Frazil export enthalpy balance ($[eq{]}^{\prime }=-f-R{e}_s$)

Let $q$ be the depth-integrated volume flux, $E$ the entrainment rate, $R$ the upward export of pure ice, and $e=c_p(T-T_{\infty })-L\varphi$. Then $q^{\prime }=E-R$, $[\overline{C}q{]}^{\prime }=C_{\infty }E$ and $[eq{]}^{\prime }=-f-R[c_p(T-T_{\infty })-L]$. Export carries negative relative enthalpy, so it is not a heat-loss term with the same sign as atmospheric cooling. Total production is the increase in advected ice plus its exported amount. When both retained ice fraction and total processed fraction are small, equilibrium salinity and conservation of energy give production $f/(L+c_{p}m{C}_{\infty })$ even when retained and exported fractions differ.