Vertical Velocity, Vertical Wind Shear, and Pressure Tendency in a Hydrostatic Atmosphere -- Potential Applications to Weather Analysis, Forecasting, and Numerical Modeling

A new mathematical relationship on the kinematic vertical velocity has been derived by combining the hydrostatic equation, the first law of thermodynamics, and the continuity equation. When the pressure tendency term is eliminated from this combined equation, a second order differential equation involving the vertical velocity may be obtained. This equation relates the vertical velocity with the horizontal divergence of the wind, the vertical wind shear, and a term involving the divergence of the vertical wind shear in addition to a couple of diabatic heating terms.

If, on the other hand, the vertical velocity is eliminated from the combined equation, a differential equation of the local pressure tendency is obtained. This equation reveals the combined effects of temperature advection, the kinematic divergence, as well as diabatic heating that are modulated by the static stability.

The derivation of these mathematical relationships will be presented along with physical interpretations of the terms involved. Potential applications to weather analysis, forecasting, and numerical modeling based on this new formulation of the vertical velocity will also be discussed.