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\subsection{Convective Velocities}  We can calculate the accelaration of a convective fluid element  \begin{equation}  \ddot{r} = - g - \frac{1}{\rho} \frac{\partial \P}{\partial r},  \end{equation}  which expanded at first order around the equilibrium state leads to an estimate of the total net force (buoyant minus gravitation) $f=-g \Delta \rho$. Along the distance $\Delta r$ a convective fluid element experience the force $f(\Delta r) = - g \Delta \rho (\Delta r)$. Here the variation of $g$ with height is considered negligible. The work done per unit volume over the distance $\Delta r$ is therefore  \begin{equation}  W (\Delta r) = \int_{0}^{\Delta r} f(x) \D x = - g \int_{0}^{\Delta  r} \Delta \rho (x) \D x = - \frac{1}{2} g \Delta \rho (\Delta r) \Delta r,  \end{equation}