A07_3eqn_impl_details.tex

    % quadratic
    \begin{equation}
	(\gamma_T c_2) S_b^2 + (\gamma_T(T_b^{n-1}-c_2 S_b^{n-1}-T)-\gamma_S \frac{-L_i}{c_p})S_b + \gamma_S\frac{-L_i}{c_p}S=0
	\end{equation}
	
	% gamma_X
	% Depends on positive melt rate and that the effect of stratification is to reduce fluxes rather than enhance fluxes as a result of buoyancy from sloping surface

	%Asay-Davis Eq. 29-30
	%\begin{equation}
	%	F_T = -c_p(\rho_{sw} \gamma_T + \rho_{fw} m)(T_b-T)
	%\end{equation}
	%$F_T > 0, m > 0, T_b-T > 0$
	
	%\begin{equation}
	%	F_S = -(\rho_{sw} \gamma_S + \rho_{fw} m)(S_b-S)
	%\end{equation}
	%$F_S > 0, m > 0, S_b-S < 0$
	
	%Substituing the melt rate equation, these equations can also be expressed as
	%\begin{equation}
	%F_T = -\rho_{sw} c_p(\gamma_T - \gamma_S \frac{S_b-S}{S_b})(T_b-T)
	%\end{equation}
	%\begin{equation}
	%F_S = -\rho_{sw} (\gamma_S - \gamma_S \frac{S_b-S}{S_b})(S_b-S)
	%\end{equation}

	General Implementation strategy
	\begin{enumerate}
		\item Initialization: 
		a. shf, sasws are assigned (to 0), 
		b. Friction velocity is initialized (to 0?). 
		c. Define buoyancy flux from shf and sasws 
			(need to have alpha and beta at the surface)
		\item Calculate exchange velocities $\gamma_T,\gamma_S$ as described in the previous section based on shf, sasws, us at previous timestep
		\item Use 3 equations to solve for $S_b, T_b$
		\item Iterate if $S_b$ is significantly different from prior $S_b$ used to define $dT_f/dS$.
		\item Calculate heat, salt fluxes from 3 equations
		\item Iterate 2-5 until the change in $S_b$ from the last iteration is less than a tolerance of XX
		% consider calculating momentum fluxes
	\end{enumerate}