强迫和耗散作用下大气运动的非线性正压不稳定

本文从Charney研究多平衡态的包含摩擦耗散、热力强迫和大地形的非线性准地转正压涡度方程出发,应用Serrin-Joseph的能量方法,利用变分原理,分别用总能景和总位涡拟能导得纬向基流的非线性正压稳定性判据。      

Variational principle of instability of atmospheric motions

Problems of instability of rotating atmospheric motions are investigated by using nonlinear governing equations and the variational principle. The method suggested in this paper is universal for obtaining criteria of instability in all models with all possible basic flows. For example, the model can be barotropic or baroclinic, layer or continuous, quasi-geostrophic or primitive equations; the basic flow can be zonal or nonzonal, steady or unsteady.Although the basic flows possess a great deal of variety, they all are the stationary points in the functional space determined by an appropriate invariant functional. The basic flow is an unsteady one if the conservation of angular momentum is included in the associated functional.The second variation, linear or nonlinear, gives the criteria of instability. Especially, the general criteria of instability for unsteady basic flow, orographically disturbed flow as well as nongeostrophic flow are first obtained by the method described in this paper.It is also shown that the difference between the criteria of instability obtained by the linear theory and our variational principle clearly indicates the importance of using nonlinear governing equations.In the appendix the theory is extended to cases such as in a β-plane where the fluid does not possess finite total energy, hence the variational principle can not be directly applied. However, a generalized Liapounoff norm can still be obtained on the basis of variational consideration.     

LINEAR AND NONLINEAR BAROTROPIC INSTABILITY

Based on a non-frictional and non-divergent nonlinear barotropic vorticity equation and its solutions oftravelling waves,the criteria for linear and nonlinear barotropic instability are gained respectively at an equilibriumpoint of the equation on a phase plane.The linear and nonlinear analytical solutions to instability waves arealso found.The computational results show that if their amplitudes are equal at the initial time,the amplitudeincrements of nonlinear instable barotropic wave are always less than those of linear instable barotropic wave.The nonlinear effects can slow down the exponential growth of linear instability.The time needed for makingthe amplitude double that of initial time by instabilities,is about 6h for linear instability and about 18h fornonlinear instability,the latter is in agreement with the observations in the real atmosphere.     

Effect of Baroclinicity on Vortex Axisymmetrization. Part Ⅰ:Barotropic Basic Vortex

The barotropic and baroclinic disturbances axisymmetrized by the barotropic basic vortex are examined in an idealized modeling framework consisting of two layers. Using a Wentzel-Kramers-Brillouin approach, the radial propagation of a baroclinic disturbance is shown to be slower than a barotropic disturbance, resulting in a slower linear axisymmetrization for baroclinic disturbances. The slower-propagating baroclinic waves also cause more baroclinic asymmetric kinetic energy to be transferred directly to the barotropic symmetric vortex than from barotropic disturbances, resulting in a faster axisymmetrization process in the nonlinear baroclinic wave case than in the nonlinear barotropic wave case.     

Effect of Baroclinicity on Vortex Axisymmetrization.Part Ⅰ：Barotropic Basic Vortex

The barotropic and baroclinic disturbances axisymmetrized by the barotropic basic vortex are examined in an idealized modeling framework consisting of two layers.Using a Wentzel-Kramers-Brillouin approach,the radial propagation of a baroclinic disturbance is shown to be slower than a barotropic disturbance,resulting in a slower linear axisymmetrization for baroclinic disturbances.The slower-propagating baroclinic waves also cause more baroclinic asymmetric kinetic energy to be transferred directly to the barotropic symmetric vortex than from barotropic disturbances,resulting in a faster axisymmetrization process in the nonlinear baroclinic wave case than in the nonlinear barotropic wave case.     

ENERGY,ENERGY FLUX AND LAGRANGIAN OF ROSSBY WAVE

The energy flux derived from the barotropic vorticity equation differs from that obtained directly from the momentum equation.We re-study this problem raised in the early 1960s.The results show that if the momentum equation is rewritten in such a way that it contains the same conditions as that for the barotropic vorticity equation,then the same form of average energy flux can be obtained for the waves with constant amplitudes.With this new momentum equation,the potential energy of Rossby wave is derived and Lagrangian of nonlinear barotropic vorticity equation can be approximately found with this potential energy.This provides a physical basis for studying the dynamics of nonlinear Rossby wave with the approach of calculus of variation.     

Nonlinear meridional scale interaction in baroclinic instability

Baroclinic instability of a zonal flow with latitudinal structure is examined using a nonlinear quasi-geostrophic, two-level β-plane model. An initially small perturbation with the structure of the linearly most unstable mode is allowed to grow to finite amplitude through nonlinear interaction. Because of latitudinal asymmetries of the basic zonal flow, a spectrum of meridional modes is generated in the perturbation. The time evolution of zonal wind and perturbation meridional structures, and their Fourier meridional mode spectra are examined. The radius of deformation is an important meridional scale in both the zonal flow and perturbation. This is especially true during the barotropic decay phase of the baroclinic wave. Time series of energy conversion terms show there is no energy accumulation.     

Criteria for the stability of steady flows induced by diabatic heating

By use of the Liapunov direct method, criteria for the stability of a general steady flow induced by heating are derived for a barotropic model with heating and dissipation. Among these criteria, necessary conditions for the instabilities are identified with the rises of supercritically high speed instability, barotropic instability and some other instabilities.     

亚洲夏季风结构和变动与大气运动流场的斜压和正压特征:正压模分析

使用ＥＣＭＷＦ１９８０—１９８６年７ａ格点资料对大气运动的正压模进行了分析。指出：对流层中大气正压运动流场显示的副热带高压带仅能反映出行星风带的季节性移动；与亚洲夏季风有关且反映季风局地性的则是副热带高压带南侧的东风带上的波状扰动；东风带上的经向风分量存在着纬向传播且传播方向和扰动幅度与印度、东亚季风子系统有关；随着北半球夏季的到来，亚洲季风区大气运动的斜压模有较大的增长且斜压运动动能占气柱内整层平均总动能的绝大部分，而在赤道附近的其他经度上则是正压运动动能的成分明显增长。     

Nonlinear energy and enstrophy transfers in a realistically stratified ocean

We discuss the nonlinear transfers possible in a quasigeostrophic fluid with a basic stratification taken from oceanic data. The energy and enstrophy conservation laws imply a cascade of energy to larger total scale (including both the horizontal scale, defined as wavelength/2π, and the deformation radius of the vertical mode). The triplet interactions among components with various horizontal scales and vertical structures, represented by the vertical mode numbers, are considered in detail for exchanges involving the barotropic and first three baroclinic modes. The initial transfer rates from one component into the other two are estimated and the most rapid transfers described as a function of the initial scale and mode number. These results suggest that barotropic motions will cascade to larger-scale barotropic motions, first baroclinic small-scale motions will transfer to first baroclinic larger scales, and first baroclinic large-scale motions will cascade to barotropic and first baroclinic motions at the deformation scale. Second and third mode motions prefer to transfer energy into small-scale (second or third mode deformation radius) first and third baroclinic mode motions.We also show the relationship of these triplet interactions to Rossby wave instabilities and resonant triads. For the latter motions, the weakness of the nonlinearity adds additional constraints which impty that the motions will tend to become zonal.     

非线性正压不稳定

本文从非线性正压涡度方程出发,考虑了行波一类的流动,并导得一个单自由度的二阶自治动力系统的常微分方程。应用常微分方程的定性理论和几何理论,在相平面的平衡点附近,对积分曲线的几何拓扑结构作了定性分析,对于两个平衡点分别导得了非线性正压不稳定的判据。还分析了整个相平面上运动的全局特征和平衡状态与参数的关系,更直观地得到与前面一致的不稳定判据。     

Variations in Wave Energy and Amplitudes along the Energy Dispersion Paths of Nonstationary Barotropic Rossby Waves

The variations in the wave energy and the amplitude along the energy dispersion paths of the barotropic Rossby waves in zonally symmetric basic flow are studied by solving the wave energy equation,which expresses that the wave energy variability is determined by the divergence of the group velocity and the energy budget from the basic flow.The results suggest that both the wave energy and the amplitude of a leading wave increase significantly in the propagating region that is located south of the jet axis and enclosed by a southern critical line and a northern turning latitude.The leading wave gains the barotropic energy from the basic flow by eddy activities.The amplitude continuously climbs up a peak at the turning latitude due to increasing wave energy and enlarging horizontal scale(shrinking total wavenumber).Both the wave energy and the amplitude eventually decrease when the trailing wave continuously approaches southward to the critical line.The trailing wave decays and its energy is continuously absorbed by the basic flow.Furthermore,both the wave energy and the amplitude oscillate with a limited range in the propagating region that is located near the jet axis and enclosed by two turning latitudes.Both the leading and trailing waves neither develop nor decay significantly.The jet works as a waveguide to allow the waves to propagate a long distance.     

Variations in Amplitudes and Wave Energy along the Energy Dispersion Paths for Rossby Waves in the Quasigeostrophic Barotropic Model

Variations in wave energy and amplitude for Rossby waves are investigated by solving the wave energy equation for the quasigeostrophic barotropic potential vorticity model.The results suggest that compared with rays in the nondivergent barotropic model,rays in the divergent model can have enhanced meridional and zonal propagation,accompanied by a more dramatic variability in both wave energy and amplitude,which is caused by introducing the divergence effect of the free surface in the quasigeostrophic model.For rays propagating in a region enclosed by a turning latitude and a critical latitude,the wave energy approaches the maximum value inside the region,while the amplitude approaches the maximum at the turning latitude.Waves can develop when both the wave energy and amplitude increase.For rays propagating in a region enclosed by two turning latitudes,the wave energy approaches the minimum value at one turning latitude and the maximum value at the other latitude,while the total wavenumber approaches the maximum value inside the region.The resulting amplitude increases if the total wavenumber decreases or the wave energy increases more significantly and decreases if the total wavenumber increases or the wave energy decreases more significantly.The matched roles of the energy from the basic flow and the divergence of the group velocity contribute to the slightly oscillating wave energy,which causes a slightly oscillating amplitude as well as the slightly oscillating total wavenumber.     

Haurwitz solutions of the nonlinear shallow-water equations for small froude number

Summary In this note we find special solutions of the nonlinear shallow-water equations. From the first integrals of the potential vorticity and energy equations for steady flow we derive a single equation in the streamfunction. In the limiting case of very small Froude number, that equation has no solutions corresponding to gravity waves. Under a nonlinear transformation of dependent variable, it becomes a linear equation whose solutions are related to Haurwitz's solutions for nondivergent barotropic flow in spherical surfaces. The distinguishing feature of these solutions is that the streamlines coincide with contours of the free surface: thus, although the height of the free surface varies, the motions of the fluid are horizontal, and the flow is nondivergent.The solutions are easily modified to correspond to Rossby waves propagating eastward or westward without change of shape.The National Center for Atmospheric Research is sponsored by the National Science Foundation.This paper is dedicated to the memory of my scientific mentor and old friend, Bernhard Haurwitz.     

北半球中高纬月平均环流正压斜压动能的年变化特征

本文利用多年月平均资料计算了北半球中高纬月平均环流正压、斜压动能的年变化特征。结果表明无论正压、斜压动能都具有明显的年变化,而且两者的变化趋势也是一致的,但是正压动能要比斜压动能对总动能的贡献大得多。环流异常具有相当正压的垂直结构。有关结论可为长期数值预报模式的简化提供依据。     

亚洲季风变动与大气正压/斜压运动动能变化的气候特征

使用ＮＣＥＰ／ＮＣＡＲ４０年（１９５８～１９９７年）月平均再分析资料,通过动力学论断研究了大气斜压／正压运动动能的变化及其相互转换,分析了亚洲季风变动与这两种动能变化的联系。指出：季风区大气运动动能的组成和变化具有独特的特征。冬季风时期大气斜压运动动能与正压运动动能具有正相关线性关系,斜压运动能向正压运动动能转换;春、秋季无论是东亚还是印度季风区斜压运动动能与正压运动动能之间转换都处于极小值,只是     

30—50 天振荡动能特性及其与平均气流正压不稳定能的转换

该文利用冬季500 hPa的欧洲中心（ECMWF）网格点逐日资料，分析了30—50天振荡的E矢量分布、动能特性及平均气流的正压不稳定能转换特征，从而得到:30—50天振荡的能量传播与西风急流的位置有密切的关系，在西风大风速区作纬向能量传播，在小风速区作指向赤道的经向传播；在急流的出口区有较强的正压能转换，低频振荡从基本流中获得能量，使这里的低频动能最大，并表现出较强的正压特性，与低纬度的斜压特性形成鲜明对照。     

南海夏季风爆发前后扰动演变及其数值研究

以球面正压涡度方程为数学模型，建立了线性和非线性的数值积分模式，通过在模式中设置不同的基本流场和初始扰动场，研究基流和初始场对扰动发展的作用，揭示在球面正压大气中扰动发展的动力学机制。数值试验结果表明:在线性模式中，扰动的移动和发展与基流的分布有着很密切的关系，基流影响着扰动纬向传播的速度和方向； 在非线性模式中，当基流稳定时，扰动的移动以及传播与线性模式的结果相同，但与线性情况的最大区别在于，此时扰动能量的增长存在上限。同时发现，扰动的发展既依赖于基本气流的分布，也依赖初始扰动的结构；南海夏季风爆发前后的基本流场是正压不稳定的，且这种不稳定在季风爆发时达到最强，这可以成为季风爆发的动力学解释。     

The effect of barotropic shear on baroclinic instability Part I: Normal mode problem

The effect of barotropic shear in the basic flow on baroclinic instability is investigated using a linear multilevel quasi-geostrophic β-plane channel model and a nonlinear spherical primitive equation model. Barotropic shear has a profound effect on baroclinic instability. It reduces the growth rates of normal modes by severely restricting their structure, confirming earlier results with a two-layer model. Dissipation, in the form of Ekman pumping and Newtonian cooling, does not change the main characteristics of the effect of the shear on normal mode instability.Barotropic shear in the basic state, characterized by large shear vorticity with small horizontal curvature, also effects the nonlinear development of baroclinic waves. The shear limits the energy conversion from the zonal available potential energy to eddy energy, reducing the maximum eddy kinetic energy level reached by baroclinic waves. Barotropic shear, which controls the level of eddy activity, is a major factor which should be considered when parameterizing the eddy temperature and momentum fluxes induced by baroclinic waves in a climate model.     

Influence of bottom topography roughness on the jet and inertial recirculation of a mid-latitude gyre

Several numerical experiments are conducted to examine the influence of mesoscale, bottom topography roughness on the inertial circulation of a wind-driven, mid-latitude ocean gyre. The ocean model is based on the quasi-geostrophic formulation, and is eddy-resolving as it features high vertical and horizontal resolutions (six layers and a 10 km grid). An antisymmetrical double-gyre wind stress curl forces the baroclinic modes and generates a strong surface jet. In the case of a flat bottom, inertia and inverse energy cascade force the barotropic mode, and the resulting circulation features strong, barotropic, inertial gyres. The sea-floor roughness inhibits the inertial circulation in the deep layers; the barotropic component of the flow is then forced by eddy-topography interactions, and its energy concentrates at the scales of the topography. As a result, the baroclinicity of the flow is intesified: the barotropic mode is reduced with regard to the baroclinic modes, and the bottom flow (constrained by the mesoscale sea-floor roughness) is decoupled from the surface flow (forced by the gyre-scale wind). Rectified, mesoscale bottom circulation induces an interfacial form stress at the thermocline, which enhances horizontal shear instability and opposes the eastward penetration of the jet. The mean jet is consequently shortened, but the instantaneous jet remains very turbulent, with meanders of large meridional extent. The sea-floor roughness modifies the energy pathways, and the eddies have an even more important role in the establishment of the mean circulation: below the thermocline, rectification processes are dominant, and eddies transfer energy toward permanent mesoscale circulations strongly correlated with topography, whereas above the thermocline mean flow and eddy generation are influenced by the mean bottom circulation through interfacial stress. The topography modifies the vorticity of the barotropic and highest baroclinic modes. Vorticity accumulates at the small topographic scales, and the vorticity content of the highest modes, which is very weak in the flat-bottom case, increases significantly. Few changes occur in surface-intensified modes. In the deep layers of the model, the inverse correlation between relative vorticity and topography at small scales ensures the homogenization of the potential vorticity, which mainly retains the largest scales of the bottom flow and the scale of β.