= Peak-temperature sea-ice survival
{title2=$h(2\pi\tau)>0$}
In a cycle starting at the atmospheric temperature maximum, positive thickness at the next maximum is a peak-temperature survival test. It is not sufficient for perennial ice: with sinusoidal forcing, surface melting continues for another quarter-cycle. If ocean heat was neglected during winter growth, the leading thickness at the next temperature maximum is
$$
\max\left\{0,h_m-\frac{\lambda_A\Delta T\tau}{\rho L}\left[1+\frac\pi2\frac{\lambda_O(T_O-T_m)}{\lambda_A\Delta T}\right]\right\}.
$$
Literal year-round survival must be tested through the end of the subsequent melt season, rather than at the temperature maximum alone. For the specified forcing with annual mean atmospheric temperature $T_m$, the zero-ocean-heat upper bound on winter growth obeys $h_m^2/2+h_m/a=2H^2$, where $a=\lambda_A/k$ and $H^2=\kappa\tau/\mathcal S$, with $\mathcal S$ the <inverse Stefan number>. Thus $h_m<2aH^2$, already less than the atmospheric melt over the full subsequent warm half-cycle. Positive ocean heat only strengthens the failure to produce literal perennial ice from an ice-free start.
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