the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 18 Jun 2026
Technical note: Evaluation of conceptual predator-prey models for the quantitative modeling of precipitating open-cell stratocumulus via feature-based Bayesian inversion of a suite of Large eddy simulations
Abstract. We consider two very different types of models of precipitating open-cell stratocumulus clouds. The first model type is a computationally expensive large eddy simulation (LES), that resolves convection and clouds at high temporal and spatial resolutions. The second model type is the nonlinear cloud and rain (C&R) equation, a scalar delay differential equation (DDE) that interprets interactions of precipitation and clouds phenomenologically by predator (rain) and prey (cloud) dynamics. We evaluate the extent to which one may use the C&R equation as a quantitative tool for representing selected aspects of an LES. Specifically, we estimate parameters of the C&R equation from a suite of LES via feature-based Bayesian inversions and track the evolution of posterior distributions over C&R model parameters under changing meteorological conditions in the LES. Our inversions show that the C&R equation can be calibrated to generate limit cycles that are quantitatively compatible with cycles of cloud growth and decay across a wide spectrum of meteorological conditions. The successful inversions reiterate the robustness of the predator-prey analogy to the dynamics of precipitating open-cell stratocumulus. When we interpret the inversions jointly, however, we observe counterintuitive and partially nonphysical shifts and changes in the posterior distributions over the C&R model parameters. Our evaluation study thus highlights the challenges one faces when mapping LES dynamics to a scalar DDE, which can stem either from structural inadequacies in the DDE model, or from the specific feature-based inversion framework, or a mixture of both.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Atmospheric Chemistry and Physics.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-2871', Anonymous Referee #1, 24 Jul 2026
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RC2: 'Comment on egusphere-2026-2871', Anonymous Referee #2, 16 Aug 2026
Review of "Technical Note: Evaluation of conceptual predator-prey models for the quantitative modeling of precipitating open-cell stratocumulus via feature-based Bayesian inversion of a suite of Large eddy simulations" by Gjini, Morzfeld, Glassmeier and Feingold, Egusphere manuscript 2026-2871
Summary:
The authors describe a method to "project" the behavior of clouds in open-cell stratocumulus onto a one-dimensional delay differential equation that models predator-prey behavior, where the predator is rain and the clouds are prey. A novel (to my understanding) predator-prey model is introduced that represents rain production by accretion, and a dynamical systems analysis of the two models is undertaken. After introducing how open-cell cases and their behavior is extracted from the LES database, a Bayesian inversion technique is described with sufficient detail for the paper to be self-contained. This method is then applied to understand cloud formation/decay cycles in open-cell stratocumulus in the LES, first separately for individual LES cases, then across the family of cases. Interestingly, opposing relationships between things like the equilibrium cloud depth and the timescale with which the equilibrium is approached are seen in individual LES (where the "expected" physical relationship is seen) and in the ensemble of LES cases (where deeper equilibrium cloud depth is associated with shorter relaxation timescales).
Assessment: Minor revisions.
The paper is very well written, given the task of introducing tools/methods from different fields as well as presenting the results. While the paper is long, my comments mainly request bits of additional material (sentences and a couple of figures) that might make the paper more approachable for the reader who is daunted by either the dynamical systems analysis or Bayesian inversion. If there is a "major comment" in the material below, it would be to check whether the cloud evolution in the individual LES timeseries are truly cloud-following or whether clouds might be blowing into and out of the chunks of the domain represented by the many LES timeseries extracted from each LES simulation. My understanding of the predator-prey models is that they are meant to represent the evolution of a single cloud or group of clouds over time in a Lagrangian frame. See comment on sec. 5.2.1 below. If this concern reflects a misunderstanding of what the LES timeseries represent or of the predator-prey models, this could be made clear in the paper.
Minor Comments (3/57 means page 3, line 57):
Suggested additional figures: This manuscript has an especially high text to figure ratio, which is reasonable given the amount of exposition needed to bring together 3-4 fields (cloud/precipitation evolution, LES, dynamical systems theory, Bayesian inversion). However, I suggest a couple of additional figures in the main manuscript to help the reader:
- sec. 2,3: With all of the talk of limit cycles, not one plot shows a closed loop in phase space. While the dynamical system is one dimensional, a new figure showing a plot of the joint evolution of cloud depth and rain rate for the three solutions shown in figure 1d would show the reader how this one-dimensional evolution corresponds to the physical evolution of the cloud and associated rainfall. Such plots are undoubtedly in the literature elsewhere, but this might be helpful here to convince people studying clouds to wade through the long paper. Ideally, this could be coupled with a similar panel showing the evolution of similar variables in the clouds extracted from the LES. If possible, the cloud depth and rain rate from the predator-prey model(s) could be dimensionalized to make them roughly comparable to the LES.
- sec. 6.2: Consider promoting figure A3 (or some pieces of it) to the main paper. If a reader doesn't search out and digest the message of Figure A3 in the supplement before reading on, they are left in suspense for too long before the Simpson's paradox explanation late in the discussion/conclusions. If a figure is this essential to the message of the paper, some part (or all) of it probably belongs in the body of the paper. I see a couple of possibilities here: First, bring A3 into the main manuscript in place of figure 7 and talk about panel A3(b) alongside A3(a,c) which seem identical to 7(a,b) and contrast with Figures 5(b) and 6(b). Similar for the H0-N relationship. Alternatively, leave A3 in the supplement and make a separate four or six panel figure that brings together the H0-tau and H0-N plots in figure 5, 6, and 7 in a way that makes the Simpson's paradox come through to the reader without flipping back and forth in the paper. The second option might focus the reader's attention on the key message while making those panels larger and possibly easier-to-digest than the smaller stamp-like format in the present triangle plots.
1/11 or 3/57: For the reader who isn't steeped in Bayesian thinking, a plain language explanation of "posterior distribution over the C&R equations' parameters" would be helpful. My suggestion: ", i.e., the combinations of C&R model parameters that produce simulations most similar to LES cloud evolution,"
7/Fig 1b: It's striking that the range of eta and D that satisfy the prior restrictions in section 5.1 occupy only a small part of the eta-D ranges shown here. This figure doesn't need to change, but this point might be worth mentioning this later in section 5.1.
9/202-203: The transient/oscillatory nature of clouds within open-cell convection is really important to what follows. Has the reader been prepared to think about a "stable ... open-cell regime" as "oscillating". Showing such limit-cycle behavior in the LES (as in the first additional figure suggested above) might prepare the reader for this. Some foreshadowing in the introduction or in this section might also prepare the reader.
12/301: Could a couple of sentences be added to say how the joint parameter space was sampled and that the prior distributions shown in figure 3 show the frequency with which those parameter combinations result in a C&R model solution where 1-3 above are satisfied. Something like "We sampled each parameter uniformly XX times along the ranges shown in figure 3, leading to a total of YY parameter combinations. The prior probability in figure 3 represents the fraction of those parameter combinations that satisfy (1-3) above for each pair of parameter values on the axes of the plot." My phrasing isn't great, but please be more explicit here.
14/322-323: Regarding "We also see how the prior constraint of limit cycles with positive cloud depth ... cuts across the much larger limit cycle/chaotic regime ...", this may not be so obvious to many readers, so help them out. Perhaps, "We also see, through the ragged edge of the bottom right of the triangles in figure 3(b,d), how the prior constraint that limit cycles have non-negative cloud depth ...". This might be the place to note that the bounds of Figures 3(b) are quite different from those of figure 1(b).
14/336: Be more explicit about the spatial smoothing and what each of these time series represent in physical space. Reading "900 time series" and looking back to see a "48x48km" domain in the horizontal, I assume that each timeseries represents the average properties of a 1.6km square in the LES domain. However, I may or not be right, and a reader with a mistaken understanding of this might not interpret the results properly. Therefore, it's important to explain how this is done to the reader, but it shouldn't take more than a sentence or two.
14-15/sec 5.2.1: My reading of the C&R model description suggests that it is meant to describe the evolution of a single cloud or an ensemble of clouds through their lifetime, following the same cloud or ensemble of clouds. A timeseries that represents a chunk of an Eulerlian LES simulation will not (in general) match the Lagrangian evolution of a cloud or set of clouds. Instead, clouds will advect into that chunk of the domain and then out of it. It's possible to perform a Gallilean transformation so that the LES domain translates to match the motion of clouds, but this is difficult in simulations with any wind shear. Small mismatches in the winds (0.2 m/s) would lead clouds to drift across a 1.6km distance in 8000 seconds, which is less than the ~150 minute limit cycles in the LES in Figure 4.
If the Hoffman et al simulations were run with no mean wind _and_ if no spontaneous mean winds arose during the simulations that would induce such cloud drift, please make clear that Hoffman et al set up their simulations like this _and_ that these 900 timeseries really represent the Lagrangian evolution of clouds over the corresponding patch of the LES domain. The reader can't be expected to have to figure this out on their own, but it seems really important to the interpretation of the Bayesian inversion.
If there is some drift of clouds in the mean wind, so that the limit cycles seen in the LES features represent longer-lived clouds blowing into and out of 1.6 km square chunks of the LES domain, this should be made clear to the reader or, alternately, the authors could consider repeating the analysis for truly Lagrangian, cloud-following timeseries for subsets of the LES domain (if possible).
15/Fig 4: I was surprised to see that every LES limit cycle starts from zero cloud depth, though this seems reasonable if the cloud fraction is low and the individual time series really are local in space.
Could the authors spend a sentence saying how the features are constructed from the many limit cycles like those in panel (a)? Are they an average, or a projection onto some smoothly varying function? Apologies if this was done clearly in the text and I missed this.
15/350: I don't think KTF (Koren-Tzipperman-Feingold??) was defined earlier in the paper. Even if it was, it's asking a lot for the reader to remember it after this much text.
17/fig 5, 19/fig 6: It's striking that the LES features and posterior samples/mode from the autoconversion C&R model are roughly symmetric in their cloud growth and decay phase, while the posterior samples/mode from the accretion C&R model are more asymmetric with more rapid cloud decay than growth. Since the two C&R models show similar behavior through much of the manuscript, perhaps a couple of sentences could ponder this difference and say something about the relative realism of the two C&R models.
20/Fig 7: If this figure is maintained, the inset in panel (h) should be explained in the caption.
Typographical/wording Suggestions:
23/523: Remove "summary and conclusions" at the start of this section.
29/refs: A couple of things looked odd to me here, so maybe read through and check the links. What I saw: capitalize "Solomon" on line 593, check page numbers in Kazarnikov, KTF and Lunderman, and, also, switch away from all caps titles for Maraia et al and Stevens et al.
Citation: https://doi.org/10.5194/egusphere-2026-2871-RC2
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Review of egusphere-2026-2871: “Technical note: Evaluation of conceptual predator-prey models for the quantitative modeling of precipitating open-cell stratocumulus via feature-based Bayesian inversion of a suite of Large eddy simulations”
In this manuscript, the authors utilize a low-complexity predator-prey model to simulate limit cycles in precipitating open-cell stratocumulus. The cloud and rain model is calibrated via feature-based Bayesian inversion to generate limit cycles similar to those of cloud growth and decay seen in the LES across varying meteorological conditions. seen in the LES across varying meteorological conditions. The authors note the skill of the C&R model in reproducing limit cycling, but also discuss that the inferred posterior distributions over the C&R model parameters shift in ways that appear physically implausible as the LES boundary conditions change, likely due to the simplified physics of the C&R equations. This is an interesting paper that presents a balanced overview of the usefulness and limitations of low-order modeling. Overall, I recommend publication after revision, as addressing the comments below would strengthen both the presentation and interpretation of the results.
Major comments:
Ln 94: Something that is not clearly stated in the text is that Ho itself implicitly depends on rain. Reevaporation of drizzle beneath the cloud base can act as a sink of turbulent kinetic energy, promoting boundary-layer decoupling and thereby influencing Ho. Could the authors discuss whether this missing feedback contributes to the nonphysical shifts in the inferred posterior parameters?
Since both formulations produce nearly identical posterior distributions and similar skill in reproducing the LES features, what additional physical insight is gained by introducing the accretion formulation? Does this suggest that the qualitative predator-prey dynamics are insensitive to the exact precipitation parameterization, or are there situations where the two models are expected to diverge?
What is the outlier in Figure 2a+b that is not labeled?
The posterior distributions exhibit substantial parameter degeneracies. Could the authors discuss whether these arise primarily because the selected feature (the average cloud growth-decay cycle) is insufficient to uniquely constrain the model, rather than only reflecting structural deficiencies in the C&R equations? Would incorporating additional observables (e.g., rain rate, cloud fraction, or LWP) help distinguish among parameter combinations?
Minor comments:
Ln 35-40: Perhaps it is worth noting the “intermediate” complexity mixed-layer model (Nicholls 1984, Bretherton & Wyant, 1997, etc) that bridges the gap between LES and the conceptual predator-prey model.
Ln 45: The introduction could more clearly define the features used in the inversion (e.g., cycle amplitude, duration, and shape), since this is not obvious until Section 5.
Eq (3), Eq (14): Citations for these?
Figure 1: It may be helpful to the reader to mark the location of the lines in Figure 1d on the phase space in Figure 1b. Additionally, in the axes labels, it would be helpful to define the nondimensional variables (i.e., D = T/tau) so that the reader does not have to flip back and forth to remind themselves.
Including representative oscillations from one or two LES in an appendix would help readers assess how well the conceptual model reproduces the original simulations.