本文選題：非靜力模式 + 譜方法　； 參考：《國防科學技術大學》2015年博士論文

【摘要】：數值天氣預報在過去幾十年的發(fā)展中一直采用一個重要的近似:靜力近似。隨著計算機集群計算能力不斷提升,數值預報模式的分辨率也越來越高,能夠為用戶提供更加精確的預報服務。一般而言,當水平分辨率優(yōu)于10km時,靜力近似已經不適用了。因此開發(fā)非靜力模式是未來數值預報系統(tǒng)發(fā)展的迫切需求。相比靜力模式,一方面非靜力模式要增加預報變量,方程的形式更加復雜;另一方面非靜力模式水平網格尺度更細,為了能夠準確地顯示高分辨率下小尺度地形引起的局部氣象要素,需要很多特定的數值處理技術。本文針對現有非靜力模式缺乏高精度垂直離散格式、模式誤差增長估計手段不足等問題,在已有的數值方法基礎上,對全球非靜力譜模式絕熱動力框架若干關鍵問題進行深入研究,主要工作包括:1.實現有限差分二維X-Z平面譜模式。二維X-Z平面模式是數值預報模式的研究模式,其方程組和數值離散要點同三維有限區(qū)域預報模式保持一致。對垂直有限差分離散格式進行了精巧設計,使得其滿足非靜力模式的垂直約束。模式采用半隱式半拉格朗日時間平流離散格式,同時也對側邊界、上下邊界條件分別進行了處理以保證計算結果的正確。在二維模式下,測試了四種山峰波測試樣例以及重力波樣例,驗證了模式采用的數值離散格式的準確性和穩(wěn)定性。2.實現垂直混合有限元有限差分二維X-Z平面模式。基于質量坐標的非靜力模式垂直離散存在一系列的約束條件,應用有限元垂直離散被證明是難以實現的。本文提出了一個高精度混合離散格式,對半隱式格式的線性部分進行有限差分離散,對非線性部分進行有限元離散。同時為提高有限差分離散的精度,增加線性部分計算的分層數量。有限元采用四階B-spline樣條基函數。為方便處理垂直邊界,先把垂直場投射到三階樣條插值函數空間形成多項式分段函數,在處理完邊界后再投射到B-spline空間。如此避免了B-spline樣條基函數直接處理邊界,使得有限元計算更加簡潔高效。對二維有限元模式也進行了多種測試樣例,并同有限差分進行了定性和定量的對比,結果表明混合有限元格式在精度上要優(yōu)于有限差分格式。3.實現三維全球非靜力譜模式。三維球面模式和二維模式的動力框架基本一致,但采用的譜方法基函數有區(qū)別,球面譜模式采用球面諧波函數為基函數。球面模式水平網格采用精簡高斯網格,模式方程的曲率項通過半拉格朗日插值時方向轉換矩陣引入。科氏力項采用的是隱式處理形式。對三維模式進行了一系列的測試,包括穩(wěn)定狀態(tài)測試、斜壓不穩(wěn)定波測試、Rossby-Haurwitz波測試、山峰波測試等。測試結果驗證了三維模式的準確性和穩(wěn)定性。4.研究混沌模型的初始誤差增長特性,發(fā)現混沌模型的平均相對誤差飽和值同初始誤差值之間存在一個簡單的對數線性關系:二者的對數和為常數。利用該關系,提出平均絕對誤差的概念,混沌系統(tǒng)的平均絕對誤差飽和值為與初始誤差無關的常數。利用這一特性,給出一個定量計算可預報期限的數學估計模型。研究了混沌系統(tǒng)模型誤差增長特性,發(fā)現模型平均絕對誤差增長在短期內呈指數增長,一段時間后達到飽和。可利用模型誤差增長指數大小和達到飽和值的時間作為衡量模型優(yōu)劣的評價手段。
[Abstract]:The numerical weather forecast has been using an important approximation in the development of the past several decades: the static approximation. With the increasing computing power of the computer cluster, the resolution of the numerical prediction model is also getting higher and higher, which can provide more accurate prediction service for the users. Generally, when the horizontal resolution is better than 10km, the static approximation has been approximated. It is not applicable. Therefore, the development of non static model is an urgent need for the development of the future numerical forecast system. Compared with the static model, the non hydrostatic model should increase the prediction variable and the form of the equation more complex. On the other hand, the scale of the non static model is finer, in order to accurately display the small scale terrain under high resolution. The local meteorological elements need a lot of specific numerical processing techniques. In this paper, in view of the lack of high precision vertical discrete schemes and the lack of model error growth estimation methods, the main problems of the key problems of the global non static spectral model thermal dynamic force framework are studied in this paper. The work includes: 1. to realize the finite difference two-dimensional X-Z plane spectrum pattern. The two-dimensional X-Z plane model is the model of the numerical prediction model. The equations and the numerical discrete points are consistent with the three-dimensional finite area prediction model. The vertical finite difference dispersion scheme is designed so that it satisfies the vertical constraint of the non static model. The semi implicit semi Lagrange time advection scheme is adopted in the formula, while the side boundary and the upper and lower boundary conditions are treated respectively to ensure the correctness of the calculation results. In the two-dimensional model, four peak wave samples and gravity wave examples are tested. The accuracy and stability of the numerical discrete scheme used in the model are verified.2. real. The vertical mixed finite difference two-dimensional X-Z plane model is presented. There is a series of constraints on the vertical discrete model of the non static mode based on the mass coordinates. The application of the finite element vertical dispersion is proved to be difficult to realize. A high precision mixed discrete scheme is proposed in this paper, and the finite difference dispersion is carried out for the linear part of the semi implicit scheme. In order to improve the precision of the finite difference separation and increase the accuracy of the finite difference separation dispersion and increase the number of layers calculated by the linear part, the finite element uses the four order B-spline spline basis function. In order to handle the vertical boundary, the vertical field is first projected to the three order spline interpolation function space to form a polynomial piecewise function, after the processing of the boundary. Then it is projected into the B-spline space, so that the B-spline spline basis function is avoided directly and the finite element calculation is more concise and efficient. A variety of test examples are also carried out for the two-dimensional finite element model, and the qualitative and quantitative comparison with the finite difference is made. The results show that the hybrid finite element scheme is better than the finite difference. The three-dimensional global non static spectral model is realized by format.3.. The three-dimensional spherical model is basically the same as the dynamic frame of the two-dimensional model, but the basis function of the spectral method is different. The spherical spectral model uses the spherical harmonic function as the base function. The spherical pattern horizontal grid uses the simplification of the Gauss grid, the curvature of the mode equation is interpolated by half Lagrange interpolation The time direction transformation matrix is introduced. The Coriolis force term is used in implicit treatment. A series of tests are carried out on the three-dimensional model, including stable state test, baroclinic unstable wave test, Rossby-Haurwitz wave test, peak wave test and so on. The test results verify the accuracy and stability of the three-dimensional model and the initial error of the chaotic model of the.4. model. It is found that there is a simple logarithmic linear relationship between the average relative error saturation value of the chaotic model and the initial error value: the logarithm and the constant of the two is a constant. Using this relation, the concept of the mean absolute error is proposed. The mean absolute error saturation value of the chaotic system is a constant independent of the initial error. A mathematical estimation model is given for the quantitative calculation of the predictable term. The error growth characteristic of the chaotic system model is studied. It is found that the average absolute error growth of the model increases exponentially in the short term and reaches the saturation after a period of time. The time of the model error growth index to the saturation value can be used as the evaluation of the model. Price means.
【學位授予單位】：國防科學技術大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：P456.7;O241.8

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