This manuscript reviews the different ways in modelling studies for estimation of heat conduction into the ice, and figures out that Holland and Jenkins (1999) best capture the variety of temperature profiles measured in boreholes. Overall, this is a well-written manuscript, however, there are several typos in the manuscript. 

1. Line 24, “solution of the thee-equation’’, typo.
2. Line 63, “temperature profiles found in Antarctic”, typo.
3. Line 100, “The Domain”, typo.

This paper presents a brief report on the representation of heat conduction into the overlying ice shelf in models of sub-ice circulation.  While that is potentially valuable, in that it makes a comparison of different approaches readily accessible, the paper is largely a reiteration of material that is already in the literature, and I struggled to see much added value.  It is a pity that what would be the main contribution of the study (a series of model runs that show the differences in computed melt rate resulting from the use of the different approaches) is mentioned only briefly with no in-depth analyses of the results.  The reader must refer to an earlier paper even to see the model setup.  That suggests that this brief summary would have been more appropriate as an Appendix or Supplement to that earlier paper.  In its present form, I don’t think it makes a sufficiently significant contribution to warrant publication as a separate paper in The Cryosphere.

In addition to the absence of significant findings, there are a number of issues that should be addressed in any rewriting:

1. In line 105 (and elsewhere) there is mention of the “error” made by two of the approximations, but I assume that “error” estimate comes from a comparison with the third approximation. That suggests an implicit, but unfounded, assumption about the correctness of the third approximation.  Likewise Figure 3 shows differences between two approximations, but neither is correct, and the evidence needed to favour one or the other isn’t shown.   Herein lies the main weakness of the study in that there is no correct answer with which any of the approximations can be compared.  If the authors really want to make a definitive statement about which approximation gives the best results, those approximations should be compared with the results of a full model of heat advection and diffusion in the ice shelf.  I realise that makes for quite a different study, but without that, nothing authoritative can be said.
2. Perhaps a partial solution would be to show and discuss in more detail the results that are briefly mentioned in lines 97-107, and figures 2b,c.  While still not a demonstration of how good or bad the various approximations are, that does at least give a demonstration of how influential the possible errors are on the results of an ocean model.  Ideally other simulations would be added to show the impact in a range of ice shelf environments.  The simulations are described as "idealised", but if real ice shelf geometry were used, it might be possible to compare results with melt rates inferred from observation.  While there could be many other causes for a model/observation mismatch, that would give an idea of how large the uncertainties are compared with other sources of error.  That might lend support to the statements in lines 113-120 that suggest the use of approximation (C) might be preferable to making other adjustments to the model,  a statement that at present is not backed up by evidence.
3. The authors seem to base their preference for approximation (C) on its ability to simulate the effect of temperature profiles observed in ice shelves.  However, the question of whether those observed profiles are in steady state with the present melt rate is not addressed.  If the profiles are not consistent with steady state vertical advection, then approximation (C) will be in error.  While the errors are likely to be small for an ice column that has experienced a long period of high or low melting, they could be significant where an ice column has recently been subjected to high melt, such as close to a grounding line.  In that key region approximation (A) or (B) might be preferable.
4. The discussion in lines 89-96 is a little misleading. When approximation (C) is used, melting will always occur when the ocean is above the pressure freezing point.  There is no possibility of freezing due to heat conduction into the ice when the water is slightly warmer than the freezing point.  The conduction term scales exactly with the melting and can never change the sign of the phase change.  Thus, using the thermal driving to determine if there will be melting or freezing is the correct procedure, and one that has been followed in all implementations of approximation (C), at least to my knowledge.
5. If this paper is really to be an authoritative summary of approaches to estimating heat conduction into ice shelf, that of Sergienko et al. (2013, J. Geophys. Res. Earth Surf., 118, 970–981, doi:10.1002/jgrf.20054), which considers lateral heat advection, should also be included, or at least discussed.
6. On line 33, kapp-sub-i is a thermal diffusivity (not a conductivity).
7. On line 73, I think you mean "when neglecting heat advection".
8. In figure 2b,c the horizontal axis label is "AW Temperature".  The meaning of "AW" is never clarified, and the reader must refer to the earlier study to make any sense of it.  A more appropriate axis label should be used.

General comments

In this brief communication, the authors examine different methods of representing the heat conduction into the ice currently in use in modeling ocean/ice shelf interactions in ocean general circulation models.  After presenting three different approaches, the authors show ice temperature profiles that are representative of the different approaches, compute the relative error over all ice shelves in Antarctica between two of the approaches (using an estimate that does not require running a circulation model), and compare the results from all three methods using an MITgcm circulation model with an idealized domain representing a fjord under a floating glacier tongue in Greenland.

I think the differences in how heat conduction into the ice is handled in current ocean GCMs with ice shelves is an important point that is worth the attention of The Cryosphere.  The manuscript was clear and easy to understand.  I would find this study more compelling if the authors had compared the different methodologies in either some realistic domain models or an idealized domain more representative of an Antarctic ice shelf (since the Antarctic ones are much more likely to be resolved in “Earth system models” (line 27)).  Perhaps though that is beyond the scope of a brief communication whose purpose is mostly to identify the problem, but not explore the entire range of differences between the methods.

I have other comments which I think would require some minor revision of the manuscript.  I am happy that someone is bringing this to the attention of the community and hope the authors are able to get this published.

Specific comments

Abstract, lines 7-8:  I think it would be helpful if the authors add the mean (or median) difference found between methods 1 and 3 in Figure 3.

Lines 39-53 and Figure 1:  I like the different temperature profiles showing dT/dz = 0 at the base (1a), vertically uniform throughout the ice shelf (1b), or much greater near the ice shelf base (1c) corresponding to the three common approximations in Eqn. 2.  However, did the authors look at estimated values of the basal melt and surface temperature at the locations of the Pine Island boreholes to see how the estimated dT/dz from Eqn. 2C actually compares to the observed gradients near the base?

Lines 76-77:  Suggest changing “overestimation is very similar” to “overestimation is often very similar” as I think there are significant parts of the Antarctic where using the linear temperature profile may not lead to a lower vertical temperature gradient than what you get including vertical heat advection (i.e. where the melt rate is low and/or the ice is not super thick).  For example, if I did the math correctly, mean values of the right hand side of Eqns. 2B and 2C for the Ross ice shelf are pretty much the same:  using mean depth 350m, mean melt rate 0.1 m/yr (Rignot et al., 2013), rho_i = 920 kg/m^3 (Holland and Jenkins, 1999), kappa_i = 1.14e-6 m^2/s (Holland and Jenkins, 1999)

Eqn. 2B RHS = (T_s – T_zd) * 3.0e-6 kg/(m^2-s)

Eqn. 2C RHS = (T_s – T_zd) * 2.9e-6 kg/(m^2-s)

Lines 82-83:  I’m not sure I agree that “most areas show a difference of around 12%” other than AmIS and RIS.  From Figure 3, most of the FRIS does look to be around 12%, but much of the smaller ice shelves are close to zero (yellow) except (significantly) the shelves in the Amundsen Embayment.  I suggest a slight modification of this text.

Line 103:  I don’t understand the phrase “In line with Fig. 2a” with respect to melt rates being very similar with Eqns. 2A and 2B since Fig. 2a only shows differences between approximations B and C.

Lines 113-114:  This implies (to me anyway) that modelers are tuning the drag coefficient just to avoid using an accurate approximation for the heat conduction into the ice and I don’t think that’s the case.  I believe it is just to get a better melt rate, regardless of how the heat conduction is being done, since the total errors are often much more than the ~ 12-15% shown here due to heat conduction.  I suggest a slight re-write of this sentence.  Also, I think it would be helpful to include a reference where tuning is mentioned.

Line 129:  If someone is assuming the ice sheet is a perfect insulator (approximation A), than I would not say that switching to approximation C comes at “no additional computation cost” because the code may have to add and keep track of a new variable (the ice shelf surface temperature).  Suggest changing “no additional” to “little additional” or “very little additional”.

Technical corrections

Equation 1 and lines 31-35:  Gamma_t (the temperature turbulent transfer parameter) in Eqn. 1 is not defined in the paragraph after the equation.

Line 38:  Why is rho_0 (density of pure water) given?

Line 66:  Missing right parenthesis to match the left one just to the left of “m < 0”.

Line 104:  Should “absolute melt rates” be “absolute melt rate differences”?

Line 106:  “<=” here should be “>=”.

Line 115:  Typo, “changes is ice shelf” should be “changes in ice shelf”.

Line 121:  Subject/verb agreement: “parameterizations ... has been identified”.