Energy- and Flux-Budget Turbulence Closure Model for Stably Stratified Flows. Part II: The Role of Internal Gravity Waves

We advance our prior energy- and flux-budget (EFB) turbulence closure model for stably stratified atmospheric flow and extend it to account for an additional vertical flux of momentum and additional productions of turbulent kinetic energy (TKE), turbulent potential energy (TPE) and turbulent flux of potential temperature due to large-scale internal gravity waves (IGW). For the stationary, homogeneous regime, the first version of the EFB model disregarding large-scale IGW yielded universal dependencies of the flux Richardson number, turbulent Prandtl number, energy ratios, and normalised vertical fluxes of momentum and heat on the gradient Richardson number, Ri. Due to the large-scale IGW, these dependencies lose their universality. The maximal value of the flux Richardson number (universal constant ≈0.2–0.25 in the no-IGW regime) becomes strongly variable. In the vertically homogeneous stratification, it increases with increasing wave energy and can even exceed 1. For heterogeneous stratification, when internal gravity waves propagate towards stronger stratification, the maximal flux Richardson number decreases with increasing wave energy, reaches zero and then becomes negative. In other words, the vertical flux of potential temperature becomes counter-gradient. Internal gravity waves also reduce the anisotropy of turbulence: in contrast to the mean wind shear, which generates only horizontal TKE, internal gravity waves generate both horizontal and vertical TKE. Internal gravity waves also increase the share of TPE in the turbulent total energy (TTE = TKE + TPE). A well-known effect of internal gravity waves is their direct contribution to the vertical transport of momentum. Depending on the direction (downward or upward), internal gravity waves either strengthen or weaken the total vertical flux of momentum. Predictions from the proposed model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulations and large-eddy simulations.     

Observations of an internal gravity wave in the lower troposphere at Halley,Antarctica

Observations of internal gravity waves in the stably-stratified atmospheric boundary layer at Halley, Antarctica are presented. These were made on 1 February, 1986 and take the form of temperature measurements from a 30 m mast and a Sodar record. The temperature record shows a clearly defined, dominant wave period of around 11 min. A high-resolution radiosonde ascent made during the period of wave activity exhibits thin layers of low Richardson number and it is suggested that these are regions of dynamic instability where the waves are generated. A linear stability analysis of the radiosonde data supports this idea. It is argued from simple theoretical ideas and by means of a numerical model that only waves with a wavelength greater than a certain critical value are likely to be observed at the surface. The observations are shown to be consistent with this hypothesis.     

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.     

A NOTE ON THE APPLICATION OF LINEAR WAVE THEORY AT A CRITICAL LEVEL

The purpose of this note is to estimate the accuracy and practical limitations of applying linear theory at a critical level over a realistic range of atmospheric stabilities for an idealized surface terrain. These estimates are made by comparing the results of a linear model with a nonlinear numerical model at a critical level. Essentially similar results are obtained from each model for wave stress, wave breaking height and wave dissipation through the critical level. Because gravity waves can be either evanescent or internal depending on the relative sizes of the Scorer parameter and the wavenumber of the ground surface disturbance, the somewhat paradoxical result develops that wave breaking and non-linearity increase with increasing bulk Richardson number. It is recommended that steady linear wave theory be used in gravity wave drag parameterizations provided near real time profiles of background velocity and temperature are available.     

A unified linear theory of wavelike perturbations under general ocean conditions

A linearized instability analysis model with five unknowns was proposed to describe disturbance motions under general oceanic background conditions, including large-scale current shear, density stratification, frontal zone, and arbitrary topography. A unified linear theory of wavelike perturbations for surface gravity waves, internal gravity waves and inertial gravity waves was derived for the adiabatic case, and the solution was then found using Fourier integrals. In this theory, we discarded the assumptions widely accepted in the literature concerning derivations of wave motions such as the irrotationality assumption for surface gravity waves, the rigid-lid approximation for internal gravity waves, and the long-wave approximation for inertial gravity waves. Analytical solutions based on this theory indicate that the complex dispersion relationships between frequency and wave-number describing the propagation and development of the three types of wavelike perturbation motions include three components: complex dispersion relationships at the sea surface; vertical invariance of the complex frequency; and expressions of the vertical wave-number (phase). Classical results of both surface waves and internal waves were reproduced from the unified theory under idealized conditions. The unified wave theory can be applied in the dynamical explanation of the generation and propagation properties of internal waves that are visible in the satellite SAR images in the southern part of the China Seas. It can also serve as the theoretical basis for both a numerical internal-wave model and analytical estimation of the ocean fluxes transported by wavelike perturbations.     

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.     

Instability of internal waves near a critical level

The three-dimensional stability problem is investigated for a family of velocity and density profiles similar in form to those expected for large-amplitude internal gravity waves near a critical level. These profiles exhibit regions of high shear and stable stratification alternating with regions of weak shear and unstable stratification. Analytical solutions are given for inviscid, neutral modes that are similar to those found under neutral conditions with stable stratification. Neutral modes form closed streamline patterns centered at locations of maximal shear, and are not strongly influenced by nearby regions of unstable stratification. Unstable modes are computed numerically. It is shown that the instability mechanism for these wave-like flows fundamentally three-dimensional in character and exhibits both shear and convective dynamics. For flows with parameter values below the neutral curves, unstable modes oriented in the streamwise direction undergo shear instability, while modes oriented orthogonally are convectively unstable. In addition to their intrinsic physical relevance, the results of this study have important implications for the physics and the numerical modeling of breaking internal gravity waves. Two-dimensional models will bias the breaking dynamics by eliminating the possibility for convection oriented in the transverse plane.     

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1...     

Turbulence in stably stratified fluids: A review of laboratory experiments

This is a review of laboratory studies of mixing in stably stratified fluids away from the direct intluence of boundary layers, and was written to introduce the session on laboratory experiments at the IUCRM Colloquium.Internal waves (Section 2) can lead to turbulence by creating regions of unstable density gradients through their mutual interactions, by individually breaking by forming rotors, and by reducing the local Richardson number until Kelvin-Helmholtz instability results. They may be important in radiating energy from turbulent or spreading regions. Critical-layer absorption of internal waves is not found to be a direct cause of turbulence in experiments with 15 Richardson number 5, although the modification of the vertical density profile may be significant.Turbulent-laminar interfaces without mean shear (Section 3) and with mean shear (Section 4) are described. The source of turbulence in these experiments is partly external, either generated by an oscillating grid or by a (relatively) moving boundary. The development of turbulence generated entirelywithin a stratified layer by Kelvin-Helmholtz instability, is described in detail in Section 5, and the results are compared with measurements in the ocean and in the atmosphere.     

二维波状地形上的流场

在一个简化的二层模式中，求解大气波动方程，得到了二维波状起伏地形上扰动流线的分析解。研究了在上、下两层大气中，不同的温度廓线和风速廓线情况下，地形引起扰动的流场形式，同时讨论了支配扰动振幅的大气因子和地形特征。分析解的结果表明:若大气低层为深厚的不稳定层，地形引起的波动很微弱;如低层大气强稳定，上层大气弱稳定，则可能产生较强的波动;而当上层稳定度增加时，可产生非陷波，有利于高层动量下传，造成较大的地面风速。     

CHARACTERISTICS OF GRAVITY WAVE PROPAGATION AND ENERGY CONVERSION IN A SUDDEN HEAVY RAINFALL EVENT

In this paper, a sudden heavy rainfall event is analyzed, which occurred over the Yellow River midstream during 5–6 August 2014. We used observational, NCEP/NCAR reanalysis, high-resolution satellite, and numerical simulation data. The main results are as follows. Under an unfavorable environmental circulation, inadequate water vapor and unfavorable dynamic conditions but sufficient energy, a local sudden heavy rainfall was caused by the release of strong unstable energy that was triggered by cold air transport into middle and lower layers and the propagation of gravity waves. The distributions of rain area, rain clusters, and 10-minute rainfall showed typical mesoscale and microscale fluctuation characteristics. In the mesoscale rain area or upstream, there was a quasi-stationary wave of mesoscale gravity waves with their propagation downstream. In the course of propagation from southwest to northeast, the wavelength became longer and the amplitude attenuated. In the various phases of gravity wave development, there were evident differences in the direction of the wave front. Wave energy was mainly in the lower layers. Unstable vertical wind shear at heights of 1–6 km provided fluctuation energy for the gravity waves. The mechanisms of heavy rainfall formation were different at Linyou and Hancheng stations. Diabatic heating was the main source of disturbed effective potential energy at Linyou. The explosive short-period strong precipitation was caused by the release of strong effective potential energy triggered by the gravity waves, and its development and propagation after that energy maximized. In contrast, the latent heat release of upstream precipitation was the main source of disturbed effective potential energy at Hancheng. This formed a positive feedback mechanism that produced continuous precipitation. In the studied event, the development of westerly belt systems had disturbed the wind field. The contribution of kinetic energy generated by this disturbance could not be ignored. The Froude number, mountain shape parameter, and ratio between mountain height and temperature inversion layer thickness had various effects of atmosphere and terrain on mesoscale and microscale mountain waves. In upper and lower layers, there were five airflows that were strengthened by the terrain. All these had important influences on local heavy rainfall at Linyou and Hancheng stations.     

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.     

Richardson number profiles through shear instability wave regions observed in the lower planetary boundary layer

Remote sensing of the lower planetary boundary layer in the vicinity of a meteorological tower on many occasions reveals the existence of shear instability (Kelvin-Helmholtz) waves. In general, such waves are found within shallow strata which are marked by strong thermal stability and large vertical wind shear. The independent and concurrent measurements of the vector wind and temperature, made on a 152-m high tower, allow the construction of wind and temperature profiles. From such measurements, the Richardson number profile is constructed as well as the instability regime according to Drazin's criterion. The results show that regions of shear-instability waves as depicted by the remote sensor (an acoustic sounder) agree well with Drazin's instability regime, and that within such regions the Richardson number is indeed 0.25.     

青藏高原东北边坡复杂地形重力波的数值模拟

基于2007年7月青海祁连站的野外加密探空资料,结合高分辨率的三维边界层模式,模拟研究了青藏高原东北边坡复杂地形条件下,边界层对流引起的干动力过程对该地区地形重力波产生及传播的影响机理。结果表明:在不同的背景场强迫下,高原东北边坡复杂地形上空对流和重力波的空间结构存在较大差异。当背景风向与山体垂直时,随着风速增加,山脊背风坡混合层顶附近大气不稳定能量加强,激发了下游区域较强的重力波信号,此时对流线组织性增强、重力波波列较长,高水汽含量的空气被波峰传输到较高的高度,为对流云发展提供了有利条件;当背景风向与山脊走向平行时,山顶上空对流发展旺盛,山脊背风坡混合层顶大气状态较稳定,激发的地形重力波信号较弱且波列较短,整个混合层顶附近水汽较少,对流云形成条件减弱;当背景大气浮力频率减小时,整个区域上空对流发展更加旺盛但组织性减弱,背风坡下游重力波向上传输的距离减小,信号不显著,混合层顶附近水汽分布均匀且变化幅度较小,有利于层状云发展。     

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.     

Radar and sodar probing of waves and turbulence in statically stable clear-air layers

This paper reviews the remote sensing of waves and turbulence in statically stable atmospheric layers, utilizing sodar and microwave radar echoes from the small-scale inhomogeneities in gaseous refractive index caused by localized fluctuations in temperature, humidity, and velocity. Scattering theory and sounding methodology are reviewed briefly, and the relative performance of typical radar and sodar systems compared. The main section of the paper takes the form of a summary and discussion of experimental progress since 1969, showing how the echo patterns obtained may be applied to the interpretation of multiple layering, gravity waves, internal fronts and the details of dynamic instability and the genesis of turbulence in stably stratified shear layers. In addition, methods for the measurement of the intensity of the small-scale ( /2) variability of wind, temperature and water vapor from the observed radar or sodar echo intensities, and the use of Doppler techniques for the measurement of mean velocity and turbulence are discussed.SODAR from SOund Detection And Ranging in analogy to RAdio Detection And Ranging.     

Observation of gravity waves in a boreal forest

In this paper we report the results of the analysis of two 60-min wave events that occurred in a boreal aspen forest during the 1994 BOREAS (Boreal Ecosystems-Atmosphere Study) field experiment. High frequency wind and temperature data were provided by three 3-D sonic anemometer/thermometers and fourteen fine-wire thermocouples positioned within and above the forest. Wave phase speeds, estimated from information revealed by spectral analysis and linear plane wave equations, are 2.2 and 1.3 m s^-1 for the two events. The wavelengths are 130 m and 65 m respectively and are much larger than the vertical wave displacements. There is strong evidence from the present analysis and from the literature supporting our postulate that these waves are generated by shear instability. We propose that wind shear near the top of the stand is often large enough to reduce the gradient Richardson number below the critical value of 0.25 and thus is able to trigger the instability. When external conditions are favorable, the instability will grow into waves.     

An energy budget for waves and turbulence within an inversion

Several features of the maintenance of breaking gravity waves and turbulence in a marine inversion are examined. A formulation is proposed for a critical Richardson number based on a mutual response of the mean and turbulent states to a wave-like disturbance. The energy balance, based on averaged aircraft soundings, is examined to ascertain the order of magnitude of the component terms in the vicinity of a contemporaneous radar echo. Some physical mechanisms are discussed which may explain some aspects of the sustained existence of the echo layers.After preparation of this paper, a considerable number of pertinent and up-dated results bearing on waves and turbulence in stable layers were published as Volume 4 ofBoundary-Layer Meteorology, April 1973. The present discussion should be considered in relation to these papers and of Gossardet al. (1973) and Metcalf and Atlas (1973) in particular.     

Evaluation and model impacts of alternative boundary-layer height formulations

We study bulk formulations for the boundary-layer height which are currently in use for atmospheric modelling. The formulations are based on various forms of the Richardson number, and these are evaluated with Cabauw field data in stable conditions. Results for both a large-eddy simulation model and anE-ε turbulence closure model for neutral boundary layers are also utilised. An updated formulation is introduced, which combines the effects of shear in the outer region of the boundary layer with surface friction. The updated formulation has a better performance for neutral boundary layers with upper level stratification. The findings are illustrated with a single-column model for a case with relatively high winds over the tropical ocean including shallow cumulus convection, and for a case with fair weather over land. We also show that for stable conditions, the updated formulation performs better than estimates on the basis of surface friction alone.     

Atmospheric-stability effect on windbreak shelter and drag

Wind speed was measured near the surface, along a line normal to a single, 50% porous windbreak, to determine its wind-reducing effect. Simultaneously, undisturbed wind and temperature profiles were measured to obtain the atmospheric stability. In some of the runs, the drag on a section of the windbreak was also measured. A systematic and significant effect of the atmospheric stability was found. The relative wind speed at any distance from the windbreak could be expressed for unstable conditions as an empirical function of the Richardson number. As an extreme example, the reduction of the surface shear, at a downwind distance 7.5 times the height of the windbreak, was 86 % in adiabatic conditions, and only 62 % when the gradient Richardson number at the windbreak height was ?1.0. The minimum relative effect was found in the neighborhood of the windbreak, where the windbreak-induced turbulence was dominant. The maximum absolute effect was found at a downwind distance of about ten times the height of the windbreak. The drag coefficient of the windbreak was found to bec _d =0.77 in a neutral atmosphere, increasing slightly with instability. This trend is contrary to the trend of the surface-shear-reduction coefficient, which decreases significantly with instability.