Chaotic behaviour of the short-term variations in ozone column observed in Arctic

Highlights

• The diurnal ozone column variations have been analysed as a component of a chaotic system.

• The dimension and Lyapunov exponents of the attractors have been evaluated.

• The recurrence plots revealed structures typical for chaotic attractors.

• The diurnal ozone variations could be forecasted knowing their past behaviour.

• Such results allow the conclusion about chaotic origin of the observed ozone variations.

Abstract

The diurnal variations observed in the ozone column at Ny-Ålesund, Svalbard during different periods of 2009, 2010 and 2011 have been examined to test the hypothesis that they could be a result of a chaotic process. It was found that each of the attractors, reconstructed by applying the time delay technique and corresponding to any of the three time series can be embedded by 6-dimensional space. Recurrence plots, depicted to characterise the attractor features revealed structures typical for a chaotic system. In addition, the two positive Lyapunov exponents found for the three attractors, the fractal Hausdorff dimension presented by the Kaplan–Yorke estimator and the feasibility to predict the short-term ozone column variations within 10–20 h, knowing the past behaviour make the assumption about their chaotic character more realistic. The similarities of the estimated parameters in all three cases allow us to hypothesise that the three time series under study likely present one-dimensional projections of the same chaotic system taken at different time intervals.

Introduction

Ozone is an atmospheric trace element, which is important for the biosphere and the climate due to its strong absorption in the ultraviolet (UV) band of the solar spectrum. The ozone columnar amount is strongly affected by dynamical processes and photochemical reactions that cause variations over different temporal scales [1]. The long-period spectral components of such variations have been carefully studied and their relationships with the atmospheric processes were well understood [2], [3]. However, the variations over the daily timescale were poorly examined and discussed. Such an occurrence could be explained by the lack of continuous time series, since the optical methods widely used to determine the total ozone column at mid- and low-latitudes, where the most of the stations are placed, usually provide data only for daylight. On the contrary, at higher latitudes, the polar day provides a good opportunity to have a long-lasting record of the ozone variations over 24 h a day. Large diurnal oscillations in the ozone column were observed in the spring 2009 at Ny-Ålesund, Norway by Petkov et al. [4] and compared with the corresponding variations in the UV irradiance measured at the ground. Such variations were likely caused by the dynamical processes characterising the Arctic atmosphere and the present study aims to examine such ozone oscillations under the hypothesis about their chaotic origin.

Aperiodic variations observed in the real world are usually attributed to numerous random factors affecting the variable under study. However, in some cases such a behaviour turns out to be described by a few parameters connected through nonlinear relationships, which determine a dynamical system called “chaotic”, while in the former case the system is named “stochastic”. If a chaotic system is defined by m parameters $P_j\text{,}(j=1\text{,}2\text{,}3\text{,}\dots \text{,}m)$, the values of $P_j(t_i)$ at each time $t_i$ form a vector $\mathbf{P}(t_i)=\left(P_1(t_i)\text{,}{P}_2(t_i)\text{,}\dots \text{,}{P}_m(t_i)\right)$ determining the state of the system at time $t_i$ in m-dimensional space. The sequence of vectors $\mathbf{P}(t_i)\text{,}(i=1\text{,}2\text{,}3\dots .)$ outline a trajectory in the state space that shapes an object named as attractor of the system. Thus, the temporal variations turn out to be represented by the trajectory in the state space and analysing the topological features of the attractor we can get information about complexity and dynamics of the system. Analysing the atmospheric phenomena, sometimes it is important to know whether the process under study is stochastic or chaotic (deterministic) that allows the construction of adequate models of the phenomena and, as a result to make realistic previsions [5].

Very often, a chaotic system is presented by a one-dimensional projection yielded from experiments or field measurements as a time series $P(t_i)\text{,}(i=1\text{,}2\text{,}3\text{,}\dots \text{,}N)$. A variety of methods aimed to detect chaos in such systems have been developed during the past few decades [6], [7], [8], [9], [10], [11], [12], [13] and applied to various problems of the atmospheric physics [14], [15], [16], [17], [18], [19], [20]. Some of these methods, used to analyse the diurnal ozone column variations observed at Ny-Ålesund, are shortly presented in the next section.