Variability features associated with ozone column and surface UV irradiance observed over Svalbard from 2008 to 2014

Abstract

The present report shortly summarises the conclusions about the ozone column and solar UV irradiance variability that were achieved through the analysis of the data obtained from the narrow-band filter radiometer UV-RAD, operating at Ny-Ålesund from 2008 to 2014. The polar summer gives the opportunity to register the solar irradiance 24 h per day during several months that provides comparatively long continuous time series as regards for the short- (diurnal) and medium-term (monthly) variations. To exclude the hypothesis about the artificial nature of the large amplitudes registered in the short-term ozone column variations, which can be due to the measurement or methodological errors, they were related to the corresponding variations in the solar UV radiation. In addition, these oscillations were studied using the methods developed for the analysis of non-linear dynamical systems that revealed a complex chaotic interaction between the ozone column and five other atmospheric factors. This approach leads to the conclusion that the short-term variations can be predicted for 10–20 h if a long history is available. The effect of a sporadic phenomenon, such as the ozone depletion event, occurred over Arctic in the spring of 2011, which can be considered an impulse perturbation of the medium-term ozone variations, on the mid-latitude ozone column and surface solar UV irradiance has been studied by analysing the data collected from six surface stations.

Appendix

1.1 Concepts of the chaos theory used in the present study

The chaos theory, or the theory of the non-linear dynamical systems that has significantly developed in the past decades together with the advancement in the computation techniques, examines the time evolution of systems determined by variables, which are connected to each other through non-linear relationships. To illustrate the main ideas of this approach, we will make use of the well-known Rössler system (Rössler 1976) defined by the following equations (map):

$$ \begin{aligned} \dot{X} = - Y - Z \hfill \\ \dot{Y} = X + 0.15 \cdot Y \hfill \\ \dot{Z} = 0.2 + Z \cdot (X - 10) \hfill \\ \end{aligned} $$

(2)

that give aperiodic solutions presented in Fig. 5a. Assuming the three solutions X(t), Y(t), and Z(t) as a basis that forms a phase space; the vectors $\text{\bf{P}}_{\varvec{i}} = \left( {X(t_{i} ),Y(t_{i} ),Z(t_{i} )} \right)$ determined by the values of the three functions at the moments t _i (i = 1, 2, 3…N) outline a specific object in the phase space named as the attractor of the system (see Fig. 5b). The Rössler attractor is a manifold embedded in three-dimensional space, and it is determined through Eq. 2 where the three variables are connected by constant coefficients that is typical for chaotic systems. In contrary, equations containing random terms define stochastic systems, whose attractors cannot be embedded in low-dimensional phase space. Despite that the chaotic (deterministic) system is defined by well-determined relationships, the further its behaviour cannot be predicted for a long time, because the initial state cannot be identified without errors, which expresses an important feature of such systems: the evolutionary patterns are very sensible to the initial conditions.

Fig. 5

The three solutions of the Rössler map (a) defined by Eq. 2, together with the corresponding attractor (b) and the recurrence plot constructed for it (c)

The temporal variations in X(t), Y(t), and Z(t) correspond to a trajectory in the attractor that traces out numerous orbits, as Fig. 5b clearly exhibits, so that the topological properties of the attractor represent the evolutionary behaviour of the system. A useful visualization of the attractor features can be achieved by constructing the recurrence matrix R _ij defined as (Marwan et al. 2007):

$$ R_{{ij}} = \left\{ {\begin{array}{*{20}c} {1,\;{\text{when}}\;\text{\bf{P}}_{\varvec{i}}\;{\text{is\;close\;to\;}}\text{\bf{P}}_{\varvec{j}}} \\ {0,\;{\text{otherwise}}} \\ \end{array} } \right. $$

(3)

that can be graphically presented by putting a dot for 1 and white space for 0 (Eckmann et al. 1987). Figure 5c shows the recurrence plot (RP) of the Rössler attractor exhibiting a specific structure composed by short lines parallel to the main diagonal, which is a typical structure of the chaotic attractors; while the stochastic system is characterised by uniformly filled RP.

The divergence of the initially close orbits is an important characteristic of the attractor, and it is quantified by the Lyapunov exponent λ that indicates how fast the orbits separate as Fig. 6 illustrates. Actually, each projection of the attractor, or each dimension of the phase space has its own Lyapunov exponent $\lambda_{k } \left( {k = 1, 2, \ldots ,m: \lambda_{1} > \lambda_{2} > \ldots > \lambda_{m} } \right)$, where m is the dimension of the space, and in case of chaotic system, few of λ _k are positive, one is zero, and others are negative. The sum of the positive exponents gives the Kolmogorov-Sinai entropy H _KS that determines how fast the system produce information and if we know the initial conditions of the system with an error ε₀, then this error will increase as $\upvarepsilon =\upvarepsilon_{0} 2^{{H_{\text{KS}} t}}$. Assuming that a doubling of the error ($\upvarepsilon = 2\upvarepsilon_{0}$) is a limit precision; this relation allows the estimation of the time T _p for which the behaviour of the system can be predicted: $T_{p} = H_{KS}^{ - 1}$. For instance, the Lyapunov exponents of the Rössler attractor were evaluated to be equal to 0.13, 0.00, and −14.10 bits/sec, respectively (Wolf et al. 1985), that gives T _p  ≈ 8 sec for X(t), Y (t), or Z(t), arbitrarily assuming the time in Fig. 5a, as being measured in seconds.

Fig. 6

Illustration of the meaning of the Lyapunov exponent defined as the separation rate of two initially close orbits

To study the chaotic systems in the real world, the inverse problem, or the capacity to reconstruct the attractor from one its projection presented as a time series, and hence, to have an idea about the system dynamics, appears to have a great importance. A powerful tool that helps to face this problem was provided by Takens (1981), who stated that the manifold of vectors x _i  = (x _i , x _i+τ , x _i+2τ ,…, x _i+(m−1)τ ), where τ is a time delay, created from an observable time series {x _i } _{i = 1, 2,…, N} , which is composed by the measurements at uniquely sampled times t _i , gives a realistic representation of the system attractor in m-dimensional embedding space. Variety of methods allowing the determination of the dimension m, Lyapunov exponents λ _k and Kolmogorov-Sinai entropy H _KS have been developed and widely used in the study of the real world systems (Eckmann and Ruelle 1985; Zeng et al. 1992; Abarbanel and Kennel 1993).