Experiments to study baroclinic waves penetrating into a stratified layer by a quasi-geostrophic potential vorticity equation

The upward penetration of the mid-latitude baroclinic waves has been well known to influence the air motion in the stratosphere. To study this in the laboratory, we present a new type of rotating fluid annulus experiment where a radial temperature difference is imposed on water near the flat bottom to produce a baroclinic wave in a lower layer and the water surface is heated to create a stratified upper layer. To test this model, we carried out experiments to study baroclinic waves penetrating into the stratified layer by a quasi-geostrophic potential vorticity equation in the atmospheric dynamics. The baroclinic waves produced in the lower fluid layer were observed to penetrate into the stratified upper layer. As expected from the equation, the vorticities of the flows for wavenumbers 3, 4 and 5 decayed exponentially with height and showed a weaker decay for lower wavenumbers.     

B平面上斜压波热力结构特性的实验研究

首次用转环实验模拟方法研究了β效应对斜压波热力结构的影响,发现β效应有抑制流动的水平混合和垂直混合的作用,使流动趋于正压;β平面上急流随高度降低而减弱,在急流的内外两侧各有一个无量纲温度值分布的突跃区,它们的空间结构与大气环流中的极锋锋区和北极锋锋区的结构相似。     

Asymptotic solutions of the axisymmetric moist Hadley circulation in a model with two vertical modes

A simplified model of the moist axisymmetric Hadley circulation is examined in the asymptotic limit in which surface drag is strong and the meridional wind is weak compared to the zonal wind. Our model consists of the quasi-equilibrium tropical circulation model (QTCM) equations on an axisymmetric aquaplanet equatorial beta-plane. This model includes two vertical momentum modes, one baroclinic and one barotropic. Prior studies use either continuous stratification, or a shallow water system best viewed as representing the upper troposphere. The analysis here focuses on the interaction of the baroclinic and barotropic modes, and the way in which this interaction allows the constraints on the circulation known from the fully stratified case to be satisfied in an approximate way. The dry equations, with temperature forced by Newtonian relaxation towards a prescribed radiative equilibrium, are solved first. To leading order, the resulting circulation has a zonal wind profile corresponding to uniform angular momentum at a level near the tropopause, and zero zonal surface wind, owing to the cancelation of the barotropic and baroclinic modes there. The weak surface winds are calculated from the first-order corrections. The broad features of these solutions are similar to those obtained in previous studies of the dry Hadley circulation. The moist equations are solved next, with a fixed sea surface temperature at the lower boundary and simple parameterizations of surface fluxes, deep convection, and radiative transfer. The solutions yield the structure of the barotropic and baroclinic winds, as well as the temperature and moisture fields. In addition, we derive expressions for the width and strength of the equatorial precipitating region (ITCZ) and the width of the entire Hadley circulation. The ITCZ width is on the order of a few degrees in the absence of any horizontal diffusion and is relatively insensitive to parameter variations.     

Stability of advective flow in an inclined plane fluid layer bounded by solid planes with a longitudinal temperature gradient 2. Stable stratification

This paper studies the stability of steady convective flow in an inclined plane fluid layer bounded by perfectly heat-conducting solid planes in the presence of a homogeneous longitudinal temperature gradient under stable stratification where the layer is inclined so that the temperature in its lower part is lower than in the upper part.     

Liquid crystal techniques of visualization in rotating annulus experiments

To the well-known rotating annulus experiments we applied liquid crystal techniques of visualization in order to obtain clear video-pictures of internal flow and temperature in the fluid. Then we developed the idea of simultaneously injecting several types of liquid crystals of different temperature ranges to observe the fluid with a wide temperature range. It was shown that with this idea it was possible to take clear video-pictures throughout the whole interior of the fluid. This revealed that the pattern of the bottom flow does not have the characteristics of the Eady type baroclinic waves. Furthermore, the typcial meridional gradient of temperature of the baroclinic wave was directly observed from isothermal lines appearing in the fluid as colour band lines.     

旋转流体斜压波三维温度结构实验研究

实验研究了负径向温度梯度下转坏系统中斜压流体的对流运动，获得了该系统中旋转流体斜压波的一般特征，通过对斜压波温度场多层次细网格测量，得到了斜压波三维温度结构和表面波状急流的温度分布，并就波－波转换的不稳定性问题作了初步分析．     

大气运动不稳定的变分原理(Ⅱ)

<正> 7 分层流模式 假设有J薄层均匀流体,其上边界面、密度和速度分别由Z_k;ρ_k和V_k表示,k=1,2,…,J(图3)。我们有如下的基本方程组 (曾庆存,1979):      

Influence of spatial modulation of the temperature field on the stability of two-dimensional steady flow in a horizontal layer of fluid

A study is made of the stability of the steady periodic regime that arises in a horizontal layer of fluid in the presence of spatial modulation of of the temperature on the solid bottom boundary. The upper free boundary of the layer is in contact with the atmosphere. The fundamental resonance values of the wave number of the modulation are found; there are five of them. If the temperature of the lower boundary of the layer is constant, and the temperature gradient is not too large, the fluid is in equilibrium. When the temperature gradient passes through the critical value, the equilibrium ceases to be stable, and steady convection develops in the fluid [1]. In the presence of spatial modulation of the temperature on the lower boundary of the layer the fluid cannot be in equilibrium, and a spatially periodic steady regime is established in it. The aim of the present paper is to find the critical values of the temperature gradient at which this fundamental steady regime becomes unstable and a secondary steady regime develops in the fluid. An analogous problem for the case when both boundaries of the layer are free surfaces and without allowance for the influence of the atmosphere has been solved by Vozovoi and Nepomnyashchii [2].     

The coupling between turbulent,penetrative convection and internal waves

Experiments aimed at exploring the coupling of penetrative convection with internal waves in the adjoining, stable layer were performed in a long convection cell. The experiments are motivated by preliminary theoretical results suggesting that an intrinsic phase instability may exist in the coupled system in which case long internal waves modulate the height and strength of convective plumes. Using a temperature-controlled, stably stratified experimental apparatus, measured temperature data reveal the presence of long internal wave modes that persist for many convective time scales. The frequencies of these waves increase linearly in time during the energy transfer between the convective and stratified regions as the depth of the stratified region diminishes and the depth of the mixed layer increases. Temporal variations in the heat flux, interface rise characteristics, and frequencies of internal wave motions are reported. A natural temporal modulation of the thickness of the transition layer separating the mixed layer from the stratified layer occurs following commencement of heating, with the amplitude and frequency of the modulation varying with the initial stratification. Temperature variance data suggest that a fairly strong interaction between convection and internal waves occurs, especially when the interface region is midway between the upper and lower boundaries of the cell and the no-slip boundary conditions play a less influential role on the dynamics of the coupling.     

Water wave interaction with a sphere in a two-layer fluid flowing through a channel of finite depth

Using linear water wave theory, we consider a three-dimensional problem involving the interaction of waves with a sphere in a fluid consisting of two layers with the upper layer and lower layer bounded above and below, respectively, by rigid horizontal walls, which are approximations of the free surface and the bottom surface; these walls can be assumed to constitute a channel. The effects of surface tension at the surface of separation is neglected. For such a situation time-harmonic waves propagate with one wave number only, unlike the case when one of the layers is of infinite depth with the waves propagating with two wave numbers. Method of multipole expansions is used to find the particular solutions for the problems of wave radiation and scattering by a submerged sphere placed in either of the upper or lower layer. The added-mass and damping coefficients for heave and sway motions are derived and plotted against various values of the wave number. Similarly the exciting forces due to heave and sway motions are evaluated and presented graphically. The features of the results find good agreement with previously available results from the point of view of physical interpretation.     

Vortex-induced vibrations of pipes conveying fluid in the subcritical and supercritical regimes

In this paper, the vortex-induced vibrations of a hinged–hinged pipe conveying fluid are examined, by considering the internal fluid velocities ranging from the subcritical to the supercritical regions. The nonlinear coupled equations of motion are discretized by employing a four-mode Galerkin method. Based on numerical simulations, diagrams of the displacement amplitude versus the external fluid reduced velocity are constructed for pipes transporting subcritical and supercritical fluid flows. It is shown that when the internal fluid velocity is in the subcritical region, the pipe is always vibrating periodically around the pre-buckling configuration and that with increasing external fluid reduced velocity the peak amplitude of the pipe increases first and then decreases, with jumping phenomenon between the upper and lower response branches. When the internal fluid velocity is in the supercritical region, however, the pipe displays various dynamical behaviors around the post-buckling configuration such as inverse period-doubling bifurcations, periodic and chaotic motions. Moreover, the bifurcation diagrams for vibration amplitude of the pipe with varying internal fluid velocities are constructed for each of the lowest four modes of the pipe in the lock-in conditions. The results show that there is a significant difference between the vibrations of the pipe around the pre-buckling configuration and those around the post-buckling configuration.     

Irregularity and singular vector growth of the differentially heated rotating annulus flow

The problem of weather prediction is to a large part determined by the large-scale atmospheric flow. This flow is irregular but not fully turbulent. The irregularity can be related to instability, nonlinear wave–wave interactions, or randomly excited singular vectors. In the present study, the latter mechanism is investigated numerically and experimentally. For this purpose, a baroclinic quasigeostrophic low-order model is adjusted to a differentially heated rotating annulus experiment, an established laboratory analog to the atmospheric circulation. We linearize the low-order model about a time-mean state to study the growth of singular vectors and compare those with singular vectors that have been derived empirically from the experimental data. Qualitative agreement of the numerically and experimentally derived singular vectors form the basis for a deeper analysis of the low-order model. In particular, for a broad range of annulus rotation frequencies and radial temperature differences, we compute the maximal growth rates of the low-order model. These growth rates are displayed as a function of the Taylor and thermal Rossby number, a frame widely used in studies of annulus regime transitions. We can show that most regime transitions defined by E.N. Lorenz in the sixties have their counterpart in the Taylor-Rossby singular value diagram. Most striking is the fact that for the irregular regime by far, the largest growth rates can be found. We suppose that irregularity in the transition region to geostrophic turbulence might for some part result from randomly excited singular vectors with unusual large growth rates. In concert with nonlinear wave–wave coupling, this process might explain the gradual broadening of the wave spectrum that has been found for the route to geostrophic turbulence in annulus flows.     

Stability of advective flow in an inclined plane fluid layer bounded by solid planes with a longitudinal temperature gradient. 1. Unstable stratification

The stability of steady convective flow in an inclined plane fluid layer bounded by ideally heat conducting solid planes is studied in the presence of a homogeneous longitudinal temperature gradient under unstable stratification conditions where the layer is inclined so that the temperature is higher in the lower part than in the upper part. It is shown that the inclination leads to the transition from critical perturbations to long-wavelength helical perturbations. Flow stability maps are given for the entire range of Prandtl numbers and inclination angles corresponding to unstable stratification.     

Simultaneous PIV and thermography measurements of partially blocked flow in a differentially heated rotating annulus

A radial barrier has been mounted in a differentially heated rotating annulus that partially blocks the azimuthal flow component. The experiment can be seen as an analog to geophysical flows with constrictions, e.g., the Antarctic Circumpolar Current. However, the experiment has been carried out without a particular natural flow in mind. The main interest was to observe a baroclinic annulus flow that does not become saturated. Hence, in contrast to the annulus flow without a barrier, the partially blocked flow remains transient and surface heat fluxes associated with baroclinic life cycles can be studied. The annulus can be subdivided into the upstream half of the barrier, where waves amplify, and the downstream half of the barrier, where waves decay. In the upstream half, the azimuthal mean flow is moderate but with a significant positive eddy radial heat flux. In the downstream half, we find a strong jet in the mean azimuthal flow and furthermore an increased radial mean temperature gradient. The latter points to a weakened or even reversed radial eddy heat flux in the lee side of the barrier. Temperature anomalies appear as large bulges in the outer part of the annulus. Moreover, an outward shift of vortex centers can be observed with respect to centers of temperature anomalies. This phase shift between pressure and temperature anomalies differs from that of classical Eady modes of baroclinic instability.     

The transient development of the flow in an impulsively rotated annular container

When a fluid-filled container is spun up from rest to a constant angular velocity the fluid responds in such a way that the fluid–container system is ultimately in a state of rigid-body rotation. The fluid can then be said to have traversed a trajectory in phase space from a simple stable equilibrium state of no motion to another stable equilibrium representing full rigid-body rotation. This simple statement belies the fact that during this process the fluid can undergo a series of transitions, from a laminar through a transient turbulent state, before attaining the stable motion that is rigid-body rotation. Using a combination of analytical and computational methods, we focus on the dynamics resulting from an impulsive change in the rotation rate of a fluid-filled annulus, specifically, the impulsive spin-up of a stationary annulus, or the impulsive spin-down of an annulus already in a state of rigid-body rotation. We explore the initial development of the impulsively generated axisymmetric boundary layer, its subsequent instability, and the larger-scale transient features within this class of flows, allowing us to look at the effect these features have on the time it takes for the system to spin up to a steady state, or spin down to rest.     

Effect of fluid microstructure on stability of plane two-layer flows

The stability of plane two-layer Couette and Poiseuille flows, where the lower layer consists of a Grad-model fluid and the upper layer is a viscous Newtonian fluid, is investigated. The disturbances are assumed to be of the long-wave type, and the analysis involves expansion in wave numbers and is limited by two approximations. Numerical calculations are made for some values of the parameters. The calculations indicate that the rotational energy of the fluid in the lower layer has a destabilizing effect on the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–127, July–August, 1978.     

Influence of porosity of the base on a plane standing wave of a heavy liquid

A study is made of the simple problem of the contact of two plane-parallel potential flows of incompressible fluid when one takes place in a layer of finite thickness and the other in a semiinfinite space of a porous medium. At the interface, which is taken to be a plane, the same conditions are used as earlier in problems of the contact of two wave flows of fluids with different densities and the contact of a wave motion in a layer of compressible fluid and wave motions in an elastic semi-infinite space. These conditions reduce to equalities of the pressures and projections of the velocity vectors onto the normal to the interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 160–163, July–August, 1984.     

Experimental investigation of turbulent entrainment in an annulus with moving sidewalls

The establishment of a turbulent mixed layer in a two-layer stratified shear flow, and the rate of entrainment into that layer were studied experimentally in a modified annulus. The modification of the conventional annulus was made by replacing the upper rotating screen with inner rotating sidewalls, extending over the upper half of the channel, so that the flow in the upper layer was nearly uniform and almost laminar, while the bottom layer was quiescent. Vertical density profile measurements were conducted using single electrode conductivity probes. The flow was visualized during the various stages of the experiment using the hydrogen bubble technique.After the start of the sidewalls rotation, the upper layer accelerates from rest, and consequently a transition process is taking place during which the initial density interface between the two layers is developed into a turbulent mixed layer. This turbulent layer is bounded by two sharp interfaces, each separating it from an outer non-turbulent zone. The generation of this five-layer structure seemed to be dominated by instabilities activated by the velocity difference between the upper and lower layer.Once a turbulent mixed layer is formed, entrainment of nonturbulent fluid into that layer is taking place causing its thickness to increase continuously. Depending on the overall Richardson number, based on the channel width, the slope of the entrainment law curve was found to have two different values, each indicating the dominance of a different source of turbulent energy production. For relatively low Richardson numbers, the slope is close to -1.8, implying that the velocity shear across each interface contributes significantly to the entrainment. On the other hand, for larger Richardson numbers the slope is about -1.25, in agreement with previous results of shear-free entrainment experiments.The measured velocity profiles indicate that as long as the mixed layer is not too thick, the radial inhomogeneities are small and the flow may be considered as nearly one-dimensional. It seems, therefore, that for the understanding of entrainment processes occurring in realistic stratified flows, the modified annulus is a more reliable tool than the conventional one.     

Effect of transverse stream velocity in an incompressible boundary layer on the stability of the laminar flow regime

The stability of the laminar flow regime in the boundary layer developed on a wall is increased considerably by the relatively slight extraction of fluid from the wall [1–4]. In the theoretical study of this phenomenon, all the investigators known to the present authors have taken into account only the increase in the fullness of the velocity profile in the boundary layer with suction. Computations of the stability characteristics have been made on the assumption that there are no transverse velocities in the laminar boundary layer.We present below an analysis of the stability of the laminar boundary layer in the presence of a constant transverse velocity in the near-wall region (suction). The calculations made show the existence for a given velocity profile in the boundary layer of a relative suction velocity v=v such that with suction velocities greater than v the flow remains stable at all Reynolds numbers, while the method used in the cited references gives a definite finite critical Reynolds number, equal in our notation to the Reynolds number at v=0, for each relative suction velocity.It was found that with suction of fluid from the boundary layer the region of instability has finite dimensions, i.e., there exist lower and upper critical Reynolds numbers. The flow is stable if its Reynolds number is less than the lower, or greater than the upper values of the critical Reynolds number.The instability region diminishes with increase in the relative suction velocity, and at a value of this velocity which is specific for each value of the velocity profile the instability region degenerates into a point-the flow becomes absolutely stable. Thus, with distributed suction it is advisable to increase the relative suction velocity only to a definite magnitude corresponding to disappearance of the instability region. The computational results presented make it possible to estimate this velocity for velocity profiles ranging from a Blasius profile to an asymptotic profile. Specific calculations were made for a family of Wuest profiles, since under actual conditions with suction there always exists a starting segment of the boundary layer [1, 2].     

Flow structure and wall layer control in a lower half heated upper half cooled enclosure with superimposed temperature deviations

This paper presents an investigation on the effects of superimposed temperature deviations as a control technique for the flows and mixing in lower half heated upper half cooled enclosures. Results show that the strength of the wall layer depends on the difference between the wall surface temperature and the fluid core temperature. The location of the head-on collision between a pair of upward/downward wall layers, which controls the mixing and fluid exchange between the two halves, is determined by the wall layer flow momentum strengths. Elevating/reducing the wall temperature by a superimposed temperature deviation is an effective control for the flow and mixing in such enclosures. Heat transfer analysis shows that the superimposed temperature deviations have minor effects on the total heat flow rate from the lower walls. Thus, this technique can be applied onto reactor vessels without modifying the reactor vessel configuration.