Sea ice simulations in this study were performed using the version 4.0.7 revision 15 731 of the Nucleus for European Modelling of the Ocean (NEMO; NEMO System Team, 2022) coupled with the Sea Ice modelling Integrated Initiative (SI3; NEMO Sea Ice Working Group, 2019), hereafter called NEMO4.0-SI3. The model represents the global ocean via a commonly used nominal 2∘ tri-polar grid (ORCA2), which has a resolution of about 85 km between 55 and 75∘ S. The ORCA2 was chosen because it is already capable of identifying features of the Southern Ocean SIC budget at this resolution (Nie et al., 2022), and considering that hundreds of experiments will be performed, using ORCA2 is computationally comparably cheap. The ORCA2 grid configuration has 31 unevenly spaced vertical layers from 10 m thick (near surface) to 500 m thick (at 5500 m depth). The vertical physics of the ocean is solved by the combination of the turbulent kinetic energy (TKE) turbulent closure scheme (Marsaleix et al., 2008), an enhanced vertical diffusion scheme applied on tracer (Madec et al., 1998), and a double diffusive mixing scheme (Merryfield et al., 1999).

The sea ice momentum equation is calculated by using the adaptive elastic–viscous–plastic (EVP) method (Kimmritz et al., 2016, 2017), which is formulated on a C grid and improves the numerical efficiency of the modified EVP scheme. The default number of sea ice thickness categories is five, with each category having two vertical layers of ice and one layer of snow on top of ice. The thermodynamic component of NEMO4.0-SI3 includes the 1D energy-conserving model (Bitz and Lipscomb, 1999) and a time-dependent vertical salinity profile (Vancoppenolle et al., 2009). The sea ice model uses the same 1.5 h time step as the ocean model.

In this study, the NEMO4.0-SI3 model is forced with the DRAKKAR Forcing Set version 5.2 (DFS5.2; Dussin et al., 2016), based primarily on the ERA-Interim with some corrections (Dee et al., 2011) and covering the time period 1979–2017. The DFS5.2 provides the atmospheric field required for the NCAR bulk formula (Large and Yeager, 2004) in NEMO4.0-SI3, which includes 2 m air temperature, 2 m specific humidity, 10 m zonal and meridional wind speeds, mean sea level pressure, downward long-wave and short-wave radiation, and the total and solid precipitation rates. In these atmospheric fields, the frequency of radiation and precipitation is 1 d and 3 h for all other surface boundary conditions. The spatial resolution of DFS5.2 is approximately 80 km, close to that of ORCA2 in the Southern Ocean. The continental discharge rates followed the climatological dataset of Dai and Trenberth (2002) and do not include ice mass loss in Antarctica. The simulations are initialized at rest via the temperature and salinity fields from the World Ocean Atlas 2018 monthly climatology (WOA18; Zweng et al., 2019), run from January 1979 to December 2017, with only the last decade of model output (2008–2017) being used for analysis.

To investigate the sensitivity of sea ice budgets, we selected 18 parameters and determined their uncertainties (Table 1), which cover a number of important processes in sea ice modelling, such as ice and snow physical properties, ocean mixing and eddies, and ice–ocean and air–ice interactions. The lower and upper bounds of the parameters were selected according to the listed references, and the uncertainty intervals were suitably extended to avoid under-sampling at the edge of the interval. The standard values of the parameters used for the control experiment (CTRL) are the default values for NEMO4.0-SI3.

The flow chart describing the procedure for obtaining the optimized model parameter values based on evaluation metrics is shown in Fig. 1. We start with the definition of the 18-dimensional parameter space (as already done in Table 1); the next steps are to sample from this parameter space and to run the NEMO4.0-SI3 model with a limited number of sampled sets of parameter values (the sampling method is described in the next section). Three sets of metrics are then calculated from the NEMO4.0-SI3 model output: (1) the area integrals of SIC budget components, (2) the area integrals of SIV budget components, and (3) the root-mean-square errors (RMSEs) between the simulated and observed SIC budgets (RMSESICB).

After the calculation of the three sets of metrics, GP emulators are trained (to be described in Sect. 2.4) to link the parameter sets with the evaluation metrics based on the NEMO4.0-SI3 simulations. Two GSA methods are used, the PAWN method (Pianosi and Wagener, 2015) and the Sobol method (Sobol, 2001; both described in Appendix A), with a large amount of input data to comprehensively explore the full parameter space covered by NEMO4.0-SI3 simulations and complemented by the GP emulators. Finally, once the key parameters have been identified by the GSA methods, parameter sets that provide results closest to the observations can be identified.

We use the Latin hypercube sampling (LHS) method with a max–min property to generate low-discrepancy sequences from the 18-dimensional parameter space (step 2 in Fig. 1) to identify parameter set values for the NEMO4.0-SI3 simulations. The LHS is a stratified sampling method that divides each dimension evenly to ensure that samples are available in all intervals and therefore allows for a more evenly drawn sample than the usual random sampling methods (Morris and Mitchell, 1995; McKay et al., 2000). Additionally, the max–min property is a space-filling criteria that aims to maximize the minimum Euclidean distance between two sampling points and thus improves the effectiveness of the GP emulation (Joseph and Hung, 2008) to be carried out after the NEMO4.0-SI3 simulations. The recommendation for the number of samples to build a GP emulator is N=10p (Loeppky et al., 2009), where p is the dimension of parameter space and is equal to 18 in this study. In practice, however, we decided to use about 20p samples in order to build the GP emulator as accurately as possible (Williamson et al., 2017). Based on this principle, and taking into account possible model run failures, we first perform a sampling of 800 points in parameter space to run the NEMO4.0-SI3, and if the number of successful experiments ends up being too little (less than 360), we will continue the sampling.

The amount of computation using NEMO4.0-SI3 required to cover comprehensively the 18-parameter space for the model evaluation remains practically too large, and the use of a much faster GP emulator is required to emulate the behaviour of NEMO4.0-SI3 given the 18 parameter values. The emulator functionality is described next.

Let Xt=(x1,x2,⋯,xN)T and Yt=(y1,y2,⋯,yN)T denote the total number of N simulations; each xi is a column vector of 18 values, sampled from the 18-dimensional parameter space by the LHS, and each yi is a real number representing the corresponding model output metric, which is assumed to be noiseless here. A GP emulator f(⋅) for a model output metric Yt=f(Xt) can generally be represented as 1f(⋅)∼GP(μ(⋅),K(⋅,⋅)), where μ(⋅) and K(⋅,⋅) are prior mean function and covariance function respectively. Then the posterior distribution for parameter values X∗ can be obtained as 2f(X∗)|f(Xt)∼N(μ∗,K∗), where 3μ∗=μ+K(X∗,Xt)K(Xt,Xt)-1(f(Xt)-μ),4K∗=K(X∗,X∗)-K(X∗,Xt)K(Xt,Xt)-1K(Xt,X∗). We used the GPy software (GPy, 2012), with the prior of the mean function set to zero by default, and the user only had to choose the covariance function K to build the GP emulator for each evaluation metric. To achieve this, we used a 10-fold cross-validation method for model selection (Geisser, 1975). The idea is to divide the dataset {Xt,Y} evenly into 10 parts, each time using 9 parts as the “training data” to train the emulator and 1 part as the “true data” for model validation and so on for 10 cycles, taking the average as a proxy for model performance. Using this approach, we traversed the linear, squared exponential, exponential, Matern 3/2 and Matern 5/2 covariance functions and their sums and products (Rasmussen and Williams, 2006) for a total of 177 different combinations and then selected the covariance function with both the minimum RMSE and the highest correlation coefficient between the simulated and emulated values.

Following the ice conservation law, the change of a sea ice state field Θ, such as SIC and SIV, can be attributed to dynamic and other processes (Leppäranta 2011, chap. 3.4): 5∂Θ∂t=-u⋅∇Θ-Θ∇⋅u+(f-r) where u is the sea ice velocity, f represents the change from freezing or melting, and r stands for any other processes (e.g. ridging and rafting). Integrating Eq. (5) over time, then the net changes in Θ over a period of time (t2-t1) can be obtained as follows: 6∫t1t2∂Θ∂tdt=-∫t1t2u⋅∇Θdt-∫t1t2Θ∇⋅udt+∫t1t2(f-r)dt, where the term on the left-hand side is the change or dadt (also referred to specifically as dC/dt and dV/dt for changes in SIC and SIV respectively), and the first term on the right-hand side represents the contribution of advection (adv), the represents the contribution of divergence (div), and the last term represents the residual (res). A positive value for each term is defined as an increase of Θ, and a negative value is defined as a decrease.

The budgets for SIC and SIV were calculated in our study, including seasonal climatologies for each SIC or SIV budget term, following the same approach as Holland and Kimura (2016). First, the daily dadt was obtained by central differencing of the ice fields on the day before and the day after; the advection and divergence were first calculated on each day and then averaged over the corresponding 3 d periods to be consistent with the daily dadt. Second, adv and div were subtracted from the dadt to obtain the daily res; and finally, all daily terms were summed over each season and averaged over the years 2008–2017.

Daily sea ice velocity observations from Kimura et al. (2013), hereafter referred to as KIMURA, and SICs from the National Oceanic and Atmospheric Administration/National Snow and Ice Data Center (NOAA/NSIDC) Climate Data Record of Passive Microwave Sea Ice Concentration, version 4 (Meier et al., 2021; hereafter referred to as CDR) were used to calculate the observed SIC budget. The ice velocity dataset KIMURA was generated from the brightness temperature of the 36 GHz channel of the Advanced Microwave Scanning Radiometer-Earth Observing System (AMSR-E) using the maximum cross-correlation technique (Kimura et al., 2013) and ultimately deriving a 60 km resolution product. Therefore, the KIMURA data share the same period as AMSR-E and its successor AMSR2, covering from 2002 to the present. Following Holland and Kwok (2012), a 3×3 grid filter was used in the calculations to smooth out the grid-scale noise present in the satellite-derived ice drift. Regarding the SIC satellite observations, the CDR SIC is a rule-based combination of the NASA Team (Cavalieri et al., 1984) and NASA Bootstrap (Comiso, 1986) ice concentration datasets in the same 25km×25km grid, covering the years from 1978 to 2021, with daily, grid-based uncertainty estimates. The other three SIC observational products are only used as references in the calculation of SIE (integral of grid cells areas where SIC >15 %) and SIA (integral of grid cells areas multiplied by the SIC in each grid cell); they are AMSR-E and AMSR2 provided by NSIDC (Cavalieri et al., 2014; Meier et al., 2018), Ocean–Sea Ice Satellite Application Facilities (OSISAF) from the Copernicus Marine Service, and CERSAT developed by the French National Institute for Ocean Science (IFREMER; Ezraty et al., 2007).

The observed SIC budget (Fig. B1) shows that the Southern Ocean sea ice is generally transported to the ice edge at lower latitudes by advection and melts there, while divergence yields open water and thus promotes freezing of ice (Holland and Kwok, 2012; Uotila et al., 2014). It is important to note that the calculated SIC budget observations were considered to be “true values” in our study despite the uncertainties and biases in the ice drift observations, such as the overall overestimation of 5 % compared to the buoy-measured velocities (Kimura et al., 2013). The simulated SIC budgets and the root-mean-square errors from the observed one were only calculated at grid points with SICs larger than 15 % and at dates where ice drift observations existed to minimize the uncertainty of results caused by missing observations and observational errors.

Out of 800 experiments, 44 % were terminated due to model instability caused by parameter combinations, resulting in an ensemble of models of size 449, which included the CTRL experiment. The seasonal cycles of SIE and SIA for the model ensemble are shown in Fig. 2. The SIE and SIA intervals for the ensemble cover the observed values fairly well, except for September, when SIA is systematically slightly overestimated. Inter-model disagreement due to parameter uncertainty is greatest in summer (ranging from 0.42 to 8.26×106 km2), when SIE and SIA are at a minimum (observed at 4.26×106 km2), while there is little disagreement between models during the autumn months. Among the members of the model ensemble, the CTRL run essentially overlaps with the ensemble mean and matches well with the observations.

In February, comparing the ensemble mean SIC (Fig. B2a and b) with the CDR observation shows that there are still challenges in the modelling of the local patterns, especially as the NEMO4.0-SI3 significantly underestimates the SIC near the East Antarctic coast. In addition, the ensemble standard deviation for February stands at a high level (around 20 %) in most regions. On the other hand, in September (Fig. B2d–f) the ensemble mean SIC is more consistent with the observations than it is in February, although differences between the ensemble members remain relatively high (around 10 %) in marginal ice areas where the SIC is low. Overall, the discrepancies between ensemble members due to parameter uncertainty are smaller in high-SIC areas (SIC>90 %) than in low-SIC areas.

Similarly to the seasonal cycles of SIE and SIA, the CTRL run's SIV remains close to the ensemble mean. However, the differences between SIVs simulated based on different parameter sets are much greater than for SIEs (Fig. 2c) – for instance, in winter, the maximum values of SIVs in the ensemble members are more than twice as large as the minimum values. Additionally, the SIV cycles show a larger spread in winter than in summer, which is opposite to that of SIE cycles. For the ensemble mean sea ice thickness, thicker sea ice of up to 2 m is maintained year-round in the western Weddell Sea (Fig. B3a and c), which appears to be higher than the previous observation-based dataset of 1.2 to 1.5 m (Haumann et al., 2016, in their Extended Data Fig. 2). However, there is a lack of observations from the same period, as this study precludes a direct comparison. The spatial pattern of ice thickness standard deviation between model ensembles (Fig. B3) is similar to that of sea ice thickness, which means thicker sea ice is usually accompanied by a larger standard deviation.

The diagnostics of the SIC and sea ice thickness of the model ensemble in the last section show that the NEMO4.0-SI3 model driven by DFS5.2 provides reasonable results. The mean states of the model ensemble being close to the CTRL experiment, particularly for SIC, matches the observations very well, which provides a good basis for the budget analysis. In this section, we first calculated the SIC budget and SIV budget for the ensemble of 449 model runs by applying the same approach as for the calculation of the observed SIC budget (see Fig. B1) and then computed the RMSESICB (step 5 in Fig. 1).

Based on the results of the last section, the area integrals of adv and res in the SIC (and SIV) budget and the RMSESICB are used as the metrics to assess the sensitivity of the model's sea ice budget to 18 parameters in this section. Before conducting the GSA, Fig. B4 shows the cross-validation results for the best GP emulator for each of the adv and res term area-integral metrics of the SIC and SIV budgets (step 6 in Fig. 1). Overall, the emulated and simulated values have very high correlation coefficients (typically greater than 0.98); thus, the built emulator is considered successful and will be used as a proxy for NEMO4.0-SI3 in the subsequent sensitivity analysis.

The sensitivity of each metric of the 18 parameters, quantified by the Sobol and PAWN methods, is illustrated in Fig. 8. It should be noted that the sensitivity scores for the two methods are independent and not comparable in absolute terms. Following Urrego-Blanco et al. (2016), the Sobol sensitivity index below 0.02 is considered insignificant, and for the Kolmogorov–Smirnov (KS) mean index in PAWN, the critical value at a confidence level of 0.05 is about 6.65×10-2. Both GSA methods show that the advection is very sensitive to ice strength (rn_pstar) outside of summer in the SIC budget. Ice–ocean drag coefficients (rn_cio) and air–ice drag coefficients (Cd_ice) have an influence on the modelled advection contribution to sea ice change from summer to autumn and spring, respectively. In summer, the snow thermal conductivity (rn_cnd_s) and two lateral-melting parameters (rn_beta and rn_dmin) also have some effect on the advection of SIC budget. The total and first-order Sobol indices are not very different, which is usually the case for both indices of the PAWN method, with the exception of the number of ice thickness categories (jpl) where the KS max is shown to be much larger than the KS mean (e.g. in autumn and summer). For other metrics, this also happens for the sensitivity assessment of some other parameters, which will be discussed further in the next section. The residual term of the SIC budget shows considerable sensitivity to the ice–ocean drag coefficient (rn_cio), which persists from autumn to spring. Meanwhile, the effect of the air–ice drag coefficient (Cd_ice) on res increases continuously from autumn to summer. Ice strength still has a weak effect, much less of an effect than that on adv. In addition, snow thermal conductivity (rn_cnd_s) and number of ice thickness categories (jpl) have a non-negligible effect on the modelling of res in winter and summer, respectively.

Among the sensitivity indices of the SIV budget, the most noticeable parameter is snow thermal conductivity (rn_cnd_s), to which both adv and res are very sensitive at all times of the year, except in the spring, when it has less impact on res (Fig. 8). Another physical parameter related to the snow on sea ice (rhos, i.e. snow density) is important for res simulations in the SIV budget, especially from autumn to winter, the period when sea ice freezes fast (Fig. 2c). Similar to the SIC budget, the air–ice and ice–ocean drag coefficients remain crucial for the SIV budget in spring and summer, while the ice strength is only important for advection in winter and spring.

The results for four RMSESICB metrics based on the best-performing GP emulators are shown in Fig. B5. The GP emulator performs well for the RMSESICB of adv, div and res, with a correlation coefficient greater than 0.998, except in summer. As can be seen in Fig. 7, in the summer months, the difference in RMSESICB for these three terms is very small compared to in other seasons, and this small difference is likely to be random and therefore difficult for the GP emulator to capture well. The GP emulator also does not perform well in terms of dC/dt RMSESICB (Fig. B5, first column), and there is also likely to be a large randomness in the difference in dC/dt between the model ensemble and the observational data. Given the poor performance of the GP emulator in terms of dC/dt RMSESICB, as well as in terms of RMSESICB over the summer, the GSA results obtained by using it instead of the NEMO4.0-SI3 dynamical ocean model are subject to uncertainty, and this should be kept in mind in the following analysis.

Figure 9 demonstrates quite clearly that for adv, div and res RMSESICB in autumn, winter and spring (which are also the terms and seasons with the largest RMSESICB values; Fig. 7), only air–ice and ice–ocean drag coefficients are the most critical parameters, while ice strength also has, but only weakly, an effect. Besides these two important drag coefficients, Fig. 9 also shows that the dC/dt RMSESICB between model and observational data might be sensitive to the snow thermal conductivity and ice category number to some extent. The analysis is more complicated in summer, as is the sensitivity of the SIC budget and SIV budget to the parameters. In addition to all the previously mentioned parameters that have an impact, Fig. 9 shows that, in summer, the RMSESICB may also be sensitive to the minimum floe diameter for the lateral-melting parameter (rn_dmin) and the magnitude of the damping on salinity (rn_deds), which is a parameter belonging to the ocean module. Further comparing Figs. 8 and 9, it can be found that, overall, both the simulation of the SIC budget by the NEMO4.0-SI3 model and its RMSESICB are most sensitive to the air–ice and ice–ocean drag coefficients, both of which belong to the coupling category in Table 1. Also important are ice strength and the thermal conductivity of snow, identified by the six metrics related to SIC budget. In summer, some thermodynamic melting-related parameters, such as rn_beta (coefficient beta in the lateral-melting parameterization scheme) and rn_dmin (minimum floe diameter in the lateral-melting parameterization scheme), are important. In contrast, the SIC budget simulated by the model is sensitive to the number of ice thickness categories (jpl), unlike the RMSESICB metrics.

As it has been identified that the RMSESICB metrics are sensitive to the two most critical parameters (air–ice and ice–ocean drag coefficients) and one relatively important parameter (ice strength), Fig. 10 illustrates the RMSESICB for all SIC budget terms and all seasons, averaged over 449 model runs, in relation to the values of these three parameters, with the top 10 combinations listed in Table 3. It can be seen in Fig. 10b that the RMSESICB broadly decreases with increasing ice–ocean drag coefficient (rn_cio) and decreasing air–ice drag coefficient (Cd_ice), such that the 10 sets of model runs with the smallest RMSESICB are concentrated in the top-left corner of the figure, where air–ice drag coefficient is from approximately 8×10-4 to 1×10-3 and where ice–ocean drag coefficient is from approximately 5.5×10-3 to 7.5×10-3 (Table 3). In contrast, the best 10 ice strength values are more dispersed and greater than 15×103 (Fig. 10a and c), and the RMSESICB does not depend linearly on it, as with air–ice and ice–ocean drag coefficients (i.e. Cd_ice and rn_cio).

Several parameters have been identified in Sect. 3.3 and 3.4 as having a significant impact on the simulated SIC and SIV budgets in the Southern Ocean. In this section, we present how these parameters specifically act on the SIC and SIV budget by looking at the impact of parameter changes on the cumulative distribution function (CDF) in the PAWN method.

Considering the performance of the GP emulator (Fig. B4), as well as the number of sensitive parameters (Fig. 8), the area integral of the res component in the SIC budget in spring and the area integral of the adv component in the SIV budget in winter have been selected here as examples to be discussed. Figures 11 and 12 show how the CDF of the model output changes as one parameter is fixed to vary across a range of values and as other parameters are varied freely.

Since the low thermal conductivity of the snow reduces the heat transfer from the bottom of the ice to the atmosphere, it reduces the ice growth rate (Fichefet et al., 2000; Lecomte et al., 2013) and therefore leads to less freezing inside the ice pack, and res moves more towards negative values (Fig. 11d). The reduction in freezing due to the reduction in snow thermal conductivity is more pronounced in winter (Fig. 8), and the SIV budget simulation is more sensitive to this parameter than the SIC budget is, as it primarily affects the vertical ice growth.

The rn_beta and rn_dmin are the two parameters that determine the minimum floe diameter of sea ice, and their decrease implies a decrease in sea ice floe sizes, which promotes the lateral melting (Lüpkes et al., 2012). Consequently, in contrast to the reduction of snow thermal conductivity (rn_cnd_s), which inhibits ice freezing, rn_beta and rn_dmin lead to more negative values of res (Fig. 11e and f) by promoting sea ice melting at low-latitude regions (Fig. 3). Furthermore, this effect is greater in summer than in spring and plays a weak role in winter (Fig. 8), which fits well with the magnitude of the SIC reduction in the res column in Fig. 3, although it is not the only process that affects SIC.

Compared to rather continuous-looking variations in CDFs of other parameters, the variation in CDFs due to changes in the number of ice thickness categories (jpl) is more dispersed (Fig. 11l), with several lines clearly being outliers, which were checked to match jpl=1. This is because the multi-category sea ice thickness takes into account the subgrid-scale variations in sea ice properties (Thorndike et al., 1975; Massonnet et al., 2019; Moreno-Chamarro et al., 2020) and is therefore significantly different from the single-thickness category (jpl=1). For instance, the presence of thin sea ice categories in multi-category sea ice schemes allows for greater melt rates compared to a single-category scheme (Uotila et al., 2017).

The ice–ocean drag coefficient and the air–ice drag coefficient should be discussed jointly, as the sea ice drift velocity is related to the Nansen number Na=ρaCa/ρwCw, where ρa/w and Ca/w are air and water density and air–ice and ice–ocean drag coefficient. The Figs. 11q and r illustrate that a decrease in Ca/Cw leads to a larger res, which has two possibilities: either sea ice melt is inhibited or freezing is intensified by assuming that sea ice deformation is comparably small (Holland and Kwok, 2012). Since the solution of free sea ice drift (Leppäranta, 2011, chap. 6.1.1) indicates that the decrease in Ca/Cw leads to a decrease in sea ice velocity, we argue that this causes a more limited transport of sea ice to low-latitude regions, leading to the inhibited melting (see spring adv and res in Fig. 3).

With the exception of snow thermal conductivity (rn_cnd_s), ice–ocean drag coefficients (rn_cio) and air–ice drag coefficients (Cd_ice), whose physical effects have been elucidated, the adv term in the winter SIV budget is also sensitive to ice strength (rn_pstar; Fig. 12a). This can be explained by the fact that the weaker ice is more easy to deform and to have ice thickness increased (Docquier et al., 2017), leading to a smaller drift speed and therefore resulting in a smaller absolute value of the area integral of adv or div. This is also true in spring (Fig. 8), as ice drift speeds are greater in winter and spring compared to in other seasons during the period of this study (not shown but similar to e.g. Holland and Kimura, 2016), making the ridging of weak ice more pronounced.

For the NEMO4.0-SI3, the snow thickness on sea ice is determined by the snow density as the solid-precipitation equivalent, which is determined by atmospheric reanalyses and other factors affecting the snow depth (e.g. wind packing and/or windblown-snow lost to leads; Petty et al., 2018) that are not included (NEMO Sea Ice Working Group, 2019). When the snow density decreases in the model, the snow thickness increases, thereby reducing the heat exchange between the ice and the atmosphere, which in turn limits the vertical increase in sea ice thickness. Thus, for the SI3 model, the effects of reducing snow thickness and of reducing snow thermal conductivity on the simulation of sea ice thickness are equivalent. This is the reason why the res term in the SIV budget shows similarly high sensitivities to snow thermal conductivity (rn_cnd_s) and ice density (rhos) (Fig. 8). These two parameters have the greatest influence on the total SIV and thus also on the area integral of the adv during autumn and winter, the seasons when sea ice vertical growth is most pronounced. When sea ice thickening is limited, the value of SIV itself becomes smaller, resulting in a smaller area integral for adv (Fig. 12b).

However, of the seven parameters discussed above that have an impact on the SIC budget, only two drag coefficients play a critical role in the RMSE of the simulated and observed SIC budgets, followed by the weak effect of sea ice strength (Fig. 9). This means that while adjusting snow thermal conductivity has an impact on the simulation of SIE (Urrego-Blanco et al., 2016) and may improve the SIE seasonal cycle to be closer to the observations (Lecomte et al., 2013), it does not make the model's simulation of the SIC budget any more realistic. In addition, although the remaining parameters display sensitivity during the summer months (bottom row in the Fig. 9), the robustness of this result is not guaranteed given the already low level of RMSE in the summer and the mediocre performance of the GP emulator (bottom row in the Fig. B5).

In addition to the sensitivity of the model to individual parameters discussed in the previous section, using the second-order sensitivity indices provided by the Sobol method, the interaction between the parameters can be further explored. We have added some vertical connector lines in Figs. 8 and 9 to indicate that a simultaneous change in two parameters has a significant impact. Not surprisingly, the interconnection of the ice–ocean and air–ice drag coefficients causes their simultaneous changes to have the greatest impact on the advection metric in both SIC and SIV budgets, especially in winter and spring, the two seasons with the largest sea ice speeds. Furthermore, for the SIV budget, the contribution of its advection term to SIV change is also sensitive to the simultaneous changes in snow thermal conductivity (rn_cnd_s) and the ice–ocean drag coefficient (rn_cio) in autumn. This makes sense, considering that sea ice starts to grow vertically in autumn and that the advection is significantly affected by the ice–ocean drag coefficient (Fig. 8). However, snow thermal conductivity (rn_cnd_s) does not interact with any drag coefficient in winter, when ice vertical growth is also rapid (Fig. 2c); thus, the interaction in autumn remains somewhat uncertain due to the fact that the GP emulator does not perform very well for adv in the autumn SIV budget (r=0.961).

The ratio between the ice–ocean and air–ice drag coefficients continues to dominate the sensitivity of the four RMSE metrics, as the sea ice velocity is controlled by Ca/Cw (Fig. 9). Although the GSA results also show some sensitivity to ice strength, there is little interaction between this parameter and the two drag coefficients in the SIV budget, except in the case of the adv term in summer. Despite this, considering that the adv RMSESICB itself fluctuates very little in summer and that the GP emulator is not a perfect performer, there is uncertainty in this result. Figure 9 also shows that the dC/dt RMSESICB is sensitive to simultaneous changes in rn_beta (coefficient beta in the lateral-melting parameterization scheme) and rn_cio (ice–ocean drag coefficient) in the autumn, which we argue may be an error introduced by the poorer-performing GP emulator (r=0.915), as the rn_beta is a parameter related to lateral melting that should not have a significant effect in the autumn.

The previous sections have shown the sensitivity of the simulated sea ice budget to parameters, and there are a number of parameter sets that are recommended (Table 3). In this section, we provide further insight into how these parameter sets perform in terms of other metrics. Figure 2 highlights the SIE, SIA and SIV seasonal cycles of the three experiments that performed best in terms of the mean RMSESICB (as listed in Table 3). An interesting thing is that, although these three experiments used rn_cio/Cd_ice values that were clearly above or below the standard values, they all exhibit SIE and SIA seasonal cycles that are very close to the model ensemble mean and the CTRL. The EXP397, which is the best-performing one, has an SIV seasonal cycle that almost overlaps with the ensemble mean, while the second and third best are both close to the CTRL. This evidence again suggests that, even if the realistic SIE is modelled, there is no guarantee of a reasonable SIC budget (Uotila et al., 2014; Nie et al., 2022).

Regionally, the recommended parameter sets match the observed SIC budgets much better in all sectors of the Southern Ocean (Fig. B6). On the other hand, even the optimal set of parameters recommended in this study (EXP397) would only reduce the dC/dt, adv, div and res RMSESICB by about 2 %, 5 %, 8 % and 10 % respectively (Figs. 7 and B6), which is a rather modest impact. This indicates that the accurate modelling of the SIC budget does not appear to be possible by simply changing the atmospheric-forcing product or tuning the ocean model's parameters, as the atmospheric forcing itself is systematically biased (Barthélemy et al., 2018). As shown in Fig. B7, all model ensembles have similarly shaped ice speed seasonal cycles that all differ significantly from observations, meaning that adjusting the parameter values alone will not correct errors caused by biases in the atmospheric forcing. Nevertheless, the parameter sets in Table 3 can be confidently recommended to NEMO4.0-SI3 modellers to optimize the southern hemispheric sea ice in the ORCA2 grid, provided that DFS5.2 is used as the atmospheric forcing.

In addition, Fig. B8 shows that the recommended parameter sets also provide some improvements in the Arctic SIE and SIA simulations compared to the default parameters, as reflected by more sea ice in summer months, which is closer to observations than in the CTRL experiment. However, given that SIE and SIA are limited metrics (Notz, 2014, 2015) and that the key parameters affecting sea ice simulations may not be the same between the Northern Hemisphere and the Southern Hemisphere due to the vast geographical differences (e.g. ocean and land locations, atmospheric and oceanic circulations), whether these parameter sets, which perform well in the Southern Ocean SIC budget, can be safely applied to the Arctic merits further investigation.