6.5 Ice to Ocean Flux Exchange

This section is only relevant when CAM 3.0 is coupled to a slab ocean. When sea ice is present, only a fraction of the melting potential from heat stored in the ocean actually reaches the ice at the base and side. The melting potential is

$$F_{max} = - h_o \rho_o C_o (T_o-T_f)$$ (6.33)

where $h_o$, $\rho_o$, $C_o$, and $T_o$ are the ocean layer thickness, density, heat capacity, and temperature and $T_f$ is the freezing temperature of the layer (assumed to be -1.8$^o$C).

Usually only a fraction of $F_{max}$ is available to melt ice at the base and side, and these fractions are determined from boundary-layer theories at the ice-ocean interfaces. However, it is critical that the sum of the fractions never exceeds one, otherwise ice formation might become unstable. Hence we compute the upper-limit partitioning of $F_{max}$, even though these amounts are rarely reached. The partitioning assumes $F_{oi}$ is dominated by shortwave radiation and that shortwave radiation absorbed in the ocean surface layer above the mean ice thickness causes side melting and below it causes basal melting:

$$\null \vcenter{\openup\jot \mathsurround=0pt \ialign{\strut\hf... ...+ (1-R)e^{-h/\zeta_2}} \cr f_{sid} = & {1-f_{bot} } \cr \crcr}}$$ (6.34)

where $R = 0.68$, $\zeta_1=1.2 m^{-1}$, $\zeta_2=28 m^{-1}$ [134] and $f_{bot}$ and $f_{sid}$ are the fractions of bottom and side melt flux available, respectively. Thus the maximum fluxes available for melt are $f_{bot} F_{oi}$ and $f_{sid} F_{oi}$. The actual amount used for bottom melting, $F_{BOT}$, is based on boundary layer theory of McPhee [127]:

$$F_{BOT} = max(-\rho_o C_o c_h u^*(T_o - T_f), f_{bot} F_{max})$$ (6.35)

where the empirical drag coefficient $c_h$=0.006 and the skin friction speed $u^* = 1$ cm/s [165].

The heat flux for lateral melt is the product of the vertically-summed, thickness-weighted energy of melting of snow and ice $E_{tot}$ with the interfacial melting rate $M_a$ and the total floe perimeter $p_f$ per unit floe area $A_f$. The interfacial melting rate is taken from the empirical expression of Maykut and Perovich [124] based on Marginal Ice Zone Experiment observations: $M_a = m_1 (T_o - T_f)^{m_2}$, where $m_1=1.6\times10^{-6}$m s$^{-1}$ deg$^{m_2}$ and $m_2=1.36$. The lead-ice perimeter depends on the ice floe distribution and geometry. For a mean floe diameter $d$ and number of floes $n_f$, $p_f = n_f \pi d$ and the floe area $A_f = \eta_{lm} d^2$ [154]. Thus the heat flux for lateral melt is $E_{tot}(p_f/A_f)M_a$, so that the actual amount used is:

$$F_{SID} = max(\frac{E_{tot} \pi}{\eta_{lm} d} m_1 (T_o - T_f)^{m_2}, f_{sid} F_{max})$$ (6.36)

where $\eta_{lm}=0.66$ [154]. Based partially on tuning and partially on the results of floe distribution measurements, the mean floe diameter of $d$=300 m was chosen. The ice area, volume, snow volume, and ice energy are all reduced by side melt in time $\Delta t$ by the fraction $R_{side} = \vert \frac{F_{SID}\Delta t}{E_{tot}} \vert$.

The heat flux that is actually used by the ice model is then:

$$F_{BOT} + F_{SID} \le F_{max}.$$ (6.37)

The net flux exchanged between ocean and ice $F_{oi}$ also includes the shortwave flux transmitted to the ocean through sea ice

$$F_{SWo} = I_{0vs} e^{-\kappa_{vs} h} +I_{0ni} e^{-\kappa_{ni} h}$$ (6.38)

(see Eq. 6.45). Hence

$$F_{oi}=F_{SWo}+F_{BOT} + F_{SID}.$$ (6.39)