MESOSCALE INSTABILITY OF A BAROCLINIC BASIC FLOW-PART Ⅱ:TRANSVERSAL INSTABILITY

This is the second part of "Mesoscale Instability of a Baroclinic Basic Flow" which discusses the instability of a basic flow against mesoscale perturbations of transversal type.A bi-mode instability spectrum is obtained by generalizing the Eady model to ageostrophic regime in an f-plane:Eady modes present at the synoptic and subsynoptic scales,while the ageostrophic baroclinic mesoscale modes present at the inertial scales of a few tens to hundreds kilometers.The mesoscale mode is featured by an asymmetric "eat eyes" pattern in the vertical cross section and by an alternative distribution of divergence and vorticity in the horizontal direction.The growth rates of the mesoscale modes are about four times larger than those of Eady modes in magnitudes for the same wind profile.The major energy source for development both Eady mode and mesoscale mode is the baroclinic available energy stored in the rotational basic flow.     

The linear steady response of a stratified baroclinic atmosphere to elevated diabatic forcing

Abstract

A parameter study is presented of the linear steady response to an elevated diabatic forcing in a Boussinesq baroclinic atmosphere. The model is two‐dimensional on the vertical plane; the basic wind is perpendicular to this plane with horizontal and vertical shear. The intensity of the circulation is sensitive to the strength of the baroclinic zone only for weak static stability. A scale analysis supports this conclusion: the importance of baroclinic effects depends on the ratio of the aspect ratio of the circulation to the aspect ratio of the baroclinic basic state. Baroclinicity also leads to a tilt of the circulation due to enhanced horizontal temperature advection, and the corresponding vertical flux of horizontal momentum. The magnitude of the latter can be large when the Richardson number approaches its critical value from above, i.e. for slightly symmetrically stable basic flows; this is true even with viscosity. The resonant limit where the Richardson number approaches the critical value for symmetric instability is also examined. For non‐zero dissipation, the critical value is smaller than that of the inviscid limit.     

Onset of convection in a rotating semi‐infinite fluid: Theory and experiment

Abstract

We look at the development of the first plumes that emerge from a convectively unstable boundary layer by modelling the process as the instability of a fluid with a time‐dependent mean density field. The fluid is semi‐infinite, rotating, dissipative ‐ characterized by the ratio of its viscosity to thermal diffusivity (Prandtl number Pr = ν/κ) ‐ and initially homogeneous. A constant destabilizing heat flux is applied at the boundary and the stability of the evolving density field is investigated both mathematically and in laboratory experiments.

Using a “natural convective” scaling, we show that the behaviour of the non‐dimensional governing equations depends on Pr and the parameter γ = f(ν/B)^1/2, where f is the Coriolis parameter, and B is the applied buoyancy flux. For the ocean, γ ≈ 0.1, whilst for the atmosphere γ ≈ 0.01. In the absence of rotation, the behaviour of the differential equations is independent of B, depending only on Pr. The boundary‐layer Rayleigh number (Ra_bl) is also independent of B. We show that Ra_bl, evaluated at the onset of rapid vertical motion, depends on the form of the perturbation.

Due to the time‐dependence of the mean density field, analytic instability analysis is difficult, so we use a numerical technique. The governing equations are transformed to a stretched vertical coordinate and their stability investigated for a particular form of perturbation function. The model predictions are, for the ocean: instability time ～2–4 h, density difference ～0.002–0.013 kg m^‐3, boundary‐layer thickness ～50–75 m and horizontal scale ～200–300 m; and for the atmosphere: instability time ～10 min, temperature difference ～2.0–3.0°C, boundary‐layer thickness ～400–500 m and horizontal scale ～1.5–2.0 km.

Laboratory experiments are performed to compare with the numerical predictions. The time development of the mean field closely matches the assumed analytic form. Furthermore, the model predictions of the instability timescale agree well with the laboratory measurements. This supports the other predictions of the model, such as the lengthscales and buoyancy anomaly.     

东亚地区南北两个风暴源地中风暴的结构和动力学差异

基于欧洲中心中期天气预报再分析资料(ERA-40),使用涡旋追踪和合成技术、多尺度子空间变换以及局地多尺度能量分析方法研究了东亚地区南北两个风暴源地中风暴的差异。结果发现,南、北两个源地风暴在结构上和内部动力过程上均存在着显著不同。南支源地(40°N以南)风暴底层比高层强,与线性斜压模式中的最不稳定模态结构相似;而北支源地(40°N以北)风暴则正好相反,与下游发展理论所描述的斜压波结构相似。并且发现,南支源地风暴的非地转风场比北支源地风暴的强。能量学诊断结果显示,南支源地风暴的能量源除了斜压不稳定外,有很大一部分来自正压不稳定,而北支源地风暴中则是存在弱的动能逆尺度传输。此外,南支源地风暴的浮力转换和非绝热做功均比北支源地风暴的强,其主要原因是南支源地风暴的垂直运动更强,风暴中的水汽更加充足。      

大气静力稳定度的铅直分布和不同风速铅直廓线对天气系统发展的影响

采用一个准地转三层模式,对于高低空不同层结及不同风速铅直廓线下的斜压不稳定性问题进行了分析和讨论。指出:(1)大气层结特征对斜压不稳定的影响不仅表现在对不稳定临界波长和临界风速切变的制约上,而且其铅直分布的不均一性决定了大气的各个层次对斜压不稳定的贡献大小;扰动的斜压发展主要取决于静力稳定度较小的那些层次内的热力、动力学条件,在这些层次内,扰动也最为明显。(2)高低空风速比中空为大的“高低空强风型”风速铅直分布最有利于扰动的不稳定发展,在这种风速铅直廓线下,不稳定波波谱较宽,不稳定波临界波长和最不稳定波波长也较短。     

Dispersion of Heavy Particles Emitted from Area Sources in the Unstable Atmospheric Boundary Layer

Reliable predictions of the daytime dispersal of heavy particles in the unstable atmospheric boundary layer are important in a variety of disciplines. For many applications, particles disperse from area sources near the ground, and corresponding theoretical solutions are desired to reveal insight into the physical processes. Here, theoretical solutions recently developed for neutral conditions are modified to include the effects of atmospheric instability. The Obukhov length L _O and convection velocity w _? are introduced to characterize the patterns of particle dispersion, in additional to friction velocity u _? and settling velocity w _s used in the neutral case. The major effects of atmospheric instability are accounted for by modifying the vertical velocity variance profile and considering the ratio of velocity scales w _?/u _?. Theoretical predictions including the mean concentration profile, plume height, and horizontal transport above the source, and ground deposition flux downwind from the source agree well with large-eddy simulation results while the particle plume is within the atmospheric surface layer. The deposition curve is characterized by a power-law decay whose exponent depends on u _?, w _s, and w _?. A second steeper power-law develops once the plume extends into the mixed layer. This effect is enhanced with increasing atmospheric instability, implying that particles disperse farther from the source.     

Passive scalar diffusion from surface sources in the convective boundary layer

We examine vertical and horizontal diffusion of a passive scalar puff from a surface point source in a convective boundary layer (CBL). Numerical results are presented from a large-eddy simulation (LES) with embedded subgrid Lagrangian particle simulation (LPS). There is good agreement in most respects with previous laboratory and numerical studies. Analytical approximations for the concentration, horizontal flux and vertical flux are found to work reasonably well; they are based on the assumption that the concentration follows a Gaussian function in the horizontal and vertical, and that the dimensionless width and height scales of the puff follow simple functions of time. Fluxes and concentration gradients are related through a continuity relationship, without the need for an eddy diffusivity assumption. The instantaneous, point-source fields can be integrated for any source geometry. We compare predictions from the LES/LPS model for a sinusoidal surface flux with previous results from an LES with sinusoidal buoyancy flux and confirm that the buoyancy perturbations diffuse like a passive scalar. We also consider a continuous point source and derive footprint functions for vertical flux measurements above the surface layer.     

The stability of buoyancy-driven coastal currents

Buoyancy-driven boundary currents were generated in the laboratory by releasing buoyant fluid from a source adjacent to a vertical boundary in a rotating container. The boundary removed the Coriolis force parallel to it, allowing the buoyant fluid to spread in a current along the boundary. Ise of a cylindrical boundary and a line source that released fluid uniformly around the circumference enabled an axisymmetric (zonal) current to be produced. With the continuous release of fluid from the source, the current grew in width and depth until it became unstable to non-axisymmetric disturbances. The wavelength and phase velocities of the disturbances were consistent with a model of baroclinic instability of two-layer flow when frictional dissipation due to Ekman layers is included. However, when the current only occupied a small fraction of the total depth, barotropic processes were also thought to be important, with the growing waves gaining energy from the horizontal shear.In other experiments, gravity currents were produced by a point source adjacent to either a zonal (circular) or a meridional (radial) vertical boundary. The currents were also observed to become unstable to the same upstream breaking waves as those on the continuous zonal current. Finally, some comparisons are made with oceanic coastal currents.     

斜压气流的中尺度稳定性 Ⅱ.横波型不稳定

本文是“斜压气流的中尺度稳定性”的第二部份,讨论基本流对横波型中尺度扰动的稳定性。 将f-平面的Eady模型推广到非地转情况,得到斜压基流的双模态不稳定谱:在天气尺度和次天气尺度上出现Eady模态,在几十至几百公里的惯性尺度上出现非地转斜压中尺度模态。在垂直剖面上中尺度模态呈非对称“猫眼”流型,在水平方向上散度与涡度交替分布。对同一线性风速廓线,中尺度模态的发展率约为Eady模态的4倍,两种扰动发展的主要能源都是旋转基流中储存的斜压有效位能。     

True gravity in ocean dynamics Part 1 Ekman transport

The direction normal to the Earth spherical (or ellipsoidal) surface is not vertical (called deflected vertical) since the vertical direction is along the true gravity g (= ig_λ+ jg_φ+ kg_z). Here, (λ, φ, z) are (longitude, latitude, depth), and (i, j, k) are the corresponding unit vectors. The spherical (or ellipsoidal) surfaces are not horizontal surfaces (called deflected-horizontal surfaces). The most important body force g (true gravity) has been greatly simplified without justification in oceanography to the standard gravity (-g₀k) with g₀ = 9.81 m/s². Impact of such simplification on ocean dynamics is investigated in this paper using the Ekman layer model. In the classical Ekman layer dynamic equation, the standard gravity (-g₀k) is replaced by the true gravity g(λ, φ, z) with a constant eddy viscosity and a depth-dependent-only density ρ(z) represented by an e-folding near-inertial buoyancy frequency. New Ekman spiral and in turn new formulae for the Ekman transport are obtained for ocean with and without bottom. With the gravity data from the global static gravity model EIGEN-6C4 and the surface wind stress data from the Comprehensive Ocean-Atmosphere Data Set (COADS), large difference is found in the Ekman transport using the true gravity and standard gravity.     

Studying the stability of the Antarctic slope front in the Commonwealth Sea

In 2004–2007 and 2011–2013, oceanography sections with the high spatial resolution were carried out on the shelf and continental slope of the Commonwealth Sea and in the Prydz Bay in the Antarctic. In combination with a fine vertical resolution, this provides an advanced interpretation of the obtained data. It becomes possible to analyze the local thermohaline structure of the baroclinic Antarctic slope front (the ASF) and to investigate its stability. The reliable determination of the ASF characteristics enables to apply the parameterizations and criteria obtained during laboratory experiments for analysis of its stability. The ASF instability is preceded by the local vertical dramatic intensification of the front, where the ASF approaches the pure baroclinic state. Therefore, the local baroclinic instability in the ASF area prevails over the barotropic one at the significant mutual impact of layers (δ = H _ASF/H _b ≈ ≈ 0.3–0.6, H _b is the total depth) and accompanied with the hydrostatic instability. Instability conditions in the frontal zone with a width of ～(1–2)Rd _L (Rd _L is local baroclinic deformation radius) relative to the wide ASF (L _ASF ≈ (3–5)Rd _L ) coincide with those of the narrow (intensified) ASF (L _ASF ≈ Rd _L* ). In the case of the runoff of dense Antarctic shelf water, on the shelf edge the realization is possible of the self-oscillatory mechanism of the ASF instability caused by the topographic beta-effect as well as the periodic generation of baroclinic vortices.     

Buoyancy in tropical cyclones and other rapidly rotating atmospheric vortices

Motivated primarily by its application to understanding tropical-cyclone intensification and maintenance, we re-examine the concept of buoyancy in rapidly rotating vortices, distinguishing between the buoyancy of the symmetric balanced vortex or system buoyancy, and the local buoyancy associated with cloud dynamics. The conventional definition of buoyancy is contrasted with a generalized form applicable to a vortex, which has a radial as well as a vertical component. If, for the special case of axisymmetric motions, the balanced density and pressure distribution of a rapidly rotating vortex are used as the reference state, the buoyancy field then characterizes the unbalanced density perturbations, i.e. the local buoyancy. We show how to determine such a reference state without approximation.The generation of the toroidal circulation of a vortex, which is necessary for vortex amplification, is characterized in the vorticity equation by the baroclinicity vector. This vector depends, inter-alia, on the horizontal (or radial) gradient of buoyancy evaluated along isobaric surfaces. We show that for a tropical-cyclone-scale vortex, the buoyancy so calculated is significantly different from that calculated at constant height or on surfaces of constant σ (σ = (p − p^*)/(p_s − p^*), where p is the actual pressure, p^* some reference pressure and p_s is the surface pressure). Since many tropical-cyclone models are formulated using σ-coordinates, we examine the calculation of buoyancy on σ-surfaces and derive an expression for the baroclinicity vector in σ-coordinates. The baroclinic forcing term in the azimuthal vorticity equation for an axisymmetric vortex is shown to be approximately equal to the azimuthal component of the curl of the generalized buoyancy. A scale analysis indicates that the vertical gradient of the radial component of generalized buoyancy makes a comparatively small contribution to the generation of toroidal vorticity in a tropical cyclone, but may be important in tornadoes and possibly also in dust devils.We derive also a form of the Sawyer–Eliassen equation from which the toroidal (or secondary) circulation of a balanced vortex may be determined. The equation is shown to be the time derivative of the toroidal vorticity equation in which the time rate-of-change of the material derivative of potential toroidal vorticity is set to zero. In analogy with the general case, the diabatic forcing term in the Sawyer–Eliassen equation is shown to be approximately equal to the time rate-of-change of the azimuthal component of the curl of generalized buoyancy.Finally, we discuss the generation of buoyancy in tropical cyclones and contrast the definitions of buoyancy that have been used in recent studies of tropical cyclones. We emphasize the non-uniqueness of the buoyancy force, which depends on the choice of a reference density and pressure, and note that different, but equivalent interpretations of the flow dynamics may be expected to arise if different reference quantities are chosen.     

斜压不稳定理论的发展历程分析

斜压不稳定理论是中高纬度地区天气尺度扰动生成和发展的机制,是继长波理论之后大气动力学上的又一重大进展,对现代天气预报具有重要的指导意义。按照斜压不稳定理论发展的时间脉络,阐述了赵九章、查尼和伊迪在斜压不稳定方面开展的研究工作,分析了3位科学家各自的研究特点及历史贡献。赵九章1946年发表在《Journal of Meteorology》上的论文,最早提出了"斜压不稳定"这一概念,给出了不稳定的临界波长,并阐述了在不稳定扰动情况下能量的转换,以及不稳定波对大气环流带来的可能影响,尽管得出的不稳定临界波长与观测差别较大,但其对波-流相互作用的讨论在当时是超前的。查尼于1947年采用滤波和尺度分析等方法,将大气扰动方程简化为一个可以求解的系统,推导出大气稳定状态的判据,建立了斜压不稳定理论,其结果与实况比较接近;并据此把准地转模式成功应用于数值天气预报实践中,促使数值天气预报获得首次成功。1949年,伊迪在查尼研究工作的基础上通过合理的简化方法,得到了更为简洁的模型。最后,通过对比他们的研究思路,重点分析了赵九章未能使得斜压不稳定理论提前一年建立的原因:由于其研究思路始终局限在大气水平运动上,忽略了斜压系统发展中散度项的贡献,因而未能抓住天气系统发生、发展的本质,致使其最终与斜压不稳定理论的成功建立失之交臂,其论文本身存在的一些亮点也因此被后人忽视。      

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.     

Barotropic and baroclinic instability in rotating stratified fluids

The linear normal mode instabilities of a parallel shear flow which varies both vertically (z) and meridionally (y) in a quasigeostrophic, rotating, stratified fluid are considered. The β effect (variation of Coriolis parameter with y) is included. Both two-layer and continuous fluids are treated. Attention is concentrated on the types of instability possible for a given shear flow. It is found that the instability can be described adequately by three nondimensional parameters: Λ, the ratio of the horizontal length scale of the shear to the internal deformation radius: δ, which is either the ratio of layer depths in the two-layer fluid or the fractional depth of variation of the stratification in the continuous fluid; and β, suitably nondimensionalized.Asymptotic analyses, confirmed by direct numerical solutions, are performed for conditions in which various parameters become large or small. The β effect is essentially quantitative, whereas Λ and δ define the type of instability as barotropic (if the kinetic energy of the mean flow feeds the growing perturbations), baroclinic (if the available potential energy of the mean flow feeds the perturbations) or mixed (a combination of the two).The case of large Λ (the most relevant for oceanographic applications) is treated in detail. It is shown that y-independent problems have only limited relevance. For a fixed deformation radius and the y scale of the mean flow increasing without limit, the asymptote is not the case of no y variation in the mean flow.     

Vertical cross-spectra of horizontal velocity components at the Boulder observatory

Cross-spectra between horizontal wind components at different levels of the Boulder Atmospheric Observatory (BAO) tower lead to the following conclusions:

1. Davenport's hypothesis is satisfied that coherence decays exponentially with the ratio of vertical separation to horizontal wave length, at least to very small values of coherence.
2. The decay coefficients increase with z/L for z/L < 0.5. For larger stabilities, irregular fluctuations with periods of order 10–20 min have considerable vertical coherence. Results at BAO are quite consistent with those elsewhere.
3. Eddy slopes in vertical planes increase with wind shear up to a point where the slope (horizontal delay over vertical separation) is just above 2. Beyond that point, the systematic increase of slopes with shear ceases. Since wind shear decreases upward, slopes tend to decrease upward. Slopes for lateral components are significantly larger than those for u-components.

     

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.     

A note on a semicircle theorem

A new semicircle theorem is derived for unstable barotropic disturbances to a class of rectilinear barotropic currents u(y) in systems with an ambient potential vorticity gradient (f/h)_y which satisfies (u—u_s)(f/h)_y ? 0 throughout the y domain, where u_s is some value of u. In the conventional semicircle theorem for this flow, the radius of the semicircle is a function of both the range of u and the maximum speed of the stable Rossby waves that can exist in the system when u = 0. In the new semicircle theorem, the radius is only a function of the former when it is small compared to the latter. Currents which do not satisfy the above condition, as well as stable modes and baroclinic systems, are also discussed.     

1215号“布拉万”台风暴雨及降水非对称性分布的成因分析

利用中尺度非静力模式WRF对2012年第15号台风“布拉万”在中国东北地区造成的暴雨过程进行了数值模拟,结合观测资料对模拟结果进行了验证,利用模式输出的高分辨率资料,对“布拉万”台风造成的强降水及其非对称性分布的成因进行了诊断分析。结果表明,模式很好地再现了台风登陆过程中的路径、强度变化和降水分布,受中纬度西风槽带来的干冷空气影响,“布拉万”台风登陆后的降水和环流结构具有明显的不对称性,降水主要集中在台风中心西北侧的能量锋区附近。水汽散度通量和水汽螺旋度能够较好地描述强降水过程的发生、发展及其非对称性分布的时空特征,在强降水区,水汽散度通量表现为正值强信号,而水汽螺旋度表现为负值强信号,在非降水区和弱降水区,两者均表现为弱信号。等熵位涡分析显示,对流不稳定只是此次台风暴雨前期和初始阶段的不稳定条件,而湿位涡(MPV)的湿斜压项(MPV₂)则是暴雨增强和出现非对称性分布的主要机制。在暴雨形成过程中,由于冷空气侵入造成了在台风环流西北侧湿等熵面的陡立倾斜和水平风垂直切变的增强,导致了气旋性涡度的显著增强,气旋性涡度增强造成的强烈上升运动将降水区东南侧输送过来的暖湿空气向上输送,从而导致了暴雨的发生,这其中条件性对称不稳定是降水得以加强的一种重要不稳定机制。     

Testing theories of the vertical stratification of the ACC against observations

Recent theories of the Antarctic Circumpolar Current (ACC) suggest that its lateral and vertical stratification is controlled by its baroclinic instability: eddies in the ACC not only feed-off the available potential energy stored in sloping isopycnals but play a central role is setting up that stratification. Simple theory makes predictions about how the depth of the thermocline in the ACC depends on the surface winds, the air–sea buoyancy flux and transfer by baroclinic eddies. By examining gridded hydrographic data, here we test some of these predictions against observations. We show that, to a remarkable degree, the buoyancy field in the ACC decays exponentially with depth beneath the mixed layer. The e-folding depth increases equatorward, from less than 500 m on the poleward flank of the ACC to greater then 1000 m on its equatorial flank, in a manner that is broadly consistent with the theory.