• Lu, Y., and R. G. Lueck, 1999a: Using a broadband ADCP in a tidal channel. Part I: Mean flow and shear. J. Atmos. Oceanic Technol.,16, 1556–1567.

  • ——, and, ——, 1999b: Using a broadband ADCP in a tidal channel. Part II: Turbulence. J. Atmos. Oceanic Technol.,16, 1568–1579.

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  • Lueck, R. G., and F. Wolk, 1999: An efficient method for determining the significance of covariance estimates. J. Atmos. Oceanic Technol.,16, 773–775.

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Rolf LueckSchool of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada

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Corresponding author address: Rolf Lueck, School of Earth and Ocean Sciences, University of Victoria, Victoria, BC V8W 2Y2 Canada.

Email: rlueck@uvic.ca

Corresponding author address: Rolf Lueck, School of Earth and Ocean Sciences, University of Victoria, Victoria, BC V8W 2Y2 Canada.

Email: rlueck@uvic.ca

The comments by van Haren (HvH, hereafter) fall into two categories: (i) the method presented by Lueck and Wolk (LW99, hereafter) is not equivalent to the one originally suggested by Barry Ruddick and (ii) LW99 do not correctly estimate the Reynolds stress from their acoustic Doppler current profiler (ADCP) data. The latter category comprises the bulk of his comments.

LW99 is a very brief article on determining the significance of covariance estimates. It is not a paper on using an ADCP to make Reynolds stress estimates. Both reviewers agreed that the method is interesting to oceanographers because of its efficiency and that the article had to be very short because of its limited scope. We readily agreed and focused on the numerical technique and not the technical details surrounding the data that was used to demonstrate the method. Specific information about the ADCP is required for the scientific interpretation of the Reynolds stress, but such an analysis is beyond the scope of LW99 and the reader was directed to two lengthy articles (Lu and Lueck 1999a,b) for technical details.

Is the method of LW99 equivalent to the original one? In paragraph 3, HvH claims that “The novelty introduced by LW99 is only equivalent to this method, if one can assume statistical independence of all estimates outside of a certain range from zero lag.” The emphasis is by HvH. Statistical independence is irrelevant. In both the technique of LW99 and its predecessor, the significance of a covariance estimate is judged by comparing this estimate against the 95-percentile width of the sample probability density function (pdf) of the covariance of two time series that are (assumed to be) uncorrelated on average. The shape of the pdf, particularly its 95-percentile width, is not affected by statistical interdependence of the samples used to form the pdf. This was pointed out in the last paragraph of LW99. Thus, the LW99 technique is not only equivalent but also superior to its predecessor because it uses all possible samples and, thus, maximizes the number of independent samples in the pdf. The original technique makes no attempt to maximize the degrees of freedom of the pdf.

Did LW99 calculate the Reynolds stress correctly? The fourth and subsequent paragraphs of HvH are almost exclusively concerned with the technique of using the ADCP to make Reynolds stress estimates, and they provide a tutorial on the use of the ADCP. These matters are beyond the scope of LW, but it is worth exploring if LW99 did (or did not) calculate the Reynolds stress correctly. HvH claims that the first equality in his Eq. (1) is the alongstream Reynolds stress, namely,
i1520-0426-17-6-887-e1
while LW99 make the same claim for the average product of the first two identities in their Eq. (4), namely,
i1520-0426-17-6-887-e2
It is trivial to show that (1) is the negative of (2). To express the Reynolds stress as a covariance, LW99 used (2). For the cross-stream stress, LW99 used
i1520-0426-17-6-887-eq1
but failed to point out that the vertical velocity, w, in this case, is based upon the scaled sum of beams 3 and 4, namely,
i1520-0426-17-6-887-eq2
Again, vw is identical to the second equality in Eq. 1 of HvH for the cross-stream Reynolds stress. We agree that the definition of w was ambiguous for the case of the cross-stream stress. As for the minus sign, that is a matter of definition of coordinate systems and is not important to demonstrating or understanding the test for statistical significance.

Paragraphs 6–9 inclusive of HvH contain rambling speculation about how the ADCP was deployed (rigid or moored, bin size, coordinate transformations, etc.), comments about appropriate averaging intervals for turbulence, and an expression for computing the stress with data that were collected in Cartesian coordinates. Such presentations and discussions may help a reader develop an understanding for the ADCP and may be required for the scientific interpretation of data collected with this instrument, but they have no place in a short article on testing the statistical significance of covariance estimates. We respond only to the charge that the cross-channel stress at middepth was found significant for a 90-min record and insignificant for a 10-min record because of “the (im)proper way of estimating these fluxes.” Because the cross-channel stress was properly estimated, the simplest explanation for the small stress is the small cross-stream shear at middepth. The three-layer secondary circulation in the channel produced a velocity maxima at middepth (Lu and Lueck 1999a,b) and, thereby, nature provided an opportunity to collect data that would fail the test for significance.

REFERENCES

  • Lu, Y., and R. G. Lueck, 1999a: Using a broadband ADCP in a tidal channel. Part I: Mean flow and shear. J. Atmos. Oceanic Technol.,16, 1556–1567.

  • ——, and, ——, 1999b: Using a broadband ADCP in a tidal channel. Part II: Turbulence. J. Atmos. Oceanic Technol.,16, 1568–1579.

    • Crossref
    • Export Citation
  • Lueck, R. G., and F. Wolk, 1999: An efficient method for determining the significance of covariance estimates. J. Atmos. Oceanic Technol.,16, 773–775.

    • Crossref
    • Export Citation
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