Direct numerical simulations are used to characterize wind-shear effects on entrainment in a barotropic convective boundary layer (CBL) that grows into a linearly stratified atmosphere. We consider weakly to strongly unstable conditions $-z_{enc}/L_{Ob}\gtrsim 4$, where $z_{enc}$ is the encroachment CBL depth and $L_{Ob}$ is the Obukhov length. Dimensional analysis allows us to characterize such a sheared CBL by a normalized CBL depth, a Froude number and a Reynolds number. The first two non-dimensional quantities embed the dependence of the system on time, on the surface buoyancy flux, and on the buoyancy stratification and wind velocity in the free atmosphere. We show that the dependence of entrainment-zone properties on these two non-dimensional quantities can be expressed in terms of just one independent variable, the ratio between a shear scale $(\unicode[STIX]{x0394}z_{i})_{s}\equiv \sqrt{1/3}\unicode[STIX]{x0394}u/N_{0}$ and a convective scale $(\unicode[STIX]{x0394}z_{i})_{c}\equiv 0.25z_{enc}$, where $\unicode[STIX]{x0394}u$ is the velocity increment across the entrainment zone, and $N_{0}$ is the buoyancy frequency of the free atmosphere. Here $(\unicode[STIX]{x0394}z_{i})_{s}$ and $(\unicode[STIX]{x0394}z_{i})_{c}$ represent the entrainment-zone thickness in the limits of weak convective instability (strong wind) and strong convective instability (weak wind), respectively. We derive scaling laws for the CBL depth, the entrainment-zone thickness, the mean entrainment velocity and the entrainment-flux ratio as functions of $(\unicode[STIX]{x0394}z_{i})_{s}/(\unicode[STIX]{x0394}z_{i})_{c}$. These scaling laws can also be expressed as functions of only a Richardson number $(N_{0}z_{enc}/\unicode[STIX]{x0394}u)^{2}$, but not in terms of only the stability parameter $-z_{enc}/L_{Ob}$.

No abstract

No abstract

Two zero-order bulk models (ZOMs) are developed for the velocity, buoyancy, and moisture of a cloud-free barotropic convective boundary layer (CBL) that grows into a linearly stratified atmosphere. The models differ in the entrainment closure assumption: in the first one, termed the “energetics-based model,” the negative and positive areas of the buoyancy flux are assumed to match between the model and the actual CBL; in the second one, termed the “geometric-based model,” the modeled CBL depth is assumed to match different definitions of the actual CBL depth. Parameterizations for these properties derived from direct numerical simulation (DNS) are employed as entrainment closure equations. These parameterizations, and hence the resulting models, are free from the potential singularity at finite wind strength that has been a major limitation in previous bulk models. The proposed ZOMs are verified using the DNS data. Model results show that the CBL depths obtained from the energetics-based model and previous ZOMs correspond to the height that marks the transition from the lower to the upper entrainment-zone sublayer; this reference height is few hundred meters above the height of the minimum buoyancy flux. It is also argued that ZOMs, despite their simplicity compared to higher-order models, can accurately represent CBL bulk properties when the relevant features of the actual entrainment zone are considered in the entrainment closures. The vertical structure of the actual entrainment zone, if required, can be constructed a posteriori using the available relationships between the predicted zero-order CBL depth and various definitions of the actual CBL depth.