Internal variability and pattern identification in cirrus cloud structure: The Fokker‐Planck equation approach

Abstract

[1] Using 35-GHz millimeter wave radar observations collected at the Southern Great Plains site of the Atmospheric Radiation Measurements (ARM) program, we study physical processes in cirrus cloud layers having different stratifications. The timedependent probability distribution functions of the backscattering cross section within different layers in the cloud describe the dynamics of the pertinent physical processes. The time-dependent tails of the probability distribution functions provide the signature of intermittency characterizing turbulent flows. In the framework of the Fokker-Planck equation approach, we derive the behavior of the drift D (1) and diffusion D (2) terms that characterize the deterministic and stochastic parts, respectively, of the dynamics of the probability distribution functions directly from the time series of observed backscattering cross section. We derive the Langevin equation that governs the temporal evolution of the backscattering cross-section signal in cloud layers having different stratification. Finally, we ascertain that the drift g and diffusion b coefficients of the backscattering cross-section signal in the upper well-mixed, neutrally stratified layer of the cirrus cloud satisfy the relationship g = 2b, which is valid for fully developed turbulence owing to the Kolmogorov À4/5 law.

Stochastic analysis of neutrally stratified cirrus layer

“…We define the time series of observations measured at the maximum height and at next height below the maximum at which observation is available at time t within each sublayer. In doing this we follow the procedure explained in detail in Ivanova et al (2006) for defining the time series at the maximum height. We briefly recall the rules of this procedure here.…”

“…In order to facilitate the interpretation of the results of any analysis of observational data from cirrus clouds, it is advantageous to design the study in such a way that one of these three properties is kept approximately constant. Toward that end we consider sublayers within the neutrally stratified layers of the cirrus clouds having their radiative properties within certain limited range of radar reflectivity values (Ivanova et al, 2006). We analyze the dynamical properties of the backscattering crosssection signal at two heights within each of the sublayers.…”

“…To study such signals we employ the Fokker-Planck equation approach which describes the evolution of the probability density function (pdf) of a signal generated by a Markov process Ivanova and Ausloos, 2002;Ivanova et al, 2006). The Fokker-Planck equation approach allows to distinguish and quantify directly from the observational data the deterministic and stochastic influences on the backscattering cross-section signal in cirrus clouds.…”

Tracking nucleation, growth, and sublimation in cirrus clouds using ARM millimeter‐wavelength radar observations

“…Because we are interested in investigating the internal structure of cirrus we consider time series of the radar reflectivity η ( t ) at various depths d = 0, 1/4, 1/3, 1/2, 2/3, 3/4, and 1 into the cloud relative to the cloud top as shown in Figures 3a and 3b for cirrus parts A and B , respectively. The approach of assessing microphysical and radiative properties of cirrus at such normalized depths into cloud has been applied previously by us [ Mace et al , 1998; Ivanova et al , 2003, 2006; Comstock et al , 2004] and by other authors [ van Zadelhoff et al , 2007].…”

“…This paper represents a case study that serves to introduce a stochastic approach to quantify the coupling between the structure of the cirrus cloud and the state of the large‐scale atmosphere. The purpose of this paper is threefold: first, to further improve the stochastic approach suggested earlier [ Ivanova et al , 2006; Ivanova and Ackerman , 2007] for analysis of radar reflectivity observations from cirrus that provides a framework suitable to tackle the two scales of processes which are characteristic of cirrus formation and maintenance; second, to relate the structure of radiative with the structure of dynamical properties of cirrus and by doing that to improve our understanding of the coupling between radiative and dynamical properties based on remote‐sensing observations; and third, to place the results into the context of the state of the large‐scale atmosphere and to improve our understanding of the structure of dynamical properties at different temporal/spatial scales from 1–2 km, to intermediate scales of 6–12 km to larger scales of 100 km.…”