## 1. Introduction

[2] A recent trend in turbulent flow research is to attempt to identify and characterize the temporal and spatial scales of coherent flow structures over smooth as well as rough beds [*Adrian et al.*, 2000; *Liu et al.*, 2001; *Shvidchenko and Pender*, 2001; *Chen and Hu*, 2003; *Roy et al.*, 2004]. Space-time correlations of high-frequency velocity time series have shown the existence of large depth-scaled coherent flow structures consisting of high- and low-speed wedges [*Nakagawa and Nezu*, 1981; *Buffin-Bélanger et al.*, 2000; *Roy et al.*, 2004]. An alternative method of investigating the spatial scales of turbulence is through the analysis of the spatial distribution of the mean turbulent parameters of interest. Following this approach, a few studies have investigated the spatial distribution of turbulence properties over and around large roughness elements such as isolated clasts or pebble clusters [*Brayshaw et al.*, 1983; *Paola et al.*, 1986; *Buffin-Bélanger and Roy*, 1998; *Lawless and Robert*, 2001; *Shamloo et al.*, 2001; *Tritico and Hotchkiss*, 2005]. In natural gravel-bed rivers, the bed is made up of poorly sorted clasts ranging in size from very coarse to fine grains, which results in abrupt roughness transitions [*Robert et al.*, 1992]. These rapid changes in roughness have a direct effect on the turbulent length scales and on the spatial patterns of the mean and turbulent flow properties [*Buffin-Bélanger and Roy*, 1998]. The mechanism responsible for these changes is the shedding of vortices in the lee of protuberant particles. Such vortices have a range of sizes and frequencies [*Kirkbride*, 1993]. This suggests a high dependence of the spatial pattern of turbulent flow structures on the distribution of large clasts and of bed forms such as clusters on the heterogeneous bed.

[3] The relationships between spatial patterns of turbulent structure and large roughness elements on the bed were investigated by *Buffin-Bélanger and Roy* [1998]. Through an intense measurement scheme around a pebble cluster, they were able to delineate six characteristic regions of the flow field (acceleration, recirculation, shedding, reattachment, upwelling, and recovering flow), and showed the relationships between these regions and the protuberant clasts. While their study provided qualitative descriptions of numerous flow variables and of their spatial patterns, it did not attempt to quantitatively explain the dependence of the flow variables on the spatial structure, and did not estimate the proportion of the variation within the flow variables explained by the spatial structures.

[4] In ecological studies, quantification of spatial structure is often obtained through trend surface analysis. This is a standard method used to explain the variance associated with spatial trends in variables measured at points in space through polynomial regressions [*Legendre and Legendre*, 1998]. The higher the polynomial order, the finer the spatial structures which can be explained. Yet the terms within the polynomial are often highly correlated with one another, which prevents the modeling of linearly independent structures at different scales [*Borcard and Legendre*, 2002]. Furthermore, trend surface analysis is devised to model large-scale spatial structures with simple shapes and cannot adequately model finer structures [*Borcard and Legendre*, 2002; *Borcard et al.*, 2004].

[5] *Borcard and Legendre* [2002] have recently developed a new statistical spatial modeling method: principal coordinates of neighbor matrices (PCNM). The method, the theory for which has been further explored by *Dray et al.* [2006], is a form of distance-based eigenvector maps (DBEM), and has been successfully applied in aquatic ecological studies to describe the dominant spatial scales at which species variation occurs [*Borcard et al.*, 2004; *Brind'Amour et al.*, 2005]. PCNM analysis resembles Fourier analysis and harmonic regression but has the advantage of providing a broader range of signals and can also be used with irregularly spaced data [*Borcard and Legendre*, 2002]. PCNM analysis is based on the orthogonal spectral decomposition of the relationships among the spatial coordinates of a sampling design [*Borcard et al.*, 2004]. The orthogonal property of the PCNM technique allows an exact partitioning (no intercorrelation) of the explained variance over different spatial scales. PCNM analysis is used in conjunction with multiple regression to study the spatial structure of a single variable, or with canonical redundancy analysis (RDA) when studying the spatial structure of multiple variables. RDA is an extension of multiple regression used to model multivariate data. It is based on the eigenvalue decomposition of the table of regressed fitted values, which reduces the large number of associated (linearly correlated) fitted vectors to a smaller composite of linearly independent variables [*Legendre and Legendre*, 1998]. With eigenvalue decomposition, most of the variability is often summarized in the first few dimensions, which facilitates interpretation. Eigenvalue analysis has been used to study turbulent coherent structures through proper orthogonal decomposition (POD) [*Liu et al.*, 2001], and is used extensively in ecology with data sets which include large numbers (hundreds, thousands) of interrelated variables.

[6] PCNM analysis bears some similarity to POD. For POD, the eigenvalue decomposition is performed directly on a two-point correlation coefficient matrix of the flow variable under investigation using Fourier modes which are sinusoidal (quasi-trigonometric) eigenvectors termed eignenfunctions [*Moin and Moser*, 1989; *Berkooz et al.*, 1993]. As POD is a direct eigenvalue decomposition of the flow variable correlation matrix, the sum of the eigenvalues is equal to the total variance of the flow variable matrix. PCNM analysis is a regression technique which identifies only the fraction of the total variance in a response variable that is spatially dependent. An advantage of the PCNM technique is that the PCNM variables represent the eigenvalue decomposition of the relationships of a specific sampling grid, and can be used to analyze irregularly spaced data with nonrectangular boundaries. PCNM analysis is, as well, a multivariate regression technique; it offers the added advantage over POD (which can only analyze a single variable at a time) of allowing the analysis of all response variables at once.

[7] *Buffin-Bélanger and Roy* [1998] investigated each flow parameter, or ratio of flow parameters individually, an approach common in studies investigating the turbulent structure of flows [*Bennett and Best*, 1995; *Lawless and Robert*, 2001]. This approach used for investigating spatial patterns of flow structure could be greatly improved using PCNM and RDA, which can identify and quantify the spatial dependence of all flow parameters at once, thus providing an efficient means of summarizing and interpreting the spatial patterns. This paper examines the potential use of PCNM analysis as a statistical tool for investigating the spatial-scale dependence of turbulent flow processes as a complement to traditional analyses. The paper revisits the turbulent flow data reported by *Buffin-Bélanger and Roy* [1998] adjacent to and overtop of a pebble cluster in a gravel-bed river. Our study furthers the previous work by explaining and partitioning the variance of the flow parameters over four spatial scales, providing a quantification of the spatially explained variance, and indicating the intercorrelations among the turbulence variables at each scale. This leads to new insights into the turbulent flow field around clusters and protuberant clasts in rivers by suggesting the appropriate scale dependence of the turbulent flow variables, and demonstrates the potential use of PCNM analysis for a wider field of application in water resources studies.