Theoretical expressions for the ascent rate of moist deep convective thermals

An approximate analytic expression is derived for the ratio lambda of the ascent rate of moist deep convective thermals and the maximum vertical velocity within them; lambda is characterized as a function of two non-dimensional buoyancy-dependent parameters y and h and is used to express the thermal ascent rate as a function of the buoyancy field. The parameter y characterizes the vertical distribution of buoyancy within the thermal, and h is the ratio of the vertically integrated buoyancy from the surface to the thermal top and the vertical integral of buoyancy within the thermal. Theoretical l values are calculated using values of y and h obtained from idealized numerical simulations of ascending moist updrafts and compared to lambda computed directly from the simulations. The theoretical values of lambda approximate to 0.4-0.8 are in reasonable agreement with the simulated lambda (correlation coefficient of 0.86). These values are notably larger than the lambda = 0.4 from Hill's (nonbuoyant) analytic spherical vortex, which has been used previously as a framework for understanding the dynamics of moist convective thermals. The relatively large values of l are a result of net positive buoyancy within the upper part of thermals that opposes the downward-directed dynamic pressure gradient force below the thermal top. These results suggest that nonzero buoyancy within moist convective thermals, relative to their environment, fundamentally alters the relationship between the maximum vertical velocity and the thermal-top ascent rate compared to nonbuoyant vortices. Implications for convection parameterizations and interpretation of the forces contributing to thermal drag are discussed.