the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 14 Apr 2025
Coriolis Recovery of Wind Farm Wakes
Abstract. Two mechanisms cause wind speed recovery in the wake of a wind farm: momentum mixing and the Coriolis force. To study these mechanisms, we use a steady linearized two-layer Fast Fourier Transform (FFT) model so that both analytical expressions and full flow fields can be derived. The model parametrizes the vertical momentum mixing as Rayleigh friction. Pressure gradient forces are computed using a two-part vertical wave number formulation in the upper layer. The Coriolis force recovery occurs by deflecting flow leftward (in the northern hemisphere). The Coriolis force, acting on this crossflow, re-accelerates the flow in the downwind direction.
The relative importance of Rayleigh versus Coriolis wake recovery depends on their two coefficients: C and f respectively, each with units of inverse time. When the coefficient ratio is large, i.e. C/f >> 1 , Rayleigh friction restores the wake before Coriolis can act. Farm size and atmospheric static stability are also important to wake recovery. The wakes of small and medium size farms will quickly approach geostrophic balance. Once balance is established, the ratio of farm size "a" to the Rossby Radius of Deformation (RRD) determines the amount of Coriolis recovery. For a small farm in a stable atmosphere (a < RRD), Coriolis acts by adjusting the pressure field to obtain geostrophic balance rather than accelerating the wind. When this occurs, only momentum mixing can restore the "inner" wake. For large farms in less stable conditions (a > RRD), the Coriolis Force significantly contributes to wake recovery. In this case, the leftward deflected flow creates "edge jets" on either side of the wake. Including the Coriolis force when modelling wind farm wakes is demonstrated to be increasingly important for larger wind farms or farm clusters.
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RC1: 'Comment on wes-2025-60', Anonymous Referee #1, 25 Jun 2025
This manuscript examines how the Coriolis force affects wind farm wake recovery using a linearized, steady-state, two-layer model with turbine drag applied to the lower layer. It extends previous work by the same author(s), which mainly focused on wind-farm-induced gravity waves and their effect on the surrounding pressure field. The strength of this approach is its computational efficiency, allowing quick analysis of large-scale flow features.However, this modeling simplicity sacrifices some interpretability and accuracy. For instance, in Figure 7, the difference between the FFT model and geostrophic theory is roughly a factor of 2, suggesting limited accuracy. Similarly, it is hard to evaluate how applicable the results are to real-world wind farm setups. In particular, the choice of model parameters and constants needs clearer justification. Explaining how specific values were chosen would increase confidence in the conclusions. Most importantly, how is the value of C, meaning f/C, determined? There should be a more thorough discussion of what values are realistic for actual wind farms. Implicitly, presenting results for f/C>1 implies these values are possible. Is that actually the case?The introduction would benefit from a stronger connection to the existing literature, especially regarding the role of the Coriolis force in wind turbine and wind farm wake recovery, several relevant studies that have not yet been discussed are:• Dörenkämper et al. (2015), J. Wind Eng. Ind. Aerodyn., 144, 146–153• Abkar & Porté-Agel (2016), Phys. Rev. Fluids, 1(6), 063701• Nouri et al. (2020), Applied Energy, 277, 115511• Englberger et al. (2020), Wind Energy Science, 5, 1359–1374• Qian et al. (2022), Energy, 239, 121876In both the abstract and introduction, the definitions of the “Rayleigh contribution” and “Coriolis contribution” to wind farm wake recovery need clarification. These terms are not common in wind farm literature, and their meanings only become clearer later in the technical sections. This could confuse readers who are unfamiliar with the specific framework used here. In particular, the “Rayleigh contribution” concept needs more explanation. I especially want to see some justification for the constant C used in this context.While focusing on these two contributions is helpful (as mentioned on line 52), other effects such as pressure effects and turbulent momentum fluxes, are also important. Including a brief explanation in the introduction that there are different approaches to analyzing the flow would be beneficial.The finding that Coriolis effects become more significant as wind turbine size and the extent of wind farms increase. The Coriolis force influences both the structure of the atmospheric boundary layer and the velocity deficit of the wind farm wake. As a result, previous studies (such as Gadde and Stevens (2025) JPCS, 1256, 012026 and Kirby & Howland, J. Fluid Mech., 1008, (2025)) showed that Coriolis-induced wake rotation can be clockwise or counterclockwise depending on atmospheric flow conditions. The latter demonstrates that wake deflection depends on the Rossby number. Recent studies showed that the Coriolis force can lead to significant wind farm wake deflection, see e.g.• Kasper et al., J. Renewable Sustainable Energy, 16, 063302 (2024)• Kirby & Howland, J. Fluid Mech., 1008, (2025)It should be noted that the Coriolis force can influence flow dynamics in multiple ways, and different studies address these effects to varying extents. The present manuscript focuses only on the direct Coriolis and Rayleigh contributions, offering a simplified representation of the broader dynamics.Overall, the manuscript's conclusion that Coriolis effects deserve more attention in future research appears reasonable. However, as noted in the manuscript, some parameters may have been selected to enhance the observed effects. Therefore, it is important to justify the chosen values clearly. For example, the wind farm size of 1600 km² exceeds the dimensions of current real-world farms, although such a size could be plausible in the future. Nonetheless, this assumption should be explicitly acknowledged and discussed. Crucially, as mentioned above, the expected f/C ratio in real situations should be discussed.Additional points• The quality of the figures is currently insufficient. For example, Figure 1 uses a smooth color bar, but the data are plotted in discrete contours, making it difficult to interpret. Also it is impossible to see where the zero line is.• The velocity deficits in the wind farms appear to develop quite slowly. Can the authors discuss this observation and comment on how this should be interpreted in the context of the model• In Figure 7, it is written that the agreement between FFT and geostrophic theory is good. While both show the same trend, I disagree on calling this a good agreement, as the amplitude obtained from both differs by nearly a factor of 2, which clearly matters considerably.• Why is the domain shown in Figures 1, 2, and 3 not symmetrically around the wind farm? Particularly, why is the vertical range in the direction in which the wind farm wake is deflected smaller?• Table 3: Using a reduced gravity value g’ = 10 m/s^2 seems unrealistically high for atmospheric boundary layer flows, as shown in Table 4. Why is this value used for the analysis?• The Coriolis parameter is sometimes given as 0.000124 and other times as 0.0001. Please use a consistent value or clarify the reason for the variation.• Line 30: Some of the referenced works concern mountain meteorology, not wind farms, as the current wording implies. These references should be removed, or this should be clarified.Citation: https://doi.org/
10.5194/wes-2025-60-RC1 -
RC2: 'Comment on wes-2025-60', Anonymous Referee #2, 30 Jun 2025
Publisher’s note: this comment was edited on 2 July 2025. The following text is not identical to the original comment, but the adjustments were minor without effect on the scientific meaning.
Review of the manuscript “Coriolis Recovery of Wind Farm Wakes” by Ronald B. Smith and Brian J. Gribben
In the manuscript “Coriolis Recovery of Wind Farm Wakes” the authors present a linearized two-layer model of the effects of the Coriolis force on wind farm wake recovery.
General remarks
Wind turbine and wind farm wakes have been studied extensively using numerical models, both engineering models (e.g., Calaf et al. 2010, Porte-Agen et al. 2020) and numerical weather prediction models (Aitken et al 2014, Rosencrans et al. 2024). The authors developed a linearized two-layer for wind farm wake recovery. The model accounts for the wake recovery by the Coriolis force. While recent work by Heck and Howland 2025 showed that the Coriolis force can play some role in the wind turbine wake recovery that effect is relatively small. Considering the length scale of (in particular) offshore wind farms it can be expect that the Coriolis effect on wind farm wake recovery is larger. However, the study presented in the manuscript does not provide a convincing argument for this hypothesis.
While the authors presented an elegant mathematical model that, for the most part, can be treated analytically, there are several assumptions that are not well articulated. First, the linearization used in the derivation does not properly account for the effects of atmospheric boundary layer (ABL) stability. Not accounting properly for the ABL stability effects likely exaggerates the impact of Coriolis force on the wake recover. Furthermore, specific application, wind farm wakes, imposes certain constraints on the problem that are not addressed in the manuscript. For example, wind turbine and wind farm wake recovery under convective atmospheric conditions is significantly faster due to energetic convective eddies, i.e. requires shorter distance from a wind turbine or a wind farm than under stably stratified conditions. The effect of Coriolis force and associated wind veering in a convective atmospheric boundary layer (ABL) are negligible and therefore the approach presented in the manuscript is likely not applicable to such cases, however, this was not considered since ABL turbulence was neglected. Furthermore, wind turbine and wind farm wakes depend also on wind speed. Under weak winds wind turbines either do not operate or generate relatively weak wakes. This means that the impact of wind farm wakes is most significant under near neutral to weakly stably stratified conditions. This is also ignored in the manuscript. The authors treat stratification through reduced gravity, giving values between 0.1 and 10, while never providing a definition of the reduced gravity. If we assume that the reduced gravity is commonly defined as: g’ = g \delta \theta / \theta_0 (e.g., Jiang, 2014, JAS), where \theta_0 is ~300, and \delta \theta potential temperature difference between the surface and the top of the boundary layer, then a reasonable value of the reduced gravity for conditions relevant for an operating wind farm is between 0 (neutral stratification) and 0.03 (weakly to moderately stable). Notice that the reduced gravity of 0.1 would mean that the potential temperature difference between the surface and the top of the boundary layer is 30 K. Such strong stability of an atmospheric boundary layer is achievable when the winds and therefore shear are weak. Under such conditions wind turbines do not generate power and therefore there are no wakes. Finally, the treatment of turbulence mixing induced by the presence of a wind farm is very simplistic and does not account for the stronger mixing and momentum entrainment induced by the shear at the top of the wake.
Taking all the above into account I do not recommend the manuscript for publication in the present form. The authors should attempt to put their work in proper context of a realistic conditions under which a wind farm operates. An analysis unconstrained by realistic conditions yields unrealistic results and leads to false conclusions.
Specific remark
- Line 109 – It is stated that “The vertical mixing process is difficult to model.” This statement should be qualified – it is difficult to model in simple models like the one presented in the manuscript.
- Line 110 – Two occurrences of the word “may” should be omitted and/or replaced with “is.”
- Line 112 – It is not clear why is Barstad (2016) cited here when the concpets are fundamental textbook concepts.
- Equation (9) – The second term on the left-hand-side should be FRR not FRC.
- Line 177 – “Understanding infinitely wide windfarms” is of no real value, since such a wind farm is unrealistic.
- Line 274 – Periodic solutions always wrap around from the exit to the entrace of the domain – the question is how the outflow impacts the inflow and the part of the domain that is of specific interest.
- Line 301 – Instead of “warm” it should be “farm.”
- Table 2 – Instead of “Inversion strength” better would be “reduced gravity.”
- Table 3 – The values of “reduced gravity” are not relevant for an operating wind farm, a more realistic values should be chosen.
- Equations (42) and (43) – instead of “Deficit(y)” a symbol representing deficit should be defined and used.
- Line 489 – However, the model does not include ABL turbulence and its effects, e.g., under convective conditions. This is a serious omission.
- Line 516 – Symbol “H” should be defined before it is used.
- Table 4 – It is not clear why would a reduced gravity value be dependent on the fram size (FS). In particular, the value of 0.1721 is likely unrealistically large for an operating wind farm.
- Line 540 – It is not clear what is meant by “low wind,” turbines do not operate below 3 m/s.
Citation: https://doi.org/10.5194/wes-2025-60-RC2
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