Review of "Sensitivity of the nocturnal and polar boundary layer to transient phenomena"

by Amandine Kaiser et al.

The manuscript analyzes transitions in the stable boundary layer (SBL) regime from weakly stable to very stable boundary layers, and vice versa. The analysis is based on a conceptual model proposed by van de Wiel et al. (2017) with a inclusion of a stochastic model to account for the effects of small-scale fluctuations of unresolved processes in the models. The study holds the potential to offer novel insights. However, I encountered some challenges while reading the manuscript due to its organization. The manuscript would require minor revisions to meet the standards for publication in EGUsphere.

Major comments

- The concept of potential was introduced at the end of section 2.2 and is only utilized in figure 2. However, this concept is not employed in the remainder of the paper. Could you please provide a more thorough explanation of the physical interpretation of this concept and how it relates to the other results?

- The legends for figures 1 and 10 are very small and incomplete.

- As a suggestion to improve the readability of the paper, I propose that section 2.3 be integrated into section 2.2 ("Impact of the choice of the stability function"). This consolidation would be beneficial since the results presented in section 2.2 involve randomized wind speeds. Alternatively, moving section 2.2 to section 2.3 would eliminate the need for readers to navigate back and forth in the manuscript.

One curiosity: What is the computational cost of the method to randomize the stability function for application in NWP? Is it necessary to perform an ensemble of simulations or just a single simulation for the climate and weather models?

Minor comments

line 120 - ODE is not defined.

lin 120 - "∆T, between a reference height (Tr) and the surface

temperature (Ts)" -> add "Temperature inversion".

line 125 - ∆T is defined in line 125, and is no longer necessary.

line 145, table 1 - units should not be in italics.

Review of “Sensitivity of the nocturnal and polar boundary layer to transient phenomena” by A. Kaiser, N. Vercauteren, and S. Krumscheid

This study considers fluctuation-induced transitions between very (vSBL) and weakly (wSBL) stable regimes in the stably stratified atmospheric boundary layer (SBL) in the context of a well-established idealized low-dimensional energy budget model. Three different classes of variability are modelled by stochastic processes: general unresolved physical processes as an additive noise, fluctuating wind speed as a multiplicative Ornstein-Uhlenbeck process (OUp), and deviations from Monin-Obukhov similarity theory (MOST) due to non-equilibrium turbulence in the VSBL as another multiplicative OUp.   It is found that combining external fluctuations with a short-tailed stability function produces more abrupt regime transitions than when a long-tailed stability function is considered.

My recommendation is that the study is acceptable for publication in Nonlinear Processes in Geophysics after major revisions. 

Major comments

1. Much of the analysis in the study contrasts the use of short-tailed and long-tailed stability functions.An essential question in the choice of stability function to model turbulent transport is the relevant horizontal scale represented by the model.  Long-tailed stability functions were introduced in the context of numerical weather prediction/Earth system modelling to account for gridboxes that are much larger than the horizontal scale of wSBL/vSBL patches, so fluxes computed over an individual grid box include contributions from regions of both states.  While use of Dome C station data to guide model development indicates that the model is to be interpreted locally, motivation for the analysis is presented in terms of NWP/ESM biases.  The revised study should carefully frame the analysis in terms of the relevant horizontal scales – and discuss the importance of this choice.
2. The results of van de Wiel et al. (2017) and Ramsey and Monahan (2022) indicate that conditions in which the PBL is bistable are relatively rare outside the very high latitudes.However, the title of the manuscript refers to “nocturnal and polar” boundary layers.  The revised manuscript should explain the emphasis on bistable conditions to the exclusion of more common conditions with single (deterministic) fixed points in which stochastic fluctuations can still induce rapid transitions.   I also recommend the title be revised to better reflect the parameter space on which the study focuses. 
3. Following on from the previous point, I note that for systems driven by multiplicative OUp noise extrema of stationary pdfs can exist that do not correspond to deterministic fixed points (cf. Monahan 2002).This effect can be understood in terms of a dynamical system driven by Brownian motion in the expanded state space corresponding to the original state variable and the OUp.  The authors may find this perspective useful for interpreting their results.
4. A novel aspect of the study is the focus on probability distributions from 24-hour simulations from fixed initial conditions, rather than the long-term statistical equilibrium stationary distributions.What motivates the choice of the 24-hour period?  If the focus is on the polar night, the effect of the diurnal cycle will be weak (other than through the Coriolis force acting on the flow above the shallow SBL, which is neglected in the model under consideration).  The revised study should provide a justification for this aspect of the experimental design.
5. Throughout the text statements are made along the lines of “outside of the bistable range low noise values are required for the system to approach the single equilibrium state”. By definition, and as is evident from the corresponding Figures, this noise level is zero.  The associated statements in the revised manuscript should be consistent with this fact.
6. LL 258-261: These results are not surprising and could have been predicted without the need of any numerical simulations.The revised manuscript should replace this text with statements regarding what new has been learned from the simulations, or clearly state that the cited results are expected from first principles.
7. Section 2.3.2: I am confused regarding the estimation of sigma_U.The text following Eqn. (4) indicates that variations in U are intended to represent submesoscale processes.  Are the Dome C data filtered to exclude processes on mesoscale and longer timescales?  If these other processes are included in the estimate of sigma_U it is perhaps not surprising that the variance is larger than had been expected. The revised manuscript should include a clearer description of the estimation of sigma_U.
8. The legend of Figure 10 is clearly wrong (or is rendering incorrectly on my version of Adobe Professional), and the caption only describes the left column.  The Figure should be corrected and the caption updated in the revised manuscript.
9. Section 2.3.3: The first set of results in this section show that on their own, multiplicative fluctuations of the stability function do not induce transitions between states. This is an interesting result that deserves further discussion.Later in the section, when both wind speed and similarity function fluctuations are considered, transitions are attributed to variations of the stability function (e.g. L 362, “However, with a randomized stability function …”). I do not follow this conclusion – in the presence of two sources of variability how do the authors attribute transitions to only one?  Finally, I do not understand how the authors conclude that “ … it follows that the probability of transitions significantly increases with the use of a randomized short-tailed stability function.” This statement seems to directly contradict what is said earlier in the section.  These issues should be addressed in the revised manuscript.

Minor comments

1. Table 1: There is a question mark in the units for c_v that should be removed.
2. LL 194-195: The text should note here that the distributions come from 24h simulations.
3. LL 201-203: Why “under-representation”?Certainly the number of vSBL states for long-tailed stability functions is smaller than for short-tailed functions but in neither case is there a clear “truth” against which to compare.  I recommend revising the text accordingly.
4. Figure 3 (and similar): The distributions presented are not densities as they are clearly not normalized.Rather these appear to be bin counts.  I recommend replotting as normalized densities or revising the label/caption.
5. The noise intensities sigma and decay timescales r both have units.These units should be included in figure axes and when values are quoted in the text.
6. The authors may consider combining Figs 6 and 9 to facilitate direct comparison of distributions.

I appreciate the importance of idealized model studies such as this one and look forward to seeing a revised manuscript.

Signed,

Adam Monahan

References

Monahan, A.H.  (2002).  Correlation effects in a simple stochastic model of the thermohaline circulation. Stochastics and Dynamics (2) 437-462.