ashkenazy-etal2013 jpo
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Ocean circulation under globally glaciated Snowball Earth
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conditions: steady state solutions
Yosef Ashkenazy
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∗
Ben-Gurion University, Midreshet Ben-Gurion, Israel
Hezi Gildor
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The Fredy and Nadine Herrmann Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel
Martin Losch
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Alfred-Wegener-Institut, Helmholtz-Zentrum für Polar- und Meeresforschung, Bremerhaven, Germany
Eli Tziperman
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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]
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ABSTRACT
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Between ∼750 to 635 million years ago, during the Neoproterozoic era, the Earth experienced
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at least two significant, possibly global, glaciations, termed “Snowball Earth”. While many
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studies have focused on the dynamics and the role of the atmosphere and ice flow over the
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ocean in these events, only a few have investigated the related associated ocean circulation,
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and no study has examined the ocean circulation under a thick (∼1 km deep) sea-ice cover,
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driven by geothermal heat flux. Here, we use a thick sea-ice flow model coupled to an ocean
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general circulation model to study the ocean circulation under Snowball Earth conditions.
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We first investigate the ocean circulation under simplified zonal symmetry assumption and
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find (i) strong equatorial zonal jets, and (ii) a strong meridional overturning cell, limited
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to an area very close to the equator. We derive an analytic approximation for the latitude-
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depth ocean dynamics and find that the extent of the meridional overturning circulation cell
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only depends on the horizontal eddy viscosity and β (the change of the Coriolis parameter
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with latitude). The analytic approximation closely reproduces the numerical results. Three-
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dimensional ocean simulations, with reconstructed Neoproterozoic continents configuration,
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confirm the zonally symmetric dynamics, and show additional boundary currents and strong
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upwelling and downwelling near the continents.
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1. Introduction
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The Neoproterozoic Snowball events are perhaps the most drastic climate events in
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Earth’s history. Between 750 and 580 million years ago (Ma), the Earth experienced at
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least two major, possibly global, glaciations (e.g., Harland 1964; Kirschvink 1992; Hoffman
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and Schrag 2002; Macdonald et al. 2010; Evans and Raub 2011). During these events (the
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Sturian and Marinoan ice ages), ice extended to low latitudes over both ocean and land. It
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is still debated whether the ocean was entirely covered by thick ice (“hard” Snowball) (e.g.,
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Allen and Etienne 2008; Pierrehumbert et al. 2011), perhaps expect very limited regions of
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sea-ice free ocean, e.g., around volcanic islands (Schrag et al. 2001) (that could have pro-
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vided a refuge for photosynthetic life during these periods), or whether the tropical ocean
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was partially ice free or perhaps covered by thin ice (“soft” Snowball) (e.g., Yang et al.
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2012c).
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The initiation, maintenance, and termination of such a climatic condition pose a first-
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order problem in ocean and climate dynamics. One may argue that the Snowball state was
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predicted by simple energy balance models (EBMs) (Budyko 1969; Sellers 1969). Snowball
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dynamics also provide a test-case for our understanding of the climate system as manifested
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in climate models. Therefore, in recent years, these questions have been the focus of nu-
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merous studies and attempts to simulate these climate states using models with different
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levels of complexity. The role and dynamics of atmospheric circulation and heat trans-
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port, CO2 concentration, cloud feedbacks, and continental configuration have been studied
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(Pierrehumbert 2005; Le-Hir et al. 2010; Donnadieu et al. 2004a; Pierrehumbert 2002, 2004;
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Le-Hir et al. 2007). Recently, the effect of clouds, as well as the role of atmospheric and
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oceanic heat transports in the initiation of Snowball Earth events was studied; these studies
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were based on atmospheric GCMs and used different setups and configurations including
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different CO2 concentrations, different continental configurations, and different sea-ice dy-
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namics (Yang et al. 2012c,b,a; Voigt and Abbot 2012; Abbot et al. 2012). It was concluded,
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e.g., that sea-ice dynamics has important role in the initiation of Snowball events (Voigt
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and Abbot 2012). Additionally, perceived difficulties in exiting a Snowball state by a CO2
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increase alone motivated the study of the role of dust over the Snowball ice cover (Abbot
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and Pierrehumbert 2010; Le-Hir et al. 2010; Li and Pierrehumbert 2011; Abbot and Halevy
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2010).
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A simple scaling calculation of balancing geothermal heat input into the ocean with heat
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escaping through the ice by diffusion leads to an estimated ice thickness of 1 km. The ice
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cover is expected to slowly deform and flow toward the equator to balance for sublimation
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(and melting at the bottom of the ice) at low latitudes and snow accumulation (and ice
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freezing at the bottom of the ice) at high latitude. The flow and other properties of such
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thick ice over a Snowball ocean (“sea glaciers”, Warren et al. 2002) were examined in quite
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a few recent studies (Goodman and Pierrehumbert 2003; McKay 2000; Warren et al. 2002;
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Pollard and Kasting 2005; Campbell et al. 2011; Tziperman et al. 2012; Pollard and Kasting
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2006; Warren and Brandt 2006; Goodman 2006; Lewis et al. 2007). Snowball Earth global
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ice cover is an extreme example within a range of multiple ice cover equilibrium states,
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which have been studied in a range of simple and complex models (e.g., Langen and Alexeev
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2004; Rose and Marshall 2009; Ferreira et al. 2011). In contrast to these many studies of
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different climate components during Snowball events, the ocean circulation during Snowball
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events has received little attention. Most model studies of a Snowball climate used an ocean
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mixed layer model only (Baum and Crowley 2001; Crowley and Baum 1993; Baum and
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Crowley 2003; Hyde et al. 2000; Jenkins and Smith 1999; Chandler and Sohl 2000; Poulsen
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et al. 2001b; Romanova et al. 2006; Donnadieu et al. 2004b; Micheels and Montenari 2008).
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The studies that used full ocean General Circulation Models (GCMs) concentrated on the
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ocean’s role in Snowball initiation and aftermath (Poulsen et al. 2001a; Poulsen and Jacob
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2004; Poulsen et al. 2002; Sohl and Chandler 2007), or other aspects of Snowball dynamics
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in the presence of oceanic feedback (Voigt et al. 2011; Le-Hir et al. 2007; Yang et al. 2012c;
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Ferreira et al. 2011; Marotzke and Botzet 2007; Lewis et al. 2007; Voigt and Marotzke 2010;
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Abbot et al. 2011; Lewis et al. 2004, 2003). Yet none of these studies employing ocean
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GCMs accounted for the combined effects of thick ice cover flow and driving by geothermal
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heating. Ferreira et al. (2011) simulated an ocean under a moderately thick (200 m) ice cover
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with no geothermal heat flux, and calculated a non steady-state solution with near-uniform
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temperature and salinity. They described a vanishing Eulerian circulation together with
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strongly parameterized eddy-induced high latitude circulation cells.
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With both the initiation (Kirschvink 1992; Schrag et al. 2002; Tziperman et al. 2011)
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and termination (Pierrehumbert 2004) of Snowball events still not well understood, and the
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question of hard vs. soft Snowball still unresolved (Pierrehumbert et al. 2011), our focus here
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is the steady state ocean circulation under a thick ice cover (hard Snowball). By examining
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ocean dynamics under such an extreme climatic state, we aim to better understand the
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relevant climate dynamics, and perhaps even provide constraints on the issues regarding soft
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vs. hard Snowball states.
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To study the 3D ocean dynamics under a thick ice cover, it is necessary to have a two-
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dimensional (longitude and latitude) ice-flow model, and this was recently developed by
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Tziperman et al. (2012), based on the ice-shelf equations of Morland (1987) and MacAyeal
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(1997), extending the 1D model of Goodman and Pierrehumbert (2003). This model is
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coupled here to the MITgcm (Marshall et al. 1997). Another challenge in studying the 3D
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ocean dynamics under a thick ice cover is that thick ice with lateral variations of hundreds of
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meters (as under Snowball conditions) poses a numerical challenge as standard ocean models
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cannot handle ice that extends through several vertical layers; we use the ice-shelf model of
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Losch (2008), which allows for this. An alternative, vertically scaled coordinates, was used
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by Ferreira et al. (2011).
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This paper expands on results briefly reported in Ashkenazy et al. (2013) (hereafter
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AGLMST), and we report the details of the steady state ocean dynamics under a thick ice
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(Snowball) cover, analytically and numerically, when both geothermal heating and a thick ice
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flow are taken into account. We find the ocean circulation to be quite far from the stagnant
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pool envisioned in some early studies, and very different from that in any other period in
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Earth’s history. In particular, the stratification is very weak as might be expected (Ferreira
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et al. 2011), and is dominated by salinity gradients due to melting and freezing of ice; we
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find a meridional overturning circulation that is confined to the equatorial region, significant
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zonal equatorial jets, and strong equatorial meridional overturning circulation (MOC).
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The paper is organized as follows. We first describe the models and configurations used
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in this study (section 2). We then present the results of the latitude-depth ocean model
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coupled to a 1D (latitude) ice-flow model when geothermal heating is taken into account
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(section 3). Analytically approximated solutions of the 2D, latitude-depth ocean model are
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then presented (section 4). Section 5 presents sensitivity runs to study the robustness of
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both the numerical results and the analytical approximations, followed by the steady state
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results of a 3D ocean model coupled to a longitude-latitude 2D ice-flow model in section 6.
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The results are discussed and summarized in section 7.
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2. Model description
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a. Ice-flow model
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The ice-flow model solves for the ice depth and velocity over an ocean as a function of
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longitude and latitude, in the presence of continents (Tziperman et al. 2012). The model
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extends the 1D model of Goodman and Pierrehumbert (2003), which was based on the Weert-
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man (1957) formula for ice shelf deformation. Because this specific formulation cannot be
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extended to ice flow in two horizontal dimensions, we instead used the ice-shelf approxima-
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tion (Morland 1987; MacAyeal 1997) that can be extended to two dimensions. The ice-shelf
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approximation implies a depth-independent ice velocity, and in addition, the vertical tem-
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perature profile within the ice is assumed to be linear (Goodman and Pierrehumbert 2003).
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The temperature at the upper ice surface and surface ice sublimation and snow accumulation
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are prescribed from the energy balance of Pollard and Kasting (2005) and are assumed to be
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constant in time. The temperature and melting/freezing rates at the bottom of the ice are
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calculated by the ocean model. The model’s spatial resolution is set to that of the ocean,
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and the model is run in either 1D (latitude only) or 2D configurations, depending on the
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ocean model used; it is typically 1-2◦ .
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b. The ocean model—MITgcm
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We used the Massachusetts Institute of Technology general circulation model (MITgcm,
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Marshall et al. 1997), a free-surface, primitive equation ocean model that uses z coordinates
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with partial cells in the vertical axis; we use a longitude-latitude grid. To account for the thick
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ice, we used the ice-shelf package of the MITgcm (Losch 2008) that allows ice thicknesses
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that span many vertical layers. Parameter values followed Losch (2008). The ocean was
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forced at the bottom with a spatially variable (but constant in time) geothermal heat flux.
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The equation of state used here (Jackett and McDougall 1995) was tuned for the present
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day ocean, while the temperature and salinity we used to simulate Snowball conditions were
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somewhat outside this range. Sensitivity tests, using mean present day salinity and mean
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salinity that is two times larger than the present day value, showed no sensitivity of the results
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for the circulation. The ocean model was run at two different configurations, including a
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zonally symmetric 2D configuration and a near-global 3D configuration, described as follows.
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1) Latitude-depth configuration
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In the 2D runs, the spatial resolution was 1◦ with 32 vertical levels spanning a depth
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of 3000 m, with vertical level thicknesses (from top to bottom) of 920, 15×10, 12, 17, 23,
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32, 45, 61, 82, 110, 148, 7×200 m; the uppermost level was entirely within the ice. The
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steady state ice thickness was calculated by the ice model to be approximately 1 km with
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lateral variations of less than 100 m. The latitudinal extent of the 2D configuration was from
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84◦ S to 84◦ N with walls specified at these boundaries to avoid having to deal with the polar
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singularity of the spherical coordinates. The bathymetry was either flat or had a Gaussian
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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)
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In the standard configuration, the ridge was located at φ0 = 20◦ N, to schematically represent
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paleoclimatic estimates of more tectonic divergence zones in the Northern Hemisphere (NH).
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We choose the bottom geothermal heat flux to have the same form of Eq. (1) such that it
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is proportional to the height of the ridge (Stein and Stein 1992). The maximal geothermal
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heating was four times larger than the background, with a spatial mean value of 0.1 W/m2 ,
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as for present day; in the standard 2D run presented below, the maximal geothermal heat
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was ∼0.3 W/m2 while the background geothermal heat, far from the ridge, was ∼0.08 W/m2 .
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The mean value of 0.1 W/m2 was based on the mean present day oceanic geothermal heat
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fluxes, given in Table 4 of Pollack et al. (1993).
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The lateral and vertical viscosity coefficients were 2×104 m2 s−1 and 2×10−3 m2 s−1 . The
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lateral and vertical tracer diffusion coefficients were 200 m2 s−1 and 10−4 m2 s−1 . To be
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conservative, the horizontal viscosity and diffusion coefficients were chosen to be larger than
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those estimated based on eddy resolving runs presented in AGLMST. Static instabilities
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in the water column were removed by increasing the vertical diffusion to 10 m2 s−1 . Their
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large values required an implicit scheme for solving the diffusion equations. We note that
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our simulations do not incorporate the effect of vertical diffusion of momentum which was
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shown to be important in atmospheric dynamics under Snowball Earth conditions (Voigt
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et al. 2012).
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For efficiency, we used the tracer acceleration method of Bryan (1984), with a tracer
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time step of 90 minutes and a momentum time step of 18 minutes. We did not expect major
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biases due to the use of this approach as time-independent forcing was used here.
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2) 3D configuration
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The domain of the 3D configuration was 84◦ S to 84◦ N, again with walls specified at these
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boundaries, with a horizontal resolution of 2◦ . The ocean depth was 3000 m, and there were
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73 levels in the vertical direction with thicknesses (from top to bottom) of: 550 m, 57 layers
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of 10 m each, 14, 20, 27, 38, 54, 75, 105, 147, and then 7 layers of 200 m each. In a steady
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state, the upper 33 levels were inside the ice — the high 10 m depth resolution was needed
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to resolve the relatively small variations in ice thickness. We used a reconstruction of the
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land configuration at 720 Ma of Li et al. (2008). The standard run used a flat ocean bottom,
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reflecting the uncertainty regarding Neoproterozoic bathymetry. To address this uncertainty,
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we showed sensitivity experiments to bathymetry using prescribed Gaussian sills and ridges
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of 1 km height.
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The average geothermal heat flux was 0.1 W m−2 , as in the 2D case. The 720 Ma config-
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uration of Li et al. (2008) also included estimates of the location of divergence zones (ocean
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ridges). In these locations, the geothermal heat flux was up to four times the background;
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we also presented sensitivity runs with uniform geothermal heat flux and with additional
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geothermal heat flux at the ocean ridges.
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The horizontal and vertical viscosity coefficients were 5×104 m2 s−1 and 2×10−3 m2 s−1 ,
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respectively. The lateral and vertical diffusion coefficients for both temperature and salinity
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were 500 m2 s−1 and 10−4 m2 s−1 . As in the 2D configuration, the implicit vertical diffusion
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scheme was used with an increased diffusion coefficient of 10 m2 s−1 in the case of statically
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unstable stratification. The tracer acceleration method (Bryan 1984) was used in these runs
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with a tracer time step of three hours and a momentum time step of 20 minutes.
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c. Initial conditions
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The initial ice thickness, both for the 2D and 3D ocean runs, was chosen with a balance
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between the geothermal heat flux of 0.1 W m−2 and the mean atmospheric temperature of
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-44◦ C in mind. As the 3D ocean model runs were highly time consuming, we choose an
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initial ice-depth that is closer to the final steady state, instead of initiating the ocean model
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with an uniform ice-depth. The initial ice depth was calculated by running the much faster
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ice-flow model for thousands of years to a steady state when assuming zero melting at its
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base. For the zonally symmetric 2D ocean runs, the initial ice depth for the ocean model
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was chosen to be uniform in space.
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Recent estimates of the mean ocean salinity in Snowball states lie somewhere between
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the present day value of ∼35 and two times this value (∼70) (although see Knauth 2005),
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based on the assumption that the ocean’s Neoproterozoic salt content prior to the Snowball
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events was similar to present day values and that the mean ocean water depth was about
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two kilometers, about half of present day values. This is based on an assumed 1 km sea level
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equivalent land ice cover (Donnadieu et al. 2003; Pollard and Kasting 2004) and 1 km ice
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cover over the ocean. We chose (somewhat arbitrarily) an initial salinity of 50. The initial
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temperature was set to be uniform and equal to the freezing temperature based on an ice
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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)
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where Sf is the freezing salinity (in our case, the initial salinity), and pb is the pressure at
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the bottom of the ice and is given in dBar. For an ice depth of 1 km and a salinity of 50, we
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obtained an initial temperature of about −3.55◦ C. For salinities of 35 and 70, we obtained
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freezing temperatures of ≈ −2.7◦ C and ≈ −4.7◦ C, respectively.
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d. Coupling the models
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The ice and ocean models were asynchronously coupled, each run for 300 years at a time.
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The ice thickness was fixed during the ocean run, at the end of which the melting rate at
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the base of the ice and the freezing temperature, calculated at each horizontal location by
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the ocean model, were passed to the ice-flow model. The ice model was then run to update
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the ice-thickness. The simulation ended after both models reached a steady state. Typically,
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more than 30 ice-flow-ocean coupling steps (9,000 years) were required.
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3. Zonally-averaged fields and MOC using a latitude-
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depth ocean model
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The ice thickness, the bottom freezing rate of the ice together with the atmospheric snow
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accumulation minus sublimation, and the ice velocity of the 2D configuration at steady
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state were already presented in AGLMST. The ice surface temperature and the net surface
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accumulation rate are symmetric about the equator (following Pollard and Kasting 2005),
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but the ice depth, the freezing rate at the bottom of the ice (calculated by the ocean model),
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and the ice velocity are not, because the enhanced geothermal heat flux over the ridge at
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20◦ N leads to thinner ice, larger melting, and a smaller ice velocity in the NH. The bottom
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ice melting rate is maximal in two locations: (i) 20◦ N due to the maximum geothermal
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heating, and (ii) at the equator due to the strong ocean dynamics (as will be shown below).
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The ice thickness is around 1150 m on average, and varies over a range of only about 80
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m. This small variation is due to the efficiency of the ice flow in homogenizing ice thickness
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(Goodman and Pierrehumbert 2003). The small variations in ice-thickness are consistent
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with previous studies (Tziperman et al. 2012; Pollard and Kasting 2005).
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The density, and the vertical derivative of the density are plotted in Fig. 1a,b while the
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oceanic potential temperature and salinity of AGLMST are presented in the top panels of
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Fig. 2. Variations in temperature, salinity, and density are ∼0.3◦ C, ∼0.5, and ∼0.3 kg/m3 ,
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respectively. The ocean temperature is low because the high pressure at the bottom of the
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(∼1 km) thick ice and the high salinity (∼49.5) reduce the freezing temperature. The small
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variations in temperature at the top of the ocean (bottom of the ice), the large variations
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in surface salinity, the similarity between the density and salinity fields, and an analysis
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based on a linearized equation of state all indicate that changes in density are dominated by
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salinity variations. The changes in salinity are brought about by melting over the enhanced
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geothermal heat flux in the NH: the warmest water is close to the warm ridge, and the
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freshest water is located above the top of the ridge.
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A notable feature of the solution is the vertically well-mixed water column, except in the
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vicinity of the geothermally heated ridge and the equator, where a very weak stratification
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exists. This weak stratification is associated with melt water at the base of the ice as a
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result of the enhanced heating there. This is also related to the zonal jets that are discussed
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below and in the next section. The nearly vertically homogeneous potential density is used
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to simplify the analytic analysis in the next section.
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The zonal, meridional, vertical velocities, and the MOC, are shown in Fig. 1c,d and in
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the top panel of Fig. 2. Surprisingly, the counterclockwise circulation is concentrated around
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the equator, while velocities away from the equator, including over the ridge and enhanced
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heating, are very weak. This result is explained in the next section. The simulated currents
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are not small, as one would naively expect from a “stagnant” ocean under Snowball Earth
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conditions (Kirschvink 1992), and the intensity of the circulation is close to that of the
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present day.
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Several additional features of the solution are worth noting: (i) there are two relatively
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strong and opposite (anti-symmetric) jets (of a few cm s−1 ) in the zonal velocity, u (top
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panel of Fig. 2). At the surface, we observe a westward current north of the equator and
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an eastward current south of the equator. The meridional velocity (Fig. 1c) is symmetric
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around the equator, with negative (southward) direction at the top of the ocean and positive
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(northward) direction at the bottom of the ocean. (ii) The zonal and meridional velocities
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are maximal (minimal) at the top and the bottom of the ocean, change sign with depth,
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and vanish at the middle of the ocean. (iii) Both the zonal and meridional velocities decay
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away from the equator where the zonal velocity decays much slower than the meridional
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and vertical velocities. (iv) The MOC (top panel of Fig. 2) stream function, implied by the
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vertical and meridional velocities, is largest at the equator and concentrated close to the
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equator. (v) The vertical velocity w (Fig. 1d) is upward (positive) north of the equator,
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downward (negative) south of the equator, vanishes at the equator and maximal at mid
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ocean depth.
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4. The dynamics of the equatorial MOC and zonal jets
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Our goal in this section is to explain the dynamical features listed in the previous section.
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We consider the steady state, zonally symmetric (x-independent) hydrostatic equations. For
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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
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viscosity terms can be neglected from the momentum equations (not shown). Apart from the
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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)
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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
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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
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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)
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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 .
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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)
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where the subscript “oe” stands for “off-equatorial”. Based on Eqs. (7), (8), it is clear that:
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(i) both the zonal (u) and meridional (v) velocities decay away from the equator, where v
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decays much faster than u; (ii) u is anti-symmetric about the equator, while v is symmetric;
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and (iii) both u and v change signs at the mid-ocean depth, z = −H/2.
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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
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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.
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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
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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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