The importance of noise and its ability to change the mean state of an ice sheet model is a relatively novel research direction and this manuscript makes important contributions to further our understanding of its role. The methods used in this manuscript are sound, but I think the conclusions in this manuscript would be strengthened with more modeling results. As it is, I do not agree that all conclusions are supported by results (see more detailed comments below). I also think that the paper would benefit from some restructuring and editing, as it appears a bit disjointed at the moment. In particular, I would move sections 3-3.3 before section 2, rather than after, and extend the abstract to be more informative.

Abstract:

To me, the abstract seemed too short and not very informative. It would be more informative if it included more details about the methods used. Also, it would be good to point out that stochastic variability in frontal ablation changes the mean state more than variability in other parameters.

Introduction: 

• The introduction is quite a bit more general than the abstract which refers to frontal ablation. It's almost as if the purpose of the paper changed from abstract to introduction. Please streamline.
• Line 21: "we show that noise-induced drift is expected in [...] any numerical modeling of ice sheets." I am pretty sure it is possible to construct counter-examples which do not show noise-induced drift, so you might want to be careful with such sweeping statements.

Section 2:

• The variability in water pressure is quite a bit smaller than the variability in calving and only specified in an unspecified "region near the glacier front", which makes a direct comparison with calving and SMB variability difficult. I recommend either including more information or leaving it out completely.
• Line 75: "the rate of drift is also approximately proportional to the amplitude of the variability in calving rate": Fig. 1 does not show this. The figure shows that the drift observed with stochastic calving appears to increase with the noise variability (interpolating from 2 data points) and that the mass change induced by stochastic noise might be reproduced by a higher deterministic calving rate. More data examples would be required to make more definite statements. If feasible, I highly recommend including more data in order to analyse the behaviour in more detail.
• It would also be interesting to see what the model behaviour for stochastic noise in calving, SMB and p_w combined is. Are drifts due to different processes simply additive, or might there be some nonlinear bahaviour?

Figure 1 legend: what does St. stand for? I think writing "stochastic" out would be clearer.

Section 3:

• Equation: personally, I would prefer f and g to be bold too, to be consistent with x as a vector
• Line 158-160: The effective pressure used in this study should be N=ρ_i g H + ρ_w g b. It would be useful to write this out here, so that readers not familiar with this particular choice of sliding law are immediately aware of the dependence of N on H. While I see how N varies linearly with changes in ice thickness, it does not necessarily follow that variations of the terminus position lead to linear variations in ice thickness or that there is a linear dependence of u on H; please rewrite this to be clearer.’
• More generally in section 3.2 it is not clear why the non-linear dependence of u on H leading to drift is fundamentally different from the case considered in 3.3. To me, the only difference is that the effect on effective pressure is located away from the terminus, which makes it less amenable to a Reynolds decomposition - and which would also explain the potentially smaller effect of these perturbations.
• Line 211: though -> through
• The analysis in section 3.3 makes specific predictions about the size of noise-induced drift depending on σ, and . It would be helpful to verify and illustrate these results with the idealized marine-terminating glacier model.

The manuscript describes ice sheet model experiments and a theoretical analysis that deal with the issue of model drift introduced by including stochastic forcing in simulations. The analysis shows that noise-induced model drift is to be expected for model setups that were initialised with deterministic forcing and then activate stochastic forcing in forward experiments. It also suggests that other setups could be impacted by the absence of realistic stochastic forcing. The paper is well written and presents a thorough analysis of the problem, albeit without offering an approach how to make meaningful projections under stochastic forcing. 

General comments

1. In the context of this study, it seems important to discuss what a 'realistic stochastic forcing' (l23, l50) actually is (in amplitude and temporal variability) and at what level that forcing is symmetric. It appears that the existence of a drift and its amplitude could crucially depend on how exactly the forcing is parameterised. In case of SMB, there is the suggestion that symmetry in temperature variability could translate through a non-linear SMB model (e.g. PDD) to an SMB forcing with an asymmetric variability. This would mean that results depend on the level at which symmetry in the forcing is prescribed. Could the same be true if instead of perturbing ocean thermal forcing or calving, a higher level ocean forcing, like advance/retreat is randomised symmetrically? While this is interesting for the interpretation of case 2 (Multiplicative noise) and case 3 (Nonlinear or asymmetric noise) in the theoretical analysis, I would think it is especially important for finding a practical approach to initialising a stochastic ice sheet model.

2. Model drift in the mean of the simulations is shown to arise by switching from deterministic forcing in the spin-up to stochastic forcing in the forward experiments. While this is an instructive example to understand something about the effect of stochastic forcing in ice sheet simulations, it clearly shows that this stochastic model setup cannot be operated for projections. In other words, what is really missing here is an approach/recommendations/proof of concept how to initialise a stochastic model in a meaningful way. How would an initialisation look like for i) an assumed steady state some time in the past, ii) a situation of mass gain, iii) the present day state (e.g. approximately matching the ongoing mass change)? Without offering a solution, I think the paper should step away from making recommendations about the use of stochastic forcing in ice sheet simulations.

Specific comments

Title "Biases in ice sheet models from missing noise-induced drift"

I find the title confusing as it suggests that missing drift in ice sheet models is a problem. I would characterise the drift that is realised in the presented experiments is an artefact of switching from deterministic forcing in the spin-up to stochastic forcing in the forward experiments. Maybe "Biases in ice sheet models from missing stochastic variability".  

l1 Abstract

The abstract is quite short and could be improved by adding more information about important aspects of the paper. I am thinking about more detail on section 3 and the three causes of noise-induced drift.

l13 "the mean state of glaciers and ice sheets"

It is not clear to me what the mean state of an ice sheet is without knowing the time scale of interest. Is that diurnal, annual, decadal, centennial or millennial? Maybe remove "mean" to avoid that complication, or introduce the time scale of interest, probably multi-decadal?

l25 "We close by arguing that all modern ice sheet models omitting variability in climate and glaciological processes produce biased estimates of the ice sheet mean state and the ice sheet response to climate change."

That's a very strong statement that I think it needs some moderation. First, it seems difficult to come to a conclusion about all models by only looking at one. Second, there is at the very least the possibility that the bias in a projection is zero, or close to zero even if we know that a model could be biased in theory. Adding a "could" in  "processes could produce" would help to mitigate my concern. 

l46 "The model domain is split into 19 glacier catchments"

Is this split effective in the implementation or only for diagnostic purposes? If the first, explain how the separate catchments communicate.

l47 "exhibits an increase in ice mass by only 0.07% in 2000 years." 

Even if small, it would be interesting to know what explains the residual drift in the control experiment and also where that drift occurs. Are changes happening in the same areas as in 2c? Are there compensating mass gains and mass losses in different regions? It is sometimes useful to calculate the integrated absolute thickness change or similar to avoid compensation of gains and losses. 

l55 "the standard deviation [...] is set to 1/3 of the mean in that catchment. This amplitude of variability is chosen for simplicity but is similar to observed variability".

How exactly do these compare to observations? It is clear that the larger the amplitude of the stochastic perturbation, the more likely a non-linear response and consequently a drift is observed. It should be shown that the (un-)desired outcome (causing drift in the stochastic simulations) is found even for conservative estimates of those amplitudes. With this in mind, is 1/6 or 1/3 of the mean closer to observations?

Figure 1 caption "deterministic but with calving rates multiplied by 2.7"

Is the spatial pattern of this experiment similar to the results shown in Figure 2c? 

Would be interesting to discuss. Could compare results in figures similar to 2a,b. 

l58 "is set to just 2% of the mean ice overburden pressure"

Could you explain a bit more? What values would you have liked to set the standard deviation to if you could freely choose? What would have motivated this number? Observations? (See point l55).

l59 "greater levels of noise lead to numerical instability in the ice sheet model"

How far away from instability is the model operated in the experiments that are deemed stable? What happens to the model when it is operated close to instability? Is the model sensitive to the sign of the perturbations, i.e. more stable with high or low water pressure levels? I think it needs some more work and words to convince the reader that none of the presented results could be influenced by running the model close to instability. I don't know of any rules of thumb, but running a model at 1% when it is known to be unstable for >2% is a precaution one could take. 

L61 "observe the immediate ice sheet response"

Not sure many people would agree 2000 years is the immediate response. Maybe "to observe which state the ice sheet evolves towards".

L75 "At the end of the 2000-year simulation with highest variability amplitude"

Along with these long-term, high-end numbers it would be instructive to give a perspective of what this would mean for a typical centennial timescale projection. Guessing from the figure, the drift in the calving ensembles is somewhere between 7 and 14 mm for a 100 year projection. If I understand well, the lower number (equal to 0.1 %) is discussed elsewhere as the criterium to define an acceptable ensemble range (l66). I think that could be compared and mentioned here. 

L77 "without stochastic variability, but with a 270% increase in the mean calving rate"

If the constant increase of the calving rate by 270% gives similar mass loss as the 1/3 mean stochastic case, what is the temporal structure of the amplitude variations in the forcing and in the simulated calving rate of the latter. Is it possible to visualise the two experiments alongside? What are the peak rates and how often are they coming out of the parametrisation and how realistic are these compared to e.g. observed speedup events for Sermeq Kujalleq? What is the time step of the stochastic variability? Is it comparable to observed (seasonal) speed changes?

L80 "A deterministic model [...] would require tuning far away from true parameter values" 

I don't think true parameter values exists. For the type of models discussed here, tuning is always a calibration with compensating effects. Maybe "original parameter values"?

L267 "start from a calibrated initial state"

Not all of the models participating in ISMIP6 have a calibrated initial state. Reword to "e.g., with many of the models participating in the recent ISMIP6 intercomparisons".

L270 "Other recent modeling studies use this same spin-up procedure"

Adjust similarly in response to comment in L267. For example "Other recent modeling studies use a calibrated initial state, but then re-calibrate"

L277 "we recommend including internal variability in the forcing of ice sheet models, both during spin-up and transient simulations" 

I disagree with this recommendation, because it does not follow from the analysis in the paper. You have shown with your modelling that including internal variability after the spin-up can cause model drift. I think the recommendation that can be clearly derived from these results is that for projections one should avoid initialising a model to a deterministic forcing and then apply stochastic forcing. I think it would be great to have some recommendation about how to use stochastic forcing during the spin-up, but the paper currently provides no analysis of that case. See also my general point #2.