Exploring the limits of ensemble forecasting via solutions of the Liouville equation for realistic geophysical models

Alejandro Hermoso,Victor Homar,Jun-Ichi Yano

doi:10.1016/j.atmosres.2020.105127

Abstract

The atmosphere is an exemplary case of uncertain system. The state of such systems is described by means of probability density functions which encompass uncertainty information. In this regard, the Liouville equation is the theoretical framework to predict the evolution of the state of uncertain systems. This study analyses the morphological characteristics of the time evolution of probability density functions for some low complexity geophysical systems by solving the Liouville equation in order to obtain tractable solutions which are otherwise unfeasible with currently available computational resources. The current and usual modest approach to overcome these obstacles and estimate the probability density function of the system in realistic weather and climate applications is the use of a discrete and small number of samples of the state of the system, evolved individually in a deterministic, perhaps sometimes stochastic, way. We investigate particular solutions of the shallow water equations and the barotropic model that allow to apply the Liouville formalism to explore its topological characteristics and interpret them in terms of the ensemble prediction system approach. We provide quantitative evidences of the high variability that solutions to Liouville equation may present, challenging currently accepted uses and interpretations of ensemble forecasts.

Highlights

Understanding the fundamentals of weather forecasting is one of the most challenging problems the scientific community currently faces, for its academic value and for its potential benefits on multiple socioeconomic assets
The probability density function (PDF) evolution for the shallow water solution illustrates the effect of divergence in phase space trajectories, which leads to a decrease in predictability
As one might intuitively expect, high probable states for the shallow water system remain compact for all times, so that a discrete sampling of the distribution function in phase space will render a satisfactorily precise representation of the PDF evolution

Summary

Introduction

Understanding the fundamentals of weather forecasting is one of the most challenging problems the scientific community currently faces, for its academic value and for its potential benefits on multiple socioeconomic assets. Multiple error sources degrade the forecasting process. Current numerical weather prediction models have essentially two sources of errors: the description of the initial state of the atmosphere, and the modeling of the physical processes and their interaction with external components (oceans, orography and cryosphere among others). The lack of infinitely precise knowledge about the state of the atmosphere, partly due to an insufficient observational sampling compared to the enormous number of degrees of freedom of the system, induce errors in the description of the system. Despite the advances in variational data assimilation methods currently used to produce the best estimate of the state of the system (Carrassi et al, 2018), initial conditions for numerical forecasts still contain relevant errors, which need to be characterized and adequately sampled

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