## Abstract

Fluctuations breaking time-reversal symmetry are common attributes of dissipative systems operating far from equilibrium. Recent developments in non-equilibrium statistical physics represent a significant step towards an understanding of how time-reversible microscopic laws can yield to inherent irreversibility on meso- or macroscopic scales. Most of the theoretical conclusions consider quantities (e.g. entropy production) that are difficult to obtain with an appropriate accuracy in real systems. Probably less-complicated measures, such as the simple step-number ratio used in this work, can also help to characterize time-asymmetric fluctuations. In the first part, we give a short summary of recent results on asymmetric daily mean temperature changes. The second part discusses total-column ozone fluctuations, where statistically significant asymmetries are also detected. A detailed correlation analysis of ozone signals and high-altitude temperature records supports the strong coupling between tropospheric dynamics and stratospheric processes on synoptic time scales.

## 1. Introduction

The asymmetry of nature under a ‘reversal of time’ appears too obvious, as it strongly affects our own form of existence. If physics is to justify the hypothesis that its laws control everything that happens in nature, it should be able to explain (or consistently describe) this fundamental asymmetry that defines what may be called a ‘direction of time’ (Zeh 2007). It is well known that the very laws of nature are in pronounced contrast to this fundamental asymmetry, since they are essentially symmetric under time reversal. This discrepancy defines the ‘reversibility paradox’ (Loschmidt 1876): how to deduce an irreversible process from time-symmetric dynamics? Both macroscopic irreversibility and microscopic time-reversal symmetry are well-accepted principles in physics, with solid observational and theoretical support, yet they seem to be in conflict; hence the paradox.

Boltzmann (1877) provided a classic explanation that the direction of time is due to the second law of thermodynamics and the growth of entropy. Entropy-decreasing processes can occur (without doing any work); they are just extremely unlikely. The evolution of a microstate in phase space is such that the entropy will most probably increase or stay constant. An apparent problem with this explanation is that it works in both directions: entropy is likely to increase towards the future and the past—the latter contrary to thermodynamics (Carroll 2010).

Over the past decade, new connections have been established that relate fluctuation with dissipation in far from equilibrium situations (Derrida *et al.* 2007; Marconi *et al.* 2008). These results shed a new light on irreversibility and on the origin of the thermodynamic time asymmetry, and open new perspectives on the implications of the second law of thermodynamics. Recent experiments in driven dissipative systems demonstrated explicit breaking of time-reversal symmetry by the non-equilibrium conditions (Andrieux *et al.* 2007, 2008). Temporal asymmetries of fluctuations in non-equilibrium systems seem to be a common attribute in very general circumstances (Marconi *et al.* 2008; Paneni *et al.* 2008), therefore it is a subject of intense interest.

Here, we present a short summary of recent empirical analyses of time-asymmetric fluctuations considering atmospheric parameters. First, we focus on daily mean temperature records because they represent the longest instrumental time series of widest spatial coverage among the fundamental parameters measured routinely. We discuss the most probable dynamical origin of the observed asymmetries based on laboratory experiments and comparison with properties of other atmospheric variables. In the second part, we show new results on the fluctuations of daily total-ozone (O_{3}) concentrations. Deviations from time reversibility are clearly present over the extratropical regions in both hemispheres, however, a possible explanation seems to be far more complicated than for the temperature fluctuations. Correlation analyses for ozone and high-altitude temperature records reveal a strong connection between tropospheric dynamics and total-ozone fluctuations.

## 2. Asymmetric fluctuations of daily temperature anomalies

Temperature-anomaly time series *T*_{a}(*t*) are computed as the difference between the actual daily mean temperature *T*(*t*) and the long-time average temperature of that particular calendar day *d* (also known as the climatological average): . Note that this procedure cannot remove slow trends from the signal, such as a background climate change, urbanization or the change of land use around the thermometer site. It is not entirely obvious that the climatological average temperature at a given measuring location has a clear physical meaning because it exhibits a large spatial variability as a function of many (primarily geographical) factors. Therefore, it is very informative to determine the *temperature response function* (Király & Jánosi 2002; Bartos & Jánosi 2005), which is the conditional mean 1 day temperature change 〈**Δ***T*_{a}|*T*_{a}〉=〈[*T*_{a}(*t*+1)−*T*_{a}(*t*)]|*T*_{a}(*t*)〉 determined in the following way. We divide the observed temperature-anomaly range into equal bins. For each day *t*, the given temperature anomaly *T*_{a}(*t*) defines the bin, where the 1 day change [*T*_{a}(*t*+1)−*T*_{a}(*t*)] is averaged over the whole time series. Clearly, the statistics at extreme anomalies is poor because the frequency of occurrences quickly breaks down for very high and very low values.

Figure 1 illustrates the temperature response function for one of the longest daily time series recorded since 1775 in Prague. In addition to many other data, it is available at the European Climate Assessment & Dataset website (http://eca.knmi.nl; Haylock *et al.* 2008). The empirical relationship for 〈**Δ***T*_{a}〉 as a function of *T*_{a} manifested in figure 1*a* exhibits the main properties of a general response function: the larger the anomaly, the more probable is a step backward on the next day. Although the statistics break down for large values, the response function is apparently asymmetric (positive excursions are hindered stronger) and clearly nonlinear. Nevertheless, this shape strongly suggests that the climatological mean temperature , where *T*_{a}≡0 by definition, properly characterizes a local *dynamical* equilibrium state of the atmosphere.

The central part of this function is magnified in figure 1*b*, where the black line illustrates a linear fit to this range. Note that the intercept is small but definitely non-zero. One might suspect that a small positive intercept is a simple consequence of statistical errors, but Bartos & Jánosi (2005) demonstrated, by an analysis of 11 827 daily temperature records, that it is statistically significant. Further investigations resulted in the explanation that the reason for the non-zero intercept is the *time asymmetry* of temperature fluctuations.

The lack of time-reversal symmetry of temperature fluctuations in *laboratory experiments* on turbulent *thermal convection* is relatively well known (Belmonte & Libchaber 1996). The statistics of time derivative is found to be an appropriate tool to characterize the dynamics of the flow. The temporal asymmetry is quantified by the skewness, the third moment around the mean normalized by the cube of the standard deviation. The observed behaviour for temperature time derivatives is that the skewness has a positive value at the cold (top) boundary. It changes sign at around the border of the cold thermal boundary layer, and is increasingly negative for larger distances (Belmonte & Libchaber 1996).

Figure 2*a* shows the central part of the histogram for 1 day temperature-anomaly changes **Δ***T*_{a} determined for the Prague record. Besides the apparent asymmetrical shape (skewness: −0.260), the cumulative sums determined separately for the negative and positive halves have a clear discontinuity at zero. This means that the total number of warming steps *N*_{w} significantly exceeds the number of cooling steps *N*_{c}. This would result in a quick heat death in the absence of a compensating effect, which is the larger mean magnitude of cooling steps 〈**Δ***T*_{c}〉 compared with the value for warming 〈**Δ***T*_{w}〉. Numerical values for Prague are *N*_{w}=43 850, *N*_{c}=40 276, 〈**Δ***T*_{c}〉=−1.795^{°}C and 〈**Δ***T*_{w}〉=1.648.

Before we proceed with an extended investigation, here we emphasize that the observed asymmetry is a consequence of higher order *nonlinear* correlations in the time series. Probably, the most effective known tool to detect nonlinearities is the *surrogate-data* technique (Kantz & Schreiber 2004). Surrogate-data testing attempts to check the hypothesis that the time series has been generated by a stationary Gaussian linear stochastic process (equivalently, an autoregressive moving average (ARMA) process). Indeed, a strictly linear temperature response function shown in figure 1*b* (with zero intercept) would be consistent with a simple first-order autoregressive (AR1) model (Bartos & Jánosi 2005). For the hypothesis testing, one generates random datasets (surrogates) that conserve the two-point autocorrelation function or, equivalently, the power spectrum. For a nonlinearity test, the single-time probability distribution (the histogram of fluctuations) must also be conserved. A detailed description of the methodology and optimized programmes are available at the site http://www.mpipks-dresden.mpg.de/~tisean/ (Hegger *et al.* 1999). An application is illustrated in figure 2*b*: surrogate data *T*′_{a}(*t*) are generated with the same histogram and power spectrum as the Prague anomaly series, then the 1 day derivatives **Δ***T*′_{a} are evaluated. The asymmetry disappeared (skewness: −0.016) demonstrating that the temperature fluctuations are related to relevant nonlinear processes.

The same conclusion arises when a generalized skewness function *Φ*(*τ*) is evaluated (Schreiber & Schmitz 1997),
2.1
where *τ* is an imposed time delay, and the normalization conserves the dimension ^{°}C. The results for the Prague time series and six different surrogates are shown in figure 2*c*. The extra information gained is that the statistically significant negative time asymmetry decays as a function of delay *τ*, and interestingly, it has a significant positive value when temperature differences at around two weeks are considered.

Bartos & Jánosi (2005) evaluated a large set of terrestrial daily temperature records, and the main results are illustrated in figure 3. Here, the simple measures of warming- and cooling-step-number ratio *N*_{w}/*N*_{c} and the ratio of average step size 〈**Δ***T*_{w}〉/〈**Δ***T*_{c}〉 are used. The scatter plots prove that the asymmetry is definitely not a finite-size effect. Unfortunately, the geographical and temporal coverage of the Global Daily Climatological Network data bank (Easterling *et al.* 2002) is quite uneven, as it provides reliable climatological information mostly over the Northern Hemisphere. The zonal average value of the asymmetry parameter *N*_{w}/*N*_{c} exhibits a linearly decreasing tendency in the range of 30^{°}–60^{°} N latitudes (figure 3*c*); the statistics are somewhat blurred outside this regime. It is remarkable that the apparent crossover is located around 60^{°} N, where the zonal mean temperature (over land) crosses the freezing point 0^{°}C (figure 3*d*).

In a comprehensive study, Ashkenazy *et al.* (2008) evaluated 59 years (from 1948 to 2006) of the National Centers for Environmental Prediction (NCEP), USA/National Center for Atmospheric Research (NCAR), USA, reanalysis data (Kalnay *et al.* 1996), including a global, multi-level coverage of temperature with a spatial resolution of 2.5^{°}×2.5^{°}. The asymmetry measure they used is also based on warming- and cooling-step numbers. However, the total number of non-zero steps served for normalization: *N*_{w}/(*N*_{w}+*N*_{c}). A comparison with the results of Bartos & Jánosi (2005) is depicted in figure 4*a*. The data reflect similar behaviour, the apparent differences are most probably owing to the different sampling densities and spatial distributions in the two analyses. Note that the cross-over ranges precisely coincide on the Northern Hemisphere.

The main findings from these works (Bartos & Jánosi 2005; Ashkenazy *et al.* 2008) can be summarized as follows.

— Apart from the vicinity of the equator, daily surface-temperature fluctuations have a significant time asymmetry on both hemispheres. The general behaviour is that large cooling steps (e.g. cold-front intrusions) are followed by slower gradual warming. This is a statistical pattern, the temperature records rarely exhibit a very clear sawtooth pattern.

— The asymmetry changes sign at latitudes around ±60

^{°}. At higher latitudes, sudden warming jumps are followed by slower gradual cooling. This behaviour is precisely reproduced in related rotating-tank laboratory experiments (Gyüre*et al.*2007), and is in agreement with the expectations close to a cold thermal boundary layer (Belmonte & Libchaber 1996).— The asymmetry fades away as a function of altitude (Ashkenazy

*et al.*(2008); figure 4*b*). This clearly indicates the role of dissipation in the atmospheric boundary layer.— The asymmetry also fades away when longer time differences are considered (Ashkenazy

*et al.*2008; figure 2*c*). This is related to the strong short-time ‘memory’ (persistence) of weather phenomena where the characteristic time scale is also 4–5 days.

The results presented above are consistent with a picture that the asymmetry is related to synoptic-scale activity, which is associated with mid-latitude fronts. The mid-latitude low-pressure eddies moving to the east are responsible for heat transport from low to high latitudes (Lindzen 1993) and are usually composed of cold and warm fronts that rotate cyclonically. Ashkenazy *et al.* (2008) demonstrate that the annual mean storm track and the transient heat-flux patterns resemble the annual mean asymmetry pattern. Furthermore, the typical lifetime of a cyclone explains the fading asymmetry for time lags larger than 5 days, and the fading asymmetry for large altitudes is because of the weakening frontal cyclone activity at pressure heights above 500 hPa.

## 3. Total-column ozone

Ozone (O_{3}) is responsible for the heating of the stratosphere and the protection of life from harmful ultraviolet radiation. Anthropogenic emissions of ozone-depleting halogen compounds have been identified as the cause for ozone losses in mid-latitudes that have been noticeable for about three decades (World Meteorological Organization 2007). In the mid-latitude lower stratosphere, where the bulk of the atmospheric ozone resides, the natural variability is rather large. Figure 5*a* illustrates 3 years of daily total-column ozone (TO) measurements recorded at the University of Houston test site (29.718^{°} N, 95.341^{°} W) participating in the National Oceanic and Atmospheric Administration (NOAA), Environmental Protection Agency (EPA), USA, Brewer Spectrophotometer UV and Ozone Network. Climatological patterns may be used to explain part of this variability, although the choice and the interpretation of the explanatory variables are not straightforward. The vertical O_{3} concentration profiles plotted in figure 5*b*,*c* illustrate that not only do the magnitudes change in short time intervals (TO is given by vertical integration), but also the changes in the stratosphere and in the troposphere might have entirely different tendencies at different days. An analysis of the correlation properties of time series might be a useful complementary tool to employing a regression analysis with explanatory variables.

In the lower stratosphere, ozone has a lifetime of months up to 1 year. Thus, variability at these altitudes is more dependent on transport processes than on photochemical ones (Rood & Douglass 1985; Stolarski & Douglass 1985). On the largest scale, the so called Brewer–Dobson circulation transports high-altitude ozone from the tropics poleward and downward to the lower stratosphere at high latitudes. At northern mid-latitudes, the typical mean annual cycle has a maximum in late winter/early spring and a minimum in late fall (figure 5*a*). This variation is different from what is observed in the upper stratosphere and from what we would expect from pure photochemical processes. The interplay of dynamical influences on ozone transport and temperature-dependent chemistry makes the explanation of ozone fluctuations rather difficult.

Figure 6 presents a map of the 1 day asymmetry parameter *N*_{a}/(*N*_{a} + *N*_{d}) (*N*_{a} and *N*_{d} denote the number of ascending and descending steps) for TO data from the Total Ozone Mapping Spectrometers (TOMS) project (see http://toms.gsfc.nasa.gov/). Records of the satellite Nimbus-7 (N7) during the period 1 November 1978 to 6 May 1993 are evaluated. In order to minimize the effects of known instrumental distortions at high solar zenith angles and missing polar night intervals, data evaluation is restricted for the band 60^{°} S–60^{°} N covering approximately 87 per cent of the Earth’s surface. For known calibration problems (Kiss *et al.* 2007*a*), results for the next equipment (TOMS Earth-Probe) are not presented here, although they are consistent with the findings for N7. Annual periodicities and the contribution of quasi-biennial oscillations are removed (Kiss *et al.* 2007*b*). However, the latter filtering did not result in detectable changes in the statistics.

Since the map in figure 6 exhibits sizeable cell-to-cell fluctuations, an extensive test of statistical significance was performed by generating 1000 randomly shuffled realizations for each time series in each geographical cell. The asymmetry parameter for the shuffled records obeys a Gaussian probability density distribution, in which the *mean* standard deviation (0.003965) was used for the estimation of 95% confidence interval. Figure 7*a* demonstrates that the zonal averages over the entire time interval (more than 14 years) indicate significant *positive* asymmetries in the latitudinal band approximately 25^{°}–40^{°} on both hemispheres, but with a large variability. ‘Positive’ asymmetry means here that the number of descending steps is larger, *N*_{d}>*N*_{a}, which means a statistical pattern of sudden TO jumps upward followed by a gradually decreasing period. (In this case, the histogram of TO changes has a positive skewness because of the larger average magnitude of ascending steps.) Figure 7*b* reveals one reason for the sizeable error bars: the location of bands has an annual periodicity by following approximately the position of the thermal equator. Notice that the bands of positive asymmetry are somewhat narrower and shallower in January. The zonal averages do not exhibit negative asymmetries in the 95% confidence range, yet there are patches around 55^{°}–60^{°} (latitude) on both hemispheres (figure 6), where the negative asymmetry is statistically significant, even over 99%. All significant asymmetries fade away in about a week when longer time differences are considered, similar to the surface-temperature records (see §2).

The tests for statistical significance were performed by the method of data shuffling, as described above. Nevertheless, we would like to emphasize that the asymmetry of TO changes is a consequence of relevant nonlinearities in the dynamics, similar to the temperature fluctuations. The surrogate-data technique (see §2) unambiguously proved that higher order correlations are responsible for the observed time asymmetry for each inspected geographical location.

Both the spotty character of the map shown in figure 6 (reflected also by the large error bars of the zonal averages in figure 7*a*) and the seasonal migration of the bands (figure 7*b*) are indicative of a dynamical origin of statistically significant time asymmetry. In order to get a hint about the determining processes, we performed a correlation analysis between TOMS-N7 TO and temperature records from the NCEP/NCAR reanalysis (Kalnay *et al.* 1996) over the same time interval. Since the spatial resolution of the latter dataset (2.5^{°}×2.5^{°}) is lower than that of the TOMS (1.0^{°}×1.25^{°} latitude/longitude), nearest-neighbour ozone and temperature grid points were considered throughout the analysis. (The temperature changes rather smoothly at the pressure levels we studied, therefore interpolation resulted in a negligible improvement in the statistics.) The temperature is chosen because its time asymmetry is related to mid-latitude tropospheric activities, as discussed in the previous section. Figure 5*b*,*c* indicates the pressure levels where temperature records were evaluated. Most of the ozone resides in the lower stratosphere between the levels 30 and 70 hPa. Additionally, the pressure height of 250 hPa is selected to check possible tropospheric impacts because this level is usually below the tropopause in the latitudinal band 45^{°} S–45^{°} N (Hoinka 1998). Furthermore, the percentage variability of ozone concentrations is the strongest at around this pressure height (Lamsal *et al.* 2004).

Figure 8 displays TO and temperature time series at a given location of significant positive asymmetry (0.476) near the Bermuda Islands. The temperature signals do not exhibit significant asymmetries at such altitudes (see figure 4*b*), therefore we evaluated the cross-correlation functions
3.1
where *τ* is an imposed time delay, TO is the total-column ozone, *T*_{h} is the temperature at a given pressure level *h* and *σ* denotes the appropriate standard deviation. Notice that the annual periodicities are not removed in equation (3.1). The average refers to the unfiltered signals plotted in figure 8*a*,*c*.

Figure 9 shows cross-correlation functions for two distant locations of different properties. The curves in figure 9*a* (gridpoint over southwestern Australia, significant *positive* asymmetry of 0.486) are characteristic for sites, where the pressure level 250 hPa is well below the tropopause (Hoinka 1998). The cross-correlation functions exhibit two ‘modes’: a slow mode with a clear phase shift between the upper tropospheric (250 hPa) and lower stratospheric (30 and 70 hPa) temperatures, and a narrow equal-time (*τ*=0) excess peak. The phase shift between the different pressure levels, the slow growth and decay of cross-correlations, together with the moderate amplitude indicate the lack of strong physical coupling between temperature and stratospheric ozone concentrations, they rather reflect correlations among different seasonal periodicities instead of general causal connections. Note the clear consequences of tropopause crossing: the cross-correlation curves become very similar in figure 9*b*, where the 250 hPa pressure level definitely resides already in the lowermost stratosphere (site in the middle of the northern Atlantic Basin with significant *negative* asymmetry of 0.512).

The situation is different for the localized peaks in figure 9, at around *τ*=0. Essentially, this is the only peak in the cross-correlation functions for the *derivative* of TO and *T*_{h} time series (not shown here). At the pressure levels of 30 and 70 hPa, the peaks are either missing (in the band of 30^{°} S–30^{°} N) or positive (for higher latitudes) with respect to the background level. Negative peaks at the pressure heights of 250 hPa (figure 9*a*) are characteristic at specific locations, figure 10 exhibits the map of the geographical distribution. The white regions in figure 10 indicate anticorrelations between the short-time fluctuations of total-column ozone and upper-tropospheric temperature, while the correlations are positive for high latitudes of approximately 60^{°} over both hemispheres (here, *T*_{250 hPa} is above the tropopause). The overlap between the bands of significant positive asymmetry (figure 6) and TO–*T*_{250 hPa} anticorrelations (figure 10) is remarkable.

## 4. Dynamical origin of time-asymmetric total-column ozone fluctuations

It is quite tempting to relate the localized peaks at *τ*=0 (figure 9) with photochemical heating effects, especially because the ozone layer is basically responsible for the existence and elevated temperature of the stratosphere. The relative importance of photochemical and transport processes can be estimated by means of a *relaxation time* of any disturbance of the photochemical equilibrium, which is approximately given by the ratio of total content of odd oxygen particles (O and O_{3}) divided by their rate of formation. It is well known (Douglass *et al.* 1985; Rood & Douglass 1985; Stolarski & Douglass 1985; Petzoldt *et al.* 1994; McLinden *et al.* 2000) that the photochemical relaxation time is around 1 h at an altitude of 70 km (oxygen concentration-limited case), and it increases very quickly with decreasing height because the ultraviolet radiation with *λ*<300 nm is depleted almost completely well above the tropopause (photon flux-limited case). At the altitude range of 20–30 km, the relaxation time is of the order of 10^{2}–10^{3} days (Rood & Douglass 1985). Therefore, the ozone distribution in the lower stratosphere is determined by air transport, that is, the seasonal and latitudinal changes of ozone concentration are governed by the general circulation. This explains the remarkable feature in figures 5*a* and 8*a*, namely that the ozone concentration is definitely higher in late winter–early spring than in the summer months over the Northern Hemisphere, in spite of the fact that the insolation flux is considerably lower in this interval.

The influence of dynamical processes on total ozone has been known for a long time (Dobson & Harrison 1926). The TO level basically varies in response to vertical and meridional transport at around the tropopause and in the lower stratosphere. Extremely low TO values are usually associated with anticyclones, and they are referred to as *ozone-miniholes* (James 1998). Preferred geographical regions of strong ozone variability on synoptic time scales (2–10 days) coincide with the storm-track belts (James 1998; Orsolini *et al.* 1998). Müller *et al.* (2008) demonstrated the importance of dynamical processes by proving that a substantial fraction of the daily minimum ozone columns occurs outside the Arctic polar vortex during winters with little chemical ozone destruction, which results in serious errors in polar ozone-loss estimates.

The dynamical origin of short-time TO fluctuations is further supported by inspecting the *volatility* of the ozone records. The volatility characterizes the magnitude or strength of instantaneous variability, appropriate measures are, for example, the absolute value or the square of daily changes, |**Δ**TO(*t*)|, or **Δ**TO(*t*)^{2}, respectively. An example is shown in figure 11*a*, where the former definition is adopted. Since numerical derivation produces very strong noise, a 21 day running average is used to smooth large amplitude fluctuations. Figure 11*b* illustrates the same signals for the temperature records at the representative pressure heights. Simple visual monitoring indicates that the seasonal pattern of the volatility is practically the same for all the records. This observation is quantitatively supported by the cross-correlation functions plotted in figure 11*c*,*d*. The most plausible explanation is that the period of strong mid-latitude dynamical activity produces an increased volatility for all the atmospheric parameters, where changes are primarily determined by transport processes on short time scales. The period of high volatility is shifted to the second half of the calendar year on the Southern Hemisphere, as expected. Finally, we note that the volatility series determined from surrogate records entirely lack the annual periodicity, which again illustrates the importance of higher order correlations in atmospheric mixing.

## Acknowledgements

This work was supported by the Hungarian Science Foundation (OTKA) under grant no. NK72037, and the European Commissions RECONCILE-226365-FP7-ENV-2008-1 project.

## Footnotes

One contribution of 13 to a Theme Issue ‘Complex dynamics of life at different scales: from genomic to global environmental issues’.

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