Eulerian perturbation of a fluid variable ($\delta q=q_{\mathrm{perturbed}}(x,t)-q_{\mathrm{background}}(x)$)

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.