Articles | Volume 25, issue 16
https://doi.org/10.5194/acp-25-9357-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/acp-25-9357-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Technical note
| 
27 Aug 2025
Technical note |
| 27 Aug 2025
| 27 Aug 2025
Technical note: General formulation for the distribution problem – prognostic assumed probability density function (PDF) approach based on the maximum-entropy principle and the Liouville equation
Jun-Ichi Yano
CORRESPONDING AUTHOR
CNRM UMR3589, CNRS and Météo-France, Toulouse, France
Vincent E. Larson
Department of Mathematical Sciences, University of Wisconsin – Milwaukee, Milwaukee, WI, USA
Pacific Northwest National Laboratory, Richland, WA, USA
Vaughan T. J. Phillips
Department of Physical Geography and Ecosystem Science, Lund University, Lund, Sweden
Related authors
Vaughan T. J. Phillips, Sanne M. H. de Jong, Robert Garvey, Martin K. Skoglund, Torbjörn Ahlström, Fredrik Charpentier Ljungqvist, Mikael Pontarp, Joacim Rocklöv, Matthew Toohey, Jun-Ichi Yano, Helene Castenbrandt, Carl-Johan Fraenkel, Adam Linder, Henrik Sjödin, and Jesper Sjolte
EGUsphere, https://doi.org/10.5194/egusphere-2026-5195, https://doi.org/10.5194/egusphere-2026-5195, 2026
This preprint is open for discussion and under review for Climate of the Past (CP).
Short summary
Short summary
We investigate how volcanic eruptions may have amplified plague pandemics through climate-driven harvest failure and malnutrition. Using a model linking climate, food production, nutrition and disease, we find that the eruptions of 536 and 540 CE could have greatly increased mortality during the Justinianic Plague. The results suggest that food shortages and human movement were important in shaping the pandemic's severity and spread.
Jun-Ichi Yano, Marta Waclawczyk, and Maarten Ambaum
EGUsphere, https://doi.org/10.5194/egusphere-2026-3530, https://doi.org/10.5194/egusphere-2026-3530, 2026
Short summary
Short summary
Inside clouds, the water droplets grow by colliding together, and as a conseqeunce, two of them merge into a larger single droplet. This process can be described by an equation involving differentials and integrals. This paper studies the growth of droplet sizes by collisions and mergers by asking how we can make it identical by rescaling time, droplet size unit, as well as the total number density, like we recreate scenes in movies by minituares.
Jun-Ichi Yano
Nonlin. Processes Geophys., 31, 359–380, https://doi.org/10.5194/npg-31-359-2024, https://doi.org/10.5194/npg-31-359-2024, 2024
Short summary
Short summary
A methodology for directly predicting the time evolution of the assumed parameters for the distribution densities based on the Liouville equation, as proposed earlier, is extended to multidimensional cases and to cases in which the systems are constrained by integrals over a part of the variable range. The extended methodology is tested against a convective energy-cycle system as well as the Lorenz strange attractor.
Vaughan T. J. Phillips, Sanne M. H. de Jong, Robert Garvey, Martin K. Skoglund, Torbjörn Ahlström, Fredrik Charpentier Ljungqvist, Mikael Pontarp, Joacim Rocklöv, Matthew Toohey, Jun-Ichi Yano, Helene Castenbrandt, Carl-Johan Fraenkel, Adam Linder, Henrik Sjödin, and Jesper Sjolte
EGUsphere, https://doi.org/10.5194/egusphere-2026-5195, https://doi.org/10.5194/egusphere-2026-5195, 2026
This preprint is open for discussion and under review for Climate of the Past (CP).
Short summary
Short summary
We investigate how volcanic eruptions may have amplified plague pandemics through climate-driven harvest failure and malnutrition. Using a model linking climate, food production, nutrition and disease, we find that the eruptions of 536 and 540 CE could have greatly increased mortality during the Justinianic Plague. The results suggest that food shortages and human movement were important in shaping the pandemic's severity and spread.
Jun-Ichi Yano, Marta Waclawczyk, and Maarten Ambaum
EGUsphere, https://doi.org/10.5194/egusphere-2026-3530, https://doi.org/10.5194/egusphere-2026-3530, 2026
Short summary
Short summary
Inside clouds, the water droplets grow by colliding together, and as a conseqeunce, two of them merge into a larger single droplet. This process can be described by an equation involving differentials and integrals. This paper studies the growth of droplet sizes by collisions and mergers by asking how we can make it identical by rescaling time, droplet size unit, as well as the total number density, like we recreate scenes in movies by minituares.
Chandra Shekhar Pant, Deepak Waman, Sachin Patade, Akash Deshmukh, Niharika Singh, Vaughan Phillips, and Aaron Bansemer
Atmos. Chem. Phys., 26, 7407–7433, https://doi.org/10.5194/acp-26-7407-2026, https://doi.org/10.5194/acp-26-7407-2026, 2026
Short summary
Short summary
Large-scale stratiform clouds play a decisive role in the Earth's radiation budget and precipitation patterns, yet global models historically exhibit major biases in their simulations. Our study addresses these gaps by implementing physically-based representations of secondary ice production pathways and advanced aerosol activation schemes, including bin-bulk microphysics. These improvements enable the robust simulation of both cloud droplet and ice formation.
Freddy P. Paul, Martanda Gautam, Deepak Waman, Sachin Patade, Ushnanshu Dutta, Christoffer Pichler, Marcin Jackowicz-Korczynski, and Vaughan Phillips
EGUsphere, https://doi.org/10.5194/egusphere-2024-3800, https://doi.org/10.5194/egusphere-2024-3800, 2025
Preprint archived
Short summary
Short summary
This study shows observations of a key mechanism for initiation of ice particles in clouds with a chamber deployed on the top of a mountain during snowfall in winter. The mechanism involves the fragmentation of snow particles in collisions with denser rimed ice precipitation, namely "graupel" or "hail". The study shows how the fragmentation can be represented in atmospheric models. An improved formulation of the mechanism is proposed in light of our observations with the chamber.
Jun-Ichi Yano
Nonlin. Processes Geophys., 31, 359–380, https://doi.org/10.5194/npg-31-359-2024, https://doi.org/10.5194/npg-31-359-2024, 2024
Short summary
Short summary
A methodology for directly predicting the time evolution of the assumed parameters for the distribution densities based on the Liouville equation, as proposed earlier, is extended to multidimensional cases and to cases in which the systems are constrained by integrals over a part of the variable range. The extended methodology is tested against a convective energy-cycle system as well as the Lorenz strange attractor.
Sachin Patade, Deepak Waman, Akash Deshmukh, Ashok Kumar Gupta, Arti Jadav, Vaughan T. J. Phillips, Aaron Bansemer, Jacob Carlin, and Alexander Ryzhkov
Atmos. Chem. Phys., 22, 12055–12075, https://doi.org/10.5194/acp-22-12055-2022, https://doi.org/10.5194/acp-22-12055-2022, 2022
Short summary
Short summary
This modeling study focuses on the role of multiple groups of primary biological aerosol particles as ice nuclei on cloud properties and precipitation. This was done by implementing a more realistic scheme for biological ice nucleating particles in the aerosol–cloud model. Results show that biological ice nucleating particles have a limited role in altering the ice phase and precipitation in deep convective clouds.
Jonas K. F. Jakobsson, Deepak B. Waman, Vaughan T. J. Phillips, and Thomas Bjerring Kristensen
Atmos. Chem. Phys., 22, 6717–6748, https://doi.org/10.5194/acp-22-6717-2022, https://doi.org/10.5194/acp-22-6717-2022, 2022
Short summary
Short summary
Long-lived cold-layer clouds at subzero temperatures are observed to be remarkably persistent in their generation of ice particles and snow precipitation. There is uncertainty about why this is so. This motivates the present lab study to observe the long-term ice-nucleating ability of aerosol samples from the real troposphere. Time dependence of their ice nucleation is observed to be weak in lab experiments exposing the samples to isothermal conditions for up to about 10 h.
Po-Lun Ma, Bryce E. Harrop, Vincent E. Larson, Richard B. Neale, Andrew Gettelman, Hugh Morrison, Hailong Wang, Kai Zhang, Stephen A. Klein, Mark D. Zelinka, Yuying Zhang, Yun Qian, Jin-Ho Yoon, Christopher R. Jones, Meng Huang, Sheng-Lun Tai, Balwinder Singh, Peter A. Bogenschutz, Xue Zheng, Wuyin Lin, Johannes Quaas, Hélène Chepfer, Michael A. Brunke, Xubin Zeng, Johannes Mülmenstädt, Samson Hagos, Zhibo Zhang, Hua Song, Xiaohong Liu, Michael S. Pritchard, Hui Wan, Jingyu Wang, Qi Tang, Peter M. Caldwell, Jiwen Fan, Larry K. Berg, Jerome D. Fast, Mark A. Taylor, Jean-Christophe Golaz, Shaocheng Xie, Philip J. Rasch, and L. Ruby Leung
Geosci. Model Dev., 15, 2881–2916, https://doi.org/10.5194/gmd-15-2881-2022, https://doi.org/10.5194/gmd-15-2881-2022, 2022
Short summary
Short summary
An alternative set of parameters for E3SM Atmospheric Model version 1 has been developed based on a tuning strategy that focuses on clouds. When clouds in every regime are improved, other aspects of the model are also improved, even though they are not the direct targets for calibration. The recalibrated model shows a lower sensitivity to anthropogenic aerosols and surface warming, suggesting potential improvements to the simulated climate in the past and future.
Tommi Bergman, Risto Makkonen, Roland Schrödner, Erik Swietlicki, Vaughan T. J. Phillips, Philippe Le Sager, and Twan van Noije
Geosci. Model Dev., 15, 683–713, https://doi.org/10.5194/gmd-15-683-2022, https://doi.org/10.5194/gmd-15-683-2022, 2022
Short summary
Short summary
We describe in this paper the implementation of a process-based secondary organic aerosol and new particle formation scheme within the chemistry transport model TM5-MP version 1.2. The performance of the model simulations for the year 2010 is evaluated against in situ observations, ground-based remote sensing and satellite retrievals. Overall, the simulated aerosol fields are improved, although in some areas the model shows a decline in performance.
Rachel L. James, Vaughan T. J. Phillips, and Paul J. Connolly
Atmos. Chem. Phys., 21, 18519–18530, https://doi.org/10.5194/acp-21-18519-2021, https://doi.org/10.5194/acp-21-18519-2021, 2021
Short summary
Short summary
Secondary ice production (SIP) plays an important role in ice formation within mixed-phase clouds. We present a laboratory investigation of a potentially new SIP mechanism involving the collisions of supercooled water drops with ice particles. At impact, the supercooled water drop fragments form smaller secondary drops. Approximately 30 % of the secondary drops formed during the retraction phase of the supercooled water drop impact freeze over a temperature range of −4 °C to −12 °C.
Cited articles
Bannister, R. N.: A review of operational methods of variational and ensemble-variational data assimilation, Q. J. Roy. Meteor. Soc., 143, 607–633, https://doi.org/10.1002/qj.2982, 2017. a
Bechtold, P., Fravalo, C., and Pinty, J. P.: A model of marine boundary–layer cloudness for mesoscale applications, J. Atmos. Sci., 49, 1723–1744, https://doi.org/10.1175/1520-0469(1992)049<1723:AMOMBL>2.0.CO;2, 1992. a
Bechtold, P., Cuijpers, J. W. M., Mascart, P., and Trouilhet, P.: Modeling of trade–wind cumuli with a low–order turbulence model – Towards a unified description of Cu and Sc clouds in meteorological models, J. Atmos. Sci., 52, 455–463, https://doi.org/10.1175/1520-0469(1995)052<0455:MOTWCW>2.0.CO;2, 1995. a
Bender, C. M. and Orszag, S. A.: Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 593 pp., ISBN 978-1-4757-3069-2, 1978. a
Bernardo, J. M. and Smith, A. F. M.: Bayesian Tehory, John Wiley & Son, Chicheter, 586 pp., ISBN 0 471 92416 4, 1997. a
Berner, J., Achatz, U., Batte, L., Bengtsson, L., de la Cámara, A., Crommelin, D., Christensen, H., Colangeli, M., Dolaptchiev, S., Franzke, C. L. E., Friederichs, P., Imkeller, P., Järvinen, H., Juricke, S., Kitsios, V., Lott, F., Lucarini, V., Mahajan, S., Palmer, T. N., Penland, C., von Storch, J.-S., Sakradžija, M., Weniger, M., Weisheimer, A., Williams, P. D., and Yano, J.-I.: Stochastic Parameterization: Towards a new view of weather and climate models, B. Am. Meteorol. Soc., 98, 565–588, https://doi.org/10.1175/BAMS-D-15-00268.1, 2017. a
Bishop, C. H.: The GIGG-EnKF: ensemble Kalman filtering for highly skewed non-negative uncertainty distributions, Q. J. Roy. Meteor. Soc., 142, 1395–1412, https://doi.org/10.1002/qj.2742, 2016. a
Bony, S. and Emanuel, K. A.: A parameterization of the cloudness associated with cumulus convection: Evaluation using TOGA COARE data, J. Atmos. Sci., 58, 3158–3183, https://doi.org/10.1175/1520-0469(2001)058<3158:APOTCA>2.0.CO;2, 2001. a
Bougeault, P.: Modeling the trade–wind cumulus boundary–layer. Part I: Testing the ensemble cloud relations against numerical data, J. Atmos. Sci., 38, 2414–2428, https://doi.org/10.1175/1520-0469(1981)038<2414:MTTWCB>2.0.CO;2, 1981. a
Butler, R. W.: Saddlepoint Approximations with Applications, Cambridge University Press, https://doi.org/10.1017/CBO9780511619083, 2007. a
Carrassi, A., Bocquet, M., Bertino, L., and Evensen, G.: Data assimilation in the geosciences: An overview of methods, issues, and perspectives, Climatic Change, 9, e535, https://doi.org/10.1002/wcc.535, 2018. a, b
Dang, C. and Xu, J.: Novel algorithm for reconstruction of a distribution by fitting its first-four statistical moments, App. Math. Model., 71, 505–524, https://doi.org/10.1016/j.apm.2019.02.040, 2019. a
Daniels, H. E.: Saddlepoint Approximations in Statistics, Ann. Math. Stat., 25, 631–650, https://doi.org/10.1214/aoms/1177728652, 1954. a
Ehrendorfer, M.: The Liouville equation and its potential usefulness for the prediction of forecast skill. Part I: Theory, J. Atmos. Sci., 122, 703–713, https://doi.org/10.1175/1520-0493(1994)122<0703:TLEAIP>2.0.CO;2, 1994a. a
Ehrendorfer, M.: The Liouville equation and its potential usefulness for the prediction of forecast skill. Part II: Applications, J. Atmos. Sci., 122, 714–728, https://doi.org/10.1175/1520-0493(1994)122<0714:TLEAIP>2.0.CO;2, 1994b. a
Ehrendorfer, M.: The Liouville equation and atmospheric predictability, in: Predictability of Weather and Climate, edited by: Palmer, T. and Hagedorn, R., Cambridge University Press, Cambridge, 59–98, https://doi.org/10.1017/CBO9780511617652.005, 2006. a
Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn., John Wiley and Sons, U. K., 509 pp., ISBN-10 0471257087, 1968. a
Fitch, A. C.: An improved double-Gaussian closure for the subgrid vertical velocity probability distribution function, J. Atmos. Sci., 76, 285–304, https://doi.org/10.1175/JAS-D-18-0149.1, 2019. a, b, c
Garratt, J. R.: The Atmospheric Boundary Layer, Cambridge University Press, U. K., 316 pp., ISBN 0 521 38052 9, 1992. a
Garret, T.: Analytical solutions for precipitation size distributions at steady state, J. Atmos. Sci., 76, 1031–1037, https://doi.org/10.1175/JAS-D-18-0309.1, 2019. a
Gentle, J. E.: Random Number Generation and Monte Carlo Methods, 2nd edn., Springer, Berlin, ISBN-10 9780387001784, 2003. a
Goldstein, H., Poole, C., and Safko, J.: Classical Mechanics, 3rd edn., Addison Wesley, San Francisco, 638 pp., ISBN-10 9780201657029, 2002. a
Goodfellow, I., Bengio, Y., and Courville, A.: Deep Learning, MIT Press, Cambridge, MA, 785 pp., ISBN 9780262035613, 2016. a
Gregory, P.: Bayesian Logical Data Analysis for the Physical Sciences, Cambridge University Press, Cambridge, 468 pp., ISBN 0 521 84150 X, 2005. a
Guiasu, S.: Information Theory with Applications, McGraw Hill, New York, 439 pp., ISBN 10: 0070251096, 1977. a
Hermoso, A., Hommar, V., and Yano, J.-I.: Exploring the limits of ensemble forecasting via solutions of the Liouville equation for realistic geophysical models, Atmos. Res., 246, 105127, https://doi.org/10.1016/j.atmosres.2020.105127, 2020. a
Jaynes, E. T.: Where do we standn on maximum entropy?, in: The Maximum Entropy Formulation, edited by: Levine, R. D. and Tribus, M., MIT Press, Cambridge, MA, 15–118, http://philsci-archive.pitt.edu/22626/1/MaximalEntropy.pdf (last access: 25 August 2025), 1978. a
Jaynes, E. T.: Probability Theory, The Logic of Science, Cambridge University Press, Cambridge, UK, 725 pp., ISBN-10 0521592712, 2003. a
Jazwinski, A. H.: Stochastic Processes and Filtering Theory, Academic Press, New York, 376 pp., ISBN: 9780080960906, 1970. a
Kapur, J. N.: Maximum–Entropy Models in Science and Engineering, John Wily & Sons, U. K., 643 pp., ISBN 9780470214596, 1989. 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 vs. bulk–microphysics, Rev. Geophys., 53, 247–322, https://doi.org/10.1002/2014RG000468, 2015. a, b, c, d, e
Klimenko, A. Y. and Bilger, R. W.: Conditional moment closure for turbulent combustion, Prog. Energ. Combust., 25, 596–687, https://doi.org/10.1016/S0360-1285(99)00006-4, 1999. a
Larson, V. E.: Prognostic equations for cloud fraction and liquid water, and their relation to filtered density functions, J. Atmos. Sci., 61, 338–351, https://doi.org/10.1175/1520-0469(2004)061<0338:PEFCFA>2.0.CO;2, 2004. a
Larson, V. E.: CLUBB-SILHS: A parameterization of subgrid variability in the atmosphere, arXiv [preprint], https://doi.org/10.48550/arXiv.1711.03675, 26 March 2022. a, b, c
Larson, V. E. and Golaz, J.: Using probability density functions to derive consistent closure relationships among higher-order Moments, Mon. Weather Rev., 133, 1023–1042, https://doi.org/10.1175/MWR2902.1, 2005. a
Larson, V. E. and Schanen, D. P.: The Subgrid Importance Latin Hypercube Sampler (SILHS): a multivariate subcolumn generator, Geosci. Model Dev., 6, 1813–1829, https://doi.org/10.5194/gmd-6-1813-2013, 2013. a
Larson, V. E., Golaz, J. C., and Cotton, W. R.: Small–scale and mesoscale variability in cloudy boundary layers: Joint probability density functions, J. Atmos. Sci., 59, 3519–3539, https://doi.org/10.1175/1520-0469(2002)059<3519:SSAMVI>2.0.CO;2, 2002. a, b
Larson, V. E., Domke, S., and Griffin, B. M.: Momentum transport in shallow cumulus clouds and its parameterization by higher-order closure, J. Adv. Model. Earth Sy., 11, 3419–3442, https://doi.org/10.1029/2019MS001743, 2019. a
Le Treut, H. and Li, Z. X.: Sensitivity of an atmospheric general circulation model to prescribed SST changes: Feedback effects associated with the simulation of cloud optical properties, Clim. Dynam., 5, 175–187, https://doi.org/10.1007/BF00251808, 1991. a
Lewellen, W. S. and Yoh, S.: Binormal model of ensemble partial cloudiness, J. Atmos. Sci., 50, 1228–1237, https://doi.org/10.1175/1520-0469(1993)050<1228:BMOEPC>2.0.CO;2, 1993. a, b
Lockwood, F. C. and Naguib, A. S.: The predictin of th efluctuations in the properties of free, round–jet, turbulent, diffusion flames, Combust. Flame, 24, 109–124, https://doi.org/10.1016/0010-2180(75)90133-9, 1975. a
Machulskaya, E.: Clouds and convection as subgrid–scale distributions, in: Parameterization of Atmospheric Convection, Volume II, edited by: Plant, R. S. and Yano, J.-I., World Scientific, Imperial College Press, U. K., 377–422, https://doi.org/10.1142/p1005, 2015. a, b, c, d
Marshall, J. S. and Palmer, W. M. K.: The distribution of raindrops with size, J. Atmos. Sci., 5, 165–166, 1948. a
Mellor, G. L.: Analytic prediction of the properties of stratified surface layers, J. Atmos. Sci., 30, 1061–1069, https://doi.org/10.1175/1520-0469(1973)030<1061:APOTPO>2.0.CO;2, 1973. a
Mellor, G. L.: Gaussian cloud model relations, J. Atmos. Sci., 34, 356–358, https://doi.org/10.1175/1520-0469(1977)034<0356:TGCMR>2.0.CO;2, 1977. a
Mellor, G. L. and Yamada, T.: A hierarchy of turbulence closure models for planetary boundary layers, J. Atmos. Sci., 31, 1791–1806, https://doi.org/10.1175/1520-0469(1974)031<1791:AHOTCM>2.0.CO;2, 1974. a
Milbrandt, J. A. and Yau, M. K.: A multimoment bulk microphysics parameterization. Part I: Analysis of the role of the spectrum shape parameter, J. Atmos. Sci., 62, 3051-3064, https://doi.org/10.1175/JAS3534.1, 2005. a, b, c
Naumann, A. K., Seifert, A., and Mellado, J. P.: A refined statistical cloud closure using double-Gaussian probability density functions, Geosci. Model Dev., 6, 1641–1657, https://doi.org/10.5194/gmd-6-1641-2013, 2013. a, b, c
Pope, S. B.: The statistical theory of turbulent flames, Philos. T. R. Soc. S.-A, 291, 529–568, 1979. a
Pope, S. B.: PDF methods for turbulent reactive flows, Prog. Energ. Combust., 11, 119–192, https://doi.org/10.1016/0360-1285(85)90002-4, 1985. a
Richard, J. L. and Royer, J. F.: A statistical cloud scheme for ruse in an AGCM, Ann. Geophys., 11, 1095–1115, 1993. a
Robert, R. and Sommeria, J.: Statistical equilibrium states for two-dimensional flows, J. Fluid Mech., 229, 291–310, https://doi.org/10.1017/S0022112091003038, 1991. a
Rogers, R. R. and Yau, M. K.: Short Course in Cloud Physics, 3rd edn., Pergamon Press, Oxford, 290 pp., ISBN: 9780750632157, 1989. a
Seifert, A. and Beheng, K. D.: A double-moment parameterization for simulating autoconversion, accretion and selfcollection, Atmos. Res., 59–60, 265–281, https://doi.org/10.1016/S0169-8095(01)00126-0, 2001. a
Seifert, A. and Beheng, K. D.: A two-moment cloud microphysics parameterization for mixed-phase clouds. Part 1: Model description, Meteorol. Atmos. Phys., 92, 45–66, https://doi.org/10.1007/s00703-005-0112-4, 2006. a
Shannon, C. E.: A mathematical theory of communication, Bell Syst. Tech. J., 27, 379–423, 623–656, 1948. a
Sommeria, G. and Deadorff, J. W.: Subgrid–scale condensation in models of nonprecipitating clouds, J. Atmos. Sci., 34, 344–355, https://doi.org/10.1175/1520-0469(1977)034<0344:SSCIMO>2.0.CO;2, 1977. a
Stull, R. B.: An Introduction to Boundary Layer Meteorology, Kluwer Academic Press, Dordrecht, 666 pp., ISBN 978-90-277-2768-8, 1988. a
Tompkins, A. M.: A prognostic parameterization for the subgrid–scale variability of water vapor and clouds in large–scale models and its use to diagnose cloud cover, J. Atmos. Sci., 59, 1917–1942, https://doi.org/10.1175/1520-0469(2002)059<1917:APPFTS>2.0.CO;2, 2002. a
Touchette, H.: The large deviation approach to statistical mechanics, Phys. Rep., 478, 1–69, https://doi.org/10.1016/j.physrep.2009.05.002, 2009. a
Verkley, W.: A maximum entropy approach to the problem of parameterization, Q. J. Roy. Meteor. Soc., 137, 1872–1886, https://doi.org/10.1002/qj.860, 2011. a
Verkley, W. and Lynch, P.: Energy and enstropy spectra of geostrophic turbulent flows derived from a maximum entropy principle, J. Atmos. Sci., 66, 2216–2236, https://doi.org/10.1175/2009JAS2889.1, 2009. a
Verkley, W., Kalverla, P., and Severijns, C.: A maximum entropy approach to the parameterization of subgrid–scale processes in two–dimensional flow, Q. J. Roy. Meteor. Soc., 142, 2273–2283, https://doi.org/10.1002/qj.2817, 2016. a
Wikle, C. K. and Berliner, L. M.: A Bayesian tutorial for data assimilation, Physica D, 230, 1–16, 2007. a
Wonnacott, T. H. and Wonnacott, R. J.: Introductory Statistics, John Wiley & Sons, New York, 402 pp., ISBN-13: 9780471615187, 1969. a
Yanai, M., Esbensen, S., and Chu, J.-H.: Determination of bulk properties of tropical cloud clusters from large-scale heat and moisture budgets, J. Atmos. Sci., 30, 611–627, 1973.
Yano, J.-I.: Fromulation structure of mass-flux convection parameterization, Dynam. Atmos. Oceans, 67, 1–28, https://doi.org/10.1016/j.dynatmoce.2014.04.002, 2014. a, b
Yano, J.-I.: Scale separation, in: Parameterization of Atmospheric Convection, Vol. I, edited by: Plant, R. S. and Yano, J.-I., World Scientific, Imperial College Press, U. K., 73–99, https://doi.org/10.1142/p1005, 2015a. a
Yano, J.-I.: Subgrid–scale parameterization problem, in: Parameterization of Atmospheric Convection, Vol. I, edited by: Plant, R. S. and Yano, J.-I., World Scientific, Imperial College Press, U. K., https://doi.org/10.1142/p1005, 2015b. a
Yano, J.-I.: What is the Maximum Entropy Principle?: Comments on “Statistical Theory on the Functional Form of Cloud Particle Size Distributions”, J. Atmos. Sci., 76, 3955–3960, https://doi.org/10.1175/JAS-D-18-0223.1, 2019. a
Yano, J.-I. and Manzato, A.: Does more moisture in the atmosphere lead to more intense rains?, J. Atmos. Sci., 79, 663–681, https://doi.org/10.1175/JAS-D-21-0117.1, 2022. a
Yano, J.-I. and Ouchtar, E.: Convective Initiation Uncertainties Without Trigger or Stochasticity: Probabilistic Description by the Liouville Equation and Bayes' Theorem, Q. J. Roy. Meteor. Soc., 143, 2015–2035, https://doi.org/10.1002/qj.3064, 2017. a, b
Yano, J.-I., Redelsperger, J.-L., Guichard, F., and Bechtold, P.: Mode decomposition as a methodology for developing convective-scale representations in global models, Q. J. Roy. Meteor. Soc., 131, 2313–2336, https://doi.org/10.1256/qj.04.44, 2005. a
Yano, J.-I., Geleyn, J.-F., Köhler, M., Mironov, D., Quaas, J., Soares, P. M. M., Phillips, V. T. J., Plant, R. S., Deluca, A., Marquet, P., Stulic, L., and Fuchs, Z.: Basic Concepts for Convection Parameterization in Weather Forecast and Climate Models: COST Action ES0905 Final Report, Atmosphere-Basel, 6, 88–147, 2014. a, b
Yano, J.-I., Heymsfield, A. J., and Phillips, V. T. J.: Size distributions of hydrometeors: Analysis with the maximum entropy principle, J. Atmos. Sci., 73, 95–108, https://doi.org/10.1175/JAS-D-15-0097.1, 2016. a, b, c
Yano, J.-I., Ziemiański, M., Cullen, M., Termonia, P., Onvlee, J., Bengtsson, L., Carrassi, A., Davy, R., Deluca, A., Gray, S. L., Homar, V., Köhler, M., Krichak, S., Michaelides, S., Phillips, V. T. J., Soares, P. M. M., and Wyszogrodzki, A.: Scientific Challenges of Convective–Scale Numerical Weather Prediction, B. Am. Meteorol. Soc., 99, 699–710, https://doi.org/10.1175/bams-d-17-0125.1, 2018. a, b
Short summary
The distribution problems appear in atmospheric sciences at almost every corner when describing diverse processes. This paper presents a general formulation for addressing all these problems.
The distribution problems appear in atmospheric sciences at almost every corner when describing...
Altmetrics
Final-revised paper
Preprint