Unstable modes of a sheared pycnocline above a stratified layer

Internal waves incident on a sheared ocean pycnocline are studied using analytic and numerical methods. Linear analysis of the unstable modes of a sheared ocean pycnocline is used to demonstrate interactions between internal waves and shear instabilities. A new analytic solution for an asymmetric shear layer over a stratified layer is presented, illustrating modes which couple to internal waves, in addition to the well-known Holmboe modes. The robustness of these solutions is demonstrated using numerical methods for realistic shear profiles. Fully nonlinear numerical simulations illustrate the growth of these modes and demonstrate the excitation of shear instabilities by incident internal waves. The results may have implications for internal wave interactions with the ocean pycnocline and the local generation of internal solitary waves.     

The instability of a horizontal five-layer jet

The instability of a symmetric jet moving horizontally, in which two shear layers with opposite shear of the same strength are separated by a central irrotational layer and are adjoined by unbounded, irrotational outer layers, is studied.First, the fluid is assumed to be homogeneous. Two unstable modes are found, the central wave one-quarter wave length out of phase with the outer wave. Mode I consists of central waves being in phase and outer waves being in phase. Mode II consists of central waves being in opposite phase and outer waves being in opposite phase. For a given width of the jet, the thicker the central irrotational layer, the stronger the shear of the shear layers, the stronger the instability. For a fixed ratio of the thickness of central layer to that of the shear layers, mode I is more unstable than mode II.Next, a density jump across the outer interface levels and another density jump across the central interface levels are introduced. The effect of these density jumps on mode I is to reduce the growth of the wave. The wave with equal density jump across every interface level grows somewhat slower than the waves with the entire density jump across outer or central interface levels. For an idealized velocity profile with isentropic layers with an overall Richardson number of 4.9, the linear theory predicts that the amplitude of the wave doubles in about 5 min and the wave-length is 241 m, which compares favorably with 320m obtained in the boundary layer by Gossard et al. (1970). For atmospheric parameters with an overall Richardson number of unity, linear theory predicts that the amplitude of the wave doubles in about % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaale% aaleaacaaIXaaabaGaaG4maaaaaaa!383C!\[2{\textstyle{1 \over 3}}\] min and the wave-length is about 510 m, which is only slightly larger than the width of the jet.A physical argument is invoked to explain the evolution of finite-amplitude waves.     

Analysis of internal solitary waves observed in Davis Strait

Abstract

Current meters and a thermistor chain deployed in the proximity of a drill‐ship over the continental shelf off Baffin Island revealed the presence of large amplitude internal waves. This paper reviews the properties of the internal waves, observed to propagate away from the coast and to coincide with the local low water phase of the tide at the drill‐ship. The observations are considered in terms of internal solitary wave models. A detailed comparison is presented of wave properties with a long‐wave model incorporating continuous stratification and shear.     

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.     

Interaction of internal waves and turbulence in the ocean

Within the framework of the semiempirical theory of turbulence for stratified fluids some aspects of the problem of internal wave-turbulence interaction in the upper layer of the ocean are discussed. The conditions of amplification and sustaining of turbulence by internal waves are investigated. Stationary distributions of turbulent energy are found for a stratified fluid with a shear flow produced, for example, by a low-frequency internal wave. The internal wave damping due to both turbulent viscosity and turbulent diffusion in the thermocline is studied. For a two-layer model damping constant is determined as a function of the wave number. The variation of surface turbulence by internal waves is estimated and the role of this process in slick formation is considered.     

Atmospheric Disturbances that Generate Intermittent Turbulence in Nocturnal Boundary Layers

Using the unprecedented observational facilities deployed duringthe 1999 Cooperative Atmosphere-Surface Exchange Study (CASES-99),we found three distinct turbulent events on the night of 18October 1999. These events resulted from a density current,solitary wave, and internal gravity wave, respectively. Our studyfocuses on the turbulence intermittency generated by the solitarywave and internal gravity wave, and intermittent turbulenceepisodes associated with pressure change and wind direction shiftsadjacent to the ground. Both the solitary and internal gravitywaves propagated horizontally and downward. During the passage ofboth the solitary and internal gravity waves, local thermal andshear instabilities were generated as cold air was pushed abovewarm air and wind gusts reached to the ground. These thermal andshear instabilities triggered turbulent mixing events. Inaddition, strong vertical acceleration associated with thesolitary wave led to large non-hydrostatic pressure perturbationsthat were positively correlated with temperature. The directionaldifference between the propagation of the internal gravity waveand the ambient flow led to lateral rolls. These episodic studiesdemonstrate that non-local disturbances are responsible for localthermal and shear instabilities, leading to intermittentturbulence in nocturnal boundary layers. The origin of thesenon-local disturbances needs to be understood to improve mesoscalenumerical model performance.     

用线性化全球原始方程谱模式研究地形强迫行星波垂直传播特征

本文用线性化全球原始方程谱模式来研究地形强迫行星波的垂直传播过程及演变特征.模式的积分结果表明:行星波的能量沿两支波导进行传播,在波从对流层向平流层传播中,极地波导起着主要作用.由于重力波破碎的阻尼作用,地形强迫的行星波无论它的波作用量或振幅都呈现振荡状态.     

Topography and the non-linear Rossby wave in the zonal shear basic flow

Under semi geostropical approximation, by means or phase angle function the non-linear ordinary differential equation is derived involving topography and zonal shear basic flow. Conditions for the existence of limited amplitude periodical and isolated wave solutions are directly obtained based on the qualitative theory of the ordinary differentical equation. Analysis is thus made of the influence of topography and zonal shear flow on the existence of wave solution. Finally, explicit wave solutions are determined by function approaching with the result that topogra-phy and zonal shear flow affect not only the existence but also the form of waves, indicating the non-linear features of waves and the effect of topography and shear basic flow on undulation.     

Nonlinear waves in barotropic model

In this paper, from the system of equation describing a barotropic atmosphere using the method of Taylor expansion for the nonlinear terms, the periodic solutions of the nonlinear inertio-surface gravity waves and Rossby waves have been obtained.The finite-amplitude nonlinear inertio-surface gravity waves and Rossby waves with horizontal divergence satisfy all the KdV equation. The solutions are all the cnoidal function, i, e, the cnoidal waves which in-clude the linear waves and form the solitary waves under certain conditions. For the finite-amplitude Rossby waves with horizontal divergence, we find the new dispersive relation including both the wave number and the amplitude parameter. In case of small amplitude it is reduced to the Yeh formula. It is shown that the larger the amplitude and width, the faster the finite-amplitude inertio-surface gravity waves and the slower the finite-amplitude Rossby waves with horizontal divergence propagate. The blocking or cut-off system in which the amplitude and width are large may be considered as Rossby solitary waves.     

Nonlinear evolution of a layered stratified shear flow

We investigate numerically and theoretically the nonlinear evolution of a parallel shear flow at moderate Reynolds number which has embedded within it a mixed layer of intermediate fluid. The two relatively thin strongly stratified density interfaces are centered on the edges of the shear layer. We are particularly interested in the development of primary and secondary instabilities. We present the results of a stability analysis which predicts that such flows may be unstable to stationary vortical disturbances which are a generalization of an inviscid instability first considered by G.I. Taylor. We investigate the behavior of these “Taylor billows” at finite amplitude through two-dimensional numerical simulations. We observe that the braid regions connecting adjacent primary Taylor billows are susceptible to secondary, inherently two-dimensional instabilities. We verify that these secondary instabilities, which take the form of small elliptical vortices, arise due to a local intensification of the spanwise vorticity in the braid region.     

Numerical simulation of air flow over breaking waves

The air flow above breaking monochromatic Stokes waves is studied using a numerical nonlinear model of the turbulent air flow above waves of finite amplitude. The breaking event (spilling breaker) is parameterized by increasing the local roughness at the downwind slope of the wave, just beyond the crest. Both moderate slope waves and steep waves are considered. Above steep breaking waves, a large increase (typically 100%) in the total wind stress — averaged over the wave profile — is found compared to nonbreaking moderate slope waves. This is due to the drastic increase of the form drag, which arises from the asymmetrical surface pressure pattern above breaking waves. Both increase of wave slope (sharpening of the crest) and increase of local roughness in the spilling breaker area cause this asymmetrical surface pressure pattern. A comparison of the numerical results with the recent experimental measurements of Banner (1990) is carried out and a good agreement is found for the structure of the pressure pattern above breaking waves and for the magnitude of enhanced momentum transfer. Also: Dept. of Applied Physics, Techn. Univ. Delft, Netherlands.     

Rossby wave propagation in a barotropic atmosphere

The ideas of ray tracing from geometrical optics and wave propagation in a slowly varying medium are applied to Rossby waves propagating in a barotropic atmosphere.The propagation of low-frequency Rossby waves in a zonally symmetric basic state is compared with that for stationary waves presented by Hoskins and Karoly (1981). These ideas are then used to study the propagation of Rossby waves in a basic state with zonally varying middle latitude or low latitude jets. Conditions which allow cross-equatorial wave propagation are presented. For a zonally varying middle latitude jet, there is weak wave convergence in regions of decreasing jet speed, However, this is not sufficient to explain the enhanced wave amplitude found in numerical-model experiments using a zonally varying basic state.     

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.     

On anomalous meridional circulations and Eliassen-Palm flux divergences in an idealized model of dissipating, non-breaking Rossby waves

Even in idealized models of steady, dissipating, non-breaking Rossby waves at small wave amplitude, and even in the absence of barotropic and baroclinic shear instabilities, there can be an anomalous Eliassen-Palm flux divergence in the sense that the divergence is positive when the background potential-vorticity gradient is also positive, implying upgradient eddy potential-vorticity transport. The phenomenon is illustrated in the simplest possible case of dissipation by Rayleigh friction and Newtonian cooling, and is shown by a more general argument not to be restricted to that case. The physical reason is that infrared radiative damping can act anti-dissipatively on potential-vorticity anomalies whenever the vertical disturbance structure is diffractive or evanescent, as with most real stratospheric synoptic and sub-synoptic-scale disturbances forced from below. Associated with this phenomenon are anomalous (eastward) phase tilts with height, and equatorward transformed Eulerian-mean (TEM) meridional velocities. It is pointed out that the latter is a clearcut example of a TEM circulation whose sense is opposite to that of the generalized Lagrangian-mean circulation (and the effective transport circulation in the sense of Plumb and Mahlman) induced by the same steady, small-amplitude disturbance.     

正压大气中的代数Rossby孤立波

本文研究了基本气流具有线性弱切变的非线性Rossby波,得到了一个非线性发展方程为Kubota方程,在一定的条件下,可变为Benjamin-Ono方程,并指出当代数Rossby孤立波的振幅越大时,代数Rossby孤立波的传播速度越小,基流切变越强,代数Rossby孤立波越慢,同时我们还指出在代数Rossby孤立波的振幅满足一定的条件下,代数Rossby孤立波才随纬度的增高(β减小)而减慢。并且代数Rossby孤立波的结构与大气中的偶极子阻塞是一致的。     

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.     

Particle dispersion and mixing induced by breaking internal gravity waves

The purpose of this paper is to analyze diapycnal mixing induced by the breaking of an internal gravity wave — the primary wave — either standing or propagating. To achieve this aim we apply two different methods. The first method consists of a direct estimate of vertical eddy diffusion from particle dispersion while the second method relies upon potential energy budgets [Winters, K.B., Lombard, P.N., Riley, J.J., D’Asaro, E.A., 1995. J. Fluid Mech. 289, 115–128; Winters, K.B., D’Asaro, E.A., 1996. J. Fluid Mech. 317, 179–193]. The primary wave we consider is of small amplitude and is statically stable, a case for which the breaking process involves two-dimensional instabilities. The dynamics of the waves have been previously analyzed by means of two-dimensional direct numerical simulations [Bouruet-Aubertot, P., Sommeria, J., Staquet, C., 1995. J. Fluid Mech. 285, 265–301; Bouruet-Aubertot, P., Sommeria, J., Staquet, C., 1996. Dyn. Atmos. Oceans 29, 41–63; Koudella, C., Staquet, C., 1998. In: Davis, P. (Ed.), Proceedings of the IMA Conference on Mixing and Dispersion on Stably-stratified Flows, Dundee, September 1996. IMA Publication]. High resolution three-dimensional calculations of the same wave are also reported here [Koudella, C., 1999].A local estimate of mixing is first inferred from the time evolution of sets of particles released in the flow during the breaking regime. We show that, after an early evolution dominated by shear effects, a diffusion law is reached and the dispersion coefficient is fairly independent of the initial seeding location of the particles in the flow.The eddy diffusion coefficient, K, is then estimated from the diapycnal diffusive flux. A good agreement with the value inferred from particle dispersion is obtained. This finding is of particular interest regarding the interpretation of in situ estimates of K inferred either from tracer dispersion or from microstructure measurements. Computation of the Cox number, equal to the ratio of eddy diffusivity to molecular diffusivity, shows that the Cox number varies within the interval [9, 262], which corresponds to the range of vertical eddy diffusivity measured in the interior of the ocean. The Cox number is found to depend on the turbulent Froude number squared.We show eventually that mixing results in a weak distortion of the initial density profile and we relate this result to observations made at small scale in the ocean.Comparisons between the analysis of the two-dimensional and high resolution (256³) three-dimensional direct numerical simulations of the primary wave were also conducted. We show that the energetics and the amount of mixing are very close when the primary wave is of small amplitude. This results from the fact that, for a statically stable wave, the dynamics of the initially two-dimensional primary wave remains mostly two-dimensional even after the onset of wavebreaking.     

三层模式中风场对背风波的影响

为了研究风场对背风波的影响,针对边界层附近为弱稳定层结的背风波,建立了一个三维三层的理论模型和线性计算模式,分析了各层中风速和风向的变化对背风波特征的影响,揭示了气流过孤立山脉产生背风波的有利风场条件。结果表明：背风波的波长、振幅等特征对各层风速和风向的变化具有相当的敏感性,波长随着低、高层风速的增大而增大,随着中层风速的增大先减小后增大;振幅随着低、中层风速的增大先增大后减小,随着高层风速的增大而增大。此外,风速和上下层风向切变的增大均使背风波的形态逐渐由横波型转为辐散型,但是上下层风向的切变对背风波形态的影响比风速更为显著。     

Over-reflection and instability

Stability problems involving parallel shear flow are considered in the context of wave-reflection problems. It is found that if an unstable solution exists and its growth rate is sufficiently small, the growth rate can be connected to the reflection coefficient through a formula as if the unstable growth were the direct result of repeated over-reflections. If the stability problem under consideration has time symmetry, then for every growing solution there exists a corresponding decaying solution. It is shown that a consistent formula can also be derived for the decaying solution, and the existence of at least one critical layer in the corresponding wave-reflection problem is needed in order to account for both the growing and decaying solutions. As an application of these concepts, the small-scale non-geostrophic instabilities found by Stone (1970) are identified to be associated with over-reflection of inertia-gravity waves.     

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。