Adiabatic fluid perturbation ($\Delta s=0,\Delta p=\gamma p\Delta \rho /\rho$)

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.