the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Impact of formulations of the nucleation rate on ice nucleation events
- 1Institute for Atmospheric Physics, Johannes Gutenberg University Mainz, Germany
- 2Zentrum für Datenverarbeitung, Johannes Gutenberg University Mainz, Germany
- anow at: DB Systel GmbH, Berlin, Germany
- bnow at: German Weather Service (DWD), Offenbach, Germany
- 1Institute for Atmospheric Physics, Johannes Gutenberg University Mainz, Germany
- 2Zentrum für Datenverarbeitung, Johannes Gutenberg University Mainz, Germany
- anow at: DB Systel GmbH, Berlin, Germany
- bnow at: German Weather Service (DWD), Offenbach, Germany
Abstract. Ice formation in cold temperature regimes is most probably dominated by homogeneous freezing of aqueous solution droplets. The nucleation rate as derived from laboratory experiments can be represented as a function of water activity. For idealized nucleation events as modelled with a state-of-the-art ice microphysics, the impact of different approximations of the nucleation rate on the resulting ice crystal number concentrations and maximum supersaturation ratios is investigated. The nucleation events are sensitive to the slope of the nucleation rate but only weakly affected by changes in its absolute value. This leads to the conclusion that details of the nucleation rate are less important for simulating ice nucleation in bulk models, if the main feature of the nucleation rate (i.e. its slope) is represented sufficiently. The weak sensitivity on the absolute values of the nucleation rate suggests that the amount of available solution droplets also does not crucially affect nucleation events. The use of just one distinct nucleation threshold for analysis and model parameterisation should be reinvestigated. The frequently used thresholds corresponding to a very high nucleation rate value is not reached in many nucleation events with low vertical updrafts. In contrast, the maximum supersaturation and thus the nucleation thresholds reached during an ice nucleation event depend on the vertical updraft velocity or cooling rate. This feature might explain some high supersaturation values during nucleation events in cloud chambers and suggests a reformulation of ice nucleation schemes used in coarse models based on a fixed nucleation threshold.
Peter Spichtinger et al.
Status: final response (author comments only)
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RC1: 'Comment on acp-2022-434', Anonymous Referee #1, 15 Jul 2022
In this paper the authors revisit formulations of the nucleation rate coefficient for homogeneous nucleation of aqueous solution droplets. The mathematics of their approaches is derived thoroughly and the results of numerical experiments are presented, so that the reader can see the effect of various simplifications.
This is a very good paper, but I think, it can still be improved.
In particular, the motivation for this work is not expressed clearly. This is my major point of critique, and I think with a minor revision the paper should be publishable.Specific comments:
"In this study we will investigate the impact of the formulation of the nucleation rate on the resulting ice crystal number concentrations."
I miss the motivation for this re-assessment. Has it turned out recently that the usual formulation (using Koop et al. 2000 essentially) is wrong?
The authors mention that the old Mishima-Stanley theory is now ruled out and that two alternative water theories have survived which give similar results for upper-tropospheric conditions. It is not clear to me, however, how this fact impacts on the Koop-etal 2000 findings and their formulation of the nucleation rate coefficient. The implications of this finding is not given in the paper, and in fact, these theories are not mentioned again. It is thus unclear why this is mentioned at all."However, the representation of these processes contains still uncertain parameters or even the (mathematical) formulation of the processes remain uncertain." Perhaps this is the point where the authors should become more concrete. What are the uncertain parameters and in which way is the mathematical formulation uncertain.
Line 75: It would be good to list the mentioned parameters.
Lines 105 ff: It is not easy to understand the statement "constant temperature and pressure is assumed" when you just before present the T and p tendency equations. My understanding is this: you do not need p and T themselves, all you need is the changing supersaturation which in turn can be prescribed, such that the proper causes for changing S, namely changing p and T, need not to be explicit. If my interpretation is correct, I suggest that the authors change their explanations accordingly. In this case, why do we still need p_env and T_env, is this because there are p and T dependent parameters in the equations (e.g. diffusion)?
Line 120: You may add that such a high value is mainly chosen in order to let the nucleation process run without consumption of the available aerosol droplets, which would complicate the interpretation of the results.
Eq. 20: As aw and Delta aw have a special value for water saturation, say Delta aw* I would prefer to have the equation in this form J_sol(Delta aw*[T])=J_pure liq(T).
Line 164: Can we unsderstand why this equality fails?
Line 168: strictly speaking, the log of the nucleation rate is shifted, which means that the nucleation rate itself is to be multiplied by a constant (obviously a constant far away from unity).
Figure 1 caption: Not completely clear. I think the blue line is for solution droplets with infinite dissolution, that is pure water droplets. This should be clear in the figure caption.
Line 176: I still don't understand this. Probably I did not follow the most recent literature, but who (reference) has found that the Koop etal 2000 parameterisation is wrong? Can the results of Koop and Murray for pure water be used as a proof for the falsehood of the older formulation when the latter is applied to solution droplets? In order to accept the shifted Koop 2000 line as a reference, one needs to know what is wrong with the unshifted original line. As far as I remember, this was based on measurements as well. Have there been indications that these measurements were incorrect? Later you demonstrate that differences between several formulations and your reference formulation are as small as typical measurement uncertainties and that it can therefore not be decided, which formulation is "correct" (I would rather say, which formulation is most appropriate, since "correct" implies that you know the truth.) So it seems, the motivation for your exercise is not that the original formulation is wrong, but rather that you seek for something quite simple that can be used in analytical studies. If so, this should be clearly stated in the beginning.
Line 202: replace "explanation" by "interpretation".
Line 211: Isn't the degree of p(x) fixed by your ansatz in Eq. 25?
Line 286: not clear, what the vertical axis here represents.
Section 3.5.2: I don't get the point here. Before you have made a best fit, but what is interesting in showing results obtained with worse fits? I think, this section can be shortend without much loss to the reader.
Section 3.6: I must admit, that I cannot follow in this section. This kind of mathematics is not in my repertoire. I fear that other readers can have problems as well with this section. I suggest that a short paragraph be added, either here or as an appendix, where this technique is shortly explained.
Section 5: Please make clearer what is new here to the developments in Section 3. In 3, the reference nucleation rate was the Koop et al 2000 formulation, but shifted such that the Koop and Murray results for pure water were matched in a certain T-range. It is not quite clear to me what the difference of this to the new fit is.
Line 572: "avoid"
Line 634: This is an important remark. Can you give any recommendation how this formulation should be changed in such models to become more realistic?
- AC2: 'Reply on RC1', Peter Spichtinger, 10 Nov 2022
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CC1: 'Comment on acp-2022-434', Bernd Kärcher, 19 Jul 2022
I take the opportunity to comment (italicized) on a few issues in the manuscript that relate to my work for clarification and in the hope to support the overall review process.
[I] Abstract line 1: "Ice formation in cold temperature regimes is most probably dominated by homogeneous
freezing of aqueous solution droplets."The first sentence in the paper is a strong statement demanding justification. Clearly, homogeneous freezing of solution droplets is a fundamental atmospheric ice formation process that operates when ice-nucleating particles are either absent or present in very low concentrations. Any assertion whether and in which circumstances this pathway dominates 'ice formation in cold temperature regimes' (an active area of research) needs to be supported by appropriate references. What about the role of mineral dust in cirrus cloud formation [Froyd et al., 2022], with dust particles long known to be very efficient heterogeneous ice-nucleating particles?
[II] line 33: in-situ vs liquid origin formationI don't quite understand the categorization in-situ vs liquid origin formation. In the absence of cloud water droplets, freezing agents include solution droplets, and also in this case, it is the liquid water that freezes. Plus, situations where cloud droplets freeze are also in-situ events. Does liquid origin formation distinguish between ice formation in convective detrainment zones and in conveyeor belts, in which either solution and cloud droplets may freeze?
[III] Abstract line 3: "... idealized nucleation events as modelled with a state-of-the-art ice
microphysics, ..."Why would the nucleation event be idealized? I think it is the modeling that is idealized. Which of the microphysics schemes referred to / used in the manuscript is 'state-of-the art' and how is this attribute defined? The numerical scheme used in Kärcher & Lohmann (2002) does not employ Lagrangian particle physics, as claimed in line 771.
In that context: could you please motivate / explain more clearly why "... details of the nucleation rate are less important for simulating ice nucleation in bulk models ..." (abstract line 6) only relates to bulk models?
[IV] line 779-785: "In the study by Kärcher and Lohmann (2002) the impact of latent heat release
on the diffusional growth is not considered. It is argued, that for cold temperature this effect
is negligible. However, we found in our investigations, that this is only true for temperatures
well below 220K. ... using parameterisations based on Kärcher & Lohmann (2002) might lead to
moderately enhanced ice crystal number concentrations in the warm temperature regime."Latent heat release enhancing temperatures during homogeneous freezing of cloud water droplets roll off around 234K depending on the CCN spectrum [Kärcher, 2017]. Here, the authors assert to find moderate (significant?) reductions of homogeneously nucleated ice number concentrations down to much colder temperatures due to latent heat release, this should be demonstrated by numerical simulations. Please analyse the individual diabatic contributions due to water phase changes to the temperature budget with the dry adiabatic tendency at 220K and show a quantitative comparison of homogeneously nucleated ice crystal number concentrations between simulations across a range of updraft speeds with and without including effects of latent release.
[V] Abstract line 11: "In contrast, the maximum supersaturation and thus the nucleation thresholds
reached during an ice nucleation event depend on the vertical updraft velocity or cooling rate. This
feature might explain some high supersaturation values during nucleation events in cloud chambers and
suggests a reformulation of ice nucleation schemes used in coarse models based on a fixed nucleation
threshold."The point that homogeneous freezing thresholds depend on cooling rate and droplet size is known [Kärcher et al., 2022]. I am aware of the difficulty to oversee the onslaught of manuscripts in the scientific literature. In Kärcher et al. [2017], we have analysed the microscale characteristics of homogeneous freezing events in detail. Both articles are relevant for the present manuscript and therefore the authors may want to update their reference list.
I express strong doubts that a w-dependence of the freezing supersaturation threshold might explain low temperature cloud chamber data; if the authors have indications for this to be a valid explanation, it should explicitly be included in the study, focussing on the region T<205 K. Also, the treatment of the whole issue in the manuscript is somewhat confusing, as the authors admit in line 631 that "This might be interpreted as a hint that the formulation by Nachbar et al. (2019) is the more appropriate formulation for the saturation vapour pressure.", i.e., invoking a low-T correction of the currently used saturation vapor pressure as the main cause of high freezing thresholds and not deviations from a fixed freezing threshold in parameterizations of cirrus cloud formation.
Also, it remains unclear why this suggests a reformulation of ice nucleation schemes used in coarse models. In cirrus, the dependence of nucleated ice numbers on the threshold supersaturation is relatively weak (less than linear, see Kärcher & Lohmann, 2002), let alone the logarithmic dependences on the cooling rate, the liquid water volume in solution droplets, and the total droplet number concentration. These dependencies are consistent with / well explained by the self-terminating homogeneous freezing-relaxtion mechanism; recently your team recognized based on numerical work that an increased nucleation threshold has little impact on ice crystal numbers (Baumgartner et al., 2022).
Please note that the purpose of cirrus parameterizations is to estimate total nucleated ice crystal number concentrations, not to resolve the region around the ice supersaturation maximum. The main differences between homogeneous freezing parameterizations (Barahona & Nenes [2008], Dinh et al. (2016), Kärcher & Lohmann (2002)) may be traced back to how the integration over the supersaturation history is approximated.
[VI] line 603: "The nucleation threshold assigned to the frequently used value j0 = 16 is completely arbitrary
chosen; there is no convincing physical justification for using this particular value; ..."I am not sure if this is actually a true statement. My understanding re Koop et al. (2000) has always been that a given choice for j0 is not 'completely arbitrary' but chosen as it relates to a nucleation timescale appropriate for analysing the laboratory experiments. This relevant observation time is roughly given by the inverse of the product of the freezing rate coefficient and the volume of supercooled liquid water in the small droplets investigated. Please check with Koop et al. In Kärcher & Lohmann (2002), we have determined T-dependent freezing thresholds by imposing j0 = 10 for (wet radius) 0.25 µm solution droplets, a typical mean size of freezing droplets in cirrus levels, ignoring the comparatively small dry aerosol core volume.
[VII] Abstract line 9: "The use of just one distinct nucleation threshold for analysis and model
parameterisation should be reinvestigated."In the light of the above , I am not aware of any cirrus parameterization scheme that uses a single value for this threshold, at least a temperature dependence is employed. Please check.
[VIII] line 290: "We use the simple bulk ice physics model as described by the set of ODEs ..."I would argue that bulk models are not the tools of choice to carry out ice nucleation studies. In the case of homogeneous freezing, early freezing large droplets form a cohort of ice crystals that grows ahead of those forming later at higher supersaturation and on smaller droplets. In this way, a size-dispersed ice crystal spectrum is generated allowing for deposition growth of early formed ice crystals to change (reduce) the supersaturation conditions for later nucleation, as growth rates of µm-crystals are very rapid. It is this self-terminating freezing-relaxation mechanism that eventually determines the total homogeneously nucleated ice numbers, irrespective of updraft speed. Due to the sensitive dependence of the freezing rate coefficient on temperature and water activity (ice supersaturation), it is key to model size-resolved homogeneous freezing, if only approximately analytically, as in our parameterization scheme (Kärcher & Lohmann, 2002). It is also needed to robustly model heterogeneous nucleation events, as ice-nucleating particles activate into ice crystals across a range of supersaturation.
The point is, of course, that, by design, bulk models (aka two-moment schemes) are not capable of resolving this competition for available water vapor and therefore do not correctly represent the nucleation pulse and may not robustly predict nucleated ice numbers. Doing so requires a size/mass bin (spectral) approach or particle-based microphysics. For the purpose of studying nucleation in clouds, why not use a detailed model describing all relevant processes properly?
Bulk models can only be expected to deliver accurate total homogeneously nucleated ice numbers if they simulate bulk water vapor uptake on freshly nucleated ice crystals accurately. Moreover, modal bulk models based on a fixed functional form of the ice crystal size spectrum (via a constant distribution width such as given in line 717) cannot reproduce the rapid change of the size spread of ice crystals during a freezing event that is ultinmately responsible for the quenching of the supersaturation and the shutting off of the freezing pulse.
If one wishes, one might view nucleation as parameterized, i.e., constrained / tuned by assumed parameters in bulk models. Arguably, the "overall good agreement of our simple model with the more sophisticated models" (line 774) might be coincidental.
[IX] Short summary: "However, the maximum supersaturation during nucleation events shows strong changes. This quantity should be used for diagnostics instead of the popular nucleation threshold."I have a hard time supporting the 'should' in the above summary statement. Rather, the question arises how accurately the bulk model used in the manuscript is able to simulate the degree of overshooting. Overshooting will always constrain the maximum supersaturation close to (within a few percent) the freezing-relaxation threshold because of the freezing-realaxtion feedback: the higher ice supersaturation overshoots (due to faster cooling), the more droplets freeze, the faster the freezing events terminates. The peak supersaturation attained during a freezing event must be distinguished from the characteristic supersaturation where freezing-relaxation sets in, see discussion in Kärcher et al. [2022].
[X] line 148: "Remark: As shown in Spreitzer et al. (2017), it is possible to determine and characterize
the steady states of the reduced system, which additionally includes sedimentation. This leads to a nonlinear
oscillator with a bifurcation diagram, depending on the updraft velocity w, and on the temperature T."I am not sure why this remark features prominently here. There is no connection made to the topic of the manuscript. Could you better motivation this insertion? Also, Spreitzer et al. (2007) seem to describe a numerical artifact (the occurrence of nucleation cycles in a nucleating air parcel triggered by sedimentation and sustained cooling) that is tied to a coarse spatial resolution (box height) relative to the shallow depth of homogeneous freezing zones. The cycles occur when the timescale of vapor loss due to sedimentation (depending on the layer depth) matches the time scale of supersaturation production that scales in proprtion to the imposed updraft speed. High resolution models (meter resolution in the vertical) show that homogeneous freezing at the top of nucleation layers is a continuous process, see e.g., Lin et al. [2005].
Additional references included in this commentary:Barahona, D. & Nenes, A. (2008).
Parameterization of cirrus cloud formation in large-scale models: Homogeneous nucleation.
J. Geophys. Res., 113.
doi: 10.1029/2007JD009355Lin, R.-F., Starr, D. O., Reichardt, J. & DeMott, P. J. (2005).
Nucleation in synoptically forced cirrostratus.
J. Geophys. Res., 110, D08208.
https://doi.org/doi:10.1029/2004JD005362.Froyd, K. D. et al. (2022).
Dominant role of mineral dust in cirrus cloud formation revealed by global-scale measurements.
Nat. Geosci. 15.
https://doi.org/10.1038/s41561-022-00901-wKärcher, B. (2017).
Homogeneous ice formation in convective cloud outflow regions.
Q. J. R. Meteorol. Soc. 143.
https://doi.org/10.1002/qj.3069Kärcher, B. & Jensen, E. J. (2017).
Microscale characteristics of homogeneous freezing events in cirrus clouds.
Geophysical Research Letters, 44.
https://doi.org/10.1002/2016gl072486Kärcher, B., DeMott, P. J., Jensen, E. J. & Harrington, J. Y. (2022).
Studies on the competition between homogeneous and heterogeneous ice nucleation in cirrus formation.
Journal of Geophysical Research, 127.
https://doi.org/10.1029/2021JD035805- AC3: 'Reply on CC1', Peter Spichtinger, 10 Nov 2022
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RC2: 'Review of acp-2022-434', Anonymous Referee #2, 15 Aug 2022
The manuscript by Spichtinger et al. revisits the impact of nucleation rate formulation on the outcome of homogeneous freezing events. These investigations are motivated by two recent studies reassessing the saturation pressure with respect to supercooled water (Nachbar et al., 2019) and the homogeneous freezing rate in pure supercooled liquid water (Koop and Murray, 2016). Without questioning the crux of Koop et al. (2000) water activity criterion (WAC) theory, the new parameterizations, if correct, would imply changes to the commonly-used formulas of homogeneous freezing rates in Koop et al (2000). More generally, these recent results emphasize uncertainties in the formulation of homogeneous freezing rates, as recognized by the authors of the present study.
In this context, the paper introduces and evaluates meaningful approximations to the nucleation rate in order to identify and discuss the key parameters for the outcome of nucleation. The authors also propose an updated formulation of the homogeneous freezing rate of aqueous solution droplets in the frame of the Koop et al. theory.
The manuscript should be a valuable contribution to ACP, but I have a few concerns about the method and presentation and recommend that the paper be reconsidered after major revisions. My comments, requests for clarification and suggestions to the authors are detailed below.
Main points :
1) Consistency between liquid saturation pressure and nucleation rate: As far as I understand, for consistency with the Koop et al. (2000) method, each change in the saturation formula (and implied change in ice water activity) requires updating the fit to the homogeneous freezing rate of pure water. Was this carried out when switching to Nachbar’s formula in section 4 ? It should be specified if yes, done if not.
Related issue: the authors have already published on the impact of Nachbar’s formulation in Baumgartner et al. (2022). Differences should be highlighted. If there are none, the whole section 4 (including Figures 10 and 11) can be removed and summarized in just a couple of sentences (in particular since the effects are small).
2) Clarity and organization: I would suggest the authors start by presenting the uncertainties regarding the nucleation rate and the new formulations before introducing their approximations and finally analyze the impact of the updated formulas. This would make the motivation clearer. I also think the manuscript would benefit from shortening. Many equations and comments are redundant (see below).
3) Suitability of the model to study details of ice nucleation: in a comment to this paper, Bernd Kärcher expressed skepticism regarding the ability of a bulk model to study ice nucleation. For such a focused study, expected limitations of the modeling approach should be discussed.
4) What is the reason for the simple adjustment to the Koop et al (2000) formula in Sect. 3.1? This seems to create a ‘third’ mixed category between the former and updated formula. I would recommend restricting the study to those two cases.
5) I had difficulties following some sections, in particular 3.6. At other places, the text sometimes falls into tautology: in section 3.4, the authors arrive at Eq. (45) which as far as I understand is exactly the same as Eq. (43) and (42) if we consider that Sc also depends on j0. Eq.(42) is repeated only a few lines below as Eq.(43) on page 11.
6) There are some ambiguities in the notations, for instance pressure and the polynomial are both represented by the same symbol (p) . I suggest clarifying and adding a table with a list of symbols.
7) I am not entirely convinced about section 6. The relevant quantity for atmospheric modeling is the ice crystal number density, not really the threshold which is mostly specific to chamber experiments. Moreover, the ‘Koop-line’ depends on the aerosols size distribution and is not supposed to represent the maximum supersaturation reached, rather an approximate ice onset.
8) Diffusion growth: it seems that it was originally intended that the paper also treats the sensitivity to diffusion growth (the summary line 644 still mentions it). While I agree this is beyond the scope of the study, could the authors comment in the text on whether the sensitivity to nucleation rate formulation they characterize holds for different growth parameters? Also, it would make sense in this case to have Part 1-Part 2 papers.
Specific comments:
Title: should mention homogeneous nucleation or homogeneous freezing since it is the only nucleation pathway considered in the paper.
P4 Model description should state that the model has two moments.
Line 111 and 132: There is an inconsistency, it is first stated that latent heat release is neglected but this term is still included later.
line 140-141: Same as above, the first term comes from considering latent heat.
Line 164-165: This disagreement at temperatures above 235 K is indeed surprising. Have you confirmed by comparing with the original Pruppacher 1995 data used by Koop et al. (2000) ?
line 181: missing index n
line 258: eq. 41 is the same as eq. 31
line 340, ‘most’: not all ?
line 701: ISO convention for natural logarithm is ln , not log
Line 747-748: Please check formula A11 and please correct if needed. Also define L.
Line 751: for which value of the eccentricity ?
- AC4: 'Reply on RC2', Peter Spichtinger, 10 Nov 2022
- AC1: 'Response to reviewer comments', Peter Spichtinger, 10 Nov 2022
Peter Spichtinger et al.
Peter Spichtinger et al.
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