Ocean-heat correction to seasonal ice growth ($F_O={\lambda }_O(T_O-T_m)$)

For $T_A=T_m+\Delta T\cos (t/\tau )$ and $F_O={\lambda }_O(T_O-T_m)$, put $r=F_O/({\lambda }_A\Delta T)$. Open-water freezing requires $r<1$ and starts at $t_1=\tau \arccos (-r)$. Neglecting ocean heat during growth is a bulk approximation when $r(1+{\lambda }_A{h}_m/k)\ll 1$. It cannot hold pointwise at the onset or maximum, where the atmospheric removal and ocean input are comparable. The actual maximum satisfies $-\cos (t_2/\tau )=r[1+{\lambda }_{A}h(t_2)/k]$; neglecting $F_O$ gives $t_2=3\pi \tau /2$.