![]() While normal diffusion describes a large set of random movements, anomalous diffusion may occur under certain conditions. In this case, the fluctuating part will have a MSD given by the previous expression while the mean position increases as V X t. If a mean flow of constant velocity V X steers the particle in a particular direction, the position of the particle will have a fluctuating component δX( t) and a deterministic part given by the mean flow. This is known as normal Brownian diffusion. ![]() According to statistical mechanics where D is the so called diffusion coefficient which depends on the temperature, the radius of the particle and the viscosity of the fluid. As this position varies erratically in time, the particle will explore a certain area which is given by the MSD. If this position is denoted X, the MSD is defined as follows: (the brackets denote an average over time t') where X( t) is the instantaneous position of the particle at time t. ![]() In the absence of flow, the particle, subject to thermal agitation of the surrounding fluid, will have a position which fluctuates in time. A classical example where this notion has gained all its importance is that of a colloidal particle in a simple fluid. The mean square displacement (MSD), a notion borrowed from statistical physics and the study of Brownian motion, is a measure of the deviation from a mean trajectory. In particular we suggest that track forecast cones available today can be linked directly to this measure of the trajectory deviations around a mean and that unless these deviations can be understood and the factors giving rise to them are fully taken into account in models and simulations, reducing such uncertainty will be a difficult task. This law appears for the random movement of generic vortices in two dimensions as has been shown in experiments and numerical simulations 9, 10, 11. This is based on recent observations suggesting that the deviations of the trajectory of generic vortices or TCs from a mean trajectory can be modeled with a universal law 9 for their so called mean square displacement. Here we show that the statistical properties of these deviations, from say a predictable simple linear track, can be used to determine possible corridors or track forecast cones for TCs. In fact, track predictions usually include an estimate of the deviation of the trajectory from the predicted track in the form of so called track forecast cones which are based on error statistics from previous hurricane tracks as compared to predictions. These deviations are difficult to predict as they are due to different factors such as the proximity of land and its topography, variations in the surrounding large scale flow, or modifications of the vortex structure itself 1. Since trajectories of TCs show large deviations from a generally predictable mean trajectory (which could be linear or recurved), prediction schemes can be imprecise, giving the statistical approaches a legitimate place. linear versus recurved) for different basins, on the mean velocity and the deviations from predicted tracks and on the landfall probability for different regions 3, 4, 5, 6, 7, 8. Different statistical analyses are also carried out on the nature of the trajectory (i.e. Some predictions are based on the knowledge of previous hurricane tracks in the geographical area of interest and others use full scale numerical simulations 1. Several schemes are used to predict their trajectories 1, 2. Tropical Cyclones (TCs), otherwise known as hurricanes or typhoons, are extreme atmospheric events which can be devastating upon landfall in populated areas.
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