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, û1 , which is defined as
uˆ1 (t )  
T /2
T /2
hˆ(t   )u2 (t )d ,
(1)
where T is a filter length that is chosen to be approximately half of the peak wave
period (Shaw and Trowbridge 2001), u2 is the velocity measured at position 2, and
ĥ is the filter weights that are determined by a least squares method:
1
hˆ   A T A  A Tu1 ,
(2)
where A 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, û1 , is estimated by convolving the
matrix A with ĥ :
uˆ 1  Ahˆ .
(3)
The velocity vector of the turbulent component at position 1 is estimated by
uˆ  u  uˆ 1
(4)
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,
(2k )1/2 , and the advection velocity scales, U and U rms (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 (2k )1/2 / U is especially
high (i.e., greater than approximately 1-5); however, it is difficult to determine clear
relations for these data with U rms because all of the data are under the condition of
(2k )1/2 / U rms  1 .
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 O(1) .
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):
Sww ( ) 
 2/3
M ww ( )
2(2 )3/2
(A1)
where   1.5 is the Kolmogorov constant, and M ww 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., M ww  M ww ( u ,cs ,  v ,cs ,  w , ucs , vcs ) .
To evaluate  , we must
numerically compute a triple integral, M ww , over three-dimensional wavenumber
space that depends on the mean current and wave motions.
Because M ww 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).
References
Bricker JD, Monismith SG (2007) Spectral wave-turbulence decomposition. Journal
of Atmospheric and Oceanic Technology 24:1479-1487
Bryan KR, Black KP, Gorman RM (2003) Spectral estimates of dissipation rate within
and near the surf zone. Journal of Physical Oceanography 33:979-993
Elgar S, Raubenheimer B, Guza RT (2005) Quality control of acoustic Doppler
velocimeter data in the surf zone. Measurement Science & Technology
16:1889-1893
Feddersen F, Trowbridge JH, Williams AJ (2007) Vertical structure of dissipation in
the nearshore. Journal of Physical Oceanography 37:1764-1777
Gerbi GP, Trowbridge JH, Terray EA, Plueddemann AJ, Kukulka T (2009)
Observations of turbulence in the ocean surface boundary layer: energetics and
transport. Journal of Physical Oceanography 39:1077-1096
Lumley JL, Terray EA (1983) Kinematics of turbulence convected by a random wave
field. Journal of Physical Oceanography 13:2000-2007
Rosman JH, Paul EL, Scotti A (2014) Interpreting shallow water turbulence
measurements: insight from numerical simulations of turbulence advected by
waves. 2014 Ocean science meeting
Shaw WJ, Trowbridge JH (2001) The direct estimation of near-bottom turbulent
fluxes in the presence of energetic wave motions. Journal of Atmospheric and
Oceanic Technology 18:1540-1557
Trowbridge JS (1998) On a technique for measurement of turbulent shear stress in the
presence of surface waves Journal of Atmospheric and Oceanic Technology
15:290-298
Fig. A1 (a) Time series of the ratios of the turbulent velocity ( (2k )1/2 ) to the advection
velocity scales of the mean current ( U ) and wave rms velocity ( U rms ). (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.