
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39227413
70868
10.1038/s41598-024-70868-8
Article
Multiscale mushy layer model for Arctic marginal ice zone dynamics
Strong Courtenay court.strong@utah.edu

1
Cherkaev Elena 2
Golden Kenneth M. 2
1 https://ror.org/03r0ha626 grid.223827.e 0000 0001 2193 0096 Department of Atmospheric Sciences, University of Utah, Salt Lake City, UT USA
2 https://ror.org/03r0ha626 grid.223827.e 0000 0001 2193 0096 Department of Mathematics, University of Utah, Salt Lake City, UT USA
3 9 2024
3 9 2024
2024
14 204369 11 2023
22 8 2024
© The Author(s) 2024
2024
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Perhaps the most dynamic component of the Arctic sea ice cover is the marginal ice zone (MIZ), the transitional region between dense pack ice to the north and open ocean to the south. It widens by a factor of four while seasonally migrating more than 1600 km poleward in the Bering–Chukchi Sea sector, impacting climate dynamics, ecological processes, and human accessibility to the Arctic. Here we showcase a transformative mathematical modeling approach to understanding changes in MIZ location and width, focusing on their seasonal cycles as observed by satellites. We view the MIZ as a liquid-solid phase transition region, or mushy zone, on the scale of the Arctic Ocean. Invoking the physics of phase changes, the MIZ is modeled as a dynamic, multiscale composite material layer; this model captures 96% of the annual cycle of MIZ location and 78% of the annual cycle of MIZ width. Temperature in the upper ocean is described by a nonlinear heat equation with effective parameters obtained using homogenization theory for a random medium of ice floes in a sea water host. Observations and simulations together indicate that MIZ location closely tracks the below-ice 273 K isotherm while the width of the MIZ follows vertical heat flux convergence, but with a three-week lag.

Subject terms

Climate sciences
Cryospheric science
Statistical physics, thermodynamics and nonlinear dynamics
http://dx.doi.org/10.13039/100000006 Office of Naval Research N00014-18-1-2552 and N00014-21-1-29 N00014-18-1-2552 and N00014-21-1-29 N00014-18-1-2552 and N00014-21-1-29 Strong Courtenay Cherkaev Elena Golden Kenneth M. National Science FoundationMS-0940249, DMS-1413454, DMS-213619, and DMS-2206171 MS-0940249, DMS-1413454, DMS-213619, and DMS-2206171 MS-0940249, DMS-1413454, DMS-213619, and DMS-2206171 Strong Courtenay Cherkaev Elena Golden Kenneth M. issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Summer Arctic sea ice extent, age, and thickness have been declining at an accelerating pace over the past few decades, and the amplitude of the seasonal cycle has abruptly increased1–4. These changes in the Arctic sea ice cover are accompanied by changes in the geometry of the marginal ice zone (MIZ), which is the dynamic and biologically active transition region between dense inner pack ice to the north and open ocean to the south5–9. MIZ width has increased by 40% over the satellite era9–13, impacting atmosphere-ocean energy exchange14, polar ecosystem dynamics and habitat selection15–19, navigability of the Arctic for humans20–22, and the penetration of waves into the ice cover which affects melting, freezing, and floe-breaking processes23. Broad interest in the MIZ underscores the need to understand the basic physics of its evolution and trends24. Fully coupled global climate models can reasonably capture aspects of MIZ variability over the sea ice annual cycle25, but the complexity of these models makes it challenging to understand the basic physical processes, from the small scale up, at work in determining MIZ dynamics.

The upper layer of the Arctic Ocean with sea ice can be thought of as a two-phase (solid and liquid), two-component (sea ice and sea water) granular composite. The interacting grains or particles range in size over many orders of magnitude, and the system exhibits rich dynamics on oceanic scales in response to mechanical and thermodynamic forcing. Modeling this macroscopic behavior is similar to statistical mechanics, which can describe complex collective behavior such as phase transitions, transport processes, and response to forcing for systems with a large number of particles or degrees of freedom. While statistical physics has seen wide-spread success in similar problems in other fields, it has been applied to the physics of sea ice in only a few contexts26. This study advances the vision26 of bringing ideas and methods of statistical physics and phase transitions to bear on sea ice science.

Our main innovation here is to consider the MIZ as a macroscale phase change front, drawing on the physics of moving boundary problems27. The melting fronts in this problem are conventionally thought of as being much smaller-scale than the MIZ itself, residing where lateral melting occurs on individual floes and basal melting occurs at the bottom interface with sea water. However, at the larger pan-Arctic spatial scales considered here, the upper ocean layer is thought of as a composite of ice floes in a sea water host, with the MIZ being a finite-width phase transition region separating the “solid” core of dense Arctic pack ice to the north from the “liquid” ocean phase to the south. It is now widely recognized that melting fronts are often transition zones rather than sharp discontinuities in phase. These effects are often difficult to deal with analytically, but insights have been gained via numerical investigation28,29. Even in pure substances with a defined melting temperature, “mushy” regions or layers can occur where solid and liquid phases coexist near the melting temperature30. We propose a multiscale mushy layer model to interpret the seasonality in width and location of the MIZ observed by satellites. We note that this task cannot be performed directly from an ocean reanalysis dataset, such as GLORYS2V4, because while solving a set of large-scale coupled equations, its underlying model employs data assimilation.

For ocean processes involving phase changes in a water-salt-ice mixture, there is theoretical and empirical support for the existence of a finite-width phase transition zone31. Sea ice itself is a two-phase (solid and liquid), two-component (ice and brine) system which can be modeled by the mathematics and physics of mushy layers32–36. Here we adapt these powerful ideas to large scale sea ice dynamics, and consider the MIZ to be an ocean-scale mushy layer separating the solid phase to the north from the liquid phase to the south. Furthermore, we employ methods of homogenization, inspired by computations in statistical mechanics, to find the large-scale effective parameters of the ice cover that enter into our MIZ model, based on smaller scale information such as sea ice concentration and floe geometry.

Results

Observed MIZ

The MIZ is defined as the region where sea ice concentration (SIC) ranges between SIC = 0.15 and SIC = 0.8 as in our prior work10,11,37. Representative examples of the analyzed MIZ are shown in Fig. 1. We focus here on the Bering–Chukchi Sea sector because it features the most dramatic MIZ seasonal cycles observed in the Arctic (e.g., Fig. 1). As sea ice extent maximizes during spring and pack ice extends into the Bering Strait, the MIZ location (central latitude, denoted s) of the Bering–Chukchi Sea MIZ minimizes close to 60∘N (Fig. 2a,b). At the other extreme, sea ice extent minimizes during fall and the MIZ migrates more than 1600 km poleward to around 75∘N before returning south by the end of the calendar year (Fig. 2a,b). This migration closely follows the latitude at which Tb=273 K, where Tb is the below-ice sea water temperature at depth zb=-5 m in the GLORYS2V4 Reanalysis. Denoting this latitude by ϕ(Tb=273K), the squared Pearson correlation r2[s,ϕ(Tb=273K)]=0.97 indicates that migration of the 273-K isotherm accounts for 97% of the MIZ location seasonal cycle.

MIZ width w fluctuates around 75 km for most of the year, but dramatically increases by a factor of four during days of year 175–225, and then returns back to around 75 km by day of year 275 (Fig. 2c). This annual cycle of w has a strong lagged correlation with the difference between the below-ice temperature and skin temperature evaluated at the MIZ location, denoted (Tb-T0)|ϕ=s, where skin temperature (T0) refers to the temperature of the very thin surface layer of the ocean or sea ice, which directly interacts with the atmosphere38. A 100-day pulse of strong positive (Tb-T0)|ϕ=s leads the MIZ widening cycle by three weeks (dotted red curves, Fig. 2d) yielding a 21-day lagged correlation r2[w,(Tb-T0)|ϕ=s]=0.92. On the physical mechanism underlying this strong lagged correlation, (Tb-T0)|ϕ=s is proportional to vertical heat flux (Jz=-kz∂T/∂z) averaged over the depth of the sea water-sea ice layer. Such averaging results in the vertical component kz of the effective thermal conductivity tensor k of this layer1 1|zb|∫zb0Jzdz=kz|zb|(Tb-T0).

The 100-day pulse of strong positive temperature difference (Tb-T0)|ϕ=s is thus a physically plausible driver of the MIZ widening cycle, with the three-week lag interpreted as the time scale for required thinning of the sea ice.

Macroscale moving boundary model

Results given above based on analysis of satellite-derived sea ice concentrations and ocean reanalyses indicate that the drivers of MIZ width and location are visible in temperatures at the surface and below the ice at depth zb. A suitable thermodynamic model forced by these boundary conditions might thus be able to capture the observed MIZ seasonal cycles. The success of such a model increases our confidence in the causal mechanisms theorized above and provides an opportunity to gain further insight into the MIZ through analysis of the model solution.

Our approach is to model the MIZ as the solution of a macroscale phase transition problem. We consider the uppermost layer of the ocean as a two-phase composite of sea ice and sea water36,39. Using ideas from moving boundary problems and simulation of phase change fronts27,29,40, we model temperature T in this layer using a heat equation with a nonlinear source term which governs the evolution of latent heat2 ρc∂T∂t=∇·(k∇T)+δH∂ψ∂t.

Here ψ is the volume fraction of ice, ρ and c are the effective density and specific heat, respectively, calculated as concentration-weighted averages, and3 δH=ρ(cl-cs)T+ρL

is the difference between liquid and solid enthalpies corresponding to a reference temperature of zero Kelvin with L being the latent heat of fusion40. The effective parameter k is generally an anisotropic tensor which accounts for the thermal conductivity of the sea ice solid phase of the composite as well as for turbulent thermal transport in the liquid phase41–45. To reduce the model to one dependent variable, we write concentration ψ as a function of temperature using a power law formulation464 ψ=f(T)=1-T-TsTl-Tsα.

Here Ts = 271.35K was prescribed to correspond to the fully solid condition, and the two free parameters α=0.8 and Tl = 271.9K were chosen to achieve a realistic seasonal cycle of MIZ properties and sea ice thickness (see Methods and Table 1). A simple rule was implemented to allow ice to float when one or more surface grid cells were fully liquid (ψ=0) with ice (ψ>0) in the underlying ice grid cell(s) (see Methods). The model is run on a two-dimensional depth-latitude domain with boundary conditions based on reanalysis data averaged with respect to longitude over the Bering–Chukchi Sea region outlined in red in Fig. 1.

Simulated MIZ

Results of numerical simulations show that the proposed model (2) produced convincing MIZ annual cycles similar to those observed by satellites, with extents and volumes being generally larger in spring than fall (Fig. 3a). The motion of the sea ice edge was realistic, with simulated MIZ location (s^; filled magenta circles, Fig. 3a) correlating very well with observed values, r2(s^,s)=0.96 (Fig. 3a,b). The model slightly underestimated the amplitude of the observed annual cycle of MIZ location, but captured the timing of its extrema well. Simulated MIZ width (w^; magenta whiskers, Fig. 3a) underwent a widening cycle aligned with observations, yielding r2(w^,w)=0.78 (Fig. 3d).

The vertical extent of solid ice (extension of ψ=1 downward from the surface; white shading, Fig. 3a) maximized in spring and minimized in fall, consistent with annual cycles of sea ice thickness estimated from satellite data47. The underlying mushy layer (0<ψ<1; blue shading, Fig. 3a) often extended below sea ice thicknesses based on remote sensing and reanalysis (yellow solid and dashed curves, Fig. 3a), and this occurred mainly at higher latitudes when the lower boundary condition decreased into the mushy temperature range (Ts<Tb<Tl). However, the simulated vertical extent of dense ice (ψ≥0.8) correlated well with sea ice thicknesses obtained from GLORYS2V4, with results from 70∘N shown as an example in Fig. 3c.

The model captured the seasonal cycles described above via simulation of heat fluxes and phase changes through the vertical extent of the simulation domain. Some insight into these processes is available by an approximate form of the model equation (2)5 ρc∂T∂t-δH∂ψ∂t≈kz∂2T∂z2,

which neglects variations in kz and uses the fact that ∂2T/∂z2 is orders of magnitude larger than its horizontal counterpart. The right side of (5) is vertical heat flux convergence (-∂Jz/∂z), and the associated energy is partitioned on the left side into warming and reduction of sea ice concentration, suggesting that positive ∂2T/∂z2 would drive melt conducive to MIZ poleward motion and widening.

Indeed, the rate of change of simulated MIZ location (∂s^/∂t) is strongly positively correlated with ∂2T/∂z2 below the ice on the southern flank of the MIZ (Fig. 4a,b). The rate of change of simulated MIZ width (∂w^/∂t) is also strongly correlated with ∂2T/∂z2 below the ice, but closer to the central latitude of the MIZ (Fig. 4c,d). The fastest rate of widening aligns well with the largest ∂2T/∂z2, and the timing of the maximum width (∂w^/∂t=0) aligns with the sign reversal of ∂2T/∂z2. Linking back to the observational results, the strong pulse of positive (Tb-T0)|ϕ=s that leads the MIZ widening cycle by three weeks (dotted red curves, Fig. 2d) makes its way into the model interior to provide heat flux convergence below the ice conducive to basal melting and MIZ widening.

Summary and discussion

In observations, the annual cycle of MIZ location tracks the below-ice 273 K isotherm well (r2=0.97), and MIZ widening follows the difference between below-ice and skin temperature at a three-week lag [r2(w,(Tb-T0)|ϕ=s)=0.92]. A moving boundary model adapted to study these cycles as a macroscale phase change front captured 96% of the annual cycle of MIZ location and 78% of the annual cycle of MIZ width. This class of models is usually applied at much smaller scales as one would see in a laboratory setting or sometimes to simulate vertical sea ice growth48. In these smaller scale applications, the mushy layer (Ts<T<Tl) is a mixture of ice and water; applied at a macroscale, the mushy layer represents a mixed phase region in the vertical and a mixture of ice floes and sea water in the horizontal directions. This model works well though it has no explicit heat advection and no dynamics beyond the rule allowing sea ice to float. Its likelihood of success is bolstered by the signals we found in its observed boundary conditions – the motion of the 273 K isotherm for MIZ position and the strengthening of the vertical temperature gradient for MIZ width. These signals imposed on the boundaries appear to drive the MIZ cycles and have embedded in them an elaborate history of oceanic advective heat fluxes, temperature changes on the upper boundary arising from net radiative and turbulent energy fluxes, and possibly also latitudinal gradients in temperature associated with sea ice advection and phase change.

The temperature range of the mushy layer in the model (Tl-Ts=0.55 K) is not large, being two orders of magnitude smaller than the range of temperatures present in the boundary conditions. We took the solid phase threshold (Ts) to be the nominal freezing temperature of sea water (-1.8∘ C) and considered the threshold for the liquid phase as a free parameter. In a region of moderate sea ice concentrations (0.20–0.40) in the MIZ north of Alaska, observed surface temperatures ranged from -1.5 to 1.5∘C, with some of this variation interpreted as stemming from proximity to individual floes49. The same study reported that surface temperatures in the lower-concentration (0.05) region of the MIZ ranged from 2.3 to 6.1∘C, indicating significant heat storage in the upper ocean. Analysis of monthly gridded sea surface temperature (SST) and sea ice concentration over the Northern Hemisphere indicate a progressive decrease of SST with increasing concentration with more rapid decreases of SST at higher concentrations, and a larger total temperature change across the transition during the warm season (∼5∘C) than the cold season (∼3∘C)50. The two parameters relating sea ice concentration to temperature in (4) could be allowed to vary over time, perhaps as functions of sea ice thickness or mixed layer depth, but we found that a constant mushy layer definition achieved acceptable correlations and mean absolute errors of MIZ width and location. Further increases in Tl tended to decrease model performance due to a nearly linear increase in the ratio of simulated to observed MIZ width.

While the MIZ is classically described as the portion of the ice pack significantly impacted by ocean wave breakage of the ice into floes6,8, a thermodynamically-driven MIZ defined precisely by the level curves of the sea ice concentration field is also very useful, particularly for multi-decadal studies of passive microwave satellite data10,18,37,51. We note that the National Ice Center also develops operational concentration-based MIZ products52. This study focused on multi-year composite seasonal MIZ cycles, but future work is planned to investigate whether the model can capture observed decadal trends in MIZ location and width10. It is also of interest to explore how the model results compare to more elaborate models when both are forced by conditions altered by future greenhouse gas concentration scenarios. For these extensions, it may be more convenient to cast the boundary conditions in terms of fluxes, which is a straightforward change to the numerical scheme. More importantly, the drivers of MIZ changes on these longer time scales may differ from the drivers of the seasonal cycles, and work is motivated to determine the extent to which this model can adequately capture such processes.

Methods

Data

For analysis of the observed MIZ, we used the climate data record (CDR) of daily sea ice concentrations at nominally 25-km horizontal resolution based on passive microwave satellite observations, archived by the National Snow and Ice Data Center (NSIDC)53. Although algorithms used in the CDR may have greater uncertainty near the ice edge, satellite-based products are widely used in MIZ research and operations because of their superior spatial coverage and continuous measurements54. Gridded sea ice thicknesses based on ICESat for spring 2004 were also obtained from NSIDC55. For boundary conditions and observational analyses, surface skin temperatures were extracted from the European Centre for Medium-range Weather Forecasts (ECMWF) ERA5 Reanalysis at hourly resolution on a 0.25∘ grid56. We also used daily mean ocean temperatures and sea ice thickness from the Global Ocean Reanalysis and Simulations (GLORYS2V4)57,58 provided on a 0.25∘ grid at depth 5.14 m.

MIZ analysis

The concentration-based MIZ was defined as the region of sea ice bounded by ψ=0.15 corresponding to the conventional ice edge59 and ψ=0.80 corresponding to “close ice” as defined by the World Meteorological Organization60. In observations, the width of the MIZ (w) was the arc length of streamlines through the solution to Laplace’s equation within the MIZ with boundary conditions ψ={0.15,0.80} following our prior work10,11,37. The location of the MIZ (s) was its area-weighted latitude. To obtain temperatures from the MIZ location, Tb and T0 were first averaged with respect to longitude over the Bering–Chukchi Sea sector (red outline, Fig. 1). These fields were smoothed using a two-dimensional Gaussian kernel with standard deviations of two weeks and 0.5 degrees of latitude (e.g., yellow contours in Fig. 2b), and then values at ϕ=s were obtained via bilinear interpolation in the time-latitude domain.

To analyze the MIZ in the model output, we took the maximum simulated volume fraction of ice in each model column6 ψmax(t,ϕ)=maxzb≤z≤0ψ(t,ϕ,z).

The edges of the MIZ at time t were then the first latitudes moving north where ψmax exceeded 0.15 and 0.8. Simulated MIZ location (s^) was the average of these two latitudes, and simulated MIZ width (w^) was their difference converted to kilometers. For the observations and model output, the MIZ width and location were converted to composite daily time series by averaging over each day of year from 2000-2004.

Model parameters

In the mixed phase region, the effective density and effective heat capacity are taken to be concentration-weighted averages7 ρ=ψρs+(1-ψ)ρl

8 c=ψcs+(1-ψ)cl,

where subscripts s and l correspond to the solid and liquid phases, respectively. The effective thermal conductivity in the mixed phase region with ice floes and sea water is anisotropic, it differs in the vertical and horizontal directions. We now estimate these key parameters. There is an extensive literature in physics and applied mathematics on the effective or homogenized transport properties of heterogeneous composite media36,61–63. The analytic continuation method61,64–68 provides integral representations as in (9) below for the effective transport coefficients of two-component composites, including69 electrical and thermal conductivity, complex permittivity, diffusivity, and magnetic permeability. Moreover, all these properties of the same material are coupled to each other through a spectral function in this representation69,70 and can be indirectly evaluated from measurements of a different parameter. These analytic representations have also been extended to elastic and viscoelastic effective properties71–74. These representations distill the complexities of the mixture geometry into the spectral properties of a self-adjoint operator like the Hamiltonian in quantum physics. They yield rigorous bounds on the homogenized transport coefficients, given partial information on the composite geometry, such as the relative area fractions or volume fractions of the components, and whether or not the medium is statistically isotropic or not. For example, the sea ice concentration (SIC) is the satellite-derived area fraction of ocean covered by sea ice.

In the absence of any motion in the liquid phase of the granular two-phase composite of ice floes and sea water, thermal transport is diffusive. In this case, we consider the effective thermal conductivity k∗ in any direction, say the vertical z direction, with component conductivities ks for sea ice and kl for sea water. Then k∗ can be written as61,64–669 F(u)=1-k∗kl=∫01dμ(λ)u-λ,u=11-ks/kl,

where μ is a spectral measure which encodes the composite geometry, with ∫01dμ=ψ. For a discretization of the system, where the self-adjoint operator becomes a matrix, μ can be calculated from the eigenvalues and eigenvectors of the matrix75,76. An upper bound on k∗ is obtained66 via a lower bound on F when μ is a Dirac point measure (or Dirac delta function) ψδ0(λ) with its mass ψ concentrated at λ=0, yielding F(u)≥ψ/u, or k∗≤ψks+(1-ψ)kl. This upper bound is optimal, in that there are actual composite geometries whose thermal conductivity, as a function of the volume fractions and component parameters, is equal to this arithmetic mean (or concentration-weighted average) expression. Layers aligned vertically are such optimal geometries. If we consider vertically aligned columns whose cross-sections coincide with our given floe geometries, then the vertical thermal conductivity kz of this composite in 3D is also exactly equal to the arithmetic mean,10 kz=ψks+(1-ψ)kl.

The lower bound on k∗ that corresponds to the upper bound on F in Eq. (9) is obtained through the same procedure, but for E(u)=1-ks/k∗, which yields E(u)≥ψ/u, or the the harmonic mean bound, k∗≥(ψ/ks+(1-ψ)/kl)-1. This bound is optimal as well, attained by parallel layers aligned perpendicular to the thermal gradient. In this case, though, the horizontal component of the effective thermal conductivity, say kx, of the granular composite is not in general equal to the harmonic mean, but is reasonably well approximated by it, particularly for matrix-particle geometries67,68,77, with separated ice floes in a sea water host. Thus as an approximation we take11 kx=ψks+1-ψkl-1.

As in the real sea ice system, heat is transported by advection as well as diffusion, we consider advection diffusion processes41–43. The analytic continuation method, as formulated by Golden and Papanicolaou66, which provides Stieltjes integral representations for the homogenized parameter, has been extended to the effective thermal diffusivity D∗ (in some direction) of advection diffusion processes41,44,45. Rigorous bounds on D∗ are obtained given the local diffusivities and information on the geometry of the incompressible velocity field. This theory is more involved than the classical two-component analytic continuation method we used above, and will be explored further in the future.

Since shear-driven mixing dominates over pure thermal diffusion in the oceanic mixed layer, to model the advective contribution to the homogenized diffusivity, we replaced the nominal thermal conductivity of sea water (0.563 W m-1K-1)78 in Eqs. (10) and (11) with an effective value to account for this mixing,12 kl=ρlclDT.

Here DT is the effective thermal diffusivity resulting from advective forcing as well as diffusion. Observationally estimated thermal diffusivities span several orders of magnitude in the Arctic79 (10-7–10-4 m2s-1). To determine appropriate values for DT and the parameters of the concentration function (4), we performed trial simulations spanning the parameter space5×10-7≤DT≤10-4Wm-1K-1271.4≤Tl≤273.0K0.2≤α≤1.4.

The values given in Table 1 were selected to achieve a reasonable seasonal cycle of MIZ location, MIZ width, and sea ice thickness.

Numerical methods

The simulation was performed for the six-year period 1999–2004 with a 24-hour time step, discarding the first year as spinup. We discretized (2) using a fully implicit (backward Euler) time integration scheme46. As a two-dimensional model, we considered a MIZ which was radially symmetric about the pole (zero variation in the longitude direction) over the Bering–Chukchi Sea sector, where the latter was defined as the region between the red meridians in Fig. 1 (166∘E and 159∘W). The model domain spanned the latitude range 50≤ϕ≤90∘N. The vertical extent of the model was set at 5 m so that the lower boundary condition represented a sea-water temperature below the climatological ice thickness in the Bering–Chukchi Sea region80,81. The horizontal grid spacing was 1/8∘ and we used a 12.5-cm vertical grid spacing.

The final term in (2) contains a nonlinearity stemming from the dependence of ψ on T via (4), so T at the current time step was obtained via a constrained iterative scheme. We write the (m+1)th iteration as13 ajTjm+1=∑nbanbTnbm+1+bjTj0+δHj0(ψjm+1-ψj0),

where the subscript nb indicates the grid points neighboring j. Using a truncated Taylor series expansion4614 ψjm+1=ψjm+dfdT[Tjm+1-f-1(gjm)],ψjm+1∈[0,1]

with df/dT evaluated at ψjm. Equation (13) was iterated until δH converged to within 10-5.

A threshold correction was applied to the concentration function (4) enforcing it to take values between 0 and 1. The grid cells with ψ>1 were reset to ψ=1 and gird cells with ψ<0 were reset to ψ=0. In addition, a simple rule was implemented to allow ice to float. In a column, if the surface grid cell(s) were liquid (ψ=0) and had ice (ψ>0) in one or more underlying grid cells, the vertical positions of the surface liquid cell(s) and underlying ice cell(s) were reversed. Without this rule, the ice layer could melt from above and below, leaving an unrealistically submerged ice tongue.

Model boundary and initial conditions

Dirichlet boundary conditions were prepared by averaging daily values with respect to longitude over the Bering–Chukchi Sea region (from 166∘E to 210∘E, outlined in Fig. 1) over the years 1999–2004. The upper boundary condition at z=0 was defined as the daily mean skin temperature of the ocean or sea ice surface obtained from the ERA5 reanalysis. The lower boundary condition at z=zb was the below-ice temperature taken at z=5 meter depth from GLORYS2V4. The lateral (depth-dependent) boundary conditions at 50∘N and 90∘N were prescribed as linear interpolations between T0 and Tb. For the initial condition, a linear interpolation was made between Tb and T0 in ice-free columns. Where the GLORYS2V4 initial sea ice thickness (h) was greater than zero on 1 January 1999, Tb was prescribed up the sea water-sea ice interface (i.e., zb≤z<-h), followed by a linear transition from Tb to T0 within the sea ice as is typical during the growth season48. Table 1 Model parameters.

Notation	Value	Name	
ρs	900 kg m-3	Density of sea ice48	
ρl	1025 kg m-3	Density of sea water48	
cs	2.1 kJ kg-1 K-1	Specific heat capacity of sea ice48	
cl	4.0 kJ kg-1 K-1	Specific heat capacity of sea water78	
ks	2.2 W m-1 K-1	Thermal conductivity of sea ice48	
DT	5×10-6 m2 s-1	Thermal diffusivity of sea water79	
kl	20.5 W m-1 K-1	Effective thermal conductivity of sea water	
Tl	271.90 K	Threshold for fully liquid state	
Ts	271.35 K	Threshold for fully solid state	
α	0.8	Exponent in equation (4)	

Figure 1 Marginal ice zone. Illustrative MIZ configurations from 15 February 2003 (light blue shading) and 15 August 2003 (dark blue shading). The simulation domain is the Bering–Chukchi Sea sector is outlined in red (166∘E and 159∘W).

Figure 2 Observed MIZ location. For observations in the Bering–Chukchi Sea sector over the period 2000-2004: (a) daily mean MIZ latitude (black curve), interquartile range of daily MIZ latitude (IQR, gray shading), and daily mean latitude of the 273 K isotherm at depth 5.14 m in the GLORYS2V4 [ϕ(Tb=273K), red curve]. (b) Observed sea ice concentration from Climate Data Record (shading), mean MIZ latitude (black curve), and 5.14-m sea water temperature from GLORYS2V4 (Tb; yellow contours, K). (c) daily mean MIZ width (black curve) and interquartile range (IQR) of daily MIZ width (gray shading). (Tb-T0)|ϕ=s denotes the below-ice temperature minus the skin temperature at the MIZ location, and is shown at zero lag (dashed red curve) and shifted to the right by 21 days (solid red curve). (d) Tb-T0 (shading), mean MIZ latitude (black curve), and Tb (gray contours, K).

Figure 3 MIZ model. (a) Simulated volume fraction of ice (ψ) averaged by month over years 2000–2004. Magenta circles and whiskers indicate the MIZ location and width, respectively, both diagnosed from the model output. Dashed yellow curves are corresponding sea ice thicknesses from GLORYS2V4, and the solid yellow curve on the March panel is sea ice thickness from ICESAT for spring 2004. (b) Mean observed MIZ location (black curve), interquartile range of simulated MIZ location (IQR, gray shading), and mean simulated MIZ location (blue curve). (c) Mean sea ice thickness from GLORYS2V4 (black curve), IQR of simulated thickness of dense sea ice (ψ>0.8; gray shading), and mean simulated thickness of dense sea ice (blue curve). (d) Mean observed MIZ width (black curve), IQR of simulated MIZ width (gray shading), and mean simulated MIZ width (blue curve).

Figure 4 MIZ model derivatives. (a) Correlation between ∂2T^/∂z2 and ∂s^/∂t. The abscissa is latitude minus the simulated MIZ location (ϕ-s^). (b) Time series of ∂2T^/∂z2 averaged within the black contour in panel a and ∂s^/∂t. (c,d) Same as a,b but for simulated MIZ width w^. A Gaussian filter was used for these calculations, with standard deviations of 14 days on the time coordinate, 0.25∘ on the latitude coordinate, and 25 cm on the vertical coordinate.

Acknowledgements

We are grateful for support from the Applied and Computational Analysis Program at the US Office of Naval Research through grants N00014-18-1-2552 and N00014-21-1-2909. We also gratefully acknowledge support from the Division of Mathematical Sciences at the US National Science Foundation (NSF) through Grants DMS-0940249, DMS-1413454, DMS-2136198, and DMS-2206171.

Author contributions

C.S., E.C., and K.M.G. conceived the model. C.S. performed the experiments and analysed the results. All authors contributed to writing the manuscript.

Data availibility

The GLORYS ocean reanalysis data analyzed here are publicly available from Mercator Ocean International (mercator-ocean.eu), and the sea ice concentration data (Climate Data Record) are pubicly available from NSIDC (nsidc.org). Sea ice concentrations simulated by the model presented here are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Stroeve JC The Arctic’s rapidly shrinking sea ice cover: A research synthesis Clim. Change 2012 110 1005 1027 10.1007/s10584-011-0101-1
Stroeve, J. C. et al. The Arctic’s rapidly shrinking sea ice cover: A research synthesis. Clim. Change 110, 1005–1027. 10.1007/s10584-011-0101-1 (2012).10.1007/s10584-011-0101-1
2. Livina VN Lenton TM A recent tipping point in the Arctic sea-ice cover: Abrupt and persistent increase in the seasonal cycle since 2007 Cryosphere 2013 7 275 286 10.5194/tc-7-275-2013
Livina, V. N. & Lenton, T. M. A recent tipping point in the Arctic sea-ice cover: Abrupt and persistent increase in the seasonal cycle since 2007. Cryosphere 7, 275–286. 10.5194/tc-7-275-2013 (2013).10.5194/tc-7-275-2013
3. Stroeve J Notz D Changing state of Arctic sea ice across all seasons Environ. Res. Lett. 2018 13 103001 10.1088/1748-9326/aade56
Stroeve, J. & Notz, D. Changing state of Arctic sea ice across all seasons. Environ. Res. Lett. 13, 103001. 10.1088/1748-9326/aade56 (2018).10.1088/1748-9326/aade56
4. Parkinson CL DiGirolamo NE Sea ice extents continue to set new records: Arctic, Antarctic, and global results Remote Sens. Environ. 2021 267 112753 10.1016/j.rse.2021.112753
Parkinson, C. L. & DiGirolamo, N. E. Sea ice extents continue to set new records: Arctic, Antarctic, and global results. Remote Sens. Environ. 267, 112753. 10.1016/j.rse.2021.112753 (2021).10.1016/j.rse.2021.112753
5. Squire VA Lepparanta M The marginal ice zone Physics of Ice-covered Seas 1998 Helsinki University Printing House 381 446
Squire, V. A. The marginal ice zone. In Physics of Ice-covered Seas (ed. Lepparanta, M.) 381–446 (Helsinki University Printing House, 1998).
6. Wadhams P Ice in the Ocean 2000 Gordon and Breach Science Publishers
Wadhams, P. Ice in the Ocean (Gordon and Breach Science Publishers, 2000).
7. Squire V Of ocean waves and sea-ice revisited Cold Reg. Sci. Technol. 2007 49 110 133 10.1016/j.coldregions.2007.04.007
Squire, V. Of ocean waves and sea-ice revisited. Cold Reg. Sci. Technol. 49, 110–133. 10.1016/j.coldregions.2007.04.007 (2007).10.1016/j.coldregions.2007.04.007
8. Weeks WF On Sea Ice 2010 University of Alaska Press
Weeks, W. F. On Sea Ice (University of Alaska Press, 2010).
9. Barber DG Selected physical, biological and biogeochemical implications of a rapidly changing Arctic marginal ice zone Progr. Oceanogr. 2015 139 122 150 10.1016/j.pocean.2015.09.003
Barber, D. G. et al. Selected physical, biological and biogeochemical implications of a rapidly changing Arctic marginal ice zone. Progr. Oceanogr. 139, 122–150. 10.1016/j.pocean.2015.09.003 (2015).10.1016/j.pocean.2015.09.003
10. Strong C Rigor IG Arctic marginal ice zone trending wider in summer and narrower in winter Geophys. Res. Lett. 2013 40 4864 4868 10.1002/grl.50928
Strong, C. & Rigor, I. G. Arctic marginal ice zone trending wider in summer and narrower in winter. Geophys. Res. Lett. 40, 4864–4868. 10.1002/grl.50928 (2013).10.1002/grl.50928
11. Strong C Foster D Cherkaev E Eisenman I Golden KM On the definition of marginal ice zone width J. Atmos. Oceanic Tech. 2017 34 1565 1584 10.1175/JTECH-D-16-0171.1
Strong, C., Foster, D., Cherkaev, E., Eisenman, I. & Golden, K. M. On the definition of marginal ice zone width. J. Atmos. Oceanic Tech. 34, 1565–1584. 10.1175/JTECH-D-16-0171.1 (2017).10.1175/JTECH-D-16-0171.1
12. Aksenov Y On the future navigability of Arctic sea routes: High-resolution projections of the Arctic Ocean and sea ice Mar. Policy 2017 75 300 317 10.1016/j.marpol.2015.12.027
Aksenov, Y. et al. On the future navigability of Arctic sea routes: High-resolution projections of the Arctic Ocean and sea ice. Mar. Policy 75, 300–317. 10.1016/j.marpol.2015.12.027 (2017).10.1016/j.marpol.2015.12.027
13. Rolph RJ Feltham DL Schröder D Changes of the Arctic marginal ice zone during the satellite era Cryosphere 2020 14 1971 1984 10.5194/tc-14-1971-2020
Rolph, R. J., Feltham, D. L. & Schröder, D. Changes of the Arctic marginal ice zone during the satellite era. Cryosphere 14, 1971–1984. 10.5194/tc-14-1971-2020 (2020).10.5194/tc-14-1971-2020
14. Zippel S Thomson J Air-sea interactions in the marginal ice zone Elementa Sci. Anthr. 2016 4 000095 10.12952/journal.elementa.000095
Zippel, S. & Thomson, J. Air-sea interactions in the marginal ice zone. Elementa Sci. Anthr. 4, 000095 (2016).10.12952/journal.elementa.000095
15. Ribic CA Ainley DG Fraser W Habitat selection by marine mammals in the marginal ice zone Antarct. Sci. 1991 3 181 186 10.1017/S0954102091000214
Ribic, C. A., Ainley, D. G. & Fraser, W. Habitat selection by marine mammals in the marginal ice zone. Antarct. Sci. 3, 181–186. 10.1017/S0954102091000214 (1991).10.1017/S0954102091000214
16. Perrette M Yool A Quartly GD Popova EE Near-ubiquity of ice-edge blooms in the Arctic Biogeosci. Discuss. 2010 7 8123 8142 10.5194/bgd-7-8123-2010
Perrette, M., Yool, A., Quartly, G. D. & Popova, E. E. Near-ubiquity of ice-edge blooms in the Arctic. Biogeosci. Discuss. 7, 8123–8142. 10.5194/bgd-7-8123-2010 (2010).10.5194/bgd-7-8123-2010
17. Post E Ecological consequences of sea-ice decline Science 2013 341 519 524 10.1126/science.1235225 23908231
Post, E. et al. Ecological consequences of sea-ice decline. Science 341, 519–524. 10.1126/science.1235225 (2013).23908231 10.1126/science.1235225
18. Williams R Counting whales in a challenging, changing environment Sci. Rep. 2014 4 4170 10.1038/srep04170 24622821
Williams, R. et al. Counting whales in a challenging, changing environment. Sci. Rep. 4, 4170. 10.1038/srep04170 (2014).24622821 10.1038/srep04170
19. Lowry KE van Dijken GL Arrigo KR Evidence of under-ice phytoplankton blooms in the Chukchi Sea from 1998 to 2012 Deep Sea Res. Part II 2014 105 105 117 10.1016/j.dsr2.2014.03.013
Lowry, K. E., van Dijken, G. L. & Arrigo, K. R. Evidence of under-ice phytoplankton blooms in the Chukchi Sea from 1998 to 2012. Deep Sea Res. Part II 105, 105–117. 10.1016/j.dsr2.2014.03.013 (2014).10.1016/j.dsr2.2014.03.013
20. Stephenson SR Smith LC Agnew JA Divergent long-term trajectories of human access to the Arctic Nat. Clim. Change 2011 1 156 160 10.1038/nclimate1120
Stephenson, S. R., Smith, L. C. & Agnew, J. A. Divergent long-term trajectories of human access to the Arctic. Nat. Clim. Change 1, 156–160 (2011).10.1038/nclimate1120
21. Schmale J Lisowska M Smieszek M Future Arctic research: Integrative approaches to scientific and methodological challenges EOS Trans. Am. Geophys. Union 2013 94 292 292 10.1002/2013EO330004
Schmale, J., Lisowska, M. & Smieszek, M. Future Arctic research: Integrative approaches to scientific and methodological challenges. EOS Trans. Am. Geophys. Union 94, 292–292. 10.1002/2013EO330004 (2013).10.1002/2013EO330004
22. Rogers TS Walsh JE Rupp TS Brigham LW Sfraga M Future Arctic marine access: Analysis and evaluation of observations, models, and projections of sea ice Cryosphere 2013 7 321 332 10.5194/tc-7-321-2013
Rogers, T. S., Walsh, J. E., Rupp, T. S., Brigham, L. W. & Sfraga, M. Future Arctic marine access: Analysis and evaluation of observations, models, and projections of sea ice. Cryosphere 7, 321–332. 10.5194/tc-7-321-2013 (2013).10.5194/tc-7-321-2013
23. Thomson J Rogers WE Swell and sea in the emerging Arctic Ocean Geophys. Res. Lett. 2014 41 3136 3140 10.1002/2014GL059983
Thomson, J. & Rogers, W. E. Swell and sea in the emerging Arctic Ocean. Geophys. Res. Lett. 41, 3136–3140. 10.1002/2014GL059983 (2014).10.1002/2014GL059983
24. Bennetts LG Bitz CM Feltham DL Kohout AL Meylan MH Marginal ice zone dynamics: Future research perspectives and pathways Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 2022 380 20210267 10.1098/rsta.2021.0267
Bennetts, L. G., Bitz, C. M., Feltham, D. L., Kohout, A. L. & Meylan, M. H. Marginal ice zone dynamics: Future research perspectives and pathways. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 380, 20210267. 10.1098/rsta.2021.0267 (2022).10.1098/rsta.2021.0267
25. Horvat C Marginal ice zone fraction benchmarks sea ice and climate model skill Nat. Commun. 2021 12 2221 10.1038/s41467-021-22004-7 33850136
Horvat, C. Marginal ice zone fraction benchmarks sea ice and climate model skill. Nat. Commun. 12, 2221. 10.1038/s41467-021-22004-7 (2021).33850136 10.1038/s41467-021-22004-7
26. Banwell AF Burton JC Cenedes C Golden K Åström J Physics of the cryosphere Nat. Rev. Phys. 2023 5 446 449 10.1038/s42254-023-00610-2
Banwell, A. F., Burton, J. C., Cenedes, C., Golden, K. & Åström, J. Physics of the cryosphere. Nat. Rev. Phys. 5, 446–449 (2023).10.1038/s42254-023-00610-2
27. Crank J Free and moving boundary problems 1987 Oxford science publications
Crank, J. Free and moving boundary problems (Oxford science publications, 1987).
28. Kim C-J Kaviany M A numerical method for phase-change problems Int. J. Heat Mass Transf. 1990 33 2721 2734 10.1016/0017-9310(90)90206-A
Kim, C.-J. & Kaviany, M. A numerical method for phase-change problems. Int. J. Heat Mass Transf. 33, 2721–2734. 10.1016/0017-9310(90)90206-A (1990).10.1016/0017-9310(90)90206-A
29. Hu H Argyropoulos SA Mathematical modelling of solidification and melting: A review Modell. Simul. Mater. Sci. Eng. 1996 4 371 10.1088/0965-0393/4/4/004
Hu, H. & Argyropoulos, S. A. Mathematical modelling of solidification and melting: A review. Modell. Simul. Mater. Sci. Eng. 4, 371 (1996).10.1088/0965-0393/4/4/004
30. Idelsohn SR Storti MA Crivelli LA Numerical methods in phase-change problems Archiv. Comput. Methods Eng. 1994 1 49 74 10.1007/BF02736180
Idelsohn, S. R., Storti, M. A. & Crivelli, L. A. Numerical methods in phase-change problems. Archiv. Comput. Methods Eng. 1, 49–74. 10.1007/BF02736180 (1994).10.1007/BF02736180
31. Grandi D A phase field approach to solidification and solute separation in water solutions Z. Angew. Math. Phys. 2013 64 1611 1624 10.1007/s00033-013-0301-9
Grandi, D. A phase field approach to solidification and solute separation in water solutions. Z. Angew. Math. Phys. 64, 1611–1624. 10.1007/s00033-013-0301-9 (2013).10.1007/s00033-013-0301-9
32. Wettlaufer JS Worster MG Huppert HE Natural convection during solidification of an alloy from above with application to the evolution of sea ice J. Fluid Mech. 1997 344 291 316 10.1017/S0022112097006022
Wettlaufer, J. S., Worster, M. G. & Huppert, H. E. Natural convection during solidification of an alloy from above with application to the evolution of sea ice. J. Fluid Mech. 344, 291–316 (1997).10.1017/S0022112097006022
33. Worster MG Wettlaufer JS Natural convection, solute trapping, and channel formation during solidification of salt water J. Phys. Chem. B 1997 101 6137 6141 10.1021/jp9632448
Worster, M. G. & Wettlaufer, J. S. Natural convection, solute trapping, and channel formation during solidification of salt water. J. Phys. Chem. B 101, 6137–6141 (1997).10.1021/jp9632448
34. Feltham, D. L., Untersteiner, N., Wettlaufer, J. S. & Worster, M. G. Sea ice is a mushy layer. Geophys. Res. Lett. 33, 10.1029/2006GL026290 (2006).
35. Wells AJ Hitchen JR Parkinson JRG Mushy-layer growth and convection, with application to sea ice Phil. Trans. R. Soc. A 2019 377 20180165 10.1098/rsta.2018.0165 30982459
Wells, A. J., Hitchen, J. R. & Parkinson, J. R. G. Mushy-layer growth and convection, with application to sea ice. Phil. Trans. R. Soc. A 377, 20180165 (2019).30982459 10.1098/rsta.2018.0165
36. Golden KM Modeling sea ice Notices Am. Math. Soc. 2020 67 1535 1555 10.1090/noti2171
Golden, K. M. et al. Modeling sea ice. Notices Am. Math. Soc. 67, 1535–1555 (2020).10.1090/noti2171
37. Strong C Atmospheric influence on Arctic marginal ice zone position and width in the Atlantic sector, February-April 1979–2010 Clim. Dyn. 2012 39 3091 3102 10.1007/s00382-012-1356-6
Strong, C. Atmospheric influence on Arctic marginal ice zone position and width in the Atlantic sector, February-April 1979–2010. Clim. Dyn. 39, 3091–3102. 10.1007/s00382-012-1356-6 (2012).10.1007/s00382-012-1356-6
38. Luo B Minnett PJ Evaluation of the ERA5 sea surface skin temperature with remotely-sensed shipborne marine-atmospheric emitted radiance interferometer data Remote Sens. 2020 12 1873 10.3390/rs12111873
Luo, B. & Minnett, P. J. Evaluation of the ERA5 sea surface skin temperature with remotely-sensed shipborne marine-atmospheric emitted radiance interferometer data. Remote Sens. 12, 1873. 10.3390/rs12111873 (2020).10.3390/rs12111873
39. Cherkaev E Golden KM Inverse bounds for microstructural parameters of a composite media derived from complex permittivity measurements Waves Random Med. 1998 8 437 450 10.1088/0959-7174/8/4/004
Cherkaev, E. & Golden, K. M. Inverse bounds for microstructural parameters of a composite media derived from complex permittivity measurements. Waves Random Med. 8, 437–450 (1998).10.1088/0959-7174/8/4/004
40. Voller VR Swaminathan CR Thomas BG Fixed grid techniques for phase change problems: A review Int. J. Numer. Meth. Eng. 1990 30 875 898 10.1002/nme.1620300419
Voller, V. R., Swaminathan, C. R. & Thomas, B. G. Fixed grid techniques for phase change problems: A review. Int. J. Numer. Meth. Eng. 30, 875–898. 10.1002/nme.1620300419 (1990).10.1002/nme.1620300419
41. Avellaneda M Majda A Stieltjes integral representation and effective diffusivity bounds for turbulent transport Phys. Rev. Lett. 1989 62 753 755 10.1103/PhysRevLett.62.753 10040324
Avellaneda, M. & Majda, A. Stieltjes integral representation and effective diffusivity bounds for turbulent transport. Phys. Rev. Lett. 62, 753–755 (1989).10040324 10.1103/PhysRevLett.62.753
42. Fannjiang A Papanicolaou G Convection-enhanced diffusion for random flows J. Stat. Phys. 1997 88 1033 1076 10.1007/BF02732425
Fannjiang, A. & Papanicolaou, G. Convection-enhanced diffusion for random flows. J. Stat. Phys. 88, 1033–1076 (1997).10.1007/BF02732425
43. Pavliotis, G. A. Homogenization theory for advection-diffusion equations with mean flow. Ph.D. thesis, Rensselaer Polytechnic Institute Troy, New York (2002).
44. Murphy NB Cherkaev E Zhu J Xin J Golden KM Spectral analysis and computation for homogenization of advection diffusion processes in steady flows J. Math. Phys. 2020 61 013102 10.1063/1.5127457
Murphy, N. B., Cherkaev, E., Zhu, J., Xin, J. & Golden, K. M. Spectral analysis and computation for homogenization of advection diffusion processes in steady flows. J. Math. Phys. 61, 013102 (2020).10.1063/1.5127457
45. Murphy NB Cherkaev E Xin J Zhu J Golden KM Spectral analysis and computation of effective diffusivities in space-time periodic incompressible flows Ann. Math. Sci. Appl. 2017 2 3 66 10.4310/AMSA.2017.v2.n1.a1
Murphy, N. B., Cherkaev, E., Xin, J., Zhu, J. & Golden, K. M. Spectral analysis and computation of effective diffusivities in space-time periodic incompressible flows. Ann. Math. Sci. Appl. 2, 3–66 (2017).10.4310/AMSA.2017.v2.n1.a1
46. Voller VR Swaminathan CR General source-based method for solidification phase change Numer. Heat Transf. Part B: Fundam. 1991 19 175 189 10.1080/10407799108944962
Voller, V. R. & Swaminathan, C. R. General source-based method for solidification phase change. Numer. Heat Transf. Part B: Fundam. 19, 175–189. 10.1080/10407799108944962 (1991).10.1080/10407799108944962
47. Kwok R Cunningham GF Variability of Arctic sea ice thickness and volume from CryoSat-2 Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 2015 373 20140157 10.1098/rsta.2014.0157
Kwok, R. & Cunningham, G. F. Variability of Arctic sea ice thickness and volume from CryoSat-2. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 373, 20140157 (2015).10.1098/rsta.2014.0157
48. Petrich C Eicken H Overview of sea ice growth and properties 2017 John Wiley & Sons 1 41
Petrich, C. & Eicken, H. Overview of sea ice growth and properties 1–41 (John Wiley & Sons, 2017).
49. Bradley AC Palo S LoDolce G Weibel D Lawrence D Air-deployed microbuoy measurement of temperatures in the marginal ice zone upper ocean during the mizopex campaign J. Atmos. Oceanic Technol. 2015 32 1058 1070 10.1175/JTECH-D-14-00209.1
Bradley, A. C., Palo, S., LoDolce, G., Weibel, D. & Lawrence, D. Air-deployed microbuoy measurement of temperatures in the marginal ice zone upper ocean during the mizopex campaign. J. Atmos. Oceanic Technol. 32, 1058–1070 (2015).10.1175/JTECH-D-14-00209.1
50. Rayner NA Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century J. Geophys. Res. 2003 108 4407 10.1029/2002JD002670
Rayner, N. A. et al. Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century. J. Geophys. Res. 108, 4407 (2003).10.1029/2002JD002670
51. Stroeve JC Jenouvrier S Campbell GG Barbraud C Delord K Mapping and assessing variability in the Antarctic marginal ice zone, pack ice and coastal polynyas in two sea ice algorithms with implications on breeding success of snow petrels Cryosphere 2016 10 1823 1843 10.5194/tc-10-1823-2016
Stroeve, J. C., Jenouvrier, S., Campbell, G. G., Barbraud, C. & Delord, K. Mapping and assessing variability in the Antarctic marginal ice zone, pack ice and coastal polynyas in two sea ice algorithms with implications on breeding success of snow petrels. Cryosphere 10, 1823–1843. 10.5194/tc-10-1823-2016 (2016).10.5194/tc-10-1823-2016
52. NIC. National Ice Center products on demand, Accessed June 2016, http://www.natice.noaa.gov (2016).
53. Meier, W. et al. NOAA/NSIDC Climate Data Record of Passive Microwave Sea Ice Concentration. Downloaded from http://nsidc.org, Boulder, Colorado USA: National Snow and Ice Data Center. Digital media (2011).
54. Peng G Meier WN Scott DJ Savoie MH A long-term and reproducible passive microwave sea ice concentration data record for climate studies and monitoring Earth Syst. Sci. Data 2013 5 311 318 10.5194/essd-5-311-2013
Peng, G., Meier, W. N., Scott, D. J. & Savoie, M. H. A long-term and reproducible passive microwave sea ice concentration data record for climate studies and monitoring. Earth Syst. Sci. Data 5, 311–318. 10.5194/essd-5-311-2013 (2013).10.5194/essd-5-311-2013
55. Stroeve, J. & Meier., W. N. Gridded observational sea ice thickness products, version 1 (2016).
56. Hersbach H The ERA5 global reanalysis Q. J. R. Meteorol. Soc. 2020 146 1999 2049 10.1002/qj.3803
Hersbach, H. et al. The ERA5 global reanalysis. Q. J. R. Meteorol. Soc. 146, 1999–2049. 10.1002/qj.3803 (2020).10.1002/qj.3803
57. Ferry, N. et al. GLORYS2V1 global ocean reanalysis of the altimetric era (1992-2009) at meso scale. Mercator Ocean Q Newslett (2012).
58. Garric, G. & Parent, L. Quality information document for global ocean reanalysis products. Accessed May 2024. https://catalogue.marine.copernicus.eu/documents/QUID/CMEMS-GLO-QUID-001-025.pdf (2017).
59. Comiso, J. C. Abrupt decline in Arctic winter sea ice cover. Geophys. Res. Lett. 3310.1029/2006GL027341 (2006).
60. WMO. World Meteorological Organization sea-ice nomenclature, terminology, codes and illustrated glossary, WMO/DMM/BMO 259-TP-145 (Secretariat of the World Meteorological Organization, 1985).
61. Milton GW Theory of Composites 2002 Cambridge University Press
Milton, G. W. Theory of Composites (Cambridge University Press, 2002).
62. Torquato S Random Heterogeneous Materials: Microstructure and Macroscopic Properties 2002 Springer-Verlag
Torquato, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer-Verlag, 2002).
63. Golden KM Climate change and the mathematics of transport in sea ice Notices Am. Math. Soc. 2009 56 562 584
Golden, K. M. Climate change and the mathematics of transport in sea ice. Notices Am. Math. Soc. 56, 562–584 (2009).
64. Bergman DJ Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material Phys. Rev. Lett. 1980 44 1285 1287 10.1103/PhysRevLett.44.1285
Bergman, D. J. Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material. Phys. Rev. Lett. 44, 1285–1287 (1980).10.1103/PhysRevLett.44.1285
65. Milton GW Bounds on the complex dielectric constant of a composite material Appl. Phys. Lett. 1980 37 300 302 10.1063/1.91895
Milton, G. W. Bounds on the complex dielectric constant of a composite material. Appl. Phys. Lett. 37, 300–302 (1980).10.1063/1.91895
66. Golden K Papanicolaou G Bounds for effective parameters of heterogeneous media by analytic continuation Comm. Math. Phys. 1983 90 473 491 10.1007/BF01216179
Golden, K. & Papanicolaou, G. Bounds for effective parameters of heterogeneous media by analytic continuation. Comm. Math. Phys. 90, 473–491 (1983).10.1007/BF01216179
67. Bruno O The effective conductivity of strongly heterogeneous composites Proc. R. Soc. London A 1991 433 353 381 10.1098/rspa.1991.0053
Bruno, O. The effective conductivity of strongly heterogeneous composites. Proc. R. Soc. London A 433, 353–381 (1991).10.1098/rspa.1991.0053
68. Golden KM Papanicolaou G The interaction of microwaves with sea ice Wave Propagation in Complex Media, IMA Volumes in Mathematics and Its Applications, 75–94 1997 Springer -Verlag
Golden, K. M. The interaction of microwaves with sea ice. In Wave Propagation in Complex Media, IMA Volumes in Mathematics and Its Applications, 75–94 (ed. Papanicolaou, G.) (Springer -Verlag, 1997).
69. Cherkaev E Inverse homogenization for evaluation of effective properties of a mixture Inverse Prob. 2001 17 1203 1218 10.1088/0266-5611/17/4/341
Cherkaev, E. Inverse homogenization for evaluation of effective properties of a mixture. Inverse Prob. 17, 1203–1218 (2001).10.1088/0266-5611/17/4/341
70. Cherkaev E Zhang D Coupling of the effective properties of a random mixture through the reconstructed spectral representation Phys. B 2003 338 16 23 10.1016/S0921-4526(03)00452-6
Cherkaev, E. & Zhang, D. Coupling of the effective properties of a random mixture through the reconstructed spectral representation. Phys. B 338, 16–23 (2003).10.1016/S0921-4526(03)00452-6
71. Kantor Y Bergman DJ Elastostatic resonances- a new approach to the calculation of the effective elastic constants of composites J. Mech. Phys. Solids 1982 30 335 376 10.1016/0022-5096(82)90005-9
Kantor, Y. & Bergman, D. J. Elastostatic resonances- a new approach to the calculation of the effective elastic constants of composites. J. Mech. Phys. Solids 30, 335–376 (1982).10.1016/0022-5096(82)90005-9
72. Ou M Cherkaev E On the integral representation formula for a two-component composite Math. Methods Appl. Sci. 2006 29 655 664 10.1002/mma.703
Ou, M. & Cherkaev, E. On the integral representation formula for a two-component composite. Math. Methods Appl. Sci. 29, 655–664 (2006).10.1002/mma.703
73. Cherkaev E Bonifasi-Lista C Characterization of structure and properties of bone by spectral measure method J. Biomech. 2011 44 345 351 10.1016/j.jbiomech.2010.10.031 21112588
Cherkaev, E. & Bonifasi-Lista, C. Characterization of structure and properties of bone by spectral measure method. J. Biomech. 44, 345–351 (2011).21112588 10.1016/j.jbiomech.2010.10.031
74. Cherkaev E Internal friction and the Stieltjes analytic representation of the effective properties of two-dimensional viscoelastic composites Arch. Appl. Mech. 2019 89 591 607 10.1007/s00419-019-01514-3
Cherkaev, E. Internal friction and the Stieltjes analytic representation of the effective properties of two-dimensional viscoelastic composites. Arch. Appl. Mech. 89, 591–607 (2019).10.1007/s00419-019-01514-3
75. Murphy NB Cherkaev E Hohenegger C Golden KM Spectral measure computations for composite materials Commun. Math. Sci. 2015 13 825 862 10.4310/CMS.2015.v13.n4.a1
Murphy, N. B., Cherkaev, E., Hohenegger, C. & Golden, K. M. Spectral measure computations for composite materials. Commun. Math. Sci. 13, 825–862 (2015).10.4310/CMS.2015.v13.n4.a1
76. Murphy NB Cherkaev E Golden KM Anderson transition for classical transport in composite materials J. Phys. Rev. Lett. 2017 118 036401 10.1103/PhysRevLett.118.036401
Murphy, N. B., Cherkaev, E. & Golden, K. M. Anderson transition for classical transport in composite materials. J. Phys. Rev. Lett. 118, 036401. 10.1103/PhysRevLett.118.036401 (2017).10.1103/PhysRevLett.118.036401
77. Orum C Cherkaev E Golden KM Recovery of inclusion separations in strongly heterogeneous composites from effective property measurements Proc. Roy. Soc. London A 2012 468 784 809
Orum, C., Cherkaev, E. & Golden, K. M. Recovery of inclusion separations in strongly heterogeneous composites from effective property measurements. Proc. Roy. Soc. London A 468, 784–809 (2012).
78. Sharqawy M Lienhard J Zubair S Thermophysical properties of seawater: A review of existing correlations and data Desalin. Water Treat. 2010 16 354 380 10.5004/dwt.2010.1079
Sharqawy, M., Lienhard, J. & Zubair, S. Thermophysical properties of seawater: A review of existing correlations and data. Desalin. Water Treat. 16, 354–380. 10.5004/dwt.2010.1079 (2010).10.5004/dwt.2010.1079
79. Shaw WJ Stanton TP Vertical diffusivity of the Western Arctic Ocean halocline J. Geophys. Res.: Oceans 2014 119 5017 5038 10.1002/2013JC009598
Shaw, W. J. & Stanton, T. P. Vertical diffusivity of the Western Arctic Ocean halocline. J. Geophys. Res.: Oceans 119, 5017–5038 (2014).10.1002/2013JC009598
80. Ricker R A weekly Arctic sea-ice thickness data record from merged CryoSat-2 and SMOS satellite data Cryosphere 2017 11 1607 1623 10.5194/tc-11-1607-2017
Ricker, R. et al. A weekly Arctic sea-ice thickness data record from merged CryoSat-2 and SMOS satellite data. Cryosphere 11, 1607–1623. 10.5194/tc-11-1607-2017 (2017).10.5194/tc-11-1607-2017
81. Wang X Key J Kwok R Zhang J Comparison of Arctic sea ice thickness from satellites, aircraft, and PIOMAS data Remote Sens. 2016 8 713 10.3390/rs8090713
Wang, X., Key, J., Kwok, R. & Zhang, J. Comparison of Arctic sea ice thickness from satellites, aircraft, and PIOMAS data. Remote Sens. 8, 713. 10.3390/rs8090713 (2016).10.3390/rs8090713
