Atmospheric and laboratory measurements of the vertical profiles of the mean buoyancy and of the root-mean-square (r.m.s.) of the buoyancy fluctuation show deviations from the predictions made according to the classical similarity theory [1, 2]. The reason for these deviations is attributed to the formation of large-scale circulations, or convection cells, identified as flow structures that extend across the whole system and interact with the flow inside the near-surface region. This interaction invalidates the basic assumption made in the classical similarity theory, namely, that near-surface properties are independent of outer-layer variables. Retaining the effect of this interaction improves theoretical predictions of near-surface properties [3, 1].

The relevance of large-scale circulations inside the near-surface region raises the following question: How
do near-surface properties depend on outer-layer features that can modify the large-scale circulations?
In this work, we address this question by investigating the influence of a linear stratification
of the free atmosphere above a convective boundary layer (CBL). We use direct numerical
simulations (DNSs) of a CBL over a smooth, horizontal surface that is forced by a constant
surface buoyancy flux. The CBL grows into fluid with a constant buoyancy gradient, N^{2} ≥ 0.
We compare two cases: a neutral stratification case, N^{2} = 0, which is representative of the
atmospheric regime in which a CBL grows into a residual layer, and a strong stratification
case, N^{2} > 0, which is representative of a CBL in the equilibrium (quasi-steady) entrainment
regime.

The first reason to consider these two regimes is that they enclose any atmospheric CBL growing into a
linearly stratified free atmosphere. With help of dimensional analysis, the cases studied in this work
represent any combination of free atmosphere stratification, surface buoyancy flux, and CBL depth, as long
as the CBL is in one of those two regimes. The second reason to consider these two regimes is that the
large-scale circulations differ from one configuration to the other. Spectral analysis shows that the
characteristic diameter of the large-scale circulations is more than three times larger in the cases with
N^{2} > 0 than in the cases with N^{2} = 0, for a given CBL depth. Thus, by comparing these two
configurations, we can study how different conditions far from the surface affect near-surface properties by
modifying the large-scale circulations.

We have used DNS to remove the uncertainty associated with turbulence models near the surface. The only atmospheric parameter that we cannot match in our simulations is the Reynolds number. However, well-known CBL properties are faithfully reproduced, and the observed dependence on the Reynolds number of the properties studied in this work is negligibly small compared to the dependence on the stratification regime.

We find that, near the surface, the vertical profiles of the mean buoyancy and buoyancy r.m.s.
are approximately independent of N^{2}, despite the strong influence of N^{2} on the large-scale
circulations. As observed in atmospheric measurements and Rayleigh-Bénard convection, these
profiles show deviations from the scaling laws derived according to classical similarity theory:
whereas the mean buoyancy gradient varies as z^{−4∕3} with respect to the distance from the
surface, z, the r.m.s. of the buoyancy fluctuation varies as z^{−0.45} instead of z^{−1∕3}. We do not
observe logarithmic variations, as it has been reported by local analyses inside the convection cells
[4].

However, we also find that the depth over which the previous scaling laws are observed depends on N^{2}:
in the stably stratified case, that depth is ≃ 25% of the CBL depth, which is more than three times the
value found in the neutrally stratified case.

We have defined the concept of the plume-merging layer (PML) to better understand these findings. This layer is conceptually different from the constant-flux (or surface) layer. According to spectral analysis, the flow structure near the surface can be understood as a hierarchy of oblate circulations that have an aspect ratio 5:2 and that remain attached to the surface. The hierarchy is established by smaller plumes merging into larger ones, starting at the surface scales and ending at the large-scale circulations. This flow structure defines the plume-merging layer.

The structure of the plume-merging layer is independent of the stratification regime, but its depth,
h_{PML } , strongly increases with stratification. The exact definition of h_{PML} has been chosen such that it
represents the depth over which the scaling laws presented above are observed. The result is
h_{PML } ≈ 0.07 z_{∗} in the neutral stratification regime and h_{PML} ≈ 0.25z_{∗} in the strong stratification regime;
z_{∗ } is an outer length scale that is commensurate with the CBL depth. The width of the large-scale
circulations is 10h_{PML}, so that it increases from 0.7z_{∗} in the neutral stratification regime to 2.5z_{∗} in the
strong stratification regime.

The implication of our results for atmospheric models is two-fold. First, the buoyancy transfer law
needed in mixed-layer and single-column models corresponds to that predicted by the classical similarity
theory, independently of the stratification in the free atmosphere, even though other near-surface properties
are inconsistent with such a theory, and despite the order-one variation of the width of the large-scale
circulations. Second, for the buoyancy transfer law to apply in the equilibrium (quasi-steady) entrainment
regime of a free convective boundary layer, the model first level can be within 20% − 25% of the CBL
depth, and not necessarily within 10%.

[1] S. S. Zilitinkevich, J. C. R. Hunt, I. N. Esau, A. A. Grachev, D. P. Lalas, E. Akylas, M. Tombrou, C. W. Fairall, H. J. S. Fernando, A. A. Baklanov, and S. M. Joffre. The influence of large convective eddies on the surface-layer turbulence. Q. J. R. Meteorol. Soc., 132:1423–1455, 2006.

[2] F. Chillà and J. Schumacher. New perspectives in turbulent Rayleigh-Bénard convection. Eur. Phys. J. E, 35(58):1–25, 2012.

[3] U. Schumann. Minimum friction velocity and heat transfer in the rough surface layer of a convective boundary layer. Boundary-Layer Meteorol., 44:311–326, 1988.

[4] G. Ahlers, E. Bodenschatz, D. Funfschilling, S. Grossmann, X. He, D. Lohse, R. J. A. M. Stevens, and R. Verzicco. Logarithmic temperature profiles in turbulent Rayleigh-Bénard convection. Phys. Rev. Lett., 109(114501):1–5, 2012.