1. Xin Xi Feb. 28

2. Basics  Convective entrainment : The buoyant thermals from the surface layer rise through the mixed layer, and penetrate (with enough momentum) across the inversion interface and into the warmer/ drier free atmosphere (more buoyant air). This overshooting causes the mixing of non-turbulent air into the turbulent region and contributes to the growth of mixed layer.  In CBL, buoyant convection can be induced by the surface heating during the daytime (positive buoyancy) and cloud-top radiative cooling (negative buoyancy).  Convective entrainment of dry air can contribute to additional evaporative cooling in clouds, which enhances the negative buoyancy.

3.  In this paper, no wind sheer (buoyancy is the source for turbulence); no clouds (buoyancy is caused only by surface heating)  Question to address is: how are the integral parameters of convective entrainment depend on the capping inversion strength and the static stability (stratification) in the free atmosphere?  Those integral or bulk parameters include entrainment rate (growth rate of mixed layer), entrainment ratio (the ratio of entrainment buoyancy flux to the surface buoyancy flux) and the relative entrainment layer depth (the ratio of the entrainment layer depth to that of CBL)

4. Bulk or slab models 1. zero-order jump model  The mean variables (wind and virtual potential temperature) are assumed constant with height in the lower layer, which incorporates the mixed layer and surface layer. (The surface layer is comparatively thin)  The inversion or interfacial layer is assumed infinitesimally thin, so a discontinuity occurs (at Zi).  Buoyancy flux decreases with height to a minimum at the discontinuity and vanishes above Geostrophic wind in FA Linearly stratified FA

5. The buoyancy budget equation: Given the buoyancy flux at the surface Bs and the buoyancy frequency N above the CBL, there are two unknowns: the zero- order buoyancy increment and the CBL depth Zi.

6. So another equation, the entrainment equation is needed to solve the problem. It assumes quasi-stationary or equilibrium entrainment, in which case the integral TKE production by the buoyancy forces is balanced by the dissipation and the entrainment at the mixed layer top. is the entrainment ratio.

7. Using N and Bs as scaling parameters gives two dimensionless equations: Analytic solutions are: Other parameters of entrainment:

8. 2. General-structure model (first-order jump model)  It’s based on a self-similar representation of the buoyancy profile within the entrainment zone, which has a finite non-zero thickness, and thus more realistic.  Upper and lower bounds are the zero-crossing height of B (Ziu) and the height where B vanishes after reaching the minimum (Zil). Geostrophic wind in FA Linearly stratified FA

9. The general-structure model contains three unknowns, which have to be solved numerically: :net increase of buoyancy B over the finite depth of inversion layer; larger than the counterpart in ZOM model. Zil :the inversion layer depth :the height of the lower bound of inversion layer, also the zero-crossing height of B

10. LES calculation The integral or bulk model quantities (CBL height, inversion layer depth and buoyancy increment at the inversion) are derived from the LES dataset based on the employed model framework, e.g., zero-order jump or general-structure models.

11. Results After the model spin-up time ( >100), the computed dimensionless CBL height for five different N is related to the dimensionless time by a 1/2 power law, which is in accordance with the analytic solution. ZOM

12. These results support the validity of the equilibrium entrainment assumption and that the entrainment ratio (C1=0.17) is constant and universal.

13.  Time evolution of Ziu has larger scatter due to the weaker criterion (heat flux changes sign or is a small portion of its value at Zi) of determining Ziu from LES data, compared to Zi and Zil.  Zi and Zil are represented very well by the ½ power law after t>100 (quasi-stationary stage).  This result show that the approximation of the inversion layer depth as 2*(Zi-Zil) is not accurate, since it actually changes with time. GSM

15.  At sufficiently large t, or Reynolds number, the calculated entrainment rate E from GSM is independent on the buoyancy frequency N.  LES results in good agreement with lab measurement.

16.  Relative entrainment layer depth decreases more sharper with growing than with other two.

17.  When the Reynolds number increases, the calculated relative entrainment layer depth from GSM and LES is larger than the lab measurement value.  It indicates that the parameterization in GSM of the energy drain due to gravity waves has to be revised.

18.  Relationships predicted by all data (GSM, LES; Lab measurements) are well within the power-exponent range in Nelson etal (1989): 1/4 ~ 1.