The first model we use is a modified version of the 2-season thermodynamic model of Thorndike , which provides an approximate analytical equation for the equilibrium ice thickness. The total energy flux for melting or growing ice is the sum of downward and upward longwave radiation (FLW and FUP), absorbed shortwave radiation ((1 − α) FSW with snow/ice albedo α), and the oceanic heat flux at the ice-ocean interface (FW). We neglect the turbulent surface fluxes of sensible and latent heat, which are well known to be much smaller than the radiative components [Maykut and Untersteiner, 1971]. Linearizing the Stefan-Boltzmann law about the bulk freezing temperature, the upward longwave radiation can be written FUP = A + BT, where T is the departure of the surface temperature of the ice from the freezing point (e.g., the temperature measured in °C). This leads to a change in thickness during winter or summer of
where τ is one half year and L is the latent heat of fusion for sea ice. Following Thorndike , we represent the seasonal cycle as a step function with melting during one half of the year (in which all the shortwave radiation occurs) and freezing during the other half of the year. During the summer melt season, in which the surface temperature is taken to be at the melting temperature (T = 0), we specify the downward longwave radiation to be FLW = FLWs and the shortwave radiation to be FSW = FSWs. In the winter freezing season, FLW = FLWw, FSW = FSWw = 0, and T is found by solving
which is derived by integrating the vertical heat diffusion equation under the quasi-stationary approximation and assuming that T is in steady state with the surface forcing. Setting summer melt −Δh equal to winter growth Δh and solving for h ≡ heq leads to a solution for equilibrium ice thickness,
with W ≡ FLWw − A and S ≡ FLWs − A + (1 − α)FSWs.
 This solution (equation 3) can be compared with equation (31) of Thorndike . The principal distinction is that here we specify FLW based on the GCM output (which implicitly contains the contribution from meridional advection of moist static energy), whereas in Thorndike's model FLW is computed as a function of T (which is itself a function of h) using a gray-body radiative balance atmosphere. We use the parameter values FLWw = 158 Wm−2, FLWs = 272 Wm−2, FSWs = 200 Wm−2, A = 320 Wm−2, B = 4.6 Wm−2 K−1, k = 2 Wm−1 K−1, Fw = 0, and α = 0.65; this produces the observed equilibrium thickness, heq = 2.8 m [Thorndike, 1992]. Note that heq represents the approximate annual mean thickness at equilibrium since it comes from assuming that the typical winter surface temperature is maintained by a flux of heat upward through the ice of kT/heq.
 The second model used in this study is the more physically complete numerical sea ice thermodynamic model of Maykut and Untersteiner , which we run to equilibrium. This single-column model simulates vertical heat diffusion throughout the depth of the snow/sea-ice system with specified surface fluxes based on observations. It produces ice thickness and temperature in good agreement with observations [e.g., Untersteiner, 1961; Wensnahan et al., 2007].