Wind-shear Effects on Entrainment in a Convective Boundary Layer

The effect of wind shear on entrainment in convective boundary layers (CBL) has often been studied during the last decades by means of atmospheric measurements, laboratory experiments, and numerical simulations. This work has shown that wind shear enhances entrainment, which thickens the entrainment zone and increases the growth rate of the CBL. The main cause for this enhancement is the shear-generated turbulence in the entrainment zone, which implies that local scales in this region become more important in sheared CBLs than in shear-free CBLs, and the boundary-layer depth and the associated convective scales are insufficient to characterize the boundary layer (see [1, 2, 4] and references therein). Nonetheless, we still miss a quantification of how wind-shear effects vary with environmental conditions, which motivates this work.

We use direct numerical simulation (DNS) and dimensional analysis to characterize wind-shear effects on entrainment in a barotropic convective boundary layer that grows into a linearly stratified atmosphere. DNS allows us to remove the uncertainty associated with turbulence models in the entrainment zone. Although the Reynolds numbers that are currently achievable are low compared to the atmospheric values, we start to observe a certain degree of Reynolds number similarity that allows us to extrapolate results, to some extent, to atmospheric conditions. Dimensional analysis allows us to characterize such a sheared CBL by a normalized CBL depth z_enc/L₀, a Froude number, Fr₀≡U₀/(NL₀), and a buoyancy Reynolds number, Re₀≡NL₀²/ν. The first two non-dimensional quantities embed the dependence of the system on time, on the surface buoyancy flux, B₀, and on the wind velocity and the buoyancy stratification in the free atmosphere, U₀ and N², respectively. The encroachment length scale z_enc is a measure of the shear-free CBL depth, and L₀=(B₀/N³)^1/2 is the reference Ozmidov length that characterizes the turbulence in the upper region of the entrainment zone.

We provide scaling laws for the CBL depth, for the entrainment-zone thickness, for the entrainment-flux ratio, and for the momentum flux at the height of the minimum buoyancy flux in terms of time, the surface buoyancy flux, the buoyancy frequency in the free atmosphere, and the velocity increment across the entrainment zone, Δu. We derive a zero-order mixed-layer model using these scaling laws.

We find that the dependence of mixed-layer and entrainment-zone properties on z_enc/L₀ and Fr₀, can be expressed in terms of just one independent variable, the ratio between a shear scale (Δz_i)_s ≡ √3 Δu/N and a convective scale (Δz_i)_c ≡1/4 z_enc. (Δz_i)_s and (Δz_i)_c represent the entrainment-zone thickness in the limits of weak instability (strong wind) and strong instability (weak wind), respectively. The ratio (Δz_i)_s / (Δz_i)_c increases as the wind velocity increases and in this study we consider the range (Δz_i)_s / (Δz_i)_c ≤ 1.5, which corresponds to the weakly-to-strongly unstable conditions -z_enc/L_Ob ≥ 3, where L_Ob is the Obukhov scale. The mean buoyancy and the mean buoyancy flux in the mixed layer follow shear-free scaling laws for all those conditions. For (Δz_i)_s / (Δz_i)_c < 0.35, convection dominates the entrainment-zone dynamics and wind-shear effects on entrainment-zone properties are negligible. Wind-shear effects on entrainment appear when (Δz_i)_s / (Δz_i)_c ≈ 0.35, which corresponds to ≈ 5 ms^-1 for typical midday conditions, and become of order one when (Δz_i)_s / (Δz_i)_c ≈ 0.6.

We rationalize the relevance of the variable (Δz_i)_s / (Δz_i)_c and the transition towards shear-free scaling laws below (Δz_i)_s / (Δz_i)_c ≈ 0.35 by analyzing the two-layer vertical structure of the entrainment zone. In particular, we obtain the entrainment-zone scale Δz_i that characterizes the lower entrainment-zone sublayer. The limit of Δz_i for vanishingly weak wind is (Δz_i)_c, and the limit for strong wind is (Δz_i)_s. The variable (Δz_i)_s can be interpreted as the asymptotic thickness of a stably stratified shear layer that would form in the limit of strong wind.

The reduction of the number of independent variables from two, i.e. z_enc/L₀ and Fr_0, to one, (Δz_i)_s / (Δz_i)_c, can help simplify the parameterization of mixed-layer and entrainment-zone properties in atmospheric models. Such a reduction from two independent variables to one independent variable can also be expressed in terms of the bulk Richardson number Ri_b ≡ N² z_enc²/Δu², but not in terms of the stability parameter -z_enc/L_Ob. Previous analyses have considered similar bulk Richardson numbers as independent variables [1, 3]. One difference in our work with respect to previous work is that, to estimate the negative area of the buoyancy flux, we use the entrainment-zone scale Δz_i instead of the CBL depth as characteristic length scale. For small values of Δu both scales are proportional to each other, and the Taylor expansion of the scaling laws for the minimum buoyancy flux of the sheared CBL compared to the shear-free CBL yields ≈ 1+ 2.8 Ri_b^-1. This result agrees with [1], once their proposed scaling law is rewritten in terms of Ri_b. For larger values of Δu, however, they differ; in particular, the scaling law derived in this work does not suffer from the possible singularity observed in previous parameterizations for large wind velocity.

[1] Conzemius, R. J. and Fedorovich, E. 2006 Dynamics of sheared convective boundary layer entrainment. Part II: Evaluation of bulk model predictions of entrainment flux. J. Atmos. Sci. 63, 1179–1199.

[2] Fedorovich, E. and Conzemius, R. J. 2008 Effects of wind shear on the atmospheric convective boundary layer structure and evolution. Acta Geophysica 56, 114–141.

[3] Pino, D., De Arellano, J. V. G. and Duynkerke, P. J. 2003 The contribution of shear to the evolution of a convective boundary layer. J. Atmos. Sci. 60, 1913–1926.

[4] Pino, D. and De Arellano, J. 2008 Effects of shear in the convective boundary layer: analysis of the turbulent kinetic energy budget. Acta Geophysica 56, 167–193.