In the present manuscript, various aspects of the blessing of dimensionality are analysed theoretically and experimentally. Specifically, high-dimensional (HD) spaces are shown to feature the phenomena of the concentration of measures and waist concentration which are explained at an abstract level, although with clear references to previous climatological research. The authors show that independent vectors in high dimensions can be taken to be orthogonal and how sample means can be treated as a constant. The extension to dependent samples is discussed by making a review of the notion of effective dimension and the mixing/decorrelation rates of a time-evolving system. Then, three HD climate data sets are studied and shown to match, generally, the earlier theoretical discussion. The CMIP5 data set was argued to posses somewhat dependent samples and thus, the blessing of dimensionality (in the sense of concentration of measures and waist concentration) did not fully apply in this case. Exploiting the orthogonality of independent vectors in HD, the author finds that the correlation between sample differences is equal to 0.5 and that correlations of sample differences and differences between ensemble mean and ensemble members is equal to 2^0.5. This calculation is reflected in the climate data. The importance of sample size is also discussed in the context of HD data, whereby the authors show that ensemble mean length converges to the true mean at a rate inversely proportional to the sample size. This fact is illustrated in the conceptual cases of Gaussian and Gamma distributions as well as two of the climate data sets.

The paper is concise and well explained, linking theoretical findings with experimental ones. I would encourage the Journal to accept this manuscript if some minor comments are taken into account.

Specific minor corrections and comments:

Line 107: are the vectors a and b sampled from the Gaussian distribution mentioned above or are they generic?

Line 119: The inequalities in Eq. (4)-(5) very much resemble a large deviation principle. Making a reference to large deviations theory will strengthen the connection of high-dimensionality with the statistical mechanics theory presented in the paragraph starting in line 99

Line 185: When presenting the climate data, it would be very useful for the reader to know the dimensions N and sample size K in each case. Certainly, this is done later when analysing each case, but I believe doing it earlier could help the reading.

Line 228: The author mentions that x^k - \bar x is orthogonal to \bar x in high dimensions. This fact follows from the concentration of measures result presented in section 2.2, so I would encourage the author to make a clear reference to the theory.

Line 240: The reference to the spikes of the unit cube is very helpful and in fact it illustrates the comment done in Line 228. The author might consider making this analogy earlier.

Clearly, the CMIP5 outputs deviated the most from having waist concentration, i.e. it provided non-orthogonal samples. This was attributed to high dependence. To what extent could it be attributed, instead, to low effective dimension of the climatological fields? From a personal perspective, I’d be interested in hearing the author’s further views on this general question.

Technical Minor Corrections:

Line 18: “This might seem…”

Figure 1: In the caption there is no reference to the right panel (two references to the “left”).

Line 171: “Other measure is…”

This paper has a strong didactic focus; much of it is a review of convergence theorems and properties when a high number of independent variables are available, i.e. high dimensionality. A lot of it is reasonably well-known, but the way it has been put together in this paper seems quite valuable. I am particularly pleased to see that basic ideas that existed for a long time in statistical mechanics (thermodynamic limits, essentially) are here being highlighted as potentially applicable to climate data. The main result seems to be that a selection of climate data is shown to be behaving as if it is drawn from a (moderately) high dimensional space. This has important repercussions for much literature that has not been highlighted in this paper, in my opinion, particularly around regime behaviour in geophysical data - NB: I am not suggesting you should now include such a discussion, but it may be something to think about as a further application.

In my view the paper is well written and can be published pretty much as is, except that I invite the author to address a few minor points/questions first. I describe them in order of apperance, below.

l.85 & l.92: "a constant" I know this phraseology is used in related literature, but I do not like it very much. I would have prefered to explicitly say something like "... a constant for differnt realizations of the sampling process," or something similar. Essentially this is a frequentists statistical argument, suggesting there is a fixed distribution mean (the "constant") which can be approximated by a sample mean.

l.116: "sub-Gaussian" I am happy with this word, but it may be useful to include a one-sentence definition of it (I am not sure how widely known the word is.)

Section 3: the discussion of effective dimensionality is a well-worn topic in geophysics, and it remains an important topic. It reminds me of a paper I wrote some years ago (doi:10.1175/2007jas2298.1) on how suggested mulimodality in a wave amplitude index for the atmosphertic circulation possibly is a statistical fluke: it hinges on exactly the high dimensionality argument you are discussing here, but interpreted slightly differently. In effect, standard statsitics (moments) from a high dimensional data set are always expected to exhibit these "blessing of dimensionality" properties, and more fancy, non-linear properties, such as multimodality, are more-or-less by definition excluded. I think this is an important application of the high dimensionality property, and I wonder whether you care to comment on it.

Section 4.2: I thought this section, despite its simplicity, was really thoughtprovoking. I tried to interpret this in light of the well-known "signal-to-noise paradox" (https://doi.org/10.1038/s41612-018-0038-4), as it seems to be germane to that problem. Does the author agree that his discsusion here may shed light in the signal-to-noise problem? It would be a rather important application.

l.333 and Abstract: Perhaps I did not catch it but the dimensionality of 25-100 seems to fall a little bit from thin air. Can you please highlight where this estimate is based on? Furthermore: Figs 2 & 3 show empirical distributions of \phi, which have a given shape for independent data (Eq. 5). I would have thought that you can fit a Gaussian to those distributions and thus estimate a value of N. Did you do this? Does it give you the 25-100 estimate?