Logarithmic variance profiles and the corresponding F⁻¹ spectra of temperature fluctuations in turbulent Rayleigh-Bénard convection

We report experimental results for the temperature variance σ²(z) and the corresponding frequency spectra P(f) in turbulent Rayleigh-Bénard convection (RBC) in a cylindrical sample of aspect ratio Γ ≡ D/L = 1.00 (D = 1.12 m is the diameter and L = 1.12 m the height). The measurements were conducted in the Rayleigh-number range 10^{11 <}_∼ Ra ^<_∼ 1.35 × 10¹⁴ and Pr ' 0.8. For Ra = 1.35×10¹⁴, σ²(z) could be described well by a logarithmic dependence on the vertical position z in a range of z₁^{∗ <}_∼ z ^<_∼ z₂^∗ with z₁^∗ ' 70λ_θ and z₂^∗ = 0.1L. Here λ_θ ≡ L/(2Nu) is the thickness of a thin thermal sublayer adjacent to the horizontal plate where the heat flux (denoted by the Nusselt number Nu) is carried mostly by thermal diffusion. In the log layer, we found that the temperature spectra had a significant frequency range over which P(f) ∼ f^−α with α close to 1. As Ra decreased, λ_θ increased so that the log layer became thinner. At Ra = 2.05 × 10¹¹, z₂^{∗ <}_∼ z₁^∗ and therefore there was no range for a log layer. Correspondingly, the temperature spectrum near the horizontal plate did not have the f⁻¹ scaling form either.