of the ice cover are shown 531 in Fig. 4. The warmest and freshest waters are found within the constricted sea area (Fig. 4), 532 due to the enhanced warming and melting in this region associated with the localized geother- 533 mal heating. Thus, the surface water is lighter in this region (bottom right panel of Fig. 4). 534 As in the 2D simulation described in section 3, temperature and salinity are almost inde- 535 pendent of depth in most areas, except very close to the ice in the constricted sea area. This 536 confirms the assumption of a vertically uniform density used in the analytic derivations of 537 section 4, as well as the assumption of density variations, mostly in the meridional direction. 538 The differences in temperature, salinity, and density in the 3D simulations are smaller than 539 those of the 2D simulations. This is a result of the zonally restricted region of enhanced 540 geothermal heating, relative to the latitudinal band of heating prescribed in the 2D case. 541 In contrast to the temperature and salinity, whose distribution can be directly linked to 542 geothermal heating, the velocities of the 3D simulations are concentrated near the equator 23 543 (Fig. 5), similar to the zonally symmetric 2D results (Figs. 1,2). The continents do not 544 inhibit the formation of strong equatorial zonal jets. Also similar to the 2D results, and as 545 predicted by the analytic expressions, the zonal and meridional velocities change signs with 546 depth and the vertical velocity does not. Yet, the latitudinal symmetry properties of the 3D 547 run are somewhat different from those of the 2D standard run shown in Fig. 1 and the top 548 panel of Fig. 2, as further discussed below. 549 Fig. 5 shows that the continents have some effect on the currents — currents, in particular 550 the equatorial zonal jets, that either encounter the continents or flow away from them lead 551 to boundary currents and to upwelling and downwelling close to the continents. The weak 552 salinity stratification over the enhanced geothermal heating regions allows some heating of 553 the deep water to occur, and the upwelling of warmer, geothermally heated, bottom water 554 near the continents. The latter can lead to enhanced melting, especially at high model 555 resolution (AGLMST). However, the coarse resolution of the current model, the absence 556 of detailed continental-shelf bathymetry, and the inability of our ice-flow model to handle 557 bottom bathymetry do not allow us to draw more specific conclusions on the implications for 558 the existence of open water (a potential refuge for photosynthetic life) due to this upwelling. 559 A very close similarity between the zonally symmetric model and the more realistic- 560 geometry 3D simulation is seen in the zonal mean temperature, salinity and velocity fields 561 of the 3D run (Fig. 6). The tracers are vertically well mixed and are almost independent of 562 depth; where the ocean is weakly stratified, there is a “cap” of fresh and warm water due 563 to the heating and melting in the vicinity of the geothermal heating. The temperature and 564 salinity range in the ocean interior are only about 0.15 ◦ C and 0.05 ppt, respectively, leading 565 to a density range of 0.06 kg m−3 . 566 The zonal mean velocities (Fig. 6) are concentrated around the equator as in the 2D case, 567 but their latitudinal symmetry properties are somewhat different from those of the standard 568 2D run, described in sections 3 and 4 and shown in Fig. 2. It is possible to see two opposite 569 zonal jets at the equator, just below the ice. However, below these jets, the zonal velocity 24 570 converges into a single symmetric jet that is similar to the one in the equatorially heated case 571 shown in Fig. 2. The zonal jet changes its sign with depth as before. The meridional velocity 572 also exhibits a different symmetry compared to the standard 2D simulations in Figs. 1,2. 573 In the 3D case, the meridional velocity is almost symmetric in latitude just below the ice 574 and becomes anti-symmetric below that, indicating the presence of two opposite MOC cells 575 with poleward velocity at the upper ocean. The meridional velocity also changes sign with 576 depth. The vertical velocity is consistent with the equatorial cells formed by the meridional 577 velocity, with rising motion at the equator. 578 The two MOC cells (Fig. 7) – a southern, counterclockwise cell, with a maximum flux of 579 15 Sv and a northern, clockwise cell, with a maximum flux of 20 Sv – are weaker than in the 580 standard 2D run (section 3 and Figs. 1,2), although the range of the stream function of 36 581 Sv is similar to that seen in the 2D standard run. The extent of the cells is several degrees 582 latitude, as for the standard 2D run, and as predicted by the analytic approximation. We 583 will show below that the presence of the two cells is a result of the presence of continents. 584 b. 3D sensitivity to bathymetry and geothermal heat flux distribution 585 The bathymetry of the Neoproterozoic is poorly constrained, and in order to examine 586 the robustness of our results with respect to this factor, we performed three additional 587 3D-ocean/2D-ice-flow sensitivity runs based on the standard 3D run described in previous 588 subsection a: Run (i) uses a uniform geothermal heat flux of 0.1 W m−2 , run (ii) has a 1 km 589 high sill between the continents around the constricted sea area, and run (iii) has the same 590 sill as run (ii) and additional zonal and meridional mid-ocean ridges that are also regions of 591 enhanced geothermal heating (the mean geothermal heat flux is again 0.1 W m−2 ). 592 A summary of the results (potential density and MOC) of the three experiments is shown 593 in Fig. 8. In experiment (i), the freshest water is not in the vicinity of the constricted sea 594 (as in the standard case shown in Fig. 4), but at the low latitudes of the open ocean, due 595 to the elimination of the enhanced melting region within the constricted sea. Because we 25 596 removed the differential geothermal heating, the difference in density is smaller compared 597 to the standard case. The zonal mean potential density is almost uniform with depth, as 598 for the 2D and 3D results presented above. The MOC is concentrated around the equator 599 as before; the details of the MOC are different though, due to the uniform heat flux. The 600 existence of two cells in both the standard 3D run and in Experiment (i) confirms that the 601 existence of two MOC cells is due to the presence of the continents rather than the locally 602 enhanced geothermal heat flux in the standard run. 603 The additional sill of 1 km height between the continents in Experiment (ii) leads to a 604 similar circulation and density pattern as for the 3D standard run (middle row of Fig. 8), 605 although the MOC is weaker because the bottom water circulation is blocked in the region 606 of the constricted sea. The presence of sills also alters the location of the freshest water. 607 One expects mid-ocean ridges to have extents that are roughly similar to those of the 608 present day. Experiment (iii), with such ridges specified, in necessarily arbitrary locations, 609 and with enhanced geothermal heat flux over these ridges, resulted in a circulation and 610 density field that are similar to the standard 3D run (bottom panels of Fig. 8). Here, 611 however, the MOC cell is stronger due to the larger heating in the NH (over the high NH 612 latitude ridge). 613 Finally, an additional 3D run, similar to the standard 2D run (discussed in section 3), 614 with no continents and with a global configuration, led to results that were almost identical 615 to those of the 2D standard run. 616 7. Summary and conclusions 617 We find that the steady circulation under a thick (∼1000 m) ice cover in a Snowball 618 Earth scenario is composed of an equatorial MOC and zonal jets. The MOC amplitude is 619 comparable to the present day North Atlantic MOC, yet is restricted to within a couple of 620 degrees latitude around the equator. These results are supported by 2D (latitude-depth) and 26 621 3D simulations with an ocean GCM. These are found to be robust with respect to geometry 622 and forcing parameters, and are consistent with analytical approximations derived from the 623 equations of motion. The analytic solution indicates that a horizontal equatorial density 624 gradient leads to a pressure gradient that, in turn, drives the MOC and zonal jets. Eddy 625 viscosity plays an important role in these dynamics, determining the meridional extent of 626 the MOC. 627 Given that the temperature, salinity and density are essentially vertically uniform in 628 nearly all locations, due to convective instability driven by the geothermal heat flux, we chose 629 not to use eddy parameterizations developed for the very different modern-day ocean (Gent 630 and McWilliams 1990). Instead, we use a simple formulation with constant strictly horizontal 631 and vertical eddy coefficients. The horizontal eddy viscosity and eddy mixing coefficients 632 are smaller than the ones predicted by a high resolution eddy resolving run (AGLMST); the 633 results of that runs confirm our results. Note that larger viscosity and diffusion coefficients 634 lead to a better agreement with the analytical prediction. An alternative approach was 635 taken by Ferreira et al. (2011) (their appendix C), who used the GM scheme and found 636 strong eddy-driven high latitude meridional cells, different from the equatorial circulation 637 found here. While their run is not at a steady state due to the lack of geothermal heat flux 638 and their ice cover is only 200 m thick, these results are very interesting and suggest that 639 further study of the role of eddies in a Snowball ocean is worthwhile. Such a study, in a 640 dynamical regime very far from that of the present-day ocean, may lead to new insights on 641 eddy dynamics that may enrich our understanding of ocean dynamics in modern conditions 642 as well. 643 An important goal of studying snowball ocean circulation is to aid geologists and geo- 644 chemists in the interpretation of the geological, geochemical and paleontological record. 645 Geochemical studies sometimes assume that the ocean was stagnant and not well mixed. 646 The first important lesson from the present study is that one expects the ocean to be well 647 mixed in the vertical nearly everywhere, as indicated by the vertically uniform tempera- 27 648 ture and salinity profiles, due to the geothermal heat flux. The second related lesson is 649 the presence of a relatively strong zonal circulation and meridional overturning circulation 650 which would have together further mixed the ocean horizontally and vertically. Ferreira 651 et al. (2011) also found a very weak stratification and strong MOC cells, although at higher 652 latitudes rather than at the equator as found here. But it does seem that the snowball ocean 653 needs to be thought of as well mixed rather than stagnant, and that one cannot assume the 654 deep water to be disconnected from the surface ocean. It is, admittedly, difficult to come 655 up with additional specific insights that are directly relevant to the observed record, and 656 it may take future geochemical studies to explore the consequences of the circulation and 657 stratification reported here. It is worth noting that much of the present study dealt with the 658 large scale ocean circulation in deep ocean basins, while the preserved geological record is 659 mostly from shelf and shallow areas that have not been subducted by now. We do note that 660 our study identifies strong tendency for near-coast upwelling and downwelling, as a result 661 of a combination of the weak stratification and the encounter of horizontal (mostly zonal) 662 currents and land masses, and this may have some geological relevance as well. 663 Acknowledgments. 664 We thank Aiko Voigt and an anonymous reviewer for their most helpful comments. This 665 work was supported by NSF Climate Dynamics, P2C2 Program, grant ATM-0902844 (ET, 666 YA) and NSF Climate Dynamics Program, grant ATM-0917468 (ET). ET thanks the Weiz- 667 mann Institute for its hospitality during parts of this work. YA thanks the Harvard EPS 668 Department for a most pleasant and productive sabbatical visit. 28 APPENDIX 669 670 Scaling of idealized 2D configuration 671 672 673 We start from the β-plane momentum equations under the assumptions of steady state (i.e., ∂t = 0 and zonal symmetry ∂x = 0) vuy + wuz − βyv = νh uyy + νv uzz , vvy + wvz + βyu = − (A1) 1 py + νh vyy + νv vzz , ρ0 (A2) 674 It is possible to switch to nondimensional variables as follows: y = (νh /β)1/3 ŷ, z = H ẑ 675 (H is the depth of the ocean), p = gHρy (νh /β)1/3 p̂, u = (gHρy )/(ρ0 β 2/3 νh )û, v = 676 (gHρy )/(ρ0 β 2/3 νh )v̂, w = (gH 2 ρy )/(ρ0 β 1/3 νh )ŵ, where the hat indicates nondimensional 677 variables. Then Eqs. (A1)-(A2) become: 1/3 1/3 678 2/3 ε1 v̂ûŷ + ε1 ŵûẑ − ŷv̂ = ûŷŷ + ε2 ûẑẑ , (A3) ε1 v̂v̂ŷ + ε1 ŵv̂ẑ + ŷû = −p̂ŷ + v̂ŷŷ + ε2 v̂ẑẑ . (A4) where ε1 = ε2 = gHρy 1, ρ0 βnuh νv 1/3 H 2 β 2/3 nuh (A5) 1, (A6) 679 are small parameters under our choice of parameters, ≈ 8 × 10−3 , ≈ 2 × 10−5 respectively. 680 Thus, it is possible to neglect the advection and vertical viscosity terms from the momentum 681 equations. 29 682 683 REFERENCES 684 Abbot, D., A. Voigt, and D. Koll, 2011: The Jormungand global climate state and implica- 685 686 687 688 689 tions for Neoproterozoic glaciations. J. Geophys. Res., 116, D18 103. Abbot, D. S. and I. Halevy, 2010: Dust aerosol important for snowball earth deglaciation. J. Climate, 23 (15), 4121–4132. Abbot, D. S. and R. T. Pierrehumbert, 2010: Mudball: Surface dust and snowball earth deglaciation. J. Geophys. Res., 115. 690 Abbot, D. S., A. Voigt, M. Branson, R. T. Pierrehumbert, D. Pollard, G. Le Hir, and D. D. 691 Koll, 2012: Clouds and Snowball Earth deglaciation. Geophys. Res. Lett., 39, L20 711. 692 Allen, P. A. and J. L. Etienne, 2008: Sedimentary challenge to snowball earth. Nature 693 694 695 Geoscience, 1, 817. Ashkenazy, Y., H. Gildor, M. Losch, F. A. Macdonald, D. P. Schrag, and E. Tziperman, 2013: Dynamics of a Snowball Earth ocean. Nature, 495, 90–93, doi:10.1038/nature11 894. 696 Baum, S. and T. Crowley, 2001: Gcm response to late precambrian (similar to 590 ma) 697 ice-covered continents. Geophys. Res. Lett., 28 (4), 583–586, doi:10.1029/2000GL011557. 698 Baum, S. and T. Crowley, 2003: The snow/ice instability as a mechanism for rapid climate 699 change: A Neoproterozoic Snowball Earth model example. Geophys. Res. Lett., 30 (20), 700 doi:10.1029/2003GL017333. 701 702 Bryan, K., 1984: Accelerating the convergence to equilibrium of ocean-climate models. J. Phys. Oceanogr., 14, 666–673. 30 703 704 Budyko, M. I., 1969: The effect of solar radiatin variations on the climate of the earth. Tellus, 21, 611–619. 705 Campbell, A. J., E. D. Waddington, and S. G. Warren, 2011: Refugium for surface life on 706 Snowball Earth in a nearly-enclosed sea? A first simple model for sea-glacier invasion. 707 Geophys. Res. Lett., 38, 10.1029/2011GL048 846. 708 Chandler, M. A. and L. E. Sohl, 2000: Climate forcings and the initiation of low-latitude ice 709 sheets during the Neoproterozoic Varanger glacial interval. J. Geophys. Res., 105 (D16), 710 20 737–20 756. 711 712 Crowley, T. and S. Baum, 1993: Effect of decreased solar luminosity on late Precambrian ice extent. J. Geophys. Res., 98 (D9), 16 723–16 732, doi:10.1029/93JD01415. 713 Donnadieu, Y., F. Fluteau, G. Ramstein, C. Ritz, and J. Besse, 2003: Is there a conflict 714 between the Neoproterozoic glacial deposits and the snowball Earth interpretation: an 715 improved understanding with numerical modeling. Earth Planet. Sci. Lett., 208 (1-2), 716 101–112. 717 Donnadieu, Y., Y. Godderis, G. Ramstein, A. Nedelec, and J. Meert, 2004a: A ’snow- 718 ball Earth’ climate triggered by continental break-up through changes in runoff. Nature, 719 428 (6980), 303–306. 720 Donnadieu, Y., G. Ramstein, F. Fluteau, D. Roche, and A. Ganopolski, 2004b: The impact 721 of atmospheric and oceanic heat transports on the sea-ice-albedo instability during the 722 Neoproterozoic. Clim. Dyn., 22 (2-3), 293–306. 723 Evans, D. A. D. and T. D. Raub, 2011: Neoproterozoic glacial palaeolatitudes: a global 724 update. The Geological Record of Neoproterozoic Glaciations, E. Arnaud, G. P. Halverson, 725 and G. Shields-Zhou, Eds., London, Geological Society of London, Vol. 36, 93–112. 31 726 727 728 729 730 731 732 733 Ferreira, D., J. Marshall, and B. E. J. Rose, 2011: Climate determinism revisited: multiple equilibria in a complex climate model. J. Climate, 24, 992–1012. Gent, P. R. and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr., 20 (1), 150–155. Goodman, J. C., 2006: Through thick and thin: Marine and meteoric ice in a ”Snowball Earth” climate. Geophys. Res. Lett., 33 (16). Goodman, J. C. and R. T. Pierrehumbert, 2003: Glacial flow of floating marine ice in ”Snowball Earth”. J. Geophys. Res., 108 (C10). 734 Harland, W. B., 1964: Evidence of late Precambrian glaciation and its significance. Problems 735 in Palaeoclimatology, A. E. M. Nairn, Ed., John Wiley & Sons, London, 119–149, 180–184. 736 Hoffman, P. and D. Schrag, 2002: The snowball Earth hypothesis: testing the limits of global 737 738 739 740 741 change. Terra Nova, 14 (3), 129–155, doi:10.1046/j.1365-3121.2002.00408.x. Hyde, W. T., T. J. Crowley, S. K. Baum, and W. R. Peltier, 2000: Neoproterozoic ’snowball earth’ simulations with a coupled climate/ice-sheet model. Nature, 405, 425–429. Jackett, D. R. and T. J. McDougall, 1995: Minimal adjustment of hydrographic profiles to achieve static stability. J. Atmos. Ocean Tech., 12 (4), 381–389. 742 Jenkins, G. and S. Smith, 1999: Gcm simulations of snowball earth conditions during the 743 late proterozoic. Geophys. Res. Lett., 26 (15), 2263–2266, doi:10.1029/1999GL900538. 744 Kirschvink, J. L., 1992: Late Proterozoic low-latitude glaciation: the snowball Earth. The 745 Proterozoic Biosphere, J. W. Schopf and C. Klein, Eds., Cambridge University Press, 746 Cambridge, 51–52. 747 748 Knauth, L., 2005: Temperature and salinity history of the Precambrian ocean: implications for the course of microbial evolution. Paleonogr. Paleoclim. Paleoecol., 219, 53–69. 32 749 Langen, P. L. and V. A. Alexeev, 2004: Multiple equilibria and asymmetric climates in the 750 ccm3 coupled to an oceanic mixed layer with thermodynamic sea ice. Geophys. Res. Lett., 751 31, L04 201. 752 753 Le-Hir, G., Y. Donnadieu, G. Krinner, and G. Ramstein, 2010: Toward the snowball earth deglaciation... Clim. Dyn., 35 (2-3), 285–297, doi:10.1007/s00382-010-0748-8. 754 Le-Hir, G., G. Ramstein, Y. Donnadieu, and R. T. Pierrehumbert, 2007: Investigating 755 plausible mechanisms to trigger a deglaciation from a hard snowball Earth. Comptes rendus 756 - Geosci., 339 (3-4), 274–287, doi:10.1016/j.crte.2006.09.002. 757 Lewis, J., M. Eby, A. Weaver, S. Johnston, and R. Jacob, 2004: Global glaciation in the 758 neoproterozoic: Reconciling previous modelling results. Geophys. Res. Lett., 31 (8), doi: 759 10.1029/2004GL019725. 760 761 762 763 764 765 766 767 768 769 770 771 Lewis, J. P., A. J. Weaver, and M. Eby, 2007: Snowball versus slushball Earth: Dynamic versus nondynamic sea ice? J. Geophys. Res., 112. Lewis, J. P., A. J. Weaver, S. T. Johnston, and M. Eby, 2003: Neoproterozoic ”snowball Earth”: Dynamic sea ice over a quiescent ocean. Paleoceanography, 18 (4). Li, D. and R. T. Pierrehumbert, 2011: Sea glacier flow and dust transport on Snowball Earth. Geophys. Res. Lett., 38, 10.1029/2011GL048 991. Li, Z. X., et al., 2008: Assembly, configuration, and break-up history of Rodinia: A synthesis. Precambrian Res., 160, 179–210. Losch, M., 2008: Modeling ice shelf cavities in a z-coordinate ocean general circulation model. J. Geophys. Res., 113, C08 043. MacAyeal, D., 1997: EISMINT: Lessons in ice-sheet modeling. Tech. rep., University of Chicago, Chicago, Illinois. 33 772 Macdonald, F. A., et al., 2010: Calibrating the Cryogenian. Science, 327 (5970), 1241–1243. 773 Marotzke, J. and M. Botzet, 2007: Present-day and ice-covered equilibrium states in a 774 comprehensive climate model. Geophys. Res. Lett., 34, L16 704. 775 Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997: A finite-volume, incom- 776 pressible Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. 777 Res., 102, C3, 5,753–5,766. 778 779 McKay, C. P., 2000: Thickness of tropical ice and photosynthesis on a snowball Earth. Geophys. Res. Lett., 27 (14), 2153–2156. 780 Micheels, A. and M. Montenari, 2008: A snowball Earth versus a slushball Earth: Results 781 from Neoproterozoic climate modeling sensitivity experiments. Geosphere, 4 (2), 401–410. 782 Morland, L., 1987: Unconfined ice-shelf flow. Dynamics of the West Antarctic Ice Sheet, 783 784 785 786 787 788 789 790 791 792 793 C. van der Veen and J. Oerlemans, Eds., D. Reidel, Boston. Pierrehumbert, R. T., 2002: The hydrologic cycle in deep-time climate problems. Nature, 419 (6903), 191–198. Pierrehumbert, R. T., 2004: High levels of atmospheric carbon dioxide necessary for the termination of global glaciation. Nature, 429 (6992), 646–649. Pierrehumbert, R. T., 2005: Climate dynamics of a hard snowball Earth. J. Geophys. Res., 110 (D1). Pierrehumbert, R. T., D. S. Abbot, A. Voigt, and D. Koll, 2011: Climate of the neoproterozoic. Ann. Rev. of Earth and Planet. Sci., 39, 417–460. Pollack, H., S. Hurter, and J. Johnson, 1993: Heat flow from the Earth’s interior: analysis of the global data set. Rev. Geophys., 31, 267–280. 34 794 Pollard, D. and J. Kasting, 2004: Climate-ice sheet simulations of Neoproterozoic glaciation 795 before and after collapse to Snowball Earth. Geophysical Monograph series, 146, 91–105. 796 Pollard, D. and J. F. Kasting, 2005: Snowball Earth: A thin-ice solution with flowing sea 797 glaciers. J. Geophys. Res., 110 (C7). 798 Pollard, D. and J. F. Kasting, 2006: Reply to comment by Stephen G. Warren and Richard 799 E. Brandt on “Snowball Earth: A thin-ice solution with flowing sea glaciers”. J. Geophys. 800 Res., 111 (C9), doi:10.1029/2006JC003488. 801 802 803 804 Poulsen, C., R. T. Pierrehumbert, and R. L. Jacobs, 2001a: Impact of ocean dynamics on the simulation of the Neoproterozoic “snowball Earth”. Geophys. Res. Lett., 28, 1575–1578. Poulsen, C. J. and R. L. Jacob, 2004: Factors that inhibit snowball Earth simulation. Paleoceanography, 19 (4). 805 Poulsen, C. J., R. L. Jacob, R. T. Pierrehumbert, and T. T. Huynh, 2002: Testing paleo- 806 geographic controls on a Neoproterozoic snowball Earth. Geophys. Res. Lett., 29 (11). 807 Poulsen, C. J., R. T. Pierrehumbert, and R. L. Jacob, 2001b: Impact of ocean dynamics 808 on the simulation of the Neoproterozoic ”snowball Earth”. Geophys. Res. Lett., 28 (8), 809 1575–1578. 810 811 812 813 814 815 816 817 Romanova, V., G. Lohmann, and K. Grosfeld, 2006: Effect of land albedo, co2, orography, and oceanic heat transport on extreme climates. Climate of the Past, 2 (1), 31–42. Rose, B. E. J. and J. Marshall, 2009: Ocean heat transport, sea ice, and multiple climate states: Insights from energy balance models. J. Atmos. Sci., 66 (9), 2828–2843. Schrag, D. P., R. A. Berner, P. F. Hoffman, and G. P. Halverson, 2002: On the initiation of a snowball Earth. Geochemistry Geophysics Geosystems, 3, doi:10.1029/2001GC000219. Schrag, D. P., P. F. Hoffman, W. Hyde, et al., 2001: Life, geology and snowball earth. NATURE-LONDON-, 306–306. 35 818 819 Sellers, W., 1969: A global climate model based on the energy balance of the Earthatmosphere system. J. Appl. Meteorol., 8, 392–400. 820 Sohl, L. E. and M. A. Chandler, 2007: Reconstructing Neoproterozoic palaeoclimates using 821 a combined data/modelling approach. Deep-Time Perspectives on Climate Change: Mar- 822 rying the Signal from Computer Models and Biological Proxies, M. M. Williams, A. M. 823 Hatwood, J. Gregory, and D. N. Schmidt, Eds., Geological Society, Micropalaeontological 824 Society Special Publication #2, 61–80. 825 826 Stein, C. A. and S. Stein, 1992: A model for the global variation in oceanic depth and heat flow with lithospheric age. Nature, 359, 123–129. 827 Tziperman, E., D. S. Abbot, Y. Ashkenazy, H. Gildor, D. Pollard, C. Schoof, and D. P. 828 Schrag, 2012: Continental constriction and sea ice thickness in a Snowball-Earth scenario. 829 J. Geophys. Res., 117 (C05016), 10.1029/2011JC007 730. 830 Tziperman, E., I. Halevy, D. T. Johnston, A. H. Knoll, and D. P. Schrag, 2011: Biologically 831 induced initiation of Neoproterozoic Snowball-Earth events. Proc. Natl. Acad. Sci. U.S.A., 832 108 (37), 15 09115 096, doi/10.1073/pnas.1016361 108. 833 834 Voigt, A. and D. S. Abbot, 2012: Sea-ice dynamics strongly promote Snowball Earth initiation and destabilize tropical sea-ice margins. Clim. Past, 8, 2079–2092. 835 Voigt, A., D. S. Abbot, R. T. Pierrehumbert, and J. Marotzke, 2011: Initiation of a Marinoan 836 Snowball Earth in a state-of-the-art atmosphere-ocean general circulation model. Clim. 837 Past, 7, 249–263, doi:10.5194/cp-7-249-2011. 838 839 840 841 Voigt, A., I. M. Held, and J. Marotzke, 2012: Hadley cell dynamics in a virtually dry snowball earth atmosphere. J. Atmos. Sci., 69 (1), 116–128. Voigt, A. and J. Marotzke, 2010: The transition from the present-day climate to a modern Snowball Earth. Climate Dynamics, 35 (5), 887–905. 36 842 Warren, S. G. and R. E. Brandt, 2006: Comment on “Snowball Earth: A thin-ice solution 843 with flowing sea glaciers” by David Pollard and James F. Kasting. J. Geophys. Res., 844 111 (C9), 10.1029/2005JC003 411. 845 846 Warren, S. G., R. E. Brandt, T. C. Grenfell, and C. P. McKay, 2002: Snowball Earth: Ice thickness on the tropical ocean. J. Geophys. Res., 107 (C10). 847 Weertman, J., 1957: Deformation of floating ice shelves. J. Glaciology, 3 (21), 38–42. 848 Yang, J., W. P. Peltier, and Y. Hu, 2012a: The initiation of modern soft and hard Snowball 849 Earth climates in CCSM4. Clim. Past, 8, 907918. 850 Yang, J., W. P. Peltier, and Y. Hu, 2012b: The initiation of modern soft snowball and hard 851 snowball climates in CCSM3. part i: The inuences of solar luminosity, CO2 concentration, 852 and the sea ice/snow albedo parameterization. J. Climate, 25, 2711–2736. 853 Yang, J., W. R. Peltier, and Y. Hu, 2012c: The initiation of modern “soft snowball” and 854 “hard snowball” climates in CCSM3. part ii: Climate dynamic feedbacks. J. Climate, 855 25 (8), 2737–2754, doi:10.1175/JCLI-D-11-00190.1. 37 856 857 List of Figures 1 (a) Density (kg m−3 ), (b) the depth derivative of the density (kg m−4 ), (c) 858 meridional velocity v (cm s−1 ), and (d) vertical velocity w (cm s−1 ), at steady 859 state of the latitude-depth standard run. The white area at the top of the plot 860 represents the ice cover and the white area at the bottom of the panels indi- 861 cates the ridge that has enhanced geothermal heating. The thick contour line 862 in panels a,b represents the zero contour line of panel b, separating the stable 863 stratification around the equator from the unstable stratification elsewhere. 864 Note that the significant circulation is confined to the equatorial regions. 865 2 41 A summary of the latitude-depth 2D profiles of the sensitivity experiments. 866 The four columns show the temperature, salinity, zonal velocity, and MOC 867 (presented between 40◦ S and 40◦ N). The contour line in the first and second 868 columns separates the vertically stable ocean regions from the unstable ones 869 while the contour line in the third column indicates the zero velocity. First 870 row: standard run, after AGLMST. Second row: same as standard run but 871 without the ridge (the geothermal heat flux is as in the standard case). Third 872 row: same as standard but with ridge and enhanced heating placed at the 873 equator. Fourth row: same as second row but without the ridge (yet with 874 an enhanced equatorial heating). Fifth row: same as standard run but with 875 ridge and enhanced heating centered at 40◦ N. Sixth row: same as standard 876 but with mean geothermal heat flux of 0.075 W/m2 instead of 0.1. 38 42 877 3 The analytic approximations vs. the numerical results for the experiments 878 described in the text (Experiment 4 of uniform geothermal heating and uni- 879 form ice-surface temperature is not presented as it resulted, as expected, in a 880 stagnant ocean). Top left: maximum zonal velocity (cm s−1 ). Top right: maxi- 881 mum meridional velocity (cm s−1 ). Bottom left: maximum MOC (Sv) Bottom 882 right: half-width of the MOC cell (degree latitude). The solid line shows the 883 linear regression where the correlation coefficients are 0.88, 0.87, 0.87, and 884 0.95, for the top-left, top-right, bottom-left, and bottom-right panels, respec- 885 tively. The dashed line indicates the “identity” line. When assuming that 886 the regression lines cross the (0,0) point the slopes of the curves are 0.56, 887 1.47, 1.63, and 0.63 for the top-left, top-right, bottom-left, and bottom-right 888 panels, respectively–the correlation coefficients are the same as the above. 889 4 Results of the 3D standard run. Ice thickness and ice velocity (top left panel), 890 potential temperature (top right panel), salinity (bottom left panel), and den- 891 sity (bottom right panel), all under the ice, at a depth of 1.2 km. The black 892 solid contour line indicates the location of geothermal heating. Ice-depth 893 temperature and salinity are after AGLMST. 894 5 panels), and vertical (bottom panels) velocities, near the ice bottom (at a 896 depth of 1.1 km, left panels) and at 2.9 km (right panels). 6 44 Circulation in the standard 3D run. Zonal (upper panels), meridional (middle 895 897 43 45 Zonal averages of the 3D standard run of potential temperature (top left), 898 salinity (middle left), density (bottom left), zonal velocity (top right), merid- 899 ional velocity (middle right), and vertical velocity (bottom right). Solid con- 900 tour lines indicate positive values while dashed contour lines indicate negative 901 values. 46 The MOC of the 3D standard run. 47 902 7 39 903 8 Results of the 3D sensitivity experiments. Density at a depth of 2.5 km (left 904 panels), zonal mean density (middle column panels), and MOC (right panels), 905 for standard run but with uniform geothermal heating (upper panels), as for 906 standard run but with sills (middle row panels), and as for standard run 907 but with sills and geothermally heated ridges (bottom panels). The dashed 908 contour lines indicate fresher water. The thick solid contour lines indicate the 909 location of the geothermal heating. 48 40 Fig. 1. (a) Density (kg m−3 ), (b) the depth derivative of the density (kg m−4 ), (c) meridional velocity v (cm s−1 ), and (d) vertical velocity w (cm s−1 ), at steady state of the latitude-depth standard run. The white area at the top of the plot represents the ice cover and the white area at the bottom of the panels indicates the ridge that has enhanced geothermal heating. The thick contour line in panels a,b represents the zero contour line of panel b, separating the stable stratification around the equator from the unstable stratification elsewhere. Note that the significant circulation is confined to the equatorial regions. 41 Fig. 2. A summary of the latitude-depth 2D profiles of the sensitivity experiments. The four columns show the temperature, salinity, zonal velocity, and MOC (presented between 40◦ S and 40◦ N). The contour line in the first and second columns separates the vertically stable ocean regions from the unstable ones while the contour line in the third column indicates the zero velocity. First row: standard run, after AGLMST. Second row: same as standard run but without the ridge (the geothermal heat flux is as in the standard case). Third row: same as standard but with ridge and enhanced heating placed at the equator. Fourth row: same as second row but without the ridge (yet with an enhanced equatorial heating). Fifth row: same as standard run but with ridge and enhanced heating centered at 40◦ N. Sixth row: same as standard but with mean geothermal heat flux of 0.075 W/m2 instead of 0.1. 42 Numerical results Max. u (cm/s) 1. 2. 10H 3. 0.1H 5. ~3∆Q 6. β/4 7. β/9 8. 16νh 1.5 1 2.14 0.5 0 A 0 1 0.5 Max. v (cm/s) A 9. νh/4 10. 4κh 11. κh/4 12. 16νh, 4κh 13. 4νh, 4κh 14. 0.1κv 15. νh/4, κh/4 1.5 1 0.5 0 0 1.5 0.69 A 1 0.5 1.5 MOC half width (degrees) Max. MOC (Sv) Numerical results 300 10 200 1.46 A 5 100 0.72 0 0 A 100 200 0 0 300 Analytical approximation 5 10 Analytical approximation Fig. 3. The analytic approximations vs. the numerical results for the experiments described in the text (Experiment 4 of uniform geothermal heating and uniform ice-surface temperature is not presented as it resulted, as expected, in a stagnant ocean). Top left: maximum zonal velocity (cm s−1 ). Top right: maximum meridional velocity (cm s−1 ). Bottom left: maximum MOC (Sv) Bottom right: half-width of the MOC cell (degree latitude). The solid line shows the linear regression where the correlation coefficients are 0.88, 0.87, 0.87, and 0.95, for the top-left, top-right, bottom-left, and bottom-right panels, respectively. The dashed line indicates the “identity” line. When assuming that the regression lines cross the (0,0) point the slopes of the curves are 0.56, 1.47, 1.63, and 0.63 for the top-left, top-right, bottom-left, and bottom-right panels, respectively–the correlation coefficients are the same as the above. 43 11111 11 11 1111 1 1 11111 Fig. 4. Results of the 3D standard run. Ice thickness and ice velocity (top left panel), potential temperature (top right panel), salinity (bottom left panel), and density (bottom right panel), all under the ice, at a depth of 1.2 km. The black solid contour line indicates the location of geothermal heating. Ice-depth temperature and salinity are after AGLMST. 44 Fig. 5. Circulation in the standard 3D run. Zonal (upper panels), meridional (middle panels), and vertical (bottom panels) velocities, near the ice bottom (at a depth of 1.1 km, left panels) and at 2.9 km (right panels). 45 Fig. 6. Zonal averages of the 3D standard run of potential temperature (top left), salinity (middle left), density (bottom left), zonal velocity (top right), meridional velocity (middle right), and vertical velocity (bottom right). Solid contour lines indicate positive values while dashed contour lines indicate negative values. 46 Fig. 7. The MOC of the 3D standard run. 47 Fig. 8. Results of the 3D sensitivity experiments. Density at a depth of 2.5 km (left panels), zonal mean density (middle column panels), and MOC (right panels), for standard run but with uniform geothermal heating (upper panels), as for standard run but with sills (middle row panels), and as for standard run but with sills and geothermally heated ridges (bottom panels). The dashed contour lines indicate fresher water. The thick solid contour lines indicate the location of the geothermal heating. 48

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