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以差分方程代替微分方程给大气原始方程组求解带来了诸多难以解决的问题, 对于(半)拉格朗日模式来说质点轨迹的计算与Helmholtz方程的求解是两大难题. 本文通过对气压变量代换, 并在积分时间步长内将原始方程组线性化, 近似为常微分方程组, 求出方程组的半解析解, 再采用精细积分法求解半解析解. 半解析方法可同时计算风、气压和位移, 无需求解Helmholtz方程, 质点的位移采用积分风的半解析解得到, 相比采用风速外推的计算方法, 半解析方法更科学合理. 非线性密度流试验检验表明: 半解析模式能够清晰地模拟Kelvin-Helmholtz 切变不稳定涡旋的发生和发展过程; 模拟的气压场和风场环流结构与标准解非常相似, 且数值解是收敛的, 同时, 总质量和总能量具有较好的守恒性. 试验初步证明了采用半解析方法求解大气原始方程组是可行的, 为大气数值模式的构建提供了一个新的思路.To solve atmospheric primitive equations, the finite difference approach would result in numerous problems, compared to the differential equations. Taking the semi-Lagrange model as an example, there exist two difficult problemsthe particle trajectory computation and the solutions of the Helmholtz equations. In this study, based on the substitution of atmosphere pressure, the atmospheric primitive equations are linearized within an integral time step, which are broadly seen as ordinary differential equations and can be derived as semi-analytical solutions (SASs). The variables of SASs are continuous functions of time and discretized in a special direction, so the gradient and divergence terms are solved by the difference method. Since the numerical solution of the SASs can be calculated via a highly precise numerical computational method of exponential matrixthe precise integration method, the numerical solution of SASs at any time in the future can be obtained via step-by-step integration procedure. For the SAS methodology, the pressure, as well as the wind vector and displacement, can be obtained without solving the Helmholtz formulations. Compared to the extrapolated method, the SAS is more reasonable as the displacements of the particle are solved via time integration. In order to test the validity of the algorithms, the SAS model is constructed and the same experiment of a non-linear density current as reported by Straka in 1993 is implemented, which contains non-linear dynamics, transient features and fine-scale structures of the fluid flow. The results of the experiment with 50 m spatial resolution show that the SAS model can capture the characters of generation and development process of the Kelvin-Helmholtz shear instability vortex; the structures of the perturbation potential temperature field are very close to the benchmark solutions given by Straka, as well as the structures of the simulated atmosphere pressure and wind field. To further test the convergence of the numerical solution of the SAS model, the 100 m spatial resolution experiment of the non-linear density current is also implemented for comparison. Although the results from both experiments are similar, the former one is better and the property of mass-energy conservation is comparatively reasonable, and furthermore, the SAS model has a convergent property in the numerical solutions. Therefore, the SAS method is a new tool with efficiency for solving the atmospheric primitive equations.
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Keywords:
- atmospheric primitive equations /
- semi-analytical method /
- precise integration method /
- density current
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[1] Ji L R, Chen J B, Zhang D M 2005 Chin. J. Atmos. Sci. 29 122 (in Chinese) [纪立人, 陈嘉滨, 张道民 2005 大气科学 29 122]
[2] Gu X Z, Zhang B 2008 Plateau Meteorol. 27 474 (in Chinese) [辜旭赞, 张兵 2008 高原气象 27 474]
[3] Gu X Z, Zhang B 2008 Plateau Meteorol. 27 481 (in Chinese) [辜旭赞, 张兵 2008 高原气象 27 481]
[4] Robert A 1982 J. Meteorol. Soc. Japan 60 319
[5] Mcdonald A 1986 Mon. Wea. Rev. 114 824
[6] Hortal M 2002 Q. J. R. Meteorol. Soc. 128 1671
[7] Chen D H, Xue J S, Yang X S, Zhang H L, Shen X S, Hu J L, Wang Y, Ji L R, Chen J B 2008 Chin. Sci. Bull. 53 2396 (in Chinese) [陈德辉, 薛纪善, 杨学胜, 张红亮, 沈学顺, 胡江林, 王雨, 纪立人, 陈嘉滨 2008 科学通报 53 2396]
[8] Staniforth A, White A, Wood N 2003 Q. J. R. Meteorol. Soc. 129 2065
[9] Staniforth A, Wood N 2008 J. Comput. Phys. 227 345
[10] Wood N, White A, Staniforth A 2010 Q. J. R. Meteorol. Soc. 136 507
[11] Zhong W X, Zhu J P 1995 Appl. Math. Mech. 16 663 (in Chinese) [钟万勰, 朱建平 1995 应用数学和力学 16 663]
[12] Zhong W X 1996 Acta Mech. Sin. 28 159 (in Chinese) [钟万勰 1996 力学学报 28 159]
[13] Sun J Q, Qin M Z 2007 Math. Numer. Sin. 29 67 (in Chinese) [孙建强, 秦孟兆 2007 计算数学 29 67]
[14] Fu M H, Lan L H, Lu K L, Zhang W Z 2012 Sci. Chin. : Phys. Mech. Astron. 42 185 (in Chinese) [富明慧, 蓝林华, 陆克浪, 张文志 2012 中国科学:物理学, 力学, 天文学 42 185]
[15] Lv H X, Yu H J, Qiu C H 2001 Appl. Math. Mech. 22 151 (in Chinese) [吕和祥, 于洪洁, 裘春航 2001 应用数学和力学 22 151]
[16] Lv H X, Cai Z Q, Qiu C H 2001 Chin. J. Appl. Mech. 18 34 (in Chinese) [吕和祥, 蔡志勤, 裘春航 2001 应用力学学报 18 34]
[17] Fan J P, Tao H, Tan C Y 2006 Acta Mech. Solid Sin. 9 289
[18] Tan S J, Gao Q, Zhong W X 2010 Chin. J. Comput. Mech. 27 752 (in Chinese) [谭述君, 高强, 钟万勰 2010 计算力学学报 27 752]
[19] Wang R Q, Li L L, Li H J 2010 Chin. J. Geophys. 53 1875 (in Chinese) [王润秋, 李兰兰, 李会俭 2010 地球 53 1875]
[20] Duan Y T, Hu T Y, Yao F C 2013 Appl. Geophys. 10 71
[21] Han M J, Ke D M, Chi X L, Wang M, Wang B T 2013 Acta Phys. Sin. 62 098502(in Chinese) [韩名君, 柯导明, 迟晓丽, 王敏, 王保童 2013 62 098502]
[22] Shi B R 2010 Chin. Phys. B 19 65202
[23] Fu M H, Zhang W Z 2011 Chin. J. Comput. Mech. 28 529 (in Chinese) [富明慧, 张文志 2011 计算力学学报 28 529]
[24] Fu M H, Zhang W Z 2010 Chin. J. Appl. Mech. 27 688 (in Chinese) [富明慧, 张文志 2010 应用力学学报 27 688]
[25] Zheng W, Xu H Z, Zhong M, Yuan M J 2009 Chin. Phys. B 193597
[26] Liu F, Chao J P, Huang G, Feng L C 2011 Chin. Sci. Bull. 56 2727
[27] Hao S F, Yang S F, Lou M Y 2014 Chin. J. Geophys. 57 2190 (in Chinese) [郝世峰, 杨诗芳, 楼茂园 2014 地球 57 2190]
[28] Hao S F, Cui X P 2012 Acta Phys. Sin. 61 039204(in Chinese) [郝世峰, 崔晓鹏 2012 61 039204]
[29] Xu Q, Xue M, Droegemerier K K 1996 J. Atmos. Sci. 53 770
[30] Xue M, Xu Q, Droegemerier K K 1997 J. Atmos. Sci. 54 1998
[31] Yang X S, Hu J L, Chen D H, Zhang H L, Shen X S, Chen J B 2008 Chin. Sci. Bull. 53 2418 (in Chinese) [杨学胜, 胡江林, 陈德辉, 张红亮, 沈学顺, 陈嘉滨, 纪立人 2008 科学通报 53 2418]
[32] Straka J M, Wilhelmson R B, Wicker L J, Anderson J R, Droegemerier K K 1993 Int. J. Num. Methods Fluids 17 1
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