Yuri N. Skiba

Elements of the mathematical modeling in the control of pollutants emissions

Abstract Two models are presented in order to describe the dispersion of a primary and secondary pollutants in the atmosphere and control their emissions. The first of them is a simple box model whose analytical solutions permit to establish easily evaluated relations between the short-term and long-term controls of pollutants emissions and the model parameters. The dispersion of a primary and secondary pollutants emitted to the atmosphere by point, line and area sources is also simulated with a more general two-dimensional diffusion–advection-reaction model. The duality principle based on using the adjoint model is derived to formulate sufficient restrictions on the emission rates of different sources allowing to maintain mean concentrations of both pollutants in a zone below maximally admissible values (sanitary norms). A short-term control strategy using the solution to the adjoint model is suggested. The control (correction of emission rates) is implemented any time when the model forecast of at least one mean pollution concentration is unfavorable. The control strategy is optimal in that it minimizes the changes in the original emission rates. The minimization process is subjected to a restriction imposed with the duality principle. The optimal control parameters are obtained with an efficient algorithm of the successive orthogonal projections.

Spectral Approximation in the Numerical Stability Study of Nondivergent Viscous Flows on a Sphere

The accuracy of calculating the normal modes in the numerical linear stability study of two-dimensional nondivergent viscous flows on a rotating sphere is analyzed. Discrete spectral problems are obtained by truncating Fouriers series of the spherical harmonics for both the basic flow and the disturbances to spherical polynomials of degrees K and N , respectively. The spectral theory for the closed operators [1], and embedding theorems for the Hilbert and Banach spaces of smooth functions on a sphere are used to estimate the rate of convergence of the eigenvalues and eigenvectors. It is shown that the convergence takes place if the basic state is sufficiently smooth, and the truncation numbersK andN of Fouriers series for the basic flow and disturbances tend to infinity keeping the ratio N/K fixed. The convergence rate increases with the smoothness of the basic flow and with the power s of the Laplace operator in the vorticity equation diffusion term. c

On the spectral problem in the linear stability study of flows on a sphere

A normal mode instability study of a steady nondivergent flow on a rotating sphere is considered. A real-order derivative and family of the Hilbert spaces of smooth functions on the unit sphere are introduced, and some embedding theorems are given. It is shown that in a viscous fluid on a sphere, the operator linearized about a steady flow has a compact resolvent, that is, a discrete spectrum with the only possible accumulation point at infinity, and hence, the dimension of the unstable manifold of a steady flow is finite. Peculiarities of the operator spectrum in the case of an ideal flow on a rotating sphere are also considered. Finally, as examples, we consider the normal mode stability of polynomial (zonal) basic flows and discuss the role of the linear drag, turbulent diffusion and sphere rotation in the normal mode stability study.

On a method of detecting the industrial plants which violate prescribed emission rates

A limited area pollution transport model and its adjoint are considered for controlling industrial emission rates and air quality. A method of detecting the industries, which ignore and violate the emission rates prescribed by an air quality control, is suggested. Some regularization techniques are considered to obtain the linear system with a well-conditioned matrix.

Air pollution estimates in Guadalajara City

A method for estimating the impact of industrial emissions is suggested and applied to the Guadalajara City Metropolitan Area (GCMA). The method is based on solutions to the pollution transport model and its adjoint. Two equivalent direct and adjoint mean pollution concentration estimates are considered for ecologically important zones of the GCMA. The dependence of these estimates on the number, positions and emission rates of industrial plants, as well as on the wind and initial pollution distribution in the GCMA is qualitatively and quantitatively examined. It is shown that the adjoint model solutions serve as the influence functions providing valuable information on the role of each of the industrial plants in polluting different zones within the GCMA. These solutions have been calculated with a balanced and absolutely stable second-order finite-difference scheme based on the splitting method. A method for an optimal allocation of a new industrial plant is considered.

DUAL OIL CONCENTRATION ESTIMATES IN ECOLOGICALLY SENSITIVE ZONES

Propagation of the oil spilling from a damaged oil tanker is considered in a limited sea area. Direct and adjoint estimates of the average oil concentration in special zones are derived by using solutions of the 2-D main and adjoint oil transport problems, respectively. The dual estimates complement each other nicely in studying the oil spill consequences. While the direct estimates are preferable to get a comprehensive idea of the oil spill impact on the whole area, the adjoint ones are specially useful and economical when the accident site-dependence or/and the oil spill rate-dependence of the oil concentration is/are studied only in a few ecologically sensitive zones. Indeed, each adjoint estimate explicitly relates the average oil concentration in a zone to the oil spill rate using the adjoint solution values at the accident site. Being independent of the two parameters (the accident site and oil spill rate), the adjoint solution can be found for each zone regardless of a concrete accident and used repeatedly for various possible values of these parameters. Several examples explain how to decide between two estimates.Thanks to special boundary conditions, the main and adjoint problems are both well-posed according to Hadamard (1923). The dual estimates can be generalized to the three dimensions. The balanced, absolutely stable and compatibie main and adjoint 3-D numerical algorithms by Skiba (1993) can easily be adapted to the problem discussed here.

Industrial pollution transport. Part 2. Control of industrial emissions

Solutions of the pollution transport problem and its adjoint are used to monitor mean pollution concentration in an ecologically important zone. Four strategies of control over pollutants released into the atmosphere by industrial plants are suggested. They differ by the restrictions imposed on the emission rate of each plant. All the strategies use solutions of the adjoint transport problem and assure the fulfillment of the sanitary norm in the zone. A linear interpolation of these strategies also brings pollution level in the zone down to the sanitary norm. A method of detecting the plants violating the prescribed emission rates is also given. A simple example is given to illustrate the strategies suggested.

Industrial pollution transport. Part 1. Formulation of the problem and air pollution estimates

A pollution transport problem is formulated in a limited area. As the pollution sources we take emissions from industrial plants. Physically and mathematically suitable conditions are prescribed on the open boundaries. We show that the problem (as well as its adjoint) is well posed in the sense that a weak solution exists, is unique and depends continuously on its data. Direct and adjoint estimates of the average pollution concentration in an ecologically important zone are given, and the sensitivity of these estimates to perturbations in model parameters is analyzed.

On dimensions of attractive sets of viscous fluids on a sphere under quasi-periodic forcing

Abstract Simple attractive sets of a viscous incompressible fluid on a sphere under quasi-periodic polynomial forcing are considered. Each set is the vorticity equation (VE) quasi-periodic solution of the complex (2n + 1)-dimensional subspace Hn of homogeneous spherical polynomials of degree n. The Hausdorff dimension of its path being an open spiral densely wound around a 2n-dimensional torus in Hn , equals to 2n. As the generalized Grashof numb G becomes small enough then the basin of attraction of such spiral solution is expanded from Hn to the entire VE phase space. It is shown that for given G, there exists an integer nG such that each spiral solution generated by a forcing of Hn with n ≥ nG is globally asymptotically stable. Thus, whereas the dimension of the fluid attractor under a stationary forcing is limited above by G, the dimension of the spiral attractive solution (equal to 2n) may, for a fixed G, become arbitrarily large as the degree n of the quasi-periodic polynomial forcing grows. Since t...

On the long-time behavior of solutions to the barotropic atmosphere model

Abstract Long-term large scale behavior and location of the attractors of the barotropic atmosphere model described by the dissipative and forced vorticity equation (VE) on a rotating sphere are studied analytically. Size of a bounded invariant set B that eventually attracts the trajectories of all the VE solutions is estimated depending on the linear drag, turbulence and spectral composition and smoothness of the forcing. If the VE forcing belongs to the set Hn of the homogeneous spherical polynomials of degree n, the solutions show quite different behavior for ideal fluid (a); nonturbulent fluid with linear drag (b), and turbulent fluid (c). For n ≫ 1, the whole space of the VE solutions is divided into sets M n + and M n − of the small and large scale fields defined by χ> n(n + 1) and χ n(n + 1) respectively (χ is the Fjortoft average spectral number of field on a sphere), and the interface M n 0:χ = n(n + 1) that includes Hn . In cases (a) and (b), M n +, M n 0, M n − and H n are invariant sets of the...