Meteorology, mathematical problems in

Problems in the area of the physics, the chemistry and the biology of the atmosphere, solved with the aid of mathematical methods. The majority of mathematical problems in meteorology are characterized by their complexity and the huge volume of information processing that is involved. Therefore for the solution of these problems, alongside analytic methods, numerical simulation on computers is widely used.

Mathematical problems of the physics of the atmosphere are primarily problems in the hydro-thermodynamics of a stratified fluid with distinctive features, created by the rotation, inhomogeneities of the relief and the surface temperature of the Earth. The theoretical basis of the mathematical models are the laws of conservation of mass, momentum and energy, which, together with the laws of thermodynamics and chemistry, describe the processes occurring in the atmosphere and the interactions of the atmosphere with the oceans and the Earth's surface. In mathematical terms this is a system of non-linear partial differential equations, which must be solved under the assumption that the external source of energy is the Sun. Besides the functions describing the state and behaviour of the atmosphere (temperature, pressure, density, wind velocities, etc.), these equations contain a number of parameters. By parameters one usually means the coefficients of the equations, and in non-stationary problems — the initial values of the functions, the state of the atmosphere, the characteristics of the Earth's surface, the external sources, etc. The initial conditions are determined as a result of measurements in the real physical system "atmosphere–Earth" . The process of mathematical modelling consists of several stages: a qualitative analysis of the mathematical model (well-posedness of the problem, its solvability in physically reasonable functions, etc.), the construction of its discrete analogue, the development of computational algorithms and programs for the computer realization of the discrete models, an analysis of the sensitivity of the model to variations of the parameters, an estimate of the parameters based on a priori information and measurements, etc. The structure of the mathematical model depends on the space-time scales of the processes being studied in the atmosphere and the method of describing them.

Numerical simulation is one of the fundamental means for the solution of problems in weather forecasting and the theory of the climate. Of particularly great significance is the problem of numerical simulation in the study of climatic variations under the influence of natural and antropogenic factors and under an estimate of the influence of the activity of man on the environment. The selection and justification of the physical models for a given class of problems is closely connected with research into the fundamental questions of stability, predictability and sensitivity of the physical system consisting of the atmosphere, the oceans, the snow cover, the continents, and the biosphere, which is usually called the climatic system. Predictability determines the possibility of a sufficiently deterministic approach to the prediction of physical processes and, at the same time, determines the possibility of constructing mathematical models for the description and prediction of the behaviour of the climatic system, or part of it. Sensitivity characterizes the degree of sensitivity of the system with respect to variations of the external influences and the internal parameters. If the influence of anthropogenic factors is interpreted as perturbations to the system, then an estimate of their influence can be considered as one of the applied aspects of sensitivity theory.

The problem of short-term weather forecasting (from several hours up to several days) is to find a non-stationary solution of the system of non-linear differential equations of the hydro-thermodynamics of the atmosphere with given boundary values and initial conditions. In problems of long-term weather forecasting certain generalized, or averaged, characteristics of the behaviour of the atmosphere are determined. Numerical experiments of the general circulation of the atmosphere consist of integrating the corresponding equations over a long period of time under idealized initial conditions. Finding a stationary solution, or a solution with an annual period, is a numerical experiment with a climatic background.

Atmospheric processes have a wave structure. The construction of different types of atmospheric waves is accomplished by methods of non-linear mechanics using asymptotic series expansions in powers of a small parameter. The mathematical theory of atmospheric waves for linearized models is well developed. Among the solutions of the hydrodynamic equations several classes of waves are isolated (acoustic, gravitational, Rossby waves). Among the non-linear waves only individual examples have been studied (Gerstner gravitational waves, Rossby non-linear waves, and others). To clarify the structure of atmospheric motions, numerical solution methods are widely used to solve eigen value and eigen function problems for the hydro-thermodynamic operators and their discrete analogues. The mathematical problem of atmospheric acoustics is the study of propagation of waves in stratified media. Here analytic (the WKB-method, etc.) and numerical methods are used. A major role, particularly in connection with the study and simulation of the dynamics of cyclones and fronts in the atmosphere, is taken by the research into hydrodynamic instability of atmospheric waves and the interaction of waves of various scales.

In atmospheric optics specific inverse problems in the class of conditionally well-posed problem arise. A typical example is the problem of recovering the parameters of the atmosphere relative to data from remote sounding by satellites. The process of multiple scattering of light in the atmosphere has been investigated by various asymptotic and numerical methods. For the solution of the radiative-transfer equation in the atmosphere, numerical methods are used. Specially effective is the Monte-Carlo method.

Many theoretical and applied problems of meteorology are connected with the problem of atmospheric turbulence. The spectrum of scales of turbulence in the atmosphere is extremely broad. Turbulence plays a determining role in the interaction of the atmosphere with the ocean and with the Earth's surface, in the diffusion of atmospheric impurities, in bumps of airplanes and other aircrafts, in the vibration of buildings under the pressure of the wind, in fluctuations of light and radio signals from terrestrial and cosmic sources, etc. There are several approaches to the creation of mathematical models of turbulence and to methods for parametrizing it.

In solving meteorological problems, a number of problems arise which are typical of the complicated problems of mathematical physics. First of all, in the preparation of the input data (objective analysis, statistical processing of time series on a network of measurements, space-time analysis and the compatibility of meteorological fields), and also the use of methods of sensitivity theory and optimization to identify the parameters of the model with respect to the measurement data. Mathematical models in meteorology have a large number of degrees of freedom, and therefore the problem of reduction arises (for example, by parametrization or by taking account of the use of informative generalized variables).

Related to the mathematical problems in meteorology are problems connected with the study and estimation of the influence of human activity on the atmosphere. These are problems of simulating the microclimate of towns and industrial areas with due regard for anthropogenic factors, of estimating the pollution of the atmosphere by industrial waste, of estimating the influence of variations in the Earth's surface on the dynamics and thermal behaviour of the atmosphere, etc. The problem of choosing an effective economic policy is formalized by methods of optimization theory, applied to meteorological problems. In particular, the question of optimal siting of economic complexes with due regard for sanitarily permissible norms of environmental pollution can be formalized mathematically as a constrained variational problem.

A complex of mathematical problems, including practically all those listed above, arises in connection with the problem of monitoring or realizing a procedure for tracking atmospheric processes on local and global scales.

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The development of numerical schemes for handling a large amount of information and for solving large systems of coupled differential equations forms the main contribution of mathematics to meteorology. It started with a numerical integration procedure by L.F. Richardson [a9] and has come to its present development at national and international institutes like the National Centre for Atmospheric Research at Boulder (Colorado, USA) and the European Centre for Medium-Range Weather Forecasts at Reading (UK), see [a3]. Recently, methods have been developed to make the data of observations consistent with the model for which the data is meant [a1]. From meteorology mathematicians have learned about an essentially non-linear phenomenon that may occur in systems of three of more coupled ordinary differential equations. That is the possible presence of a strange attractor, which was first observed by E.N. Lorenz [a7] in a heavily truncated spectral model of the local vertical circulation of the atmosphere. Spectral models of the global circulation have since then been studied intensively, see [a4] and [a2] for a survey. The general theory of the physics of the atmosphere is found in [a5] and [a6], while the introduction of J. Pedlosky [a8] is more mathematically oriented. The references [a10] and [a11] deal with boundary-layer phenomena and turbulence in relation with meteorological problems.

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