Statistical Derivation of the Evolution Equation of Liquid Water Path Fluctuations in Clouds

TL;DR

This paper derives a stochastic evolution equation for liquid water path fluctuations in clouds, revealing strong correlations across time scales and providing a detailed probabilistic model consistent with empirical data.

Contribution

It presents a first-principles derivation of the Fokker-Planck and Langevin equations for LWP fluctuations, highlighting the role of strong correlations rather than hierarchical energy cascades.

Findings

Fokker-Planck equation accurately models LWP distribution tails

Kramers-Moyal coefficients match empirical estimates

LWP increments follow a Langevin equation with strong correlations

Abstract

How to distinguish and quantify deterministic and random influences on the statistics of turbulence data in meteorology cases is discussed from first principles. Liquid water path (LWP) changes in clouds, as retrieved from radio signals, upon different delay times, can be regarded as a stochastic Markov process. A detrended fluctuation analysis method indicates the existence of long range time correlations. The Fokker-Planck equation which models very precisely the LWP $fluctuation$ empirical probability distributions, in particular, their non-Gaussian heavy tails is explicitly derived and written in terms of a drift and a diffusion coefficient. Furthermore, Kramers-Moyal coefficients, as estimated from the empirical data, are found to be in good agreement with their first principle derivation. Finally, the equivalent Langevin equation is written for the LWP increments themselves. Thus rather than the existence of hierarchical structures, like an energy cascade process, {\it strong correlations} on different $time$ $scales$, from small to large ones, are considered to be proven as intrinsic ingredients of such cloud evolutions.