the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 12 Dec 2024
Brief communication: A note on the variance of wind speed and turbulence intensity
Abstract. This note addresses the issue that several papers in the peer-reviewed literature of wind energy applications have used an incorrect equation that equals the variance of wind speed (σ2U) to the sum of the variances of the wind components. This incorrect equation is often used to calculate turbulent intensity (TI), which, as a consequence, is often incorrectly estimated too. While exact analytical equations do not exist, here two approximate analytical equations are derived for σ2U and TI, both functions of the variances of the wind components. Both formulations are validated with samples from a prior field campaign and perform satisfactorily.
Competing interests: I am a member of the editorial board of Wind Energy Science.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this preprint. The responsibility to include appropriate place names lies with the authors.- Preprint
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RC1: 'Comment on wes-2024-159', Anonymous Referee #1, 12 Dec 2024
This paper deals with the difference between the variance of the wind component along the mean wind vector and the variance of the length of the wind vector, also called the wind speed. It is well known that those quantities are under most circumstances (i.e. not too high turbulence intensity) almost equal (e.g. L.. Kristensen 1998, JTech, vol 5, p6). The transverse component enters only the speed variance to second order in the turbulence intensity (see eq 8 in the mentioned paper). These observations do not change if the coordinate system is not aligned with the wind.
The other subject paper is an apparent mistake in the literature. The author states that the variance of the wind speed is sometimes mistakingly said to be equal to the sum of the variances of the two horizontal components. This is obviously wrong, as the author clearly states, but I’m am unaware of these mistakes in the literature. The author does not provide evidence for these mistakes, which makes the need for this paper limited. The author might be wary to point out mistakes in specific papers, but this is unfortunately what has to be done in order to advance science. You cannot leave it to the readers to find documentation for this possible mistake in the literature.
I therefore think, that the paper is not suited for publication in Wind Energy Science. The problem in the literature has to be documented. Once the issue is confirmed with a thorough list of papers containing this mistake, and thus proving its relevance, the paper should be made more succinct, as there are no substantially new derivations.
Citation: https://doi.org/10.5194/wes-2024-159-RC1 -
AC1: 'Reply on RC1', Cristina Archer, 16 Dec 2024
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RC2: 'Reply on AC1', Anonymous Referee #1, 08 Jan 2025
The authors are right that the listed references, to my surprise, get the definition of turbulence intensity wrong. This strengthens the relevance of this paper. The author does not want to reference these work explicitly in her paper, which I think she should do. A compromise could be to reference the oldest reference, Joffre and Laurila (1988), where the mistake is very explicit. One could then mention that this definition has been used later without making explicit references.
Citation: https://doi.org/10.5194/wes-2024-159-RC2 -
AC3: 'Reply on RC2', Cristina Archer, 23 Jan 2025
Thank you, I like your idea and I have incorporated it in the manuscript as follows:
" ... and often treated, incorrectly, as an exact definition (see for example Eq. 6 in Joffre and Laurel (1988))."
Citation: https://doi.org/10.5194/wes-2024-159-AC3
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AC3: 'Reply on RC2', Cristina Archer, 23 Jan 2025
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RC2: 'Reply on AC1', Anonymous Referee #1, 08 Jan 2025
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AC1: 'Reply on RC1', Cristina Archer, 16 Dec 2024
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EC1: 'Comment on wes-2024-159', Etienne Cheynet, 17 Dec 2024
General Remarks
Dear Cristina, Thank you for addressing an important issue in wind energy science. I would like to expand on and clarify certain points that I believe could further strengthen the scope and implications of this work.
My comments below may go beyond the topic of wind turbine wakes. Here, I aim to provide a slightly different perspective with the hope of creating greater alignment across wind energy, wind engineering, and meteorology. Bridging the differences in practices and definitions between these domains could lead to a more coherent and standardized approach, ultimately benefiting both research and practical applications in wind energy.
I would go beyond the statement that the variance of the wind speed is often miscalculated. I would argue that using the variance of the wind speed itself—rather than treating the variance of the along-wind and cross-wind velocity components separately—is fundamentally problematic. In wind engineering and micrometeorology, these components are considered separately due to their distinct characteristics. The design of wind turbines, particularly for structural and turbulent loading considerations, is based on the variances of the along-wind and cross-wind components, not the wind speed. The continued use of wind speed variance might be a legacy of outdated practices.
Specific Comments:
Line 15-20: The distinction between aligning the x-axis with the wind direction or with the East-West coordinate system is not unique to wind energy; it largely depends on the spatial and temporal scales of interest. In boundary-layer meteorology, particularly micrometeorology, the x-axis is typically aligned with the mean wind direction due to the focus on turbulence, as detailed in Kaimal and Finnigan (1994). In mesoscale meteorology, where the emphasis is on mean wind speed, the x-axis is, indeed, often aligned with the East-West direction. To avoid conflating discipline-specific conventions, I recommend acknowledging this broader context.
Conflict of Definitions in Different Fields: There may be conflicting definitions of "turbulence" between mesoscale and microscale meteorology that require clarification. In micrometeorology, turbulence is typically considered a three-dimensional process occurring within temporal scales of up to one hour and spatial scales smaller than a few kilometres. In micrometeorology, the variance of the along-wind and across-wind components differs significantly. Motions exceeding these scales are often classified as “non-turbulent motion,” consistent with the concept of the spectral gap. However, mesoscale meteorology may occasionally describe such motions as “2D turbulence.” These differences reflect divergent focuses and terminologies across disciplines and should be recognized explicitly.
Line 28-29: The statement “turbulence intensity is a function of the standard deviation of wind speed” could be misleading. From micrometeorology and wind engineering perspectives, turbulence intensity is typically defined based on the individual velocity components (along-wind, cross-wind, and vertical), not wind speed. Defining turbulence intensity based on wind speed lacks physical relevance. In my humble opinion, its continued use in wind energy science is puzzling.
Line 31: While it is true that mesoscale meteorology often simplifies wind velocity as a 2D vector, this approach does not hold in micrometeorology or wind energy, where the vertical velocity component significantly contributes to turbulence kinetic energy (TKE). I understand that the inclusion of TKE in this discussion depends on the desired level of detail. If brevity is prioritized, this aspect could be omitted.
Table 1: The two examples in Table 1 effectively demonstrate the value of the proposed equation. However, it is unclear whether the statistics are based on six hours of data or shorter sub-samples. If turbulence is the focus, time averaging over six hours is not appropriate, especially since the second panel shows clear non-stationary fluctuations. If the table uses shorter intervals (e.g., 10 minutes to 30 min), I recommend expanding the analysis to include all samples from Figure 1. Comparing the wind speed variance estimated by the older equation with that from your proposed equation would strengthen the analysis. A scatter plot of these comparisons across the full dataset would complement Table 1. This visualization would make it easier to assess the overall performance and accuracy of the new equation relative to the older one.
Citation: https://doi.org/10.5194/wes-2024-159-EC1 - AC2: 'Reply on EC1', Cristina Archer, 03 Jan 2025
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AC4: 'Reply on EC1', Cristina Archer, 24 Jan 2025
I found some high-frequency data and I will be able to prepare the scatter plots that you recommended. Stay tuned!
Citation: https://doi.org/10.5194/wes-2024-159-AC4
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RC3: 'Comment on wes-2024-159', Anonymous Referee #2, 20 Jan 2025
Dear Christina,
Thank you for adressing this misconception. In my opinion, this relates to 1) split disciplines between wind systems engineering and micrometeorology, and 2) a confusion between timescales (and intrisically linked space scales) in the literature and current practice. There I would first like to mention that while wind engineering is my background, I know only little about meteorology.
Wind turbine structures are typically only concerned by microscale, and 10-minutes load cases are tradiationally used. There it is assumed that turbines would yaw to align with an assumed constant wind direction. Fluctuations are then represented around this direction and separated into along-wind and cross-wind drections. In this case the mean cross-wind component v_bar is always zero, and the original equation to compute TIs is valid. An exception may be for wake steering applications where a yaw misalignment with mean flow is intentionally created, but the coordinate system used to represent the wind speed is still relative to the slowly-varying wind direction.
However, longer load cases may be of interest when for instance looking at slowly-varying motions of floating substructures or power fluctuations from wind farms heavily influenced by mesocale fluctuations. In this case, the concept itself of characterising wind fluctuations by turbulence intensities covering all timescales (integrated over the entire width of the wind spectrum) is discussable, and might be outdated practice. Strictly speaking, TIs should only be used to describe microscale fluctuations (i.e what is commonly referred to as turbulence, to the right of the spectral gap in the wind spectrum), while mesoscale fluctuations (to the left of the spectral gap in the wind spectrum) should be described by a distinct quantity. Assuming "mesoscale turbulence intensities" are used for this purpose, I agree that they should be calculated using the method you suggest.
To improve the quality and impact of your manuscript, I would suggest to 1) make this distinction between scales, disciplines and applications clearer, particularly through the role of turbine yawing, 2) add references to where you claim erroneous formulations have been used (coming from a different field, this statement looks superficial without examples)
Citation: https://doi.org/10.5194/wes-2024-159-RC3 -
AC5: 'Reply to all comments', Cristina Archer, 14 Feb 2025
Please use attached document for my final reply to all the reviewers comments. My initial replies should be replaced with this document because I made many changes to the paper in response to subsequent reviews and therefore I also had to change some of my previous replies.
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