equation of continuity

(Or continuity equation.) A hydrodynamical equation that expresses the principle of the conservation of mass in a fluid. It equates the increase in mass in a hypothetical fluid volume to the net flow of mass into the volume. The equation of continuity is usually written in either of two forms:

or

where ρ is the fluid density and u the velocity vector. Under the quasi-static approximation with pressure as vertical coordinate, the equation takes the form

where p is the pressure, v is the horizontal velocity, ω = Dp/Dt the vertical p velocity, and ∇_p the del operator in the isobaric surface.