“Statistical properties of a stochastic model of eddy hopping” by Saito et al. 2021

This paper discussed a Lagrangian stochastic model for supersaturation fluctuations driven by vertical velocity fluctuations in clouds. This model has been used in the past for simulating the interactions of unresolved supersaturation fluctuations and Lagrangian super-particles (in the super-droplet method). The paper discussed the scaling relationship of supersaturation variance in large and small length scale limits. The authors suggest that the current “eddy hopping model” has limitations in representing the correct scaling at small scales. A correction to the model is introduced (an additional drift term representing the turbulent mixing). A further modification is introduced by multiplying the timescales with constant coefficients that adjust the magnitude of the supersaturation variance. One of the corrections was already included in earlier papers (additional drift term from turbulent mixing), although the evaluation of the supersaturation variance scaling was not presented before. The new simplified model for the super-droplet method is good. But it might not provide much computational benefit if the subgrid-scale transport of super-droplets is also needed. Overall, the content of this paper is important, and it would help model the subgrid-scale supersaturation fluctuations in a correct and consistent way. Derivation steps provided by the authors in different sections seem to be accurate, and the results are convincing.  The results are also presented clearly and concisely. Apart from the above general comment, I have the following specific comments:

L120-125 and Eq. 19: I agree the drift term due to turbulent mixing is necessary for correctly representing the supersaturation fluctuations. In fact, it was included in some of the past studies. However, a corresponding complementary diffusion term (the Wiener increment term) representing small-scale fluctuations/mixing would also be required in the corrected model.

L132-136: Are the drift coefficients introduced just to scale the magnitude of the supersaturation fluctuations to a correct value, or are there any other physical reasons? More explanation would be helpful for the readers.

A figure showing sample supersaturation trajectories from all three models (original, corrected, and simplified) could be informative (probably in the appendix section). 

It would also be good to discuss the limitation of the current approach in representing the supersaturation fluctuation generation from scalar mixing (e.g., during the turbulent entrainment-mixing ).

The submitted manuscript presents a study of the statistical properties of the stochastic model of eddy hopping. The Authors validated the model (in the statistically stationary state) against reference data from direct numerical simulations (DNS) and Large-eddy simulation (LES). However, the Authors claim that they have "corrected" the stochastic model of eddy hopping. I do not agree with this statement.

The stochastic model of eddy hopping was designed to mimic supersaturation fluctuations at unresolved scales in LES Lagrangian cloud models. The model has been presented in two articles cited in the References: Grabowski and Abade (2017) (GA17 hereafter) and Abade

et al. (2018) (AGP18). GA17 presented the first version the model, which is valid in the regime of large Damkholer numbers (or fast microphysics). For fixed microphysical characteristics (such as the droplet integral radius), the Damkholer number depends on the resolution length scale "L" of the LES. Typical values for "L" used in cloud-scale flow simulations range from several meters to several tens

of meters, then yielding a Damkholer number of at least O(10). In this regime, the GA17 is a useful model.

AGP18 extends the eddy hopping model by introducing a term accounting for the relaxation of supersaturation fluctuations due to turbulent mixing. The extended model by AGP18 mimics supersaturation fluctuations in the whole range of Damkholer numbers, from fast to slow microphysics. However, the Authors of the submitted manuscript claim to have corrected the eddy hopping model of GA17 by including a relaxation term that have been already included and studied by AGP18.

The Authors claim to have introduced a second correction of the model by introducing tuning parameters (the factors c₁ and c₂

multiplying the two time-scales involved: the phase relaxation time and the turbulent integral time). In GA17 and AGP18 the constants c₁

and c₂ are assumed to be of order unity and the results reported in the submitted manuscript confirms this assumption. Thus, the

introduction of tuning parameters to fit the model to reference data cannot be regarded as a "correction" to the model (particularly when

those fitting parameters turn out to be O(1)), unless theoretical expressions for the parameters and a rationale are provided.

The phase relaxation time encapsulates intricate interactions among growing droplets by condensation. Also, the Lagrangian integral

time-scale requires a priori knowledge of the Lagrangian correlation function, which we do not possess. Thus, a better theoretical

parametrization of the two time-scales involved is a difficult task and this task has not been tackled in the submitted study.

Finally, the Authors suggest a simplification of the model that eliminates the vertical velocity fluctuations w' from the list of stochastic variables attached to computational particles (superdroplets) in LES Lagrangian cloud models. The discussion and procedure of model simplification is instructive and gives valuable insights into the model. However, this simplification does not necessarily imply in reduction of computational cost. This is because the velocity fluctuations of superdroplets are usually necessary to resolve the subgrid-scale transport of superdroplets. Also, the simplified model accurately reproduces time correlations in the Lagrangian supersaturation in the regime of large Damkholer numbers. This is exactly the regime of applicability of the GA17 model.

Overall, the submitted manuscript is instructive and gives valuable insights into the statistical properties of the eddy hopping model. However, the Authors should state differently their contribution to the field.