\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*}






