Nonlinear Error Dynamics And Predictability Study

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.