We use the Community Earth System Model version 1 Large Ensemble (CESM1-LE; Kay et al., 2015) to explore relationships between Arctic wintertime conductive heat fluxes and sea ice and snow thickness fields. The CESM1-LE consists of 40 simulations forced with historical forcing from 1920–2005 and then the Representative Concentration Pathway 8.5 (RCP8.5) forcing from 2005–2100 (the no-mitigation scenario with a top-of-the-atmosphere radiative forcing of 8.5 W m-2 by 2100; Meinshausen et al., 2011). The sea ice model component (the Los Alamos Sea Ice Model, CICE; Hunke et al., 2017) in the CESM1 uses a sub-grid-scale ice thickness distribution (ITD) in which thermodynamics are calculated over five discrete sub-grid-cell thickness categories with minimum thickness bounds of 0, 0.64, 1.39, 2.47, and 4.57 m. The presence of the ITD influences both the mean climate state and climate feedbacks (Holland et al., 2006). The simulated ITD and sea ice concentrations within each category evolve through ice growth and melt, ridging due to mechanical forcing, and ice transport (e.g., Thorndike et al., 1975). The resulting concentrations of sea ice can range from 0 % to 100 % (although values of 100 % are rare due to lead formation) in the discrete thickness layers. Snowfall can accumulate on sea ice and be affected by snowmelt, ice ridging, and transport. Effectively this means that a different snow depth is present across the different ice thickness categories. Ice–atmosphere fluxes are calculated separately in each sea ice thickness category, weighted by the concentration in each category, and passed as grid-cell means to the flux coupler for use in the atmospheric model. CICE uses a multi-layer thermodynamic scheme (Bitz and Lipscomb, 1999; BL99) that includes the effects of a prescribed vertical salinity profile. Comparisons with the Pan-Arctic Ice-Ocean Modeling and Assimilation system (PIOMAS; Zhang and Rothrock, 2003) sea ice thickness reanalysis product show that the CESM1-LE and PIOMAS agree well for both summer and winter rates of sea ice volume change (Labe et al., 2018a). The CESM1-LE and PIOMAS also tend to agree on regional mean thicknesses and variabilities throughout much of the Arctic with the exception of the Canadian Archipelago and coastal Greenland, where the CESM1-LE overestimates sea ice volumes compared to PIOMAS (and PIOMAS underestimates volumes compared to buoy and submarine data).

Not all thermodynamic variables for individual ice thickness categories were saved as part of the CESM1-LE output. In order to disentangle the relative influences of sea ice and snow thicknesses and their distributions, we use the available CESM1 output along with the zero-layer thermodynamic model of Semtner (1976) to estimate the conductive heat flux. This simple zero-layer model – developed originally to minimize computational costs associated with ice thickness calculations in climate models – assumes a linear temperature gradient through the sea ice and snow, and that the conductive heat flux through the ice + snow layer is 1Fcond=KsTb-Tshs+KsSITKi, where Ks and Ki are the snow and ice conductivities of heat (0.3 Wm-1 K-1 and 2.0 Wm-1 K-1), hs and SIT are the snow and ice thicknesses, and Tb and Ts are temperatures at the bottom (ocean–ice) and surface (ice–atmosphere) of the ice.

The conductive heat flux reduces to 2ΔFcond=KiΔTheff, where ΔT=Tb-Ts and heff= (SIT + (Kratio⋅hs)), which is a measure of the effective thickness from an insulating perspective, with Kratio=Ki/Ks.

In the Arctic winter, surface temperatures are cold – well below the melting point of ice and snow – and the net surface energy budget (the sum of the net short- and longwave radiation and sensible and latent heat fluxes) in ice-covered regions is balanced by the conductive heat flux from the ocean through the sea ice and snow. Where open water is present, such as leads in the sea ice, direct atmosphere–ocean exchange also occurs. Climate simulations that prescribe constant sea ice thickness such as those used in Atmospheric Model Intercomparison Project (AMIP) and most of the Polar Amplification Model Intercomparison Project (PAMIP; Smith et al., 2019) experiments only allow for changes in conductive heat fluxes to occur through changes in temperatures and possibly through snow depth changes resulting from changing snowfall. This results in inaccurate estimates for changes in surface heat fluxes (and thus also temperatures) which can be quite sizable and of the wrong sign. For example, in response to 20 ∘C surface warming, changes in conductive heat fluxes from a “typical” 20th-century mid-Arctic ocean (with 2.0 m thick ice covered by 0.1 m thick snow with a surface to base ΔT of 40 ∘C) will be roughly halved if SIT and snow thicknesses are held constant but doubled if the ice thins to 0.5 with 0.02 m of snow (e.g., Table 1). In CESM2 PAMIP (100-member ensembles comparing present-day and future changes from 1850s; Smith et al., 2019; Sun et al., 2022) and AMIP (10-member ensembles investigating historical changes; Hurrell et al., 2008; Simpson et al., 2020) simulations in which ice thickness is prescribed and unchanging, net surface heat fluxes increase outside of sea-ice-covered areas but decrease over sea-ice-covered areas as the surface temperatures warm (Fig. 1). For reference, Arctic sea ice volume decreased by ∼66 % from the mid-20th century to the present (Kwok, 2018; Lindsay and Schweiger, 2015). Thus, keeping SIT constant during this period, in direct contrast to observations, artificially introduces errors in surface heat flux calculations – and thus also temperature changes over sea-ice-covered regions. These differences in wintertime conductive heat fluxes are not insignificant contributions to the surface energy budget (e.g., Huwald et al., 2005). Notably, many simulations that have been used to diagnose Arctic amplification rely on AMIP-type simulations which specify the fractions of open water and sea ice yet neglect ice thickness changes (e.g., Smith et al., 2019). Even when ice thickness anomalies are applied (e.g., Lang et al., 2017; Labe et al., 2018b; Sun et al., 2018; Smith et al., 2019), they are specified for a mean grid-cell value, missing the potential role of ice thickness heterogeneity which could be important given that the conductive flux has a nonlinear dependence on ice and snow thickness. AMIP and PAMIP protocols do not specify snow depths as part of the sea ice boundary conditions, and different modeling systems use different approaches to modify (or not) the snow over sea ice in perturbed climate simulations.

We are interested in wintertime Arctic heat fluxes during a period when wintertime central Arctic SICs remain relatively high and changing sea ice and snow thickness will play a dominant role in changing surface fluxes. Thus, we focus initially on 1950–2070 and on the month of February to explore in detail relationships between conductive heat fluxes, snow and sea ice thicknesses, and thickness distributions. Time series presented are area averages over the Arctic Ocean (68–90∘ N from 100–243∘ E and 80–90∘ N elsewhere – see inset in Fig. 2b), thereby reducing the influence of changes in winter sea ice concentrations (SIC) that are seen in the simulations during this time period in the neighboring regions. Changes in conductive heat fluxes due to thinning sea ice dominate the surface heat budget until the sea ice thins enough to subsequently start retreating, exposing open ocean. To highlight this contrast between ice-covered regions and time periods and those with increasing open-water areas, we also show time series from the Kara and Barents seas (see inset in Fig. 5) and some results from October, when SICs are already changing rapidly at the beginning of the 21st century. Our regional definitions are those used by the National Snow & Ice Data Center (https://nsidc.org/, last access: 7 April 2022) and discussed in Cavalieri and Parkinson (2008; 2012) and Parkinson et al. (1999).

We then explore the relative and changing importance of sea ice conductive heat fluxes to the total surface heat fluxes in a warming world in the cold season (October through March) in the 21st century. In the high-latitude Arctic winter, when there is no solar radiation and the surface air temperature is well below the freezing point, the net surface heat flux over the sea ice (total surface short- and longwave radiation and sensible and latent heat fluxes) is balanced by the conductive heat flux through the ice and snow. In regions not 100 % covered by sea ice, there will also be heat flux from the ocean to the atmosphere. The relative contributions of heat fluxes from ocean and sea ice areas will change both as ice and snow thin and as sea ice concentrations decrease. We ask how these relative contributions change in winter months as the projected climate warms and compare ice-covered regions with two regions (Barents Sea and Kara Sea) that are experiencing significant wintertime sea ice concentration losses from 2010–2070.

We also assess the role of sub-grid-cell sea ice thickness distributions in simulated conductive heat fluxes. Model output of conductive heat flux is available on a grid-cell level and is computed as the area-weighted average of the sub-grid-cell conductive heat fluxes. This explicitly accounts for changing ice and snow thicknesses, surface temperatures, and areal concentrations for the different ice thickness categories. Although sub-grid-cell conductive heat flux information that would enable us to directly relate changes in the sub-grid-cell ITD to the net flux was not saved and is not available, sub-grid-cell ice and snow thicknesses are available. Comparisons between the model net grid-cell conductive heat fluxes and those calculated using the sub-grid-cell ice and snow thicknesses, the grid-cell surface temperatures, and the zero-layer model demonstrate that the zero-layer model gives a good approximation for this analysis (Supplement). We then compare the model conductive heat fluxes which account for the sub-grid-scale ice thickness distribution (CESM1-LE) with those calculated from the grid-cell mean SIT and snow thicknesses using the zero-layer model (MNthick) to investigate the influence of snow and sea ice heterogeneity on conductive heat flux calculations.

Over the Arctic Ocean, simulated February surface temperatures and conductive heat fluxes, which are equivalent to the ice–atmosphere heat exchange, increase by ∼8 ∘C and 9 W m-2 between the late 20th century and 2070 in the CESM1-LE (Fig. 2a). The turbulent heat flux exchange between the ocean and the sea ice increases during this time as well – however by less than 1 W m-2 and therefore is not a dominant contribution to the increased net surface heat exchange. Increases in conductive heat flux are driven by decreases in sea ice and snow thickness, since the increase in surface temperatures alone would result in a decrease in conductive heat fluxes (Eq. 2). During this time, sea ice concentrations in the Arctic Ocean remain high, yet sea ice and snow thin dramatically (Fig. 2b), with a mean total effective thickness (heff) decreasing from a peak near 6 m in the 1970s to 1.5 m by 2070 (Fig. 2c). Snow thicknesses averaged over the Arctic Ocean region are typically less than 0.5 m thick – much thinner than sea ice – however snow, as well as changes in snow, makes a significant contribution to both the total effective thickness and the changes in total effective thickness, due to the much larger insulating capacity of snow (Fig. 2c). The decreasing wintertime snow depths over the 21st century in the CESM1-LE are due primarily to sea ice forming later in the fall, which in turns leads to a later start of the accumulation of snow on the sea ice (e.g., Webster et al., 2014; 2018; 2020). This mechanism has also been found in projections of snow on sea ice (Hezel et al., 2012) and in observations (Webster et al., 2014; 2018). In addition, as the temperatures warm, the rain/snow seasons increase/decrease (e.g., Webster et al., 2020).

Conductive heat fluxes increase first over the East Siberian and Chukchi seas (Fig. 3). By the 2050s, increases in conductive heat fluxes of 9–12 W m-2 are seen not only in the Chukchi and East Siberian seas but also extending into the Beaufort Sea and the central Arctic Ocean. Surface temperature increases likewise show the earliest and greatest increases on the Pacific side of the Arctic Ocean. However, by the 2050s the areas of greatest warming (>10 ∘C) are present over comparatively larger sections of the Arctic Ocean and Beaufort Sea and extend as far as northern Greenland. Sea ice concentrations remain above 95 % in the Arctic Ocean even by the 2050s. As such, changes in heat fluxes from the relatively warmer surface to the colder atmosphere primarily result from thinning sea ice and snow rather than increases in open water (see Sect. 3.2). Changes in sea ice thicknesses in the CESM1-LE tend to be largest from the Canadian Archipelago, across the central Arctic Ocean, and onto the East Siberian Sea (Supplement), whereas changes in snow thickness are greatest near the Canadian Archipelago and northern and eastern Greenland. Changes in both sea ice and snow thicknesses are important contributors to the changes in effective thickness (Fig. 3). Although snow depth changes are small compared to changes in SIT, they contribute 40 % or more to the changes in effective thicknesses from the central Arctic Ocean to the Canadian Archipelago, to Greenland, and into the Atlantic sector of the Arctic Ocean (Fig. 3 and Supplement). It is important to note that changes in conductive heat fluxes during this time are mitigated by changes in surface temperatures: the roughly 13 ∘C surface warming over this period would lead to a ∼5 W m-2 decrease in conductive heat flux if the sea ice and snow did not thin (and as they do in the AMIP and PAMIP atmosphere only runs when SITs are held constant – e.g., Fig. 1).

Changes in conductive heat fluxes due to thinning sea ice dominate the surface heat budget until the sea ice thins enough to subsequently start retreating, whereupon the surface heat flux quickly becomes dominated by fluxes from exposed open water (Figs. 4 and 5). Ensemble mean February net surface heat fluxes over the Arctic ocean increase modestly from the 1950s to the 2010s (∼3–6 W m-2), then by ∼10–18 W m-2 by the 2050s. Arctic Ocean SICs remain above 98 % – increases in surface heat flux are primarily due to changes in snow and ice thicknesses, with open-water areas contributing less than 10 %–25 % to the net surface heat flux budget. In contrast, outside of the ice-covered areas, increases in surface heat fluxes are larger than 25 W m-2 and predominantly due to fluxes from open water. In February, these areas expand from the Sea of Okhotsk and the Bering, Kara, and Barents seas in the 2010s to include most of the Arctic Ocean by the end of the 21st century. Rates of surface temperature warming reflect this as well – with the highest February Arctic amplifications occurring initially over the Chukchi and East Siberian seas, where sea ice is thinning; over the Barents and Kara seas by the 2050s; and finally over large sections of the Arctic Ocean as SICs fall by the 2090s (Fig. 4). These patterns of relative contributions of sea ice and ocean to the net surface heat fluxes, as well as corresponding rates of AA, are also seen in the fall (October; Supplement), when observed AAs are highest (e.g., Chung et al., 2021) and underscore the transient and seasonal nature of AA which may increase in regions experiencing sea ice loss before decreasing again after becoming ice free (e.g., Holland and Landrum, 2021). Regionally averaged time series show how contributions from open ocean water to the net surface heat fluxes tend to be greater than those from ice once SICs fall below ∼90 % – highlighting how changes in surface heat fluxes will be dominated by those over sea ice (due to thinning sea ice and snow) until sea ice itself starts retreating (Fig. 5). Variability in heat fluxes from the ocean tends to be very high compared to those from sea ice, particularly in the Barents and Kara seas.

Given the nonlinear response of the conductive heat flux to ice and snow thickness, the changing distribution of ice at the sub-grid-cell level may play an important role. We investigate the influence of thickness distributions on conductive heat fluxes by calculating conductive heat fluxes using the Semtner zero-layer model, daily mean sea ice, and snow thicknesses over the ice-covered areas in each grid cell along with daily grid-cell average surface temperatures (MNthick) and then averaging over the month of February. Relative to an estimate of the fluxes computed by the full model (CESM1-LE), the MNthick conductive heat fluxes underestimate heat fluxes throughout the Arctic Ocean (Fig. 6a and Supplement). Ensemble mean Arctic Ocean average conductive heat flux is underestimated by ∼6–9 W m-2 using MNthick (Fig. 6). This highlights the importance of resolving thin ice and snow within the sub-grid-scale ice thickness distribution.

Differences between these wintertime conductive heat fluxes throughout the Arctic basin are larger in the beginning of the 21st century (by ∼35 %) when considerable thick ice is present. Discrepancies between MNthick conductive heat flux estimates and the CESM1-LE are reduced by the 2070s (to ∼16 %) when almost all the February sea ice in the Arctic Ocean has thinned considerably to less than 1 m over the Arctic Ocean and the ITD lies within the two thinnest sea ice categories (Fig. 6b). Although grid-cell average thicknesses lead to an underestimation of mean conductive heat fluxes, they result in an overestimation in the changes in conductive heat fluxes (by ∼36 % by 2070; Fig. 6a) and thus influence feedbacks in the warming climate.