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At least 19 records

Multi‐Model Ensembles in Ecosystem Modeling: Challenges and Best Practices for Decision‐Making

Ecosystem models are increasingly central to the decision-making for environmental policy, conservation planning, and climate-related investments. Yet, the growing reliance on Multi-Model Ensembles (MMEs) of ecosystem models by practitioners and policymakers, sometimes under tight timelines and imperfect information, has frequently outpaced the scientific rigor required to ensure ensemble reliability. Here, MMEs refer to approaches that combine targeted predictions from multiple models with the expectation of improving robustness and quantifying predictive uncertainty. Poorly designed MMEs may create a false sense of confidence and lead to suboptimal policy and market decisions. This perspective argues that robust decision-making-relevant MMEs must be grounded on two pillars: (1) rigorous Model Intercomparison Projects (MIPs), which identify inter-model agreement and disagreement, characterize model uncertainties, and evaluate robustness with observationally based benchmarks—MIPs' diagnostic evaluation is so critical that it must be needed to drive MME's decision in model selection and weighting, especially when only a limited number of models available; and (2) co-design by both stakeholders and scientists to ensure that scenarios, metrics and uncertainty requirements provide decision-relevant information. Building upon the past success and lessons from the existing MIPs-MMEs efforts (e.g., climate/Earth system/crop), we derived the theoretical basis for MMEs, addressed their specific challenges in ecosystem modeling, and highlighted proper consideration of model numbers and diversity, risk of model inter-dependence, effective calibration of model parameters, possible overdue of some ecosystem model development, critical roles of open benchmark data across a wide range of conditions, and suggested use of Artificial Intelligence to support MIPs-MMEs. We highlighted the under-recognized opportunity for MIPs and MMEs to drive scientific progress and innovation through identifying better performing models, systematic benchmarking, feedback loops, and targeted model improvement. By following actionable best practice guidelines, MMEs can evolve from ad hoc aggregation of models into a trusted backbone of environmental policy and decision-making.

ecosystem modeling↗

Regional and Model-Specific Response Types in A Global Gridded Crop Model Ensemble

Crop models are often employed to project crop yields under changing conditions such as global warming and associated management change for adaptation. Multi-model ensembles are promoted to enhance the robustness of projections, but questions remain on what causes often large differences between projections of individual models. Global Gridded Crop Models (GGCMs) are especially exposed to this question when applied for assessing climate change impacts, adaptation, environmental impacts of agricultural production, because their results are used in downstream analyses, such as in integrated assessment or economic modeling for projecting future land-use change. Even though global gridded crop models are often based on detailed field-scale models or have implemented similar modeling principles in other ecosystem models, global-scale models are subject to substantial uncertainties from both model structure and parametrization as well as from calibration and input data quality. AgMIP’s Global Gridded Crop Model Intercomparison (GGCMI) has thus set out to intercompare GGCMs in order to evaluate model performance, describe model uncertainties, identify inconsistencies within the ensemble and underlying reasons, and to ultimately improve models and modeling capacities. In phase 2 of the GGCMI activities, 12 modeling groups followed a modeling protocol that asked for up to 1404 31-year global simulations at 0.5 arc-degree spatial resolution to assess models’ sensitivities to changes in carbon dioxide (C; 4 different levels) temperature (T; 7 different offset levels), water supply (W; 9 levels), and nitrogen (N; 3 levels), the so-called CTWN experiment (Franke et al. 2020; http://dx.doi.org/10.5194/gmd-13-2315-2020). We here present analyses of model response types using impact response surfaces along the C, T, W, and N dimensions, respectively and collectively. Doing so, we can understand differences in simulated responses per driver rather than aggregated changes in yields. We find that models’ sensitivities to the individual driver dimensions are substantially different and often more different across models than across regions. A cluster analysis finds regional and model-specific patterns. There is some agreement across models with respect to the spatial patterns of response types but strong differences in the distribution of response type clusters across models suggests that models need to undergo further scrutiny. We suggest establishing standards in model process evaluation not only against historical dynamics but also against dedicated experiments across the CTWN dimensions.

crop models↗

Multi-Model Ensemble Wake Vortex Prediction

Several multi-model ensemble methods are investigated for predicting wake vortex transport and decay. This study is a joint effort between National Aeronautics and Space Administration and Deutsches Zentrum fuer Luft- und Raumfahrt to develop a multi-model ensemble capability using their wake models. An overview of different multi-model ensemble methods and their feasibility for wake applications is presented. The methods include Reliability Ensemble Averaging, Bayesian Model Averaging, and Monte Carlo Simulations. The methodologies are evaluated using data from wake vortex field experiments.

Koerner, Stephan↗

Multi-Model Ensemble Wake Vortex Prediction

Several multi-model ensemble methods are investigated for predicting wake vortex transport and decay. This study is a joint effort between National Aeronautics and Space Administration and Deutsches Zentrum fuer Luft- und Raumfahrt to develop a multi-model ensemble capability using their wake models. An overview of different multi-model ensemble methods and their feasibility for wake applications is presented. The methods include Reliability Ensemble Averaging, Bayesian Model Averaging, and Monte Carlo Simulations. The methodologies are evaluated using data from wake vortex field experiments.

Koerner, Stephan↗

The North American Multi-Model Ensemble (NMME): Phase-1 Seasonal to Interannual Prediction, Phase-2 Toward Developing Intra-Seasonal Prediction

The recent US National Academies report "Assessment of Intraseasonal to Interannual Climate Prediction and Predictability" was unequivocal in recommending the need for the development of a North American Multi-Model Ensemble (NMME) operational predictive capability. Indeed, this effort is required to meet the specific tailored regional prediction and decision support needs of a large community of climate information users. The multi-model ensemble approach has proven extremely effective at quantifying prediction uncertainty due to uncertainty in model formulation, and has proven to produce better prediction quality (on average) then any single model ensemble. This multi-model approach is the basis for several international collaborative prediction research efforts, an operational European system and there are numerous examples of how this multi-model ensemble approach yields superior forecasts compared to any single model. Based on two NOAA Climate Test Bed (CTB) NMME workshops (February 18, and April 8, 2011) a collaborative and coordinated implementation strategy for a NMME prediction system has been developed and is currently delivering real-time seasonal-to-interannual predictions on the NOAA Climate Prediction Center (CPC) operational schedule. The hindcast and real-time prediction data is readily available (e.g., http://iridl.ldeo.columbia.edu/SOURCES/.Models/.NMME/) and in graphical format from CPC (http://origin.cpc.ncep.noaa.gov/products/people/wd51yf/NMME/index.html). Moreover, the NMME forecast are already currently being used as guidance for operational forecasters. This paper describes the new NMME effort, presents an overview of the multi-model forecast quality, and the complementary skill associated with individual models.

NOAA CPC↗

Crop Model Improvement Reduces the Uncertainty of the Response to Temperature of Multi-Model Ensembles

To improve climate change impact estimates and to quantify their uncertainty, multi-model ensembles (MMEs) have been suggested. Model improvements can improve the accuracy of simulations and reduce the uncertainty of climate change impact assessments. Furthermore, they can reduce the number of models needed in a MME. Herein, 15 wheat growth models of a larger MME were improved through re-parameterization and/or incorporating or modifying heat stress effects on phenology, leaf growth and senescence, biomass growth, and grain number and size using detailed field experimental data from the USDA Hot Serial Cereal experiment (calibration data set). Simulation results from before and after model improvement were then evaluated with independent field experiments from a CIMMYT worldwide field trial network (evaluation data set). Model improvements decreased the variation (10th to 90th model ensemble percentile range) of grain yields simulated by the MME on average by 39% in the calibration data set and by 26% in the independent evaluation data set for crops grown in mean seasonal temperatures greater than 24 C. MME mean squared error in simulating grain yield decreased by 37%. A reduction in MME uncertainty range by 27% increased MME prediction skills by 47%. Results suggest that the mean level of variation observed in field experiments and used as a benchmark can be reached with half the number of models in the MME. Improving crop models is therefore important to increase the certainty of model-based impact assessments and allow more practical, i.e. smaller MMEs to be used effectively.

damage assessment↗

Multi-Wheat-Model Ensemble Responses to Interannual Climate Variability

We compare 27 wheat models' yield responses to interannual climate variability, analyzed at locations in Argentina, Australia, India, and The Netherlands as part of the Agricultural Model Intercomparison and Improvement Project (AgMIP) Wheat Pilot. Each model simulated 1981e2010 grain yield, and we evaluate results against the interannual variability of growing season temperature, precipitation, and solar radiation. The amount of information used for calibration has only a minor effect on most models' climate response, and even small multi-model ensembles prove beneficial. Wheat model clusters reveal common characteristics of yield response to climate; however models rarely share the same cluster at all four sites indicating substantial independence. Only a weak relationship (R2 0.24) was found between the models' sensitivities to interannual temperature variability and their response to long-termwarming, suggesting that additional processes differentiate climate change impacts from observed climate variability analogs and motivating continuing analysis and model development efforts.

uncertainty↗

Using a numerical cumulus ensemble model as a tool for studying cloud processes

A numerical cumulus ensemble model is presented, which covers a large area, and resolves individual cumulus clouds. The clouds produced by the model simulation are illustrated. Scatter plots of cloud amount vs relative humidity are examined. Plans for a cumulus ensemble model study on the effects of boundary layer processes on cloud formation are discussed.

Krueger, Steven K.↗

“Which Projections Do I Use?” Strategies for Climate Model Ensemble Subset Selection Based on Regional Stakeholder Needs

Climate model (or earth system model) projections are increasingly used for climate adaptation planning and impact assessments. As part of this process, many end‐users evaluate a subset of downscaled climate projections without being aware of the implications of downscaling methodology for statistics or event outcomes. Approaches for determining a subset of global climate models to use often focus on values from the raw models, rather than from their downscaled counterparts, in other words assuming that the statistical distribution of the multi‐model ensemble does not change post downscaling. This study demonstrates that a downscaled ensemble will typically retain the change distribution as a raw ensemble, but individual models can differ dramatically post‐downscaling. We recommend that subset‐selection methods account for this possibility and that decision‐relevant downscaled climate projections provide proper descriptions of fitness‐for‐purpose and essential caveats, so that non‐specialists can interpret the results with an appropriate level of confidence.

54 ENVIRONMENTAL SCIENCES↗

The tropical water and energy cycles in a cumulus ensemble model. Part 1: Equilibrium climate

A cumulus ensemble model is used to study the tropical water and energy cycles and their role in the climate system. The model includes cloud dynamics, radiative processes, and microphysics that incorporate all important production and conversion processes among water vapor and five species of hydrometeors. Radiative transfer in clouds is parameterized based on cloud contents and size distributions of each bulk hydrometeor. Several model integrations have been carried out under a variety of imposed boundary and large-scale conditions. In Part 1 of this paper, the primary focus is on the water and heat budgets of the control experiment, which is designed to simulate the convective - radiative equilibrium response of the model to an imposed vertical velocity and a fixed sea surface temperature at 28 C. The simulated atmosphere is conditionally unstable below the freezing level and close to neutral above the freezing level. The equilibrium water budget shows that the total moisture source, M(sub s), which is contributed by surface evaporation (0.24 M(sub s)) and the large-scale advection (0.76 M(sub s)), all converts to mean surface precipitation bar-P(sub s). Most of M(sub s) is transported verticaly in convective regions where much of the condensate is generated and falls to surface (0.68 bar-P(sub s)). The remaining condensate detrains at a rate of 0.48 bar-P(sub s) and constitutes 65% of the source for stratiform clouds above the melting level. The upper-level stratiform cloud dissipates into clear environment at a rate of 0.14 bar-P(sub s), which is a significant moisture source comparable to the detrained water vapor (0.15 bar-P(sub s)) to the upper troposphere from convective clouds. In the lower troposphere, stratiform clouds evaporate at a rate of 0.41 bar-P(sub s), which is a more dominant moisture source than surface evaporation (0.22 bar-P(sub s)). The precipitation falling to the surface in the stratiform region is about 0.32 bar-P(sub s). The associated latent heating in the water cycle is the dominant source in the heat budget that generates a net upward motion in convective regions, upper stratiform regions (above the freezing level), and a downward motion in the lower stratiform regions. The budgets reveal a cycle of water and energy resulted from radiation-dynamic-convection interactions that maintain equilibrium of the atmosphere.

Sui, C. H.↗

Data from: Coupled machine learning-ecosystem ensemble models substantially improve predictions of nitrous oxide (N 2 O) fluxes from US croplands

Nitrous oxide (N₂O) is a potent and persistent greenhouse gas, with rising atmospheric concentrations driven in part by inefficient use of synthetic nitrogen (N) fertilizers in agriculture. Predicting soil N₂O emissions is challenging due to high spatial and temporal variability arising from complex soil biogeochemical processes. Process-based ecosystem models and standalone machine learning (ML) approaches without extensive site-specific calibration often miss high emission episodes. Here, we show how an Ensemble Modeling System (EMS) based on outputs from an ensemble of ecosystem models coupled to an ensemble of ML models can improve predictions and understanding of N2O fluxes from US cropland. Trained and validated on approximately 12,000 N2O chamber measurements at 17 U.S. Midwest sites (six crops, 35 management practices), the EMS accurately predicted daily fluxes of N2O at both training (R² = 0.84, RMSE = 16.4 g N ha⁻¹ d⁻¹) and held-out testing sites (R² = 0.84, RMSE = 6.2 g N ha⁻¹ d⁻¹). Analyses identified six dominant N₂O drivers: soil organic carbon (SOC), NH₄⁺, NO₃⁻, water-filled pore space (WFPS), soil temperature, and biomass production. Wet, warm soils produced large N₂O peaks only with sufficient SOC and mineral N; in low-SOC soils, fluxes remained low. Incorporating these drivers into process-based models might significantly improve their predictive capacity. The EMS demonstrates a strong potential to predict N₂O fluxes at unseen sites, enabling more reliable regional inventories, improved gap-filling where measurements are sparse, and enhanced understanding of mechanisms to advance targeted mitigation strategies in food, feed, and bioenergy crops.

agricultural sciences↗

Coupled machine learning–ecosystem ensemble models substantially improve predictions of nitrous oxide (N 2 O) fluxes from US croplands

Nitrous oxide (N 2 O) is a potent and persistent greenhouse gas, with rising atmospheric concentrations driven in part by inefficient use of synthetic nitrogen (N) fertilizers in agriculture. Predicting soil N 2 O emissions is challenging due to high spatial and temporal variability arising from complex soil biogeochemical processes. Process-based ecosystem models and standalone machine learning (ML) approaches without extensive site-specific calibration often miss high-emission episodes. Here, we show how an Ensemble Modeling System (EMS) based on outputs from an ensemble of ecosystem models coupled to an ensemble of ML models can improve predictions and understanding of N 2 O fluxes from US cropland. Trained and validated on ~12,000 N 2 O chamber measurements at 17 US Midwest sites (six crops, 35 management practices), the EMS accurately predicted daily fluxes of N 2 O at both training (R 2 = 0.84, RMSE = 16.4 g N ha −1 d −1 ) and held-out testing sites (R 2 = 0.84, RMSE = 6.2 g N ha −1 d −1 ). Analyses identified six dominant N 2 O drivers: soil organic carbon (SOC), NH 4 + , NO 3 - , water-filled pore space, temperature, and aboveground biomass production. Wet, warm soils produced large N 2 O peaks only with sufficient SOC and mineral N; in low-SOC soils, fluxes remained low. Incorporating these drivers into process-based models might significantly improve their predictive capacity. The EMS demonstrates a strong potential to predict N 2 O fluxes at unseen sites, enabling more reliable regional inventories, improved gap-filling where measurements are sparse, and enhanced understanding of mechanisms to advance targeted mitigation strategies in food, feed, and bioenergy crops.

AI↗

Parameterization-Induced Uncertainties and Impacts of Crop Management Harmonization in a Global Gridded Crop Model Ensemble

Global gridded crop models (GGCMs) combine agronomic or plant growth models with gridded spatial input data to estimate spatially explicit crop yields and agricultural externalities at the global scale. Differences in GGCM outputs arise from the use of different biophysical models, setups, and input data. GGCM ensembles are frequently employed to bracket uncertainties in impact studies without investigating the causes of divergence in outputs. This study explores differences in maize yield estimates from five GGCMs based on the public domain field-scale model Environmental Policy Integrated Climate (EPIC) that participate in the AgMIP Global Gridded Crop Model Intercomparison initiative. Albeit using the same crop model, the GGCMs differ in model version, input data, management assumptions, parameterization, and selection of subroutines affecting crop yield estimates via cultivar distributions, soil attributes, and hydrology among others. The analyses reveal inter-annual yield variability and absolute yield levels in the EPIC-based GGCMs to be highly sensitive to soil parameterization and crop management. All GGCMs show an intermediate performance in reproducing reported yields with a higher skill if a static soil profile is assumed or sufficient plant nutrients are supplied. An in-depth comparison of setup domains for two EPIC-based GGCMs shows that GGCM performance and plant stress responses depend substantially on soil parameters and soil process parameterization, i.e. hydrology and nutrient turnover, indicating that these often neglected domains deserve more scrutiny. For agricultural impact assessments, employing a GGCM ensemble with its widely varying assumptions in setups appears the best solution for coping with uncertainties from lack of comprehensive global data on crop management, cultivar distributions and coefficients for agro-environmental processes. However, the underlying assumptions require systematic specifications to cover representative agricultural systems and environmental conditions. Furthermore, the interlinkage of parameter sensitivity from various domains such as soil parameters, nutrient turnover coefficients, and cultivar specifications highlights that global sensitivity analyses and calibration need to be performed in an integrated manner to avoid bias resulting from disregarded core model domains. Finally, relating evaluations of the EPIC-based GGCMs to a wider ensemble based on individual core models shows that structural differences outweigh in general differences in configurations of GGCMs based on the same model, and that the ensemble mean gains higher skill from the inclusion of structurally different GGCMs. Although the members of the wider ensemble herein do not consider crop-soil-management interactions, their sensitivity to nutrient supply indicates that findings for the EPIC-based sub-ensemble will likely become relevant for other GGCMs with the progressing inclusion of such processes.

Folberth, Christian↗

An Alternative Ensemble Streamflow Prediction Approach Using Improved Subseasonal Precipitation Forecasts from the North America Multi-Model Ensemble Phase II

In this article, streamflow forecasting at a subseasonal time scale (10–30 days into the future) is important for various human activities. The ensemble streamflow prediction (ESP) is a widely applied technique for subseasonal streamflow forecasting. However, ESP’s reliance on the randomly resampled historical precipitation limits its predictive capability. Available dynamical subseasonal precipitation forecasts provide an alternative to the randomly resampled precipitation in ESP. Prior studies found the predictive performance of raw subseasonal precipitation forecast is limited in many regions such as the central south of the United States, which raises questions about its effectiveness in assisting streamflow forecasting. To further assess the hydrologic applicability of dynamical subseasonal precipitation forecasts, we test the subseasonal precipitation forecast from North America Multi-Model Ensemble Phase II (NMME-2) at four watersheds in the central south region of the United States. The subseasonal precipitation forecasts are postprocessed with bias correction and spatial disaggregation (BCSD) to correct bias and improve spatial resolution before replacing the randomly resampled precipitation in ESP for streamflow predictions. The performance of the resulting streamflow predictions is benchmarked with ESP. Evaluation is conducted using Kling–Gupta Efficiency (KGE), continuous ranked probability score (CRPS), probability of detection (POD), false alarm ratios (FARs), as well as reliability diagrams. Our results suggest that BCSD-corrected subseasonal precipitation forecasts lead to overall improved streamflow predictions due to added skills in winter and spring. Our results also suggest that BCSD-corrected subseasonal precipitation forecasts lead to improved predictions on the occurrence of high-percentile streamflow values above 75%. Overall, BCSD-corrected subseasonal precipitation has shown promising performance, highlighting its potential broader applications for river and flood forecasting.

54 ENVIRONMENTAL SCIENCES↗

A multi model ensemble reveals net climate benefits from regenerative practices in US Midwest croplands

Process-based cropping systems models (CSMs) are key components of measurement, monitoring, reporting, and verification frameworks of carbon markets, but model-specific differences limit their applicability across diverse pedo-climatic conditions and agronomic practices. Multi-model ensemble (MME) provides an opportunity to better estimate changes in soil organic carbon (SOC) and nitrous oxide (N 2 O) emissions from agronomic practices at scale. We used an MME across 46 million hectares of US Midwest cropland at a resolution of 4-km 2 to assess the aggregate ability of different regenerative practices to sequester SOC and N 2 O emissions compared to their counterfactual dynamic baselines. MME was validated against long-term trials and compared to its constituent CSMs, showing greater accuracy and lower uncertainty. The results show that adopting no-till combined with cover crops increased SOC stocks by 0.36 ± 0.12 Mg ha -1 yr -1 , corresponding to a net regional SOC gain of 16.4 Tg C yr -1 compared to business-as-usual baselines. These benefits are halved when each management is practiced individually, and the SOC gains are only fully realized with low initial carbon stock. By including N 2 O emissions, we can assess the overall climate mitigation potential, specifically, the extent to which carbon sequestration can offset direct N 2 O emissions. The magnitude of this potential varies depending on management practices and geographic location with net climate benefits on average ranging from 0 to 3 Mg CO 2 -eq ha -1 yr -1 . High-resolution MME results allow for robust estimates of climate mitigation, reducing barriers to carbon market participation and supporting regenerative agriculture initiatives at scale.

carbon credits↗

The Effect of Future Ambient Air Pollution on Human Premature Mortality to 2100 Using Output from the ACCMIP Model Ensemble

Ambient air pollution from ground-level ozone and fine particulate matter (PM(sub 2.5)) is associated with premature mortality. Future concentrations of these air pollutants will be driven by natural and anthropogenic emissions and by climate change. Using anthropogenic and biomass burning emissions projected in the four Representative Concentration Pathway scenarios (RCPs), the ACCMIP ensemble of chemistry climate models simulated future concentrations of ozone and PM(sub 2.5) at selected decades between 2000 and 2100. We use output from the ACCMIP ensemble, together with projections of future population and baseline mortality rates, to quantify the human premature mortality impacts of future ambient air pollution. Future air-pollution-related premature mortality in 2030, 2050 and 2100 is estimated for each scenario and for each model using a health impact function based on changes in concentrations of ozone and PM(sub 2.5) relative to 2000 and projected future population and baseline mortality rates. Additionally, the global mortality burden of ozone and PM(sub 2.5) in 2000 and each future period is estimated relative to 1850 concentrations, using present-day and future population and baseline mortality rates. The change in future ozone concentrations relative to 2000 is associated with excess global premature mortality in some scenarios/periods, particularly in RCP8.5 in 2100 (316 thousand deaths per year), likely driven by the large increase in methane emissions and by the net effect of climate change projected in this scenario, but it leads to considerable avoided premature mortality for the three other RCPs. However, the global mortality burden of ozone markedly increases from 382000 (121000 to 728000) deaths per year in 2000 to between 1.09 and 2.36 million deaths per year in 2100, across RCPs, mostly due to the effect of increases in population and baseline mortality rates. PM(sub 2.5) concentrations decrease relative to 2000 in all scenarios, due to projected reductions in emissions, and are associated with avoided premature mortality, particularly in 2100: between 2.39 and 1.31 million deaths per year for the four RCPs. The global mortality burden of PM(sub 2.5) is estimated to decrease from 1.70 (1.30 to 2.10) million deaths per year in 2000 to between 0.95 and 1.55 million deaths per year in 2100 for the four RCPs due to the combined effect of decreases in PM(sub 2.5) concentrations and changes in population and baseline mortality rates. Trends in future air-pollution-related mortality vary regionally across scenarios, reflecting assumptions for economic growth and air pollution control specific to each RCP and region. Mortality estimates differ among chemistry climate models due to differences in simulated pollutant concentrations, which is the greatest contributor to overall mortality uncertainty for most cases assessed here, supporting the use of model ensembles to characterize uncertainty. Increases in exposed population and baseline mortality rates of respiratory diseases magnify the impact on premature mortality of changes in future air pollutant concentrations and explain why the future global mortality burden of air pollution can exceed the current burden, even where air pollutant concentrations decrease.

Silva, Raquel A.↗

The macroscopic behavior of cumulus ensembles simulated by a cumulus ensemble model

A number of simulation experiments are performed with the 2D UCLA cumulus ensemble model to study the macroscopic behavior of cumulus convection under a variety of different large-scale and underlying surface conditions. Specifically, the modulation of cumulus activity by the imposed large-scale processes and the eddy kinetic energy (EKE) budget are investigated in detail. In all simulations, cumulus convection is rather strongly modulated by large-scale advective processes in spite of the existence of some nonmodulated high-frequency fluctuations. The modulation exhibits some phase delays, however, when the basic wind shear is strong. This is presumably associated with the existence of mesoscale convective organization. The EKE budget analysis shows that the net eddy buoyancy generation rate is nearly zero for a wind range of cumulus ensembles.

Xu, Kuan-Man↗