Peak-temperature sea-ice survival ($h(2\pi \tau )>0$)

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 isLiteral 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 $h_m<2a{H}^2$, 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.