Logarithmic Mean Temperature Profiles and Their Connection to Plume Emissions in Turbulent Rayleigh-Bénard Convection

Two-dimensional simulations of Rayleigh-Bénard convection at $\mathrm{Ra}=5\times 10^{10}$ show that vertical logarithmic mean temperature profiles can be observed in regions of the boundary layer where thermal plumes are emitted. The profile is logarithmic only in these regions and not in the rest of the boundary layer where it is sheared by the large-scale wind and impacted by plumes. In addition, the logarithmic behavior is not visible in the horizontal average. The findings reveal that the temperature profiles are strongly connected to thermal plume emission, and they support a perception that parts of the boundary layer can be turbulent while others are not. The transition to the ultimate regime, in which the boundary layers are considered to be fully turbulent, can therefore be understood as a gradual increase in the fraction of the plume-emitting (“turbulent”) regions of the boundary layer.

Figure 1

(a) Instantaneous and (b) time-averaged temperature fields. A corresponding movie can be found in the Supplemental Material [38]. Examples of the three different regions are indicated in the bottom panel. In the panels, red and blue indicate hot and cold fluid, respectively. (c) The Richardson number Ri, the log amplitude $-A$, the fit residual $R$, and the log quality $P$ as a function of the horizontal coordinate $x$. The plotted quantities are normalized by their maxima. It can be seen that there is a strong correlation between plume-ejecting regions and logarithmic mean-temperature profiles.

Figure 2

Mean-temperature profiles for three distinct boundary-layer regions, horizontally averaged. The black dashed line is a fit to the logarithmic part of the profile. There is a logarithmic-like profile visible in the ejecting mean-temperature profile. The region is approximately one decade in width, namely, for $0.02\le z\le 0.2$. A zoom can be seen in the inset.

Figure 3

(a) Logarithmic diagnostic function $\mathrm{\Xi }$ as a function of $z$ for horizontal regions where $P$ is smaller and larger than 0.4. This function is constant if $⟨\theta ⟩$ is logarithmic in $z$. This is the case in the one-decade log regime in the range $0.02\le z\le 0.2$, delimited by the dashed black lines. The blue dashed line indicates the logarithmic fit from Fig. 2. (b) $⟨{u_z}^{\prime }{\theta }^{\prime }{⟩}_t$ as a function of $z$ for horizontal regions where $P$ is smaller and larger than 0.4. The statistics used in these plots are extracted from both the top and bottom boundary layers.