## 1. Introduction

[2] Flow properties, such as velocity and surface shear stress, in turbulent boundary layers (e.g., the atmospheric boundary layer (ABL)) exhibit considerable variability over a broad range of spatial and temporal scales. Understanding the multiscale interactions between these flow properties is essential to improving parameterizations used in high-resolution numerical models of turbulent transport. For instance, an important parameterization in large-eddy simulation (LES) is the surface boundary condition which relates the surface shear stress to the “resolved” velocity. Typical boundary condition formulations consist of using similarity theory to compute the surface shear stress as a linear function of the horizontal velocity (spatially filtered with a filter of size equal to or slightly larger than the grid size). However, similarity theory is valid only for mean quantities and over homogeneous surfaces. This approach is therefore questionable for general unsteady conditions and over heterogeneous natural surfaces (in the ABL). Given the importance of the boundary condition on the dynamics of the flow near the surface, obtaining physically more realistic boundary condition formulations has been recognized as one of the most urgent challenges that needs to be met to make LES a more reliable tool in simulations of high-Reynolds-number boundary layers in both engineering and environmental applications [*Piomelli et al.*, 1999; *Piomelli and Balaras*, 2002]. In order to achieve this goal, we need a better understanding of the relation between the surface shear stress and the velocity at different scales, particularly at the grid/filter scale in a simulation. To this end, wavelets [e.g., see *Mallat*, 1989a, 1989b; *Daubechies*, 1992] provide a convenient and powerful framework to probe into the scales of complex interactions between these two processes.

[3] Over the last decade, starting with the works of *Meneveau* [1991] and *Farge* [1992], wavelet transforms have been successfully applied to study atmospheric turbulence. The studies have ranged from identifying coherent structures from time series of velocity and temperature in the atmospheric boundary layer [e.g., *Hagelberg and Gamage*, 1994; *Howell and Mahrt*, 1994; *Brunet and Collineau*, 1994], to investigating intermittency [*Katul et al.*, 1994], to exploring energy cascading in the dynamic sublayer [*Katul and Chu*, 1998], to studying the effect of topography on the scales of response of a simulated convective boundary layer [*Roy and Avissar*, 2000]. Correlation measures in the wavelet domain were used by *Meneveau and Lund* [1994] to show that propagation of kinetic energy from a particular scale to smaller scales can be characterized by a peak in the correlation coefficient between local kinetic energies at different scales. Along similar lines, *Arnéodo et al.* [1998] used space-scale correlations obtained from wavelet transforms of turbulent velocity signals to identify what kinds of multiplicative cascades (correlated weights or not) can be used to statistically model turbulence.

[4] The purpose of this work is to quantify the multiscale interactions between surface shear stress and velocity over a homogeneous surface and discuss the implications of our results toward characterizing the surface boundary condition in high-resolution models. The spatial and temporal information about the flow properties in a boundary layer needed to study the aforementioned multiscale interactions cannot be obtained in the field, even with state-of-the-art measurement techniques. On the other hand, a wind tunnel offers an optimal setting for controlled experiments, wherein simultaneous high-frequency measurements of the surface shear stress and the velocity field can be obtained using special hot wire and hot film sensors mounted on the surface of the tunnel [see *Chew et al.*, 1998; *Marusic et al.*, 2001]. Such data have been used in our analysis and are described in section 2.

[5] In section 3 a theoretical background on wavelet-based analysis techniques is presented. The results of applying these methodologies to decompose the cross correlation between shear stress and velocity series at multiple scales are presented in section 4. The implications of our findings in improving surface boundary condition parameterization are elaborated in section 5. Finally, section 6 summarizes our findings and discusses future research.