The hydrostatic approximation, used in most large-scale atmospheric models, replaces the full vertical momentum equation with a simple balance between the vertical pressure gradient force and gravity, effectively filtering out the vertical acceleration term. This eliminates high-frequency, small-scale phenomena that fundamentally depend on vertical acceleration, most notably vertically propagating acoustic (sound) waves.
The hydrostatic primitive equations retain the large-scale balanced dynamics relevant to synoptic-scale weather systems (where vertical velocities are indeed roughly an order of magnitude smaller...
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