From AMS Glossary
The form assumed by the vertical component of the vector equation of motion when all Coriolis, earth curvature, frictional, and vertical acceleration terms are considered negligible compared with those involving the vertical pressure force and the force of gravity.
where p is the pressure, ρ the density, g the acceleration of gravity, and z the geometric height. For cyclonic-scale motions the error committed in applying the hydrostatic equation to the atmosphere is less than 0.01%. Strong vertical accelerations in thunderstorms and mountain waves may be 1% of gravity or more in extreme situations.