Linear theory of shallow convection in deep, vertically sheared atmospheres

Abstract

The linear theory of vertically sheared convection is extended to deep‐atmosphere flows with arbitrary wind, stability, and diffusion profiles. Consistent with previous findings, reference single‐layer vertical channel flows show fastest growth for shear‐parallel roll circulations and much weaker growth for shear‐transverse circulations, causing the former to dominate. In a more realistic three‐layer setting, where the cloud layer lies between a mixed layer below and a stable free troposphere aloft, shear‐parallel rolls also dominate. However, shear‐transverse rolls grow much faster than before, which degrades the convective organization. An analysis focused on the vertical perturbation phase tilt leads to a novel interpretation of these results. Vertical shear imparts a downshear tilt, which acts to weaken the convective growth driven by dynamic and non‐hydrostatic buoyant vertical perturbation pressure gradients (VPPGs). Whether these VPPGs can maintain growth in the face of the shear depends largely on the Richardson number (Ri), with becoming a necessary condition for (inviscid) growth of shear‐perpendicular rolls in the short‐wave limit. In deeper, three‐layer atmospheres, longer vertical wavelengths are admitted, which fosters less tilted and faster growing perturbations. This effect, however, is partially offset by differential tilting between kinematic and thermal anomalies. Numerical simulations are used to verify the linear results and to explore the evolution of the convection into the nonlinear regime. As the nonlinearities grow, an initial preference for smaller‐scale, shear‐parallel circulations is ultimately overwhelmed by larger‐scale perturbations with no preferred orientation. Thus, the linear findings are most applicable to the early evolution of cloud layers undergoing turbulent transition.