Sea Ice Biogeochemistry: A Guide for Modellers

Sea ice is a fundamental component of the climate system and plays a key role in polar trophic food webs. Nonetheless sea ice biogeochemical dynamics at large temporal and spatial scales are still rarely described. Numerical models may potentially contribute integrating among sparse observations, but available models of sea ice biogeochemistry are still scarce, whether their relevance for properly describing the current and future state of the polar oceans has been recently addressed. A general methodology to develop a sea ice biogeochemical model is presented, deriving it from an existing validated model application by extension of generic pelagic biogeochemistry model parameterizations. The described methodology is flexible and considers different levels of ecosystem complexity and vertical representation, while adopting a strategy of coupling that ensures mass conservation. We show how to apply this methodology step by step by building an intermediate complexity model from a published realistic application and applying it to analyze theoretically a typical season of first-year sea ice in the Arctic, the one currently needing the most urgent understanding. The aim is to (1) introduce sea ice biogeochemistry and address its relevance to ocean modelers of polar regions, supporting them in adding a new sea ice component to their modelling framework for a more adequate representation of the sea ice-covered ocean ecosystem as a whole, and (2) extend our knowledge on the relevant controlling factors of sea ice algal production, showing that beyond the light and nutrient availability, the duration of the sea ice season may play a key-role shaping the algal production during the on going and upcoming projected changes.


S0. Model code
The sea ice biogeochemistry model used in this work is part of the BFM package and it is written in FORTRAN90. The model code, physical forcing data, reference simulation and all scenarios discussed in this work are available for download from the BFM web site (http://www.bfm-community.eu). After registration, go to http://bfm-community.eu/download and download the version including the sea ice. Follow the instruction for setting up the pelagic model and compile with the following command: ./bfm configure.sh -gcd -p STANDALONE SEAICE S1. Equations of the sea ice thermo-halodynamic model used for the forcing computation This is a selection of the equations of the sea ice halo-thermodynamic model described in [1,2] that are used in this work to compute the physical forcing data of the Biologically Active Layer (BAL) and that are needed by the biogeochemical model.
For the continuity of the heat conduction at the snow/ice interface [3], we have: that can be approximated as: where K i is the sea ice thermal conductivity (2.0 W m K −1 ), T f r is the seawater freezing temperature (-1.728 o C), T i is the temperature at the snow/ice interface, h i is the sea ice thickness, K s is the snow thermal conductivity (0.2 W m K −1 ), T 0 is the surface temperature (prescribed, between T f r and -20 o C in S0), and h s is the snow thickness. As the sea ice is continuously in thermal equilibrium, the sea ice bulk temperature (T bk ) is always equal to brines temperature (T br ) and the latter is related to brines salinity (S br ) by the following relationship: where µ is 0.054 if temperature is in o C [4]. Brine volume (V br ) is related to brine salinity (S br ) as: where S bk is the sea ice bulk salinity.
As sea ice is isosaline, we can easily derive the brine temperature and salinity at the boundary with the BAL, considering that the BAL is defined as the sea ice part where the relative brine volume is 0.05: For the continuity of the heat conduction we will then have: to derive the thickness of the BAL. Assuming the same continuity of heat conduction: we are able to calculate the ice/brines temperature within the BAL (T BAL ). And for the same thermal equilibrium the BAL brine salinity (S brBAL ) and relative brine volume (V BAL ) within the BAL can be known: The remaining physical variable that is needed is the average amount of Photosynthetically Available Radiation (PAR) found in the BAL. We use a simple parameterization based on the Bouguer-Lambert law (e.g. [5]) to calculate the amount of solar radiation penetrating below the surface: where I 0 is the fraction of the incident irradiance transmitted through the surface, α x is the albedo either for snow or ice (see Table S1 for values), Q s is the incoming short-wave radiation (prescribed, as described in the main paper), and κ x is the extinction coefficient for either snow or ice (see Table S1 for values).

S2. Equations of the simplified sea ice biogeochemistry model and coupling fluxes with the ocean
This section presents the equations of the sea ice biogeochemical model used in this work, a simplified version of the model described in [6]. Model variables are listed in Table S2 and depicted in Fig. 6 of the main paper and parameter values are given in Table S3.
The rate of change of a sea ice constituent like for instance sea ice algae concentration (A i , expressed in concentration of carbon (c), Chl (l) and silica (s) integrated over the biological layer) can be described as the sum of the rate of change due to the local biogeochemical processes (bio) and the one due to the exchange with the ocean (oce): The biological time-rate of change for sea ice algae components considers the processes of photosynthesis (gross primary production= gpp), respiration (rsp), mortality/excretion (lysis =lys and exudation = exu), acclimation and silicate uptake (upt): Gross primary production and respiration involve the uptake and release of dissolved carbon dioxide F (3) . Production is controlled by temperature, light and silicate concentration as where Q A 10 is the characteristic doubling temperature parameter based on the 10 o C reference temperature, h s A is the half-saturation parameter for dissolved silicate uptake, E P AR is the Photosynthetic Available Radiation (PAR) and E K is the light saturation parameter, the ratio between the realized maximum Chl-a specific photosynthetic rate P m and the maximum light utilization coefficient (α 0 A ): Respiration is parameterized as the sum of temperature-controlled basal rate and net growth energy requirement (gross production minus exudation) as where b A is a constant specific rate of respiration and γ A is a fraction of the assimilated production. The exudation losses are also linearly proportional to gross primary production Cell mortality in sea ice algae is parameterized as a first order process dependent on the internal physiological state (the nutrient quota) and a constant maximum specific lysis rate d 0 A . Since in this case the only limiting nutrient is silicate which is not stored internally in the cell, the term is simplified to Net Chl-a synthesis is a function of acclimation to light conditions, nutrient availability and turnover rate. The rate of change of net photosynthesis in (13) is controlled by the dynamical Chl:C ratio θ chl originally proposed by [7], which regulates the amount of Chl-a in the cell according to a non-dimensional ratio between the realized photosynthetic rate in (15) and the maximum potential photosynthesis: where θ 0 chl is the maximum quotum of Chl-a:C.
Silicate uptake for sea ice diatoms in (14) is also derived from pelagic diatoms, linearly coupling net carbon growth by means of an optimal cellular quota Si:C s opt A as dA s dt upt The rate of biogenic silica release in (14) is directly derived from the carbon-based mortality rate (17).
The exchange rate with the ocean in eq. (11) has to be carefully specify to ensure mass conservation with pelagic constituents. When an ice-free ocean is freezing, the first term on the right-hand side of eq.
(11) will be zero and the initial concentration of sea ice algae will only be due to the ocean boundary flux with the variable concentration in seawater. Let X si and X sw be the concentration of a generic variable in sea ice and sea water, then the flux can be written as a function of sea ice growth velocity v si : where h si is the thickness of the growing sea ice where biology can be found. Note that the thickness parameter depends on the choice of the model vertical layout and thus h si can be either constant or the variable thickness of the layer in contact with the ocean. This boundary flux is assumed to exist over the entire growth season. At the end of the ice season the flux is generally reverted as there will be a certain amount of X si left in sea ice that will be transferred to seawater. The concentration loss from melting sea ice, which is equivalent to the concentration growth in seawater may be written similarly to (21): A further note must be added when integrated sea ice concentration is used. The equation will return a concentrations of X in mmol m −3 that will be added at the first level of the ocean. In the case of the Biologically Active Layer used in this work we specify the velocity of the boundary fluxes in eq. (21) and eq. (22) as a function of the growth/melt rate dhi dt [6,8]. The rate of change of dissolved silica is then: where N (5) is the silicate concentration in seawater, and A s is the content of silica in sea ice diatoms computed by the eq. (20). The silicate boundary flux is written as: At sea ice formation, the uptake term in eq. 23 will be zero and the initial concentration of silicate in sea ice will only be due to the boundary flux with the ocean, i.e.: The boundary flux is positive as long as sea ice grows, and negative as sea ice melts. At the very end of the ice season, the concentration of dissolved silica left in sea ice will be transferred to seawater silicate as: The sea ice enrichment in living and non living particulate organic matter differs from that of dissolved components as it is only a function of the seawater concentration, the ice growth rate and the actual available space in the sea ice matrix, i.e. the relative brine volume, V BAL . Thus the corresponding boundary flux for e.g. the biogenic silica content of sea ice algae is: and similar considerations as done in eq. 25 and eq. 26 for dissolved silicate are done for the particulate forms.

Supplementary Tables
Supplementary Table 1

0.03
Optimal Si:C ratio in sea ice diatoms mmol Si mg C −1