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
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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.
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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.
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various momentum dissipation terms (
friction) as well as diabatic heating were selectively turned off over the tropics for the range of the latitudes from 20° S to 20° N.
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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...
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