In the manuscript the authors study the development of a wind turbine wake is studied in a moving reference frame. To be specific, the mean velocity and turbulence in the fixed field of reference are broken down into different terms, which are analyzed in the moving reference frame. In the present work the authors show that including the cross-terms improves the results. The effect of the cross terms is relatively modest given that analytical wake models generally have various simplifications. It is argued that this analysis is relevant to extend models like the DWM, which is submitted as a second paper, but not discussed here. Overall, such modeling efforts are relevant for the community to get improved insights. In any case the presented analysis does provide additional insights into the development of a wake. Although the statistical convergence of the data seems limited and uncertainty analysis is limited. These aspects should be improved.

- Line 228-230: "are not taken into account in this work, nor is the sub-grid turbulence. The latter is negligible in the unstable and neutral cases but can reach more than 10% in the stable case." --> How did you determine the sub-grid turbulence, and what day you mean by negligible? That seems a bit of a strong statement. 

- Line 314-315: "I wonder how accurate that is". How reproduce-able are your simulations? The turbulence in turbulence simulations is not necessarily exactly reproducible. The statistics can of course be reproduced, but not necessarily its instantaneous realizations. 

- Figure 8,9,10,12: "Your results do not seem to be that well converged". Additional discussion on the uncertainty in your data is required. 

- Given that you want to use the results for model development, should you not determine the developments of the different terms further downstream the wind turbine. Now you only go to x/D=8.

-- 375: the statement seems rather bold. Only 3 cases are presented, and the added turbulence intensity for the three cases is not really the same. Looking at figure 12; A difference of 3 to 4 percentage points on 16 percent turbulence intensity. 

- Line 391: How about figure 10? It is not quite clear what the statement "that it is reachable given the shapes." Just before you state "in the wake with the term (V) could be found yet." I tend to agree with the latter statement. I do not see any particular patterns

- Figure 12: How does the rotor added turbulence in the moving reference frame compare to what you would get in the flxed reference frame. What is your uncertainty in these results? There namely seems to be significant rotor added turbulence outside the wake.

* Typos

- Line 94: translation "opertor" --> "operator"; "time-dependant" --> "time-dependent"

- Velocity in the wake --> you are not actually showing velocities, but streamwise turbulence intensity.

General comments:

The paper presents an in-depth investigation into the specific mechanisms behind the mean velocity and turbulence following a wind turbine in a fixed frame of reference. The study breaks down the contributions and discusses the importance of each term within a stable, neutral, and unstable regime. By discussing the relative importance of each component’s contribution to the mean velocity and the turbulence, conclusions on what needs to be considered when modeling can be made and can be discussed specific to the stability regime.

The importance of the terms that contribute to the mean velocity and turbulence are found based on a large eddy simulation with an actuator line method. The study investigates the wake of a single wind turbine from near-field x/D=1 to far-field x/D=8 locations.  The referee believes the current work is of interest to those modeling turbine wakes and should be considered for publication after revisions. Comments are specified below.

Specific comments:

There are a points in the introduction when a citation should be included to validate the statement of the authors, for example, line 27 sentence beginning ‘Most analytical models..’

The coordinate system should be explicitly stated, perhaps placed in figure 1, to orient the reader initially.

Why does the study only look at the wake until 8D downstream (Computational cost, based on current farm arrangements, etc.?)

Line 267 – The authors discuss errors in the unstable case only at x/D = 8 (at 6%) but leave out the error at x/D = 3, which is touched on as a location where the neutral boundary layer flow case shows reasonable overestimation at 2%. Comparisons should be discussed at this location as well as farther downstream at 8D because this effects turbine placement downstream.

Lines 225-229 – If velocity is stored at a rate of 1Hz, then are the first and second order statistics averaged over, for example for the stable case, only 60 snapshots? Uncertainty of the turbulence breakdowns and convergence is not discussed.

Lines 225-229 – More information needs to be included on how the contributions of sub-grid scales are quantified and in turn negligible as the authors suggest.

Figure 7 shows the RSME of the maximum axial turbulence, is there an explanation for the max RSME value to be at x/D = 5 for the neutral case (is it also observed in other cases)? Also, no quantitative information is provided for the other two stability cases, only trends of the data and order of magnitude.

Technical corrections:

Line 94 – opertor => operator

Line 96 – …U_MF allows to re-write Eq. 1… => …U_MF allows one to re-write Eq. 1…

Line 134 – …in Sect. 2 is applied… => …in Sect. 2 are applied…

Line 321 - …turbulence is going back its unperturbed value.. => …turbulence is going back to its unperturbed value..