Is it correct to think of the Laplacian as the divergence of a gradient field? Factoring out the notation, I see that 
$$\nabla^2(\phi) = \nabla \cdot \nabla(\phi) = \nabla \cdot (\nabla(\phi)) $$
which looks something like the divergence of the gradient of phi.
Is it actually true?
Thanks,

Yes, it is correct. Indeed, let $\phi \colon \mathbb{R}^N \to \mathbb{R}$ be of class $C^2$, then 
\begin{align}
\operatorname{div}(\nabla\phi) := \sum_{i = 1}^N\frac{\partial}{\partial x_i}(\nabla\phi)_i = \sum_{i = 1}^N\frac{\partial^2 \phi}{\partial x_i^2} =: \Delta\phi.
\end{align}
