The Two-layer Structure of the Entrainment Zone in the Convective Boundary Layer

Incomplete understanding of the entrainment zone (EZ), even for the simplest convective boundary layer (CBL), has led to difficulty in modeling the evolution of entrainment rate parameters, as reflected by the disagreement among the different parameterizations proposed in the literature.

We therefore study the entrainment zone of the simplest case, a dry, shear-free convective boundary layer growing into a linearly stratified fluid, within the equilibrium entrainment regime by using direct numerical simulation (DNS). We use DNS in particular to eliminate uncertainty stemming from turbulence models at this relatively thin region where effects of stratification are significant. We show that this approach is justified since most of the statistics of interest already exhibit Reynolds number independence, and for other statistics, Reynolds number effects are at most 15% for a factor of three increase in the reference buoyancy Reynolds number.

Regarding the vertical structure of the entrainment zone, we find that contrary to previous considerations, the entrainment zone is better described by two sub-layers: an upper EZ sub-layer dominated by the penetrating thermals that are directly affected by the stratification N², and a lower EZ sub-layer dominated by the troughs of mixed fluid that are only indirectly affected by the stratification N² through its effect on the CBL thickness. Consequently, the two regions are characterized by different length scales.

For the upper EZ sub-layer, we argued that a local length scale δ, defined as the gradient thickness based on the maximum buoyancy gradient, is the characteristic vertical length, since scaling with it leads to a self-similar behavior of the mean and r.m.s. buoyancy profiles within that part of the entrainment zone, whereas scaling with the CBL thickness z_i does not. Physically, we interpret δ as the mean penetration depth of an overshooting thermal, and such interpretation is supported by the agreement of δ's evolution in time with the prediction from parcel theory, namely, δ is proportional to (w_*/N), where w_* is the convective velocity. We also found that δ is at the same time the integral length scale of the turbulence inside those crest regions, since the viscous dissipation rate ε at the height of maximum gradient z_i,g scales proportionally with [w_rms(z_i,g)]³/δ. Within the lower EZ sub-layer, the characteristic length scale is transitioning from δ to z_i as one approaches the mixed layer. Correspondingly, different buoyancy scales are found, which reflects on the buoyancy fluctuations being a combination of the buoyancy increment associated with the penetrating thermal, and the buoyancy increment associated with the non-thermal regions that mainly retain the original stratification N². Parameterizations for the characteristic scales are provided, which allows for the reconstruction of the vertical profiles of the mean and variance of the buoyancy within the EZ at any time within the equilibrium entrainment regime.

These findings justify the consideration for a second turbulence length scale in turbulence models at the EZ, one that is different from z_i and behaves according to the parcel theory prediction. This multiplicity of scales inside the EZ also explains difficulties found in previous analyses that considered the entrainment zone as a single layer with vertical profiles characterized by a single set of characteristic scales.