涡旋流自发辐射惯性重力波的初步解析研究
A Preliminary Analytical Study on the Spontaneous Emission of Inertia gravity Waves from Vortical Flows
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摘要: 平衡流(或粗略地说涡旋流)调整目前被认为是惯性重力波产生的一种新的机制并被称为自发辐射问题。基于f平面上正压模式,该文对涡旋流自发辐射惯性重力波这一问题进行了初步解析研究。首先,通过该模式讨论了慢流形和平衡流的概念并强调了平衡流的涡旋运动特征。假定Froude数F<<1且Rossby数ε=O(1)(即近似为平衡流的涡旋流包括了梯度风和其他非地转成分),于是基本方程可以写为有关惯性重力波的非齐次波动方程,其非齐次项与涡旋流的非平衡性质有关,而对于严格平衡的涡旋流,非齐次项消失, 故涡旋流的非平衡性实际上提供了惯性重力波的强迫源, 通过找出该方程的格林函数给出了反映自发辐射的非齐次解。在远离波源的区域,该非齐次解可以展开为远场形式,包括单极、偶极和四极辐射, 另一方面,在波源区域以内及其附近,解的近场形式可以看作是慢流形的近似表达式。结果表明:与涡旋流相伴随的辐合/辐散运动主要由两部分组成,即由它自发辐射出的惯性重力波以及从属于平衡流的缓慢变化的辐合/辐散运动。与Ford的匹配渐近展开方法相比较,尽管不能给出更高阶近似,但格林函数法得到的非齐次解形式,在描写自发辐射方面更具物理直观性。Abstract: The emission of inertia gravity waves from the adjustment of balanced flows (or, basically, vortical flows) is currently regarded as a new mechanism of the pro duction of inertia gravity waves and referred as the spontaneous emission. Base d on f plane barotropic model, a preliminary analytical study on the spontaneou s emission of inertia gravity waves from vortical flows is conducted. First of all, the concept of slow manifold and balanced flow are discussed via this model and vortical property of the balanced flow is emphasized. Then, by assuming tha t the Froude number F and Rossby number satisfy and (implying that vortic al flow approximately balanced flow that includes gradient wind and other ageost rophic contributions), the basic equations are simplified to an inhomogeneous wa ve equation of inertia gravity wave, while the inhomogeneous term is related to the imbalance of the vortical flow. This inhomogeneous term vanishes when vorti cal flow is exactly balanced. So the imbalance of the vortical flow provides a f orcing or source to the inertia gravity wave. The Green function of this equa tion is found to give the inhomogeneous solution standing for the spontaneous em ission. In the field far from the wave "source", this inhomogeneous solution c an be expanded into far field form, including wave emission of monopole, dipole and quadrupole types. On the other hand, within or near to the "source" field, the near field form can serve as an approximate expression of the slow manifo ld. These results indicate that the convergence/divergence fields accompanied wi th vortical flows is composed of two main parts, i.e., the spontaneous emission o f inertia gravity waves from vortical flows and slow varying convergence/diver gence filed slaved to the adjustment of balanced flow. Although higher order ap proximation solution has already been given by Ford (2000) using method of match ed asymptotic expansions, the Green function solution can depict the spontaneo us emission in a more physical way.
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图 2 反映格林函数时空结构的等值线图
(已经进行了无量纲化,仅在负位相区域 (阴影区) 画出等值线0.000,-0.025,-0.050,-0.075,-0.125,-0.20,-0.30,-0.40,-0.50)
Fig. 2 The contour map of the non-dimensio nalizedspatio-tem poral struc ture of the Green function
(contour lines are drawn only in the negative phase (shadows) at 0.000, -0.025, -0.050, -0.075, -0.125, -0.20, -0.30, -0.40, -0.50)
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[1] Dunkerton T J.The role of gravity waves in the quasi-biennial oscillation.Journal of Geophysical Research, 1997, 102 :26053-26076. doi: 10.1029/96JD02999 [2] Holton J R.Role of gravity-wave-induced drag and diffusion in the momentum budget of the mesosphere.Journal of the Atmospheric Sciences, 1982, 39 :791-799. doi: 10.1175/1520-0469(1982)039<0791:TROGWI>2.0.CO;2 [3] 王晓芳, 崔春光, 胡伯戚.与水平风切变强度不均匀相联系的CISK惯性重力波.应用气象学报, 2007, 18(6):760-768. http://qikan.camscma.cn/jams/ch/reader/view_abstract.aspx?file_no=200706116&flag=1 [4] 吴洪, 林锦瑞.垂直切变基流中东西向地形对惯性重力波稳定性的影响.应用气象学报, 1997, 8(2):242-246. http://qikan.camscma.cn/jams/ch/reader/view_abstract.aspx?file_no=19970233&flag=1 [5] 王永中, 夏友龙.CISK影响下的线性和非线性惯性重力波.应用气象学报, 1996, 7(1):82-88. http://qikan.camscma.cn/jams/ch/reader/view_abstract.aspx?file_no=19960111&flag=1 [6] 赵南, 沈新勇, 丁一汇.大气运动的慢流形概论.地球科学进展, 2007, 22(4):331-340. http://www.cnki.com.cn/Article/CJFDTOTAL-DXJZ200704000.htm [7] Lorenz E N.Attractor sets and quasi-geostrophic equilibrium.Journal of the Atmospheric Sciences, 1980, 37 :1685-1699. doi: 10.1175/1520-0469(1980)037<1685:ASAQGE>2.0.CO;2 [8] Lorenz E N.On the existence of a slow manifold.Journal of the Atmospheric Sciences, 1986, 43 :1547-1557. doi: 10.1175/1520-0469(1986)043<1547:OTEOAS>2.0.CO;2 [9] Lorenz E N, Krishnamurthy V.On the nonexistence of a slow manifold.Journal of the Atmospheric Sciences, 1987, 44 :2940-2950. doi: 10.1175/1520-0469(1987)044<2940:OTNOAS>2.0.CO;2 [10] Vautard R, Legras B.Invariant manifolds, quasi-geostrophy and initialization.Journal of the Atmospheric Sciences, 1986, 43 :565-584. doi: 10.1175/1520-0469(1986)043<0565:IMQGAI>2.0.CO;2 [11] Lorenz E N.The slow manifold-Whet is it.Journal of the Atmospheric Sciences, 1992, 49 :2449-2451. doi: 10.1175/1520-0469(1992)049<2449:TSMII>2.0.CO;2 [12] Ford R, McIntyre M E, Norton W A.Balance and the slow quasimanifold:Some explicit results.Journal of the Atmospheric Sciences, 2000, 57 :1236-1254. doi: 10.1175/1520-0469(2000)057<1236:BATSQS>2.0.CO;2 [13] McIntyre M E, Balance.Potential-vorticity Inversion, Lighhill Radiation, and the Slow Quasimanifold∥Proc IUTAM/IUGG/ Royal I rish Academy Symposium on Ad vanced in Math ematical Modelling of Atmos phere and Ocean.Ireland, 2000. [14] Warn T, Bokhove O, Shepherd T G, et al.Rossby number expansions, slaving principles, and balance dynamics, Quarterly.Journal of the Royal Meteorological Society, 1995, 121 :723-739. doi: 10.1002/(ISSN)1477-870X [15] Vallis G K.Potential vorticity and balanced equation of motion for rotating and stratified flows.Quarterly Journal of the Royal Meteorological Society, 1996, 122 :291-322. doi: 10.1002/(ISSN)1477-870X [16] Lighthill M J.On Sound Generated Aerodynamically, I:General Theory∥Proceedings of the Royal Society of London.1952, A211 :564-587. [17] 郭敦仁.数学物理方法.北京:人民教育出版社, 1979. [18] 路季平.积分变换及在物理海洋学中的应用.北京:海洋出版社, 1984. [19] 刘式适, 刘式达.特殊函数.北京:气象出版社, 1988.