Anelastic approximation ($\nabla \cdot ({\rho }_0\mathbf{u})=0$)

An anelastic model filters acoustic density fluctuations while retaining a prescribed stratified reference mass density. For ${\rho }_0\propto e^{-z/H}$, continuity is $u_x+v_y+w_z-w/H=0$. Momentum and buoyancy conventions must be consistent: with reduced pressure $\Phi =p^{\prime }/{\rho }_0$, a pressure acceleration $-\nabla \Phi$ contains a vertical density-gradient term that is absent from $-{\rho }_0^{-1}\nabla p^{\prime }$. The two expressions cannot be interchanged when deriving waves in a stratified atmosphere.