The effect of barotropic shear on baroclinic instability Part II: The initial value problem

The effect of barotropic shear on baroclinic instability has been investigated using both a linear quasi-geostrophic β-plane channel model and a multilevel primitive equation model on the sphere when a nonmodal disturbance is used as the initial perturbation condition. The analysis of the initial value problem has demonstrated the existence of a rapid transient growth phase of the most unstable mode. The inclusion of a linear barotropic shear reduces initial rapid transient growth, although at intermediate times the transient growth rates of the sheared cases can be larger than in the unsheared case owing to downgradient eddy momentum fluxes. Certain disturbances can amplify by factors of 4.5–60 times (for the L₂ norm), or 3–30 times (for the perturbation amplitude maximum), as large as disturbances based on the linear normal modes. However, linear horizontal shear always reduces the amplification factors. The mechanism is that the shear confines the disturbance meriodionally and therefore limits the energy conversion from the zonal available potential energy to eddy energy. The effect of barotropic shear on the transient growth is not changed much in the presence of either thermal damping or Ekman pumping. Nonmodal integrations of baroclinic wave lifecycles show that the energy level reached by eddies is not very sensitive to the structure of the initial disturbance if the amplitude of the initial disturbance is small. Although in some cases the eddy kinetic energy level reached by the wave integrated from nonmodal disturbance can be 25–150% larger than the normal mode integrations, barotropic shear, characterized by large shear vorticity with small horizontal curvature, always reduces the eddy kinetic energy level reached by the wave, confirming the results of normal mode studies.     

The dynamics of baroclinic and barotropic solitary eddies

We derive exact nonlinear solutions to the inviscid quasigeostrophic two-layer equations. The solutions represent isolated eddies with an amplitude that decays rapidly away from the eddy. They translate steadily either to the east (if there is a barotropic component in the fringes of the eddy) or either to the east or rapidly to the west (if the exterior field has only a baroclinic mode structure). They are strong in the sense that all of the fluid within a finite radius from the center is carried along with the eddy as it translates. The solutions have a characteristic north-south antisymmetry with the nonlinear vortex pair interactions being an essential feature of the dynamics. However, we also show that radially symmetric perturbations of a special form but arbitrary amplitude can be superimposed on the basic solutions; thus the antisymmetric structure may to some extent be masked. We show nondimensional dispersion relationships: atmospheric and oceanic planetary wave phenomena do fall in the range of parameters where theory predicts solitary eddy solutions can exist.     

Bounds on the eigenvalues of the planetary-scale baroclinic instability problem

In the framework of the linear baroclinic instability problem for planetary geostrophic flows, bounds on the related complex eigenvalues are deduced. The main feature of this result is the independence of these bounds from the latitude and the density stratification, so that it generalizes the results of previous work on the subject.     

The effects of latitudinal asymmetries on baroclinic instability

Abstract

Baroclinic instability of zonal flows with different latitudinal structures is examined, using a linear, quasi‐geostrophic two‐level ß‐plane model. The flows have different amounts of skew, with respect to the channel centre, at different vertical levels. The results are interpreted in terms of the instability of the baroclinic components of the zonal flows. Because of the presence of latitudinal asymmetries, a spectrum of meridional modes is generated in the perturbation. In general, the meridional spectrum has two peaks: a primary peak at the planetary basic flow scale, and a secondary peak near the radius of deformation. As neutral stability is approached, the latter scale becomes more important, i.e. there is a tendency for more small‐scale structure near neutral stability. The perturbation zonal scale is close to the radius of deformation. The eddy amplitudes and momentum fluxes are also examined. The case that best applies to the atmosphere is also discussed.     

The effects of friction and heating of convective condensation in the baroclinic instability problem

By using the β-plane, two-layer quasi-geostrophic baroclinic model, this paper discusses the baroclinic in-stability problem concerning the effects of friction and heating of convective condensation. By Linear analysis it is shown that the combination of β effect, friction and convective heating brings about the asymmetric phenom-enon of margin curves. The convective heating plays a role in the increased baroclinic instability. As the heating increases (m*→1), the short wave cutoff can increase infinitely. Besides, the numerical integration of the finite-amplitude equations shows that the trajectory on the phase plane oscillates periodically in the case of non-dissipation. When the friction dissipation is considered, the trajectory of phase decays and oscillates to the equilibrium. The stronger convective heating not only makes the unstable wave length shorter and the amplitude of the equilibrium decrease, but also makes multiple equilibrium into single equilibrium.     

The impact of meridonal wind shear on baroclinic life cycle

In this study, initial-value calculations are performed with a primitive equation model to examine whether the stabilizing effect of the horizontal zonal wind shear in the background state can account for the observed variability in baroclinic life cycles. The life cycle calculations show that a greater maximum eddy energy is attained for the observed basic state with anomalously weak horizontal zonal wind shear, suggesting that the horizontal zonal wind shear indeed plays the dominant role in determining the eddy amplitude. In addition, under this weak shear, the life cycle produces a more pronounced poleward jet shift. Because model simulations of warmer climates tend to show both a poleward jet shift and more intense zonally localized tropical convection which tends to produce the weak shear state, the result of this study provides a mechanism whereby the strengthening of tropical convection can contribute toward the poleward jet shift in warm climates.     

An experimental study of nonlinear baroclinic instability and mode selection in a large basin

This paper reports on two-layer rotating liquid experiments designed to study the behavior of non-linear baroclinic waves under conditions where the Rossby radius of deformation R_d is much smaller than the geometric length scale L imposed by the size of the laboratory apparatus. The apparatus is constructed to consistently simulate f-plane dynamics. When F = L²/R_d² > > 1, it is found that the unstable waves first encountered as friction is decreased have high frequencies, in accord with linear theory. As the friction parameter $\text{Q = 0.7 E}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{/R}{}_0$ (where E is the Ekman number and R₀ the Rossby number) is further decreased into the non-linear region, singlewave amplitude vacillation is observed. Generally, as Q decreases lower frequencies (and low wavenumbers) dominate the response, which ultimately becomes turbulent at values of Q of the order 0.1. This is contrary to the result expected from an extrapolation of linear theory. Further observations show that the finite-amplitude state is not unique: multi-equilibria are possible depending on the initial conditions.     

维向切变流中的非线性对称不稳定

讨论了维向切变流中的非线性对称不稳定问题。中采用绝热无粘的非线性对称扰动方程组，利用多尺度摄动方法分析其不稳定波动的有限振幅特性。研究结果表明：不稳定波的有限振幅在强度上呈现出振荡趋势。无论是超临界切变情况，还是次临界切变情况，对称扰动振幅都随时间呈现出周期性的变化，振荡周期的大小不仅与基本场稳定度参数及波的特性有关，而且还与初始扰动的振幅及其增长率有关。     

A note on baroclinic instability of a source-sink driven flow

The stability properties of a source-sink driven flow in a two-layer, rotating fluid is considered. A steady flow in an annulus has been determined previously (Maeland, 1982) by Ekman-layer dynamics at the interface of the two layers; in this paper, the stability will be studied by linear, inviscid theory. For a given configuration, the two parameters which govern the stability of the basic flow are the internal Froude numbers of each layer. The growth rates for the instabilities are calculated by a simple numerical method (matrix method).     

Phillips模式的斜压不稳定的非线性饱和问题:能量情形

A conservation law for the Phillips model is derived. Using this law, the nonlinear saturation of purely baroclinic instability caused by the vertical velocity shear of the basic flow in the Phillips model-the case of energy-is studied within the context of Arnold's second stability theorem. Analytic upper bounds on the energy of wavy disturbances are obtained. For one unstable region in the parameter plane, the result here is a second-order correction in ε to Shepherd's; For another unstable region, the analytic upper bound on the energy of wavy disturbances offers an effective constraint on wavy (nonzonal) disturbances φ'i at any time.     

有限区域湿有效能量的层结、斜压、正压分量及其收支方程

本文探讨了将湿有效能量分解成层结、斜压、正压分量的方法,并讨论了它们的收支方程式.其中层结分量与实际大气中对流不稳定有较好的对应关系,但在干有效位能中分解不出这一项.在斜压、正压分量的收支方程中,除附加项外都与干有效位能的方程类似.由三个分量的收支方程看出,湿有效能量向动能的转换只可能发生在层结不稳定或(与)有斜压存在的大气中.这与天气分析经验以及经典的马古列斯(M.Margules)模式是一致的.     

阻塞过程中正斜压涡度拟能相互转换机制的重要性

使用正斜压涡度拟能方程对一次乌拉尔山阻塞过程的研究表明，虽然正、斜压涡度拟能向阻塞区内的净输送、阻塞区域内正斜压涡度拟能的净生成以及β效应是阻塞环流建立、维持和崩溃的基本能源，但正斜压涡度拟能相互转换机制则是阻塞过程得以形成的根本原因。如果没有正斜压涡度拟能相互转化机制，则阻塞环流不但不能形成，反而使大气的斜压性不断增强，正压性不断减弱。     

垂直风切变对螺旋波不稳定的影响

通过建立一系列局地准二维直角坐标系,用以研究垂直风切变对热带气旋中螺旋波不稳定的影响,并计算了螺旋波不稳定增长率在螺旋线上的分布,得到了以下结论:不稳定增长率开始随热带气旋半径增大而增大,在最大风速圈处达最大;然后随半径减小,以后再随半径增大,出现一个局地极值,再以后又减小,即在螺旋线上有两个极值中心;这与实际热带气旋的情况相一致。不稳定螺旋波的增长率与基流的垂直切变和层结参数密切有关,基流垂直切变越大,层结参数越小,则增长率越大;也即Richardson数越小,则不稳定增长率越大。     

Normal mode theory of lee cyclogenesis and the initial value problem

Summary An initial value problem is proposed to study the influence of mountains upon atmospheric flows. The normal mode, quasigeostrophic theory of lee cyclogenesis is revisited; in order to model Alpine cyclogenesis, the flow, upon which the mountain acts, should represent a cyclone wave propagating eastward in a sheared zonal current. Two different models are considered in order to describe cyclone waves: an unstable Eady normal mode and a finite amplitude, neutral Rossby mode in the two layer model. It is shown that the topographic modifications proper of normal modes emerge very rapidly from the integration of the initial value problem. It is also shown that an enhanced deepening of the upper level trough can occur when finite amplitude effects are taken into account.With 8 Figures     

The effect of topography on the stability of a barotropic coastal current

We consider small non-divergent perturbations to a barotropic current flowing parallel to a straight coastline. Sufficient conditions for stability and a semi-circle theorem are established for general current profiles and topography. An asymptotic expansion for long waves is described. Some analytic solutions are obtained when the topography is piece-wise constant and the current is piece-wise linear. Two kinds of instability, a shear instability and a topographic instability, are identified. Our results are applied to a model of the East Australian current.     

The instability of the vortex sheet along the shear line

The traditional Kelvin-Helmholtz notion of studying the shear instability is not suitable for the case associated with shear line with the strong wind shear in the vortex sheet. Since then, the shear instability be-comes the instability of the vortex sheet. If the velocity is induced by the vortex sheet, the inequalities (1 – Rv + Rid)> 0 and U (y,t)> U{A(t)} become the criterion of the vortex sheet instability. This criterion indicates that 1) the disposition of environment field restrains the disturbance developing along the shear line. 2) There exist multi-scale interactions in the unstable process of the shear line. The calculation of the necessary condition for the instability is also presented in this paper.     

大地形对切变基流上罗斯贝波稳定性的影响

利用WKB近似讨论大地形对切变基流上Rossby波稳定性的影响.北半球东西走向地形有利导式长波槽辐合能量于地形北坡附近,对曳式长波槽有使能量从地形南坡区域辐散的趋势.南半球地形的影响正好相反.散度顶有利于导式长波槽在东西风带过渡区域的能量辐射,有利于曳式长波槽能量在这里辐合.