Persistent impact of winter atmospheric circulation anomalies on Arctic sea ice

According to previous studies, the Arctic Oscillation (AO) in winter has an impact on the evolution of sea ice conditions from winter until the following September. This study explores and compares the sea ice responses to two climatic modes of variability in the winter, the positive phase of the AO and the negative phase of the Arctic Dipole (AD) patterns. This study for the first time verifies that the AD-induced sea ice thickness change is comparable in magnitude to the AO-induced change by the end of the melting season. It proves that the negative AD in winter has a persistent impact on sea ice thickness comparable to the AO has. Furthermore, the AO and AD in the winter impact sea ice conditions in different ways. The AO cannot decrease sea ice thickness until the melting season. Most decreases in sea ice thickness are found over the Beaufort and Chukchi Seas starting in May. On the other hand, the negative AD mostly inhibits sea ice growth during the growing season thermodynamically over the Atlantic sector of the Arctic Ocean. The ensemble means of 22 CMIP6 models can reasonably catch the response of sea ice thickness change to the AO and AD in spatial distribution and temporal evolution. The CMIP6 model ensemble is better at reproducing the AD-induced response of sea ice than the AO-induced response, while the spread of results from individual models is extensive.


Introduction
The sea ice cover in the Arctic has been decreasing since 1979 (Stroeve et al 2007, Comiso et al 2008, Vihma 2014) .The atmospheric circulations are known to make impacts on sea ice in both dynamic and thermodynamic pathways (Zhang et al 2000).The Arctic Oscillation (AO; Thompson and Wallace 1998) and Arctic Dipole (AD; Watanabe et al 2006) are the two dominant modes of climatic variability in the Arctic.The AO was known to force sea ice changes dynamically.Previous studies have come up with the mechanisms that during the positive phase of the AO, the cyclonic atmospheric circulation pattern at the surface leads to an anomalous convergence of sea ice, making the sea ice cover shrink on a large scale and a higher ratio of first-year sea ice along the coastal area of the Arctic Ocean during winter (Thompson andWallace 1998, Rigor et al 2002).The first-year ice melts much more quickly than multi-year ice in spring, causing decreases in sea ice cover in the following summer (Rigor andWallace 2004, Overland andWang 2010).The AO is stronger during winter than summer, and studies have suggested a relatively long memory time of the wintertime AO that its effect could be persistent until the following fall (Rigor et al 2002, Ogi et al 2016).
Up until the first decade of the 21st century, the winter AO explains most of the interannual variation in sea ice concentration in March (Deser et al 2000, Stroeve et al 2008).The observational results show that the linkage between the AO index and the interannual variation of the September sea ice extent seems to have weakened after 2007 (Ogi et al 2016).On the other hand, the importance of AD in forcing sea ice change has been attracting more attention since the sea ice plummets in 2007.A strong negative AD and the associated heat-moisture intrusion originating from the North Atlantic could result in increases in downward longwave radiation flux at the surface over the Atlantic sector of the Arctic Ocean and even the central Arctic, inhibiting sea ice growth in the growing season (Park et al 2015, Alexeev et al 2017, Park et al 2018).These studies including our previous article (Cai et al 2020) explored the thermodynamic impacts of AD on sea ice growth on a daily to weekly time scale, while there has not been any study exploring the persistence of the impact of the winter AD to Arctic sea ice, and how it compares to that of the winter AO.
In summary, previous studies have not yet compared the winter AO and AD in terms of their impacts on sea ice in winter through the following summer and their different physical mechanisms.One cannot help but ask which one of the two modes of climatic variability plays the dominant role in each season and what are their roles in sea ice dynamics and thermodynamics?To answer these research questions, we compare the spatial distribution and temporal evolution of sea ice changes in response to the AO and AD in both dynamic and thermodynamic aspects.Our analysis emphasizes the persistence of the impact of the AO and AD on sea ice conditions throughout the growing and melting seasons of sea ice.We analyze CMIP6 model results for an intercomparison with each other and with reanalysis data results to verify the proposed physical processes.

Data sources
We rely on the Pan-Arctic Ice Ocean Modeling and Assimilation System (PIOMAS; Zhang and Rothrock 2003) for its sea ice thickness/concentration (SIT/SIC) and sea ice velocity data in this study.The PIOMAS data is based on an ocean-sea ice coupled model forced by atmospheric conditions and with the observed SIC and sea surface temperature data assimilated, offering sea ice conditions with a daily frequency from 1979 to the present (Zhang and Rothrock 2003).The PIOMAS takes the NCEP/NCAR Reanalysis Product (NNRP, Kistler et al 2001) as atmospheric forcing so that we employ the NNRP data to calculate the AO/AD index (see supplemental material for method).If not specifically mentioned, the AO and AD hereafter refer only to the positive phase of the AO and the negative phase of the AD respectively when analyzing and discussing the sea ice responses.
Aside from the PIOMAS reanalysis product, we examine sea ice response to AO and AD in 22 CMIP6 models in addition to the reanalysis product.We employ the all-coupled experiment in the historical period (with the tag 'historical') from 1950 to 2015.Note that the PIOMAS data does not involve any coupling between the atmosphere and the ocean/sea ice: sea ice dynamics and thermodynamics in PIOMAS therefore do not have influences on atmospheric variables, which is the opposite of the CMIP6 output.Limited by the data availability, we only examine the monthly mean model output including SIT, SIC, and SLP for CMIP6 data.The CMIP6 model results are interpolated onto the sea ice grids of CESM2 before calculating the ensemble mean and standard deviation.

Sea ice thickness changes
This study defines the total change in SIT as the SIT time derivative.We attribute the total change of SIT to either dynamic or thermodynamic reasons as in previous studies (Park et al 2015, Cai et al 2020).The calculation and discussion of this approach are in the supplemental material.For convenience, we apply the term 'ΔSIT' for the total SIT change in the growing or melting season in this section hereafter.Correspondingly, the 'dynamic ΔSIT', 'thermodynamic ΔSIT' and 'total ΔSIT', refer respectively to the magnitude of SIT change in the whole seasons due to the dynamic and thermodynamic forcing, and their sum.More details of the calculation of SIT are included in the supplemental material.
Since the Arctic sea ice has been declining since the 1970s, we filter the global warming signal to emphasize the daily to seasonal SIT anomalies.SIT anomalies are calculated based on a 21-year moving average climatology.For example, the sea ice thickness anomaly for the year 2000 is based on the climatology of sea ice thickness in the years from 1990 to 2010.In this study, the growing season of sea ice is defined as from December to March (DJFM) as sea ice cover reaches its annual peak in March.Defining such a period aims to emphasize the synchronous response of sea ice to the winter AO and AD.The melting season is defined as from April to August (AMJJA), to explore the lagged sea ice response to the winter AO and AD.
For the temporal evolution of sea ice response, we focus on the accumulation of ΔSIT anomalies by integrating the anomalies of daily ΔSIT rate through time.We check this temporal evolution via composite analysis.We pick the years with an AO index greater than 0.5 as the 'positive AO years' and the years with the AD index lower than −0.5 as the 'negative AD years'.We also check the responses on the regional scale by splitting the study area into four sectors, which are the 'Eurasian sector' (69-90°N, 30-140°E), the 'Pacific sector' (69-90°N , 140-230°E), the 'North American sector' (69-90°N, 230-330°E), and the 'Atlantic sector' (69-90°N, 330-30°2 E) to further examine and compare the regional sea ice responses associated with the positive AO and negative AD.

Results
The behavior of AO/AD indices in DJFM and the observed September sea ice extents suggest that a positive AO and/or a negative AD could be associated with a local maximum/minimum of the September sea ice extent.The correlation coefficients between the AO/AD indices and the sea ice extent anomaly from March to November exhibit a maximum in September for both modes of variability, with the correlation coefficients being significant at a 95% confidence level (figure 1).The sea ice extent anomaly is calculated after removing the longterm trend to eliminate the global warming effect.This result is consistent with earlier studies, which motivates us to explore and compare the physical mechanisms and relative contributions of the winter AO and AD to sea ice conditions in both winter and the following summer (Rigor et al 2002, Watanabe et al 2006, Park et al 2018).
Examining the spatial distribution of sea ice response shows that in the growing season, the anomaly pattern of the AO-induced ΔSIT displays negative anomalies along the Arctic coast of Eurasia and positive anomalies along the Arctic coast of North America (figure S2).The AD-induced ΔSIT, on the other hand, shows negative anomalies in the Atlantic sector and positive anomalies in the Pacific sector.The AO drives a cyclonic movement of sea ice in the Arctic in the growing season.The corresponding ice divergence flux, in turn, leads to a negative ΔSIT anomaly, and the greatest difference is present along the coast of the East Siberian Sea, the Beaufort Sea, and the Chukchi Sea.Earlier studies have suggested that the positive AO in winter and its associated cyclonic wind anomaly can favor the precondition for abrupt sea ice melt along the Arctic coast during the melting season (Wang and Ikeda 2000, Rigor et al 2002, Day et al 2012).
On the contrary, the sea ice movement induced by the winter AD is generally transpolar, from the Barents/ Kara Sea to the central Arctic, and the velocity anomaly of which is slightly smaller than that induced by the AO.The AD-induced dynamic ΔSIT anomaly, therefore, is negative in the Atlantic sector and positive in the Pacific sector in the growing season, and the magnitude of the difference is not as big as those induced by the AO.The thermodynamic ΔSIT anomaly induced by the AO is generally positive over a large area in the central Arctic and in the north of North America, and it does not have any apparent positive anomalies in its spatial pattern.The negative ΔSIT anomaly can be found along the sea ice edge in the Atlantic sector.
During the melting season, the total ΔSIT induced by the AO and AD generally compensates for those in the growing season (figure S3).In response to the positive AO, the total ΔSIT presents negative anomalies in the central Arctic, as well as the non-coastal regions of the Beaufort and Chukchi Seas.Positive ΔSIT anomalies are found along the arctic coast of Eurasia.In response to the negative AD, negative ΔSIT anomalies are in the Laptev, East Siberia, and Chukchi Seas, while positive anomalies are mostly along the Atlantic edge of sea ice.
Regarding the dynamic ΔSIT, there are not many grids with significant sea ice velocity anomaly compared to the growing season (figure S3).The regressed sea ice velocity fields do not show any noticeable patterns.The AOinduced ΔSIT in general shows positive anomalies over the Eurasia side while negative anomalies over the North America side.The AD-induced ΔSIT shows negative anomalies with small magnitudes over the central Arctic.Thermodynamically, the winter AO induces negative ΔSIT anomalies over a majority of the Arctic Ocean, while the AD does not show any significant responses in the central Arctic.It does have some positive anomalies in the north of the Barents Sea.
Averaged over the pan-Arctic, the temporal evolution of the AO-induced accumulated ΔSIT remains positive until May.The positive AO in winter favors sea ice growth during the growing season (figure 2).Starting in late May, the AO-induced ΔSIT starts to drop rapidly till the end of September, resulting in almost 1 cm less of SIT on average for the whole Arctic.Such an abrupt sea ice thickness change associated with the winter AO is consistent with the mechanism suggested by Rigor et al (2002), which is that the winter AO causes sea ice convergence and extra formation of new sea ice, the faster melting of first-year sea ice compared to multi-year sea ice would cause an abrupt decrease in sea ice cover.On the contrary, the AD-induced ΔSIT anomaly is negative and keeps decreasing during the growing season, making nearly 2 cm of ΔSIT anomaly by March.The anomaly starts to recover during the melting season, also leaving nearly 1 cm of ΔSIT anomaly by the end of September.The AO and AD in winter eventually led to a comparable magnitude of ΔSIT from December to the following September, both of which tested significantly at a 95% confidence level.Furthermore, in thinning sea ice, the positive AO takes most of the effect in the melting season, while the negative AD results in most of the sea ice thinning in the growing season.This result is consistent with previous studies that have examined the persistent impacts of the AO and AD on sea ice, while this study for the first time quantifies that the impacts from winter AO and AD are comparable in magnitude.
Both the AO and AD in winter change ΔSIT with regional heterogeneities.The AO-induced decreases of SIT are found in the regions of the Eurasia and Pacific sectors, while increases are found in North America and Atlantic sectors of the Arctic Ocean.On the other hand, the AD-induced decreases are over Eurasia and Atlantic sectors, while increases are over the Pacific sector.The winter AD does not change the SIT over the North American sector.
A detailed look at the dynamic and thermodynamic ΔSIT anomalies reveals the different physical processes the AO and AD hold on changing SIT.The dynamic ΔSIT by both the AO and AD presents negative anomalies throughout both seasons.On the thermodynamic side, the two climatic variabilities present the opposite impacts on sea ice.The impact of the positive AO leads to anomalously thick SIT, while the negative AD mostly leads to thinner SIT.Specifically for the AD in its negative phase, the thermodynamic effect remains positive in December, and meanwhile the dynamic effect takes the dominant effect in thinning sea ice.Starting in February, the thermodynamic effect of the negative AD overweighs the dynamic effect.The temporal evolutions verify that in making a thinner sea ice cover, the positive AO makes the effect dynamically while the negative AD does so thermodynamically.
In three sectors of Eurasia, the Pacific, and Atlantic, the dynamic and thermodynamic changes of SIT compensate for each other in general.Among the three sectors, the anomalies are positive for the dynamic ΔSIT while negative for the thermodynamic ΔSIT, induced by both the AO and AD.Meanwhile, for the Eurasia sector, both the AO and AD lead to negative dynamic ΔSIT anomalies and positive thermodynamic ΔSIT anomalies, noting that the AO-induced thermodynamic ΔSIT anomaly is close to zero and tested insignificant statistically.Compared to the three sectors, the North American sector shows much smaller magnitudes of both dynamic and thermodynamic ΔSIT anomalies.
We compare the accumulated anomalies of ΔSIT induced by the winter AO and AD in the CMIP6 models with those in the PIOMAS data.The spatial patterns of the CMIP6 ensemble-mean show similar features compared to those derived from the PIOMAS data.In the growing season, the positive AO results in negative anomalies of sea ice thickness change are found along the Eurasia coast of the Arctic while positive anomalies are along the Arctic coast on the North American side.On the other hand, the negative AD produces negative anomalies of sea ice change on the Atlantic edge while positive anomalies on the Pacific side over the Beaufort and Chukchi seas.During the melting season, both climatic variabilities fail to produce distinctive anomalies in ΔSIT.CMIP6 models present little agreement with each other in the spatial distribution of ΔSIT anomaly.The standard deviation of 22 CMIP6 models is higher along the coastal area than in the central Arctic, and it is higher in the melting season than in the growing season (figure 3).The ΔSIT response in the growing season represents a synchronous impact, while that in the melting season represents the impact with a lag time.It is normal for different CMIP6 models to have a larger deviation from each other in modeling lagged responses than synchronous responses.Specifically for the melting season, the standard deviation is about twice as big as the anomalies in ΔSIT.This result suggests that the CMIP6 models still vary extensively from each other in retrieving the sea ice response to the atmospheric forcing.
Comparing the temporal evolutions of ΔSIT anomalies in PIOMAS data and CMIP6 model data reveals that in the growing season, the CMIP6 models agree on the effect of AO that slightly favors sea ice growth and the effect of negative AD on suppressing sea ice growth remains distinctive throughout the whole growing season, which is also consistent with the results of PIOMAS data (figure 4).In the melting season, the AD-induced ΔSIT from the CMIP6-model ensemble also agrees with that from the PIOMAS data that the ΔSIT anomaly recovers in the melting season, causing up to 1 cm thinner sea ice averaged on the pan-Arctic region by the end of the melting season.On the other hand, the averaged AO-induced ΔSIT anomaly remains positive in the melting season, suggesting an overall thickening of sea ice by the end of the melting season under the effect of positive AO.The results from CMIP6 models show significant differences among each other in the magnitude of accumulated ΔSIT anomalies.They also vary distinctively from the relative contributions of the AO and AD in thinning sea ice.Even so, there are CMIP6 models that reproduce the rapid decrease of ΔSIT anomaly, ending up with negative anomalies in September.It verifies that some CMIP6 models can show agreement with the PIOMAS data in reproducing the persistence of sea ice conditions in response to the winter AO and AD.
A majority of the 22 CMIP6 models have similar performance with each other in reproducing the AO-and AD-induced ΔSIT anomalies by the end of growing and melting seasons respectively, showing clusters on the scatter plots (figure 5).Several models are dramatically biased from other models and the reanalysis data.In the growing season, a majority of CMIP6 models tend to have positive biases in AO-induced ΔSIT anomaly while negative biases in AD-induced ΔSIT anomaly, sharing the same feature as the in the PIOMAS data.Most models agree on the inhibition of sea ice growth induced by the negative AD, while models have a greater extent of disagreement with each other on AO's effect on SIT [figure 5 In summary, the CMIP6 model ensemble has a smaller spread in the sea ice response during the growing season than during the melting season.The ensemble means of the 22 CMIP6 models produce a better performance of the AD-induced ΔSIT than the AO-induced ΔSIT with the PIOMAS result as the reference.The CMIP6 models present a wide spread of sea ice responses compared to each other.This comparison calls for  further studies on the performance of CMIP6 models in simulating sea ice dynamics and thermodynamics in response to atmospheric forcing.

Discussion
Our previous study (Cai et al 2020) concentrated on the details of sea ice responses to the AO/AD in the growing season.In this study, on the other hand, we explore the accumulated impact of positive AO and negative AD patterns in winter and their effects on changing sea ice conditions in both the growing and melting seasons.Previous studies have emphasized the effect of winter AO on sea ice and discussed its physical mechanisms, especially for its preconditioning effect during the melting season (Rigor et al 2002, Park et al 2015).In comparison, this study for the first time highlights that the winter AD in its negative phase can cause a similar, if not stronger, magnitude of impact on decreasing sea ice thickness in the following September relative to the winter AO.Furthermore, this study verifies that the AO and AD in winter cause similar magnitudes of change in SIT via different mechanisms, with different temporal evolutions of ΔSIT.The findings above offer new insights into the persistence of impact on sea ice of both climatic variabilities over the Arctic.
The spatial distribution of ΔSIT response is by regressing the ΔSIT anomaly onto the index of the AO/AD.Regions with a noticeable ΔSIT response are typically associated with a statistical significance (figures S2 and S3), verifying that the ΔSIT response is generally linear to the AO and AD.It is worth noticing, however, that the AO and AD together explain less than 50% variance of the atmospheric circulation anomaly pattern by definition.Multiplying the regression factor by the indices of AO/AD of a certain year obtains the relative contributions of the positive AO and/or negative AD (figure S4).As a result, the positive AO and negative AD together account for slightly larger than 40% of the sea ice thickness change anomaly at most by the end of the melting season.Synoptic weather events such as the storms in the North Atlantic (Alexeev et al 2017)   on sea ice evolution.The sea ice is also influenced by the dynamic and thermodynamic forcing of the ocean (Kwok et al 2013, Liang et al 2019, Dai et al 2020).Considering these contributing factors, we speculate that some inconsistencies found on the association of AO/AD and sea ice in some specific years could result from factors other than AO/AD.
Previous studies have revealed that both the surface wind and surface radiation flux anomaly patterns associated with the AD resemble those during heat and moisture intrusion events that happened in the North Atlantic, however with a seasonal time scale (Woods et al 2013, Woods andCaballero 2016).This study confirms that the magnitude of the thermodynamic effect of moisture intrusion events is so extensive that the associated SIT decrease could not be fully recovered in the following summer, resulting in ΔSIT with a comparable magnitude induced by the winter AO.Considering the AO has been regarded as an important contributing factor to the sea ice decline, the result of this study highlights that on the annual scale, winter AD in its negative is an as important contributing factor to the sea ice thinning as the AO is.
To focus on the temporal evolution of winter climate variability signals, we do not discuss how these two modes of variability behave during spring and summer and how they affect the sea ice conditions.There have been studies discussing the importance of a strong AD in its positive phase in summer (Overland et al 2012, Wu et al 2012).The positive AD in summer and its transpolar drift corresponds to an abnormally low cloud cover over the Beaufort and Chukchi Seas, accelerating sea ice melting over the Pacific sector of the Arctic and being partially responsible for the minimum of September sea ice cover in the 2000s (Kay et al 2008, Kay et al 2011).The positive AD also accelerates the sea ice export on the Atlantic edge via the Fram Strait (Watanabe et al 2006).Exploring the combined effect of the negative AD in winter and the positive AD in the following summer is worth a separate study, and it is beyond the scope of this research.
Since the generation of CMIP5 models, the earth system models have shown an improved performance in reproducing a reasonable AO activity that is typically better than modeling the AD (Cai et al 2018).This study further highlights that regarding the air-sea ice interaction, the models perform better in reproducing the ADinduced sea ice response than the AO-induced response.We suggest that the fully coupled earth system models are more competent in reproducing the thermodynamic air-ice interaction than the dynamic air-ice interaction.The magnitude of disagreement between the CMIP6 models in reproducing sea ice response to the atmospheric forcing is still remarkable.It is consistent with the evaluation studies of sea ice simulation for the CMIP6 models that most models fail to simulate a temporal evolution of sea ice area compared to observations (e.g., Notz and Community 2020).Some of the biases could be related to the ocean forcing that lacks the capability of wellreproducing seasonal sea ice changes in the marginal seas (Watts et al 2021).Considering the systematic improvements from its previous generation, it is also worth comparing the performance of sea ice modeling and response to a relatively higher frequency (seasonal to annual) variability of atmospheric forcings in the CMIP5 and CMIP6 models to check if there are significant improvements in the CMIP6 model results.Other than the fully-coupled option, the model results from SIMIP (Sea-Ice Model Intercomparison Project, Notz et al 2016) simulation would be specifically helpful in understanding the difference in the sea ice modeling outcomes between models.

Conclusions
This study examines and compares the effect of the positive AO and negative AD in winter on changing sea ice conditions both synchronously in winter through the following summer.This study has found that on an annual scale, the positive AO and negative AD in winter result in impacts of similar magnitude on changing sea ice conditions by the end of the melting season in the following summer, verifying that the AO and AD in winter are equally important in driving sea ice anomalies that persist until the next September.The AO and AD in winter have their impact on sea ice thickness through different mechanisms.Averaged for the pan-Arctic, the dynamic change of sea ice takes the dominant effect in the AO-induced thinning of sea ice.In contrast, the AD in its negative phase takes most of its effect via the thermodynamic forcing during the sea ice growing season.
The ensemble means of 22 CMIP models can reasonably present the SIT response to the climatic variabilities as present in the reanalysis data.It performs better in reproducing the sea ice response to the winter AD than to the winter AO.Meanwhile, the spreads between each model are high on both the spatial distribution and temporal variation of sea ice responses.The differences between CMIP6 models are higher in the Arctic coastal regions along the edge of sea ice cover than in the central Arctic and are higher during the melting season than during the growing season.This comparison offers a possibility that the CMIP6 models better represent thermodynamic interactions than dynamic interactions regarding the atmosphere-sea ice coupling in the model framework.This study offers new insights into the persistence of the impact of Arctic climatic variabilities on sea ice conditions, calling for further studies on the physical mechanisms of the persistence of interaction between the atmosphere and sea ice.

Figure 1 .
Figure 1.(a) The timeseries of the AO (red line) and AD (blue line) index in winter, as well as the sea ice extent in the following September (gray line).(b) The linear correlations between the AO (red line) and AD (blue line) and the monthly sea ice extent from March to November in the following year.

Figure 2 .
Figure2.The accumulated total (solid lines in the upper panels), dynamic (solid lines in the lower panels), and thermodynamic (dotted lines in the lower panels) ΔSIT anomalies induced by the positive AO and negative AD for the whole pan-Arctic and four subregions from December to September.Thick lines indicate that the accumulated anomalies tested significantly at a 95% confidence level.
(a)].By the end of the melting season, on the other hand, the magnitudes of AO-and AD-induced ΔSIT anomalies in each model tend to change synchronously.Models with a positive AO-induced ΔSIT anomaly tend to have a positive AD-induced anomaly, and vice versa.There are a fair number of models with ΔSIT anomalies close to zero [figure 5(b)].

Figure 3 .
Figure 3.The regression of sea ice thickness change (m) in the growing and melting seasons following the winters with a positive AO and negative AD along with the standard deviation of the regressed sea ice thickness change in 22 CMIP6 models.The black contours outline the region in which the results from 22 CMIP6 models have an agreement with each other with a 95% significance level.

Figure 4 .
Figure 4.The accumulated monthly anomalies (thick lines) of total ΔSIT induced by the AO (red lines) and AD (blue lines) in the 22 CMIP6 models.The shades indicate the range of one standard deviation of results in the 22 CMIP6 models.The thin lines indicate the same changes in the PIOMAS data, which are the same as in figure 2.
and in the Arctic (Simmonds and Keay 2009, Itkin et al 2017, Peng et al 2021) have been known to have a significant impact

Figure 5 .
Figure 5.The comparison of the AO-and AD-induced sea ice thickness changes (cm) by the end of (a) growing season and (b) melting season and their RMSEs (c and d) between each model and the PIOMAS reanalysis data.