# Scale Properties of Anisotropic and Isotropic Turbulence in the Urban Surface Layer

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## Abstract

The scale properties of anisotropic and isotropic turbulence in the urban surface layer are investigated. A dimensionless anisotropic tensor is introduced and the turbulent tensor anisotropic coefficient, defined as *C*, where \(C = 3d_{3}\,+\,1 (d_{3}\) is the minimum eigenvalue of the tensor) is used to characterize the turbulence anisotropy or isotropy. Turbulence is isotropic when \(C \approx 1\), and anisotropic when \(C \ll 1\). Three-dimensional velocity data collected using a sonic anemometer are analyzed to obtain the anisotropic characteristics of atmospheric turbulence in the urban surface layer, and the tensor anisotropic coefficient of turbulent eddies at different spatial scales calculated. The analysis shows that *C* is strongly dependent on atmospheric stability \(\xi = (z-z_{\mathrm{d}})/L_{{\textit{MO}}}\), where *z* is the measurement height, \(z_{\mathrm{d}}\) is the displacement height, and \(L_{{\textit{MO}}}\) is the Obukhov length. The turbulence at a specific scale in unstable conditions (i.e., \(\xi < 0\)) is closer to isotropic than that at the same scale under stable conditions. The maximum isotropic scale of turbulence is determined based on the characteristics of the power spectrum in three directions. Turbulence does not behave isotropically when the eddy scale is greater than the maximum isotropic scale, whereas it is horizontally isotropic at relatively large scales. The maximum isotropic scale of turbulence is compared to the outer scale of temperature, which is obtained by fitting the temperature fluctuation spectrum using the von Karman turbulent model. The results show that the outer scale of temperature is greater than the maximum isotropic scale of turbulence.

### Keywords

Anisotropy Outer length scale Stability Reynolds-stress tensor Temperature fluctuations Turbulent kinetic energy Urban surface layer## Notes

### Acknowledgements

This study was supported by the National Key Research and Development Program under Grant No. 2016YFC0203306, and the National Natural Science Foundation of China (41475012). We also thank two anonymous reviewers for their constructive and helpful comments.

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