Experiments on high Richardson number instability of a rotating stratified shear flow

A recently discovered mechanism for producing step-like structure in a fluid's density field is explored under controlled laboratory conditions. Measurements of the conditions under which these “layers” exist, their direction, wavelength, and growth rate, are made and compared to the predictions of instability theory. The good agreement indicates that the layers are a result of a diffusive instability of a geostrophic shear flow. The instability can occur at higher and higher Richardson numbers, as the Schmidt number (ratio of diffusivities of momentum and salt) increases. Comparison of the density and velocity fields in the presence and absence of fully developed layers indicates that the instability functions mainly to transport momentum. Extrapolation of the measured wavelength to midlatitude oceanic conditions suggests that the momentum-density diffusive instability may play a role in producing meter-size oceanic microstructure.     

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.     

The dependence of the bulk Richardson number on stability in the surface layer

Relationships between the bulk Richardson number, B, and other stability parameters were derived for the atmospheric surface layer. Nomograms were constructed, relating the Monin-Obukhov stability parameter, z/L, to B. The nomograms and the graph of Golder (1972) were used to establish various schemes for classifying z/L, B and Ri, in terms of the Pasquill stability classes.     

Richardson number,vertical wind shear and storm occurrences at Kano,Nigeria

Radiosonde data for thirteen summer months have been used to relate Richardson under Ri and vertical wind shear to storm occurrences at Kano. It is shown that thunderstorms occur most frequently in association with low-level shears, ΔU_L, below the Africal Easterly Jet (surface to 700 mb) within −20⩽ΔU_L⩽−5 ms⁻¹ and for the 700-400 layer, ΔU_m, in the range 0<ΔU_m<10 m s⁻¹. The Richardson number with which storm occurrences are most common is bi-modal in both lower and middle troposphere: −2⩽Ri⩽0 and Ri⩽ −10 in the boundary layer (surface to 900 mb); 1⩽Ri⩽4 and Ri⩾15 in the inflow region (origin) of the downdraft air between 800 and 600 mb.Storms rarely occur (one in every seven cases) if boundary layer Ri>0 and virtually no storm should be expected if the boundary layer Ri>0 and for DDR (the regional origin of down draft air) Ri satisfying 4<Ri<15 simultaneously. The lower cut-off (Ri ⩾ 1) for the inflow air is close to the value (Ri⩾2) obtained by Moncrieff and Miller (1976) for propagating tropical storms.     

Inertial instability of large Rossby number horizontal shear flows in a thin homogeneous layer

A horizontal shear flow having a Rossby number, Ro, greater than unity on a rotating plane can become unstable when its shear value is less than −f, the Coriolis frequency. In this paper, this instability is investigated for an O(10 km) submesoscale, sinusoidal shear flow in a thin homogeneous fluid layer as in an oceanic mixed layer or a shallow sea. The most unstable mode is shown by a linear analysis to occur in a narrow localized region centered around the maximum anticyclonic current shear. However, nonlinear numerical calculations show that the instability can grow to encompass both unstable and stable regions of the current. A consequence of this finite-amplitude evolution is the formation of surface convergence/shear fronts. The possibility that inertial instability mechanism is a source of some surface convergence/shear features seen in remote sensing images of the sea surface is discussed. A comparison is made with the shear-flow instability that can occur concurrently in a sinusoidal shear current, and inertial instability is shown to be the dominant instability mechanism in the immediate range above Ro=2.     

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.     

Stability of stratified shear flows

The concept of stability is discussed and results of the linear inviscid theory are reviewed. Examples are given to illustrate the point that stable stratification can, in some circumstances, be destabilizing due to the vorticity generated by non-homogeneity. The linear initial value problem is discussed and related to the more usual normal mode approach. Finally, the weakly non-linear theory and the non-linear critical layer theory are outlined.     

Internal gravity waves in a stably stratified boundary layer

Often, a combination of waves and turbulence is present in the stably stratified atmospheric boundary layer. The presence of waves manifest itself in the vertical profiles of variances of fluctuations and in low-frequency contributions to the power spectra. In this paper we study internal waves by means of a linear stability analysis of the mean profiles in a stably stratified boundary layer and compare the results with observed vertical variance profiles of fluctuating wind and temperature along a 200 m mast. The linear stability analysis shows that the observed mean flow is unstable for disturbances in a certain frequency and wavenumber domain. These disturbances are expected to the detectable in the measurements. It is shown that indeed the calculated unstable frequencies are present in the observed spectra. Furthermore, the shape of the measured vertical variance profiles, which increase with height, is explained well by the calculated vertical structure of the amplitude of unstable Kelvin-Helmholtz waves, confirming the contribution of waves to the variances. Because turbulence and waves have quite distinct transport properties, estimates of diffusion from measurements of variances would strongly overestimate this diffusion. Therefore it is important to distinguish between them.     

Stably stratified flow in a marine atmospheric surface layer

Data from the marine atmospheric surface layer have been analysed. The data set consists of about two weeks with tower measurements up to 31 m of mean profiles of wind, temperature, and humidity, together with 20 Hz turbulence data. Mean wind, temperature, and humidity profiles up to 2000 m are also available from pibal trackings and radio soundings. Wave height was measured at 2 Hz, using an inverted echo-sounder.It was found from pibal wind profiles that low level jets were present during 2/3 of the measurements, having their maxima in the height interval 40 to 300 m. Here only data from the remaining 1/3 of the measurements, without low level jets, have been analysed.Non-dimensional wind and temperature gradients agree with results over homogeneous land surfaces as regards stability dependence during stable conditions that prevailed during this experiment. Linear regression gave _m = 1 + 6.8z/L and _m = 1 + 8.3z/L. No significant sea wave influence was found. The same was vrue for me dimensionless standard deviations of the three wind components, except for the vertical component. The expected wind speed dependence was found for the neutral drag coefficient, givingC _dN = 0.109U + 0.33 at 10 m, and a dependence on the wave parameter,C/u _*, was confirmed. Note, however, that the data set was restricted to low and moderate wind speeds and that stratification was mainly stable.Power spectra, non-dimensionalized according to suface-layer theories, do not follow the expected stability dependence. It was shown that this may be a consequence of the presence of gravity waves in the stable marine boundary layer. Indicators of gravity waves were found in most runs. The TKE budget agrees with findings over homogeneous land areas. The pressure transport term was found to be a source of energy also for near neutral conditions.     

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.     

A mass flux closure function in a GCM based on the Richardson number

A mass flux closure in a general circulation model (GCM) was developed in terms of the mean gradient Richardson number (GRN), which is defined as the ratio between the buoyancy and the shear-driven kinetic energy in the planetary boundary layer. The cloud resolving model (CRM) simulations using the tropical ocean and global atmosphere-coupled ocean–atmosphere response experiment forcing show that cloud-base mass flux is well correlated with the GRN. Using the CRM simulations, a mass flux closure function is formulated as an exponential function of the GRN and it is implemented in the Arakawa–Schubert convective scheme. The GCM simulations with the new mass flux closure are compared to those of the GCM with the conventional mass flux closure based on convective available potential energy. Because of the exponential function, the new closure permits convective precipitation only when the GRN has a sufficiently large value. When the GRN has a relatively small value, the convection is suppressed while the convective instability is released by large-scale precipitation. As a result, the ratio of convective precipitation to total precipitation is reduced and there is an increase in the frequency of heavy precipitation, more similar to the observations. The new closure also improves the diurnal cycle of precipitation due to a time delay of the large GRN with respect to convective instability.     

三维层结流中地形上空湍流边界层结构特征研究

利用中尺度数值模式ARPS,在理想条件下研究了旋转、地表摩擦的共同作用下,均匀来流条件下,三维层结大气中地形上空及其边界层结构特征.重点讨论了不同的山地流动Fr数、不同上游来流方向情况下,山地流动及其边界层的特征.在大气稳定层结条件下,山地流动具有上游阻挡、下游急流加速以及重力波破碎等流动特征,这些山地流动特征在柯氏力的影响下,均表现出了南北非对称性.当山地流动的Fr数增加(增加地形高度)时,山地流动的南北急流增强,由于柯氏力的影响,南侧急流的增幅更为显著,流动的南北非对称性增强.在干过程山地流动中,由急流所导致的强烈的垂直风切变,是湍流边界层产生和发展的直接决定因素.     

Rapid vertical sampling in stably stratified turbulent shear flows

A new method for obtaining instantaneous vertical profiles of two components of velocity and temperature in thermally stratified turbulent shear flows is presented. In this report, the design and construction of the traversing system will be discussed and results to date will be presented. The method is based on rapid vertical sampling whereby probe sensors are moved vertically at a high speed such that the measurement is approximately instantaneous. The system is designed to collect many measurements for the calculation of statistics such as vertical wave number spectra, mean square vertical gradients, and Thorpe scales. Results are presented for vertical profiles of temperature and compared to vertical profiles measured by single-point Eulerian time averages. The quality of the vertical profiles is found to be good over many profiles. Some comparisons are made between vertical measurements and standard single-point Eulerian measurements for three cases of stably stratified turbulent shear flow in which the initial microscale Reynolds number, Re_λ≈30. In case 1, the mean conditions are characterized by a gradient Richardson number, Ri_g=0.015, for which the flow is “unstable”, meaning the spatially evolving turbulent kinetic energy (E_k) grows. In case 2, Ri_g=0.095, for which the evolving turbulent kinetic energy is almost constant. In case 3, the flow is highly stable, where Ri_g=0.25 and E_k decays with spatial evolution. The measurements indicate anisotropy in the small scales for all cases. In particular, it is found that the ratio grows initially to a maximum and then decays with further evolution. Maximum Thorpe displacements are measured and compared to single-point measures of the vertical scales. It is found that vertical length scales derived from single-point measurements, such as the Ozmidov scale, L_O=(ε/N³)^1/2 and the overturn scale, L_t=θ′/(dT/dz), do not represent well the wide range of overturning scales which are actually present in the turbulence.     

层结流体中具有纬向基本剪切流的Rossby波振幅的mKdV方程

In this paper,modified Korteweg-de Vries (mKdV) equations for the amplitude of solitary Rossby waves in stratified fluids with a zonal shear flow are derived by using a weakly nonlinear method.It is found that the coefficients of mKdV equations depend not only on the β effect and the Visl-Brunt frequency,but also on the basic shear flow.     

On the Scale-dependence of the Gradient Richardson Number in the Residual Layer

We present results of a technique for examining the scale-dependence of the gradient Richardson number, Ri, in the nighttime residual layer. The technique makes use of a series of high-resolution, in situ, vertical profiles of wind speed and potential temperature obtained during CASES-99 in south-eastern Kansas, U.S.A. in October 1999. These profiles extended from the surface, through the nighttime stable boundary layer, and well into the residual layer. Analyses of the vertical gradients of both wind speed, potential temperature and turbulence profiles over a wide range of vertical scale sizes are used to estimate profiles of the local Ri and turbulence structure as a function of scale size. The utility of the technique lies both with the extensive height range of the residual layer as well as with the fact that the sub-metre resolution of the raw profiles enables a metre-by-metre ‘sliding’ average of the scale-dependent Richardson number values over hundreds of metres vertically. The results presented here show that small-scale turbulence is a ubiquitous and omnipresent feature of the residual layer, and that the region is dynamic and highly variable, exhibiting persistent turbulent structure on vertical scales of a few tens of metres or less. Furthermore, these scales are comparable to the scales over which the Ri is less than or equal to the critical value of Ri _c of 0.25, although turbulence is also shown to exist in regions with significantly larger Ri values, an observation at least consistent with the concept of hysteresis in turbulence generation and maintenance. Insofar as the important scale sizes are comparable to or smaller than the resolution of current models, it follows that, in order to resolve the observed details of small Ri values and the concomitant turbulence generation, future models need to be capable of significantly higher resolutions.     

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.     

On an instability mechanism in a stably stratified atmospheric layer over a moistened surface

In air stratified by a specific humidity gradient, the vertical motions result in variations in specific humidity (mixing ratio) near the underlying surface. This, in turn, causes a variation of evaporation from the surface, resulting in horizontal thermal inhomogeneities on the surface, which under certain conditions can strengthen the initial vertical motions. The linear problem of the stability of the system under consideration is solved in this paper, boundaries of the unstable region are defined, and specific values of growth rates of disturbances are investigated. The estimates show that even in a density-stratified atmospheric layer over a moist surface, rapid development of disturbances with horizontal scales of several hundred metres is possible. The horizontal sizes of the most rapidly growing modes, as a rule, are an order of magnitude larger than the vertical sizes. The possibility of observing this instability under natural conditions is discussed.     

应用里查逊数判别中尺度波动的不稳定

本文采用非地转Eady模式通过摄动展开,讨论了由于低空强风切变导致的里查逊数Ri的减小与对称斜压不稳定的关系.证明了对中尺度扰动,在Ri<5.2的情况下,就会发生对称斜压不稳定.     

The temperature structure in a stably stratified internal boundary layer over a cold sea

Data from the Öresund experiment are used to investigate the structure of the stably stratified internal boundary layer (SIBL) which develops when warm air is advected from a heated land surface over a cooler sea. The present study is based on a theory developed by Stull (1983a, b, c). He proposed that the turbulence and the mean structure of the nocturnal boundary layer is controlled by the time-integrated value of surface heat flux and that the instantaneous heat flux is of less importance.Dimensional arguments are used to define simple, physically consistent, temperature, velocity and length scales. The dimensionless surface heat flux has a high value immediately downwind of the shoreline and it decreases rapidly in magnitude with increasing distance from the coast. Farther away, it is essentially constant. The dimensionless potential temperature change exhibits an exponential profile. It is estimated that turbulence accounts for 71% of boundary-layer cooling while clear-air radiational cooling is responsible for the remaining 29%.Finally it is found that theoretical predictions for the height of the SIBL are in a good agreement with observations.     

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.