\newcommand{\e}[1]{\cdot 10^{#1}} \subsection{List of Constants and Symbols} \label{locs} \begin{tabular*}{\textwidth}{l@{\extracolsep\fill}lll} Symbol & Definition & Value & Unit \\ \hline \\ $a$ & earth radius & $\rm 6371\e{3}$ & $\rm m$ \\ $A$ & $=D+\vec{V} \cdot \nabla \ln p_s $ & & $\rm -$ \\ ${\cal A}$ & absorptivity/emissivity & & $\rm -$ \\ ${\cal A}_S$ & surface emissivity & &$\rm -$ \\ $B(T)$ & Planck function & &$\rm Wm^{-2}$ \\ $cc$ & cloud cover & & $\rm -$\\ $C_{char}$ & Charnock constant & 0.018 & $\rm -$ \\ $C_h$ & transfer coefficient for heat & & $\rm -$ \\ $C_m$ & drag coefficient for momentum & & $\rm -$ \\ $c_p$ & specific heat of moist air at constant pressure & & $\rm J\, kg^{-1}\, K^{-1}$ \\ $c_{pd}$ & specific heat of dry air at constant pressure & 1005.46 & $\rm J\, kg^{-1}\, K^{-1}$ \\ $c_{pv}$ & specific heat of water vapor at constant pressure & 1869.46 & $\rm J\, kg^{-1}\, K^{-1}$ \\ $c_{p_i}$ & specific heat of sea ice & 2070 & $\rm W\, s\, kg^{-1}\, K^{-1}$ \\ $c_{p_s}$ & specific heat of snow & 2090 & $\rm W\, s\, kg^{-1}\, K^{-1}$ \\ $c_{p_w}$ & specific heat of sea water & 4180 & $\rm W\, s\, kg^{-1}\, K^{-1}$ \\ $c_w$ & coefficient for the deep ocean heat flux & 4 & $\rm W\, m^{-2}\, K^{-1}$ \\ $C_w$ & wetness factor & & $\rm -$ \\ $D$ & scaled divergence & & $\rm -$ \\ $E$ & evaporation & & $\rm m\, s^{-1}$ \\ $E_0$ & extrateristical solar flux density & & $\rm W\, m^{-2}$ \\ $f$ & Coriolis parameter $=:2\Omega\sin\varphi$ & & $\rm s^{-1}$ \\ $F_p$ & tendency of the first moment$=:\frac{d R_{1}}{d t}$ & & $\rm K\, m^2\, s^{-1}$ \\ $F_q$ & tendency of the zeroth moment$=:\frac{d R_{0}}{d t}$ & & $\rm K\, m\, s^{-1}$ \\ $F_q$ & surface moisture flux & & $\rm kg\, m^{-2}\, s^{-1}$ \\ $F_T$ & surface sensible heat flux & & $\rm W\, m^{-2}$ \\ $F_u$ & surface zonal wind stress & & $\rm Pa$ \\ $F_v$ & surface meridional wind stress & & $\rm Pa$ \\ $F^{LW}$ & long wave radiation flux density & &$\rm W m^{-2}$ \\ $F^{SW}$ & short wave radiation flux density & &$\rm W m^{-2}$ \\ $g$ & gravitational acceleration & 9.81 & $\rm m^{-2}$ \\ $h_{mix}$ & mixed layer depth & & $\rm m$ \\ $h_{mix_c}$ & climatological mixed layer depth & & $\rm m$ \\ $H_q$ & effective mixed layer depth $=:\frac{R_{0}}{T_{mix}-T_{ref}}$ & & $\rm m$ \\ $H_p$ & reduced center of gravity $=:\frac{R_{1}}{R_0}$ & & $\rm m$ \\ $J_q$ & vertical turbulent moisture flux & & $\rm kg\, m^{-2}\, s^{-1}$ \\ $J_T$ & vertical turbulent temperature flux & & $\rm K\, m^{-2}\, s^{-1}$ \\ $J_u$ & vertical turbulent flux of zonal momentum & & $\rm Pa$ \\ $J_v$ & vertical turbulent flux of meridional momentum & & $\rm Pa$ \\ $k$ & von Karman constant & 0.4 & $\rm -$ \\ $K_h$ & exchange coefficient for heat & & $\rm -$ \\ $K_m$ & exchange coefficient for momentum & &$\rm -$ \\ $L$ & latent heat & & $\rm J\, kg^{-1}$\\ \end{tabular*} \newpage \begin{tabular*}{\textwidth}{l@{\extracolsep\fill}lll} $L_f$ & latent heat of fusion = $L_s - L_v$ & $\rm 3.28\e{5}$ & $\rm J\, kg^{-1}$ \\ $l_h$ & mixing length for heat & & $\rm m$ \\ $l_m$ & mixing length for momentum & & $\rm m$ \\ $L_s$ & latent heat of sublimation & $\rm 2.8345\e{6}$ & $\rm J\, kg^{-1}$ \\ $L_v$ & latent heat of vapourization & $\rm 2.5008\e{6}$ & $\rm J\, kg^{-1}$ \\ $P_c$ & convective precipitation & &$\rm m s^{-1}$ \\ $P_l$ & large scale precipitation & &$\rm m s^{-1}$ \\ $P^m_n(\mu)$ & associated Legendre function of the first kind & & $\rm -$ \\ $p$ & pressure \hspace*{\fill} & & $\rm Pa$ \\ $p_S$ & surface pressure & & $\rm Pa$ \\ $p_s$ & scaled surface pressure & & $\rm -$ \\ $q$ & specific humidity & & $\rm kg\, kg^{-1}$ \\ $Q$ & total heat flux through sea ice & & $\rm W\, m^{-2}$ \\ $\tilde{Q}$ & flux correction heat flux through sea ice & & $\rm W\, m^{-2}$ \\ $Q_a$ & total atmospheric heat flux & & $\rm W\, m^{-2}$ \\ $Q_c$ & conductive heat flux through sea ice & & $\rm W\, m^{-2}$ \\ $Q_f$ & heat flux available for freezing sea ice & & $\rm W\, m^{-2}$ \\ $Q_g$ & heat flux into the soil & & $\rm W m^{-2}$ \\ $Q_m$ & snow melt heat flux & & $\rm W m^{-2}$ \\ $Q_o$ & oceanic heat flux & & $\rm W\, m^{-2}$ \\ $q_S$ & surface specific humidity & & $\rm kg\, kg^{-1}$ \\ $q_{sat}$ & saturation specific humidity & & $\rm kg\, kg^{-1}$ \\ ${\cal R}$ & refexivity/albedo & &$\rm -$ \\ ${\cal R}_S$ & surface albedo & & $\rm -$ \\ $R_d$ & gas constant for dry air & 287.05 & $\rm J\, kg^{-1}\, K^{-1}$ \\ $R_l$ & surface long wave radiation & & $\rm W\, m^{-2}$ \\ $R_s$ & surface short wave radiation & & $\rm W\, m^{-2}$ \\ $R_v$ & gas constant for water vapor & 461.51 & $\rm J\, kg^{-1}\, K^{-1}$ \\ $R_{0}$ & zeroth moment of the temperature distribution & & $\rm K\, m$ \\ $R_{1}$ & first moment of the temperature distribution & & $\rm K\, m^2$ \\ $Ri$ & Richardson number & & $\rm -$ \\ $S_w$ & salinity of sea water & 34.7 & $\rm psu$ \\ $t$ & time & & $\rm s$ \\ $t$ & scaled time step & & $\rm -$ \\ ${\cal T}$ & transmissivity & &$\rm -$ \\ $T$ & temperature & & $\rm K$ \\ $T'$ & temperature anomaly $=:T-T_0$ & & $\rm -$ \\ $T_d$ & deep ocean temperature (at 400m) & & $\rm K$ \\ $T_i$ & sea ice surface temperature & & $\rm K$ \\ $T_f$ & freezing temperature & 271.25 & $\rm K$ \\ $T_s$ & surface temperature & & $\rm K$ \\ $T_{sea}$ & sea surface temperature & & $\rm K$ \\ $T_{melt}$ & melting point & 273.16 & $\rm K$ \\ $T_{mix}$ & mixed layer temperature & & $\rm K$ \\ $T_{mix_c}$ & climatological mixed layer temperature & & $\rm K$ \\ $T_{ref}$ & asymptotic reference temperature & & $\rm K$ \\ \end{tabular*} \newpage \begin{tabular*}{\textwidth}{l@{\extracolsep\fill}lll} $T_w$ & oceanic temperature profile & & $\rm K$ \\ $T_0$ & reference temperature profile & 250.0 & $\rm K$ \\ $U$ & scaled zonal wind $=:u\cdot\cos\varphi$ & & $\rm -$ \\ $u$ & zonal wind & & $\rm m\, s^{-1}$ \\ $u_*$ & friction velocity & & $\rm m\, s^{-1}$ \\ $V$ & scaled meridional wind $=:v\cdot\cos\varphi$ & & $\rm -$ \\ $v$ & meridional wind & & $\rm m\, s^{-1}$ \\ $\vec{v}$ & horizontal wind vector & & $\rm m\, s^{-1}$ \\ $W_L$ & cloud liquid water path & & $\rm g m^2$ \\ $W_{snow}$ & mass of snow water & & $\rm kg $\\ $W_{soil}$ & soil water & & $\rm m $\\ $z$ & height & & $\rm m$ \\ $z_0$ & roughness length & & $\rm m$ \\ $\Delta t $ & time increment & & $\rm s$ \\ $\Delta z $ & height increment & &$\rm m$ \\ $\alpha$ & thermal expansion coefficient $\frac{1}{\rho} \frac{d\rho}{dT}$ & $\rm 2.41\e{-4}$ & $\rm K^{-1}$ \\ $\beta$ & back scattering coefficient & &$\rm - $\\ $\beta_d$ & diffusivity factor & 1.66 &$\rm - $ \\ $\zeta$ & scaled vorticity & & $\rm -$ \\ $\theta$ & potential temperature & & $\rm K$ \\ $\kappa$ & $R_d/C_{pd}$ & & $\rm -$ \\ $\bar{\kappa}$ & mean heat conductivity in ice and snow & & $\rm W\, m^{-1}\, K^{-1}$ \\ $\kappa_i$ & heat conductivity in ice & 2.03 & $\rm W\, m^{-1}\, K^{-1}$ \\ $\kappa_s$ & heat conductivity in snow & 0.31 & $\rm W\, m^{-1}\, K^{-1}$ \\ $\lambda_h$ & asymptotic mixing length for heat & &$\rm m $\\ $\lambda_m$ & asymptotic mixing length for momentum & &$\rm m $\\ $\lambda$ & longitude & & $\rm -$ \\ $\mu$ & $\sin\varphi$ & & $\rm -$ \\ $\mu_0$ & cosine of the solar zenith angle & & $\rm -$ \\ $\rho$ & density of air & & $\rm kg\, m^{-3}$ \\ $\rho_i$ & density of sea ice & 920 & $\rm kg\, m^{-3}$ \\ $\rho_s$ & density of snow & 330 & $\rm kg\, m^{-3}$ \\ $\rho_w$ & density of sea water & 1030 & $\rm kg\, m^{-3}$ \\ $\rho_o$ & density of fresh water & 1000 & $\rm kg\, m^{-3}$ \\ $\sigma$ & normalized pressure coordinate $=: p/p_s$ & & $\rm -$ \\ $\dot{\sigma}$ & vertical velocity in $\sigma$ system & & $\rm -$ \\ $\sigma_{SB}$ & Stefan-Bolzmann constant & $ 5.67 \e{-8}$ &$\rm W m^{-2}K^{-4}$ \\ $\tau_N$ & cloud optical depth & &$\rm -$\\ $\tau_{F}$ & time scale for RF & & $\rm -$ \\ $\tau_{R}$ & time scale for NC & & $\rm -$ \\ $\tau_{T}$ & time scale for temperature flux correction & & $\rm s$ \\ $\tau_{h}$ & time scale for depth flux correction & & $\rm s$ \\ $\phi$ & geopotential height $:=g\cdot{z}$ & & $\rm m^2\, s^{-2}$ \\ $\phi$ & scaled geopotential height & & $\rm -$ \\ $\varphi$ & latitude & & $\rm -$ \\ $\chi$ & scaled velocity potential & & $\rm -$ \\ \end{tabular*} \newpage \begin{tabular*}{\textwidth}{l@{\extracolsep\fill}lll} $\psi$ & scaled streamfunction & & $\rm -$ \\ $\Omega$ & angular velocity of the earth & $\rm 7.292\e{-5}$ & $\rm s^{-1}$ \\ $\tilde{\omega_0}$ & single scattering albedo & &$\rm- $\\ \end{tabular*}