Articles | Volume 15, issue 9
https://doi.org/10.5194/gmd-15-3879-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/gmd-15-3879-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Development and technical paper
| 
13 May 2022
Development and technical paper |
| 13 May 2022
| 13 May 2022
On numerical broadening of particle-size spectra: a condensational growth study using PyMPDATA 1.0
Michael A. Olesik
CORRESPONDING AUTHOR
Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Kraków,
Poland
Jakub Banaśkiewicz
Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Piotr Bartman
Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Manuel Baumgartner
Zentrum für Datenverarbeitung, Johannes Gutenberg University Mainz, Mainz, Germany
Institute for Atmospheric Physics, Johannes Gutenberg University Mainz, Mainz, Germany
Simon Unterstrasser
Institute of Atmospheric Physics, German Aerospace Center (DLR), Oberpfaffenhofen, Germany
Sylwester Arabas
CORRESPONDING AUTHOR
Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
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Simon Braun, Magdalena Pühl, Andreas Marsing, Daniel Sauer, Gregor Neumann, Elena de la Torre Castro, Valerian Hahn, Michael Lichtenstern, Paul Stock, Christopher Heckl, Dennis Hillenbrand, Anke Roiger, Christiane Voigt, Simon Unterstraßer, Arne Vasenden, Roberta Vasenden, Jean Cammas, Florian Thorey, Catherine Mackay, Charles Renard, and Tina Jurkat-Witschas
EGUsphere, https://doi.org/10.5194/egusphere-2026-2887, https://doi.org/10.5194/egusphere-2026-2887, 2026
This preprint is open for discussion and under review for Atmospheric Measurement Techniques (AMT).
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In this work, we share the results of first in-flight emission and contrail measurements behind a fuel cell exhaust emulator, with the goal of quantifying the contrail properties for this new aviation propulsion concept. An artificial dilution tracer system was used to assess plume dilution and derive emission indices. At ambient temperatures close to the homogeneous freezing threshold, a strong temperature-dependence of the number of contrail ice crystals, due to droplet freezing, was observed.
Sylwester Arabas, Agnieszka Żaba, Daria Klimaszewska, Emma C. Ware, Clare E. Singer, Manhal Alhilali, Jason Barr, Daniel G. Partridge, Shin-ichiro Shima, and Robert Wood
EGUsphere, https://doi.org/10.5194/egusphere-2026-3906, https://doi.org/10.5194/egusphere-2026-3906, 2026
This preprint is open for discussion and under review for Geoscientific Model Development (GMD).
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The paper focuses on the use of Python and its package ecosystem for modeling the dynamics of atmospheric aerosol and cloud droplets. We highlight that concise code and high-level abstractions (incl. programmatic expression of physical units) can be used without performance tradeoffs. The paper features two cloud-microphysics classics: an adiabatic parcel model and a Monte-Carlo coagulation solver. The presented code constitutes a single Jupyter notebook that can be executed "in the cloud".
Wenhan Tang, Sylwester Arabas, Jeffrey H. Curtis, Daniel A. Knopf, Matthew West, and Nicole Riemer
Atmos. Chem. Phys., 26, 9221–9255, https://doi.org/10.5194/acp-26-9221-2026, https://doi.org/10.5194/acp-26-9221-2026, 2026
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We studied how aerosol particles help form ice in clouds. Using new theory and detailed computer simulations, we found that the way different materials are mixed within these particles has a strong impact on how much ice forms. When ice-forming material is spread across all particles, more droplets freeze than when it is only in a few. This result means that to better predict clouds and climate, models need to account for how particle materials are mixed.
Hannes Bruder, Robin N. Thor, Malte Niklaß, Katrin Dahlmann, Roland Eichinger, Florian Linke, Volker Grewe, Sigrun Matthes, and Simon Unterstrasser
Geosci. Model Dev., 19, 3551–3567, https://doi.org/10.5194/gmd-19-3551-2026, https://doi.org/10.5194/gmd-19-3551-2026, 2026
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We develop an easy-to-use tool to estimate the per-flight climate effect of CO2 and non-CO2 emissions, based only on aircraft size as well as origin and destination airports. The implemented model results from a comparison of Multiple and Symbolic Regression approaches and exhibits a mean relative error of 21 % with respect to climate response model results. The simplified method is designed for climate footprint assessment and covers jet-powered passenger aircraft with over 20 seats.
Tim Lüttmer, Sylwester Arabas, and Peter Spichtinger
EGUsphere, https://doi.org/10.5194/egusphere-2026-2217, https://doi.org/10.5194/egusphere-2026-2217, 2026
Short summary
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Homogeneous freezing describes the spontaneous freezing of supercooled droplets, with the probability depending on temperature, droplet size, and elapsed time. Droplets freeze across a range of temperatures rather than all at once, and faster updrafts and larger droplets lead to freezing at warmer temperatures. Once some droplets freeze, the growing ice crystals deplete the surrounding vapour, causing the remaining droplets to evaporate. This affects the number of ice crystals in the cloud.
Josef Zink, Cornelius Weiß-Rehm, Simon Unterstrasser, and Ulrike Burkhardt
EGUsphere, https://doi.org/10.5194/egusphere-2026-1088, https://doi.org/10.5194/egusphere-2026-1088, 2026
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The study introduces a novel parameterization for the number of contrail ice crystals formed on ambient aerosols entrained into the exhaust plume behind aircraft burning hydrogen. This parameterization can be integrated into large-scale models to assess the radiative impacts of contrails originating from a fleet of aircraft with hydrogen combustion.
Vanessa Santos Gabriel, Luca Bugliaro, Mara Montag, Sabrina Ries, Ziming Wang, Kai Widmaier, Matteo Arico, Simon Unterstrasser, Johanna Mayer, Deniz Menekay, Andreas Marsing, Elena de la Torre Castro, Liam Megill, Monika Scheibe, and Christiane Voigt
Earth Syst. Sci. Data, 18, 2397–2412, https://doi.org/10.5194/essd-18-2397-2026, https://doi.org/10.5194/essd-18-2397-2026, 2026
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We provide observations of the geostationary Meteosat satellite with contrails labeled by three people complemented with detailed cloud information. Contrails influence climate but are hard to identify in satellite imagery. With this study, we support contrail detection development and evaluation, stress the subjectivity of human labeling and reveal which meteorological conditions highlight or hide contrails. This dataset contributes to a better understanding of aviation’s climate impact.
Josef Zink and Simon Unterstrasser
Atmos. Chem. Phys., 26, 3145–3165, https://doi.org/10.5194/acp-26-3145-2026, https://doi.org/10.5194/acp-26-3145-2026, 2026
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The climate impact of aviation-induced contrail cirrus clouds is strongly influenced by the number of ice crystals that form in the wake of an aircraft under certain conditions. In this study, we investigate how engine-related aspects influence the number of ice crystals formed for hydrogen combustion. We derive suitable (scaling) relations that can be integrated into large-scale models used to estimate the climate impact of contrail cirrus clouds.
Josef Zink, Simon Unterstrasser, and Ulrike Burkhardt
Atmos. Chem. Phys., 26, 3125–3143, https://doi.org/10.5194/acp-26-3125-2026, https://doi.org/10.5194/acp-26-3125-2026, 2026
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The climate impact of aviation-induced contrail cirrus is strongly influenced by the number of ice crystals that form in an aircraft's exhaust plume. In this study, we systematically investigate the key microphysical processes of contrail formation for hydrogen combustion. A large simulation data set provides the basis for a data-driven parameterization of ice crystal number that can be integrated into large-scale models.
Gregor Neumann, Andreas Marsing, Theresa Harlass, Daniel Sauer, Simon Braun, Magdalena Pühl, Christopher Heckl, Paul Stock, Elena De La Torre Castro, Valerian Hahn, Anke Roiger, Christiane Voigt, Simon Unterstraßer, Jean Cammas, Charles Renard, Roberta Vasenden, Arnold Vasenden, and Tina Jurkat-Witschas
Atmos. Meas. Tech., 18, 6795–6816, https://doi.org/10.5194/amt-18-6795-2025, https://doi.org/10.5194/amt-18-6795-2025, 2025
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This study presents the first successful in-flight emission characterization of a turboprop engine using a fully autonomous airborne measurement platform, offering new insights into the atmospheric impacts of regional aviation. By equipping the high-altitude Grob G 520 Egrett with a suite of custom and modified commercial instruments, we demonstrate precise, high-resolution measurements of aerosol particles, trace gases, and contrail ice in the engine exhaust plume at cruise altitudes.
Annemarie Lottermoser and Simon Unterstrasser
Atmos. Chem. Phys., 25, 7903–7924, https://doi.org/10.5194/acp-25-7903-2025, https://doi.org/10.5194/acp-25-7903-2025, 2025
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Contrail cirrus significantly contributes to aviation's overall climate impact. As hydrogen combustion and fuel cell use are emerging technologies for aircraft propulsion, we simulated individual contrails from hydrogen propulsion during the first 6 min after exhaust emission, termed the vortex phase. The ice crystal loss during that stage is crucial, as the number of ice crystals has a large impact on the further evolution of contrails into contrail cirrus and their radiative forcing.
Birte Klug, Ralf Weigel, Konrad Kandler, Markus Rapp, Manuel Baumgartner, Thomas Böttger, Klaus Dieter Wilhelm, Harald Rott, Thomas Kenntner, Oliver Drescher, and Anna Hundertmark
EGUsphere, https://doi.org/10.5194/egusphere-2025-510, https://doi.org/10.5194/egusphere-2025-510, 2025
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The nuclei onto which noctilucent clouds (NLC) form are largely unknown. We investigated the development of an inertia-based particle collector allowing for sampling NLC particles during a sounding rocket flight for off-line single particle physico-chemical analyzes. Computational fluid dynamics simulations (for Mach numbers 1.31 and 1.75) support the design and development process in reference to a basic mechanical concept of particle sampling and sample storage, which is also presented here.
Andreas Bier, Simon Unterstrasser, Josef Zink, Dennis Hillenbrand, Tina Jurkat-Witschas, and Annemarie Lottermoser
Atmos. Chem. Phys., 24, 2319–2344, https://doi.org/10.5194/acp-24-2319-2024, https://doi.org/10.5194/acp-24-2319-2024, 2024
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Using hydrogen as aviation fuel affects contrails' climate impact. We study contrail formation behind aircraft with H2 combustion. Due to the absence of soot emissions, contrail ice crystals are assumed to form only on ambient particles mixed into the plume. The ice crystal number, which strongly varies with temperature and aerosol number density, is decreased by more than 80 %–90 % compared to kerosene contrails. However H2 contrails can form at lower altitudes due to higher H2O emissions.
Emily de Jong, John Ben Mackay, Oleksii Bulenok, Anna Jaruga, and Sylwester Arabas
Geosci. Model Dev., 16, 4193–4211, https://doi.org/10.5194/gmd-16-4193-2023, https://doi.org/10.5194/gmd-16-4193-2023, 2023
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In clouds, collisional breakup occurs when two colliding droplets splinter into new, smaller fragments. Particle-based modeling approaches often do not represent breakup because of the computational demands of creating new droplets. We present a particle-based breakup method that preserves the computational efficiency of these methods. In a series of simple demonstrations, we show that this representation alters cloud processes in reasonable and expected ways.
Peter Spichtinger, Patrik Marschalik, and Manuel Baumgartner
Atmos. Chem. Phys., 23, 2035–2060, https://doi.org/10.5194/acp-23-2035-2023, https://doi.org/10.5194/acp-23-2035-2023, 2023
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We investigate the impact of the homogeneous nucleation rate on nucleation events in cirrus. As long as the slope of the rate is represented sufficiently well, the resulting ice crystal number concentrations are not crucially affected. Even a change in the prefactor over orders of magnitude does not change the results. However, the maximum supersaturation during nucleation events shows strong changes. This quantity should be used for diagnostics instead of the popular nucleation threshold.
Andreas Bier, Simon Unterstrasser, and Xavier Vancassel
Atmos. Chem. Phys., 22, 823–845, https://doi.org/10.5194/acp-22-823-2022, https://doi.org/10.5194/acp-22-823-2022, 2022
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We investigate contrail formation in an aircraft plume with a particle-based multi-trajectory 0D model. Due to the high plume heterogeneity, contrail ice crystals form first near the plume edge and then in the plume centre. The number of ice crystals varies strongly with ambient conditions and soot properties near the contrail formation threshold. Our results imply that the multi-trajectory approach does not necessarily lead to improved scientific results compared to a single mean trajectory.
Manuel Baumgartner, Christian Rolf, Jens-Uwe Grooß, Julia Schneider, Tobias Schorr, Ottmar Möhler, Peter Spichtinger, and Martina Krämer
Atmos. Chem. Phys., 22, 65–91, https://doi.org/10.5194/acp-22-65-2022, https://doi.org/10.5194/acp-22-65-2022, 2022
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An important mechanism for the appearance of ice particles in the upper troposphere at low temperatures is homogeneous nucleation. This process is commonly described by the
Koop line, predicting the humidity at freezing. However, laboratory measurements suggest that the freezing humidities are above the Koop line, motivating the present study to investigate the influence of different physical parameterizations on the homogeneous freezing with the help of a detailed numerical model.
Julia Schneider, Kristina Höhler, Robert Wagner, Harald Saathoff, Martin Schnaiter, Tobias Schorr, Isabelle Steinke, Stefan Benz, Manuel Baumgartner, Christian Rolf, Martina Krämer, Thomas Leisner, and Ottmar Möhler
Atmos. Chem. Phys., 21, 14403–14425, https://doi.org/10.5194/acp-21-14403-2021, https://doi.org/10.5194/acp-21-14403-2021, 2021
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Homogeneous freezing is a relevant mechanism for the formation of cirrus clouds in the upper troposphere. Based on an extensive set of homogeneous freezing experiments at the AIDA chamber with aqueous sulfuric acid aerosol, we provide a new fit line for homogeneous freezing onset conditions of sulfuric acid aerosol focusing on cirrus temperatures. In the atmosphere, homogeneous freezing thresholds have important implications on the cirrus cloud occurrence and related cloud radiative effects.
Ralf Weigel, Christoph Mahnke, Manuel Baumgartner, Martina Krämer, Peter Spichtinger, Nicole Spelten, Armin Afchine, Christian Rolf, Silvia Viciani, Francesco D'Amato, Holger Tost, and Stephan Borrmann
Atmos. Chem. Phys., 21, 13455–13481, https://doi.org/10.5194/acp-21-13455-2021, https://doi.org/10.5194/acp-21-13455-2021, 2021
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In July and August 2017, the StratoClim mission took place in Nepal with eight flights of the M-55 Geophysica at up to 20 km in the Asian monsoon anticyclone. New particle formation (NPF) next to cloud ice was detected in situ by abundant nucleation-mode aerosols (> 6 nm) along with ice particles (> 3 µm). NPF was observed mainly below the tropopause, down to 15 % being non-volatile residues. Observed intra-cloud NPF indicates its importance for the composition in the tropical tropopause layer.
Ralf Weigel, Christoph Mahnke, Manuel Baumgartner, Antonis Dragoneas, Bärbel Vogel, Felix Ploeger, Silvia Viciani, Francesco D'Amato, Silvia Bucci, Bernard Legras, Beiping Luo, and Stephan Borrmann
Atmos. Chem. Phys., 21, 11689–11722, https://doi.org/10.5194/acp-21-11689-2021, https://doi.org/10.5194/acp-21-11689-2021, 2021
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In July and August 2017, eight StratoClim mission flights of the Geophysica reached up to 20 km in the Asian monsoon anticyclone. New particle formation (NPF) was identified in situ by abundant nucleation-mode aerosols (6–15 nm in diameter) with mixing ratios of up to 50 000 mg−1. NPF occurred most frequently at 12–16 km with fractions of non-volatile residues of down to 15 %. Abundance and productivity of observed NPF indicate its ability to promote the Asian tropopause aerosol layer.
Cited articles
Abade, G., Grabowski, W. W., and Pawlowska, H.: Broadening of Cloud Droplet
Spectra through Eddy Hopping: Turbulent Entraining Parcel Simulations, J.
Atmos. Sci., 75, 3365–3379, https://doi.org/10.1175/JAS-D-18-0078.1, 2018. a
Arabas, S. and Pawlowska, H.: Adaptive method of lines for multi-component aerosol condensational growth and CCN activation, Geosci. Model Dev., 4, 15–31, https://doi.org/10.5194/gmd-4-15-2011, 2011. a
Arabas, S. and Shima, S.-I.: Large-Eddy Simulations of Trade Wind Cumuli Using
Particle-Based Microphysics with Monte Carlo Coalescence, J. Atmos. Sci., 70, 2768–2777,
https://doi.org/10.1175/JAS-D-12-0295.1, 2013. a
Arabas, S. and Shima, S.: On the CCN (de)activation nonlinearities, Nonlin. Processes Geophys., 24, 535–542, https://doi.org/10.5194/npg-24-535-2017, 2017. a
Arabas, S., Pawlowska, H., and Grabowski, W.: Effective radius and droplet
spectral width from in-situ aircraft observations in trade-wind cumuli during
RICO, Geophys. Res. Lett., 36, L11803, https://doi.org/10.1029/2009GL038257, 2009. a
Bartman, P., Banaśkiewicz, J., Drenda, S., Manna, M., Olesik, M., Rozwoda, P.,
Sadowski, M., and Arabas, S.: PyMPDATA v1: Numba-accelerated implementation
of MPDATA with examples in Python, Julia and Matlab, J. Open Source Soft.,
https://joss.theoj.org/papers/10e7361e43785dbb1b3d659c5b01757a (last access: April 2022), in review,
2022a. a, b
Bartman, P., Banaśkiewicz, J., Drenda, S., Manna, M., Olesik, M. A.,
Rozwoda, P., Sadowski, M., and Arabas, S.: PyMPDATA 1.0, Zenodo [code], https://doi.org/10.5281/zenodo.6329303, 2022b. a
Bartman, P., Banaśkiewicz, J., Drenda, S., Manna, M., Olesik, M. A.,
Rozwoda, P., Sadowski, M., and Arabas, S.: PyMPDATA-examples 1.0.1, Zenodo [code], https://doi.org/10.5281/zenodo.6471494, 2022c. a
Beason, C. W. and Margolin, L. G.: DPDC (double-pass donor cell): A
second-order monotone scheme for advection, in: Fifth Nuclear Code
Developers' Conference, Boulder, CO, 11–14 October 1988,
LLNL report
UCRL-99731, https://www.osti.gov/servlets/purl/7049237 (last access: April 2022), 1988. a
Brown, R.: A numerical study of radiation fog with an explicit formulation of
the microphysics, Q. J. Roy. Meteor. Soc., 106, 781–802, https://doi.org/10.1002/qj.49710645010,
1980. a, b
Chandrakar, K. K., Cantrell, W., and Shaw, R. A.: Influence of Turbulent
Fluctuations on Cloud Droplet Size Dispersion and Aerosol Indirect Effects,
J. Atmos. Sci., 75, 3191–3209, https://doi.org/10.1175/JAS-D-18-0006.1, 2018. a, b
Courant, R., Isaacson, E., and Rees, M.: On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Differences, Comm. Pure Appl.
Math., 5, 243–255, https://doi.org/10.1002/cpa.3160050303, 1952. a
Cristiani, E.: Blending Brownian motion and heat equation, J. Coupled Syst.
Multiscale Dyn., 3, 351–356, https://doi.org/10.1166/jcsmd.2015.1089, 2015. a
Crowley, W. P.: Numerical advection experiments, Mon. Weather Rev., 96, 1–11,
https://doi.org/10.1175/1520-0493(1968)096<0001:NAE>2.0.CO;2, 1968. a
Devenish, B. J., Bartello, P., Brenguier, J.-L., Collins, L. R., Grabowski,
W. W., IJzermans, R. H. A., Malinowski, S. P., Reeks, M. W., Vassilicos,
J. C., Wang, L.-P., and Warhaft, Z.: Droplet growth in warm turbulent clouds,
Q. J. Roy. Meteor. Soc., 138, 1401–1429, https://doi.org/10.1002/qj.1897, 2012. a
Dhaniyala, S. and Wexler, A. S.: Numerical schemes to model condensation and
evaporation of aerosols, Atmos. Env., 30, 919–928, https://doi.org/10.1016/1352-2310(95)00288-X,
1996. a
East, T. W. R. and Marshall, J. S.: Turbulence in clouds as a factor in
precipitation, Q. J. Roy. Meteor. Soc., 80, 26–47, https://doi.org/10.1002/qj.49708034305,
1954. a
Feingold, G. and Chuang, P. Y.: Analysis of the Influence of Film-Forming
Compounds on Droplet Growth: Implications for Cloud Microphysical Processes
and Climate, J. Atmos. Sci., 59, 2006–2018,
https://doi.org/10.1175/1520-0469(2002)059<2006:AOTIOF>2.0.CO;2, 2002. a
Field, P., Heymsfield, A., Shipway, B., DeMott, P., Pratt, K., Rogers, D.,
Stith, J., and Prather, K.: Ice in Clouds Experiment–Layer Clouds. Part II:
Testing Characteristics of Heterogeneous Ice Formation in Lee Wave Clouds, J.
Atmos. Sci., 69, 1066–1079, https://doi.org/10.1175/JAS-D-11-026.1, 2012. a
Gettelman, A. and Morrison, H.: Advanced Two-Moment Bulk Microphysics for
Global Models. Part I: Off-Line Tests and Comparison with Other Schemes, J.
Climate, 28, 1268–1287, https://doi.org/10.1175/JCLI-D-14-00102.1, 2015. a
Grabowski, W. and Wang, L.-P.: Growth of Cloud Droplets in a Turbulent
Environment, Annu. Rev. Fluid Mech., 45, 293–324,
https://doi.org/10.1146/annurev-fluid-011212-140750, 2013. a
Grabowski, W., Andrejczuk, M., and Wang, L.-P.: Droplet growth in a bin
warm-rain scheme with Twomey CCN activation, Atmos. Res., 99, 290–301,
https://doi.org/10.1016/j.atmosres.2010.10.020, 2011. a
Grabowski, W. W.: Comparison of Eulerian Bin and Lagrangian Particle-Based Microphysics in Simulations of Nonprecipitating Cumulus, J. Atmos. Sci., 77, 3951–3970,
https://doi.org/10.1175/JAS-D-20-0100.1, 2020. a
Hall, W. D.: A Detailed Microphysical Model Within a Two-Dimensional Dynamic
Framework: Model Description and Preliminary Results, J. Atmos. Sci., 37, 2486–2507,
https://doi.org/10.1175/1520-0469(1980)037<2486:ADMMWA>2.0.CO;2, 1980. a
Hernandéz Pardo, L. H., Morrison, H., Machado, L. A. T., Harrington, J. Y.,
and Lebo, Z. J.: Drop Size Distribution Broadening Mechanisms in a Bin
Microphysics Eulerian Model, J. Atmos. Sci., 77, 3249–3273,
https://doi.org/10.1175/JAS-D-20-0099.1, 2020. a, b, c
Hill, A., Shipway, B., and Boutle, I.: How sensitive are aerosol-precipitation
interactions to the warm rain representation?, J. Adv. Model. Earth Syst., 7, 987–1004,
https://doi.org/10.1002/2014MS000422, 2015.
Hirt, C. W.: Heuristic stability theory for finite-difference equations, J.
Comput. Phys., 2, 339–355, https://doi.org/10.1016/0021-9991(68)90041-7, 1968. a
Howell, W.: The growth of cloud drops in uniformly cooled air, J. Meteorol., 6, 134–149,
https://doi.org/10.1175/1520-0469(1949)006<0134:TGOCDI>2.0.CO;2, 1949. a
Hulburt, H. and Katz, S.: Some problems in particle technology: A statistical
mechanical formulation, Chem. Eng. Sci., 19, 555–574,
https://doi.org/10.1016/0009-2509(64)85047-8, 1964. a
Jaruga, A., Arabas, S., Jarecka, D., Pawlowska, H., Smolarkiewicz, P. K., and Waruszewski, M.: libmpdata++ 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations, Geosci. Model Dev., 8, 1005–1032, https://doi.org/10.5194/gmd-8-1005-2015, 2015. a, b, c
Jeffery, C. A., Reisner, J. M., and Andrejczuk, M.: Another Look at Stochastic
Condensation for Subgrid Cloud Modeling: Adiabatic Evolution and Effects, J.
Atmos. Sci., 64, 3949–3969, https://doi.org/10.1175/2006JAS2147.1, 2007. a
Khain, A. P., Beheng, K. D., Heymsfield, A., Korolev, A., Krichak, S. O.,
Levin, Z., Pinsky, M., Phillips, V., Prabhakaran, T., Teller, A., van den
Heever, S. C., and Yano, J.-I.: Representation of microphysical processes in
cloud-resolving models: Spectral (bin) microphysics versus bulk
parameterization, Rev. Geophys., 53, 247–322, https://doi.org/10.1002/2014RG000468, 2015. a
Kostinski, A. B. and Jameson, A. R.: On the Spatial Distribution of Cloud
Particles, J. Atmos. Sci., 57, 901–915,
https://doi.org/10.1175/1520-0469(2000)057<0901:OTSDOC>2.0.CO;2, 2000. a
Kostoglou, M. and Karabelas, A. J.: Evaluation of numerical methods for simulating an evolving particle size distribution in growth processes, Chem.
Eng. Comm., 136, 177–199, https://doi.org/10.1080/00986449508936360, 1995. a
Kovetz, A.: An Analytical Solution for the Change of Cloud and Fog Droplet
Spectra Due to Condensation, J. Atmos. Sci., 26, 302–304,
https://doi.org/10.1175/1520-0469(1969)026<0302:AASFTC>2.0.CO;2, 1969. a
Kovetz, A. and Olund, B.: The Effect of Coalescence and Condensation on Rain
Formation in a Cloud of Finite Vertical Extent, J. Atmos. Sci., 26, 1060–1065,
https://doi.org/10.1175/1520-0469(1969)026<1060:TEOCAC>2.0.CO;2, 1969. a, b, c
Kühnlein, C. and Smolarkiewicz, P. K.: An unstructured-mesh finite-volume
MPDATA for compressible atmospheric dynamics, J. Comp. Phys., 334, 16–30,
https://doi.org/10.1016/j.jcp.2016.12.054, 2017. a
Kuo, H.-C., Leou, T.-M., and Williams, R. T.: A study on the high-order
Smolarkiewicz methods, Comput. Fluids, 28, 779–799,
https://doi.org/10.1016/s0045-7930(98)00036-x, 1999. a
Lange, R.: ADPIC: a three-dimensional computer code for the study of pollutant dispersal and deposition under complex conditions,
LLNL report no. UCRL-51462, https://doi.org/10.2172/4308175, 1973. a
Lange, R.: ADPIC–A Three-Dimensional Particle-in-Cell Model for the Dispersal
of Atmospheric Pollutants and its Comparison to Regional Tracer Studies, J.
Appl. Meteorol., 17, 320–329, https://doi.org/10.1175/1520-0450(1978)017<0320:ATDPIC>2.0.CO;2,
1978. a
Lee, H., Fridlind, A., and Ackerman, A.: An Evaluation of Size-Resolved Cloud
Microphysics Scheme Numerics for Use with Radar Observations. Part II:
Condensation and Evaporation, J. Atmos. Sci., 78, 1629–1645,
https://doi.org/10.1175/JAS-D-20-0213.1, 2021. a, b, c
Li, X.-Y., Brandenburg, A., Haugen, N. E. L., and Svensson, G.: Eulerian and
Lagrangian approaches to multidimensional condensation and collection, J.
Adv. Model. Earth Syst., 9, 1116–1137, https://doi.org/10.1002/2017MS000930, 2017. a, b
Liu, Q., Kogan, Y. L., Lilly, D. K., and Khairoutdinov, M. P.: Variational
Optimization Method for Calculation of Cloud Drop Growth in an Eulerian
Drop-Size Framework, J. Atmos. Sci., 54, 2493–2504,
https://doi.org/10.1175/1520-0469(1997)054<2493:VOMFCO>2.0.CO;2, 1997. a, b, c
Lu, M.-L. and Seinfeld, J.: Effect of aerosol number concentration on cloud
droplet dispersion: A large-eddy simulation study and implications for
aerosol indirect forcing, J. Geophys. Res., 111, D02207, https://doi.org/10.1029/2005JD006419,
2006. a
Margolin, L. G. and Shashkov, M.: MPDATA: gauge transformations, limiters and
monotonicity, Int. J. Numer. Methods Fluids, 50, 1193–1206, https://doi.org/10.1002/fld.1070, 2006. a, b, c
Margolin, L. G. and Smolarkiewicz, P. K.: Antidiffusive Velocities for
Multipass Donor Cell Advection, SIAM J. Sci. Comput., 20, 907–929,
https://doi.org/10.1137/S106482759324700X, 1998. a, b, c, d
Margolin, L. G., Shashkov, M., and Smolarkiewicz, P. K.: A discrete operator
calculus for finite difference approximations, Comput. Methods Appl. Mech.
Eng., 187, 365–383, https://doi.org/10.1016/S0045-7825(00)80001-8, 2000. a
Morrison, H., van Lier-Walqui, M., Fridlind, A. M., Grabowski, W. W.,
Harrington, J. Y., Hoose, C., Korolev, A., Kumjian, M. R., Milbrandt, J. A.,
Pawlowska, H., Posselt, D. J., Prat, O. P., Reimel, K. J., Shima, S.-I., van
Diedenhoven, B., and Xue, L.: Confronting the Challenge of Modeling Cloud and
Precipitation Microphysics, J. Adv. Model. Earth Syst., 12, e2019MS001689,
https://doi.org/10.1029/2019MS001689, 2020. a
Onishi, R., Sugimura, T., and Takahashi, K.: CIP-CSLR Scheme for Condensation
and Evaporation Calculations of Water Droplets, J. Environ. Eng., 5, 1–14,
https://doi.org/10.1299/jee.5.1, 2010. a
Ramkrishna, D.: Population Balances: Theory and Applications to Particulate
Systems in Engineering, Academic Press, ISBN 978-0-12-576970-9,
https://doi.org/10.1016/B978-0-12-576970-9.X5000-0, 2000. a
Rauber, R., Stevens, B., Ochs III, H., Knight, C., Albrecht, B., Blyth, A.,
Fairall, C., Jensen, J., Lasher-Trapp, S., Mayol-Bracero, O., Vali, G.,
Anderson, J., Baker, B., Bandy, A., Burnet, F., Brenguier, J.-L., Brewer, W.,
Brown, P., Chuang, P., Cotton, W., Di Girolamo, L., Geerts, H., Gerber, H.,
Göke, S., Gomes, L., Heikes, B., Hudson, J., Kollias, P., Lawson, R.,
Krueger, S., Lenschow, D., Nuijens, L., O'Sullivan, D., Rilling, R., Rogers,
D., Siebesma, A., Snodgrass, E., Stith, J., Thornton, D., Tucker, S., Twohy,
C., and Zuidema, P.: Rain in Shallow Cumulus Over the Ocean: The RICO
Campaign, B. Am. Meteorol. Soc., 88, 1912–1928, https://doi.org/10.1175/BAMS-88-12-1912,
2007. a
Roberts, K. V. and Weiss, N. O.: Convective Difference Schemes, Math. Comput.,
20, 272–299, https://doi.org/10.2307/2003507, 1966. a
Schneider, T. Teixeira, J., Bretherton, C. S., Brient, F., Pressel, K. G.,
Schär, C., and Siebesma, A. P.: Climate goals and computing the future of
clouds, Nat. Clim. Change, 7, 3–5, https://doi.org/10.1038/nclimate3190, 2017. a
Shipway, B. and Hill, A.: Diagnosis of systematic differences between multiple
parametrizations of warm rain microphysics using a kinematic framework, Q. J.
Roy. Meteor. Soc., 138, 2196–2211, https://doi.org/10.1002/qj.1913, 2012. a, b, c, d
Slawinska, J., Grabowski, W., Pawlowska, H., and Morrison, H.: Droplet
Activation and Mixing in Large-Eddy Simulation of a Shallow Cumulus Field, J.
Atmos. Sci., 69, 444–462, https://doi.org/10.1175/JAS-D-11-054.1, 2012. a
Smolarkiewicz, P. and Margolin, L.: MPDATA: A Finite-Difference Solver for
Geophysical Flows, J. Comp. Phys., 140, 459–480, https://doi.org/10.1006/jcph.1998.5901, 1998. a, b
Smolarkiewicz, P. K.: Multidimensional positive definite advection transport algorithm: An overview, Int. J. Numer. Methods Fluids, 50, 1123–1144,
https://doi.org/10.1002/fld.1071, 2006.
a, b
Smolarkiewicz, P. K. and Clark, T. L.: The Multidimensional Positive Definite Advection Transport Algorithm: Further Development and Applications, J. Comp.
Phys., 67, 396–438, https://doi.org/10.1016/0021-9991(86)90270-6, 1986. a, b, c, d
Smolarkiewicz, P. K. and Grabowski, W. W.: The Multidimensional Positive Definite Advection Transport Algorithm: Nonoscillatory Option, J. Comp.
Phys., 86, 355–375, https://doi.org/10.1016/0021-9991(90)90105-A, 1990. a, b, c, d
Smolarkiewicz, P. K. and Margolin, L. G.: On Forward-in-Time Differencing for
Fluids: Extension to a Curvilinear Framework, Mon. Weather Rev., 121, 1847–1859,
https://doi.org/10.1175/1520-0493(1993)121<1847:OFITDF>2.0.CO;2, 1993. a
Smolarkiewicz, P. K. and Margolin, L. G.: MPDATA – A Multipass Donor Cell Solver for Geophysical Flows, in: Godunov methods: Theory and applications,
edited by: Toro, E., Springer, https://doi.org/10.1007/978-1-4615-0663-8_81, 2001. a
Smolarkiewicz, P. K. and Rasch, P. J.: Monotone Advection on the Sphere: An
Eulerian Versus Semi-Lagrangian Approach, J. Atmos. Sci., 48, 793–810,
https://doi.org/10.1175/1520-0469(1991)048<0793:MAOTSA>2.0.CO;2, 1991. a, b, c
Smolarkiewicz, P. K. and Szmelter, J.: MPDATA: An edge-based unstructured-grid
formulation, J. Comp. Phys., 206, 624–649, https://doi.org/10.1016/j.jcp.2004.12.021, 2005. a
Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, 3 edn., https://doi.org/10.1007/b79761, 2009. a
Tsang, T. H. and Brock, J. R.: Simulation of Condensation Aerosol Growth by
Condensation and Evaporation, Aerosol Sci. Tech., 2, 311–320,
https://doi.org/10.1080/02786828308958637, 1982. a, b
Tsang, T. H. and Korgaonkar, N.: Effect of Evaporation on the Extinction
Coefficient of an Aerosol Cloud, Aerosol Sci. Tech., 7, 317–328,
https://doi.org/10.1080/02786828708959167, 1987. a
Tsang, T. H. and Rao, A.: Comparison of Different Numerical Schemes for
Condensational Growth of Aerosols, Aerosol Sci. Tech., 9, 271–277,
https://doi.org/10.1080/02786828808959214, 1988. a, b, c
Tsang, T. H. and Rao, A.: A moving finite element method for the population balance equation, Num. Meth. Fluids, 10, 753–769, https://doi.org/10.1002/fld.1650100704, 1990. a
Waruszewski, M., Kühnlein, C., Pawlowska, H., and Smolarkiewicz, P. K.:
MPDATA: Third-order accuracy for variable flows, J. Comput. Phys., 359, 361–379,
https://doi.org/10.1016/j.jcp.2018.01.005, 2018. a, b, c
Wei, L., Sun, J., Lei, H., Dong, L., and Hu, W.: A Lagrangian Advection Scheme
for Solving Cloud Droplet Diffusion Growth, Atmosphere, 11, 632,
https://doi.org/10.3390/atmos11060632, 2020. a, b
Williams, M. M. R. and Loyalka, S. K.: Aerosol Science: Theory and Practice: With
Special Applications to the Nuclear Industry, Oxford, Pergamon Press, 1991. a
Yang, F., Kollias, P., Shaw, R. A., and Vogelmann, A. M.: Cloud droplet size distribution broadening during diffusional growth: ripening amplified by deactivation and reactivation, Atmos. Chem. Phys., 18, 7313–7328, https://doi.org/10.5194/acp-18-7313-2018, 2018. a
Short summary
In systems such as atmospheric clouds, droplets undergo growth through condensation of vapor. The broadness of the resultant size spectrum of droplets influences precipitation likelihood and the radiative properties of clouds. One of the inherent limitations of simulations of the problem is the so-called numerical diffusion causing overestimation of the spectrum width, hence the term numerical broadening. In the paper, we take a closer look at one of the algorithms used in this context: MPDATA.
In systems such as atmospheric clouds, droplets undergo growth through condensation of vapor....
