Electronic Supplementary Material(ESM) for
Vertical structure of turbulence within a depression surrounded by coral-reef colonies
Appendix A. Turbulence decomposition method
The quality of the raw velocity data was controlled, and the outliers were removed using the guideline proposed by Elgar et al. (2005). Several useful turbulence decomposition methods have been proposed to filter the wave motions(e.g., Shaw and Trowbridge 2001; Bricker and Monismith 2007). The differencing technique with the adaptive least-squares filter proposed by Shaw and Trowbridge (2001) was used to separate the components of the wave-induced velocities and the turbulent velocities. This method was modified from Trowbridge (1998) to minimize the possible contamination of wave bias on the TSS. This technique assumes that the incoherent signals between the two ADVs represent turbulence, whereas the coherent signals represent wave motions. The wave-induced velocity at position 1 is estimated from the filtered velocity, , which is defined as
,(4)
where is a filter length that is chosen to be approximately half of the peak wave period (Shaw and Trowbridge 2001), is the velocity measured at position 2, and is the filter weights that are determined by a least squares method:
,(5)
where is a windowed data matrix of the velocity at position 2 that includes three velocity components to increase the number of degrees of freedom by a factor of 3. The coherent wave velocity vector at position 1, , is estimated by convolving the matrix with :
.(6)
The velocity vector of the turbulent component at position 1 is estimated by
(7)
Appendix B. Advection velocity scale versus data quality
Two tests were used to control the data quality in the analysis: tests of the presence of the inertial subrange (QC1) and the ogive curve of the cospectra of TSS (QC2). Because turbulence is advected by waves and currents, it could be useful to determine whether these quality controls are related to the turbulent velocity scale, , and the advection velocity scales, and (Fig. A1). The results show that most of the samples that fail both tests occur during low-wave and low-current conditions due to the increasing instrument noise (Fig. A1b). Most of the data that pass QC1 but fail QC2 occur when the ratio is especially high (i.e., greater than approximately 1-5); however, it is difficult to determine clear relations for these data with because all of the data are under the condition of . In general, the good data of the turbulence measurements that pass the two tests mostly occur at sufficiently high advection velocities and at ratios of the turbulent velocity to the advection velocity scales that are smaller than .
Appendix C. Validation and uncertainties of turbulent dissipation rate estimates
The turbulent dissipation rate in unsteady advection for multidirectional waves was estimated using the method reported by Gerbi et al. (2009):
(A1)
where is the Kolmogorov constant, and is an integral over three-dimensional wavenumber space that depends on the mean current and waves and is a function of the standard deviations of the wave velocity and the mean velocity; i.e., . To evaluate , we must numerically compute a triple integral, , over three-dimensional wavenumber space that depends on the mean current and wave motions. Because is numerically integrated, our computational scheme is tested by comparing the numerical integral to the analytic solution under no-current conditions (Gerbi et al. 2009). The analytic solution agreed well with our numerical solution. In addition, the method proposed by Bryan et al. (2003) was used to estimate the turbulent dissipation rate. Good agreement between the results calculated from the two methods was obtained. Rosman et al. (2014) showed that for isotropic turbulence that is advected by wave motions, the estimates of from the inertial subrange technique (Lumley and Terray 1983; Feddersen et al. 2007) agree with the direct gradient-based estimates of the dissipation rate tensor if the assumptions for the inertial subrange technique hold. Additional details of the derivations of the model can be found in Feddersen et al. (2007) and Gerbi et al. (2009).
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Fig. A1 (a) Time series of the ratios of the turbulent velocity () to the advection velocity scales of the mean current () and wave rms velocity (). (b, c) Scatter plots of the advection velocities and velocity ratios with marks showing the data quality controls that are used, where QC1 and QC2 denote the controls using the tests of the presence of the inertial subrange and the ogive curve of the Reynolds shear stress, respectively, and the symbols F and T denote the data that fail and pass the tests, respectively.