This study investigates relationships between vertical velocity, perturbation pressure, updraft size, and dimensionality for cumulus convection. Generalized theoretical expressions are derived from approximate analytic solutions of the governing momentum and mass continuity equations for both two-dimensional (2D) and axisymmetric quasi-three-dimensional (3D) steady-state updrafts. These expressions relate perturbation pressure and vertical velocity to updraft radius R, height H, and thermal buoyancy. They suggest that the vertical velocity at the level of neutral buoyancy is reduced from perturbation pressure effects by factors of and in 2D and 3D, respectively, where is a nondimensional length, with somewhat different scalings lower in the updraft (α is a parameter equal to the ratio of vertical velocity horizontally averaged across the updraft to that at the updraft center). They also indicate that updrafts are weaker in 2D than 3D, all else being equal, with a difference of up to a factor of 2 in vertical velocity for as a direct result of differences in mass continuity between 2D and axisymmetric 3D flow. Differences between these expressions and other analytic solutions, including those derived from single normal mode Fourier/Fourier–Bessel expansion of the buoyant perturbation pressure Poisson equation, are discussed. Part II of this study compares the theoretical expressions with numerical solutions of the buoyant perturbation pressure Poisson equation for a wide range of thermal buoyancy profiles representing shallow-to-deep moist convection and also with fully dynamical 2D and 3D updraft simulations.
