Please see the attached for review comments.

I found this to be a useful contribution to the literature on  the structure of coherent anticyclonic eddies in the ocean. The authors essentially fit idealized horizontal and vertical structure functions to five coherent anticyclonic eddies. A key novelty, compared with earlier studies are the use of potential density, rather than velocity, fields in fitting the models to the observations. The authors find that heaving anomalies dominate over spiciness anomalies - which is perhaps not surprising, but nevertheless valuable to confirm. There is also an interesting finding regarding the horizontal structure of the eddies, where the exponential power is higher than Gaussian or even cubic exponential - indeed I thought that this point could have been highlighted in the conclusions. Overall, the manuscript is well written  and should be published after minor revisions. 

I note that the first reviewer has commented extensively on the parameter fitting/optimization process and, in particular, number of free parameters as well as the detailed methodology. I  will not dwell on these issues here, on which I am not as expert, except to note that I agree with the points raised by the reviewer. 

Specific points: 

1. The first few sentences of the abstract need tightening. Firstly, I disagree with the statement "In situ observations and Lagrangian analyses have shown that most eddies are materially coherent", indeed the work of Abernathey and Haller (doi: 10.1175/JPO-D-17-0102.1) reaches a very different conclusion. The easiest solution, since this point is of tangental relevance to the present manuscript, is to revert to the text in the introduction about coherent vortices being long-lived and playing an important role in ocean circulation and transport.  There is also the separate issue raised by Abernathey and Haller of whether "eddy" is a noun or adjective, which again I would encourage the authors to sidestep by inserting the adjective "coherent".  Secondly, I don't understand the statement "laboratory experiments indicate that eddies locally modify stratification in accordance with thermal wind balance, regardless of whether they trap a water mass". It is not obvious that stratification needs to altered at all in a heaving mode, but merely that the isopycnals and raised or lowered. And at low Rossby number, away from boundaries, any flow should be close to thermal wind balance, but I don't believe it follows that this causes eddies to modify stratification, only that lateral density gradients and vertical shear will co-vary in accordance with thermal wind balance. 

2. I understand that it is convenient to use potential density referenced to the surface, but is there a good reason not to have used neutral density for this study, given that the amount of data involved does not seem particularly prohibitive? 

3. Line 84: I think you need to add that the stratification is assumed to be stable for invertibility. 

4. Figure 1: I suggest reducing the magnitude of the sea surface displacement in the upper panel. 

5. I understand the rationale for invoking the quasigeostrophic approximation for analytical tractability, but you should comment on the extent to which the quasigeostrophic  assumptions are (not) satisfied in the eddies under consideration, and whether this is a significant limitation, in your view, or otherwise (and why). 

6. I must confess I spent some time to derive equation (38) and wonder if you can provide some pointers to help the reader? Also, is there a good reason that the zero is moved to the left hand side of (37) compared with the original equation (34)? 

7. Line 215: I understand what you mean by the "nonlinear" term, but technically this remains a linear equation. It is nonlinear only in the vertical coordinate.

8. Diffusion arguments" - what specifically do you mean by diffusion? Is this physical, based on an oceanographic process? I have no idea from what is written, and this must be explained and justified.