The initial-value problemweather prediction is a classical deterministic problem in classical physics,there are a lot of difficulties in the method of initial-value problemto predict long-rangeweather and short-range climate.Three characteristics of atmospheric motion are very important:nonlinearity,being forced and dissipation.It is necessaryto combine the dynamical and statistical methods.The prediction problemcould be looked as the inverse problemofthe initial-value problem,i.e. an evolution problem,sothe multiple-time data in the past could be used to modify initial values and/or numerical prediction model.The inverse problems are divided into there kinds,there are corresponding solutions.After the inverse problemof the initial-value problem is introduced,the global analysis to the atmospheric dynamical equations(ADE) are reviewed. The global analysis is anothermethod of climate research different fromthe simple initialvalue problem. The global analysis is also called global asymptotic analysis.It is used to study the long-range behavior of deterministic systems.It is unnecessary to solve the equations,the properties ofADE are researched by means of studying the equations themselves.With the aid of the direct geometric pictures(the concept of phase space is introduced),the
qualitative global analysis to the limit solution set of ADE could be done.It is proved that ADE is a special operator equation in Hilbert Space.The properties of operators must be maintainedwhenADE is simplified or discretized.It is revealed that the nonlinear adaptation process of a system to external source,so the system could be described by a small number of degrees of freedomsustaining attractors because of dissipative process.Cell-to-cell mapping is a powerful tool to
study nonlinear systems,by use of the concept and method of cell-to-cell mapping,the global analysis to the general characteristics of numerical model of atmospheric system could be done.The global analysis to round-off errors and observational errors could be done.With the aid of the language of probabilitytheory,the statistical description of the characteristics could be given.It is revealed that the indeterminacy of individual state and the determinacy of global state.The restrict mathematical definition of the vague concept of“CLIMATE”is:climate could be defined as probability distribution function of state variables in chaotic attractors determined by controlling variables,and it is proved that it exists chaotic state in climate systemreally.The approach of quantitative studies to the predictability of the atmosphere could be given,the predictability is the time before the state of a system arrive the probability distribution of chaotic attractors.