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.     

线性和非线性正压不稳定

本文从无摩擦的、无辐散的非线性正压涡度方程出发,利用行波解,在相平面上对方程的平衡点分别导得线性和非线性正压不稳定判据,与传统的不稳定判据作了比较分析。还求得了线性和非线性不稳定波的解析解,计算结果表明,若初始时刻两者波幅相等,则以后非线性不稳定波幅的增长值总小于线性不稳定波幅的增长值。非线性效应有使线性不稳定指数型增长减缓的作用。不稳定使得振幅比初始时刻增大一倍所需要的时间,对于线性不稳定大约是6h,对于非线性不稳定大约是18h,后者与实际大气中的观测事实是一致的。     

GENERALIZED ENERGY CONSERVATION AND UNSTABLE PERTURBATION PROPERTY IN BAROTROPIC VORTEX

Based on a barotropic vortex model, generalized energy-conserving equation was derived and two necessary conditions of basic flow destabilization are gained. These conditions correspond to generalized barotropic instability and super speed instability. They are instabilities of vortex and gravity inertial wave respectively. In order to relate to practical situation, a barotropic vortex was analyzed, the basic flow of which is similar to lower level basic wind field of tropical cyclones and the maximum wind radius of which is 500 km. The results show that generalized barotropic instability depending upon the radial gradient of relative vorticity can appear in this vortex. It can be concluded that unstable vortex Rossby wave may appear in barotropic vortex.     

两层正压涡旋稳定性问题分析

本文从柱坐标中线性化的两层均质流体的正压原始方程组出发，从广义能量的角度，研究了该模型中涡旋波的稳定性问题，发现此时不仅可存在广义正压不稳定和超高速不稳定，当上下层扰动厚度场反相时还有新的不稳定类型发生。     

The generalized energy equation and instability in the two-layer barotropic vortex

The linear two-layer barotropic primitive equations in cylindrical coordinates are used to derive a gen- eralized energy equation, which is subsequently applied to explain the instability of the spiral wave in the model. In the two-layer model, there are not only the generalized barotropic instability and the super high- speed instability, but also some other new instabilities, which fall into the range of the Kelvin-Helmholtz instability and the generalized baroclinic instability, when the upper and lower basic flows are different. They are perhaps the mechanisms of the generation of spiral cloud bands in tropical cyclones as well.     

Charney's Model—the Renowned Prototype of Baroclinic Instability—Is Barotropically Unstable As Well

The Charney model is reexamined using a new mathematical tool, the multiscale window transform(MWT), and the MWT-based localized multiscale energetics analysis developed by Liang and Robinson to deal with realistic geophysical fluid flow processes. Traditionally, though this model has been taken as a prototype of baroclinic instability, it actually undergoes a mixed one. While baroclinic instability explains the bottom-trapped feature of the perturbation, the second extreme center in the perturbation field can only be explained by a new barotropic instability when the Charney–Green number γ 1, which takes place throughout the fluid column, and is maximized at a height where its baroclinic counterpart stops functioning.The giving way of the baroclinic instability to a barotropic one at this height corresponds well to the rectification of the tilting found on the maps of perturbation velocity and pressure. Also established in this study is the relative importance of barotropic instability to baroclinic instability in terms of γ. When γ 1, barotropic instability is negligible and hence the system can be viewed as purely baroclinic; when γ 1, however, barotropic and baroclinic instabilities are of the same order;in fact, barotropic instability can be even stronger. The implication of these results has been discussed in linking them to real atmospheric processes.     

On the origin of upper tropospheric cyclonic vortices in the South Atlantic ocean and adjoining Brazil during the summer

Summary The origin of upper tropospheric cyclonic vortices over the South Atlantic and adjoining Brazil is discussed. Calculations of the barotropic energy exchange term based on observations indicated that in the mean, and during some days, zonal kinetic energy gets converted into eddy kinetic energy. This suggests the occurrence of barotropic instability. Examination of zonal winds for barotropic instability, however, revealed weak growth rates. Thus other mechanisms such as condensation heating, the formation of cyclonic centers downstream of the Bolivian high, and middle latitude coupling might also be important.With 3 Figures     

Non-modal baroclinic instability

A class of non-linear instabilities of a vertically sheared zonal flow is discussed. This is a type of baroclinic instability that lies outside the purview of a linear eigenmode analysis of baroclinic instability problems. The form taken by the instability is that of an ensemble of three neutral Rossby waves whose amplitudes are slowly modified by their mutual non-linear interactions. For a triad of small amplitude, these interactions introduce a weak, vertical variation of phase to the structure of the individual waves. This allows the generation of rectified heat fluxes and an exchange of energy with the mean flow.This instability exhibits explosive growth and spans a range of horizontal wavenumbers that exceeds the range that is unstable in the corresponding linear model. It is shown that the type of instability discussed can only occur when the model used admits unstable eigenmodes as well as neutral Rossby waves.The mechanism for the non-linear instability discussed here is believed to be fairly general and should exist also in the context of a horizontally sheared flow where it would take the form of a barotropic instability.     

Snowstorm over the Southwestern Coast of the Korean Peninsula Associated with the Development of Mesocyclone over the Yellow Sea

This study investigates the characteristics of a heavy snowfall event over the southwestern part of the Korean Peninsula on 4 December 2005. The snowstorm was a type of mesoscale maritime cyclone which resulted from barotropic instability, and diabatic heating from the warm ocean in continental polar air masses. Based on surface observations, radiosonde soundings, MTSAT-1R satellite data and the 10-km grid RDAPS （Regional Assimilation and Prediction System based on the PSU/NCAR MM5） data, the evolution of the mesocyclone is explained by the following dynamics; （1） In the initial stage, the primary role in the cyclogenesis process of the mesocyclone is a barotropic instability in the horizontal shear zone. （2） In the developing stage, the mesocyclone moves and deepens into a baroclinic zone corresponding to the surface heating and moistening. （3） In the mature stage, it is found that the mesocyclone is intensified by the destabilization caused by enhanced low-level heating and condensation, the moisture flux convergence, and the interaction between upper and lower-level potential vorticity anomalies. We suggest that a checklist with stepwise indicators responsible for development be prepared for the forecasting of heavy snowfall over the southwestern part of the Korean Peninsula.     

远距离台风暴雨过程的正/斜压不稳定

利用正／斜压联合不稳定计算方法，分析了由9711号台风引起的远距离台风暴雨(实况及MM4模拟结果)过程中不同时段、不同位置的正／斜压项。结果表明：台风暴雨初始时段的降水主要由台风系统本身正压不稳定扰动引起，随着台风与西风带系统的结合，其南北部出现了2个雨区，均是正／斜压联合不稳定的产物；北部雨区的斜压不稳定较明显，且有随高空急流的非纬向性增强而增强的趋势。     

Eddy fluxes and mean flow tendencies in open-ocean baroclinic instability

Linear normal-mode instabilities of a mean flow varying over many internal radii of deformation are considered. Attention is devoted not to the instabilities themselves, but to the second-order effects of such instabilities on the mean flow. These are derived analytically from solutions by Killworth. The effects on symmetric profiles are very different from those on asymmetric profiles. Specifically, the tendencies are much larger in the latter case with the strongest tendency being on the barotropic part of the mean flow; this tendency apparently attempts to make the profile more symmetric.     

A BAROTROPIC QUASI-GEOSTROPHIC MODEL WITH LARGE-SCALE TOPOGRAPHY, FRICTION AND HEATING

Based on the barotropic equations including large-scale topography, friction and heat factor, a barotropic quasi-geostrophic model with large-scale topography, friction and heating is obtained by means of scale analysis and small parameter method. It is shown that this equation is a basic one, which is used to study the influence of the Tibetan Plateau on the large-scale flow in the atmosphere. If the friction and heating effect of large-scale topography are neglected, this model will degenerate to the general barotropic quasi-geostrophic one.     

The influence of the coast on the dynamics of upwelling fronts: Part II. Numerical simulations

We investigated the dynamics of upwelling fronts near a coast. This work was first motivated by laboratory experiments [Bouruet-Aubertot, Linden, Dyn. Atmos. Oceans, 2002] in which the front is produced by the adjustment of a buoyant fluid initially confined within a bottomless cylinder. It was shown that cyclonic eddies consisting of coastal waters are enhanced when the front is unstable near the coast (the outer vertical boundary). The purpose of this paper is to provide further insights into this process. We reproduced the experimental configuration using a three-dimensional model of the primitive equations. We first show that for coastal fronts more potential energy, in terms of the maximum available potential energy, is released than for open-ocean fronts. Therefore, waves of larger amplitude are generated during the adjustment and the mean flow that establishes has a higher kinetic energy in the former case. Then as baroclinic instability starts and wave crests reach the boundary, cyclonic eddies are enhanced as in the laboratory experiments and in a similar way. However, in contrast to the laboratory experiments, offshore advection of cyclonic eddies can occur in two stages, depending on the spatial organization of the baroclinic wave. When the baroclinic wave consists of the sum of different modes and is thus highly asymmetric, the offshore advection of cyclonic eddies occurs just after their enhancement at the boundary, as in the laboratory experiments. By contrast, when a single-mode baroclinic wave develops, neighboring cyclonic eddies first merge before being advected offshore. Very different behavior is observed for open-ocean fronts. First a mixed baroclinic–barotropic instability grows. Then the eddies transfer their energy to the mean flow and the barotropic and baroclinic instabilities start again. An excellent agreement is obtained with the main result obtained in the laboratory experiments: the ratio between growth rates of surface cyclonic and anticyclonic vorticity increases as the instability develops nearer to the coast.     

一次闽南暖区暴雨的多尺度动力过程

作为气象研究中的一个难点问题,暖区暴雨的动力学一直为学界所关注。基于多尺度子空间变换(MWT)以及基于MWT的局地多尺度能量学分析和正则传输理论,对2018年5月7日的一次闽南暖区暴雨进行研究以了解其多尺度动力过程。首先将原始物理量场重建到三个尺度子空间:背景流子空间、天气尺度子空间和暴雨子空间。重构场上可以很好地看出背景环流尺度的高低空急流,以及暴雨尺度上的垂直环流。以往的研究普遍认为暖区暴雨的动力过程具有弱斜压性这一特征,而就此次事件而言,正压失稳和斜压失稳都起着很关键的作用,暴雨主要落区内既发生了正压失稳,也发生了斜压失稳。研究表明,对流层不同高度上的动力学存在差异,低层主要表现为正压失稳,天气尺度子空间与背景流子空间向暴雨子空间传输的动能相当; 中层主要是混合失稳,除正压失稳外,斜压正则传输也将有效位能从背景流子空间传输到了暴雨子空间,再通过浮力转换将有效位能转为动能,从而维持暴雨在中层的动力过程; 高层则与低层相似,但只存在背景流子空间向暴雨子空间的能量传输。      

热带对流层低频振荡过程的能量学初步分析

本文利用线性化的能量方程和交叉谱方法分别研究了准两周振荡和准40天振荡的能量来源及其转换过程。发现:在热带对流层中,对于低频振荡过程来说,来自中高纬度的侧向强迫作用和水汽凝结的加热作用是非常重要的,为扰动的产生和维持提供了重要能源。而正、斜压不稳定作用对扰动发展的作用是极小的。并且指出,在东亚热带对流层上部,准两周振荡通过与平均气流的相互作用为基本气流提供许多能量,对于维持这里的高空东风急流有重要作用。准40天振荡向平流层输送能量,对平流层平均环流的演变可能有重要贡献。      

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。     

青藏高原热力作用下的非绝热Rossby波

从含非绝热项的准地转运动方程组出发，分析了青藏高原大尺度热力作用下非绝热Rossby波的一些性质，从理论上证明当背景西风气流为正压时，冬季高原冷却作用有利于Rossby波的经向传播，夏季高原大尺度热力作用不利于波动的经向传播。非绝热Rossby波的频率方程说明冬季高原的热力作用是中纬季节内振荡的重要激发机制。同时，在背景西风气流为纯斜压条件下，求解了高原热力作用下非绝热Rossby波的频率，并由频率方程说明冬季高原热力作用有利于波动向不稳定方向发展，而夏季高原的大尺度热力作用对波动稳定性的影响存在临界值。     

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.     

Some effects of truncation on topographic instability

Abstract

The topographic stability of forced planetary waves in α β‐channel is investigated using a barotropic model. The equilibrium forced waves are the result of the interaction of a constant mean zonal wind over finite‐amplitude surface orography. Small‐amplitude perturbations of the equilibrium flows are considered that have a wavy part with the same zonal wavenumber as the forcing but an arbitrary meridional structure. The mean zonal part of the perturbations is also taken to be arbitrary. This configuration allows us to (1) isolate those instabilities that depend crucially on topography through form drag and (2) investigate non‐topographic effects on topographic instability that arise from the convergence of Reynolds stresses. A numerical stability analysis is then performed wherein the effects of truncation are emphasized.

This numerical approach casts doubts about the results obtained from some earlier studies involving various ad hoc assumptions. We find, in particular, that unstable long waves (i.e. waves with the zonal wavelength greater than the meridional wavelength) exist under superresonant conditions. This contradicts some previous results that suggest long waves are unstable only when the flow is subresonant. Further, our model reveals the existence of some interesting travelling instabilities. The latter are shown to depend on both form drag and Reynolds stresses in that these two mechanisms alternate in time in supplying the perturbation with the required energy to maintain the exponential growth.     

一个正压和斜压不稳定的充要条件

提出一个新的李雅普诺夫函数，首次得到了正压和斜压不稳定的充要条件，大大改善了热知的Ｒａｙｌｅｉｇｈ－郭晓岚定理和Ｆｊｏｒｔｏｆｔ定理，解决了Ｒａｙｌｅｉｇｈ（１８８０）提出的无粘切变流正压稳定性问题。