Turbulent exchanges may be parameterized through the flux of a quantity at the interface between the ocean and the sea ice, calculated as the product of a scale velocity and the change in the quantity from the boundary to some reference level (McPhee, 2008): 1w′S′=αsu∗Sw-So, where 〈w′S′〉 represents the averaged co-variance of the turbulent fluctuations of interface vertical velocity (m s-1) and salinity, αs is an interface salt–nutrient exchange coefficient (dimensionless), u∗ is the friction velocity (m s-1), and So and Sw are interface and far-field salinities, respectively.

We calculate nutrient exchanges using a similar approach: 2FN=-αsu∗Nw-N0. This is an extension of the concept used for heat and salt by McPhee (2008; see p. 112, Fig. 6.3). The minus sign used in Eq. (2) is for compatibility with the CICE + Icepack convention that upward fluxes be negative (e.g., Hunke et al., 2015). αs varies from 8.6×10-5 during the melting season to 0.006 during winter (McPhee et al., 2008).

Before explaining how Eq. (2) was implemented in the CICE + Icepack we describe the model vertical biogeochemical grid (biogrid), the tracer equation and the bottom boundary conditions. The biogrid is the non-dimensional grid used for discretizing the vertical transport equations of biogeochemical tracers, defined between the brine height (h), which takes the value 0, and the ice–ocean interface, which takes the value 1 (Jeffery et al., 2016). The Icepack tracer equation (without biogeochemical reaction terms for the sake of simplicity) may be written as follows (for more details, see Jeffery et al., 2011, 2016): 3φ∂N∂t=x-1h∂zt∂t-xh∂zb∂t∂∂xφN+1h∂∂xwfN+∂∂xDMLD+φDmh2∂N∂x, where 0≤x≤1 is the relative depth of the vertical domain of the biogrid, zt and zb are vertical positions of the ice top and bottom (m), respectively, ϕ is sea ice porosity, wf is the Darcy velocity due to the sea ice flushing of tracers (m s-1), Dm is the molecular diffusion coefficient, and DMLD is the mixed length diffusion coefficient (m2 s-1). DMLD is detailed in Jeffery et al. (2011), and it is 0 when the brine vertical density gradient is stable, otherwise (when density increases towards the ice top) it is calculated as follows: 4DMLD=gkμΔρel, where g is the acceleration of gravity (9.8 m s-2), k is sea ice permeability, μ is dynamic viscosity (2.2 kg m-1 s-1), ρe is the equilibrium brine density, and l is a length scale (7 m). The values shown here are the default ones in CICE + Icepack.

The bottom boundary condition of Eq. (3) is based on values of N at the sea ice bottom interface (N0 at x=1) and in the ocean (Nw) (Jeffery et al., 2011). Therefore, the last term of Eq. (3) at the bottom boundary may be written as follows: 5DMLD+φDmh2∂xN0-Nw. In CICE + Icepack, diffusion timescales are calculated separately for later usage in Eq. (3) as follows: 6τ=Dmh2s-1, and 7τ=DMLDh2s-1. A similar timescale for the turbulent process described by Eq. (2) may be calculated from the following equation: 8τ=αsu∗hs-1. Therefore, in the Los Alamos Sea Ice Model the implementation of turbulent diffusion nutrient exchanges at the ice–ocean interface is quite straightforward. In other models, other approaches may be required.

The usage of h in these timescales merely implies the way they are normalized in the code before the actual diffusive fluxes are calculated considering the distance between the points (h∂x; see Eq. 3) where variables are calculated along the layers of the biogrid. The product h⋅x corresponds to the actual distance of a given point from the top of the biogrid.

In the simulations using turbulent diffusion, we perform the same calculations, except that the molecular diffusion term φDmh2 is replaced with a turbulent diffusion term αsu∗h at the interface between the last model layer and the ocean. This exchange process takes place “outside” the sea ice where φ=1, directly affecting only the tracer concentration at the ice–ocean interface.

From Eqs. (6)–(8) it turns out that the product αsu∗ by distance (z) has the same dimensions as Dm or DMLD, corresponding to a turbulent diffusion coefficient. Assuming z≈0.01 m, turbulent diffusion induced by velocity shear becomes comparable with molecular diffusion only for u∗<0.0012 m s-1 considering the lower end of the αs range (8.6×10-5; see above) or u∗<1.7×10-5 m s-1 considering the upper end of the αs range (0.006). If we instead assume z≈0.00054 m (the average thickness of the molecular sublayer reported in Lavoie et al., 2005), the calculated u∗ values increase by 1 to 2 orders of magnitude (depending on αs) but are still low (0.0004–0.03 m s-1). In fact, such low friction velocities would require low “stream” velocities, i.e., relative ice–ocean velocities. For an account of the relationship between stream and friction velocities under the sea ice, see Supplement 3 of Olsen et al. (2019) and references therein. These authors show that stream velocities of only a few centimeters per second lead to friction velocities that are an order of magnitude lower but are still on the order of 0.001 m s-1, i.e., smaller only than the highest u∗ values estimated above. Considering current velocities relative to the sea ice observed during the N-ICE2015 cruise (Granskog et al., 2018; Fig. 2d of Duarte et al., 2017), with most values between 0.05 and >0.2 m s-1, it is rather likely that friction velocities under the ice are frequently above the thresholds calculated above and that turbulent diffusion will dominate over molecular diffusion. Dalman et al. (2019) provided experimental evidence for such turbulent nutrient fluxes to the ice bottom, leading to increased chlorophyll concentrations at the bottom ice in a strait with strong tidal currents. The mechanism treated here as turbulent diffusion seems analogous to “forced convection” in the lowermost parts of the brine network, which is driven by pressure differences caused by the shear under the sea ice (Vancoppenolle et al., 2013).

We used the Los Alamos Sea Ice Model, which is managed by the CICE Consortium with an active forum (https://bb.cgd.ucar.edu/cesm/forums/cice-consortium.146/, last access: 26 January 2022), and a Git repository (https://github.com/CICE-Consortium, last access: 26 January 2022). It includes two independent packages: CICE and Icepack. The former computes ice dynamic processes, and the latter computes ice column physics and biogeochemistry. Their development is handled independently with respect to the GitHub repositories (https://github.com/CICE-Consortium). All of the changes described below were implemented in two forks to the above repository, one for Icepack and another for CICE, and they may be found in Duarte (2021a) and Duarte (2021b), respectively.

Our simulations may be run using only Icepack, since they are focused on ice column physics and biogeochemistry, without the need to consider ice dynamic processes. However, we used both CICE + Icepack together to allow for use of a netCDF-based input/output not included in Icepack. Therefore, we defined a 1D vertically resolved model with 1 snow layer and 15 ice layers and 5×5 horizontal cells. This is the minimum number of cells allowable in CICE due to the need to include halo cells (only the central “column” is simulated). Therefore, ice column physics and biogeochemistry were calculated by Icepack, but CICE was the model driver. The input file (ice_in) used in this study was included in our CICE fork and it lists all parameters used in the model and described in Hunke et al. (2015), Jeffery et al. (2016), and Duarte et al. (2017) and in Tables S1 and S2. Any changes in “default” parameters or any other model settings will be specified.

We made several modifications in CICE to allow using forcing time series collected during the Norwegian young sea ice (N-ICE2015) expedition (Granskog et al., 2018) and described in Duarte et al. (2017) (see Fig. 2 of the cited authors). These modifications were meant to allow reading of forcing data at higher frequencies than possible with the standard input subroutines in the CICE file ice_forcing.F90.

When the dynamical component of CICE is not used, u∗ is set to a minimum value instead of being calculated as a function of ice–ocean shear stress (Hunke et al., 2015). Duarte et al. (2017) implemented shear calculations from surface current velocities (one of the models forcing functions) irrespective of the use of the CICE dynamics code. These modifications were also incorporated in the current model configuration so that shear can be used to calculate friction velocity and thereafter influence heat and tracer–nutrient exchanges, following Eqs. (2) and (8) and parameters described in McPhee et al. (2008). When the parameter kdyn is set to 0 in ice_in, ice dynamics are not computed, but shear is calculated in the modified subroutine icepack_step_therm1, file icepack_therm_vertical.F90. If kdyn is not 0, these calculations are ignored since shear is already calculated in the dynamical part of the CICE code.

A Boolean parameter (Bottom_turb_mix) was added to the input file, which is set to “false” or “true” when the standard molecular diffusion approach or the new turbulence-based diffusion approach is used, respectively. Another Boolean parameter (Limiting_factors_file) was added to the ice_in file. When set to true, limiting factor values for light, temperature, nitrogen, and silicate are written to a text file every model time step. These are calculated by Icepack biogeochemistry, according to Jeffery et al. (2016), but there is no writing output option in the standard code.

Simulations were run for a refrozen lead (RL) without snow cover and for second-year sea ice (SYI) with ∼40 cm snow cover monitored in April–June during the N-ICE2015 expedition (Granskog et al., 2018, and Fig. 1 of Duarte et al., 2017). Details of model forcing with atmospheric and oceanographic data collected during the N-ICE2015 expedition, including citations and links to the publicly available datasets, are given in Fig. 2 and Sect. 3 of Duarte et al. (2017) and in the Supplement. These datasets include wind speed, air temperature, precipitation, and specific humidity in Hudson et al. (2015); incident surface short and longwave radiation in Hudson et al. (2016); ice temperature and salinity in Gerland et al. (2017); sea surface current velocity, temperature, salinity, and heat fluxes from a turbulence instrument cluster (TIC) in Peterson et al. (2016); sea surface nutrient concentrations in Assmy et al. (2016); and sea ice biogeochemistry in Assmy et al. (2017). Ocean forcing is based on measurements within the surface 2 m that provide the boundary condition for the sea ice model. Model forcing files may be found in Duarte (2021c).

RL simulations started with zero ice, whereas SYI simulations started with initial conditions described in the Supplement (Table S3).

We ran simulations with the standard formulations for biogeochemical processes described in Jeffery et al. (2016) and settings described in Duarte et al. (2017) using mushy thermodynamics and vertically resolved biogeochemistry and including freezing, flushing, brine mixing length, and molecular diffusion within the ice and at the interface between the ocean and the sea ice as nutrient exchange mechanisms (Jeffery et al., 2011, 2016). We contrasted the above simulations against others that replaced brine molecular and mixed-length diffusion of nutrients at the interface between the ocean and the sea ice with diffusion driven by current velocity shear (Table 1) calculated similar to heat exchanges and following the parameterization described in McPhee et al. (2008) and detailed above (Eqs. 2–7). This contrast provides insight into the effects of velocity shear on nutrient diffusion, ice algal production (mg C m-2 d-1), chlorophyll standing stocks (mg Chl a m-2), and vertical distribution of chlorophyll concentration (mg Chl a m-3) (note that CICE model output for algal biomass in mmol N m-3 was converted to mg Chla m-3 as in Duarte et al., 2017, using 2.1 mg Chl a mmol N-1 and following Smith et al., 1993). However, due to the concurrent effects of algal biomass exchange between the ocean and ice, such a contrast is not enough to explicitly test our hypothesis and conclude about the effects of turbulence-driven nutrient supply on ice algal nutrient limitation. Therefore, simulations were also run contrasting the same model setups, as described above but restarting from similar algal standing stocks and vertical distributions within the ice and switching off algal inputs from the water to the ice. This was done by nullifying the variable algalN, defining the ocean surface background ice algal concentration, in file icepack_zbgc.F90, subroutine icepack_init_ocean_bio, and the restart files. In the case of the RL simulations that started with zero ice, first a simulation was run until the 12 May, and then the obtained ice conditions were used to restart new simulations without algal inputs from the ocean (algalN = 0 mmol N m-3). Therefore, when the simulations restarted there was already an ice algal standing stock necessary for the modeling experiments developed herein. The SYI simulations were by default “restart simulations” beginning with observed ice physical and biogeochemical variables. Therefore, there was already an algal standing stock in the ice from the onset (Sect. S1 and Table S3).

McPhee et al. (2008) estimated different values for αs depending on whether the sea ice is growing (highest value) or melting (lowest value) (Table 1). When running simulations for the RL, in some cases we used only the minimum or the maximum values for αs to allow for a more extreme contrast between molecular and turbulent diffusion parameterizations. This was done since the former value will tend to minimize differences, whereas the latter will tend to emphasize them. We also completed simulations for the RL and for SYI changing between the maximum and the minimum values of αs when ice was growing or melting, respectively, and following McPhee et al. (2008) (see Table 1 for details). This parameterization with variable αs is likely the most realistic one, accounting for double diffusion during ice melting (McPhee et al., 2008).

Apart from contrasting the way bottom-ice exchanges of nutrients were calculated, some simulations contrasted different parameters related to silicate limitation (Table 1). This approach follows Duarte et al. (2017), where simulations were tuned by changing the Si:N ratio and the half-saturation constant for silicate uptake because silicate limitation was leading to an underestimation of algal growth. From this exercise we were able to assess if such tuning was still necessary after implementing turbulent diffusion at the interface between the ocean and the sea ice, driven by velocity shear. Moreover, we repeated simulations with varying snow heights to further investigate the interplay between light and nutrient limitation under contrasting nutrient diffusion parameterizations (Table 1).

All simulations with turbulent diffusion (RL_Sim2–RL_Sim5, Table 1) predict higher bottom chlorophyll a (Chl a) concentration than with the standard molecular diffusion formulation (RL_Sim1) (Fig. 1a). Simulations RL_Sim2–4 grossly overestimate observations. Simulation RL_Sim3, using the lowest value for αs, is closer both to observations and to RL_Sim1, as well as RL_Sim5, with the latter having the same αs values of RL_Sim4 but a half-saturation constant for silicate limitation increased from its tuned value in Duarte et al. (2017) of 2.2 to 5.0 µM and algalN reduced (Table 1) to bring model results closer to observations. Patterns between simulations for the whole ice column and considering both standing stocks and net primary production are similar to those observed for the bottom ice (Fig. 1b). Algal biomass is concentrated at the bottom layers (Fig. 2). Concentrations in the layers located between the bottom and the top of the biogrid, defined by the vertical extent (brine height) of the brine network (green lines in the map plots) (Jeffery et al., 2011), are <10 mg Chl a m-3. Ice thickness, temperature, and salinity profiles are extremely similar among these simulations (Figs. S1 and S2).

Results for the silicate- and nitrogen-limiting factors are based on brine concentrations. Limiting factors exhibiting lower values (more limitation) in RL simulations are silicate, followed by light (Figs. 3, S3–S5). Limiting values for silicate range between 0 (maximum limitation) and 1 (no limitation), with stronger limitation after 13 May in all simulations (Fig. 3). The most severe silicate limitation is for RL_Sim1, where values drop to near zero around middle May. Despite the high average bottom Chl a concentration predicted in all simulations, the bottom layer is where silicate limitation is less severe after 13 May. This is more evident in simulations with turbulent bottom diffusion, where light limitation at the bottom ice becomes more severe than silicate limitation around the end of May (Fig. S6).

Results obtained with RL_Sim6–9 without algal exchanges between the ocean and the ice (see Table 1) show similar patterns of those observed with RL_Sim1–5, respectively (Fig. 4 versus Fig. 2, Fig. S9 versus Fig. 3, Figs. S7 and S8 versus Figs. S1 and S2, and Figs. S10–S12 versus Figs. S3–S5).

Interface diffusivity (one of CICE diagnostic variables, corresponding to the diffusion coefficient between adjacent biogeochemical layers and between the bottom layers and the ocean) for simulations with turbulent exchanges (αsu∗H) are up to 2 orders of magnitude higher at the bottom (diffusivity between the bottom layer and the ocean) than for the RL_Sim1 simulation with only molecular diffusion (Dm) + the mixed-length diffusion coefficient (DMLD) (see Sect. 2.1 and Fig. 5).

Simulations with turbulent diffusion (SYI_Sim2 and 3) predict only slightly higher standing stocks and net primary production than with the standard molecular diffusion formulation (SYI_Sim1) (Fig. 6). The visual fit to the standing stock observations is comparable between the various simulations. Changing the half-saturation constant for silicate limitation from 2.2 to 4.0 µM has no impact on model results. This is confirmed by analyzing the evolution of Chl a concentration as a function of time and depth in the ice (Fig. 7), with only minor differences being apparent towards the end of the simulation, when Chl a increases at the bottom layers in the simulations with turbulent diffusion (SYI_Sim 2 and 3). Ice thickness, temperature, and salinity profiles are extremely similar among these simulations (Fig. S13).

The dominant limiting factor in these simulations is light, followed by silicate (compare Fig. 8a, c, and e with 8b, d, and f and with Fig. S14). Light limitation is less severe after the onset of snow and ice melting at the beginning of June. Silicate limitation is very strong above the bottom ice. Nitrogen limitation is highest at a depth range between ∼0.4–0.7 m below the ice top, with a large overlap with the depth range where a Chl a maximum is observed (Fig. 7). Maximal Chl a concentration predicted for the RL_Sim1 and RL_Sim5 simulations – those closer to observations – are 2 orders of magnitude higher than those predicted for SYI (Fig. 2a and e versus Fig. 7). However, standing stocks predicted for RL_Sim1 and RL_Sim5 simulations are smaller than for SYI simulations, as confirmed by the observations (Figs. 1b and 6). Opposite to what was described for the RL simulations, silicate limitation becomes more severe than light limitation at the bottom layer only in SYI_Sim_1 at the beginning of June close to the end of the simulation (Fig. S15).

Results obtained without algal exchanges between the ocean and the ice (SYI_Sim4 and 5; see Table 1), show the same patterns as those observed with SYI_Sim1 and 2, respectively (Fig. 9 versus Fig. 7, Fig. S17 versus Fig. 8, Fig. S18 versus Fig. S14a–d, and Fig. S16 versus Fig. S13a–d).

Interface diffusivity (one of CICE diagnostic variables; see Sect. 3.1) for simulations with turbulent bottom exchanges are up to 4 orders of magnitude higher at the bottom ice than for simulations with only molecular diffusion (Fig. S19, showing a comparison between SYI_Sim1 and SYI_Sim2).

SYI_Sim1 and 2 were repeated with varying snow thickness (Table 1 and Figs. 10 and 11). In the former simulation (Fig. 10a), as snow height decreases, there is a reduction in light limitation and a sharp increase in silicate limitation, overtaking light limitation (values becoming lower) as early as mid-May. In the latter simulation (Fig. 10b), light limitation prevails irrespective of snow height, except in the case of the lower snow height of 15 cm where silicate becomes more limiting towards the end of the simulation. With the decrease in snow height, there is an increase in Chl a concentration in all simulations. The highest values for SYI_Sim2 are around an order of magnitude larger than those for SYI_Sim1. Moreover, the decrease in snow heights is followed by an earlier and more intense bottom ice algal bloom.