Eulerian and Lagrangian fluid perturbations

For a background scalar field $f$ and fluid displacement $\mathbf{\xi }$, the Lagrangian perturbation follows the displaced fluid element while the Eulerian perturbation compares fields at one fixed position. To first order,

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 ${\mathbf{v}}_1={\partial }_t\mathbf{\xi }$.

Moving a uniform-density boundary produces a delta-function sheet in the Eulerian mass density perturbation. For a slab displaced vertically by ${\eta }_{\pm }$, ${\rho }^{\prime }={\rho }_0{\eta }_{+}\delta (z-H)-{\rho }_0{\eta }_{-}\delta (z+H)$. Poisson equation then gives a continuous gravitational potential with derivative jumps fixed by those sheets.

The Lagrangian pressure perturbation is the change in pressure experienced by a displaced fluid element: $\Delta p=\delta p+\mathbf{\xi }\cdot \nabla p_0$. For a free surface against fixed external pressure, $\Delta p=0$, rather than $\delta p=0$ at the original surface.