Theoretical Understanding of the Linear Relationship between Convective Updrafts and Cloud-Base Height for Shallow Cumulus Clouds. Part I: Maritime Conditions

Zheng and Rosenfeld found linear relationships between the convective updrafts and cloud-base height z(b) using ground-based observations over both land and ocean. The empirical relationships allow for a novel satellite remote sensing technique of inferring the cloud-base updrafts and cloud condensation nuclei concentration, both of which are important for understanding aerosol-cloud-climate interactions but have been notoriously difficult to retrieve from space. In Part I of a two-part study, a theoretical framework is established for understanding this empirical relationship over the ocean. Part II deals with continental cumulus clouds. Using the bulk concept of mixed-layer (ML) model for shallow cumulus, I found that this relationship arises from the conservation law of energetics that requires the radiative flux divergence of an ML to balance surface buoyancy flux. Given a certain ML radiative cooling rate per unit mass Q, a deeper ML (higher z(b)) undergoes more radiative cooling and requires stronger surface buoyancy flux to balance it, leading to stronger updrafts. The rate with which the updrafts vary with z(b) is modulated by Q. The cooling rate Q manifests strong resilience to external large-scale forcing that spans a wide range of climatology, allowing the slope of the updrafts-z(b) relationship to remain nearly invariant. This causes the relationship to manifest linearity. The physical mechanism underlying the resilience of Q to large-scale forcing, such as free-tropospheric moisture and sea surface temperature, is investigated through the lens of the radiative transfer theory (two-stream Schwarzschild equations) and an ML model for shallow cumulus.