摘要
本文针对过去动力系统线性误差增长理论研究的不足,利用非线性动力系统的理论和方法引入了非线性误差增长理论;然后利用实际观测资料,结合大气的动力学特征,给出了利用非线性误差增长理论确定大气可预报期限的计算方法;最后分别研究了大气中不同变量场天气可预报性和气候可预报性的时空分布,天气可预报性的年代际变化,以及海温可预报性的时空分布等问题。主要结论如下:
(1)、对非线性系统的误差发展方程不作线性化近似,直接用原始的误差发展方程来研究初始误差的发展,定义了非线性局部Lyapunov指数的概念,引入非线性误差增长理论。根据非线性混沌动力系统的性质和概率理论,证明了混沌系统误差增长的饱和值定理;根据这个定理,对于一个非线性混沌动力系统而言其平均相对误差增长到达饱和后系统的演变进入了随机运动状态,可预报性丧失,因此,非线性动力系统平均相对误差增长达到饱和的时间即为非线性系统可预报性时间。利用这个定理可以定量地确定混沌系统整体的可预报期限,也可以定量地确定混沌系统单个分量的可预报期限。此外,混沌系统可以通过系统局部平均的相对误差增长定量地确定出局部可预报期限的大小。
(2)、一些简单混沌系统的可预报期限与初始误差的对数存在线性关系,线性系数与全局最大Lyapunov指数有关,最大Lyapunov指数越大,则可预报性期限随着初始误差对数的增大线性减少的速率也越大。
(3)、初始误差和参数误差对非线性系统可预报性的影响所起的作用大小主要取决于初始误差和参数误差的相对大小,当初始误差远大于参数误差时,系统的可预报期限主要由初始误差决定;反过来,当参数误差远大于初始误差时,系统的可预报期限主要由参数误差决定;当初始误差和参数误差大小相当时,两者都对系统的可预报期限起重要作用。
(4)、利用实际观测资料,结合大气的动力学特征,给出了利用大气的实际观测资料估计大气可预报期限的计算方法;利用NCEP/NCAR再分析资料,对大气高度场、温度场、纬向风场、经向风场以及垂直速度场等要素场的可预报性的时空分布进行了研究,结果表明对于不同的要素场,其可预报期限的大小以及时空分布规律都不一样;对于同一高度层而言,全球大部分地区高度场和温度场的可预报期限相对较大,纬向风场和经向风场次之,垂直速度场的可预报期限最小。
(5)、在500 hPa高度场上,全年平均的月和季节尺度可预报期限的空间分布都存在明显的南北纬向性差异,其中在热带地区月和季节尺度可预报期限都为最大,而从热带地区到南北半球的中高纬度地区,月和季节尺度可预报期限迅速减少。
(6)、在全球的大部分地区天气可预报性都存在着明显的年代际变化特征。具体来说,在对流层各层(850 hPa、500 hPa和200 hPa)的热带太平洋地区,天气可预报性在1980、90年代,与1950、60年代相比较明显增加,而在热带非洲和大西洋地区天气可预报性在1980、90年代,与1950、60年代相比较明显减小;此外,北半球中高纬度的大部分地区从1950年代至1990年代天气可预报性明显减小,南半球中高纬度的大部分地区天气可预报性明显增加。在平流层的下层(50 hPa),在整个热带地区(30oS~30oN)从1950年代至1990年代可预报期限都明显增加。
(7)、从海温全年平均可预报期限的空间分布来看,可预报期限的最大值出现在热带中东太平洋地区即发生ENSO现象的区域,可预报期限基本上都在8个月以上,最大值达到了11个月以上。此外,在热带印度洋和热带大西洋的大部分地区,可预报期限也相对较高。北太平洋地区、北大西洋地区以及南半球中高纬度海区可预报期限较小。
全球大部分海洋可预报期限都随季节有明显变化,在热带中东太平洋地区和热带东南印度洋地区,可以分别发现春季和冬季可预报性障碍对可预报性有很大的影响。
Because of the limitations of linear error dynamics, nonlinear error dynamics is introduced. By applying the nonlinear error dynamics to the predictability analysis of atmospheric and ocean observation data, some questions including the temporal-spatial distribution of the weather and climate predictability limit of different variables such as geopotential height, temperature etc., the decadal change of the predictability limit, and the temporal-spatial distribution of predictability limit of monthly sea surface temperature, are studied respectively. The major results and conclusions of this study are summarized as follows:
(1) By applying nonlinear error growth equations of nonlinear dynamical systems instead of linear approximation equations to discuss the evolution of initial perturbations, a novel concept of nonlinear local Lyapunov exponent (NLLE) is proposed and nonlinear error dynamics is developed. According to the chaotic dynamical system theory and probability theory, the saturation theorem of mean relative growth of initial error (RGIE) derived by the mean NLLE is proved, that is, for a chaotic dynamic system, the mean RGIE will necessarily reach a saturation value in a finite time interval. Once the mean RGIE reaches the saturation, at the moment almost all predictability of chaotic dynamic systems is lost. Therefore, the predictability limit can be defined as the time at which the mean RGIE reaches its saturation level. By use of this theorem, the average predictability of whole system or its single variable may be obtained quantitatively. In addition, the local average predictability limit of a chaotic system could be quantitatively determined by examining the evolution of the local mean relative growth of initial error (LRGIE).
(2) There exists a linear relationship between the predictability limit of some simple chaotic systems and the logarithm of initial error. Linear coefficient is relevant to the largest global Lyapunov exponent of chaotic system. Greater is the largest global Lyapunov exponent, faster decreases linearly the predictability limit as the logarithm of initial error becomes greater.
(3) The influences of initial error and parameter error on the predictability of chaotic systems depend on their relative sizes. When the size of initial error is far greater than that of parameter error, the predictability limit of chaotic systems mainly depends on initial error. On the contrary, when the size of parameter error is far greater than that of initial error, the predictability limit of chaotic systems mainly depends on parameter error. When the size of initial error is close to that of parameter error, initial error and parameter error together contribute to the predictability limit of chaotic systems.
(4) Based on the atmospheric dynamic features, a reasonable algorithm is provided to obtain the predictability limit of atmosphere by use of atmospheric observation data. By applying the NCEP/NCAR reanalysis data, the temporal-spatial distributions of predictability limit of different variables including geopotential height, temperature, zonal wind, meridional wind, and vertical velocity, are studied respectively. The results showed that for different variables, the temporal-spatial distribution characteristics of predictability limit are also different. For the same pressure level, the predictability limit of geopotential height and temperature is largest in the most regions, that of zonal wind and meridional wind second, and that of vertical velocity is smallest.
(5) At the 500 hPa geopotential height field, the predictability limit of monthly and seasonal scales shows obvious differences between the tropics and middle-high latitudes of southern and northern hemispheres. The predictability limit of monthly and seasonal scales is largest in the tropics, and decreases quickly from the tropics to middle-high latitudes of southern and northern hemispheres.
(6) There exists obvious decadal change for weather predictability in most of the globe. Specifically, at the several pressure levels of troposphere (850 hPa、500 hPa and 200 hPa) in the tropical Pacific, the predictability limit increases obviously in the 1980’s-90’s compared with that in the 1950’s-60’s; on the contrary, in the tropical Africa and tropical Atlantic, the predictability limit decreases obviously in the 1980’s-90’s compared with that in the 1950’s-60’s. In addition, the predictability limit decreases obviously from the 1950’s to the 1990’s in the most regions of northern middle-high latitudes, while opposite change occurs in the most regions of southern middle-high latitudes. In the tropical low stratosphere, predictability limit increases obviously from the 1950’s to the 1990’s.
(7) The predictability limit of monthly sea surface temperature in the tropical central-eastern Pacific has very large value. The predictability limit there is beyond 8 months and the maximum value exceeds 11 months. In addition, the predictability limit in the most regions of tropical Indian and Atlantic oceans has also relatively large value. The predictability limit in the north Pacific, north Atlantic and the middle-high latitude oceans of southern hemisphere has minimum value.
In the most regions of global oceans, the predictability limit varies obviously with season. In the tropical central-eastern Pacific and east-southern Indian Ocean, spring predictability barrier and winter predictability barrier can be found respectively.
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