What does $\nabla\left(\nabla \phi_{1}\right)$ mean?

Question

I found this notation in a paper: $\nabla\left(\nabla \phi_{1}\right)$ where $\phi_{1}$ is a scalar. I can understand $\nabla \times \left(\nabla \phi_{1}\right)$ or $\nabla \cdot \left(\nabla \phi_{1}\right)$. But what manipulation is this?

Answer 1

The problem term is $$ (\xi_1 + \alpha_1\times\mathbf{X})\cdot\nabla(\nabla \phi_1). $$ You're parsing it as a vector dotted with a..something: $$ (\xi_1 + \alpha_1\times\mathbf{X})\cdot[\nabla(\nabla \phi_1)], $$ but it's actually the material derivative in the direction $\xi_1+\alpha_1\times\mathbf{X}$ of the vector field $\nabla\phi_1$: $$ [(\xi_1 + \alpha_1\times\mathbf{X})\cdot\nabla](\nabla \phi_1). $$ That said, you will probably come across the notation $\nabla\nabla\phi_1$ to mean the rank-2 tensor of second derivatives of $\phi_1$, also known as the Hessian. In this case it works out that $$ [(\xi_1 + \alpha_1\times\mathbf{X})\cdot\nabla](\nabla \phi_1)= (\xi_1 + \alpha_1\times\mathbf{X})\cdot(\nabla\nabla \phi_1). $$ However, be careful when authors use this type of notation. For a general vector $\mathbf{x}$ and tensor $\mathbf{T}$, $\mathbf{x}\cdot \mathbf{T} = x_iT_{ij}\ne x_iT_{ji} = \mathbf{T}\cdot\mathbf{x}$, so you should be wary of which tensor contraction is intended. This gets even worse for higher-rank constructions like $\nabla \mathbf{T}$ or $\nabla\nabla\nabla \phi_1$, both of which I have encountered in plasma dynamics.