Lagrangian displacement (fluid mechanics, $\mathbf{\xi }$)

A Lagrangian displacement maps each equilibrium fluid element to its perturbed position. Its time derivative is the velocity perturbation when the equilibrium is static.

The Eulerian perturbation compares field values at the same spatial location, rather than following a fluid element. To first order it satisfies $\delta q=\Delta q-\xi \cdot \nabla q$. Thus a field advected unchanged by individual elements can still have a nonzero Eulerian perturbation in an inhomogeneous background.

The Lagrangian perturbation compares a variable on the same moving fluid element before and after a displacement. Taylor expansion of the background value at the displaced position gives the displayed relation to an Eulerian perturbation. For an adiabatic ideal gas, $\Delta (p/{\rho }^{\gamma })=0$ even when the equilibrium entropy varies spatially.

An adiabatic fluid perturbation preserves the specific entropy of each displaced parcel. For a perfect gas, mass conservation gives $\Delta \rho =-\rho \nabla \cdot \mathbf{\xi }$ and the adiabatic equation of state gives $\Delta p=\gamma p\Delta \rho /\rho$. The corresponding Eulerian perturbation of a fluid variable is $\delta p=-\gamma p\nabla \cdot \mathbf{\xi }-\mathbf{\xi }\cdot \nabla p$. An isothermal equilibrium may still have adiabatic perturbations, because the thermodynamic constraint on displaced parcels differs from the equilibrium temperature profile.