Ice shelf basal melting removes more mass from the Antarctic Ice Sheet than iceberg calving does, but the three largest ice shelves, Filchner-Ronne, Ross, and Amery, contribute only 18 % of the net meltwater flux (Rignot et al., 2013). That is because the seawater-filled cavities beneath those ice shelves are dominated by high-salinity shelf water that has a potential temperature at or near the surface freezing point. Ice shelf basal melting occurs at depth because the freezing point temperature is lower under elevated pressure, and results in the formation of ice-shelf water (ISW), characterized by potential temperatures below the surface freezing point. When the buoyant ISW ascends along the ice-shelf base, the pressure relief causes it to become supercooled in situ, a necessary condition for ice crystals to persist in suspension. Those disk-shaped frazil-ice crystals accumulate under the ice shelves, leading to the formation of marine ice that is thicker and more localized than would be possible through direct freezing at the ice-shelf base (Morgan, 1972; Oerter et al., 1992; Fricker et al., 2001; Holland et al., 2007, 2009). Occasionally, frazil-ice crystals bathed in supercooled ISW are also carried out beyond the ice-shelf front and precipitated under adjacent sea ice, forming an unconsolidated, porous, sub-ice platelet layer (SIPL) (Gow et al., 1998; Hunkeler et al., 2016; Langhorne et al., 2015; Leonard et al., 2006; Robinson et al., 2014). SIPL not only harbours some of the highest concentrations of sea-ice algae on Earth (Arrigo et al., 2010) but also contributes to the sea-ice thickness when the water within the pores of SIPL freezes, due to heat loss to the atmosphere, to become incorporated platelet ice (Smith et al., 2001). Therefore, SIPL should not be ignored when investigating sea-ice thickness near an ice-shelf front.

Owing to the paucity of direct observation, our understanding of the evolution of frazil-ice-laden ISW relies heavily on numerical models. Those models are mostly derived from plume theory (Holland and Feltham, 2006; Jenkins and Bombosch, 1995; Rees Jones and Wells, 2018; Smedsrud and Jenkins, 2004) but include three-dimensional ocean circulation models (Galton-Fenzi et al., 2012) and have been widely applied to assess the marine ice beneath Filchner-Ronne (Bombosch and Jenkins, 1995; Holland et al., 2007; Smedsrud and Jenkins, 2004), Larsen (Holland et al., 2009) and Amery ice shelves (Galton-Fenzi et al., 2012), and SIPL under the sea ice in McMurdo Sound (Hughes et al., 2014, hereinafter HU14). To date, all the ISW plume models mentioned above have been depth-integrated, and all the scalar quantities, i.e. potential temperature, salinity, and frazil concentration in those models, are treated as vertically uniform. The well-mixed potential temperature and salinity have been validated by borehole observations beneath the Amery Ice Shelf (Herraiz-Borreguero et al., 2013) and under the sea ice in McMurdo Sound (Robinson et al., 2014; HU14). Although there are no observations of the vertical profile of frazil-ice concentration, it is unlikely to be vertically uniform because the buoyant rise of the crystals will counteract the turbulent diffusion that tends to homogenize the other properties. Recently, Cheng et al. (2017) showed that adopting an approach in which the frazil-ice growth is calculated using a vertically uniform frazil concentration results in substantial underestimation of marine-ice production underneath the western side of Ronne Ice Shelf. Idealized one-dimensional models confirm that the vertical distribution of frazil concentration cannot remain well mixed in the upper layers of the ocean (Svensson and Omstedt, 1998) and beneath the ice shelves (Holland and Feltham, 2005). Consequently, earlier assessments of either marine-ice or SIPL production in the aforementioned areas may need to be re-evaluated.

McMurdo Sound, located in the south-western Ross Sea (Fig. 1), is characterized by significant ISW outflow, arguably one of the most comprehensively observed ISW plumes available (HU14; Langhorne et al., 2015; Robinson et al., 2014). A prominent SIPL forms in the central-western sound (Dempsey et al., 2010); the maximum (area-averaged) observational first-year sea ice and SIPL are 2.5 (2) and 8 (3) m as determined from drill-hole measurements adjacent to the McMurdo Ice Shelf front between late November and early December in 2011 (Fig. 9 in HU14). The thin (∼20 m) McMurdo Ice Shelf front allows the ISW outflow to be delivered to the ocean surface without mixing with warmer ambient waters (Robinson et al., 2014). The study documented in HU14 was the first to apply the steady, one-dimensional frazil-ice-laden ISW plume model developed by Smedsrud and Jenkins (2004) to McMurdo Sound, although a constant ISW plume thickness was used. McMurdo Sound therefore seems an ideal setting in which to apply and evaluate the new vertically modified ISW plume model proposed by Cheng et al. (2017), which includes time dependence and two horizontal dimensions. The main objective is to explore the possibility of finding the quantitative relationship between the SIPL thickening rate and ISW supercooling. Establishing such a relationship is of significance to the assessment of total sea-ice thickness and thus the oceanic heat flux associated with SIPL in McMurdo Sound and elsewhere.

Here we first analyse the combined non-linear effects of the vertical distributions of supercooling and frazil concentration on the suspended frazil-ice growth rate in a supercooled ISW plume and compare results with those obtained with a commonly used, depth-averaged formulation. Then, we evaluate the performance of the vertically modified ISW plume model in reproducing the observed ISW supercooling and SIPL distribution to show the importance of considering the combined non-linear effects. Finally, we conduct 211 sensitivity simulations with the purpose of quantitatively establishing the response of the SIPL thickening rate as well as the frazil-ice concentration to variations in ISW supercooling in McMurdo Sound.

The growth rate of suspended frazil-ice controls both the dynamic and thermodynamic evolution of ISW plumes and the accretion of ice crystals beneath ice shelves (Cheng et al., 2017; Holland and Feltham, 2006; Smedsrud and Jenkins, 2004) and sea ice (HU14). The frazil-ice growth rate is found to be proportional to the following integral expression once a number of physical parameters within the commonly used formulation of Jenkins and Bombosch (1995) are merged: Igr=∫01TSCciσdσ,TSC=Tfσ,S-T, where σ∈[0,1] is the relative vertical coordinate, with 0 and 1 respectively corresponding to the upper-ice-plume and lower-plume–ambient-water interfaces. T and S are respectively the plume's potential temperature and salinity, vertically well mixed within the plume, ci is the vertically distributed (in this study) volumetric frazil concentration within the plume, and TSC and Tf are respectively the supercooling level (positive for supercooling) and local freezing point. Because of the well-known linear decrease in Tf with increasing water depth, TSC also varies linearly with depth, transitioning from supercooling to overheating as σ increases (Figs. 2a and 3). The corresponding transition height at which TSC=0 is defined by supercooled thickness, DSC=σSCD, where σSC and D are respectively the supercooled fraction and total ISW plume thickness.

In earlier ISW plume models, because ci is treated as vertically uniform, the integral of Eq. (1) can be represented by the product of the depth-averaged values TSC0.5 (0.5 means at mid-depth) and Ci. Thus, we refer to these ISW plume models as vertically uniform (VU). It is worth mentioning that, in order to take the supercooling into account when σSC<0.5, HU14 integrated TSC over the supercooled part only without introducing any frazil-ice melting. However, in this study, we will demonstrate that the important role of frazil-ice melting in the lower, overheated part of the plume cannot be ignored.

The vertical distribution of frazil concentration, in reality, much like the concentration of suspended sediment (Cheng et al., 2013, 2016), should be vertically non-uniform, with higher concentrations near the ice-shelf/sea-ice base. Considering only the balance between the buoyant-rise-induced vertical advection and turbulent diffusion terms, the governing equation for frazil concentration can be written as ddσKDdcidσ+widcidσ=0, where wi is the frazil-ice rise velocity, determined by ice crystal size. K is the vertical frazil concentration diffusion coefficient, which can be parameterized as vertically constant (Cheng et al., 2013, 2016): K=16κu∗D, where κ=0.4 is von Karman's constant, u∗=CdU is the friction velocity, related to the turbulent intensity within the ISW plume, Cd is the basal drag coefficient, U=Up+Ua2+Vp+Va2+Ut2 is the total flow speed, UpUa and VpVa are the depth-averaged ISW plume (ambient current) speed in the x and y directions respectively, Ut is the root-mean square tidal speed. Using a zero net flux condition in the equilibrium state at the lower boundary of the plume, i.e. KDdcidσ+wici=0,forσ=1 and a Dirichlet boundary condition at the upper boundary, i.e. ci=ci,b,forσ=0, where ci,b is the frazil concentration at the ice-shelf/sea-ice base, the vertical exponential profile for the equilibrium frazil concentration can be readily obtained (Cheng et al., 2017): ciσci,b=exp⁡-6Z∗, where Z∗=wi/κu∗ is the suspension index, otherwise known as the Rouse number. Integrating this exponential profile from σ=0 to 1, we finally obtain the relation between ciσ and Ci as ciσCi=6Z∗exp⁡-6Z∗σ1-exp⁡-6Z∗. As shown in Fig. 2a, the vertical distribution of frazil concentration is strongly controlled by Z∗. The gradient of the vertical distribution becomes greater with increasing Z∗, and a vertically uniform frazil concentration distribution can only be achieved as Z∗ approaches 0. While low values of Z∗ are attainable with strong currents, those conditions also reduce the tendency for frazil to precipitate and contribute to SIPL formation (see Eq. 3 below). Therefore, we expect a non-uniform vertical distribution of frazil wherever there is active formation of SIPL. Accordingly, Cheng et al. (2017) introduced Eq. (2) into Eq. (1), and as a result, significantly improved the simulated pattern of marine-ice growth under the western side of Ronne Ice Shelf, compared with the VU and satellite-derived (Joughin and Padman, 2003) results. Hereinafter, we refer to this vertically modified ISW plume model as VM. To conclude, the only difference between VM and VU models is whether the vertical distribution of frazil-ice concentration is introduced.

The dependence of the integral value of Igr on Z∗ under specified conditions of supercooling (Fig. 2a) is shown in Fig. 2b, where DSC=50 m (a value within the calculated range for the standard run, Fig. 1) in all the cases. It can be seen that the integral value increases non-linearly with Z∗. The critical Z∗ that represents the transition from frazil-ice melting (Igr<0) to freezing (Igr>0) decreases as the supercooled part of ISW plume increases. In contrast, owing to the neglect of vertical variation in ci, the integral values calculated using the VU formulation are constant, leading to transitions from overestimation of frazil-ice growth to underestimation, compared with VM, as Z∗ increases. Only if the ISW plume is fully supercooled (σSC=1) and Z∗ is close to 0 are the integral values of Igr calculated by VU and VM formulations equal (star in Fig. 2b). These features are illustrated in Fig. 2a: for given supercooling, if Z∗ becomes larger, there is higher (lower) frazil concentration in the upper (lower), supercooled (overheated) part of the ISW plume. Owing to the assumption that thermohaline exchanges between frazil crystals and ambient water occur only at the crystal edge for freezing, but over the whole crystal surface for melting (Jenkins and Bombosch, 1995), the integral values of Igr for the lower overheated part can be of much greater magnitude (Fig. 2b). It is therefore necessary to limit the mass loss due to frazil melting in one model time step, such that it does not exceed the frazil concentration in the lower, overheated part of the plume. Overall, the frazil concentration and frazil growth rate distributions in the VM model show physically reasonable and desirable characteristics that are absent from the VU model, and the impacts will be demonstrated by evaluation of the VM model in McMurdo Sound.

The unsteady VM and VU models used in this study are described in detail by Cheng et al. (2017). The governing equations for ISW properties and frazil concentration in both VM and VU models remain as they were in the depth-integrated, two-dimensional ISW plume model developed by Holland and Feltham (2006), except for the different treatments of the specific terms associated with the frazil-ice growth rate, described above, in the frazil concentration and potential temperature transport equations of the VM model. Both VM and VU models combine the same commonly used parameterizations of thermohaline exchanges across the ice–water interfaces, specifically a three-equation formulation (Holland and Jenkins, 1999) for the sea-ice base and a two-equation formulation for frazil ice (Galton-Fenzi et al., 2012), with a multiple size-class frazil dynamics model (Smedsrud and Jenkins, 2004) to calculate basal freezing (f′) and frazil melting/freezing (w′), secondary nucleation (N′), and precipitation (p′). These processes are summarized in Fig. 3. Rather than repeat all the equations here, we recall some of them and present how we set up our ISW plume models on the McMurdo Sound domain.

The model domain (Fig. 1) is delimited by a 45×40 km rectangle in the x–y plane with an ISW outflow from beneath McMurdo Ice Shelf. The base of the sea ice in McMurdo Sound is assumed to be horizontal and rough, owing to the presence of SIPL. The drag coefficient of the ice underside is therefore 6–30 times larger than that typically applied in ice–ocean interaction models (Robinson et al., 2017). The parameterization of the sea-ice thermodynamics, the assumption of no entrainment of ambient water into the ISW plume, and the boundary conditions at the ISW outflow follow HU14. The initial thickness of the ISW outflow (indicated by blue arrow in Fig. 1) from underneath McMurdo Ice Shelf is set equal to that of the supercooled layer; i.e. D=DSC, and the discharge per unit width is set to 0.02 m2 s-1. The addition of both an ambient circulation and tides follow HU14: the former, which represented the only source of momentum in the study of HU14, is assumed to be parallel to the Victoria Land coast, in the negative y direction, and to be constant throughout the model domain; the latter is calculated using root-mean square tidal speeds from Padman and Erofeeva (2005). Because ISW persists in McMurdo Sound for at least the 8–9 months of the ice growth season (Robinson et al., 2014), all runs are integrated for 240 days. The model resolution and time step (t) are 1 km and 25 s, respectively. The frazil-ice size distribution is represented by five crystal size classes, and the transfer processes, induced by frazil freezing and melting, between different size classes are calculated using the scheme proposed by Smedsrud and Jenkins (2004). Sensitivity experiments with more crystal size classes yielded qualitatively similar results. The ice concentration at the ISW outflow is evenly distributed among the classes (Holland and Feltham, 2005, 2006; Smedsrud and Jenkins, 2004).

We treat the frazil-ice precipitation rate p′ as inverted sedimentation and follow the parameterization of McCave and Swift (1976): p′=wiCi1-U2Uc2×He1-U2Uc2, where Uc is a critical velocity, above which precipitation cannot occur, determined by Jenkins and Bombosch (1995): Uc2=θiρ0-ρig2reρ0CD, where θi is the Shields criterion, ρ0 and ρi are reference seawater and ice densities, respectively, g is gravity, and re is the equivalent radius of a sphere with the same volume as the frazil disk. The frazil-ice rise velocity, wi, is calculated by Morse and Richard (2009): wi=2.025Di1.621ifDi≤1.27mm-0.103Di2+4.069Di-2.024if1.27<Di≤7mm, where Di=2ri is the diameter of a frazil crystal in millimetres. The inclusion of the Heaviside function He means that negative precipitation (i.e. erosion of previously deposited frazil ice) is not permitted. Because we have no idea about how cohesive the ice crystals are once they have settled, the estimation of an erosion rate would entail additional uncertainties.

The complex processes after the frazil-ice precipitates onto the sea-ice base are simplified in our model. In order to calculate SIPL thickness DP at the nth time interval, we adopt the assumptions of HU14 that solid ice fraction within SIPL in McMurdo Sound is 0.25 based on the observational estimation from Gough et al. (2012) and that the ice crystals, on average, double in volume after precipitation: DP=10.25×2×∑k=1np′k×t. It should be noted that the volume change factor is a broad estimate, with almost no supporting evidence in the literature to guide it. Coupling our VM model with a model focusing on the processes associated with platelet ice accretion within the sea ice (Buffo et al., 2018) would be necessary to improve on that rough estimate but is beyond the scope of the present study.

In this study, we demonstrated how the vertical distributions of supercooling and frazil ice concentration within an ISW plume jointly determine the growth of suspended frazil ice, and thus the rate of SIPL formation under sea ice and marine ice beneath ice shelves. A vertically modified, frazil-ice-laden ISW plume model, which encapsulates these combined non-linear effects, was applied to the McMurdo Sound region, and reproduced the observed ISW supercooling and SIPL distributions in two horizontal dimensions. Using multiple model runs, the relationship of ISW supercooling to SIPL thickening rate and frazil concentration in McMurdo Sound was explored and shown to be dependent on the suspension index that controls the vertical distribution of frazil concentration within the ISW plume. Moreover, when the thickness of a supercooled layer of ISW is large enough, the efficiency of converting ISW supercooling into frazil concentration, and thus SIPL growth is determined by the suspension index. These findings highlight the need for further observations in McMurdo Sound, particularly focused near the ISW outflow region in the western sound, where the supercooled ISW plume and SIPL are prominent, and more general observations that help to constrain the frazil size spectrum within the sea-ice–ocean boundary layer. In addition, the performance of the VM model in providing reliable estimates of supercooling and frazil ice flux at the SIPL base makes it an attractive tool for coupling with sea-ice models, focusing on microscale processes within the bottom layer of the ice (Buffo et al., 2018).

It would be straightforward for the next step to investigate the relationship between supercooling and marine-ice thickening rate underneath ice shelves using the VM model. Quantifying this relationship would be the key to parameterizing the process in more complex three-dimensional, primitive equation ocean models, which frequently neglect details of the ice-shelf–ocean boundary layer and processes associated with an evolving suspension of frazil ice crystals (Liu et al., 2017, 2018; Mueller et al., 2012, 2018). Results may differ from those discussed above, because of the subtly different environments beneath sea ice and ice shelves. Beneath a SIPL, supercooling is produced by the pressure drop experienced by ISW as it emerges from beneath an ice shelf and rises towards the sea surface, while supercooling that drives marine-ice accretion beneath ice shelves is produced as the ISW ascends a very gentle basal slope. The in situ supercooling level beneath ice shelves is therefore likely to be much smaller than that observed in McMurdo Sound, while the differing slopes also yield differing buoyancy forcings on the flow. Furthermore, after experiencing the step change in pressure as it ascends the ice front, the supercooled plume in McMurdo Sound is in the process of adjustment, through the formation of suspended frazil and direct freezing onto the accreted SIPL, towards an equilibrium that is presumably attained beyond the region of observations. At the base of an ice shelf, typically several hundred metres thick, the vertical temperature gradient is comparatively small, so the deposited crystals form a slushy layer (Engelhardt and Determann, 1987) that slowly consolidates, possibly as much through compaction as freezing. The ice–ocean interface and the associated drag coefficient are therefore likely to be very different to those observed in McMurdo Sound, where SIPL appears to comprise a more open matrix of ice and water that consolidates by freezing as heat is lost to the atmosphere. In addition, the vastly different timescales over which crystal accretion occurs (about 1–3 years in McMurdo Sound vs. tens of hundreds of years beneath ice shelves) could lead to further differences in the internal structure of the crystal layers and hence in the physical boundaries they present to the ISW plume. Therefore, the VM model would need to be re-evaluated against observations of sub-ice-shelf ISW plumes and the ice-shelf–ocean boundary layer. Finally, further process studies, including the influence of the vertical current structure within either the ice-shelf or sea-ice–ocean boundary layer (Jenkins, 2016; Robinson et al., 2017) could also contribute to improving our understanding of marine-ice and SIPL formation.