This work revisits the mixing in the stratosphere using on a metric based on the gradient of the so-called M-function which has already been used in several previous studies by the authors to visualize the Lagrangian coherent structures (LCS) in the flow. More precisely the module of the gradient of M is a metric for the local shear/strain along a Lagrangian trajectory and is akin to a measure of the horizontal isentropic mixing due to this shear/strain. This metric is here compared to effective diffusivity and another metric based on the age of air. This work is interesting and should eventually be publishable but it suffers from several issues in the present form that call for a major revision.

The main issue is a sort of reversal of the concept of proof. From an over simplified and a priori concept of what should be the mixing in the stratosphere, the results provided by the standard effective diffusivity are criticized to the benefit of the new metric. This cannot be a valid method in a system as complex as the atmosphere. The basic concept that less waves equal less mixing during summer is not necessarily entirely true. First it can be that summer waves are less intense but more efficient at mixing as stated in d’Ovidio et al., 2009, and Shuckburgh et al., 2009. Technically it is related to the fact that stable and instable manifold are generating more mixing when they intersect at larger angle. Then during summer period in the northen hemisphere, the permanent Asian Monsoon anticyclone is a sort of turnstile that generates a large amount of mixing between subtropical and mid-latitude lower stratosphere with southward transport on the east side and northward transport on the west side. There is a quite abundant literature on the topic (see eg., Dethof et al., QJRMS, 1999, doi: 10.1002/qj.1999.49712555602 and Ploeger et al., JGR, 2013, doi: 10.1002/jgrd.50636). Other monsoons have similar albeit smaller effects. Therefore, it is not implausible that such reasons and others make summer stratospheric mixing more efficient than a naive expectation based on wave activity alone. In any case, the validity of a new measure of mixing should be proven on a controlled case and not claimed from weakly funded a priori expectations.

There are actually a number of good reasons to use equivalent latitudes in the stratosphere, especially for the polar regions (see e.g. Allen & Nakamura, 2003) and this cannot be discarded for weak reasons. I tend to think of effective diffusivity as an optimal method to characterize isentropic mixing properties in a zonal mean. The Gupta’s method based on the age of air, although largely heuristic, takes into account vertical mixing as well.  There is a need to characterize longitudinal variations of mixing, not manageable by effective diffusivity and this is where LCS based methods have some room.

Besides this issue, the manuscript is often lacking of precision on many points. It is for instance very difficult to understand what is really shown as the LCS metric. L. 115-116 mention a maximum of the normalized gradient of M. If it is normalized and how, what is the meaning of the maximum? Then it is said that a PDf is calculated at each latitude, longitude and altitude but nothing is mentioned on the range of elements that enter this PDF nor of its normalization (density of PDF?). Then figure 2 is said to show the zonal average of this PDF but an averaged PDF is still a PDF not a single value. There are several possible ways to correct these statements so that they make sense but this is not the job of the reviewer. This is not helped by the fact that in general figures show quantities without unit and figure 1 lacks even a scale. In several instances of the manuscript the so-called density of LCS is quantitively compared to the effective diffusivity but we do not have the smallest idea of what should be the functional relation between these two quantities. There is no reason to expect a simple linear law.

Other more minor points

• The function M and its gradient is not really a Lagrangian invariant as it would be modified by an arbitrary rotation on the sphere. This is probably of minor concern here but one should not claim a mathematical property which is not satisfied.
• It is not possible to see the barrier effect of the polar jets in graphs that range from 60°S to 60°N in latitude.
• The WACCAM model is used here at very low resolution and there is no surprise in the fact that the PV estimate is noisy. PV is also very badly conserved by the dynamics in such configuration. Such resolution is usually imposed by a whole set of chemical components and complex chemistry but here only velocities are used. It is hard to understand the rational of this choice when reanalysis at much higher resolution and with good PV conservation properties are available.
• The limitations of the model are perhaps a reason of the discrepancies observed here. The differences between the diagnosed effective diffusivity shown in fig. 2 and that shown in Abalos et al., 2016 are quite significant and impact on the discussion.
• The text on l. 215-217 does not match what I see on figure 3. There is a strong seasonal cycle in the effective diffusivity in particular at 800K. It is difficult to appreciate its amplitude at 450K due to the saturated color bar. It looks smaller than for the LCS density but again, how can the two quantities be compared?
• Some hints on the sensitivity of the LCS density to the time parameter tau should be mentioned.
• The whole discussion starting on line 246 seems to me misleading and useless. There is no mystery that the effective diffusivity is large where the relation between equivalent latitude and PV (or tracer) fluctuates. This is in some way a built-in property of the method. There is no problem either in the fact that the relation between equivalent latitude and PV displays a seasonal cycle. In any case, the area enclosed within an equivalent latitude is the same as the area enclosed in this latitude. In passing, I m a but surprised that a Q contour with negative value is located at 40°N.
• The paragraphs starting on l. 107 and 115 are somewhat repetitive.