# 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.

## References

- Andrews LC, Phillips RL (2005) Laser beam propagation through random media. SPIE Press, Bellingham, 790 ppGoogle Scholar
- Arad I, L’Vov VS, Procaccia I (1999) Correlation functions in isotropic and anisotropic turbulence: the role of the symmetry group. Phys Rev E 59(6):6753–6765CrossRefGoogle Scholar
- Banerjee S, Krahl R, Durst F, Zenger C (2007) Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches. J Turbul 8(32):1–27Google Scholar
- Biferale L, Procaccia I (2005) Anisotropy in turbulent flows and in turbulent transport. Phys Rep Rev Sect Phys Lett 414(2–3):43–164Google Scholar
- Biferale L, Vergassola M (2001) Isotropy vs. anisotropy in small-scale turbulence. Phys Fluids 13(8):2139–2141CrossRefGoogle Scholar
- Borgnino J (1990) Estimation of the spatial coherence outer scale relevant to long base-line interferometry and imaging in optical astronomy. Appl Opt 29(13):1863–1865CrossRefGoogle Scholar
- Choi KS, Lumley JL (2001) The return to isotropy of homogeneous turbulence. J Fluid Mech 436:59–84CrossRefGoogle Scholar
- CMA (1996) Observation methods for meteorological radiation. China Meteorological Press, Beijing, p 165Google Scholar
- Consortini A, Innocenti C, Paoli G (2002) Estimate method for outer scale of atmospheric turbulence. Opt Commun 214(1–6):9–14CrossRefGoogle Scholar
- Corrsin S (1958) Local isotropy in turbulent shear flow, NACA RM58B11Google Scholar
- Courault D, Drobinski P, Brunet Y, Lacarrere P, Talbot C (2007) Impact of surface heterogeneity on a buoyancy-driven convective boundary layer in light winds. Boundary-Layer Meteorol 124(3):383–403CrossRefGoogle Scholar
- Darbieu C, Lohou F, Lothon M, de Arellano JV-G, Couvreux F, Durand P, Pino D, Patton EG, Nilsson E, Blay-Carreras E, Gioli B (2015) Turbulence vertical structure of the boundary layer during the afternoon transition. Atmos Chem Phys 15(17):10071–10086CrossRefGoogle Scholar
- Fernando HJS (2010) Fluid dynamics of urban atmospheres in complex terrain. Annu Rev Fluid Mech 42:365–389CrossRefGoogle Scholar
- Foken T, Gockede M, Mauder M, Mahrt L, Amiro B, Munger J (2004) Post-field data quality control. In: Lee X et al (eds) Handbook of micrometeorology A guide for surface flux measurements. Kluwer, New YorkGoogle Scholar
- Frisch U (1995) Turbulence—the legacy of A. N. Kolmorogorov. Cambridge University Press, Cambridge, 296 ppGoogle Scholar
- Gurvich AS (1997) A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere. Ann Geophys Atm Hydr 15(7):856–869CrossRefGoogle Scholar
- Hocking A, Hocking WK (2007) Turbulence anisotropy determined by wind profiler radar and its correlation with rain events in Montreal, Canada. J Atmos Ocea Technol 24(1):40–51CrossRefGoogle Scholar
- Ishihara T, Yoshida K, Kaneda Y (2002) Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow. Phys Rev Lett 88(15):154501CrossRefGoogle Scholar
- Kaimal JC, Finnigan JJ (1992) Atmospheric boundary layer flows. Oxford University Press, New York, 304 ppGoogle Scholar
- Kaimal JC, Izumi Y, Wyngaard JC, Cote R (1972) Spectral characteristics of surface-layer turbulence. Q J R Meteorol Soc 98(417):563–589CrossRefGoogle Scholar
- Klipp C (2014) Turbulence anisotropy in the near-surface atmosphere and the evaluation of multiple outer length scales. Boundary-Layer Meteorol 151(1):57–77CrossRefGoogle Scholar
- Lukin VP (2005) Outer scale of atmospheric turbulence. Proc SPIE 5981:598101CrossRefGoogle Scholar
- Lumley JL (1965) Interpretation of time spectra measured in high-intensity shear flows. Phys Fluids 8(6):1056–1062CrossRefGoogle Scholar
- Lumley JL, Newman GR (1977) The return to isotropy of homogeneous turbulence. J Fluid Mech 82:161–178CrossRefGoogle Scholar
- Maire J, Ziad A, Borgnino J, Martin F (2008) Comparison between atmospheric turbulence models by angle-of-arrival covariance measurements. Mon Notic R Astronom Soc 386(2):1064–1068CrossRefGoogle Scholar
- Newsom R, Calhoun R, Ligon D, Allwine J (2008) Linearly organized turbulence structures observed over a suburban area by dual-Doppler lidar. Boundary-Layer Meteorol 127(1):111–130CrossRefGoogle Scholar
- Pan Y, Chamecki M (2016) A scaling law for the shear-production range of second-order structure functions. J Fluid Mech 801:459–474CrossRefGoogle Scholar
- Panchev S (1971) Random functions and turbulence. Headington Hill Hall, Oxford, 458 ppGoogle Scholar
- Poggi D, Porporato A, Ridolfi L (2003) Analysis of the small-scale structure of turbulence on smooth and rough walls. Phys Fluids 15(1):35–46CrossRefGoogle Scholar
- Roth M (1993) Turbulent transfer relationships over an urban surface. II: integral statistics. Q J R Meteorol Soc 119(513):1105–1120CrossRefGoogle Scholar
- Saddoughi SG (1997) Local isotropy in complex turbulent boundary layers at high Reynolds number. J Fluid Mech 348:201–245CrossRefGoogle Scholar
- Saddoughi SG, Veeravalli SV (1994) Local isotropy in turbulent boundary-layers at high Reynolds number. J Fluid Mech 268:333–372CrossRefGoogle Scholar
- Stull RB (1988) An introduction to boundary layer meteorology. Springer, Dordrecht, 666 ppGoogle Scholar
- Tatarskii VI (1961) Wave propagation in a turbulent medium. McGraw-Hill, New York, 285 ppGoogle Scholar
- Tennekes H, Lumley JL (1972) A first course in turbulence. MIT Press, Cambridge, 300 ppGoogle Scholar
- Toschi F, Leveque E, Ruiz-Chavarria G (2000) Shear effects in nonhomogeneous turbulence. Phys Rev Lett 85(7):1436–1439CrossRefGoogle Scholar
- von Kármán T (1948) Progress in the statistical theory of turbulence. Proc Natl Acad Sci USA 34(11):530–539CrossRefGoogle Scholar
- Wilczak JM, Oncley SP, Stage SA (2001) Sonic anemometer tilt correction algorithms. Boundary-Layer Meteorol 99(1):127–150CrossRefGoogle Scholar
- Wyngaard JC, Clifford SF (1977) Taylors hypothesis and high-frequency turbulence spectra. J Atmos Sci 34(6):922–929CrossRefGoogle Scholar
- Yuan R, Sun J, Luo T, Wu X, Wang C, Fu Y (2014) Simulation study on light propagation in an anisotropic turbulence field of entrainment zone. Opt Expr 22(11):13427–13437CrossRefGoogle Scholar
- Yuan R, Luo T, Sun J, Liu H, Fu Y, Wang Z (2016) A new method for estimating aerosol mass flux in the urban surface layer using LAS technology. Atmos Meas Tech 9:1925–1937CrossRefGoogle Scholar
- Ziad A, Conan R, Tokovinin A, Martin F, Borgnino J (2000) From the grating scale monitor to the generalized seeing monitor. Appl Opt 39(30):5415–5425CrossRefGoogle Scholar
- Ziad A, Schock M, Chanan GA, Troy M, Dekany R, Lane BF, Borgnino J, Martin F (2004) Comparison of measurements of the outer scale of turbulence by three different techniques. Appl Opt 43(11):2316–2324CrossRefGoogle Scholar
- Zou J, Liu G, Sun J, Zhang H, Yuan R (2015) The momentum flux-gradient relations derived from field measurements in the urban roughness sublayer in three cities in China. J Geophys Res Atmos 120(20) 10797–10809Google Scholar