Analytical solution to a thermodynamic model for the sensitivity of afternoon deep convective initiation to the surface Bowen ratio

The tendency of convective rainfall to initiate over a wetter or drier land surface is a critical feedback process in the climate system, influencing the hydrological cycle on a variety of spatial scales, especially in parts of the world where water is limited. A simple algebraic solution is derived from fundamental physical equations, to predict the sign of this convective rainfall feedback with the surface. The tendency for convection to occur is evaluated by the rate at which the convective boundary‐layer top approaches the level of free convection. Well‐known integral models predict the rate of ascent of the boundary‐layer top, which tends to be faster over a dry surface. The associated changes in equivalent potential temperature in the boundary layer determine the rate at which the level of free convection descends, typically faster over a wet surface, as a function of the ambient profile, the thermodynamic forcing and the surface Bowen ratio. The resulting system is controlled by three parameters. Two nondimensional parameters determine whether there is wet or dry “advantage”; the Bowen ratio at the boundary‐layer top and a “convective instability parameter,” defined as the ratio of the vertical gradient of saturated equivalent potential temperature at the level of free convection to the profile stability just above the boundary layer. A dimensional function, dependent on the surface fluxes, the boundary‐layer depth, and the profile stability, provides the magnitude of the response. In comparison with previous work, the solution is both rigorously derived from physical principles and encapsulated in a simple algebraic form. A first evaluation of the theoretical framework has been made using data from a convection‐permitting numerical model simulation over India, and this indicates that the equations successfully determine the conditions under which convection is triggered over dry surfaces. We analyse the evolution of a convective boundary layer and the conditions for initiation of deep convection. Considering the rate at which the boundary‐layer top and the level of free convection converge, we find an equation determining whether deep convection will occur preferentially over a wet or a dry surface. The nondimensional inversion Bowen ratio and a “convective instability parameter” determine whether there is wet or dry “advantage,” but a third, dimensional parameter gives the amplitude of the advantage.