On the Settling Depth of Meltwater Escaping from Beneath Antarctic Ice

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1 On the settling depth of meltwater escaping from beneath Antarctic ice

2 shelves

∗ 3 Constantin W. Arnscheidt and John Marshall

4 Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of

5 Technology, Cambridge, MA, USA

6 Pierre Dutrieux

7 British Antarctic Survey, Natural Environment Research Council, Cambridge, UK and

8 Lamont-Doherty Earth Observatory of Columbia University, Palisades, NY, USA

9 Craig D. Rye

10 Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of

11 Technology, Cambridge, MA, USA and Goddard Institute for Space Studies, New York City, NY,

12 USA

13 Ali Ramadhan

14 Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of

15 Technology, Cambridge, MA, USA

∗ 16 Corresponding author: Constantin W. Arnscheidt, [email protected]

1 ABSTRACT

17 Antarctic glacial meltwater likely plays an important role in determining large-scale Southern

18 Ocean climate trends, yet recent modeling eﬀorts have proceeded without a good understanding

19 of how its vertical distribution in the water column is set. To rectify this, here we conduct new

20 large-eddy simulations of a buoyant meltwater plume escaping from underneath an Antarctic ice

21 shelf. We ﬁnd that the meltwater’s settling depth is primarily a function of the buoyancy forcing per

22 unit width and the ambient stratiﬁcation, consistent with the classical theory of turbulent buoyant

23 plumes and in contrast to previous work that suggested an important role for centrifugal instability.

24 Our results further highlight the signiﬁcant role played by the variability in stratiﬁcation; this helps

25 explain observed interannual variability in the vertical meltwater distribution near Pine Island

26 Glacier. Because of the vast heterogeneity in mass loss rates from diﬀerent Antarctic ice shelves,

27 a dynamic parameterization of meltwater settling depth may be crucial for accurately simulating

28 high-latitude climate in a warming world; the work presented in this study is a ﬁrst step towards its

29 development.

2 30 1. Introduction

31 A notable failure of the global coupled climate models included in the Coupled Model Intercom-

32 parison Project Phase 5 (CMIP5, Taylor et al. 2012) has been their inability to hindcast important

33 observed Southern Ocean climate trends such as surface cooling, surface freshening, and sea-ice

34 expansion (Turner et al. 2013; Jones et al. 2016; Kostov et al. 2018). Recent work suggests that

35 the increase in Antarctic meltwater anomaly over this period may have played an important role in

36 driving the observed trends. Climate models typically neglect the freshwater ﬂux due to net mass

37 loss from the Antarctic ice sheet: this has increased over the past few decades to around 500 Gt/yr

38 (Paolo et al. 2015; Rignot et al. 2019). Recent work suggests that the incorporation of this process

39 into climate models could help to explain the observed trends, resolving the discrepancy (Bintanja

40 et al. 2013; Rye et al. 2014; Bintanja et al. 2015; Rye et al. 2020). The incorporation of Antarctic

41 glacial meltwater also has a signiﬁcant impact on projections of future climate. (Bronselaer et al.

42 2018; Golledge et al. 2019). Although there remains some disagreement about the magnitude of

43 the climate impacts due to meltwater (Swart and Fyfe 2013; Pauling et al. 2016), understanding

44 how to correctly represent this process in global climate models is clearly of importance.

45 In climate modeling studies, the meltwater has generally been represented as an externally

46 imposed freshwater ﬂux; this requires a starting assumption about where in the water column the

47 glacial meltwater is situated. In many studies, glacial meltwater has been introduced at or near the

48 surface (Bintanja et al. 2013; Swart and Fyfe 2013; Rye et al. 2014; Bintanja et al. 2015; Hansen

49 et al. 2016; Pauling et al. 2016; Bronselaer et al. 2018), or over a constant depth (Rye et al. 2020).

50 Even though most of the melting occurs at depth, the meltwater might be expected to rise to the

51 surface due to its relatively low density; however, this assumption is not supported by observations.

52 For example, measurements of noble gas concentrations in the Ross Sea (Loose et al. 2009) and

3 53 in the Amundsen Sea (Kim et al. 2016; Biddle et al. 2019) reveal vertical meltwater distributions

54 centered at around 300m-400m depth. Near Pine Island Glacier, which is the source of a large

55 fraction of the total Antarctic melt, Dutrieux et al. (2014) found large interannual variability in

56 meltwater settling depth, with meltwater settling close to the surface in some years and hundreds

57 of meters at depth in other years. A better understanding of what determines the settling depth

58 of Antarctic glacial meltwater may greatly improve our understanding of ice-ocean interactions as

59 well as their representation in climate models.

60 Aspects of glacial meltwater dynamics have been studied previously. In the Antarctic context,

61 the priority has been to determine the rate and spatial distribution of sub-ice-shelf melting for

62 given boundary conditions and forcings. To this end, studies have employed one-dimensional

63 plume models (MacAyeal 1985; Jenkins 1991, 2011; Lazeroms et al. 2018), box models (Olbers

64 and Hellmer 2010; Reese et al. 2018), and three-dimensional ﬂuid dynamics simulations on the

65 ice-shelf scale (Losch 2008; De Rydt et al. 2014; Mathiot et al. 2017). In an Arctic context, where

66 glaciers generally exhibit a near-vertical ice face for the entire depth of the water column instead

67 of an ice shelf cavity, meltwater dynamics have been studied using high-resolution numerical

68 simulations of individual (Xu et al. 2012, 2013; Sciascia et al. 2013) and distributed (Carroll et al.

69 2015; Slater et al. 2015) plumes. Again, it was largely the spatial distribution of melting that was

70 emphasized, together with implications for the fjord-scale circulation. Finally, Naveira Garabato

71 et al. (2017) have studied the small-scale ﬂuid dynamics of meltwater escaping from underneath

72 an Antarctic ice shelf, with an explicit focus on meltwater settling depth. They employed a

73 two-dimensional simulation of the meltwater outﬂow to argue that centrifugal instability, which

74 contributes to lateral mixing of the rising meltwater plume, plays a dominant role in controlling

75 the settling depth.

4 76 In this study, we revisit the small-scale ﬂuid dynamics of meltwater escape from underneath an

77 ice shelf. First, we describe an idealized glacial meltwater outﬂow, and introduce simple models for

78 the meltwater’s settling depth. Second, we describe new three-dimensional large-eddy simulations

79 of the meltwater outﬂow, and compare the results to the predictions of the simpler models. Third,

80 we use our models to address observed interannual variability in meltwater settling depth near

81 Pine Island Glacier. Finally, we discuss why a dynamic parameterization of meltwater settling

82 depth could be crucial for accurately simulating high-latitude climate, and outline how such a

83 parameterization could be implemented building on the work in this study.

84 2. Theory and Methods

85 The object of this study is described schematically in Figure 1. Much of the total mass loss from

86 the Antarctic ice sheet stems from a small number of rapidly-melting ice shelves; here, we focus

87 on Pine Island Glacier, which is the source of a large fraction of the total mass loss. The meltwater

88 outﬂow from underneath the Pine Island ice shelf is concentrated in a narrow km-scale ﬂow at its

89 western edge (Thurnherr et al. 2014; Naveira Garabato et al. 2017). This narrow meltwater outﬂow

90 is likely a generic feature of many Antarctic ice shelves, as it is a straightforward consequence

91 of a typical sub-ice-shelf circulation (e.g. Grosfeld et al. 1997; Losch 2008). We investigate the

92 dynamics of such a meltwater outﬂow by idealizing it as a ﬁxed buoyancy source , with width !,

93 applied to the bottom of our model domain. In this section, we outline the hierarchy of theoretical

94 and modeling approaches that we will use.

95 a. Simple scaling relationships

96 The glacial meltwater escaping from underneath the ice shelf undergoes turbulent buoyant

97 convection in a stratified ambient fluid. The theory of such processes was first developed by

5 98 Morton et al. (1956). For plumes originating from a point source, far from any walls, this theory

99 has yielded robust scaling laws for the plume’s rise height in terms of the buoyancy source and

100 the background buoyancy frequency #. These scaling laws have been repeatedly conﬁrmed in

101 laboratory and experimental work (Turner 1986; Helfrich and Battisti 1991; Speer and Marshall

102 1995; Fabregat Tomàs et al. 2016). As described, for example, by Speer and Marshall (1995), as

103 long as # is substantially larger than the Coriolis parameter 5 , the only two parameters that could

4 3 104 physically control the rise height are (m /s , consider an area-integrated buoyancy ﬂux) and #

−1 105 (s ). Dimensional analysis then yields a vertical scale

1 4 ℎ = . (1) # #3

106 The real rise height ℎ is proportional to this vertical scale:

ℎ = 0ℎ# , (2)

107 where 0 is a constant. Numerical experiments consistently yield a value of 0 ' 2.6 (e.g. Speer and

108 Marshall 1995; Fabregat Tomàs et al. 2016).

109 In the case of the glacial meltwater outﬂow, however, the meltwater plume does not originate

110 from a point source: it is rather in the shape of a line, where the total buoyancy forcing is

111 distributed over some width ! (see Figure 1). Therefore, we modify equation (1) by assuming that

112 the two parameters exerting control over the rise height are the buoyancy source per unit width,

3 3 −1 113 /! (m /s ), and the background buoyancy frequency, # (s ). Dimensional analysis now yields

114 a vertical scale of 1 3 1 ℎ = . # ! # (3)

115 Again, the real rise height is proportional to this scale:

ℎ = 0ℎ# , (4)

6 116 where the constant of proportionality could be expected to match the value observed for plumes

117 originating from a point source (0 ' 2.6).

118 b. One-dimensional line plume model

119 The scaling theory described above cannot account for the eﬀects of non-uniform stratiﬁcation

120 (i.e. # = # (I)), and provides only limited physical insight. To improve upon it, we can follow

121 Morton et al. (1956) in constructing a one-dimensional vertical steady-state model of the buoyant

122 plume. The original model of Morton et al. (1956) describes a point buoyancy source, and has

123 been previously adapted to consider a point source of meltwater next to a vertical wall (Carroll

124 et al. 2015). One-dimensional models of buoyant line plumes rising underneath a sloping interface

125 have also been widely applied to the study of sub-ice-shelf meltwater dynamics (MacAyeal 1985;

126 Jenkins 1991, 2011; Lazeroms et al. 2018; Pelle et al. 2019). These models generally consider

127 explicit ﬂuxes of heat and salt instead of a generic buoyancy ﬂux, as well as interactions across the

128 ice-ocean interface.

129 Throughout this study we will assume that the dominant contribution to meltwater production is

130 made below the ice shelf and that thermodynamic interactions between the plume and the ice shelf

131 front itself (see Figure 1) are negligible. For a buoyant plume originating from a line source next

132 to a vertical wall, these assumptions lead to the following system of coupled ordinary diﬀerential

133 equations (see Appendix A): 3& " = U 3I & (5) 134 3" & = 3I " (6) 135 3 = −. 3I (7)

136 Here &, ", and are vertical ﬂuxes per unit length of volume, momentum, and buoyancy,

137 respectively. # (I) is the buoyancy frequency, 6 is the acceleration due to gravity, and U is a

7 138 non-dimensional entrainment coeﬃcient. The model is solved for a given buoyancy forcing /!

139 by setting = /! at the bottom of the domain and integrating upwards. The meltwater’s settling

140 depth is then given by the level of neutral buoyancy, which is where (I) = 0. Since /! and

141 # are the only dimensional input parameters, a characteristic vertical scale is again given by

1/3 142 ℎ# = (/!) /#.

143 Example solutions of this one-dimensional model are shown in Figure 2, for a range of buoyancy

−3 −1 144 forcings /!. Here, the background buoyancy frequency # = 3 × 10 s , a realistic value for

145 Pine Island Bay. Values used for the entrainment coeﬃcient vary across the literature; here,

146 we use U = 0.15, which is consistent with eﬀective entrainment coeﬃcients calculated from past

147 numerical simulations of hydrothermal plumes (Jiang and Breier 2014; Fabregat Tomàs et al.

148 2016). We integrate our model equations using an eighth-order Runge-Kutta method (Prince and

149 Dormand 1981).

150 c. Three-dimensional large-eddy simulations

151 To accurately study the behavior of the buoyant plume, and to evaluate the utility of the sim-

152 pler theories described above, we need to conduct high-resolution simulations of the underlying

153 small-scale ﬂuid dynamics. Many previous studies have simulated the dynamics of hydrothermal

154 plumes rising far from any walls (e.g. Lavelle 1995; Speer and Marshall 1995; Jiang and Breier

155 2014; Fabregat Tomàs et al. 2016). In the Arctic context, two-dimensional and three-dimensional

156 simulations of glacial meltwater plumes have been conducted (Xu et al. 2012, 2013; Sciascia

157 et al. 2013; Carroll et al. 2015); these studies emphasized the spatial distribution of melting and

158 the fjord-scale circulation. Meltwater escape from underneath an Antarctic ice shelf has been

159 previously simulated in two dimensions by Naveira Garabato et al. (2017), who also based their

160 simulations on an idealization of the outﬂow from below the Pine Island ice shelf.

8 161 All of these numerical simulations of glacial meltwater plumes have used the Massachusetts

162 Institute of Technology general circulation model in a non-hydrostatic conﬁguration (MITgcm,

163 Marshall et al. 1997). Here, we conduct new three-dimensional large-eddy simulations of a line

164 glacial meltwater plume rising next to a wall using the software package Oceananigans.jl (Ramad-

165 han et al. 2020). Oceananigans.jl is written in the high-level Julia programming language (Bezanson

166 et al. 2017), simulates the rotating non-hydrostatic incompressible Boussinesq equations using a

167 ﬁnite volume discretization similar to that of the MITgcm, and is optimized to run on Graphical

168 Processing Units (GPUs). The equations are integrated using a second-order Adams-Bashforth

169 scheme with adaptive time stepping. The eﬀects of sub-grid scale processes are parameterized

170 via an eddy viscosity and eddy diﬀusivity modeled using the anisotropic minimum dissipation

171 (AMD) large-eddy simulation closure (Rozema et al. 2015). The AMD formalism was reﬁned by

172 Verstappen (2018) and validated for ocean-relevant scenarios by Vreugdenhil and Taylor (2018).

173 Our model domain follows the schematic in Figure 1. The horizontal widths !H and !G are

174 both set to 5 km, while the depth of the ice shelf front !I is set equal to 400m (approximately

175 consistent with Pine Island Glacier, see Jenkins et al. 2010). The domain is re-entrant in the zonal

176 G-direction. We use 512 grid cells in each horizontal direction and 96 grid cells in the vertical:

177 this corresponds to a horizontal resolution of 9.77 m and a vertical resolution of 4.17 m. We

178 consider the evolution of temperature, salinity, and a passive tracer representing meltwater. Glacial

179 meltwater escaping from underneath the ice shelf is represented as a constant buoyancy source

180 applied to a horizontal area of length ! next to the southern edge of the domain (see Figure

181 1). ! is set to 1 km; as previously discussed, the meltwater from beneath Pine Island glacier is

182 concentrated in a narrow outﬂow broadly consistent with this scale (Naveira Garabato et al. 2017).

183 The buoyancy source is implemented as a constant volume-conserving “virtual salinity ﬂux"

9 −4 −1 184 (Huang 1993; see Appendix B for details). The Coriolis parameter, 5 , is set to −1.4 × 10 s ,

185 appropriate for the latitude of Pine Island.

186 3. Results

187 a. The simulated meltwater plume

188 The basic behavior of the simulated glacial meltwater plume is demonstrated in Figure 3; here,

−2 3 3 189 /! = 10 m /s . As in Figure 2, the initial condition is a uniform buoyancy frequency of

−3 −1 190 # = 3 × 10 s ; this yields #/ 5 ' 20, similar to the meltwater plume simulations of Naveira

191 Garabato et al. (2017). For now, the stratiﬁcation is implemented through a linear vertical salinity

192 gradient, ﬁxed temperature, and a linear equation of state with haline contraction coeﬃcient

−4 −1 193 V = 7.8×10 psu (Vallis 2017). Following the evolution of the passive meltwater tracer, we see

194 that the turbulent plume initially rises rapidly, and then moves northward once it reaches neutral

195 buoyancy. The northward ﬂow is deﬂected to the left by the Coriolis force, resulting in a strong

196 westward jet; this is consistent with the observations and two-dimensional simulations of Naveira

197 Garabato et al. (2017).

198 Next, we consider the time evolution of the horizontally averaged meltwater distribution over

199 one day of simulation. To quantify the eﬀect that the Earth’s rotation may play in determining the

200 plume’s settling depth (e.g. Fabregat Tomàs et al. 2016; Naveira Garabato et al. 2017), we conduct

−4 −1 201 two simulations: one where the Coriolis parameter 5 has a realistic value −1.4×10 s , and one

202 where 5 has been set to zero. The results of these experiments are shown in Figure 4. We observe

203 that, for this realistic choice of #/ 5 , the meltwater’s settling depth is essentially determined on a

−1 204 timescale # : the non-rotating and rotating experiments both yield similar mean settling depths.

205 However, the rotating experiment shows a distinct broadening of the vertical meltwater distribution

10 206 as we approach a timescale of 1 day, suggestive of rotational eﬀects playing an important mixing

207 role.

208 Interestingly, these results conﬂict with those of Naveira Garabato et al. (2017), who used two-

209 dimensional simulations to argue that centrifugal instability is a dominant mechanism controlling

210 the settling depth of the meltwater. As the northward-moving meltwater is deﬂected to the left by

211 the Coriolis force, a strong zonal jet develops (Figure 3); centrifugal instability can occur if the

212 resulting anticyclonic vorticity is large enough (Z/ 5 < −1, Haine and Marshall 1998), promoting

213 lateral export and mixing of the meltwater. In their two-dimensional simulations, Naveira Garabato

−4 −1 214 et al. (2017), observed over the same timeframe of 1 day that setting 5 = −1.4 × 10 s was

215 suﬃcient to deepen the peak of the meltwater distribution by ∼ 50 m compared to the case with 5

216 = 0, a substantial eﬀect that is not present in Figure 4. In Appendix C we address this discrepancy

217 using additional two-dimensional simulations: those results suggest that the eﬀect observed in the

218 simulations of Naveira Garabato et al. (2017) may be related to their use of a restoring buoyancy

219 source formulation rather than a constant buoyancy source formulation as implemented in this

220 study. Here, we simply conclude that, for realistic values of #/ 5 , rotational eﬀects seem to play a

221 negligible role in determining the meltwater’s settling depth.

222 b. Vertical meltwater distribution: uniform stratiﬁcation

223 Now, we can evaluate how the meltwater’s settling depth depends on the buoyancy source and

224 the background stratiﬁcation. We conduct a set of simulations where /! and # are separately

225 varied: the vertical meltwater distributions after 6 hours of integration are shown in Figure 5.

226 We choose this timescale because by this point the settling depth has been essentially determined

−2 3 3 −3 −1 227 (Figure 4). The default values of /! and # in Figure 5 are 10 m /s and 3 × 10 s . For

228 the case of varying /! we included as an additional x-axis an estimate of the corresponding

11 229 glacial mass loss due to melt, divided by the outﬂow width; details are described in Appendix D.

230 On top of the distributions obtained from the simulations we also plot predictions from the simple

231 scaling solution (eq. 3, 0 = 2.6) and the one-dimensional line plume model presented above. Both

232 show excellent agreement with the high-resolution simulations, suggesting that they parametrize

233 the settling depth extremely well in these idealized conditions.

234 c. Vertical meltwater distribution: non-uniform stratiﬁcation

235 In the real world, the buoyancy frequency # is non-uniform in time and space. For example,

236 observations from Pine Island Bay show that vertical proﬁles of temperature, salinity, and meltwater

237 fraction show signiﬁcant interannual variability (Dutrieux et al. 2014). In Figure 6 we demonstrate

238 this variability by plotting temperature and salinity proﬁles collected next to the meltwater outﬂow

239 from Pine Island Glacier in 2009 and 2014 (Jacobs et al. 2011; Heywood et al. 2016), together

240 with estimates of the corresponding meltwater fractions. Notably, in 2009 meltwater was primarily

241 centered at a depth of 400m, while in 2014 it was able to rise to the surface. This diﬀerence appears

1/3 242 too dramatic to be explained purely by interannual variability in melt rates (consider the ℎ ∝

243 scaling); hence, we propose that the variability in stratiﬁcation played a major role.

244 We investigate the eﬀect of the diﬀerent background conditions in 2009 and 2014 by using the

245 top 400m of the observed temperature and salinity proﬁles as our initial conditions in our high-

246 resolution simulations. From these, Oceananigans.jl calculates a density proﬁle using the idealized

247 nonlinear equation of state proposed by Roquet et al. (2015), optimized for near freezing. We

−3 3 3 −2 3 3 248 consider two diﬀerent buoyancy sources, /! = 10 m /s and /! = 10 m /s . The vertical

249 meltwater distributions after 6 hours are shown in Figure 7. We additionally plot an estimate of the

2 6 3d 250 strength of the stratiﬁcation as a function of depth; this is obtained by calculating # = − on a d0 3I

251 point-wise basis and applying a moving average with a 20m window to isolate larger-scale trends.

12 −2 3 3 252 For the case of /! = 10 m /s , we see that there is little diﬀerence in the vertical meltwater

−3 3 3 253 distribution between 2009 and 2014 conditions. However, the simulations with /! = 10 m /s

254 show a marked diﬀerence: in the 2009 case, meltwater settles at ∼350 m depth, while in the

255 2014 case it rises around 100m shallower. The qualitative trend is consistent with the observations

256 (Figure 6). Finally, we have also plotted the settling depths predicted by the one-dimensional plume

257 model: there is near-perfect agreement with the peaks of the meltwater distributions obtained from

258 our high-resolution simulations.

−3 3 3 259 We suggest that this diﬀerence in behaviors for the /! = 10 m /s case can be simply explained

2 260 by changes in the background stratiﬁcation. Namely, in 2009 there was a marked peak in # at

261 around 350m depth that was not present in 2014. The meltwater settling at this very depth in the

262 simulations is thus an indication that it was “trapped" by the local maximum in stratiﬁcation. When

−2 3 3 263 the buoyancy source was larger (/! = 10 m /s ), the meltwater was able to “break through" the

264 stratiﬁcation maximum, and ended up with a vertical distribution very similar to the corresponding

265 2014 stratiﬁcation proﬁle.

266 4. Discussion

267 The potency of Antarctic glacial meltwater as a driver of regional and global climate trends

268 likely depends strongly on its settling depth or vertical distribution after exiting the ice shelf cavity.

269 Speciﬁcally, it seems feasible that meltwater could only explain the signs of the observed Southern

270 ocean trends (surface cooling, surface freshening, and sea-ice expansion) as long as it rises close

271 enough to the surface to shoal the mixed layer and to yield a measurable surface salinity anomaly.

272 Pauling et al. (2016), who considered the eﬀects of releasing freshwater at diﬀerent depths, found

273 that the depth of meltwater release had no signiﬁcant eﬀect on the magnitude of sea-ice expansion;

274 however, they also found a much weaker response of sea-ice expansion to freshwater forcing

13 275 than other studies (Bintanja et al. 2013, 2015; Rye et al. 2020). The causes of these inter-model

276 diﬀerences are poorly understood. This issue was highlighted by the recent work of Naveira

277 Garabato et al. (2017), which uses observations and an idealized model to suggest that centrifugal

278 instability acts to rapidly distribute melt water laterally, reducing its potency to the surface climate;

279 however, other observations (e.g. Dutrieux et al. 2014) suggest that meltwater can settle at a

280 range of depths in the Subpolar Sea, implying that time-varying environmental conditions and

281 the properties of the meltwater plume play important roles in determining the vertical distribution

282 of settled meltwater in the Shelf Seas and therefore the climate impact of meltwater anomaly

283 production. The role that the depth of meltwater release plays remains very far from settled, and

284 deserves further study.

285 In Figure 8, we identify two diﬀerent paradigms for introducing glacial meltwater ﬂuxes into

286 high-latitude oceans for simulations of global climate. In paradigm A, the ﬂuxes calculated by a

287 melt rate model are inserted into the ocean model at some ﬁxed vertical level. This paradigm has

288 dominated the literature: as described earlier, most climate modeling studies have introduced all

289 of the meltwater ﬂux at the surface. In other studies, the meltwater has been uniformly distributed

290 over a ﬁxed range of depths below the ice shelf front (Beckmann and Goosse 2003; Mathiot et al.

291 2017). Given the likely climatic importance of glacial meltwater, the strong dependence of settling

292 depth on buoyancy release (e.g. as explored in this study), and the vast heterogeneity in the

293 observed mass loss rates from diﬀerent ice shelves (Rignot et al. 2019), any such "one-size-ﬁts-all"

294 solution risks missing substantial aspects of the climate response to Antarctic mass loss. However,

295 an alternative approach is possible: in paradigm B, the melt rate model is coupled to a dynamic

296 plume model that describes the small-scale dynamics of buoyant meltwater plumes and accurately

297 calculates the vertical distribution of meltwater. The meltwater is then inserted into the ocean

298 model in accordance with this distribution.

14 299 The work done in this study serves as a ﬁrst step towards developing such a dynamic meltwater

300 plume parameterization, although some issues still remain to be solved. As indicated in Figure

301 7, the simple one-dimensional plume model accurately predicts the peak of the vertical meltwater

302 distribution even for complex non-uniform stratiﬁcation. The fact that many sub-ice-shelf melt-

303 rate parameterizations are based on similar models of a sloping plume (Jenkins 2011; Lazeroms

304 et al. 2018; Pelle et al. 2019) suggests that a single appropriately speciﬁed model could potentially

305 calculate both the melt rates and the meltwater settling depth. One challenge with this kind of

306 formulation would be dealing with the large discontinuous jump in slope that occurs at the bottom of

307 the ice shelf front; here, we have considered only the part of the plume next to the front, idealizing

308 the meltwater outﬂow from below as a constant buoyancy source. Another issue relates to a

309 fundamental limitation of the one-dimensional view; it neglects the along-shelf dynamics, which

310 have been shown to signiﬁcantly aﬀect total melt rates in the Arctic context (Jackson et al. 2020).

311 The most signiﬁcant limitation with respect to computing meltwater settling depths, however, is

312 that these one-dimensional parameterizations can only output a single meltwater settling depth

313 ((I) = 0). Meanwhile, observed vertical meltwater distributions can have complex, possibly

314 multi-modal shapes. Short of explicitly resolving the small-scale ﬂuid dynamics of the meltwater

315 plume next to and below the entire ice shelf, it may be possible to extend upon the one-dimensional

316 plume model, perhaps by introducing a time dependence, to explicitly include a passive meltwater

317 tracer that would allow for the calculation of a vertical distribution rather than just its peak.

318 5. Conclusion

319 Antarctic glacial meltwater is likely an important driver of observed Southern Ocean climate

320 trends (Bintanja et al. 2013; Rye et al. 2014; Bintanja et al. 2015; Rye et al. 2020), and will have

321 a signiﬁcant impact throughout the twenty-ﬁrst century (Bronselaer et al. 2018; Golledge et al.

15 322 2019). Nevertheless, the factors determining the vertical distribution of meltwater in the water

323 column remain poorly understood. Here, we have used a hierarchy of approaches, spanning simple

324 scaling laws to high-resolution large-eddy simulations of the meltwater outﬂow from beneath an

325 ice shelf, to gain a fundamental understanding of the most important controls on the meltwater’s

326 settling depth. We found that the settling depth is primarily a function of the buoyancy forcing per

327 unit width and the ambient stratiﬁcation, consistent with the classical theory of turbulent buoyant

328 plumes and in constrast to previous suggestions that centrifugal instability plays an important role

329 (Naveira Garabato et al. 2017). Our simulations also provide insight into the observed interannual

330 variability in meltwater settling depth, using Pine Island Glacier as an example; the role of the non-

331 uniform background stratiﬁcation is highlighted. Because the focusing of sub-ice-shelf meltwater

332 into a narrow outﬂow is a basic consequence of a generic sub-ice-shelf circulation (Grosfeld et al.

333 1997; Losch 2008; De Rydt et al. 2014), we expect that the results of this study are relevant to a

334 wide range of Antarctic ice shelves. The work presented in this study is the ﬁrst step towards a

335 dynamic parameterization of meltwater settling depth for simulations of global climate. Because

336 of the likely climatic importance of glacial meltwater, the strong dependence of mass loss rates

337 on buoyancy forcing, and the vast heterogeneity in observed mass loss rates from diﬀerent ice

338 shelves, such a parameterization could be crucial for the accurate simulation and forecasting of

339 high-latitude climate in a warming world.

340 Acknowledgments. We thank Alberto Naveira Garabato for discussions that helped motivate

341 this work, and Gianluca Meneghello and Jean-Michel Campin for discussions that aided in its

342 execution. J. M. and C. D. R. acknowledge support from the NASA MAP program and the MIT-

343 GISS cooperative agreement. A. R. was supported through the CliMA initiative funded by the Eric

16 344 and Wendy Schmidt foundation. P. D. was supported by NSF OPP award 1643285 and his Center

345 for Climate and Life Fellowship from the Earth Institute of Columbia University.

346 Data availability statement. This study generated no new data. Code for the one-dimensional

347 line plume model and the two- and three-dimensional large-eddy simulations is available at

348 https://github.com/arnscheidt/antarctic-meltwater-settling-depth.

349 APPENDIX A

350 One-dimensional line plume model

351 We construct a 1-dimensional vertical line plume model in the spirit of Morton et al. (1956). Here,

352 the rate of turbulent entrainment of ambient ﬂuid into the rising buoyant plume is parametrized as

353 proportional to the plume’s vertical velocity via an entrainment coeﬃcient, U. We assume that the

354 vertical velocity F is uniform within the plume and zero outside, and that the plume is rising next

355 to a wall (so that entrainment can only occur from one side). We can then write down volume,

356 momentum, and mass conservation equations within the plume:

3 (F) = UF 3I (A1)

357 3 (FdF) = 6(d − d) 3I 0 (A2)

358 3 (Fd) = UFd . 3I 0 (A3)

359 Here, d(I) is the density of the plume, d0 (I) is the ambient density, is the width of the plume

360 perpendicular to the wall, and U is the entrainment coeﬃcient. Assuming that d(I) diﬀers only

361 slightly from the reference density d0, we can rewrite Equation (A2) as

3 6 2 (F ) = (d0 − d). (A4) 3I d0

17 362 Following the reasoning in Morton et al. (1956), we can use Equation (A1) to rewrite Equation

363 (A3) as 3 3 3 3 (Fd) = d (F) = (Fd ) − F d , 3I 0 3I 3I 0 3I 0 (A5)

364 such that 3 3d (F(d − d)) = F 0 . 3I 0 3I (A6)

2 (d0−d) 365 Now, writing F = & (volume ﬂux), F = " (momentum ﬂux) and F6 = (buoyancy d0

366 ﬂux), we obtain the three coupled ODEs

3& " = U 3I & (A7) 367 3" & = 3I " (A8) 368 3 6 3d = & 0 = −. (A9) 3I d0 3I

369 These equations are similar but not equivalent to the corresponding equations for point plumes.

370 Furthermore, each of the three governing equations has implicitly been divided by a factor of !

371 (x-width of the plume); thus, all of the quantities &, ", are ﬂuxes per unit width.

372 APPENDIX B

373 Buoyancy source implementation

4 3 374 We implement the buoyancy source (m /s ) in our high-resolution simulations as a volume-

375 conserving “virtual salinity ﬂux" (Huang 1993). The conservation law for an arbitrary tracer 2 in

376 Oceananigans.jl is m2 + · ∇2 = −∇ · + , mC u qc 2 (B1)

377 where qc is a diﬀusive ﬂux and 2 is an external source term. In our simulations, we introduce the

378 buoyancy uniformly across a volume that extends width ! in the G-direction, 10 grid cells in the

18 379 H-direction (∼ 100 m), and one grid cell in the I-direction (∼ 4 m), Deﬁning this volume as +, we

380 can write ¹ 31 3+ = , 3C (B2) + source

31 381 where 3C source refers only to the term within the full buoyancy conservation equation that comes

382 from the external buoyancy source. Now, recall that

6 1 = − (d − d0), (B3) d0

383 and that, to ﬁrst order,

d = d0(1 − U() −)0) + V(( − (0)). (B4)

384 Thus, if no temperature forcing is introduced,

31 31 3d 6 3d 3( = = − = −6V , (B5) 3C source 3d 3C source d0 3C source 3C source

385 and, by (B2): ¹ 3( = − 3+6V ≡ −6V , 3C ( (B6) + source

3 386 where ( is the volume-integrated salinity ﬂux (psu m /s). For a chosen we therefore obtain a

387 corresponding ( by (B6). Then, in our simulations, we distribute ( uniformly across +.

388 APPENDIX C

389 Restoring buoyancy sources may exaggerate the importance of rotational eﬀects in

390 determining the meltwater’s settling depth

391 Our results conﬂict with those of Naveira Garabato et al. (2017). Using a two-dimensional

392 model, they found that including realistic rotation deepened the peak of the observed meltwater

393 distribution by ∼ 50 m compared to a non-rotating case, after one day of integration. To clarify why

19 394 there is a discrepancy, we conduct additional two-dimensional simulations with Oceananigans.jl

395 that are designed to closely replicate those of Naveira Garabato et al. (2017).

396 The model domain spans 5km × 300m and is zonally re-entrant. Our resolution is 512 × 96, i.e.

397 ∼10m×3m. The initial stable stratiﬁcation is implemented using a linear equation of state and a

◦ ◦ 398 linear temperature gradient from 1 C at the bottom to 3 C at the top. At the northern boundary,

399 we continuously relax back to the stable initial condition. At the base of the southern boundary we

400 introduce meltwater via an unstable restoring region that extends 160m in the y-direction. In the

401 unstable restoring region, temperature is relaxed to a temperature )A (H), which is set following a

◦ ◦ 402 linear gradient: its value is 2 C at H = 0 m and 1 C at H = 160 m. For clarity, in the buoyancy

403 source region: 3) = ( ) +_() (H) −)), 3C other terms A (C1)

−1 −4 −1 404 where _ = 1/20 s . This experiment is conducted twice, once with 5 = −1.4 × 10 s (realistic

405 rotation) and once with 5 = 0 (no rotation). We then conduct an additional set of simulations using

406 a constant buoyancy source, which is set to approximately yield the same settling depth.

407 Figure 9 shows the vertical distribution of glacial melt in the water column after 1 day, for

408 both rotating and non-rotating cases, and for a restoring formulation and a constant buoyancy

409 source formulation. When a restoring formulation is used, in the rotating case the peak is ∼ 50 m

410 deeper than in the non-rotating case, consistent with the results of Naveira Garabato et al. (2017).

411 However, when a constant buoyancy source is used, rotation appears to have no eﬀect on the peak

412 of the meltwater distribution. Since the magnitude of the buoyancy source is a primary control

413 on the meltwater’s settling depth, the importance of any other parameters can only be accurately

414 investigated by holding the buoyancy source constant. This suggests that the bottom results in

415 Figure 9 are more physical, and that the use of restoring non-constant buoyancy sources may

416 exaggerate the eﬀect of rotation on the settling depth.

20 417 APPENDIX D

418 Estimating the glacial mass loss due to melt that corresponds to a given buoyancy source

419 For the second G-axis included in Figure 5, we have estimated the glacial mass loss due to melt

4 3 420 that corresponds to a given buoyancy source (m /s ). For this conversion, we have assumed that

421 the input of a given volume of freshwater into the system is equivalent to the removal of that same

422 volume of water at a reference salinity (0 (set to 34.6). This can be justiﬁed rigorously by noting

423 that, if we add a small volume of water Δ+ with salinity 0 to a large volume of water + with salinity

424 (, the new salinity is given by

+( Δ+ ( + Δ( = ' ((1 − ) (D1) + + Δ+ +

425 i.e.

+Δ( ' −(Δ+. (D2)

426 Deﬁning an equivalent mass loss ﬂux " (kg/s), we have, following (B6),

( d " ' d0 = . (D3) (0 6V(0

427 The quantity measured by the second G-axis in Figure 5 is " divided by the outﬂow width !.

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28 579 LIST OF FIGURES

580 Fig. 1. A schematic describing the object of study. Much of the total mass loss from the Antarctic

581 ice sheet stems from a small number of rapidly-melting ice shelves; here we highlight the

582 Pine Island ice shelf as an example. The meltwater escaping from underneath the ice shelf is

583 concentrated in a narrow km-scale outﬂow at its western edge; this is likely a generic feature

584 of many Antarctic ice shelves. We idealize this meltwater outﬂow as a constant buoyancy

585 source , with width !, applied to the bottom of our model domain...... 31

586 Fig. 2. Example solutions of the one-dimensional line plume model for diﬀerent buoyancy forcings

587 /!. ℎ = 0 represents the base of the ice shelf front. In each case, the black dot highlights

588 the meltwater’s settling depth; this is the level of neutral buoyancy, i.e. where (I) = 0 . . . 32

589 Fig. 3. Evolution of a simulated meltwater plume, after 6 hours and after one day. The meltwater

590 concentrations are in arbitrary units. The upper two rows depict a HI-plane with G = 0 (i.e.

591 perpendicular to the ice shelf front). The bottom row depicts an GI-plane with H = 0 (i.e.

592 along the ice shelf front). A strong zonal ﬂow develops: this is consistent with observations

593 of the outﬂow from beneath the Pine Island ice shelf. The zonal ﬂow is responsible for the

594 transport of meltwater in the G-direction that can be observed in the bottom right plot: the

595 meltwater outﬂow is deﬂected to the west by the Coriolis force, and eventually re-enters the

596 domain at the eastern boundary...... 33

597 Fig. 4. The evolution of the horizontally averaged vertical meltwater distribution over 1 day of

598 simulation, for a realistic value of the Coriolis parameter 5 and for a case where 5 = 0. −2 3 3 599 Here, /! = 10 m /s . We see that the primary role of rotational eﬀects is to broaden the

600 distribution of meltwater over a wider range of depths, suggesting that they play an important

601 mixing role. However, they do not appear to have a substantial eﬀect on the mean settling

602 depth...... 34

603 Fig. 5. The horizontally averaged vertical meltwater distribution after 6 hours of simulation, for

604 varying buoyancy source /! and buoyancy frequency #. For the case of varying /!,

605 we have also estimated an equivalent mass loss ﬂux (see text). On top of the distributions

606 we plot the settling depths predicted by the simple scaling relationship (dashed) and the

607 one-dimensional line plume model (solid): both show excellent agreement with the high-

608 resolution simulations...... 35

609 Fig. 6. Observed 2009 and 2014 temperature and salinity proﬁles next to the meltwater outﬂow

610 from Pine Island Glacier, as well as estimated meltwater fractions. In 2009, meltwater was

611 primarily centered at a 400m depth, while in 2014 it was able to rise to the surface. . . . . 36

−3 3 3 −2 3 3 612 Fig. 7. Simulated vertical meltwater distributions for /! = 10 m /s and /! = 10 m /s ,

613 with initial conditions set by observed temperature and salinity proﬁles for 2009 and 2014.

614 Horizontal dashed lines indicate the settling depths predicted by the one-dimensional line

615 plume model for the same conditions; notably, the line plume model accurately predicts the

616 peak of the simulated meltwater distribution in all cases. We also plot depth proﬁles of 2 −2 3 3 617 stratiﬁcation strength in terms of # (see text). For /! = 10 m /s we see that there

618 is little diﬀerence in the vertical meltwater distribution between 2009 and 2014 conditions. −3 3 3 619 However, the simulations with /! = 10 m /s show a marked diﬀerence: the qualitative

620 trend is consistent with observations (Figure 6). Here, we propose that the rising meltwater

621 was “trapped” by the notable local stratiﬁcation maximum at around 350m depth in the 2009

622 conditions...... 37

29 623 Fig. 8. Schematic describing two diﬀerent paradigms for meltwater ﬂuxes in simulations of global

624 climate. In paradigm A, the ﬂuxes from a melt rate model are inserted into the ocean model at

625 some ﬁxed vertical level; this approach has dominated the literature. In paradigm B, the melt

626 rate model is coupled to a dynamic plume model that describes the small-scale dynamics of

627 buoyant meltwater plumes and accurately calculates the vertical distribution of meltwater for

628 insertion into the ocean model. Given the potential climatic importance of glacial meltwater,

629 the strong dependence of settling depth on the buoyancy forcing, and the vast heterogeneity in

630 the observed mass loss rates from diﬀerent ice shelves, this approach would likely represent

631 a signiﬁcant improvement over the “one-size-ﬁts-all" approach of paradigm A...... 38

632 Fig. 9. Vertical meltwater distributions, for rotating and non-rotating cases, in a two-dimensional

633 domain. On the top, we have introduced meltwater via a restoring buoyancy source (following

634 Naveira Garabato et al. (2017), see text), while on the bottom we have used a constant

635 buoyancy source (as in the simulations described in the main text). When a constant buoyancy

636 source is employed, the peak of the vertical distribution is not noticeably inﬂuenced by the

637 eﬀects of rotation. However, when a restoring buoyancy source is employed, rotation deepens

638 the peak by ∼ 50 m, consistent with the simulations of Naveira Garabato et al. (2017). Since

639 the magnitude of the buoyancy source is a primary control on the meltwater’s settling depth,

640 the importance of any other parameters can only be accurately investigated by holding the

641 buoyancy source constant; therefore, these results show that the use of restoring non-constant

642 buoyancy sources may exaggerate the eﬀect of rotation on the settling depth...... 39

30 Ly

model domain z L c hl front shelf ice

F glacier L L x meltwater z ice shelf cavity x y

643 Fig. 1. A schematic describing the object of study. Much of the total mass loss from the Antarctic ice sheet

644 stems from a small number of rapidly-melting ice shelves; here we highlight the Pine Island ice shelf as an

645 example. The meltwater escaping from underneath the ice shelf is concentrated in a narrow km-scale outﬂow at

646 its western edge; this is likely a generic feature of many Antarctic ice shelves. We idealize this meltwater outﬂow

647 as a constant buoyancy source , with width !, applied to the bottom of our model domain.

31 400 F/L = 0.005 m3/s3 F/L = 0.01 m3/s3 3 3 300 F/L = 0.015 m /s

200 h (m)

100

0 0 5 10 15 0 2 4 0.01 0.00 0.01 − Q (Volume flux, m2/s) M (Momentum flux, m3/s2) B (Buoyancy flux, m3/s3)

648 Fig. 2. Example solutions of the one-dimensional line plume model for diﬀerent buoyancy forcings /!.

649 ℎ = 0 represents the base of the ice shelf front. In each case, the black dot highlights the meltwater’s settling

650 depth; this is the level of neutral buoyancy, i.e. where (I) = 0

32 Time = 6 hours Time = 1 day 0.27 0.27 100 0.18 0.18 − 0.09 0.09 200 0.00 0.00

z (m) − 0.09 0.09 300 −0.18 u (m/s) −0.18 u (m/s) − −0.27 −0.27 − − 3000 2500 2000 1500 1000 500 0 3000 2500 2000 1500 1000 500 0 y (m) y (m)

72 72 100 60 60 − 48 48 200 36 36

z (m) − 24 24 300 Meltwater Meltwater − 12 12 0 0 3000 2500 2000 1500 1000 500 0 3000 2500 2000 1500 1000 500 0 y (m) y (m)

108 108 100 90 90 − 72 72 200 54 54

z (m) − 36 36 300 Meltwater Meltwater − 18 18 0 0 2000 1000 0 1000 2000 2000 1000 0 1000 2000 − − x (m) − − x (m)

651 Fig. 3. Evolution of a simulated meltwater plume, after 6 hours and after one day. The meltwater concentrations

652 are in arbitrary units. The upper two rows depict a HI-plane with G = 0 (i.e. perpendicular to the ice shelf front).

653 The bottom row depicts an GI-plane with H = 0 (i.e. along the ice shelf front). A strong zonal ﬂow develops:

654 this is consistent with observations of the outﬂow from beneath the Pine Island ice shelf. The zonal ﬂow is

655 responsible for the transport of meltwater in the G-direction that can be observed in the bottom right plot: the

656 meltwater outﬂow is deﬂected to the west by the Coriolis force, and eventually re-enters the domain at the eastern

657 boundary.

33 4 1 f = -1.4 10− s− f = 0 × 0.060 0.060 50 50 − − 100 0.048 100 0.048 − − 150 150 − 0.036 − 0.036 200 200

z (m) − z (m) − 0.024 0.024 250 250 − − 300 300 − 0.012 − 0.012 350 350 − − 0.000 Normalized meltwater concentration 0.000 Normalized meltwater concentration 0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00 Time (days) Time (days)

658 Fig. 4. The evolution of the horizontally averaged vertical meltwater distribution over 1 day of simulation, for

−2 3 3 659 a realistic value of the Coriolis parameter 5 and for a case where 5 = 0. Here, /! = 10 m /s . We see that

660 the primary role of rotational eﬀects is to broaden the distribution of meltwater over a wider range of depths,

661 suggesting that they play an important mixing role. However, they do not appear to have a substantial eﬀect on

662 the mean settling depth.

34 Mass loss (Gt/km/yr) 1 0 1 10− 10 10 0.120 h (F/L)1/3 50 ∝ − 1-D model 0.105 100 − 0.090 150 − 0.075 200 0.060

z (m) − 250 0.045 − 300 0.030 − 0.015 350

− Normalized meltwater concentration 0.000 4 3 2 1 10− 10− 10− 10− Buoyancy source (F/L, m3/s3)

1 0.120 h N − 50 ∝ − 1-D model 0.105 100 − 0.090 150 − 0.075 200 0.060

z (m) − 250 0.045 − 300 0.030 − 0.015 350

− Normalized meltwater concentration 0.000 0.002 0.004 0.006 0.008 0.010 1 N (s− )

663 Fig. 5. The horizontally averaged vertical meltwater distribution after 6 hours of simulation, for varying

664 buoyancy source /! and buoyancy frequency #. For the case of varying /!, we have also estimated an

665 equivalent mass loss ﬂux (see text). On top of the distributions we plot the settling depths predicted by the simple

666 scaling relationship (dashed) and the one-dimensional line plume model (solid): both show excellent agreement

667 with the high-resolution simulations.

35 0

200 − 400 −

z (m) 600 − 2009 2009 2009 800 − 2014 2014 2014 1000 − 0.000 0.002 1 0 1 34.0 34.5 − Meltwater T(◦C) S

668 Fig. 6. Observed 2009 and 2014 temperature and salinity proﬁles next to the meltwater outﬂow from Pine

669 Island Glacier, as well as estimated meltwater fractions. In 2009, meltwater was primarily centered at a 400m

670 depth, while in 2014 it was able to rise to the surface.

36 3 3 3 2 3 3 Stratiﬁcation F/L = 10− m /s F/L = 10− m /s 0 2009 2009 2014 2014 100 −

200

z (m) −

300 − 2009 2014 400 − 0 1 2 0.00 0.05 0.00 0.02 0.04 2 5 2 N (10− /s ) Meltwater Meltwater

−3 3 3 −2 3 3 671 Fig. 7. Simulated vertical meltwater distributions for /! = 10 m /s and /! = 10 m /s , with initial

672 conditions set by observed temperature and salinity proﬁles for 2009 and 2014. Horizontal dashed lines indicate

673 the settling depths predicted by the one-dimensional line plume model for the same conditions; notably, the line

674 plume model accurately predicts the peak of the simulated meltwater distribution in all cases. We also plot

2 −2 3 3 675 depth proﬁles of stratiﬁcation strength in terms of # (see text). For /! = 10 m /s we see that there is little

676 diﬀerence in the vertical meltwater distribution between 2009 and 2014 conditions. However, the simulations

−3 3 3 677 with /! = 10 m /s show a marked diﬀerence: the qualitative trend is consistent with observations (Figure

678 6). Here, we propose that the rising meltwater was “trapped” by the notable local stratiﬁcation maximum at

679 around 350m depth in the 2009 conditions.

37 A

ﬁxed level ice shelf

melt ocean rate model model

ice shelf depth

meltwater ocean dynamic melt model plume rate model model

680 Fig. 8. Schematic describing two diﬀerent paradigms for meltwater ﬂuxes in simulations of global climate.

681 In paradigm A, the ﬂuxes from a melt rate model are inserted into the ocean model at some ﬁxed vertical

682 level; this approach has dominated the literature. In paradigm B, the melt rate model is coupled to a dynamic

683 plume model that describes the small-scale dynamics of buoyant meltwater plumes and accurately calculates the

684 vertical distribution of meltwater for insertion into the ocean model. Given the potential climatic importance of

685 glacial meltwater, the strong dependence of settling depth on the buoyancy forcing, and the vast heterogeneity

686 in the observed mass loss rates from diﬀerent ice shelves, this approach would likely represent a signiﬁcant

687 improvement over the “one-size-ﬁts-all" approach of paradigm A.

38 Restoring buoyancy source

0 4 1 f = -1.4 10− s− × 50 f = 0 − 100 − 150

z (m) − 200 − 250 − 300 − 0.00 0.01 0.02 0.03 Normalized meltwater concentration Constant buoyancy source

0 4 1 f = -1.4 10− s− × 50 f = 0 − 100 − 150

z (m) − 200 − 250 − 300 − 0.000 0.005 0.010 0.015 0.020 0.025 Normalized meltwater concentration

688 Fig. 9. Vertical meltwater distributions, for rotating and non-rotating cases, in a two-dimensional domain. On

689 the top, we have introduced meltwater via a restoring buoyancy source (following Naveira Garabato et al. (2017),

690 see text), while on the bottom we have used a constant buoyancy source (as in the simulations described in the

691 main text). When a constant buoyancy source is employed, the peak of the vertical distribution is not noticeably

692 inﬂuenced by the eﬀects of rotation. However, when a restoring buoyancy source is employed, rotation deepens

693 the peak by ∼ 50 m, consistent with the simulations of Naveira Garabato et al. (2017). Since the magnitude of the

694 buoyancy source is a primary control on the meltwater’s settling depth, the importance of any other parameters

695 can only be accurately investigated by holding the buoyancy source constant; therefore, these results show that

696 the use of restoring non-constant buoyancy sources may exaggerate the eﬀect of rotation on the settling depth.