大气地转静力平衡的方差分析与L分布

Variance analysis of geostrophic-static equilibrium process and L probability distribution function

在对大气地转静力系统平衡问题的研究中,首次引出一种概率分布函数L分布.通过分析得出L分布函数具有明确的数字特征:①它的方差为1/9初始振幅的平方,均方差为1/3初始振幅,数学期望为0;②4阶中心矩为1/25初始振幅的4次方,峰度系数等于正0.24,偏度系数为0;③L分布的m阶矩存在,在正负1/e初始振幅区间内,它的概率是2/e(约74.04%);④在一个均方差区域内的概率是70.0%,在0值附近分布概率最大;⑤L分布比正态分布更集中在它数学期望值附近;峰度系数比正态分布高0.24;⑥最后应用到大气中推出方差无量纲数W.

The new probability function, which is named as L distribution, has been pointed out in this paper by comparing with another famous distribution such as normal （Gaussian） distribution, exponential distribution, uniform distribution, student distribution and so on. The main numerical characteristics of L distribution have been deduced in the paper. For example, mean value and variance etc have already been inferred here. Most physical system of being deviating from its equilibrium state, when in damping process generally should be identified as falling into L distribution. The total average of variable of the distribution is equivalent to zero, at this time this is defined as absolute equilibrium state. But since the function has infinite result at the point θ = 0, which is supposed that the absolute equilibrium state essentially does not occur in this kind of system or in this kind of physical processes. In addition, continuous random variables concentrically （thickly） scatter in area near to zero θ= 0, this is just an obvious character （quality） of L distribution and peculiar of L function differentiating from other renowned distribution. In a word, the new distribution function f（0） = ln（ θM/θ）^2/（4θM） has been discovered in detail by author of this article, its mean square deviation is （θM/3）^2, its mathematical expectation is zero and the coefficient of kurtosis of the distribution function is 0.24. the fourth moment v4 = （ θM）^4/25, finally third moment and coefficient of skew are both zero. Also M-th moment exist, this article has several extension which follows below： The probability is equal 2/e = 74.04% within coverage of 0〈 θ≤θM/el. The probability will be certainly nonexistent in the context of absolute Equilibrium θ= 0. The probability will be biggest within the realm（area） in the vicinity of Quasi- Equilibrium State （ θ→0 = mathematical expectation）. It was new trial that Quasi- Equilibrium state is put into probability for discussion in spite of general analyzed traditionally in dynamical balance way. Along with its application into atmospheric field, the non - dimensional number W is surfaced.