Spin-up of a two-layer fluid in a rotating cylinder

Abstract

we report the results of experiments on the spin-up of two layers of immiscible fluid with a free upper surface in a rotating cylinder over a wide range of internal Froude numbers. Observations of the evolution of the velocity field by particle tracking indicates that spin-up of the azimuthal velocity in the upper layer take much longer than in a homogeneous fluid. Initially, spin-up occurs at a rate comparable to that of homogeneous fluid but, at high internal Froude number, a second phase follows in which the remaining lative motion decays much more slowly. Quantitative comparison of these measurements to the theory of Pedlosky (1967) shows good agreement.

Visualization of the interface displacement during spin-up detected the presence of transient azimuthal variations in the interface elevation over a wide range of Froude (F), Ekman (E), and Rossby (ε) number. nalysis of the occurrence of the asymmetric variations using the parameter space (Q, F), where Q = E ^1/2/ε, suggested by the baroclinic instability theory and experiments of Hart (1972), showed that the flow was stable for Q > 0.06 with no discernable dependence on F. This result, together with the prediction of Pedlosky's theory that radial gradient of potential vorticity in the two layers have opposite signs, suggests at the baroclinic instability mechanism was responsible for the asymmetries. The location and timing of these instabilities may account for the discrepancies between the observations and the Pedlosky (1967) theory.     

Influence of condensation and latent heat release upon barotropic and baroclinic instabilities of vortices in a rotating shallow water f-plane model

Analysis of the influence of condensation and related latent heat release upon developing barotropic and baroclinic instabilities of large-scale low Rossby-number shielded vortices on the f-plane is performed within the moist-convective rotating shallow water model, in its barotropic (one-layer) and baroclinic (two-layer) versions. Numerical simulations with a high-resolution well-balanced finite-volume code, using a relaxation parameterisation for condensation, are made. Evolution of the instability in four different environments, with humidity (i) behaving as passive scalar, (ii) subject to condensation beyond a saturation threshold, (iii) subject to condensation and evaporation, with three different parameterisations of the latter, are inter-compared. The simulations are initialised with unstable modes determined from the detailed linear stability analysis in the “dry” version of the model. In a configuration corresponding to low-level mid-latitude atmospheric vortices, it is shown that the known scenario of evolution of barotropically unstable vortices, consisting in formation of a pair of dipoles (dipolar breakdown) is substantially modified by condensation and related moist convection, especially in the presence of surface evaporation. No enhancement of the instability due to precipitation was detected in this case. Cyclone-anticyclone asymmetry with respect to sensitivity to the moist effects is evidenced. It is shown that inertia-gravity wave emission during the vortex evolution is enhanced by the moist effects. In the baroclinic configuration corresponding to idealised cut-off lows in the atmosphere, it is shown that the azimuthal structure of the leading unstable mode is sensitive to the details of stratification. Scenarios of evolution are completely different for different azimuthal structures, one leading to dipolar breaking, and another to tripole formation. The effects of moisture considerably enhance the perturbations in the lower layer, especially in the tripole formation scenario.     

Nonlinear stability of baroclinic flows over topography

Abstract

A new nonlinear stability criterion is derived for baroclinic flows over topography in spherical geometry. The stability of a wide class of exact three-dimensional nonlinear steady state solutions subject to arbitrary disturbances is established. The resonance condition, at the highest total wavenumber, for the steady state solutions and the stability criteria for baroclinic flow in the absence of topography provide the boundaries of the regions of stability in the presence of topography. The analogous results for flow on periodic or infinite beta planes incorporating non-orthogonal function large scale flows are also discussed.     

The role of boundary conditions in the simulation of rotating,stratified turbulence

Abstract

In this paper we use the CASL method to explore the role of boundary conditions in determining the long-time behaviour of rotating, stratified, quasi-geostrophic turbulence. We find that initially two-dimensional (sufficiently tall) columns of potential vorticity (PV) break down through three-dimensional instability to give a fully three-dimensional flow consisting of ellipsoidal structures. This is the case both for rigid-lid (isothermal) vertical boundary conditions and for vertically periodic boundaries. However, the rigid boundary case gives rise to semi-ellipsoids at both the top and bottom boundaries, and, for sufficient domain depths, preferred depths for the formation of ellipsoids in the interior. By contrast, in the vertically periodic case, the distribution of ellipsoids is homogeneous in depth.

The role of the horizontal boundaries is indirect, but still significant. In all cases doubly periodic horizontal boundary conditions are imposed. We consider a range of initial conditions where in each case equal numbers of two-dimensional columns of positive and negative vorticity are used, taking up a fixed, but relatively small fraction of the domain (approximately 5%). Thus when there is only a small number of vortices, they have larger radius. When the initial number of vortices is small enough (i.e., when the radius is not small compared with the horizontal domain width), at long time there is a two-dimensionalisation giving rise to a single column of positive PV and a single column of negative PV, as has been observed in some previous simulations. We find the same phenomenon for both vertically periodic and rigid lid boundary conditions, but it occurs over a broader range of initial conditions in the vertically periodic case. However, in all cases fully three-dimensional final states are regained when the number of vortices is increased while keeping the fraction of the domain occupied by vortices fixed, i.e., when the vortex radius/domain width ratio is sufficiently small.     

Laboratory and numerical model studies of a negatively-buoyant jet discharged horizontally into a homogeneous rotating fluid

Abstract

The results of laboratory experiments and numerical model simulations are described in which the motion of a round, negatively-buoyant, turbulent jet discharged horizontally above a slope into a rotating homogeneous fluid has been investigated. For the laboratory study, flow visualisation data are presented to show the complex three-dimensional flow fields generated by the discharge. Analysis of the experimental data indicates that the spatial and temporal developments of the flow field are controlled primarily by the lateral and vertical discharge position of the jet (with respect to the bounding surfaces of the container of width W) and the specific momentum (M ₀) and buoyancy (B ₀) fluxes driving the jet. The flow is seen to be characterised by the formation of (i) a primary anticyclonic eddy (PCC) close to the source, (ii) an associated secondary cyclonic eddy (SCE) and (iii) a buoyancy-driven bottom boundary current along the right side boundary wall. For the parameter ranges studied, the size L _{p, s} and spatial location x _{p, s} of the PCC and SCE (and the nose velocity u _N of the boundary current) are shown to be only weakly-dependent upon the value of the mixed parameter M ₀Ω/B ₀, where Ω is the background rotation rate. Both L _p and x _p are shown to scale with the separation distance y?/W of the right side wall (y = 0) from the source (y = y?), both L _s and x _s scale satisfactorily with the length scale l _M (= M ₀ ^3/4/B ₀ ^½) and u _N is determined by the appropriate gravity current speed [(g']₀ H]^½ and the separation distance y?/W.

Numerical model results show good qualitative agreement with the laboratory data with regard to the generation of the PCC, SCE and boundary current as characteristic features of the flow in question. In addition, extension of the numerical model to

diagnose potential vorticity and plume thickness distributions for the laboratory cases allow the differences in momentum-and buoyancy-dominated flows to be clearly delineated. Specifically, the characteristic features of the SCE are shown to be strongly dependent upon the value of M ₀Ω/B ₀ for the buoyant jet flow; not least, the numerical model data are able to confirm the controlling role played by the boundary walls in the laboratory experiments. Quantitative agreement between the numerical and laboratory model data is fair; most significantly, the success of the former model in simulating the dominant flow features from the latter enables the reliable extension of the numerical model to be made to cases of direct oceanic interest.     

An experimental study of global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder

Abstract

Broad band secondary instability of elliptical vortex motion has been proposed as a principal source of shear-flow turbulence. Here experiments on such instability in an elliptical flow with no shear boundary layer are described. This is made possible by the mechanical distortion in the laboratory frame of a rotating fluid-filled elastic cylinder. One percent ellipticity of a 10 cm diameter cylinder rotating once each second can give rise to an exponentially-growing mode stationary in the laboratory frame. In first order this mode is a sub-harmonic parametric Faraday instability. The finite-amplitude equations represent angular momentum transfer on an inertial time scale due to Reynolds stresses. The growth of this mode is not limited by boundary friction but by detuning and centrifugal stabilization. On average, a generalized Richardson number achieves a marginal value through much of the evolved flow. However, the characteristic flow is intermittent with the cycle: rapid growth, stabilizing momentum transfer from the mean flow, interior re-spin up, and then again. Data is presented in which, at large Reynolds numbers, seven percent ellipticity causes a fifty percent reduction in the kinetic energy of the rotating fluid. In the geophysical setting, this tidal instability in the earth's interior could be inhibited by sub-adiabatic temperature gradients. A near adiabatic region greater than 10 km in height would permit the growth of tidally destabilized modes and the release of energy to three-dimensional disturbances. Such disturbances might play a central role in the geodynamo and add significantly to overall tidal dissipation.     

A comparison of numerical simulations of eddy generation performed with a two- and a three-layer quasi-geostrophic model of oceanic mesoscale eddies

Abstract

The generation of eddies by a large-scale flow over mesoscale topography is studied with the help of two- and three-layer nonlinear quasi-geostrophic models of the open ocean. The equations are integrated forward in time with no eddies present initially. For a given time, the displacement of the interface between layers two and three (ζ) tends to a well-defined limit (function of the horizontal spatial coordinates) as ρ ₃ - ρ ₂ → 0 (ρ_r is the density of layer r). Even for values of α[= (ρ ₃ - ρ ₂)/(ρ ₂ - ρ ₁)] ^as small as 0.01 the potential energy due to ζ is not negligible and it can reach, in some cases, a considerable fraction of the total eddy energy.     

Convection in a rotating cylinder with its sidewall having a finite thermal conductance

Abstract

An investigation is made of steady thermal convection of a Boussinesq fluid confined in a vertically-mounted rotating cylinder. The top and bottom endwall disks are thermal conductors at temperatures T_t and T_b with δT = T_t ? T_b >0. The vertical sidewall has a finite thermal conductance. A Newtonian heat flux condition is adopted at the sidewall. The Rayleigh number of the fluid system is large to render a boundary layer-type flow. Finite-difference numerical solutions to the full Navier-Stokes equations are obtained. The vertical motions within the buoyancy layer along the sidewall induce weak meridional flows in the interior. Because of the Coriolis acceleration, the meridional flows give rise to azimuthal flows relative to the rotating container. Strong vertical gradients of azimuthal flows exist in the regions near the endwalls. As the stratification effect increases, concentration of flow gradients in thin endwall boundary layers becomes more pronounced. The azimuthal flow field exhibits considerable horizontal gradients. The temperature field develops horizontal variations superposed on the dominant vertical distribution. As either the sidewall thermal conductance or the stratification effect decreases, the temperature distribution tends to the profile varying linearly with height. Comparisons of the sizes of the dynamic effects demonstrate that, in the bulk of flow field, the vertical shear of azimuthal velocity is supported by the horizontal temperature gradient, resulting in a thermal-wind relation.     

Sensitivity experiments with a quasi-geostrophic model of stratospheric sudden warming

We describe a series of sensitivity experiments with a quasi-geostrophic model of the interaction of stationary planetary waves with the mean zonal flow in the stratosphere and mesosphere. The model is of the Matsuno type, which neglects wave-wave interaction and includes only a single zonal harmonic of the planetary wave spectrum in each simulation. We employed the model to investigate the source of the double-layer structure previously obtained by several authors for the stratospheric sudden warming with wavenumber one. Our results suggest that this characteristic of the model-produced warming is a property only of models without damping. When reasonable dissipation is included in the model, the double-layer structure disappears. This implies the importance of the drag parameterization in properly simulating warming events and, since the actual drag very probably is effected by breaking internal waves, it suggests that future analysis should include a specific representation of this effect. We also investigated the dependence of stratospheric warming on the structure of the zonal wind field. Our analyses show in particular that substantial reduction of the height of the polar night jet mitigates strongly against the occurrence of a sudden warming event.     

一个三维斜压水动力模型的建立及其在日本琵琶湖中的应用

水温的空间分布对很多湖泊水文现象关系甚大,本文建立了日本琵琶湖的三维斜压水动力数值模式。模拟计算结果表明：１）在不同风速驱动下,湖的上、下两层可形成不同的环流型;２）除形成环流处外,其它部分流速、流向在上、下层表现出明显的垂直切变;３）闭合式温度场储存位能,即使在无风和初始静止流场情况下,也可驱动湖流并形成若干环流特征;４）风对水的驱动作用是形成北湖北部稳定环流的决定性因子,并且本身可构造出温度场     

Jet instability and the stabilizing effect of topography on jets in two-layer rotating systems

Abstract

Laboratory experiments concerning azimuthal jets in two-layer rotating systems in the absence and presence of bottom topography aligned along the jets have been conducted. The jets were forced by the selective withdrawal of fluid from the upper layer of a two-fluid system contained in a circular dishpan geometry. The principal parameters measured in the experiments were the jet Rossby number, Ro, and a stratification parameter F = r ₁/(λ₁λ₂)^1/2 where r ₁ is the radius of the circular disc used for the selective withdrawal (i.e., r ₁ is the approximate radius of curvature of the jet) and λ₁,λ₂ are the internal Rossby radii of deformation in the upper and lower fluids, respectively.

The no-topography experiments show that for a sufficiently small F, the particular value depending on Ro, the jet is stable for the duration of the experiment. For sufficiently large F, again as a function of Ro, the jet becomes unstable, exhibiting horizontal wave disturbances from modes three to seven. An Ro against F flow regime diagram is presented.

Experiments are then conducted in the presence of a bottom topography having constant cross-section and extending around a mid-radius of the dishpan. The axis of the topography is in the vicinity of the jet axis forced in the no-topography experiments and the crest of the topography is in the vicinity of the interface between the two fluids (i.e., the front associated with the jet). The experiments show that in all cases investigated the jet tends to be stabilized by the bottom topography. Experiments with the topography in place, but with the interface between the fluids being above the topography crest, are shown to be unstable but more irregular than their no-topography counterparts.

Various quantitative measurements of the jet are presented. It is shown, for example, that the jet Rossby number defined in terms of the fluid withdrawal rate from the tank. Q, can be well correlated with a dimensionless vorticity gradient, V_G , across the upper layer jet. This allows for an assessment of the stability characteristics of a jet based on a knowledge of V_G (which can be estimated given a jet profile) and F.     

Experimental determinations of the transition between the symmetrical and wave regimes in a rotating differentially heated annulus of fluid

Summary The transition between axisymmetric and wave convection in a rotating, cylindrical annulus of fluid subjected to a horizontal temperature gradient is usually determined in laboratory experiments by visually observing the motion of tracer particles at the top surface of the fluid. More recent transition determinations by means of small transducers suspended within the body of the fluid give evidence of quantitative disagreement with the visual method. The dgree of disagreement and experimental details are discussed in this note.Contribution No. 20 of the Geophysical Fluid Dynamics Institute.     

Instability of planetary waves in a high vertical resolution model

Abstract

A high vertical resolution model is used to examine the instability of a baroclinic zonal flow and a finite amplitude topographically forced wave. Two families of unstable modes are found, consisting of zonally propagating most unstable modes, and stationary unstable modes. The former have time scale and spatial structure similar to baroclinic synoptic disturbances, but are localized in space due to interaction with the zonally asymmetric forcing. These modes transport heat efficiently in both the zonal and meridional directions. The second family of stationary unstable modes has characteristics of modes of low frequency variability of the atmosphere. They have time scales of 10 days and longer, and are of planetary scale with an equivalent barotropic vertical structure. The horizontal structure resembles blocking flows. They are maintained by available potential energy of the basic wave, and have large zonal heat fluxes. The results for both families of modes are interpreted in terms of an interaction between forcing and baroclinic instability to create favoured regions for eddy development. Applications to baroclinic planetary waves are also considered.     

地壳中流体动力学模型研究

较全面介绍了近年国内外地壳地下流体动力学研究的最新成果。综合分析前人研究成果并将其系统化为地完尺度上地下流体动力学系统的概念模型和数学模型。模型根据影响地下流体动力特性的主要因素和介质渗透特征,把地壳渗流场分为深度小于6km的饯部带、6～17km的过渡带和大于17km的深部带。数学处理上把驱动流体运动的各种地质作用归一化为“地质力”,以地下流体运动方程中的源汇项表示。分析了不同地质环境各种地质作用对地下流体系统的作用特征。对地壳中普遍存在的异常孔隙压力现象,从流体动力学的角度进行了解释,并认为周期性地震活动是形成异常高孔隙压力的主导因素。     

A numerical model for shore-normal sediment size variation on a macrotidal beach

This paper describes a computer simulation model which is designed to predict the selective shore-normal sorting of grain sizes in the nearshore environment. The model simulates wave shoaling, wave height attenuation due to frictional losses and breaking, using linear theory up to the break point and a breaker decay model in the surf zone. Peak horizontal orbital velocities at the bed are calculated from Stokes second-order wave theory. The peak onshore and offshore velocities are used with the threshold expression of Komar and Miller (1975) to generate a spatial pattern of size variation of threshold grain diameter along a profile normal to the shore from deep water to the swash zone. The predicted grain size is used in an hydraulic interpretation of grain size distribution on the intertidal profile, based on the hydrodynamic variations over a tidal cycle on a macrotidal beach. The model is successful in predicting the broad pattern of increasing grain size in the onshore direction which has been observed in nature. Comparisons between measured and predicted grain size distributions indicate that the predictions of the model are better than those of previous models, but the model is more successful at predicting sediment size distributions than at predicting mean sizes on a beach profile.     

Effects of dissipation on internal waves in a contained rotating stratified fluid

Abstract

Boundary layer techniques are used to examine the modifications due to dissipation in the normal modes of a uniformly rotating, density stratified, Boussinesq fluid in a rigid container. Arbitrary relative influence of rotation and stratification is considered. The existence of critical regions of the container boundary is discussed. In cylindrical geometry a formula is derived for the decay factor on the homogeneous “spin-up” time scale which reveals how the dominant dissipation varies as a function of several parameters. For the situation where the buoyancy and inertial frequency are exactly equal, all boundaries are everywhere critical. In this case the method of multiple time-scales is employed to investigate the confluence inertial-gravity mode which is shown to persist until the diffusive time-scale is achieved.