23 Jan 2024
 | 23 Jan 2024
Status: this preprint is open for discussion and under review for Atmospheric Chemistry and Physics (ACP).

Technical Note: General Formulation For the Distribution Problem: Prognostic Assumed PDF Approach Based on The Maximum–Entropy Principle and The Liouville Equation

Jun-Ichi Yano, Vince Larson, and Vaughan T. J. Phillips

Abstract. A general formulation for the distribution problem is presented, which is applicable to frequency distributions of subgrid-scale variables, hydrometeor size distributions, as well as to probability distributions characterizing data uncertainties. The general formulation is presented based upon two well-known basic principles: the maximum-entropy principle and Liouville equation. The maximum-entropy principle defines the most likely general distribution form, if necessary constraints are specified. This paper proposes to specify these constraints as the output variables to be used in a host model. Once a general distribution form is defined, the problem of temporal evolution of the distribution reduces to that of predicting a small number of parameters characterizing it. This paper derives prognostic equations for these parameters from the Liouville equation. The developed formulation, which is applicable to a wide range of atmospheric modelling problems, is specifically applied to condensation growth of cloud droplets as a demonstration.

Jun-Ichi Yano, Vince Larson, and Vaughan T. J. Phillips

Status: open (until 05 Mar 2024)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on egusphere-2023-2278', Anonymous Referee #2, 08 Feb 2024 reply
    • CC1: 'Reply on RC1', Jun-Ichi Yano, 20 Feb 2024 reply
Jun-Ichi Yano, Vince Larson, and Vaughan T. J. Phillips
Jun-Ichi Yano, Vince Larson, and Vaughan T. J. Phillips


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Short summary
The distribution problems appear in atmospheric sciences at almost every corner for describing diverse processes. This manuscript presents a general formulation for addressing all these problem.