This manuscript is a pure modelling study (following a recent study of the same type by the same author) of the diffusion through vein and grain boundary in addition to slow diffusion in ice. Initially, larger than expected diffusion (excess diffusion) has been observed in several cases ; those cited in this study are GRIP Holocene, WAIS deglaciation and EPICA Dome over Marine Isotopic Stage 9. Studies of grain boundaries and vein contributions to the diffusion were already performed, e.g. in the book chapter of Johnsen et al. (2000) cited in the present study. The present study wants to go one step further compared to Johnsen et al. (2000) and following his recent paper to show the expected isotopic patterns to be observed at the grain scales for a range of parameters.

The main problem is that no observation is provided and even if laser ablation techniques are progressing, we are still far to the point where such observation can be done and it is also not clear that observations will actually be possible in a near future (see following comments). As it stands now, this study is focused on the resolution of the differential equations for diffusion through ice, grain boundary and vein in cylindrical coordinates. This is a serious calculation work giving the expected changes of patterns with changing parameters. I do not see how it can really be used by others as long as no observation is provided and there are no clear other scientific perspectives for this work. But the study is seriously conducted and well detailed.

I concentrate below on comments along the text.

- Introduction : the introduction is well written and summarizes previous findings on diffusion. It is interesting that the author mention the 3 cases where excess diffusion has been identified but I am wondering if the origin of the excess diffusion is the same for all three cases. Indeed in the case of EPICA Dome C, the ice is very deep and old, with relatively large ice crystals and the origin of the excess diffusion may be different than for WAIS and GRIP Holocene. It would be nice to have a discussion on these differences and perhaps discuss the possible mechanisms in each cases.

- 2.1- the system is clearly described

- 2.2- Figure 2 is complicated to understand and it was only much later during the reading of the manuscript that I could really understand what « high », « medium-high », …, « low » mean. It should be explained in this section as well as in the caption of figure 2.

Also, in some places the description is not accurate, e.g. « departures from the formulas by a few times », « Pre-melting occurs at high temperature », « ice-core samples can be very variable in these », … and should be precised.

1. 160 : the author refers to the hypothesis of liquid diffusion at -32°C for Db and say that this estimate will be ruled out but they do not give the value for Db in this case and we can not really see it in Table 1.

In general, it would be nice to make a more clear link between the hypotheses listed in p. 7 and the values for Db explored in this manuscript and listed in Table 1. Perhaps the authors could take in Table 1 the values of Db from the litterature (or explain how the range of chosen values are linked to previous estimates) and refer to the different papers in Table 1 so that we can make the link between the present study and the previous studies.

- The fractionation explanation is not very clear from l. 162. Which fractionation does the author refer to ? No data for fractionation is given nor its dependency to temperature.

- 2.3 – The formulation and resolution follow a classical mathematical approach. Still, because many parameters and variables are involved in this resolution, it would help to have somewhere a table gathering the different parameters and variables, explaining their meaning and giving their values. As it is now, it is difficult to read. It is also certainly possible to have the details on the resolution (e.g. the sections 2.4, 2.5 and 2.6) in an appendix and go directly to the results once the problem and the way to solve it has been defined.

3-

-l. 369 : no reference nor explanation for the choice of the fractionation coefficients are given

- l. 373 : I do not see why studying signal wavelength of 5 mm ? Such signal is not detectable in ice core, even in CFA because of mixing in the system – at best the resolution could be 1 cm which will not enable capturing such signal. Also the following discussion (section 3.1) and display of the results (figure 4 and all figures after figures 4) with a wavelength of 2 cm is not very realistic. Also because firn diffusion occurs with a diffusion length of several cm, it would be more realistic to discuss signals of wavelength 10 cm at least. This is a major limitation of this study and it is important to address it for more realism and potential use of this study for ice core people. Some figures are shown in the supplement (showing much less excess diffusion) but not really discussed in the main text.

- l. 376 : intermediate

-  l. 408 and after : some terms need to be better explained : « vertical stretches », « stretch transition », « transitions » (from what to what ?), « hole type», « spoke type»

- l. 447 : The sentence lacks a word.

- l. 464 : when should we have vein-water flow – please explain this case a little bit

- when discussing enhancement factors in this study (e.g. figure 6), it should be compared to what has been measured (if possible on signals with similar wavelength in the data and modeling approach).

- When showing isotopic patterns in the figure, the unit should be provided

- l. 634 : I do not really see how a signal of 2 cm wavelength can be well determined by CFA, see comment above.

- l. 670 and below: the author correctly notes that it is really difficult to find ice to test this effect since the isotopic signal usually measured has a wavelength much higher than 2 cm and for short wavelength, the amplitude of the signal is accordingly small.

- l. 729 : a sensitivity of 0.1 per mill is required but for which isotopic delta ? d18O or dD ?

Dear Author and Editor,

Please find attached my review for the manuscript.