Eulerian fluid perturbation ($f_1=\Delta f-\mathbf{\xi }\cdot \nabla f$)

An Eulerian fluid perturbation compares a perturbed field with its equilibrium value at one fixed spatial point. It differs from the perturbation following a displaced material element: $\Delta f=f_1+\mathbf{\xi }\cdot \nabla f$. Linearizing a static fluid velocity gives ${\mathbf{v}}_1={\partial }_t\mathbf{\xi }$.