Pierre S. Farrugia

Comparative analysis of estimators for wind direction standard deviation

Wind direction is a circular variable. This makes the algorithms used to find its standard deviation different from that of the linear variables. In particular, the requirement for storing all the data points before the standard deviation can be computed limits the storage capacity and puts great strain on remote data acquisition systems. Various algorithms have therefore been developed to estimate the standard deviation in order to reduce the number of terms stored. The following work consists of a comparative analysis of such estimators together with the parameters used. It emerges that some of the assumptions adopted to produce the equations being analysed do not hold in practice, even though this does not affect significantly the performance of the estimators that depend on them. On the other hand, the parameter that has the best trend with the algorithm adopted is the magnitude of the vector to the centre of gravity of the system. However, such a result gives rise to some concerns since it does not account for the ‘vectorial’ nature of the angle being treated. Copyright

On the Algorithms Used to Compute the Standard Deviation of Wind Direction

Abstract The standard deviation of wind direction is a very important quantity in meteorology because in addition to being used to determine the dry deposition rate and the atmospheric stability class, it is also employed in the determination of the rate of horizontal diffusion, which in turn determines transport and dispersion of air pollutants. However, the computation of this quantity is rendered difficult by the fact that the horizontal wind direction is a circular variable having a discontinuity at 2π radians, beyond which the wind direction starts again from zero, thus preventing angular subtraction from being a straightforward procedure. In view of such a limitation, this work is meant to provide new mathematical expressions that simplify both the computational and analytical work involved in handling the standard deviation of wind direction. This is achieved by deriving a number of Fourier series and Taylor expansions that can represent the minimum angular distance and its powers. Using these expr...

N-body gravity and the Schrödinger equation

We consider the problem of the motion of N bodies in a self-gravitating system in two spacetime dimensions. We point out that this system can be mapped onto the quantum–mechanical problem of an N-body generalization of the problem of the H+2 molecular ion in one dimension. The canonical gravitational N-body formalism can be extended to include electromagnetic charges. We derive a general algorithm for solving this problem, and show how it reduces to known results for the 2-body and 3-body systems.

Vectorial statistics for the standard deviation of wind direction

The standard deviation of wind direction is an important parameter in atmospheric pollution management. It can be used to calculate the rate of horizontal diffusion and from this the transport and dispersion of air contaminants can be determined. The standard deviation of wind direction cannot be calculated directly from customary linear statistics, mainly because of its periodic nature which makes the zero position arbitrary. Various algorithms have been proposed to estimate its value. The methodologies adopted in meteorology implicitly assume that the wind angle can be treated independently of the wind speed. Such an assumption might not be appropriate in some instances, as will be shown in this work by means of an example. To overcome this limitation, a new algorithm that takes into account both the periodic and the vectorial nature of the wind direction will be proposed. This is done by weighing each sample value with the corresponding wind speed. The results obtained from the new method were compared to those determined from algorithms available in literature using measured data. The comparison indicates that while the behavior is similar, differences do exist. Further investigation indicated that while the differences can be small, they might be physically important.

Series solutions for turbulent plumes evolving in a natural environment

Closed form solutions to the boundary layer equations for turbulent point and line thermal plumes evolving in natural convection have been obtained in the form of power series for the case when the turbulent viscosity is assumed to be proportional to the vertical height above the source. The initial values needed to generate the coefficients of the power series for different turbulent Prandtl number have been obtained numerically. To compliment these values the constants of proportionality between the turbulent viscosity and the height were determined using different methodologies including analysis of experimental data, computational fluid dynamics and numerical considerations. Evaluation of the results is primarily carried out by comparison to experimentally determined profiles of the temperature and velocity that are found in the literature. The best agreement was obtained when the Reynolds analogy—giving the turbulent Prandtl number as unity—was adopted. While the range of validity of the power series is limited, its radius of convergence can be extended using a suitable transformation.

Power series solutions for laminar plumes in a natural environment

Power series solutions to the boundary layer equations for laminar point and line thermal plumes in natural convection have been derived in terms of recurrent relations. These together with the initial conditions constitute closed-form solutions for any Prandtl number in the region where the series converge. The starting conditions are related to the maximum values of the temperate and velocity profiles. Their values together with those for the radius of convergence of the series have been obtained numerically for different Prandtl numbers, and best-fitting functions have been proposed for the variation observed. The validity of the approach has been tested against the known closed-form solutions giving identical results in the region of convergence. While the utility of the equations does not extend to infinity, the tests conducted indicate that the range of convergence can be potentially extended by using the Euler transform. This is especially true for results involving point heat sources, where it has been shown that, for all Prandtl numbers, the nearest singularity is found in the complex plane and, hence, has no physical significance.