Thank you for your appraisals and very useful review of the manuscript, for pointing out the uncertainty in the diffusivities and fractionation coefficient and their potential dependence on other factors, and for making a list of the key aspects with reference information to help me.

Emphasising parameter uncertainty is important and timely. This has not been done in the submitted manuscript, nor in the thread of key glaciological papers modelling excess diffusion.

Following your suggestion, I plan to extend a passage in the Discussion (probably Lines 574-580) to outline the issue, alongside the other model limitations already raised there. I probably will focus on the overall idea and not give much detail. However, I should be able to elaborate on a few details in Appendix A (probably not as many as you gave in your reivew), where the diffusivity parameterisations are described. The main message is that better knowledge of the parameters (notably the diffusivities) is necessary for reliable prediction/simulation of signal evolution in ice cores.

In terms of the degree of parameter variations/error, it is difficult to guess the amount as long as the parameters have not been comprehensively and rigorously characterised by laboratory experimenters. However, in the revision, I can try to comment on the potential impact on model results in general terms. A broad handle on the sensitivity can be gauged from Figure 2c (showing how the liquid-to-solid diffusivity contrast, ß, varies with temperature) and Figure 7 (showing how the functional surface of the enhancement factor f compresses or dilates in the w-direction as temperature varies). These figures together show that changes/correction of ß by a factor of two could compress or dilate the surface of f by as much as or a higher degree than seen in Figure 7. This consideration shows that how much a given f value changes depends on the signal wavelength and the vein-flow velocity. (The baseline decay rate also changes if the solid diffusivity D_s is corrected.) Howevever, the parameter changes affect the numerical results, not the underlying mathematical model. The shape of the surface f at a given temperature remains similar. I will briefly outline these ideas in the Discussion (if space allows) or in Appendix A. 

I thank both reviewers for evaluating the manuscript and providing useful suggestions. 

As described in my recent replies to them, I am happy to extend parts of the Discussion section of the manuscript to emphasise (i) the caveats around parameter uncertainty (probably describing a few of the details in Appendix A) and (ii) the importance for future work to test whether the process occurs, by sketching some ideas of using ice-core isotopic records for such tests (I have reservations about how far this can go before the process has been studied more at the grain-scale with laboratory experiments on ice samples, and given the current challenges of constraining/measuring vein sizes and vein-network connectivity in the ice cores, so these ideas will be tentative). 

Thank you for your appraisals and valuable comments.

It is highly reassuring to know that you find those experiments applying the model to the GRIP core and EPICA Dome C core to be useful.

Equation (10) is a Bessel Equation of zero order, with the general solution being a linear combination of J₀(sr) and Y₀(sr) (Bessel functions of the first kind and second kind) and thus having the form in (14). The results in (14) and (15) can be checked against the problem in (10), (11) and (12) by using the differentiation formulas J₀'(x) = -J₁(x) and Y₀'(x) = -Y₁(x) and using the equation

       (2α/a) F'(a) + (s² - βk_z² - ik_zw/D_s) F(a) = 0,              (*1)

which is found by subtracting the boundary condition (12) from (10) evaluated at r = a. Specifically, differentiating the solution (14) with respect to r gives 

       F'(r) = s [ -J₁(sr) + Y₁(sr)*J₁(sb)/Y₁(sb) ],                  (*2)

which satisfies the boundary condition in (11) at r = b. To check the other boundary condition, take (*1) and substitute for F'(a) and F(a) in it -- respectively, by using (*2) and (14) evaluated at r = a. We find 

       (2α/a) / (s² - βk_z² - ik_zw/D_s) = - F(a)/F'(a) = [J₀(sa) - Y₀(sa)*J₁(sb)/Y₁(sb)] / s[J₁(sa) - Y₁(sa)*J₁(sb)/Y₁(sb)] ,

which gives (15) after a minor reorganisation. These workings essentially illustrate the solution method. Including them here in this reply may be useful to some readers.

>>> Brief discussion at the end of the paper indicates Ng is well aware of the issue, but perhaps the most interesting issue looking forward is whether the phenomenon invoked here exists at all. (Or, to cast it in a different light: (1) whether the expected downward flux through the ice might be overwhelmed by variable local flows related to grain evolution, and (2) whether the vein diameters are large enough to allow for isotopically-significant fluxes.) It would be great to see a more thorough discussion of ways in which extant ice core records could be used to identify behaviors predicted herein, and possibly even invert for effective velocities of intergranular water advection.

I absolutely agree that establishing/testing whether (and to what extent, and where) the theorised process occurs in ice cores to cause excess diffusion is important. I will try to extend the Discussion to elaborate on this avenue. Probably I won't be able to describe firmly how existing isotopic records can be used for this purpose. I can only make tentative suggestions, because factors such as vein connectivity/disconnection (which is probably not simply predictable from impurity concentration) and vein-size /grain-size variations (even if one disregards the geometrical idealisation of the current model) are key unknowns that need to be measured to inform such tests. Their measurement is not trivial. Thus I think that while ice-core isotopic records should certainly be analysed systematically to map excess diffusion (and lack thereof), this work can only take us so far at present. Fundamental breakthrough in imaging the vein system and water percolation will be necessary. Critical efforts probably need to come (first) from laboratory studies of controlled ice samples to ascertain the grain- and vein-scale isotopic interactions and test the theory, as mentioned in the present Discussion. Hopefully these will advance our physical understanding of the process and the roles of different factors sufficiently for refined modelling and the results of ice-core analyses to meet in the middle. 

>>> Ng highlights the predicted distortion of frequency spectra relative to the case of pure white noise altered by standard diffusion. One finding that strikes me as useful is the potentially rapid rates of signal migration velocity. If those have occurred, the net translation will be significant in some cores. It should be possible to conduct analyses that compare water-isotopic measurements to measurements of impurities that are less mobile and, assuming initial coincidence of changes, identify separations that can be explained by this phenomenon. I also suggest that the portions of ice core records that lack any excess diffusion are also worth focusing on, as they should also be predicted by the present theory. Perhaps transition regions between impure ice and pure ice would be interesting to focus on as well, as the fluxes of intergranular water should be smoothed out across such transitions by pressure gradients, perhaps creating deviations from patterns observed elsewhere.

These ideas of harnessing (i) the migration and displacement between different records and (ii) differences in signal behaviour across the transition between pure and impure ice are really interesting. They may provide tentative tests of the theory's predictions, although again subject to the caveats described above. I will look for a way to mention them when revising the Discussion section.

I thank both reviewers for evaluating the manuscript and providing useful suggestions. 

As described in my recent replies to them, I am happy to extend parts of the Discussion section of the manuscript to emphasise (i) the caveats around parameter uncertainty (probably describing a few of the details in Appendix A) and (ii) the importance for future work to test whether the process occurs, by sketching some ideas of using ice-core isotopic records for such tests (I have reservations about how far this can go before the process has been studied more at the grain-scale with laboratory experiments on ice samples, and given the current challenges of constraining/measuring vein sizes and vein-network connectivity in the ice cores, so these ideas will be tentative).