the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 26 Jun 2025
Evaluation of Semi-Implicit and Explicit Sedimentation Approaches in the Two-Moment Cloud Microphysics Scheme of ICON
Abstract. In the ICOsahedral Nonhydrostatic (ICON) model, the Seifert-Beheng two-moment microphysics scheme is one approach to simulate clouds with different hydrometeor classes. In this bulk description, sedimentation is modeled by advecting the first two moments (number and mass densities) of the hydrometeor size distributions with velocities derived from fitting a generalized gamma distribution to the moments. This method implicitly relies on the diffusive properties of the numerical advection schemes to obtain results in closer agreement with the exact spectral solution. The implementation in ICON offers both a semi-implicit and largely untested explicit method for sedimentation. Currently, the semi-implicit scheme is substantially slower on graphics processing units (GPUs), which is particularly relevant considering the recent rise of GPUs in supercomputing; this raises the question of whether the explicit scheme is a viable alternative.
We provide a detailed examination of both sedimentation schemes, their differences, and underlying assumptions. Using idealized one-dimensional experiments, we identify a minor issue in the default semi-implicit scheme (flux limiter artifacts) and propose a solution. Additionally, we show that the explicit scheme exhibits less numerical diffusion, though some diffusion is crucial for accurate bulk sedimentation. We caution that in the future, finer grid resolutions may result in insufficient diffusion, especially for the explicit scheme. An analysis of six case studies with thunderstorms reveals that the explicit scheme gives rise to more jagged patterns in the hydrometeor profiles, although without concerning instabilities. Furthermore, some differences in hail and graupel precipitation rates can be attributed to different ways of considering the microphysical source terms (e.g., hydrometeor interactions) during the sedimentation step.
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- CC1: 'Comment on egusphere-2025-2804', Ted Mansell, 26 Jun 2025
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RC1: 'Comment on egusphere-2025-2804', Anonymous Referee #1, 10 Jul 2025
The paper compares the two numerical schemes to solve sedimentation that are implemented in the two-moment scheme of ICON: a semi-imcplicit one and an explicit one. The overall quality of the paper is good: the topic is interesting, the experiments are well designed and the conclusions are well sustained. I only have some doubts about the reference solution presented in the second experiment (see below). Another important issue is the lack of information in several parts of the text. Some important information is only provided in the figure captions, which makes it difficult to read the paper.
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 Detailed comments:
- The authors should explain why the spectral solution in the second experiment (Figures 2 and 3) is considered the reference. While spectral methods are clearly better at representing changes in the size distribution, the sedimentation scheme typically suffers from diffusion like the standard two-moment scheme. This is seen in the leading edge of the spectral solution in Figure 3, where the diameter smoothly goes to zero. The authors should explain how the spectral solution is calculated (which resolution, which numerical scheme, which dt). One idea to show that diffusion is negligible for the spectral scheme would be to show how the spectral scheme deals with the linear advection case of the first experiment.
- Please describe the linear advection experiment in the text and not only in the caption. It would help if it were written that no two-moment scheme is used and it is just the sedimentation of a tracer with a prescribed velocity.
- I suggest the authors to write the parameterization for the hail velocity and for the mass-diameter relationship in the appendix and not to cite an ICON file. This can help for a better understanding and better reproducibility. It is not that long.
- In the abstract and conclusion, the authors state: finer grid resolutions may result in insufficient diffusion, especially for the explicit scheme. I disagree with this sentence in the NWP context. Finer grid resolutions in the horizontal are usually not matched by similar refinements in the vertical, while dt decreases with dx/dy. For example, ICON runs operationally with the same vertical grid for 2km and 500m resolutions. A more relevant comparison would be by changing dt, while keeping dz constant. In the explicit scheme this is similar to sub-stepping, and I would guess that more diffusion is expected.
- Related to the previous point. Do you know what happens to the semi-implicit scheme when dt is reduced? (for the same dz). This could help to understand what happens at the higher resolution models.
- Figure 7 shows differences when using a very small threshold (10-9 g/m-3), which is almost negligible. What happens when using a more-relevant threshold like 10-6 g/m-3?
- Figure 4: do you keep a constant dz / dt ratio like in Figure 3? Please clarify it.
- In several parts of the paper, it is stated that turbulent diffusion can smooth the profiles of L and N. In ICON NWP there is no turbulent diffusion of hail and graupel.
- Page 10, line 240. What do you mean with there is no case distinction?
- This is just a suggestion. If you want to show that comparable results in NWP are obtained with both schemes you could show the one-hour precipitation map, one hour after the simulation start for one case study. This could enhance the confidence in the explicit scheme and show that the differences in roughness are not relevant for typical NWP applications.
- Why do you write the approximate symbol in Equation (6)?
- I think there is a mistake in Equation (12). It should be 0.5(vkn + vk-1n+1).
- Why do you  write the super index lim and lim2 in the right-hand side of Equations (13) and (14)?
Citation: https://doi.org/10.5194/egusphere-2025-2804-RC1 -
CEC1: 'Comment on egusphere-2025-2804', Astrid Kerkweg, 23 Jul 2025
Dear authors,
to meet all requirements w.r.t. code provision, you need to provide also the code modification in an open way in a longterm archive. The ETHZ gitlab is neither open nor does it fulfill the requirements for a longterm archive.
Best regards,Â
Astrid Kerkweg (GMD executive editor)
Â
Citation: https://doi.org/10.5194/egusphere-2025-2804-CEC1 -
RC2: 'Comment on egusphere-2025-2804', Ulrich Blahak, 02 Oct 2025
This paper analyses and compares two numerical hydrometeor sedimentation schemes which are offered as options in the ICON model's two-moment bulk cloud microphysical scheme. As the original author of the herein examined explicit scheme, first I would like to thank the authors for this overall good and useful paper. I especially like the graphical explanation of the explicit FFSL method. The experiments are well designed and the conclusions are well founded. In detail, I see some room for improvement, which I would consider as minor revisions: for example, the nature of the analytic (?) / numerical (?) reference solution for the idealized sedimentation experiments need more explanation, and the paper and its conclusions would profit from adding more informations on the overall effect on practically relevant model outputs like 1-h rain accumulation, 1-h hail accumulation and radar reflectivity at relevant heights above ~500 m AGL.
Before my detailed comments, I would like to give some historical information which might be useful background and might even be mentioned somehow in the introduction. As stated in my 2020 documentation of the explicit schemes, the explicit "boxtracking" scheme was meant as a first improvement over the traditional explicit scheme, which was at that time the only sedimentation option for the two-moment scheme and which showed worse "spiky" and "rough" behaviour at Courant numbers > 1, especially close to the ground, where the model layers are very thin. It's main flaw was the approximation of the fallspeed v_k-l == v_k+1/2. The explicit boxtracking scheme has been developed in the COSMO-model framework and takes into account the "true" fallspeed of each box k-l (therefore I called it "boxtracking"). The implicit solver existed only for the one-moment scheme, in both the COSMO and ICON implementation of the two-moment microphysics. Lateron, Axel Seifert implemented the implicit scheme also for the two-moment microphysics in the ICON framework, and this became our standard scheme during ICON-RUC development. We already noticed the high diffusivity, but as we did not find a significant impact on practically relevant fields in real-case weather forecasting, we favored the implicit scheme due to its conceptual advantages concerning the hydrometeor interactions at large Courant numbers, which are also mentioned in the present paper.Â
Detailed comments:
1) Abstract: "... we caution that in the future, finer grid resolution may result in insufficient diffusion, ..."
I think this statement cannot be made in general, as it depends on adaptions of vertical layers and model time step which are done simultaneously alongside a reduction of horizontal grid spacing. For example, if just the horizontal grid spacing and the time step are reduced, but the vertical levels are not touched (which is sometimes the case), then I would expect more diffusion from the sedimentation scheme. Similar statements are also made elsewhere in the manuscript. You could relax the statement in a way that you discuss that a change of diffusion depends on changes to the vertical resolution and the model time step and may change in the one or other direction, when model resolution is enhanced in the future.
2) Line 48: "... shown by runtime measurements in Table 1." Add a remark that this behaviour will be explained/examined in more detail in later section 2.2
3) Line 91: "... to account for the increasing air density rho ..." --> better: " ... to account for the dependence of the terminal fall speed on air density rho ..."
4) Line 108: "... deliberate use of diffusive numerical schemes ..." While I completely agree with the statement, to me it is unclear what an "optimal" amount of diffusion might be, if at all, as this depends on many things. So how do we decide what amount of diffusion is appropriate? Maybe you could add some brief discussion of your view on this point and the associated uncertainty, because the paper compares schemes with different amount of diffusion.
5) Line 110: please explain also Q as shortcut for Q_N and Q_L.
6) Line 120: "v_k^{(n+1)}" --> "\tilde{v}_k^{(n+1)"?
7) Line 147: "This accounts for nearly all the runtime difference ..." To support this statement, please add the timing numbers for the semi-implicit sequential scheme in comparison to the full semi-implicit and explicit scheme in a suitable way.
8) Figure 1: Some sub- and superscripts are missing, at least in my pdf version: "z - z" should be "z_{up} - z_{low}", "v_2^{()}" should be "v_2^{(n)}"
9) Line 193 ff: Please add in a suitable way the information that the substepping is optionally implemented by an internal code switch, which is disabled by default because of reproducability problems in current ICON. You should clearly state in the following paragraph that you found the reason for the reproducability bug (missing global mpi max reduction of N_sub), fixed it in your code version and reported it to the ICON developers (I'm one of them, so I know now :-) Thank you for finding it!).
10) Line 2019: Please define the "clamp()" operator, because not everyone from the meteorological community will be familiar with it.
11) Line 247: Please explain the "spectral solution" in more detail. Was it obtained by a bin scheme or by an analytic solution of Eq. (1)?
12) Line 249: Please mention the initial condition with it's single block of hydrometeors. This is different to your next test with 3 initial blocks (Figure 3).
13) Line 252: "... future resolution increases in accordance ..." --> "... future resolution increases (also in the vertical) in accordance ..."
14) Line 254: "... showing initial conditions, ..." --> "... showing initial conditions with 3 peaks, ..."
15) Line 254: Is the spectral solution the same method as for figure 2? Please explain.
16) Captions of Fig. 3 and Fig. A1: "... graupelhail_cosmo5 in the file src/atm_phy_schemes/..." Please do not reference the code, but put the relevant parameter values into a table.
17) Caption to Fig. 4: "... for spatial resolutions ranging from \delta z = 5m ... in 2x increments" --> "... for spatial resolutions \delta z = 5, 10, 20, 40, ..., 1280m. "
18) Caption to Fig. 4: "... and further details are provided by Fig. 3." --> "... and further details have already been provided in Fig. 3"Â
19) Line 275: "... show, ..." --> "... show up, ..."
20) Line 279: Which individual summer days? Please list the dates for reproducability of your work.
21) Section 4.1: The motivation and illustration for investigating the roughness would benefit from an exemplary figure. You could add a zoom-in to a precip rate field from one of your experiments to illustrate it by spatial noise, because noise in time (=roughness) translates into noise in space under horizontal transport.
22) Line 386: The amount of numerical diffusion depends also on the time step (= how often the scheme is called)
23) Line 409: "This inevitably results in reduced numerical diffusion" (See above point 1)
24) Conclusions:
I agree fully to your statement in line 406ff that "... in full model simulations those issues seem to be mitigated by ..." and "... we find that the explicit scheme can be safely used on GPUs ...".
Since this is one of your main messages for the community, please add a new section right before the conclusions section, where you qualitatively show/compare real-case results for practically relevant forecast fields like 1-h precipitation accumulation, 1-h hail accumulation, and radar reflectivity at relevant heights > 500 m AGL, e.g., dbz_850 (because model verification and radar data assimilation is usually based on observations from those and larger heights). This should be not too complicated and time-consuming, because I would think that you already have these fields on disk from your real-case experiments. You could plot zoom-ins to these fields of one example time step of one of your real-case experiments, and compare the fully semi-implicit, the sequential semi-implicit and the explicit scheme. My expectation would be that the three methods show some differences, but that these differences are negligible compared to other meteorological forecast uncertainties, and this would be a clear illustration of your point.
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Citation: https://doi.org/10.5194/egusphere-2025-2804-RC2
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Just a quick comment for the authors. I highly recommend adding a profile of the reflectivity moment to Figure 3. The mean diameter alone does not tell us about excessive size sorting, but Z definitely does. Â Relying on low-order diffusion is really not a great solution, and as you show, vertical grid spacing and the Courant number play a strong role. It can smooth out the shock, but that is only part of the problem because the leading edge is still there. Whatever is done, however, showing Z is important so that the reader can at least see whether it increases (i.e., sorts excessively) or not. Having excessive sorting in the result doesn't necessarily distort rain rates etc., but it can be detrimental for assimilation of radar reflectivity by causing biases.
The common strategy of placing limits on the slope parameter or mean size only treats excessive sorting at the point where reflectivity is becoming unrealistic, but doesn't fix the underlying problem. Of course, I would advocate using a temporary Z moment in the sedimentation to adaptively adjust N while conserving mass. But all that is really needed here is to show what the given schemes are doing, and showing Z would give a more complete picture of that.
yours,
Ted Mansell