Estimation of the energy dissipation rate, ε

Turbulent energy dissipation rates (ε) were estimated using the relationship, ε = (2π)⁴k_B⁴νκ², where ν represents the kinematic viscosity (m² s⁻¹), κ represents the molecular thermal diffusivity (m² s⁻¹), and k_B represents the Batchelor wavenumber of the temperature spectrum (cpm). We estimated k_B by fitting the universal spectrum to the observed temperature gradient spectrum^33,34. The data processing was the same as in a previous study²¹, based on the maximum likelihood estimation method³⁵. Each spectrum was determined from a profile segment within 1 s and corrected using the double-pole low-pass filter function³⁶. Considering that each thermistor was not calibrated, the time constant, which represents the effect of smoothing the microstructures due to relatively slow sensor response, was fixed at 3 ms²⁰.

Time constant dependence of the sensor fall rates has been demonstrated in some previous studies^36,37. The faster the sensor fall rate, the smaller the time constant becomes; thus, the required correction is also smaller. However, making corrections considering this characteristic could result in underestimation in areas with strong turbulence²¹. When the sensor falls with a higher speed, higher frequencies are necessary to determine the Kraichnan spectrum. These higher frequencies could be significantly attenuated, making it difficult to attain full correction, regardless of the use of double- or single-pole functions. Consequently, the smaller correction associated with the higher speed results in underestimation. Accordingly, the dependence of the time constant on the sensor speed was not considered in this study.

To estimate k_B, we only used the spectrum in the lower frequency domain avoiding using the high frequencies dominated by electrical noise, which was determined by comparing the noise spectrum obtained with dummy probes in our laboratory. The form of the fitted universal spectrum was the one³⁴ with the fixed universal constant, q_K = 5.26^38,39. After the fitting, automatic rejection criteria⁴⁰ based on the shape of the observed spectrum were applied on each spectrum to eliminate poorly fitted data.