Weak Pressure Gradient Approximation and Its Analytical Solutions

Abstract

Abstract A weak pressure gradient (WPG) approximation is introduced for parameterizing supradomain-scale (SDS) dynamics, and this method is compared to the relaxed form of the weak temperature gradient (WTG) approximation in the context of 3D, linearized, damped, Boussinesq equations. It is found that neither method is able to capture the two different time scales present in the full 3D equations. Nevertheless, WPG is argued to have several advantages over WTG. First, WPG correctly predicts the magnitude of the steady-state buoyancy anomalies generated by an applied heating, but WTG underestimates these buoyancy anomalies. It is conjectured that this underestimation may short-circuit the natural feedbacks between convective mass fluxes and local temperature anomalies. Second, WPG correctly predicts the adiabatic lifting of air below an initial buoyancy perturbation; WTG is unable to capture this nonlocal effect. It is hypothesized that this may be relevant to moist convection, where adiabatic lifting can reduce convective inhibition. Third, WPG agrees with the full 3D equations on the counterintuitive fact that an isolated heating applied to a column of Boussinesq fluid leads to a steady ascent with zero column-integrated buoyancy. This falsifies the premise of the relaxed form of WTG, which assumes that vertical velocity is proportional to buoyancy.