Moisture Statistics in Free Convective Boundary Layers

Surface processes and cloud formation critically depend on the moisture field in the planetary boundary layer. Measurement campaigns and numerical simulations have provided considerable insights into moisture properties during the last decades. However, we often lack the parametrizations that allow us to extrapolate those results, quantitatively, to different environmental conditions. In this talk, we present a recent analysis that provides these parametrizations for the equilibrium (quasi-steady) entrainment regime of a free convective boundary layer (CBL) that grows into a linearly stratified atmosphere [1]. This regime is representative of strongly convective conditions over land, e.g., during the late morning and early afternoon period. In particular, we parametrize the first three moments and the mean vertical flux of specific humidity for arbitrary combinations of surface fluxes and free-atmosphere lapse-rates of buoyancy and specific humidity.

The analysis is based on a top-down--bottom-up decomposition. Appropriately combining the buoyancy and top-down scalar, we construct the specific humidity field for arbitrary combinations of surface flux and lapse rate of buoyancy, {B₀, -N²}, and surface flux and lapse rate of specific humidity, {F_q,0, γ_q}. Direct numerical simulations are used to ascertain the coefficients of the scaling laws. The sensitivity of the results to the Reynolds number is ≈5% or less in mixed-layer properties, and ≈20% or less in entrainment-zone properties, despite changes in Reynolds number by a factor of 3. Although the Reynolds numbers that are currently achievable are low compared to the atmospheric values, this degree of Reynolds number similarity allows us to extrapolate results, to some extent, to atmospheric conditions.

We first study the transition between drying and moistening regimes. For typical atmospheric conditions, entrainment tends to dry the CBL whereas evapotranspiration tends to moisten it. These two counteracting processes are characterized by the entrainment moisture flux and the surface moisture flux. When the former dominates, the CBL is in the entrainment-drying regime; when the latter dominates, the CBL is the surface-moistening regime [e.g., 2,3]. In the equilibrium entrainment regime of a CBL, however, the entrainment flux is a priori unknown. We show that the entrainment flux of specific humidity in the pure drying regime is commensurate with (γ_qL₀)(L₀N), where the first factor provides a scale for the variation of specific humidity in the entrainment zone, and the second factor provides a velocity scale. The length scale L₀=(B₀/N³)^1/2 is the reference Ozmidov length, which characterizes the turbulence in the upper sublayer of the entrainment zone, a strongly stratified region that serves as a transition between the CBL and the free atmosphere. The reference flux (γ_qL₀)(L₀N) can be written in terms of the original control parameters as γ_qB₀N^-2. Based on this result, we demonstrate that the flux-ratio parameter φ=2F_q,0/(F_q,0+γ_qB₀N^-2) proves convenient to characterize the statistical properties of the specific humidity: it varies between 0 for the pure drying regime and 2 for the pure moistening regime, and the transition between drying and moistening regimes occurs at φ_cr≈1.15. This condition corresponds to F_q,0≈1.35γ_qB₀N^-2, a relationship that allows us predict the moisture regime of the shear-free CBL for given environmental conditions. Based on these results, we construct a zero-order model for the specific humidity.

We also characterize the specific-humidity variance. Previous work has shown that, for typical atmospheric conditions, the moisture variance peaks in the entrainment zone and is not characterized by the convective scales that characterize the variance in the mixed layer [e.g., 2,4]. We show that, in the mixed layer, the convective scale q_*=(1/2)(F_q,0+γ_qB₀N^-2)/w_*, where w_* is the convective velocity scale, explains more than 80% of the variation of the specific-humidity r.m.s. with φ. The specific-humidity r.m.s. in the entrainment zone is parametrized based on entrainment-zone scales.

Last, we investigate the skewness. In agreement with previous work, we observe that the skewness of q is negative in the mixed layer and becomes positive slightly below the height of minimum mean gradient of q (or maximum r.m.s.). In addition, we show that this behavior is independent of φ, i.e., independent of the surface fluxes and lapse rates in the free atmosphere. In the lower 20% of the CBL depth, however, the skewness depends strongly on both the vertical distance from the surface and the environmental conditions. Between z≈0.1z_enc and z≈0.2z_enc, where z_enc is the encroachment CBL depth, the skewness is negative in half of the moistening regime (φ≥φ_cr≈1.15); below that region, the skewness is positive in half of the drying regime (φ≤φ_cr≈1.15). Hence, using a positive skewness near the surface as an indicator of the moistening regime, as conjectured before based on single-case studies, is not robust.

One application of the expressions provided in this work could be the parametrization of cloud formation. The variance and skewness near the entrainment zone can be used to better model the probability density function of specific humidity, and hence better predict condensation once the positive tail of the density function surpasses the saturation vapor humidity.

[1] J. P. Mellado, M. Puche, and C. C. van Heerwaarden. Moisture statistics in free convective boundary layers growing into linearly stratified atmospheres. Q. J. R. Meteorol. Soc., 2017.
[2] L. Mahrt. Boundary-layer moisture regimes. Boundary-Layer Meteorol., 152:151–176, 1991.
[3] F. Couvreux, F. Guichard, V. Masson, and J.-L. Redelsperger. Negative water vapour skewness and dry tongues in the convective boundary layer: observation and large-eddy simulation budget anaylsis. Boundary-Layer Meteorol., 123:269–294, 2007.
[4] Z. Sorbjan. Statistics of scalar fields in the atmospheric boundary layer based on large-eddy simulations. Part 1: Free convection. Boundary-Layer Meteorol., 116:467–486, 2005.