This article provides a comprehensive review of the research progress on atmospheric vertical motion equations, which are crucial for the formation and development of weather systems, particularly small and medium-scale systems. Vertical motion can be calculated by integrating continuity equations that conserve mass, but this method requires the accurate computation of divergence, which is challenging to implement. The adiabatic method for calculating vertical motion is also imprecise, as the underlying assumption of adiabatic internal atmospheric changes does not align with actual thermal variations in the atmosphere. Vertical motion diagnosis is associated with atmospheric scales, and large-scale systems are primarily governed by vortex motion, adhering to quasi-horizontal motion. Such systems often give little consideration to buoyancy and wind shear effects. Consequently, a vertical motion equation that incorporates the first law of thermodynamics, the atmospheric state equation, hydrostatic equilibrium, and quasi-geostrophic conditions is better suited for large-scale motion applications.

The equation for vertical motion in mesoscale systems is more complex owing to the equally significant relationship between convergent or divergent motion and rotational motion. Additionally, vertical motion cannot be disregarded. However, the forcing term still consists of vorticity advection variations with height and the Laplacian of temperature advection, which is fundamentally similar to the ω equation in large-scale systems. In strong small-scale storm motions, convergent and divergent motions are predominant, with buoyancy and wind shear playing critical roles, making the equation for vertical motion even more intricate. Nonetheless, vertical motion forced by the background field persists, as strong small- or medium-scale convective systems cannot occur independently from large- or medium-scale background fields. Consequently, a comprehensive vertical motion analysis should consider the vertical velocity generated by the adjustment of weather systems across different scales. This implies that a combination of vertical motion equations should be used for calculations. The novel vertical motion equation incorporating multiple effects can enhance the precision of diagnostic analysis of vertical velocity in small-scale, strong convective systems.