Using the following code I got no overfull hbox error.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\frac{\partial M_u}{\partial z} = E_u - D_u \\
\frac{\partial (M_u s_u)}{\partial z} = E_u \overline{s} - D_u s_u + L_v
\overline{\rho} c_u \\
\frac{\partial (M_u q_u)}{\partial z} = E_u \overline{q} - D_u q_u -
\overline{\rho} c_u \\
\frac{\partial (M_u l)}{\partial z} = - D_u l + \overline{\rho} c_u -
\overline{\rho} G_p \\
\end{equation}
\end{document}
Maybe the problem is because your defined margins are shorter with respect to the default margins in the minimal document class.
As Au101 said, you should not use \\ on a equation environment. If you want the equations to be in different lines and numbered, you could use the align (or align* to appear without numbers) environment:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align}
\frac{\partial M_u}{\partial z}
&= E_u - D_u \\
\frac{\partial (M_u s_u)}{\partial z}
&= E_u \overline{s} - D_u s_u + L_v \overline{\rho} c_u \\
\frac{\partial (M_u q_u)}{\partial z}
&= E_u \overline{q} - D_u q_u - \overline{\rho} c_u \\
\frac{\partial (M_u l)}{\partial z}
&= - D_u l + \overline{\rho} c_u - \overline{\rho} G_p
\end{align}
\end{document}
\\within anequationenvironment: http://tex.stackexchange.com/questions/8936/how-to-break-a-long-equation – Au101 Mar 20 '17 at 00:09