The relaxation algorithm consists of two main parts: initialization and application. The initialization routines are executed during the model setup phase, prior to the start of time integration. These routines identify ocean grid points where nudging will be applied, based on the specified relaxation rates, where the rate is greater than zero. The set of grid points selected based on the relaxation mask (defined as an input field and described in Sect. 4.3) is fixed prior to the start of the simulation and remains unchanged during time integration.

In the application step, the input ice concentration and thickness from external fields are first converted into the model-defined state variables before relaxation is applied. The target fields represent cell-mean ice concentration and thickness, whereas SIS2 defines variables per ice category. Therefore, the target ice variables must be distributed across the model's ice thickness categories in a way that conserves total ice mass.

The relationship between the target ice concentration (aT) and the model ice concentration for thickness category n (acat(n)) is as follows 7aT≡atot=∑n=1Ncacat(n), where atot is the aggregated (grid-cell mean) ice concentration and Nc is the total number of ice thickness categories.

Similarly, the relationship between the target ice thickness (hT) and the model thickness-category field is 8hT≡vtot=∑n=1Ncacat(n)⋅hcat(n), where vtot is the total ice volume per unit grid-cell area (m³ m⁻²), equivalent to the cell-mean ice thickness provided in the target fields, and hcatn denotes the mean ice thickness over the ice-covered area of category n. Once the target thickness is distributed across the categories, it is converted to the SIS2 units of mass per unit ice area (kg m⁻²) for each category n as 9Mcat(n)=ρicehcat(n), where ρice is sea ice density (Table 2).

To ensure physical consistency of the prescribed target state, the target ice concentration and thickness are first remapped onto the SIS2 thickness categories in a manner that conserves total ice area and volume (Eqs. 7, 8). Because only bulk ice concentration and volume are prescribed, the problem is inherently underdetermined, and many different category distributions can satisfy the same constraints. Consequently, multiple redistribution strategies can be used to remap the target ice state onto the SIS2 thickness categories.

A simplest approach is to assign the entire target ice state to a single thickness category (the primary thickness category, n0, defined below). This initialization conserves the prescribed bulk ice concentration and volume but leaves the remaining categories empty. Although formally valid, such a configuration represents a highly singular ice thickness distribution (ITD) and may require substantial redistribution during the first few model time steps as the model re-establishes a physically realistic multi-category structure. In particular, numerical formulations that depend on category ice area fractions can become sensitive when one or more categories contain little or no ice. In our experiments, single-category initializations were generally successful, but a small number of simulations failed during the early redistribution stage. The exact cause of these failures was not investigated in detail.

To provide a smoother initial ITD and avoid empty thickness categories, ice area and volume are redistributed across thickness categories up to the primary category (i.e., only categories thinner than n0), while strictly conserving the prescribed bulk ice properties. This initialization strategy was used in all experiments presented here and exhibited stable behavior in our tests. Specifically, at each grid cell where the relaxation time scale is nonzero, the target ice state is distributed across the thickness categories (Hcat) using the procedure outlined below. i.

Ice sufficiency constraint

Redistribution is performed only when the target ice state is non-negligible, such that10hT≥ϵTaT≥ϵa,where hT and aT denote the target ice thickness and concentration defined in Eqs. (7) and (8), respectively. The thresholds ϵT=1×10-10 m³ m⁻² and ϵa=1×10-10 exclude infinitesimal ice states.

During relaxation, ice volume is artificially modified as it is adjusted toward the target ice concentration and thickness. Each ice layer within each thickness category is characterized by salinity (Si) and enthalpy (qi). Because ice volume is altered during the relaxation procedure, salinity and enthalpy must be maintained within physically realistic bounds, particularly in grid cells transitioning from ice-free to ice-covered conditions. After the ice state has been relaxed to the target concentration and thickness, salinity and enthalpy are adjusted as follows.

For each ice layer in every thickness category, salinity is checked to ensure Si≥Sbulk, where Sbulk is the prescribed bulk sea ice salinity (3.0 psu in this study). If this condition is not satisfied, salinity in all layers within that ice column is reset to Sbulk. Next, the ice enthalpy of each layer is evaluated to ensure that qi≤qmax, where qmax=q(T̃) is the enthalpy of sea ice with salinity Si at a temperature T̃ defined as 20T̃=min{TfrzSi-0.1,Tocn},

where TfrzSi is the freezing point of sea ice of salinity Si, and Tocn is ocean surface temperature beneath the ice. The freezing point is offset by 0.1 °C to prevent rapid melting of newly formed ice immediately following the relaxation step. The use of the minimum operator ensures that newly initialized ice is not warmer than the underlying ocean surface temperature, preventing the ice enthalpy from being too high. If the condition is violated, the layer enthalpy is reset to qmax. For grid cells transitioning from ice-free to ice-covered conditions, Si is initialized to Sbulk and qi to qmax.

A multi-decadal NEP10k hindcast spanning the period 1993–2024 was analyzed (experiment HCST_BGC_NEP; Table 3). This simulation employed a fully coupled MOM6-SIS2 system with biogeochemistry (COBALT). The model configuration, forcing fields, and general hindcast setup follow Drenkard et al. (2025) (Tables 1 and 2) with the following differences: (i) SIS2 was configured with explicit ice ridging (mechanical redistribution); (ii) ocean temperature and salinity fields in the interior domain were nudged toward GLORYS12v1 ocean reanalysis (Lellouche et al., 2021) with a 15 d relaxation time scale; and (iii) the sea ice relaxation algorithm described in Sect. 4 was applied to nudge ice concentration and thickness. Sea ice was relaxed toward monthly PIOMASv2.1 fields using the relaxation mask shown in Fig. 3c, with a minimum (i.e., strongest) relaxation time scale of 12 h. The hindcast was initialized from a three-year spinup simulation.

To isolate the effects of sea ice nudging and eliminate corrections to ocean fields imposed by nudging toward GLORYS reanalysis, a second NEP10k multi-decadal hindcast with disabled ocean nudging was analyzed. This simulation employed the coupled MOM6-SIS2 system without biogeochemistry and covered the period 1993–2019. Sea ice relaxation was applied using the same relaxation mask shown in Fig. 3c. All other parameters and model settings were identical to those used in HCST_BGC_NEP.

A third hindcast employed the coupled MOM6-SIS2 system (physics only), with ocean temperature and salinity nudged toward GLORYS but without applying sea ice relaxation. The simulation covered the period 1993–2019. All parameters and settings were consistent with those in HCST_BGC_NEP.

The NEP10k short-term test experiments analyzed in this study were initialized from the HCST_BGC_NEP hindcast simulation. All experiments were started on 1 January 2001, and integrated forward for one year as free-running simulations, forced by hourly ERA5 atmospheric fields. Ocean LBCs were prescribed from daily GLORYS12 ocean reanalysis fields.

A control experiment (IRLX0_NEP) was conducted without sea ice relaxation. In experiment IBND001_NEP, sea ice concentration and thickness were relaxed toward PIOMASv2.1 at the lateral boundaries using a 1 h relaxation time scale (Fig. 3a). In the remaining experiments, sea ice concentration and thickness were relaxed toward PIOMASv2.1 monthly fields using the same spatially varying relaxation mask applied over the polar portion of the domain (Fig. 3c), with increasing minimum (strongest) relaxation time scales: 1 h (IRLX001_NEP), 24 h (IRLX024_NEP), 120 h (IRLX120_NEP), and 360 h (IRLX360_NEP). These experiments are used to assess the impact of lateral boundary versus interior ice relaxation on sea ice state, edge position, and surface fields. Results are presented in Sect. 6.

The ARC10k sensitivity experiments were initialized on 1 January 1995 from an existing hindcast simulation. As with the NEP10k experiments, the ARC10k simulations were integrated forward for one year. Atmospheric forcing was provided by hourly JRA-55 reanalysis fields, and ocean LBCs were prescribed from the GLORYS reanalysis. A control experiment (IRLX0_ARC) was conducted without sea ice relaxation. In the remaining experiments, sea ice state was nudged toward target fields derived from monthly PIOMASv2.1 data. The relaxation time scales used in these experiments were similar to those in the NEP10k simulations. However, unlike NEP10k, sea ice relaxation was applied uniformly across the entire Arctic north of 60° N (Fig. 3d). The experiments are analyzed in Sect. 7.

The impact of closed ice boundaries on the northern NEP10k domain is illustrated using the control simulation without ice relaxation (IRLX0_NEP; Table 3). Sea ice concentration and thickness differences are computed relative to three-month mean PIOMAS fields (Figs. 4a, 5a). During the first three months (JFM), the model develops positive sea ice concentration anomalies over the Bering Sea (Fig. 4b), while concentration anomalies in the Arctic Ocean remain small during winter, when sea ice coverage is near complete (Fig. 4a). In spring (AMJ) and summer (JAS), pronounced negative concentration anomalies emerge along the northeastern boundary north of Alaska. During the freeze-up season (OND), the model tends to overestimate ice concentration in the northern and northeastern Bering Sea and in the Chukchi Sea.

The effect of closed boundaries is more evident in sea ice thickness (Fig. 5b), with large positive anomalies rapidly emerging in the western Chukchi Sea. These arise from ice accumulation along the lateral boundary driven by prevailing anticyclonic Arctic winds, while the same circulation removes ice from the northeastern boundary. Ice continues to pile up throughout the integration, and by the end of the one-year simulation, thickness in the western Chukchi Sea exceeds the PIOMAS reference (Fig. 5a) by more than 2 m.

These results are supported by a quantitative comparison of the control simulation with PIOMAS, performed separately for the Bering Sea and Arctic subdomains (Fig. 1c). In the Bering Sea, the control run exhibits large differences in ice concentration (blue line with bullets in Fig. 6a, c, e). The results closely overlap with those from IBND001_NEP, as explained in Sect. 6.2. The differences are particularly large during January–April, when sea ice extent and thickness peak. During this period, ice area and volume are overestimated (Fig. 6a, b), primarily due to positive biases in both ice concentration and thickness (Fig. 6d, e). In summer, when the Bering Sea is largely ice-free, differences in all sea ice metrics are substantially reduced (Fig. 6c–f).

Differences are more pronounced in the Arctic portion of the NEP10k domain (Fig. 7), where the statistics exhibit a strong seasonal cycle. Differences in ice concentration are relatively small during winter and late fall, when the region is nearly fully ice-covered (Fig. 7a, c, e). In contrast, during summer the control simulation poorly reproduces the spatial distribution of ice concentration, with RMSE values exceeding 0.25 and a systematic underestimation of partial ice area, leading to an underprediction of total ice area (Fig. 7a, c, e). The simulation also shows substantial differences in ice thickness (Fig. 7d, f), resulting in a marked overestimation of ice volume beginning in April and increasing through the end of the integration (Fig. 7b).

In the IRLX0_NEP control simulation, the simulated ice edge closely matches PIOMAS during periods when the ice edge lies within the Bering Sea (January–March and November–December; Fig. 8). From May through October, as the ice edge retreats into the Arctic Ocean, the simulation exhibits larger discrepancies. This part of the NEP10k domain is strongly influenced by the presence of closed boundaries, which contribute to substantial uncertainties in simulated ice characteristics and lead to increased errors in predicting the ice edge location.

Overall, the NEP10k control experiment exhibits substantial errors in sea ice characteristics across the Arctic portion of the NEP10k domain and the northern Bering Sea, primarily due to the proximity of ice closed boundaries. The most prominent issues include unrealistic spatial distribution of ice and excessive ice thickness.

The IBND001_NEP experiment was designed to reduce the impact of LBCs on the interior ice state in the NEP10k domain. Compared to the control run, differences in ice concentration between IBND001_NEP and PIOMAS show improvement over the Arctic portion of the domain from January through September (Fig. 4c). In particular, relaxation along the boundaries reduces the strong negative anomaly near the northeastern boundary in the Beaufort Sea present in IRLX0_NEP during summer (Fig. 4b). No notable improvement is observed during October–December. Boundary relaxation has no discernible impact on ice concentration in the Bering Sea, where differences remain nearly identical to those in IRLX0_NEP. This supports the assumption that the Bering Sea ice state is weakly sensitive to Arctic LBCs.

Sea ice thickness in the Arctic Ocean is also affected by boundary relaxation, with reduced disagreement between IBND001_NEP and PIOMAS (Fig. 5c). Differences are zero within the relaxation region along the northwestern and northeastern boundaries. Outside the relaxation region, two unrealistic features present in IRLX0_NEP, ice pileup along the northwestern boundary and ice depletion along the northeastern boundary (Fig. 5b), are eliminated. However, Arctic interior ice remains too thick, with differences exceeding 2 m during July–September. Consistent with the ice concentration results, sea ice thickness in the Bering Sea is largely unchanged relative to IRLX0_NEP.

In the Bering Sea, performance metrics for IBND001_NEP are nearly identical to those from the control run (Fig. 6), reflecting the low sensitivity of the regional ice state to Arctic LBCs. In contrast, IBND001_NEP shows substantial improvement across all Arctic ice metrics compared to IRLX0_NEP (Fig. 7).

The sea ice edge position in IBND001_NEP is similar to IRLX0_NEP during winter and late fall (Fig. 8), when the ice edge lies in the Bering Sea and the influence of LBCs is minimal. During June–August, IBND001_NEP produces a more realistic ice edge, closer to the PIOMAS-derived contour, consistent with improved ice concentration in the Arctic Ocean. Overall, lateral boundary relaxation modestly improves the Arctic ice state in NEP10k. To further evaluate the approach, relaxation is next applied over the entire polar domain (Fig. 3c) to reduce misfits in ice concentration and thickness.

Seasonal ice concentration differences between the NEP10k experiments and PIOMAS show clear improvement during April–September when relaxation is applied over the Arctic portion of the domain. Among the nudged experiments, IRLX001_NEP (strongest relaxation; Fig. 4d) exhibits the smallest concentration differences. In contrast, improvements relative to the control run (IRLX0_NEP; Fig. 4b) are less evident during the cold seasons (January–March and October–December). During January–March, concentration differences of approximately 0.1–0.4 persist in the Bering Sea, consistent with the weaker relaxation applied in that region. During October–December, the differences are smaller (approximately 0.1–0.2) but extend beyond the Bering Sea into the Chukchi and coastal Beaufort Seas, with magnitudes comparable to those in the non-nudged experiment. As the relaxation time scale increases, concentration differences become more pronounced during April–June and July–September, as illustrated by the weakest-relaxation experiment, IRLX360_NEP (Fig. 4e). By contrast, the sensitivity of ice concentration to the relaxation time scale is less apparent during January–March and October–December.

The impact of ice relaxation is more pronounced for ice thickness, as demonstrated by the IRLX001_NEP and IRLX360_NEP experiments (Fig. 5d, e). Throughout the integration period, all nudged experiments show substantial improvement relative to both the control run (Fig. 5b) and the lateral-boundary relaxation experiment (IBND001_NEP; Fig. 5c). In IRLX001_NEP, thickness differences are generally close to zero over most of the domain. The magnitude of the differences increases as the relaxation weakens, becoming noticeable in IRLX360_NEP (Fig. 5e), but remaining substantially smaller than in IRLX0_NEP and IBND001_NEP. Notably, even the weakest relaxation constrains ice thickness to within approximately 0.5 m of PIOMAS across most of the domain, whereas substantially larger differences persist in IBND001_NEP (Fig. 5c).

Because the relaxation rate is weaker in the Bering Sea than in the Arctic (Fig. 3c), a reduced sensitivity to nudging is expected there. In the Bering Sea, all NEP10k experiments overestimate total ice area during months with ice cover (Fig. 6a), reflecting a persistent positive bias in ice concentration relative to PIOMAS (Fig. 6e). Compared to the control run, nudging improves ice area and volume estimates from January to May, with IRLX001_NEP (strongest relaxation) showing the best agreement with PIOMAS (Fig. 6a, b). During the freeze-up period (October–December), however, all experiments similarly overestimate ice area and volume, indicating limited sensitivity to the relaxation strength. The RMSE and bias statistics further demonstrate the benefits of nudging, particularly for ice thickness (Fig. 6c–f). Ice-thickness RMSE decreases monotonically with increasing relaxation strength, and even weak nudging (IRLX360_NEP) substantially reduces errors relative to both the control and lateral-boundary relaxation experiments. Ice-thickness biases remain small (< 0.03 m) in all nudged simulations, with the smallest biases occurring in IRLX001_NEP.

In the Arctic portion of the domain, where relaxation is applied uniformly over most of the region, the impact of nudging is evident across all metrics (Fig. 7). Nudged simulations closely reproduce PIOMAS ice area and volume from January through September, representing a marked improvement over the control (IRLX0_NEP) and IBND001_NEP, both of which diverge from PIOMAS after April (Fig. 7a, b). Ice concentration RMSE and bias are reduced in all nudging experiments, with IRLX001_NEP consistently exhibiting the highest skill (Fig. 7c, e), although errors increase during October as rapid ice formation begins. In contrast, ice thickness RMSE and bias remain low throughout the simulation in all nudged experiments (Fig. 7d, f), with error magnitudes again decreasing with increasing relaxation strength.

The IRLX001_NEP experiment outperforms all other NEP10k simulations in predicting the ice edge location, exhibiting the lowest MHD score throughout the integration period (Fig. 8). The remaining experiments rank consistently with relaxation strength, with the control run generally exhibiting the poorest skill. The spread across the simulations is small from January to April and from November to December but increases during May to October. Again, the impact of nudging is less pronounced during winter and late fall, when the ice edge lies in the Bering Sea, where relaxation is substantially weaker than in the Arctic portion of the domain.

The ARC10k control simulation (IRLX0_ARC) has a closer match with the PIOMAS reanalysis (Fig. 9a) than the IRLX0_NEP experiment, because the lateral boundaries are located far from the ice edge. During winter and spring, the IRLX0_ARC ice concentration fields show good agreement with the PIOMAS reanalysis except for the Bering Sea (Fig. 9b). However, the model has difficulty reproducing the ice edge along the East Greenland Current and within the MIZ of the Barents Sea, where ARC10k overestimates ice concentration by more than 0.5. During summer (JAS), the model underestimates ice concentration on the Eurasian shelf by ∼ 0.1–0.3. During the freeze-up season (OND), differences increase over the shelf seas. Ice thickness differences remain significant in IRLX0_ARC throughout the year (Fig. 10b). The model consistently overestimates ice thickness, by more than 0.8 to 1 m relative to PIOMAS (Fig. 10a), in regions such as the East Greenland Current, the Beaufort Sea, and the Chukchi Sea.

The ARC10k control simulation consistently overestimates both sea ice area (Fig. 11a) and volume (Fig. 11b) relative to PIOMAS. The error in ice concentration remains high throughout the year, exceeding 0.13–0.15 (Fig. 11c), and is positive (Fig. 11e). The IRLX0_ARC experiment also exhibits substantial errors in ice thickness, with deviations reaching up to 0.5 m compared to PIOMAS (Fig. 11d). The mean thickness errors are predominantly positive, except in March and April, indicating that the model tends to overestimate ice production relative to the reference data (Fig. 11f).

The position of the sea ice edge in the IRLX0_ARC experiment is relatively close to the PIOMAS derived estimate during the cold months (January through March and December) (Fig. 12). However, during the melt and freeze-up seasons, the accuracy of the simulated ice edge declines, consistent with the NEP10k control run.

Applying relaxation reduces the overestimation of ice concentration in the marginal ice zone (MIZ) observed in IRLX0_ARC (Fig. 9). Summer ice concentration is substantially improved in the relaxation experiments, with IRLX001_ARC exhibiting the smallest differences relative to PIOMAS (Fig. 9c). Even the weakly nudged experiment (IRLX360_ARC; Fig. 9d) shows improvement, although overestimation of ice concentration in the MIZ remains noticeable. Consistent with the NEP10k experiments, nudging produces only modest improvements over the shelf seas during OND. During this period, ice concentration differences over the shelf seas remain evident, although relatively small (approximately 0.1–0.2), and are comparable to those in the control run without relaxation. These persistent autumn differences may reflect a colder simulated ocean state, likely arising from differences between the JRA-55 atmospheric forcing used here and the NCEP/NCAR Reanalysis 1 forcing used in PIOMAS, which are not directly corrected by sea ice nudging.

As in the NEP10k experiments, the effect of nudging is more pronounced in the ice thickness fields (Fig. 10c, d). Regions south of the 60° N relaxation boundary show weaker improvements, partially retaining the discrepancies seen in the control run. Relaxation effectively constrains ice thickness toward PIOMAS, resulting in consistently small thickness differences throughout the year across all nudging experiments.

The ARC10k relaxation experiments show improved ice area estimates relative to the control run (IRLX0_ARC) from January through July, with closer agreement to PIOMAS (Fig. 11a). During the freeze-up months (October–December), however, relaxation has little impact on ice area. From January to September, RMSE and bias in ice concentration (Fig. 11c, e) behave as expected, with stronger nudging generally yielding better performance. Consistent with the NEP10k simulations, the impact of relaxation during freeze-up is reduced, and all nudging experiments exhibit increased RMSE and bias in ice concentration during October–December.

Ice-thickness RMSE is substantially lower in all nudging experiments compared to the control run, indicating an overall improvement throughout the year (Fig. 11d). However, contrary to expectations, the strongest relaxation experiment (IRLX001_ARC) does not achieve the lowest RMSE. Instead, IRLX024_ARC, which applies slightly weaker relaxation, performs best. Ice-thickness bias is generally negative in the nudged simulations, whereas the control run exhibits a positive bias for most of the year, except in March and April when it becomes slightly negative and near zero (Fig. 11f). Overall, IRLX024_ARC shows the best ice-thickness performance in terms of both RMSE and bias, outperforming the strongest-relaxation configuration.

Relaxation of ice concentration and thickness improves the accuracy of the simulated ice edge position, as indicated by notably lower MHD values in the nudging experiments compared to the control run, except for IRLX360_ARC in October (Fig. 12). Overall, model skill aligns with expectations: the experiment with the strongest nudging achieves the lowest MHD, and skill generally decreases as the strength of the nudging weakens.