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Use a quasistatic surface energy balance with negligible heat storage in the surface or ice, a linear ice temperature profile, and a fixed freezing temperature at the ice-ocean interface. These are the assumptions behind the single-category sea-ice thermodynamic model used below. Write $\theta =t/\tau$, $F_O={\lambda }_O(T_O-T_m)>0$ andHere ${\lambda }_A$ and ${\lambda }_O$ are heat transfer coefficients, $k$ is thermal conductivity, $\kappa =k/(\rho c_p)$ is thermal diffusivity, and $\mathcal{S}$ is the inverse Stefan number. Here $L$ is latent heat per unit mass. The thermal $k$ denotes conductivity; the previous questions use $k$ for permeability.

While there is open water, the upward ocean heat supply balances the loss to air: ${\lambda }_O(T_O-T_s)={\lambda }_A(T_s-T_A)$. ThusFreezing begins when this surface reaches $T_m$ on the cooling part of the annual cycle, which requires $r<1$ and givesFor $r\ge 1$ the surface never cools strictly below freezing and no positive ice thickness develops in this model. The limit $r\to 0$ gives $t_1\to \pi \tau /2$.

With ice present and a cold surface, thermal conduction and Newton's law of cooling give $k(T_m-T_s)/h={\lambda }_A(T_s-T_A)$. The surface temperature isDuring surface melting the temperature is capped at $T_s=T_m$; the ice is not permitted to follow the above formula to a temperature higher than its melting point. The upward conductive heat flux during the cold phase isThe Stefan condition at the ocean interface gives $\rho L\dot{h}=F_C-F_O$. Under the permitted approximation of negligible ocean heat during growth,Starting from $h(t_1)=0$, integration yields the implicit growth lawThe leading maximum occurs when the air returns to freezing from below:The equality for $h_m$ is within the ocean-heat-neglected growth model, with the freezing time set by the open-water balance. If $aH\gg 1$, conductive resistance through the ice dominates the atmospheric boundary resistance at maximum thickness. ConsequentlyThis reproduces the displayed large-conductance limit. A sufficient condition for the quasistatic temperature assumption is $h_m^2/\kappa \ll \tau$; large $\mathcal{S}$ ensures this in the thick-ice scaling.

The approximation concerning ocean heat needs care. Its ratio to atmospheric heat removal during the cold phase isIt is uniformly small away from the beginnings and ends of winter providedFor the thick-ice limit this requires both $r\ll 1$ and $raH\ll 1$. This also makes the integrated ocean heat small relative to the latent heat of the maximum ice column. It is not possible to neglect $F_O$ pointwise throughout the closed growth interval: at the actual freezing onset it balances the atmospheric removal, and at the true maximum the growth rate is zero because the two fluxes again balance. The ocean-heat correction to seasonal ice growth therefore gives narrow endpoint corrections. In the full model the maximum is slightly earlier than $3\pi \tau /2$, withThus $t_2=3\pi \tau /2-O(\tau r[1+a{h}_m])$ in the stated approximation, rather than an exact maximum time for finite ocean heat.

For $3\pi \tau /2\le t\le 2\pi \tau$, while ice survives, both the warm air and ocean melt it. The surface is at $T_m$, and the total Stefan condition isIntegrating from the leading maximum givesuntil its first zero. The physical thickness is then zero for the rest of this warm interval and the open-water temperature applies. Therefore at the end of the first year,Positive ice at that time is equivalent, in the specified approximation, toThis establishes the algebraic criterion printed in the paper.

There is, however, an actual qualification to the paper's word “perennial.” This inequality tests peak-temperature sea-ice survival, since $t=2\pi \tau$ is the next atmospheric temperature maximum. Surface melting continues until at least $t=5\pi \tau /2$, so positive thickness at the maximum does not guarantee survival throughout the summer. In the same approximation, survival through that full warm half-cycle would require the stronger inequalityIn fact, starting ice-free under the specified zero-mean sinusoidal forcing, even the zero-ocean-heat upper bound on winter growth cannot meet it. In that limit $t_1=\pi \tau /2$ andsince $h_m^2/2>0$. The expression on the right is already the atmospheric melt over the entire subsequent warm half-cycle. Positive ocean heat can only reduce the winter ice and increase summer melt. Thus the printed condition is a first-year peak-temperature survival test, not a sufficient criterion for literal year-round ice in this model. For example, with $aH=1/2$ and very small positive $r$, it gives $h_m/H\simeq 2(\sqrt{2}-1)\simeq 0.828$, while first-year melt is approximately $0.5H$ but full warm-season melt is approximately $H$. Ice is present at the year's endpoint and disappears later that same summer. No reinterpretation of the printed inequality as guaranteed perennial cover is justified without an additional seasonal assumption.