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Regular network model for the sea ice-albedo feedback in the Arctic
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23. Thermodynamic processes in sea ice are distinctly different from those in fresh water ice, since sea ice grows from a solution of water and various ions. Many of the physical variables depend on temperature, salinity or the age of the ice (Ref. 22). The phase transition does not occur suddenly, that is, a volume of sea ice at the generally accepted freezing point of about does contain a considerable fraction of brine, which will solidify at lower temperatures (Refs. 24 and 25) Nonetheless, the core process of freezing and melting is similar to that of a pure water system in that during freezing latent heat is released and during melting latent heat is absorbed. Under such strong abstraction and simplification of the phase transition it is common practice for simple models to assume T = 273 K as the freezing and melting point of ice (Ref. 14 and 17).
26. The volumetric latent heat of fusion is calculated as , with l the latent heat of fusion, h the dimension of a cubic cell, and the densities of ice/water (Ref. 27). Temperature dependent density changes are ignored, except averaging over the density of water and the density of ice during the release of latent heat L. for a single cell (Ref. 27).
27. Physical constants and fixed system parameters used in this paper: size of ice-ocean layer: N = 16, M = 100; dimension of cubic cell h = 0.3 m; sharpness factor ; density of water ; density of ice ; latent heat of fusion ; volumetric heat capacity of water ; volumetric heat capacity of ice ; coupling constant ; thermal conductivity of water ; thermal conductivity of ice , atmospheric transmittance , albedo of open water , latitude .
28. The global coupling constant (with a cell dimension of h = 0.3 m and a time step of ) used for the coupled map lattice in this paper corresponds to (but must not be limited to) the discretized energy diffusion equation (PDE) for constant diffusivities.
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33. Parameterizations for saturation vapor pressure over ice and water differ, but they are negligibly small in the temperature range considered here (Ref. 32). We further assume that the atmosphere is saturated with water vapor at the interface between atmosphere and ice-ocean layer.
34. A weak local minimum in RLW around the freezing temperature T0 was also reported by Thorndike (Ref. 14) in a significantly different model. Instead of an atmospheric spatiotemporal drive acting on an extended ice-ocean layer, Thorndike uses an atmospheric model coupled to a one-dimensional ice-ocean model.
35. A qualitative argument for the stable state in the liquid phase is presented. If solar radiation is assumed to be constant in time (e.g. replaced with its yearly average), the total drive is given by a shift in RLW [Fig. 3(b)] such that it vanishes at three discrete energies. In the spatially homogeneous case a zero drive yields in Eq. (3). The fixed point with is stable due to the negative slope of the shifted RLW-curve in this E-regime. Simulations of Eq. (3) confirm a stable asymptotic cycle for OW in the “ vs E” phase space ( day). An analogous argument and simulation holds for the stable state in the solid phase.
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37. The spectral albedo for a given wave length is defined as the ratio of reflected to incoming shortwave solar radiation, . The albedo used throughout the paper is defined as the integral of the spectral albedo over the solar shortwave radiation spectrum.
38. Diffuse solar radiation received during twilight () and corresponding corrections to the formula for the extraterrestrial solar flux density 0 are neglected. In addition, the error in omitting the eccentricity e in the extraterrestrial solar flux density, , is considered small since .
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The Arctic Ocean and sea ice form a feedback system that plays an important role in the global climate. The complexity of highly parameterized global circulation (climate) models makes it very difficult to assess feedback processes in climate without the concurrent use of simple models where the physics is understood. We introduce a two-dimensional energy-based regular network model to investigate feedback processes in an Arctic ice-ocean layer. The model includes the nonlinear aspect of the ice-water phase transition, a nonlinear diffusive energy transport within a heterogeneous ice-ocean lattice, and spatiotemporal atmospheric and oceanic forcing at the surfaces. First results for a horizontally homogeneous ice-ocean layer show bistability and related hysteresis between perennial ice and perennial open water for varying atmospheric heat influx. Seasonal ice cover exists as a transient phenomenon. We also find that ocean heat fluxes are more efficient than atmospheric heat fluxes to melt Arctic sea ice.
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