Shadowing the rotating annulus. Part I: Measuring candidate trajectory shadowing times

TL;DR

This paper develops a method to measure how long high-dimensional model trajectories can stay close to observed data, using a rotating annulus model, to evaluate forecast reliability in complex systems.

Contribution

It introduces a novel technique for quantifying shadowing times in high-dimensional models, bridging the gap between low-dimensional theory and complex climate models.

Findings

Shadowing times match divergence observations.

Method's results align with simple distance metrics.

Empirical relationship confirms method validity.

Abstract

An intuitively necessary requirement of models used to provide forecasts of a system's future is the existence of shadowing trajectories that are consistent with past observations of the system: given a system-model pair, do model trajectories exist that stay reasonably close to a sequence of observations of the system? Techniques for finding such trajectories are well-understood in low-dimensional systems, but there is significant interest in their application to high-dimensional weather and climate models. We build on work by Smith et al. [2010, Phys. Lett. A, 374, 2618-2623] and develop a method for measuring the time that individual "candidate" trajectories of high-dimensional models shadow observations, using a model of the thermally-driven rotating annulus in the perfect model scenario. Models of the annulus are intermediate in complexity between low-dimensional systems and global atmospheric models. We demonstrate our method by measuring shadowing times against artificially-generated observations for candidate trajectories beginning a fixed distance from truth in one of the annulus' chaotic flow regimes. The distribution of candidate shadowing times we calculated using our method corresponds closely to (1) the range of times over which the trajectories visually diverge from the observations and (2) the divergence time using a simple metric based on the distance between model trajectory and observations. An empirical relationship between the expected candidate shadowing times and the initial distance from truth confirms that the method behaves reasonably as parameters are varied.