Topological Entropy : A Lagrangian Measure of the State of the Free Atmosphere

Topological entropy is shown to be a useful characteristic of the state of the free atmosphere. It can be determined as the stretching rate of a line segment of tracer particles in the atmosphere over a time span of about 10 days. Besides case studies, the seasonal distribution of the average topological entropy is determined in several geographical locations. The largest topological entropies appear in the mid- and high latitudes, especially in winter, owing to the greater temperature gradient between the pole and the equator and the more intense stirring and shearing effects of cyclones. The smallest values can be found in the trade wind belt. The local value of the topological entropy is a measure of the chaoticity of the state of the atmosphere and of how rapidly pollutants and contaminants spread from a given location

Bevezetés a környezeti áramlások fizikájába

A környezettudomány és azon belül a környezetfizika egyik legnagyobb jelentőségű területét a globális környezeti áramlások vizsgálata jelenti. A környezeti áramlások (akár a légkörben vagy vizekben zajlanak, akár globális vagy lokális kiterjedésűek) környezetünk alakulásának legfontosabb mozzanataihoz tartoznak. Ide sorolhatjuk még a Földünk belsejében, elsősorban a folyékony köpenyben zajló áramlási jelenségeket is, amelyek a Föld kialakulása óta eltelt évmilliárdok alatt markánsan alakították és pillanatnyilag is alakítják a felszín viszonyait. Erről a közegről azonban a későbbiekben nem lesz szó, mert tárgyunk a Nap energiája által hajtott légköri és óceáni áramlások, amelyek nem geológiai, hanem „emberi” időskálán befolyásolják környezetünket. Az áramlási jelenségeknek környezetünk szempontjából alapvetően meghatározó a jelentősége. Gondoljunk csak a napi időjárás változásainak áramlási vonatkozásaira, a Golf-áramlatra, amely lakhatóvá teszi Európa északi övezeteit, a világtengerek más, jelentős áramlataira, vagy kisebb skálán azokra a jelenségekre, amelyek a bioszféra egy-egy szegmensének benépesülését és létezését lehetővé tették és teszik. A modern világ viszonyait kétség kívül alapvetően befolyásolják az emberi társadalmak gazdasági tevékenységét kísérő nem kívánt mellékhatások, elsősorban a környezet szennyezése. Ezek a szennyezések az áramlási folyamatok során mind a légkörben, mind a vizekben távoli területekre is eljutnak. Nem kérdéses, hogy a káros hatások megelőzésének, és a károk csökkentésének előfeltétele a kapcsolódó folyamatok megismerése, megértése, és ezek után esetleges befolyásolása

Quantifying nonergodicity in nonautonomous dissipative dynamical systems: an application to climate change

In nonautonomous dynamical systems, like in climate dynamics, an ensemble of trajectories initiated in the remote past defines a unique probability distribution, the natural measure of a snapshot attractor, for any instant of time, but this distribution typically changes in time. In cases with an aperiodic driving, temporal averages taken along a single trajectory would differ from the corresponding ensemble averages even in the infinite-time limit: ergodicity does not hold. It is worth considering this difference, which we call the nonergodic mismatch, by taking time windows of finite length for temporal averaging. We point out that the probability distribution of the nonergodic mismatch is qualitatively different in ergodic and nonergodic cases: its average is zero and typically nonzero, respectively. A main conclusion is that the difference of the average from zero, which we call the bias, is a useful measure of nonergodicity, for any window length. In contrast, the standard deviation of the nonergodic mismatch, which characterizes the spread between different realizations, exhibits a power-law decrease with increasing window length in both ergodic and nonergodic cases, and this implies that temporal and ensemble averages differ in dynamical systems with finite window lengths. It is the average modulus of the nonergodic mismatch, which we call the ergodicity deficit, that represents the expected deviation from fulfilling the equality of temporal and ensemble averages. As an important finding, we demonstrate that the ergodicity deficit cannot be reduced arbitrarily in nonergodic systems. We illustrate via a conceptual climate model that the nonergodic framework may be useful in Earth system dynamics, within which we propose the measure of nonergodicity, i.e., the bias, as an order-parameter-like quantifier of climate change

Escape rate : A Lagrangian measure of particle deposition from the atmosphere

Due to rising or descending air and due to gravity, aerosol particles carry out a complicated, chaotic motion and move downwards on average. We simulate the motion of aerosol particles with an atmospheric dispersion model called the Real Particle Lagrangian Trajectory (RePLaT) model, i.e., by solving Newton's equation and by taking into account the impacts of precipitation and turbulent diffusion where necessary, particularly in the planetary boundary layer. Particles reaching the surface are considered to have escaped from the atmosphere. The number of non-escaped particles decreases with time. The short-term and long-term decay are found to be exponential and are characterized by escape rates. The reciprocal values of the short-term and long-term escape rates provide estimates of the average residence time of typical particles, and of exceptional ones that become convected or remain in the free atmosphere for an extremely long time, respectively. The escape rates of particles of different sizes are determined and found to vary in a broad range. The increase is roughly exponential with the particle size. These investigations provide a Lagrangian foundation for the concept of deposition rates

Chaotic Explosions

We investigate chaotic dynamical systems for which the intensity of trajectories might grow unlimited in time. We show that (i) the intensity grows exponentially in time and is distributed spatially according to a fractal measure with an information dimension smaller than that of the phase space,(ii) such exploding cases can be described by an operator formalism similar to the one applied to chaotic systems with absorption (decaying intensities), but (iii) the invariant quantities characterizing explosion and absorption are typically not directly related to each other, e.g., the decay rate and fractal dimensions of absorbing maps typically differ from the ones computed in the corresponding inverse (exploding) maps. We illustrate our general results through numerical simulation in the cardioid billiard mimicking a lasing optical cavity, and through analytical calculations in the baker map.Comment: 7 pages, 5 figure

Memory effects in chaotic advection of inertial particles

A systematic investigation of the effect of the history force on particle advection is carried out for both heavy and light particles. General relations are given to identify parameter regions where the history force is expected to be comparable with the Stokes drag. As an illustrative example, a paradigmatic two-dimensional flow, the von Kármán flow is taken. For small (but not extremely small) particles all investigated dynamical properties turn out to heavily depend on the presence of memory when compared to the memoryless case: the history force generates a rather non-trivial dynamics that appears to weaken (but not to suppress) inertial effects, it enhances the overall contribution of viscosity. We explore the parameter space spanned by the particle size and the density ratio, and find a weaker tendency for accumulation in attractors and for caustics formation. The Lyapunov exponent of transients becomes larger with memory. Periodic attractors are found to have a very slow, ${{t}^{-1/2}}$ type convergence towards the asymptotic form. We find that the concept of snapshot attractors is useful to understand this slow convergence: an ensemble of particles converges exponentially fast towards a snapshot attractor, which undergoes a slow shift for long times

Dispersion of aerosol particles in the free atmosphere using ensemble forecasts

The dispersion of aerosol particle pollutants is studied using 50 members of an ensemble forecast in the example of a hypothetical free atmospheric emission above Fukushima over a period of 2.5 days. Considerable differences are found among the dispersion predictions of the different ensemble members, as well as between the ensemble mean and the deterministic result at the end of the observation period. The variance is found to decrease with the particle size. The geographical area where a threshold concentration is exceeded in at least one ensemble member expands to a 5-10 times larger region than the area from the deterministic forecast, both for air column "concentration" and in the "deposition" field. We demonstrate that the root-mean-square distance of any particle from its own clones in the ensemble members can reach values on the order of one thousand kilometers. Even the centers of mass of the particle cloud of the ensemble members deviate considerably from that obtained by the deterministic forecast. All these indicate that an investigation of the dispersion of aerosol particles in the spirit of ensemble forecast contains useful hints for the improvement of risk assessment

Climate change in mechanical systems: the snapshot view of parallel dynamical evolutions

We argue that typical mechanical systems subjected to a monotonous parameter drift whose timescale is comparable to that of the internal dynamics can be considered to undergo their own climate change. Because of their chaotic dynamics, there are many permitted states at any instant, and their time dependence can be followed—in analogy with the real climate—by monitoring parallel dynamical evolutions originating from different initial conditions. To this end an ensemble view is needed, enabling one to compute ensemble averages characterizing the instantaneous state of the system. We illustrate this on the examples of (i) driven dissipative and (ii) Hamiltonian systems and of (iii) non-driven dissipative ones. We show that in order to find the most transparent view, attention should be paid to the choice of the initial ensemble. While the choice of this ensemble is arbitrary in the case of driven dissipative systems (i), in the Hamiltonian case (ii) either KAM tori or chaotic seas should be taken, and in the third class (iii) the best choice is the KAM tori of the dissipation-free limit. In all cases, the time evolution of the chosen ensemble on snapshots illustrates nicely the geometrical changes occurring in the phase space, including the strengthening, weakening or disappearance of chaos. Furthermore, we show that a Smale horseshoe (a chaotic saddle) that is changing in time is present in all cases. Its disappearance is a geometrical sign of the vanishing of chaos. The so-called ensemble-averaged pairwise distance is found to provide an easily accessible quantitative measure for the strength of chaos in the ensemble. Its slope can be considered as an instantaneous Lyapunov exponent whose zero value signals the vanishing of chaos. Paradigmatic low-dimensional bistable systems are used as illustrative examples whose driving in (i, ii) is chosen to decay in time in order to maintain an analogy with case (iii) where the total energy decreases all the time

Signatures of fractal clustering of aerosols advected under gravity

Aerosols under chaotic advection often approach a strange attractor. They move chaotically on this fractal set but, in the presence of gravity, they have a net vertical motion downwards. In practical situations, observational data may be available only at a given level, for example at the ground level. We uncover two fractal signatures of chaotic advection of aerosols under the action of gravity. Each one enables the computation of the fractal dimension $D_{0}$ of the strange attractor governing the advection dynamics from data obtained solely at a given level. We illustrate our theoretical findings with a numerical experiment and discuss their possible relevance to meteorology.Comment: Accepted for publication in Phys. Rev. E (Rapid Communications