This analysis requires a sea ice thickness dataset, a sea ice motion estimation dataset, and a thermodynamic sea ice growth retrieval.

2.3 Stefan's Law Integrated Conducted Energy

The SLICE methodology drives Stefan's law (Stefan, 1891; Lepparanta, 1993) with snow–ice interface temperature retrieved with passive microwave brightness temperatures (Kilic et al., 2019) in order to retrieve an instantaneous thermodynamic sea ice thickness growth rate. Stefan's law of simple sea ice thermodynamics states that

$$\begin{array}{}\text{(1)}& f(t,H,\mathbf{x})={\displaystyle \frac{{\mathit{\kappa }}_{\mathrm{eff}}}{{\mathit{\rho }}_{i}LH}}\left(T_{\mathrm{f}}-T_{\mathrm{si}}\right)-{\displaystyle \frac{F_{\mathrm{w}}}{{\mathit{\rho }}_{\mathrm{i}}L}},\end{array}$$

where $f(t,H,\mathbf{x})$ is the thermodynamic growth rate, ρ_i is the density of sea ice, L is the latent heat of fusion, κ_eff is the effective thermal conductivity of sea ice, H is sea ice thickness, T_f is the freezing point of seawater, T_si is the snow–ice interface temperature, and F_w is the basal heat flux from the liquid seawater to the solid sea ice. Latent heat of fusion and effective thermal conductivity are calculated using a set of equations that accounts for the multi-phase properties of sea ice (Feltham et al., 2006). In the present analysis, sea ice thickness is taken from the CS2SMOS thickness field, sea ice density is taken to be 917 kg m⁻³, and basal flux is a constant 2 W⁻². SLICE is available daily and basin-wide but does carry three assumptions – heat conduction in the horizontal direction is assumed to be negligible, it is assumed that there is no thermal inertia present in the ice, and it is assumed that there is no internal heat source, such as the absorption of shortwave radiation. These assumptions and the snow–ice interface temperature retrieval are only valid during winter, constraining the analysis to 1 November through 31 March. Additionally, SLICE is only viable in areas with greater than 95 % sea ice concentration due to the effects of open water on passive microwave emissivity. SLICE utilizes daily level 3 passive microwave brightness temperatures and sea ice concentration from JAXA AMSR2 and AMSR-E on a 25 km north polar stereographic grid, as provided by the NSIDC (Cavalieri et al., 2014; Markus et al., 2018). We linearly interpolate these data from the north polar stereographic grid to the 25 km EASE-Grid 2.0 for our analysis. More information regarding SLICE can be found in Anheuser et al. (2022).

Sea ice thickness is affected by thermodynamic processes and dynamic processes. Thermodynamic processes serve to change sea ice thickness through molecular phase change, and dynamic processes serve to change local sea ice thickness through the mechanical processes of advection and deformation (ridging or lead formation). An Eulerian governing equation for sea ice thickness sums thermodynamic and dynamic processes:

$$\begin{array}{}\text{(2)}& {\displaystyle \frac{\partial H}{\partial t}}=f(t,H,\mathbf{x})-\mathrm{\nabla }\cdot \left(\mathit{u}H\right),\end{array}$$

where H is plane slab sea ice thickness; t is time; f is a function of time, thickness, and position x describing thermodynamic sea ice thickness increase; and u is the ice motion vector. The second term on the right-hand side represents dynamic sea ice thickness processes.

We aimed to partition weekly basin-wide observations of overall changes to the sea ice thickness field, $\frac{\partial H}{\partial t}$, into its components thermodynamic growth, $f(t,H,\mathbf{x})$, and dynamic effects, $-\mathrm{\nabla }\cdot \left(\mathit{u}H\right)$. The result is basin-wide estimates of thermodynamic and dynamic process effects on sea ice thickness. At each weekly time step, $\frac{\partial H}{\partial t}$ was calculated in centered-difference fashion as half the difference between the CS2SMOS sea ice thickness field from the previous time step and that from the following time step. We estimated dynamics, $-\mathrm{\nabla }\cdot \left(\mathit{u}H\right)$, as the residual difference between total growth, $\frac{\partial H}{\partial t}$, and expected thermodynamic growth, $f(t,H,\mathbf{x})$, retrieved using the SLICE methodology with the CS2SMOS sea ice thickness from the previous time step as the initial thickness. Due to the 95 % or greater sea ice concentration constraint on SLICE, only grid cells that meet this condition at a given time step per an AMSR-E/AMSR2 passive microwave sea ice concentration product (Cavalieri et al., 2014; Markus et al., 2018) are included in this analysis. The results shown are time averages over various time periods. Grid cells that do not contain 95 % sea ice concentration for over 40 % of the time steps within the averaging time period are discarded and not considered. Figure 1a shows the portion of total time steps that each grid cell shows over 95 % sea ice concentration. Also shown in Fig. 1 is a map of regions used in Sect. 4. Regions are defined similar to those shown in Ricker et al. (2021) to aid comparison.

Figure 1Plots showing (a) percent of total time steps with 95 % or greater sea ice concentration and (b) location, extent, and corresponding name of regions used in Section 4.

The dynamic term within Eq. (2) can be further decomposed to form

$$\begin{array}{}\text{(3)}& \mathrm{\nabla }\cdot \left(\mathit{u}H\right)=\left(\mathrm{\nabla }H\right)\cdot \mathit{u}+H(\mathrm{\nabla }\cdot \mathit{u}),\end{array}$$

where the first term on the right represents changes to local sea ice thickness due to advection, i.e., the movement of ice transporting ice of a new thickness into a grid cell, and the second term on the right represents sea ice thickness changes due to deformation processes caused by divergence of the ice motion vector field. The deformation effect does not include advection and therefore can be considered Lagrangian dynamics – i.e., the dynamic effect as observed by a Lagrangian drifter. Our estimates of dynamic effects were partitioned into advection effects and residual effects using the sea ice motion vector product. At every weekly time step, the first term on the right-hand side of Eq. (3) is calculated as the mean of the three time steps centered on the current time step (in order to maintain temporal resolution with the centered-difference scheme used to calculated overall thickness change) using the CS2SMOS ice thickness field and sea ice motion vectors and is taken to be advection effect. The residual difference of the overall dynamic effect and this advection effect includes the effects of ice deformation and any other effects that are not accounted for in SLICE or the calculation of advection. These additional effects include lateral growth, snow ice formation, and any frazil or new ice growth that occurs above 95 % sea ice concentration and is not captured within SLICE. The uncertainty in this residual is a summation of the uncertainties in CS2SMOS, SLICE, and the advection calculation. This approach was taken over calculating the deformation effect using the motion vectors and advection effect as a residual because motion vector divergence was found to be significantly more noisy than the motion vector fields themselves. As motion vectors are not available within the Canadian Arctic Archipelago, advection effects and the residual effects are not available in this region.

In summary, the governing equation expressed in Eq. (2) is conceptually reconstructed as follows:

$$\begin{array}{}\text{(4)}& {\displaystyle \frac{\partial \left(\mathrm{CS}\mathrm{2}\mathrm{SMOS}\right)}{\partial t}}=\mathrm{SLICE}+\mathrm{dynamic}\mathrm{effect},\end{array}$$

and the dynamic effect is further decomposed into advection and residual effects with Eq. (3) conceptually reconstructed as follows:

$$\begin{array}{}\text{(5)}& \begin{array}{rl}\text{dynamic\hspace{0.17em}effect}& =\left(\mathrm{\nabla }\mathrm{CS}\mathrm{2}\mathrm{SMOS}\right)\cdot \text{ice\hspace{0.17em}motion\hspace{0.17em}vector}\\ & +\text{deformation\hspace{0.17em}effect}.\end{array}\end{array}$$

The results are presented monthly rather than weekly to reduce noise. Weekly integrations of the terms within Eqs. (4) and (5) were summed to monthly temporal resolution. The dataset covers November through April, due to SLICE assumptions described in Sect. 2.3 and availability of CS2SMOS data, spanning the 10 winters beginning in the years 2010 and 2012–2020. The 2010 data begin on 15 November rather than 1 November along with the availability of CryoSat-2 data, and the winter beginning in 2011 is not included due to a gap between the availability of passive microwave data from the earlier AMSR-E and later AMSR2. At each weekly time step, i, the following steps were completed at each point on the 25 km EASE-Grid 2.0:

1. $$\begin{array}{}\text{(6)}& \begin{array}{rl}{\displaystyle \frac{\partial \left(\mathrm{CS}\mathrm{2}\mathrm{SMOS}\right)}{\partial t}}& ={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\cdot (\mathrm{CS}\mathrm{2}{\mathrm{SMOS}}_{i+\mathrm{1}}\\ & -\mathrm{CS}\mathrm{2}{\mathrm{SMOS}}_{i-\mathrm{1}})\end{array}\end{array}$$

   
2. $$\begin{array}{}\text{(7)}& \text{thermodynamic\hspace{0.17em}growth}={\mathrm{SLICE}}_{i-\mathrm{1}}\end{array}$$

   
3. $$\begin{array}{}\text{(8)}& \begin{array}{rl}\text{dynamic\hspace{0.17em}effect}& ={\displaystyle \frac{\partial \left(\mathrm{CS}\mathrm{2}\mathrm{SMOS}\right)}{\partial t}}\\ & -\text{thermodynamic\hspace{0.17em}growth}\end{array}\end{array}$$

   
4. $$\begin{array}{}\text{(9)}& \begin{array}{rl}\text{advection}& \mathrm{effect}=\sum _{n=i-\mathrm{1}}^{i+\mathrm{1}}\\ & {\displaystyle \frac{\left(\mathrm{\nabla }\mathrm{CS}\mathrm{2}{\mathrm{SMOS}}_n\right)\cdot {\text{ice\hspace{0.17em}motion\hspace{0.17em}vector}}_n}{\mathrm{3}}}\end{array}\end{array}$$

   
5. $$\begin{array}{}\text{(10)}& \mathrm{\text{residual\hspace{0.17em}effect}}=\text{dynamic\hspace{0.17em}effect}-\text{advection\hspace{0.17em}effect}\end{array}$$

   

3.1 Uncertainty

Uncertainty in the individual weekly observations of thermodynamic, dynamic, advection, and residual effect can be calculated using a general formula for uncertainty in a function of several variables (Taylor, 1982):

$$\begin{array}{}\text{(11)}& {\mathit{\delta }}_q=\sqrt{{\left({\displaystyle \frac{\partial q}{\partial x}}{\mathit{\delta }}_x\right)}^{\mathrm{2}}+\mathrm{\cdots }+{\left({\displaystyle \frac{\partial q}{\partial z}}{\mathit{\delta }}_z\right)}^{\mathrm{2}}},\end{array}$$

where q is the computed value, $x,\mathrm{\cdots },z$ are independent and random inputs to that computed value, and ${\mathit{\delta }}_x,\mathrm{\cdots },{\mathit{\delta }}_z$ are those inputs' associated uncertainties. Applying Eq. (11) to the terms as described in Sect. 3 and adding covariance terms to the uncertainty of the space and time derivatives of CS2SMOS, we have

$$\begin{array}{}\text{(12)}& {\displaystyle }{\mathit{\delta }}_{\mathrm{thm}}=\sqrt{{\mathit{\delta }}_{\mathrm{SLICE}}^{\mathrm{2}}+{\left({\displaystyle \frac{\text{thermodynamic\hspace{0.17em}growth}}{\mathrm{CS}\mathrm{2}\mathrm{SMOS}}}{\mathit{\delta }}_{\mathrm{CS}\mathrm{2}\mathrm{SMOS}}\right)}^{\mathrm{2}}},\text{(13)}& {\displaystyle }\begin{array}{rl}{\mathit{\delta }}_{\mathrm{dyn}}& =\sqrt{{\displaystyle \frac{\mathrm{1}}{\mathrm{\Delta }{t}^{\mathrm{2}}}}\left({\mathit{\delta }}_{\mathrm{CS}\mathrm{2}\mathrm{SMOS},i-\mathrm{1}}^{\mathrm{2}}+{\mathit{\delta }}_{\mathrm{CS}\mathrm{2}\mathrm{SMOS},i+\mathrm{1}}^{\mathrm{2}}-\mathrm{2}\cdot \mathrm{0.6}\right.}\\ & \stackrel{\mathrm{‾}}{\left.\cdot {\mathit{\delta }}_{\mathrm{CS}\mathrm{2}\mathrm{SMOS},i-\mathrm{1}}\cdot {\mathit{\delta }}_{\mathrm{CS}\mathrm{2}\mathrm{SMOS},i+\mathrm{1}}\right)+{\mathit{\delta }}_{\mathrm{thm}}^{\mathrm{2}}},\end{array}\text{(14)}& {\displaystyle }\begin{array}{rl}{\mathit{\delta }}_{\mathrm{adv}}& =\sqrt{{\left({\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}\right)}^{\mathrm{2}}\sum _{n=i-\mathrm{1}}^{i+\mathrm{1}}{\left({\displaystyle \frac{u_n}{\mathrm{\Delta }x}}\sqrt{\mathrm{0.8}}{\mathit{\delta }}_{\mathrm{CS}\mathrm{2}\mathrm{SMOS},n}\right)}^{\mathrm{2}}}\\ & \stackrel{\mathrm{‾}}{+{\left({\displaystyle \frac{\partial \mathrm{CS}\mathrm{2}{\mathrm{SMOS}}_n}{\partial x}}{\mathit{\delta }}_u\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{v}{\mathrm{\Delta }y}}\sqrt{\mathrm{0.8}}{\mathit{\delta }}_{\mathrm{CS}\mathrm{2}\mathrm{SMOS},n}\right)}^{\mathrm{2}}}\\ & \stackrel{\mathrm{‾}}{+{\left({\displaystyle \frac{\partial \mathrm{CS}\mathrm{2}{\mathrm{SMOS}}_n}{\partial y}}{\mathit{\delta }}_v\right)}^{\mathrm{2}}},\end{array}\text{(15)}& {\displaystyle }{\mathit{\delta }}_{\mathrm{res}}=\sqrt{{\mathit{\delta }}_{\mathrm{dyn}}^{\mathrm{2}}+{\mathit{\delta }}_{\mathrm{adv}}^{\mathrm{2}}},\end{array}$$

where δ_thm, δ_dyn, δ_adv, δ_res, δ_SLICE, δ_CS2SMOS, δ_u, and δ_v are uncertainties in the thermodynamic growth, dynamic effect, advection effect, residual effect, SLICE, CS2SMOS thickness, x direction component of the sea ice motion vector, and y direction component of the sea ice motion vector, respectively; u is the x direction component of the sea ice motion vector; v is the y direction component of the sea ice motion vector; Δt is time step size; and Δx and Δy are the grid box size.

The uncertainty in the space and time derivatives of CS2SMOS contains covariance terms. CS2SMOS uncertainty is a significant source of uncertainty within the uncertainty framework above, but some portion of this uncertainty would cancel when a difference between time steps or neighboring grid points is taken. Though these covariances have not been explored in the literature in relation to CS2SMOS, we look to Fig. 7 within Ricker et al. (2017b) for guidance on correlation between grid cells within a single CS2SMOS field. For the example region depicted in this figure, correlations between thickness observations at grid points located less than 100 km apart are nearly always greater than 0.6. This 100 km radius includes neighboring grid points, which are separated by 25 km, and displacement during the 2 weeks between time steps in Eq. (6), which typically will not exceed 100 km. Based on this figure, we assume a correlation between time steps or neighboring grid points of 0.6 as a conservative estimate. Other than in this instance, our uncertainty formulas do not account for covariances between the input terms. Though covariances may be present across the input data, inclusion of their effects on uncertainty is outside the scope of this work.

The uncertainty in SLICE is taken from Anheuser et al. (2022), who report SLICE to have a thermodynamic growth mean bias of $\mathrm{4}\times {\mathrm{10}}^{-\mathrm{4}}$ m d⁻¹ and standard deviation bias of $\mathrm{2.2}\times {\mathrm{10}}^{-\mathrm{3}}$ m d⁻¹ when compared against ice mass balance buoy data. Here we use this standard deviation as SLICE uncertainty. The analysis presented in Anheuser et al. (2022) does not include the effect of uncertainty in initial sea ice thickness, so we add the second term on the right side of Eq. (12) to account for the uncertainty in CS2SMOS sea ice thickness. The uncertainty in CS2SMOS is calculated for each week and available in the data product. Tschudi et al. (2020) list a maximum ice motion vector error of 0.7 cm s⁻¹, which we use here for the uncertainty in the ice motion vector components. Lastly, the time step is 1 week and grid cell size is 25 000 m. Using these inputs, we calculate uncertainty in the thermodynamic growth, dynamic effect, advection effect, and residual effect terms at each time step and grid cell location.

Monthly uncertainty is calculated by summation of the weekly uncertainties within each month using Eq. (11). When the terms are averaged across time, the uncertainty of the mean is reduced through the averaging. Applying Eq. (11) to an averaging operation, we have the following:

$$\begin{array}{}\text{(16)}& {\mathit{\delta }}_{\mathrm{mean}}=\sqrt{{\left({\displaystyle \frac{\mathrm{1}}{N}}{\mathit{\delta }}_{\mathrm{1}}\right)}^{\mathrm{2}}+\mathrm{\cdots }+{\left({\displaystyle \frac{\mathrm{1}}{N}}{\mathit{\delta }}_N\right)}^{\mathrm{2}}},\end{array}$$

where δ_mean is the uncertainty of the mean, N is the number of samples, and δ₁, ⋯, δ_N are the individual uncertainties of each sample.

Figure 2 shows the wintertime mean total sea ice thickness growth, dynamic effects, and thermodynamic growth for Arctic sea ice thickness across the entire 10-year analysis period. Total growth and thermodynamic effects are always positive, though with varying magnitudes, while the magnitude and sign of dynamic effect vary across the Northern Hemisphere sea ice. As expected, thermodynamic thickness growth is highest in the seasonal ice zones, in some areas greater than 0.3 m per month, and inversely proportional to the climatological sea ice thickness, leading to less than 0.1 m per month of thermodynamic growth in much of the central Arctic. Dynamic effects decrease sea ice thickness over 63 % of the area exhibiting ice during the study period and increase sea ice thickness over the remaining area. An increase in sea ice thickness due to dynamics occurs off the Siberian coast in the Chukchi Sea, where the Beaufort Gyre tends to deposit advected ice, and similarly north of the Canadian Arctic Archipelago, where both the Beaufort Gyre and Transpolar Drift tend to deposit advected ice. The highest positive dynamic effects of greater than 0.1 m per month occur just north of the central Canadian Arctic Archipelago and in the Chukchi and East Greenland seas. A decrease in sea ice thickness due to dynamics, in some areas less than −0.2 m per month, occurs in the coastal regions of the Kara and Laptev seas, from where the Transpolar Drift tends to remove ice, and similarly in the coastal regions of the Beaufort Sea due to a similar effect of the Beaufort Gyre.

Figure 2Wintertime mean from late 2010 through early 2021 (except the winter of 2011–2012) of (a) overall sea ice thickness change, (b) thermodynamic sea ice thickness effects, and (c) dynamic sea ice thickness effects. Mean sea ice motion vectors from the same period are also plotted with dynamic effect, which follows spatial patterns suggested by the ice motion vectors.

Figure 3 shows the wintertime mean advection effect and residual effects across the 10-year analysis period. Negative advection effect dominates the Arctic sea ice, covering 70 % of the study area. The exception primarily occurs where the Beaufort Gyre advects thick ice from north of the Canadian Arctic Archipelago to the Beaufort Sea. Here, advection effects greater than 0.2 m per month can be found. The most significant negative advection effects, less than −0.2 m per month, occur in the coastal Laptev Sea. Residual effects are negative over 48 % of the study area. The largest residual effects of greater than 0.1 m per month occur where the ice motion tends to deposit and form ice ridges, north of the central Canadian Arctic Archipelago in the Chukchi Sea and north of Greenland in the Transpolar Drift. The Barents and Kara seas are dominated by strong negative deformation effects, some areas with less than −0.3 m per month. Coupled with the positive advection effects in the Beaufort Sea are negative deformation effects in this region.

Figure 3Wintertime mean from late 2010 through early 2021 (except the winter of 2011–2012) of (a) advection sea ice thickness effects and (b) residual sea ice thickness effects. Residual effects include those from ice deformation processes.

Figure 4 shows the uncertainty in the mean effects shown in Figs. 2 and 3. Per Eq. (16), uncertainties for mean values across time are reduced through the averaging operation. When reduced in this manner, uncertainty in the thermodynamic effect is small. The dynamic effect uncertainty, which includes the effects of uncertainty in both thermodynamic and CS2SMOS overall thickness change, is greater than thermodynamic growth uncertainty and highest in the lower latitudes. The uncertainty in CS2SMOS is the larger source of uncertainty in the dynamics term because the differencing of thickness between two separate time steps means overall uncertainty from both time steps are included. This is the case even when the covariance between time steps is included. Advection uncertainty is affected by a similar mechanism, as it includes the effects of spatial derivatives in the CS2SMOS field in addition to the motion vector uncertainty. Finally, residual effects have the highest uncertainty, as they include the uncertainty of overall change, thermodynamic growth, and advection summed in quadrature. In some areas, uncertainty has a similar magnitude to the effects themselves.

Figure 4Uncertainty calculated per Sect. 3.1 for each grid cell, during wintertime, from late 2010 through early 2021 (except the winter of 2011–2012) of sea ice thickness changes due to (a) thermodynamic growth, (b) dynamic effect, (c) advection effects, and (d) deformation effects. Uncertainty increases with a decrease in latitude as the number of weeks with ice cover and number of satellite overpasses decrease.

To investigate how the budgetary terms interact on a regional scale and facilitate comparison with Ricker et al. (2021) and Keen et al. (2021), Fig. 5 depicts mean total monthly sea ice volume changes due to each effect over the regions shown in Fig. 1 and summed across the entire Arctic. Also included in this figure is the mean monthly volume contribution from areas with less than or equal to the 95 % sea ice concentration threshold required for inclusion in the other terms to illustrate the effect of this condition. Each budgetary term was summed across each region and each month before a mean across these months was taken. The ≤95 % sea ice concentration contribution was calculated by a centered-difference scheme similar to Eq. (6) but with the factor of its sea ice concentration included. The uncertainty shown on Fig. 5 is a sum of uncertainties from all grid cells within each region for a given effect, which are then binned and summed by month using Eq. (11) and averaged over time using Eq. (16) to yield a monthly uncertainty for each effect and region. Applying Eq. (11) to the uncertainty summation across each region yielded unreasonably low uncertainty, so a basic sum was chosen.

Across the Arctic, thermodynamic growth accounts for 1936±45 km³ per month of volume growth. The dynamic effect across the Arctic is a volume sink of $-\mathrm{583}\pm \mathrm{359}$ km³ per month, which is −30 % of thermodynamic growth. Advection accounts for $-\mathrm{126}\pm \mathrm{229}$ km³ per month of this dynamic effect, while residual effects are $-\mathrm{457}\pm \mathrm{397}$ km³ per month. Volume changes in areas that do not have more than 95 % ice concentration and therefore are not included in the other terms are 247±52 km³ per month or 13 % that of thermodynamic growth. In all regions, thermodynamic growth is the strongest effect. The central Arctic has the highest thermodynamic volume growth at 320±12 km³ per month and is the only region with a positive dynamic effect at 56±101 km³ per month. The regions with the strongest dynamic effects are the Laptev Sea, the Kara Sea, and Baffin Bay, all regions with significant polynyas, with $-\mathrm{135}\pm \mathrm{24}$, $-\mathrm{142}\pm \mathrm{22}$, and $-\mathrm{162}\pm \mathrm{19}$ km³ per month, respectively. The remaining regions all have negative dynamic effects, but they are nearer to zero. The central Arctic has the strongest advection and residual effects among the regions, with $-\mathrm{106}\pm \mathrm{77}$ km³ per month advection effect and 163±127 km³ per month residual effect, and it is the only region with a positive residual effect. The only regions with a positive advection effect are the Beaufort and East Greenland seas and Baffin Bay, with 46±34, 53±39, and 3±13 km³ per month, respectively. The remaining regions have negative advection effects that are closer to zero. The strongest negative residual effect is in Baffin Bay, with $-\mathrm{165}\pm \mathrm{20}$ km³ per month. In the Barents and East Greenland seas, volume contributions in grid cells with less than or equal to 95 % sea ice concentration are not insignificant. Other regions have similar contributions from these grid cells below the sea ice concentration threshold, but they make up a smaller portion of the total volume changes.

Figure 5Mean monthly volumetric thermodynamic growth (blue), dynamic effect (green), advection effect (orange), deformation effect (pink), and ≤95 % sea ice concentration volume changes (yellow) for (a) the entire Arctic, (b) central Arctic, (c) Beaufort Sea, (d) Chukchi Sea, (e) East Siberian Sea, (f) Laptev Sea, (g) Kara Sea, (h) Barents Sea, (i) East Greenland Sea, (j) Baffin Bay, and (k) Canadian Archipelago. Over the entire Arctic, dynamic effects are −30 % that of thermodynamic growth.

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Figure 6 shows mean dynamics and mean residual effects in relative terms as ratios to thermodynamic growth. To calculate these metrics, the ratio of dynamics and residual to thermodynamics was calculated at each time step, and then a time mean of these ratios was taken across all time steps in this study. These plots look similar to those in Fig. 2, as the scaling quantity, thermodynamic growth, is fairly uniform across most of the Arctic. Nevertheless, the relative importance of dynamics and other residual effects to thermodynamic growth is an important result. Much of the Arctic shows a slightly negative impact of total dynamics relative to thermodynamic growth. The areas north of the Canadian Arctic Archipelago and Greenland show the highest relative impact of dynamics, with some grid cells showing dynamics with over twice the impact relative to thermodynamics. The coastal regions of the Kara and Laptev seas show significant negative impacts of dynamics, with magnitudes nearly equal to thermodynamics. The greatest relative importance of residual effects is also found in the areas north of the Canadian Arctic Archipelago and Greenland, though skewing more towards the central Arctic. In these regions, residual effects can be twice that of thermodynamics. The largest positive relative impact of residual effects is found between Svalbard and Greenland, where the Transpolar Drift causes ridging in thick ice that is not experiencing large thermodynamic growth.

Figure 6Wintertime mean from late 2010 through early 2021 (except the winter of 2011–2012) of the relative impact of (a) dynamic effects over thermodynamic sea ice thickness growth and (b) residual effects (excluding advection) over thermodynamic sea ice thickness growth. Alternatively, the figure can be viewed as (a) Eulerian dynamics and (b) Lagrangian dynamics.

Figure 7 shows the monthly mean overall dynamic effect, thermodynamic growth, advection effect, and residual effects across the 10 winters of data. The thermodynamic growth field, inversely proportional to sea ice thickness, remains consistently low across the central Arctic through most of the winter and higher in the perennial ice zones, though growth decreases as thickness increases with time in these areas. The residual and advection fields sum to an overall dynamic effect that increases with time in winter, dominated by the residual effect, potentially as ridging effects increase with overall ice thickness. The monthly residual effect fields depict a negative residual effect in the westward leg of the Beaufort Gyre that peaks in December, decreases in January, and is nearly absent in February and March. February and March do, however, depict residual effect maxima north of Svalbard and eastern Siberia. Positive advection of ice thickness by the Beaufort Gyre similarly peaks in the early winter, though the broad pattern of the advection field remains consistent throughout the winter.

Figure 7Monthly mean (a–e) dynamic effect, (f–j) thermodynamic effect, (k–o) residual effect, and (p–t) over the analysis period. Dynamic and residual effects increase through the growth season.

A detailed investigation of interannual variability upon mean thermodynamic, dynamic, advection, and deformation effects is outside the scope of this work. However, Fig. 8 shows yearly averages for each of these effects during the study period.

Figure 8Mean thermodynamic, dynamic, advection, and residual effects for the winters beginning in 2010–2020, except 2011. The patterns of dynamic, advective, and residual effects broadly follow the ice motion vectors.

4.1 2019–2020 winter and MOSAiC

The MOSAiC field experiment offers an opportunity to drill down further into an individual season's results and compare against similar studies of thermodynamic and dynamic effects along the MOSAiC drift track (Nicolaus et al., 2022). We begin with basin-wide monthly results from the 2019–2020 winter shown in Fig. 9. While the time step of our analysis is 1 week, the weekly results are noisy and have high uncertainty. Therefore, we present the 2019–2020 results on a monthly timescale. The results show that thermodynamic growth decreases in the peripheral seas throughout the winter, while thermodynamic thickness growth in the central Arctic remains consistently below 0.1 m per month throughout the year, with both effects due to thickness' inverse relationship to growth rate. The dynamic fields show a shifting area of positive dynamics. In November, positive dynamic effects are climatologically located north of the Canadian Archipelago. This region of positive dynamics shifts towards Greenland through February, when it increases to over 0.3 m per month in some areas. In March, a large region of greater than 0.3 m per month is centered in the Beaufort and Chukchi seas, while more sparse regions of 0.3 m per month are still located near the Fram Strait. These patterns are dominated by residual effects during this time. Advection effects appear relatively climatological when compared against Fig. 7, with an area of positive advection effect in the Beaufort Sea beginning in November that decreases in magnitude through the year as negative advection effects grow in the Laptev and Kara seas, where the Transpolar Drift originates.

Figure 9Monthly dynamic effect, thermodynamic effect, advection effect, and residual effect for (a–d) November 2019, (e–h) December 2019, (i–l) January 2020, (m–p) February 2020, and (q–t) March 2020. Thermodynamic effect decreases while dynamic and residual effects increase through the growth season.

Cumulative residual effects (i.e., Lagrangian dynamics), thermodynamics, and their sum and relative magnitudes, calculated using the methodology described herein, as experienced by the grid cell nearest to the MOSAiC drift station at each time step are depicted in Fig. 10 in order to compare these results with those reported by von Albedyll et al. (2022) and Koo et al. (2021). The analysis period is 1 November 2019 through 1 April 2020. Uncertainty in the dynamic and thermodynamic effects is calculated by summing in quadrature the weekly uncertainties within each month using Eq. (11) and then reporting uncertainty by cumulatively summing in quadrature each monthly uncertainty using the same equation. Thermodynamic growth during the period is steady and consistent, ranging between 0.11 and 0.20 m per month. Cumulative thermodynamic growth at the end of the period is 0.72±0.03 m. As expected, Lagrangian dynamic effects were more variable, ranging from −0.06 to 0.28 m per month. Lagrangian dynamic effects have local maxima in November and February and a cumulative total of 0.67±0.16 m. Total growth steadily rises due to the thermodynamic component and follows the shape of the Lagrangian dynamics component. The highest total cumulative growth of 1.39 m is found at the end of the period. As a percentage of total growth, Lagrangian dynamics end the season at 48 % of the total growth. Over a similar study area, Koo et al. (2021) found Lagrangian dynamics to account for 42.6 % of mean total growth.

Figure 10Cumulative Lagrangian dynamics (blue line), thermodynamic (green line), and total sea ice thickness growth (orange) from our results on the primary vertical axis and cumulative Lagrangian dynamics over cumulative total sea ice thickness growth during MOSAiC drift track from our results (pink circle) and by Koo et al. (2021), who used ICESat-2 to determine dynamics vs. thermodynamics along the MOSAiC drift track. Lagrangian dynamics account for roughly half of all thickness growth by 1 April 2020.

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The climatology of ice motion during the CryoSat-2 era as plotted in Fig. 2 suggests that the patterns of dynamic effect determined here are sound. The Beaufort Gyre and Transpolar Drift both transport ice towards the Canadian Arctic Archipelago, where high positive dynamic effects are found. The high residual effects in this region are likely explained by ridging. Where the Transpolar Drift originates, in the coastal region of the Laptev Sea, strong negative dynamic effects dominate. In these regions, lead formation likely explains the strongly negative residual effects. Between these regions and the Canadian Arctic Archipelago, a couplet of negative advection effect and positive residual effect characterizes a region where the Transpolar Drift tends to move thinner ice towards thicker ice, all the while experiencing ridging and other effects that increase thickness. Where the motion vectors show the Beaufort Gyre transporting ice westward from north of the Canadian Arctic Archipelago, a similar but reversed couplet of high positive advection effect and strong negative residual effect is found. The ice is transported from a region of climatologically thicker ice north of the Canadian Arctic Archipelago to a region of climatologically thinner ice in the Beaufort Sea, leading to a positive advection effect. In this same region, the Beaufort Gyre flow pattern is diverging and accelerating westward. This divergence leads to lead formation and a reduction in mean overall thickness, as reflected by negative residual effects in this region. This matches previous work suggesting high divergence and lead formation in this region (Kwok, 2006; Willmes and Heinemann, 2016; Hoffman et al., 2019). As this ice is further advected westward around the Beaufort Gyre, the advection and residual effects return to near zero before reaching areas north of eastern Siberia, where ridging may explain positive dynamic and residual effects. The location and magnitude of these patterns vary year to year with sea ice flow patterns and likely atmospheric conditions, as demonstrated in Fig. 8.

Our results add spatial detail to previous studies and agree well with previous studies when summed to similar scales. When summed across the Arctic, our results show that dynamic effects are a volume sink with a magnitude that is 30 % of thermodynamic growth. This similar to the CMIP6 multi-model mean of 30 % as reported by Keen et al. (2021). Volume changes that occurred in grid cells containing sea ice concentration beneath the 95 % threshold for inclusion in our results account for 13 % of total volume growth, potentially containing much of the frazil growth making up 19 % of total growth in the CMIP6 models (Keen et al., 2021). When our results are summed to the regional scale, they agree well with the previous regional-scale results from Ricker et al. (2021), though their results go back to 2002, whereas ours begin in 2018. For each of the six regions included in Ricker et al. (2021), mean thermodynamic growth from our results is within 50 km³ per month. Of these six regions, our results also agree on which have the strongest dynamic effects (Laptev and Kara seas), while the rest of the regions show dynamic effects closer to zero – though sometimes with differing signs, perhaps due to the differing time frames.

Kwok (2006) reported on the spatial and seasonal characteristics of Arctic sea ice deformation in the ice motion vector fields using high-resolution RGPS data from 1997–2000. Though the years in question do not overlap with the analysis period shown here, their results offer context for understanding dynamic sea ice effects. Their analysis showed divergence in the Beaufort Sea and convergence north of eastern Siberia, a pattern reflected here by the residual effects in these same regions. They show that the fraction of deformed ice in these regions decreases over the course of the growth season, a phenomenon also shown by a lessening of the negative deformation effects from November through March in Fig. 7. These points support our supposition that residual effects are comprised mostly of the effects of deformed ice on sea ice thickness, whether it be through ridging (positive residual effects) or lead formation (negative residual effects). The ice motion vectors used here are not suited for vector calculus calculations of deformation terms, but a future comparison between concurrent observations of vector deformation fields and dynamics would be fruitful for model improvement. We set a 95 % sea ice concentration threshold, at or below which a grid cell is not considered for partitioning of thermodynamic and dynamic changes. In Fig. 5, we report sea ice volume changes that occurred in grid cells with below this threshold to be 266 km³ per month, or 13 % that of thermodynamic growth. Volume changes due to changes in sea ice concentration between time steps will be at most 5 % but would be included in the residual effects. It should be noted that new ice formation in leads that have opened up due to divergence in the flow field will not be included in the thermodynamic effect; rather, the balance of the new ice thickness and the thickness of the remaining ice in the grid cell will be quantified as a negative residual effect. This type of effect has the potential to be greater than 5 %, as the leads may grow to arbitrary size and not be apparent in the passive microwave sea ice concentration product if new ice fills the lead prior to the next passive microwave brightness temperature observation. It is this initial closing of the lead with new ice that will not be captured as thermodynamic growth here.

The mean relative dynamic and residual effects over thermodynamic growth shown in Fig. 6 are useful for understanding the relative importance of these processes and are useful for an eventual comparison across time periods. The highest relative impact of dynamics is found north of the Canadian Arctic Archipelago, where the Beaufort Gyre and Transpolar Drift both deposit ice and thermodynamic growth is limited due to the high thickness of the ice. The relative residual plot tells a different story. With advection removed, this plot shows that a drifting observer would experience strong negative residual deformation effects relative to thermodynamic growth in the Beaufort Sea and increasingly strong positive residual deformation effects relative to thermodynamics in the Transpolar Drift and eastward leg of the Beaufort Gyre. In the areas north of the Canadian Arctic Archipelago, both dynamics and residual effects are equal to and greater than thermodynamic effects, consistent with Kwok and Cunningham (2016), who report that 42 %–56 % of mean thickness change is due to deformation. In some areas, the effects of dynamics dominate thermodynamics by a factor of 3. These results confirm the findings of Holland and Kimura (2016) and Ricker et al. (2021) that dynamics play an important role in shaping the climatological sea ice thickness patterns in the Arctic.

Within the methodology used here, the dynamic effects term represents Eulerian dynamics – i.e., a spatially stationary observer of sea ice thickness would observe changes due to thermodynamics and changes captured by the dynamics term, which includes advection. A Lagrangian observer, who is advecting as described by sea ice motion vector, would not experience changes due to this advection. The Lagrangian observer would only experience changes due to the residual in the framework here. In our interpretation, this residual effect is dominated by deformation effects. In this way, our Eulerian deformation term can be considered Lagrangian dynamics. Two Lagrangian studies of dynamics observed along the MOSAiC drift track offer useful context for validating our Eulerian results (von Albedyll et al., 2022; Koo et al., 2021). The comparison is necessarily between our monthly Eulerian deformation term – i.e., Lagrangian dynamics – from closest in space and time to the MOSAiC drift track and dynamics, as described within those studies.

A 10 % dynamic sea ice thickness growth relative to total growth along the MOSAiC drift was reported by von Albedyll et al. (2022), lower than the 48 % reported here and the 42.6 % reported by Koo et al. (2021). A likely primary cause of this discrepancy is related to temporal resolution. AEM sea ice thickness distributions were analyzed by von Albedyll et al. (2022) across the 50 km buoy network at the beginning of the growth season and at the end of the growth season, estimating a cumulative thermodynamic growth during the season using buoy thicknesses. Through a phenomenon acknowledged by the authors, this does not account for dynamics affecting thermodynamic growth throughout the growth season. If dynamics were to increase thickness, as our results and those reported by Koo et al. (2021) show did indeed occur along the MOSAiC drift track, thermodynamic growth rate would decrease, as shown in Eq. (1). Without accounting for this effect, thermodynamic growth is overestimated and dynamically driven growth is underestimated, due to the latter being calculated as a residual. The higher temporal resolutions in this work and Koo et al. (2021) greatly improve – though do not eliminate – this issue. Indeed, von Albedyll et al. (2022) report a cumulative thermodynamic growth of 1 m, whereas we report total thermodynamic growth of 0.72 m. This is significant, especially given von Albedyll et al. (2022) measured growth between 14 October and 17 April, whereas our analysis period is from 1 November to 1 April (this time discrepancy alone may cause differences as well).

Another potential reason for discrepancy between our results and those of von Albedyll et al. (2022) is the higher overall thickness growth measured by CryoSat-2 relative to the AEM measurements. We report a total mean growth along the MOSAiC drift track of 1.39 m relative to 1.1 m from the AEM survey. Indeed, CS2SMOS shows a mean sea ice thickness of 2.52 m on 1 April 2020 at the MOSAiC location relative to 2.2 m from the AEM surveys. This would manifest as an increase in dynamic effect in our analysis, as dynamics are calculated as a residual when thermodynamics are subtracted from total growth. That there are differences here is not surprising. The satellite measurements are gridded and taken from the nearest grid cell to MOSAiC, while the AEM surveys are centered on the MOSAiC buoy array. On the other hand, though the AEM has higher spatial resolution, the coverage over the 50 km buoy network is not complete. The satellite samples a larger area, although not centered exactly on the MOSAiC buoy array. That our results agree better with Koo et al. (2021) is not surprising, given that both studies have used satellite sea ice thickness as the primary dataset. Given that our study aims at a temporally and spatially larger scale while these studies are more focused on a single drift track, we can expect differences in results while using these more localized studies to provide context for our larger-scale study.

The calculations and analyses carried out here are all performed using satellite data on a 25 km EASE-Grid 2.0. Whereas sea ice processes can occur on much smaller scales, the results on a satellite scale are useful for deciphering patterns and trends on an Arctic-basin-wide basis. The CS2SMOS sea ice thickness dataset represents mean sea ice thickness within each grid cell. In actuality, thickness over the grid cell is defined by a distribution rather than a single value. However, without having observed local thickness distributions available at each time step, we have omitted thickness distributions and applied the SLICE retrieval using the mean thickness provided by CS2SMOS. It is likely that implementing thickness distributions would augment our results. Given the non-linear and inverse relationship between thickness and thermodynamic growth rate present in Eq. (1), a distribution favoring ice which is thinner than the mean thickness over that which is thicker than the mean within a grid cell would increase thermodynamic growth and decrease dynamic effects (and vice versa). Though the snow–ice interface temperature is not expected to vary as greatly as sea ice thickness across a 25 km grid cell, passive microwave snow–ice interface temperature retrieval also represents the mean across each grid cell. The dynamic, thermodynamic, advection, and residual effects then necessarily represent mean effects over the grid cell area. While few areas within a grid cell will have experienced exactly the effects described by our results, the cell will have experienced these effects on the mean.

Moving through the methodology described in Sect. 3 and uncertainty calculations shown in Sect. 3.1, it is apparent that the uncertainties associated with the various input products stack in such a way that uncertainty increases moving from thermodynamic growth with the lowest uncertainty through the dynamic effect, advection effect, and residual effect with the highest uncertainty. The uncertainty in thermodynamic growth is a summation in quadrature of SLICE uncertainty and that of the input thickness from CS2SMOS. As the dynamic effect is a residual between thermodynamic growth and overall sea ice thickness change from CS2SMOS, uncertainty in this term is a summation in quadrature of uncertainties in these terms. The first term under the radical is uncertainty in overall thickness change rate from CS2SMOS, as calculated in Eq. (6). The uncertainty in the advection calculation is an application of Eqs. (11) to (9), with uncertainties in the spatial derivatives of CS2SMOS appearing similar to that of the time derivative in Eq. (6). Residual effects, calculated as the difference between the dynamic effect and advection effect, are a summation in quadrature of the uncertainties in these terms, meaning the effects of uncertainty in SLICE, CS2SMOS, and CS2SMOS temporal and spatial derivatives and ice motion are all included.

When compared against buoy data and using the buoy thickness as the a priori initial thickness, Anheuser et al. (2022) report SLICE to have a thermodynamic growth mean bias of $\mathrm{4}\times {\mathrm{10}}^{-\mathrm{4}}$ m d⁻¹ and a standard deviation bias of $\mathrm{2.2}\times {\mathrm{10}}^{-\mathrm{3}}$ m d⁻¹. The assumption of 2 W m⁻² of basal flux from liquid water to solid sea ice leads to additional uncertainty from SLICE. Assuming a density of 917 kg m⁻² and a latent heat of fusion of 3.32×10⁵ J kg⁻¹, each 1 W m⁻² of basal sensible heat flux from the liquid seawater to solid sea ice is equivalent to a sea ice thermodynamic growth rate of $\mathrm{2.84}\times {\mathrm{10}}^{-\mathrm{4}}$ m d⁻¹. If the assumed 2 W m⁻² of basal sensible heat flux were removed, sea ice growth would increase by $\mathrm{5.67}\times {\mathrm{10}}^{-\mathrm{4}}$ m d⁻¹, and an increase from 2 to 10 W m⁻² would decrease thermodynamic sea ice thickness growth by $\mathrm{2.27}\times {\mathrm{10}}^{-\mathrm{3}}$ m d⁻¹. The SLICE thermodynamic growth retrieval also does not account for lateral melt and freeze processes or any new or frazil ice growth that occurs above 95 % sea ice concentration.

Because Anheuser et al. (2022) used buoy thickness as the initial a priori thickness when comparing against buoys, the uncertainty associated with initial input thickness was not accounted for. To account for this, we have added the second term under the radical in Eq. (12), which accounts for CS2SMOS uncertainty as per Eq. (11). The AWI CS2SMOS product merges sea ice thickness retrievals from CryoSat-2 and SMOS into a product that contains reduced uncertainties relative to each instrument's products independently (Ricker et al., 2017b). CryoSat-2 uncertainties are highest over thin ice, while SMOS uncertainties are highest over thick ice, creating the opportunity for synergy. CryoSat-2 uncertainties are made up of observational uncertainties or noise and systemic uncertainties or bias (Ricker et al., 2014). Observational uncertainties are reduced through spatial averaging on the grid and optimal estimation methodology used to create the CS2SMOS product. While systemic uncertainties affect estimates of absolute thickness, the differencing of thickness between time steps removes them from the estimations of various thickness effects calculated in this work. SMOS uncertainties are caused by uncertainties in the input parameters to the energy budget used to estimate sea ice thickness and are especially high over multi-year ice (MYI), results from which are removed from the optimal interpolation. The AWI CS2SMOS product provides an uncertainty value that estimates observational uncertainties for each individual estimate at each time step and grid cell, which allows our calculations of thermodynamic growth, dynamic effect, advection effect, and deformation effect to also have associated uncertainties at each time step and grid cell.

A potential mechanism for error occurs in the relationship between lead frequency and the snow–ice interface retrieval results. Leads and areas of lower sea ice concentrations contain open seawater exposed at the surface. Seawater has significantly lower emissivity in the microwave band than sea ice, therefore reducing passive microwave brightness temperatures in these regions. To the extent that leads or open water cover a grid cell, these lower brightness temperatures would then artificially reduce the retrieved snow–ice interface temperature and cause erroneously large thermodynamic growth rates. Via Eq. (4), erroneously high thermodynamic growth without a change to the CS2SMOS estimates leads to erroneously lower dynamic effects. This phenomenon is difficult to spot because negative dynamic effects are expected in regions with high lead frequency. We restrict our analysis to sea ice concentrations of greater than 95 %, as retrieved by established passive microwave methods. As such, the highest possible open water fraction within a grid cell is 5 %. Assuming that emissivity of a satellite field of view is a linear sum of scene type emissivities weighted by area fraction, the effect of 5 % open water by area on satellite-retrieved snow–ice interface temperature can be approximated. Using an approximate open water emissivity at 6.9 GHz of 0.56 and a sea ice emissivity of 0.98, the emissivity of a 95 % sea ice concentration is 0.959. This reduction in emissivity from 0.98 for a 100 % sea ice concentration scene equates to a 5.25 K reduction in brightness temperatures for a 250 K snow–ice interface temperature. Propagating this difference through the retrieval algorithm per Kilic et al. (2019) leads to a reduction of the retrieved snow–ice interface temperature of 6 K. In a scenario with thin ice and a small temperature gradient across the ice, this difference could be significant.

Data used in the creation of all figures are available at https://doi.org/10.5281/Zenodo.7987917 (Anheuser et al., 2023). Code for the creation of data and figures is available at https://doi.org/10.5281/Zenodo.7987926 (Anheuser, 2023) and https://github.com/janheuser/thmdyn (last access: 30 May 2023). The following auxiliary datasets were used and are available at these locations: AMSR-E and AMSR2 brightness temperatures, https://doi.org/10.5067/AMSR-E/AE_SI25.003 (Cavalieri et al., 2014) and https://doi.org/10.5067/TRUIAL3WPAUP (Markus et al., 2018); AMSR-E and AMSR2 SIC, https://doi.org/10.5067/AMSR-E/AE_SI25.003 (Cavalieri et al., 2014) and https://doi.org/10.5067/TRUIAL3WPAUP (Markus et al., 2018); AWI CS2SMOS v203, https://spaces.awi.de/display/CS2SMOS/CryoSat-SMOS+Merged+Sea+Ice+Thickness (Kaleschke et al., 2019); sea ice motion vectors, https://doi.org/10.5067/INAWUWO7QH7B (Tschudi et al., 2019); and MOSAiC drift track, https://doi.org/10.1594/PANGAEA.937193 (Nicolaus et al., 2021).

JA completed all analyses and wrote the first draft under guidance from YL and JRK. All authors worked together towards a final draft.

The contact author has declared that none of the authors has any competing interests

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. The views, opinions, and findings contained in this report are those of the author(s) and should not be construed as an official National Oceanic and Atmospheric Administration or U.S. Government position, policy, or decision.

This research has been supported by the National Oceanic and Atmospheric Administration (grant no. NA20NES4320003).

This paper was edited by Vishnu Nandan and reviewed by two anonymous referees.