Structures of dynamically unstable shear flows and their implications for shallow internal gravity waves part II: Nonlinear response

Summary The nonlinear response of a dynamically unstable shear flow with critical level to an initial temperature anomaly is investigated using a nonlinear numerical model. Both nonconstant and constant shear profiles of the basic flow are considered. Effects of the solid lower boundary on the dynamically unstable, nonlinear flow are also studied. It is found that in a dynamically unstable, linear flow with a hyperbolic tangent wind profile, the updraft is tilted upshear. The result in consistent with that of a linear stability model (LC). The upshear tilt can be explained by the Orr mechanism (1907) and the energy argument proposed by LC. In a dynamically unstable, nonlinear flow, the updrafts produced by a sinusoidal initial temperature perturbation are stronger in the lower layer and are more compact and located further apart compared to the corresponding linear flow. In addition, the perturbed wave energy is slightly smaller than the linear case. It is found that the growth rate is smaller during the early stage and much larger during the later stage. For a localized initial temperature perturbation in a dynamically unstable flow, a stronger updraft with two compensated downdrafts are produced. Gravity waves are produced in a dynamically stable flow with both a hyperbolic tangent wind profile and a linear wind profile. For a linear shear flow with Richardson number less than 1/4, the disturbance grows in the early stage and then decays algebraically at later times, similar to that found in other linear theoretical studies. The influence of the solid lower boundary is to suppress the shear instability in a nonlinear flow with a hyperbolic tangent wind profile ofRi<1/4.With 9 Figures     

Enhanced response of an atmospheric flow to a line-type heat sink in the presence of a critical level

Summary Response of the atmosphere to a line-type heat sink, which represents a cold pool induced by evaporative cooling in falling precipitation, is investigated. Two-dimensional, steady-state, linear perturbations forced by the diabatic cooling in the presence of a critical level are solved analytically. The ambient flow is assumed to have a three-layer structure which is the same as that in Lindzen and Tung (1976, denoted as LT hereafter) except that an isolated diabatic cooling is specified in the lower layer. The shear layer with a critical level is assumed to be either dynamically stable or unstable. A steady-state solution is possible even for a dynamically unstable flow in the middle layer because the wave energy is allowed to propagate to infinity in the upper layer.Thermally induced wave response near the critical level depending on the stability and wind shear in terms of the Richardson number exhibits the same behavior as the eigenvalue problem solved by Jones (1967) and LT because the source mechanism does not change the critical level behavior. That is, when the shear layer is dynamically stable, almost all of the wave energy is absorbed near the critical level, while for dynamically unstable case waves can be partially-or over-reflected from the critical level depending on the Richardson number. Waves can be over-reflected and over-transmitted simultaneously as they travel through the critical level in the process of the energy extraction from the unstable mean shear flow.Using the duct condition proposed by LT, it is found that the magnitude of the perturbation vertical velocity in the lower layer is 6 times larger than that for a nonducting case with the same cooling rate. This implies that under a proper choice of the critical level height and stability profile, the response of the atmosphere to a line-type heat sink can be significantly enhanced through the over-reflection process.With 7 Figures     

"96.1"高原暴雪过程三维条件性对称不稳定的数值研究

利用“96．1”青藏高原东北部暴雪过程的较成功的中尺度模拟的输出资料,用非纬向非平行基流中的对称不稳定模式,对“96．1”暴雪发生发展过程的动力学机制进行了三维基流中二维非线性对称不稳定数值模拟试验。结果表明,ψ场和ω场的三维配置与切变线的演变过程相当一致,说明用三维基流中二维非线性对称不稳定可能能较好地描述SI在切变线东伸南移过程中所起的作用。     

两层正压流体涡旋中螺旋波的不稳定

利用线性化的两层正压原始方程模型，对有水平和垂直切变基流的圆形涡旋中螺旋波的不稳定作了研究。结果表明，当基流失稳时，涡旋中不稳定扰动的厚度场、速度场在上、下两层都具有明显的螺旋结构，下层的螺旋结构要较上层复杂。基流垂直切变越大则越易失稳。失稳时上、下层扰动的配置接近反位相，故该螺旋波结构相应于斜压模。此时螺旋波上的扰动中心在切向是逆基流传播的，在径向则基本没有传播，而螺旋臂的整体运动缓慢。失稳的螺旋波其散度场要较涡度场明显，物理量的配置也大体符合重力惯性波的情况，故可认为其是重力惯性波的不稳定所致。本模型中该螺旋波的形态与实际热带气旋中的螺旋云(雨)带很相象。     

Wave/wave interaction producing horizontal mean flows in stably stratified fluids

We show that internal wave/wave interactions in stratified fluids are able to produce strong horizontal mean currents. A simple analytical model allows us to estimate the amplitude of the time-periodic horizontal mean flow induced by the interaction of two monochromatic waves. This model shows that in some cases, the mean flow velocity can overgo a threshold beyond which critical layers and intense energy transfers from the waves to the mean flow are expected. This prediction is confirmed by direct pseudo-spectral simulations of the Navier–Stokes equations under the Boussinesq approximation. Such interactions may help to further understand the presence of strong vertical shear observed in the final stage of stratified flows in oceans and atmospheres.     

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.     

三维非静力平衡大气中平行切变流上重力内波的不稳定及半椭圆定理

本文应用守恒型非弹性模式研究了三维非静力平衡大气中平行切变流上重力内波的不稳定，推广了Miles定理和半圆定理，得到了有限深度气层的半椭圆定理。该定理指出不稳定波的复波速被限制在相速复平面上一个内接于霍华德（Howard）半圆的半椭圆内。该半椭圆长轴重合于半圆的直径，而短轴则取决于最小Ri数、波长、波宽等因素。     

风速垂直切变下非均匀层结大气中重力惯性内波的稳定性

本文应用WKB方法研究了在弱非均匀层结大气中,当基本气流具有弱垂直切变时,重力惯性内波的稳定性问题.由导得的波能量方程出发,分析了风速垂直切变及非均匀大气层结对重力惯性内波波能变化率的影响.     

Properties and stability of a meso-scale line-form disturbance

By using the 3D dynamic equations for small- and meso-scale disturbances, an investigation is performed on the heterotropic instability (including symmetric instability and traversal-type instability) of a zonal line-like disturbance moving at any angle with respect to basic flow, arriving at the following results: (1) with linear shear available, the heterotropic instability of the disturbance will occur only when flow shearing happens in the direction of the line-like disturbance movement or in the direction perpendicular to the disturbance movement, with the heterotropic instability showing the instability of the internal inertial gravity wave; (2) in the presence of second-order non-linear shear, the disturbance of the heterotropic instability includes internal inertial gravity and vortex Rossby waves. For the zonal line-form disturbance under study, the vortex Rossby wave has its source in the second-order shear of meridional basic wind speed in the flow and propagates unidirectionally with respect to the meridional basic flow. As a mesoscale heterotropic instable disturbance, the vortex Rossby wave has its origin from the second shear of the flow in the direction perpendicular to the line-form disturbance and is independent of the condition in the direction parallel to the flow; (3) for general zonal line-like disturbances, if the second-order shear happens in the meridional wind speed, i.e., the second shear of the flow in the direction perpendicular to the line-form disturbance, then the heterotropic instability of the disturbance is likely to be the instability of a mixed Rossby–internal inertial gravity wave; (4) the symmetric instability is actually the instability of the internal inertial gravity wave. The second-order shear in the flow represents an instable factor for a symmetric-type disturbance; (5) the instability of a traversal-type disturbance is the instability of the internal inertial gravity wave when the basic flow is constant or only linearly sheared. With a second or nonlinear vertical shear of the basic flow taken into account, the instability of a traversal-type disturbance may be the instability of a mixed vortex Rossby – gravity wave.     

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.     

MESOSCALE NONLINEAR GRAVITATIONAL WAVES AND THEIR INTERACTION IN VERTICAL SHEAR FLOW*

Interaction equations of two nonlinear gravitational waves in baroclinic atmosphere are presented via multi-scale perturbation method,which can be classified into coupling nonlinear Schrodinger equations.In particular,the interaction course of two nonlinear gravitational waves of basic flow in vertical linear and quadratic shear is illustrated.Numerical calculation displays that wave amplitude enlarges and wave width narrows when two solitary gravitational waves meet and chase;that basic flow with single shear is more beneficial than that with quadratic shear to the interaction of two nonlinear wave packets;and that the interaction of two wave packets makes wave shape change more greatly and energy more dispersive,which contributes to the occurrence of changeable weather.Therefore,one of the probable mechanisms for the appearance of strong convection weather is the interaction between mesoscale nonlinear gravitational waves.     

A model of turbulence spectra in the atmospheric surface layer

The spectral equations of turbulent kinetic energy and temperature variance have been solved by using Onsager's energy cascade model and by extending Onsager's model to closure of terms that embody the interaction of turbulent and mean flow.The spectral model yields the following results: In a stably stratified shear flow, the peak wave numbers of the spectra of energy and temperature variance shift toward larger wave numbers as stability increases. In an unstably stratified flow, the peak wave numbers of energy spectra move toward smaller wave numbers as instability increases, whereas the opposite trend is observed for the peak wave numbers of temperature variance spectra. Hence, the peak wave numbers of temperature spectra show a discontinuity at the transition from stable to unstable stratification. At near neutral stratification, both spectra reveal a bimodal structure.The universal functions of the Monin-Obukhov similarity theory are predicted to behave as _m ~ _H ~ (- Z/L)^-1/3 in an extremely unstable stratification and as _m ~ _H ~ z/L in an extremely stable stratification. For a stably stratified flow, a constant turbulent Prandtl number is expected.     

Substructure layers and modes in the stratified boundary layer

Geophysical flows include small-scale substructures that support shear instabilities where the smoothed mean profiles indicate only stability. The substructures must draw their energy from the mean flow, probably through wave interactions, and it is shown that the substructures themselves give rise to a type of mode that is well suited for nonlinear interactions with the flow in and around them. The study indicates that substructures and their associated modes form a dynamically interacting system that may contribute to the chaotic nature of a stably stratified region.     

最敏感扰动的演变与夏季乌拉尔地区的持续性异常环流

通过数值实验对初始扰动不稳定发展建立异常环流型的过程进行了分析。选择了东亚夏季风异常的１９９１年和１９８５年两个典型年份的平均环流作为基流,以共轭敏感性分析得到的最敏感扰动作为初始扰动,分析其在不同的环流背景下的发展情况。结果说明,１９９１年欧亚地区的基流不稳定性较强,且初始扰动结构有利扰动发展,因而通过能量频散在乌拉尔地区激发扰动并强烈发展,建立起阻塞形势。而１９８５年的扰动在乌拉尔地区形成正涡度距平,不利于阻高的建立。从而导致１９９１年和１９８５年夏季乌拉尔地区几乎相反的异常环流。一系列对比实验说明,异常环流型的建立,不仅依靠能量的频散,更依靠扰动通过正压不稳定过程从基流吸收能量而发展。它既有赖于基流自身的不稳定,又取决于初始扰动的结构以及相对于基流的位置。基流选择了具有特定结构的初始扰动型,只要在有利位置上给予扰动,就能够激发出扰动波列,建立强的持续异常环流型。不论初始波列的位相在一定范围内如何改变,扰动总是倾向于在基流的特定不稳定区域发展。     

重力波与对流耦合作用在一次山地突发性暴雨触发中的机理分析

利用欧洲中期天气预报中心（ECMWF）开发的新一代ERA5再分析资料、中国自动站与CMORPH降水产品融合的逐小时降水资料以及国家卫星气象中心FY-2G卫星云图资料,对2018年5月21～22日发生在四川盆地西南部的一次山地突发性暴雨过程中的重力波特征进行天气动力学分析。得到以下结果:此次山地突发性暴雨受到了波长约为150 km,周期为5 h的重力波活动的影响,是典型的β中尺度天气系统诱发的暴雨事件。此过程中的重力波主要是在地形和切变不稳定的共同作用下触发的。切变不稳定先于重力波的传播出现在下游降水区域,可表征切变不稳定的理查逊数对重力波传播方向及降水落区有很好指示作用。此次暴雨发生前,重力波中的上升支气流输送低层水汽到高空助力对流发展,而下沉支气流使得低层不稳定能量不断累积。随着东北低空急流的发展,在大气低层（700～800 hPa）东西风切变的过渡带内形成临界层,临界层不断吸收高空波动能量造成重力波能量下传,触发低层不稳定能量释放,促使对流不断加强,最终引发此次山地突发性暴雨。     

The Stability and Existence Condition of Nonlinear Travelling Wave Solution in Basic Uniform Flow

Summary  Nonlinear waves in a stratified atmosphere with basic uniform flow are studied by adopting geostrophic momentum approximation (cf. Hoskins, 1975) and the method of travelling wave solution. A nonlinear equation containing a unique variable of vertical p velocity is derived from a complete system of equations considering frictionless and adiabatic fluid. The stability of the solution for the nonlinear equation is discussed then. Furthermore, the approximate cnoidal wave solution, solitary wave solution and their existence condition are obtained. Received July 8, 1998 Revised December 1, 1998     

斜压切变基流中横波型扰动的特征波动 Ⅱ:谱函数

“斜压切变基流中横波型扰动的特征波动Ⅰ谱点分布”一文中分析了斜压切变基流中横波型扰动的谱点分布，这里又对其谱函数进行了分析讨论。结果表明当基流在垂直方向存在切变时，重力惯性波与涡旋波的谱函数在垂直方向上均可出现临界层，临界层的高度随频率σ而变化，即重力惯性波与涡旋波都存在连续谱，但涡旋波与重力惯性波连续谱的结构却不同；对天气尺度扰动，两支重力惯性波和1支涡旋波的连续谱不重叠，此时每支波动仅有1个临界层；而对次天气尺度的扰动，重力惯性波与涡旋波的连续谱区会发生重叠，在连续谱的重叠区，重力惯性波仍只有1个临界层，但涡旋波则可以有2个或3个临界层。无论是涡旋波还是重力惯性波其连续谱的波包随时间都是衰减的，但涡旋波波包比重力惯性波波包衰减得慢。     

赤道两层海洋模式中基本流的切变不稳定性

设计了一个热带赤道β-平面的两层海洋模式,在准长波近似下,应用最大截断模分析赤道波的基本形态,指出无论是正压模或斜压模Kelvin波、Rossby波及基本流所对应的“地形Rossby波”是最基本的波系,在基本流的一定切变条件下,它们之间可以耦合出一类不稳定波。在浅混合层近似和“快波近似”下,正压模和斜压模是可以分离的,因此可以分别分析它们的色散特征,由于它们的特征量不同,在同样波长(扰动的纬向尺度)下,扰动的增长率也不同,通过分析得出在一定参数下,斜压模扰动增长率为正压模的2倍。近似分析表明,混合层中流场的增长要快于温跃层,但温跃层的温度增长要比混合层明显。     

Numerical experiments on internal gravity waves in an accelerating shear flow

The influence of an accelerating shear flow on the propagation of an internal gravity wave in a continuously stratified fluid is studied by means of two-dimensional numerical simulations. These are motivated by earlier laboratory experiments [Thorpe, S.A. 1978b. On internal gravity waves in an accelerating shear flow, Vol. 88. J. Fluid Mech. pp. 623–639]. In these experiments the mean flow is an accelerated Couette flow and the mean density profile is linear. The laboratory experiments revealed the striking effect of the unsteady shear flow in the evolution of an internal gravity wave leading to the wave focusing in a region where the flow is extremum. This phenomenon is associated with the growth of small scale density fluctuations. As a result density overturns are sometimes observed. This behaviour is well reproduced by the numerical simulations. We provide insights on the flow dynamics in particular on the possible occurrence of wavebreaking. We show that the dynamics is characterized by two competitive mechanisms that is a damping of the wave and a local enhancement of its steepness leading sometimes to density overturns. The budget for the energy of the wave reveals that the initial damping of the wave results from wave-mean flow interactions. These interactions lead to the development of a fine scale vertical density structure which is associated with high vertical shear. We find that in some cases wavebreaking occurs as a result of shear instability. The value of the acceleration of the mean flow is very likely to influence the onset of the instability. The scaling laws of the wave evolution, in particular the rate of decrease of its energy, are determined. From these laws the lifetime of the wave is found as a function of the acceleration of the shear. It may be expected that, in the ocean, this development will result in the largest fluctuations derived from wave-flow interactions occurring where the mean flow in the wave direction is greatest. Waves travelling normal to a two-dimensional shear flow will be unchanged. Waves travelling parallel will be damped. This may have particular application at the continental shelf where flow, mainly parallel to the isobaths, will damp waves travelling along-slope, but allows waves travelling normal to the isobaths (e.g., directly across the shelf-break) to be transmitted without attenuation. Similar effects are expected for the evolution of a high frequency wave interacting with a lower frequency (e.g., near inertial) motion.     

The interaction of ellipsoidal vortices with background shear flows in a stratified fluid

We show that an arbitrarily oriented ellipsoid of uniform potential vorticity, embedded in a background flow described by a quadratic streamfunction, is an exact solution of the quasigeostrophic equations governing motion in a uniformly stratified, unbounded fluid. This type of flow includes plane horizontal shear and strain as well as uniform vertical shear of a unidirectional horizontal flow. We derive ordinary differential equations describing the motion of such a vortex and discuss some aspects of their solutions. We note the existence of steady states (solutions in which the vortex is in equilibrium with the background flow), of periodic solutions near these steady states, of non-periodic trajectories which nervertheless remain in the vicinity of the steady states, and of solutions which represent the shearing out of the vortex by the background flow. We try to use this information to propose partial answers to the question of when a given horizontal or vertical shear flow is likely to destroy a vortex and when a vortex might survive external shear and strain.