Experimental investigation of stratorotational instability using a thermally stratified system: instability,waves and associated momentum flux

Stratorotational instability (SRI) has been proposed as a mechanism for outward angular momentum transport in Keplerian accretion disks. A particular designed Taylor–Couette laboratory experiment with axial stratification is suitable for studying the instability. Bottom endplate is cooled and top endplate is heated to achieve axial stratification. Due to constructive constraints, endplates are visually unamenable and quantitative measurement techniques in the co-rotating frame can only be done by looking through the outer cylinder. For this purpose, we built a co-rotating mini-PIV (Particle Image Velocimetry) system with a camera having a tilted viewing angle regarding the horizontal laser sheet. The aim of this study is (i) to quantify the uncertainty of the mini-PIV together with the used calibration technique and (ii) to compare experimental findings on SRI with theoretical predictions.

We perform measurements of the azimuthal and radial component of the velocity in axial stably stratified Taylor–Couette flows, consider velocity profiles and do frequency-filtering and flow decomposition. The absolute error of the mini-PIV system is 2% and we realised that stratified Taylor–Couette flows have smaller Ekman endwall effects than homogeneous ones. Still, Ekman pumping has an impact of the flow and might be responsible for differences between the data and theoretical models ignoring the endwalls. Here we focus on the flow structure during transition to SRI, the drift rate of SRI modes and the radial momentum flux as a function of the Reynolds number. Whereas the structure in form of trapped boundary Kelvin modes and the drift rate corresponds well with earlier predictions, the momentum flux shows a nonlinear dependency with respect to the Reynolds number. Away from the region of transition, theoretical models show a linear relationship. Several possible reasons for the mismatch between the experimental and theoretical models are discussed. Most important, we experimentally demonstrated that in the Rayleigh stable flow regime the SRI can provide a significant amount of outward momentum flux which makes this instability interesting in the context of accretion disks and also of atmospheric vortices where rotation and stratification also play a significant role.     

Experiments on instability of columnar vortex pairs in rotating fluid

We present results from a new series of experiments on the geophysically important issue of the instability of anticyclonic columnar vortices in a rotating fluid in circumstances such that the Rossby number exceeds unity. The vortex pair consisting of a cyclonic and an anticyclonic vortex is induced by a rotating flap in a fluid which is itself initially in a state of solid-body rotation. The anticyclonic vortex is then subject to either centrifugal or elliptical instability, depending on whether its initial ellipticity is small or large, while the cyclone always remains stable. The experimental results demonstrate that the perturbations due to centrifugal instability have a typical form of toroidal vortices of alternating sign (rib vortices). The perturbations due to elliptical instability are of the form of sinuous deformation of the vortex filament in the plane of maximal stretching which corresponds to the plane of symmetry for the vortex pair. The initial perturbations in both cases are characterized by a definite wave number in the vertical direction. The characteristics of the unstable anticyclone are determined by the main nondimensional parameter of the flow - the Rossby number. The appearance of both centrifugal and elliptical instabilities are in accord with the predictions of theoretical criteria for these cases.     

Diffusive destabilisation of the baroclinic circular vortex

Abstract

It is shown that, even for vanishingly small diffusivities of momentum and heat, a rotating stratified zonal shear flow is more unstable to zonally symmetric disturbances than would be indicated by the classical inviscid adiabatic criterion, unless σ, the Prandtl number, = 1. Both monotonic instability, and growing oscillations ("overstability") are involved, the former determining the stability criterion and having the higher growth rates. The more σ differs from 1, the larger the region in parameter space for which the flow is stable by the classical criterion, but actually unstable.

If the baroclinity is sufficiently great for the classical criterion also to indicate instability, the corresponding inviscid adiabatic modes usually have the numerically highest growth rates. An exception is the case of small isotherm slope and small σ.

A single normal mode of the linearized theory is also, formally, a finite amplitude solution; however, no theoretical attempt is made to assess the effect of finite amplitude in general. But, in a following paper, viscous overturning (the mechanism giving rise to the sub‐classical monotonic instability when σ > 1) is shown to play an important role at finite amplitude in certain examples of nonlinear steady thermally‐driven axisymmetric flow of water in a rotating annulus. Irrespective of whether analogous mechanisms turn out to be identifiable and important in large‐scale nature, it appears then that a Prandtl‐type parameter should enter the discussion of any attempt to make laboratory or numerical models of zonally‐symmetric baroclinic geophysical or astrophysical flows.     

An experimental study of Stewartson layer separation

Abstract

The separation of sidewall boundary layers in a rotating annulus of homogeneous fluid is studied experimentally. The flow is driven by a differentially rotating lid, and a very small uniform slope of the bottom causes a weak mountain vortex pair to form in the interior, away from the sidewalls. A necessary condition for aerodynamic separation of the sidewall boundary layers is derived and compared with the experimental results. The laboratory flow separates for parameters that are just slightly more inviscid than those required by the necessary condition for the existence of adverse pressure gradients at the wall. As the bottom friction is decreased further, the flow becomes unsteady and chaotic. The most interesting aspect of this problem is that chaotic interior behavior, associated with the separated boundary layer, is observed for parameter values for which the interior topographically forced flow is, by itself, essentially linear.     

On the transition from the laminar to disordered flow in a precessing spherical-like cylinder

We investigate, through both asymptotic analysis and direct numerical simulation, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses about a different axis that is fixed in space. A particular emphasis is placed on a spherical-like cylinder whose diameter is nearly the same as its length. At this special aspect ratio, the strongest direct resonance occurs between the spatially simplest inertial mode and the precessional Poincaré forcing. An asymptotic analytical solution in closed form describing weakly precessing flow is derived in the mantle frame of reference for asymptotically small Ekman numbers. We also construct a nonlinear three-dimensional finite element model – which is validated against both the asymptotic solution and a constructed exact solution – for elucidating the nonlinear transition leading to disordered flow in the precessing spherical-like cylinder. Properties of both weakly and strongly precessing flows are investigated with the aid of a complete inertial-mode decomposition of the fully nonlinear solution. Despite a large effort being made, the well-known triadic resonance is not found in the precessing spherical-like cylinder. The energy contained in the precessionally forced inertial mode is primarily transferred, through nonlinear effects in the viscous boundary layers, to the geostrophic flow that becomes predominant when the precessional Poincaré force is sufficiently large. It is found that the nonlinear flow evolutes gradually and progressively from the laminar to disordered as the precessional force increases.     

A 3-D coastal tidal rectification process: observations, theory and experiments

In homogeneous rotating fluid, when there is an oscillating forcing in the interior fluid with a period long enough for an Ekman layer to develop, there is an interaction between the oscillatory Ekman layer and the vertical wall, since the latter imposes an alternating adjustment flow confined near the wall. As a result, this coastal rectification process leads to a Lagrangian transport along the coast. The Ekman number, the Rossby number and the temporal Rossby number of the forcing flow are the governing parameters of that mechanism which can be described by a simplified analytical model taking into account both the vertical time-dependent structure of the current and the presence of the wall. The model shows that the residual (rectified) current flowing with the coast to its right results from the strong nonlinear interaction between along- and cross-shore tidal currents leading to asymmetrical momentum exchanges between the Ekman bottom layer and the coastal boundary layer. The model provides simple scaling laws for the maximum intensity and width of the residual current. The latter is significantly larger than the friction (Stokes) lateral boundary layer of the forcing flow. A comprehensive set of experiments is performed in the 13 m diameter rotating tank by oscillating an 8 m×2 m horizontal plate and vertical wall in a homogeneous fluid at rest in solid-body rotation and measuring the two horizontal components of the current at several locations and depths above the central part of the plate. The predicted and experimentally measured maximum intensity and width of the residual current are in very good agreement, within the range of validity of the model, i.e. when the Ekman number is sufficiently small. However experiments also show that the residual current still occurs when the Ekman layer thickness is of the same order as the fluid depth, but it is then confined to a narrower band along the vertical wall. The flow structure found experimentally is also correctly described by a numerical model developed by Zhang et al. (1994). Current measurements in the Eastern part of the English Channel near the French coast reveal a significant coastal residual current flowing Northward and the coastal rectification process described here may account for part of it.     

Comparison between measured and calculated tidal ellipses in the German Bight

Using a three-dimensional non-linear shelf model, the elliptical properties (ellipticity, inclination of the ellipse, major and minor semi-axis and phase) of the M₂ tide in the German Bight were calculated and compared with CODAR measurements. A series of barotropic and baroclinic calculations were carried out to investigate the influence of geometry, stratification and particularly inputs of freshwater on these parameters. The elliptical properties undergo stronger changes in zones of influence of embayments and in the deepening of the old Elbe Valley. Friction effects in the shallow areas are responsible for robust vertical variations of the ellipticity. The island of Helgoland induces wakes on its western and eastern sides. The discharge of freshwater of the rivers Elbe, Weser and Ems induced in general negative ellipticity. Although primarily determined by geography, baroclinic effects significantly modified the inclination of the ellipses. The calculated ellipses pattern of anticlockwise and clockwise tidal current rotation agrees quite well with CODAR measurements. The elliptical properties give a general idea of the interaction of tidal waves with coastal geometries.Responsible Editor: Hans Burchard     

Nonlinear dynamical modeling of solar cycles using dynamo formulation with turbulent magnetic helicity

Abstract

We describe a sequence of two-dimensional numerical simulations of inflection point instability in a stably stratified shear flow near the ground. The fastest growing Kelvin-Helmholtz modes are studied in detail; in particular we investigate the growth inhibiting effect of the ground which is predicted by linear theory and the Reynolds number dependence of the process of growth to finite amplitude. We consider flows which are both above and below the critical Reynolds number (Re = 300) which has been reported by Woods (1969) to mark the boundary between flows which have turbulent final states and those which do not. A global energy budget reveals a fundamental difference in character of the finite amplitude billows in these two Reynolds number regimes. However, for relatively high Reynolds numbers (Re = 10³) we do not find any explicit evidence for secondary instability. Above the transition Reynolds number the modified mean flow induced by wave growth is characterized by a splitting of the original shear layer and of the in version in which it is embedded.     

The instability of precessing flow

Abstract

An explanation is put forward for the instability observed within a precessing, rotating spheroidal container. The constant vorticity solution for the flow suggested by Poincaré is found to be inertially unstable through the parametric coupling of two inertial waves by the underlying constant strain field. Such resonant couplings are due either to the elliptical or shearing strains present which elliptically distort the circular streamlines and shear their centres respectively. For the precessing Earth's outer core, the shearing of the streamlines and the ensuing shearing instability are the dominant features. The instability of some exact, linear solutions for finite precessional rates is established and used to corroborate the asymptotic analysis. A complementary unbounded analysis of a precessing, rotating fluid is also presented and used to deduce a likely upperbound on the growth rate of a small disturbance. Connection is made with past experimental studies.     

On the normal mode instability of harmonic waves on a sphere

Abstract

The normal mode instability of harmonic waves in an ideal incompressible fluid on a rotating sphere is analytically studied. By the harmonic wave is meant a Legendrepolynomial flow αP_n(μ) (n ≥ 1) and steady Rossby-Haurwitz wave of set F ₁ ⊕ H_n where H_n is the subspace of homogeneous spherical polynomials of the degree n(n ≥ 2), and F ₁ is the one-dimensional subspace generated by the Legendre-polynomial P₁(μ). A necessary condition for the normal mode instability of the harmonic wave is obtained. By this condition, Fjörtoft's (1953) average spectral number of the amplitude of each unstable mode must be equal to . It is noted that flow αP_n (μ) is Liapunov (and hence, exponentially and algebraically) stable to all the disturbances whose zonal wavenumber m satisfies condition |m| ≥ n. The bounds of the growth rate of unstable normal modes are estimated as well. It is also shown that the amplitude of each unstable, decaying or non-stationary mode is orthogonal to the harmonic wave.

The new instability condition can be useful in the search of unstable perturbations to a harmonic wave and on trials of numerical stability study algorithms. For a Legendre-polynomial flow, it complements Kuo's (1949) condition in the sense that while the latter is related to the basic flow structure; the former characterizes the structure of a growing perturbation.     

Improved bounds on linear instability of barotropic zonal flow within the shallow water equations

Here we develop mathematical results to describe the location of linear instability of a parallel mean flow within the framework of the shallow water equations; growth estimates of near neutral modes (for disturbances subcritical with respect to gravity wave speed) in the cases of non-rotating and rotating shallow water. The bottom topography is taken to be one-dimensional and the isobaths are parallel to the mean flow. In the case of a rotating fluid, the isobaths and the mean flow are assumed to be zonal. The flow is front-like: there is a monotonic increase of mean flow velocity. Our results show that for barotropic flows the location of instabilities will be a semi-ellipse region in the complex wave velocity plane, that is based on the wave-number, Froude number, and depth of the fluid layer. We also explore the instability region for the case of spatially unbounded mean velocity profiles for non-rotating shallow water.     

Tides and mixing in the northwestern East China Sea Part I: Rotating and reversing tidal flows

Bottom-mounted ADV and ADCP instruments in combination with CTD profiling measurements taken along the Chinese coast of the East China Sea were used to study the vertical structure of temperature, salinity, and velocity in reversing tidal currents on a shallow inner shelf and in rotating tidal flows over a deeper sloping bottom of the outer shelf. These two regimes of barotropic tide affect small-scale dynamics in the lower part of the water column differently. The reversing flow was superimposed by seiches of ∼2.3 h period generated in semienclosed Jiaozhou Bay located nearby. As the tidal vector rotates over the sloping bottom, the height of the near-bottom logarithmic layer is subjected to tidal-induced variations. A maximum of horizontal velocity U_max appears at the upper boundary of the log layer during the first half of the current vector rotation from the minor to the major axis of tidal ellipse. In rotating tidal flow, vertical shear generated at the seafloor, propagated slowly to the water interior up to the height of U_max, with a phase speed of ∼5 m/h. The time-shifted shear inside the water column, relative to the shear at the bottom, was associated with periodically changing increases and decreases of the tidal velocity above the log layer toward the sea surface. In reversing flows, the shear generated near the bottom and the shear at the upper levels were almost in phase.     

On the nonlinear evolution of columnar vortices in a rotating environment

We examine the three-dimensional, nonlinear evolution of columnar vortices in a rotating environment. As the initial vorticity distribution, a wavetrain of finite amplitude Kelvin-Helmholtz vortices in shear is employed. Through direct numerical simulation of the Navier-Stokes equations we seek to better understand the process of maturation of the various three-dimensional modes of instability to which such vortical flows are subject, especially those which exist as a consequence of the action of the Coriolis force. In the absence of rotational influence, we thereby demonstrate that the nonlinear evolution of columnar vortices is most strongly controlled by one or the other of two mechanisms. One mechanism of instability is identifiable as a so-called elliptical instability, which promotes the initial bending of vortex tubes in a sinusoidal fashion, while the other is a hyperbolic mode, which is responsible for the development of streamwise vortex streaks in the "braids" between adjacent vortex cores. In the rotating case, anticyclonic vortices are strongly destabilized by weak background rotation, while rapid rotation stabilizes both the cyclones and anticyclones. The strong anticyclones are subject to two distinct forms of instability, namely a Coriolis force modified elliptical instability and an inertial (centrifugal) instability. The former instability is very similar to the nonrotating form of the elliptical instability as it promotes bending of vortex tubes, while the latter instability grows on the edge of the vortex core and generates streaks of vorticity, which surround the vortex core itself. These results of direct numerical simulation fully verify the results of previous linear stability analyses. Taken together, they provide a simple explanation for the broken symmetry that is often observed to be characteristic of the von Karman vortex streets that develop in the atmospheric lee of oceanic islands.     

利用超导重力仪观测资料检测地球近周日共振

简要介绍了地球近周日摆动（ＮＤＦＷ）及其在周日重力潮汐中共振的理论背景,利用武汉和Ｂｒｕｓｓｅｌｓ两台站超导重力仪的潮汐观测结果,采用造积方法（Ｓｔａｃｋｉｎｇｍｅｔｈｏｄ）检测ＮＤＦＷ,根据ＮＤＦＷ在４个周日潮波的共振,得到近周日摆动的本征频率和品质因子．考虑ＮＤＦＷ,可精化地球模型,更客观的理解和分析固体潮观测值和其它一些地球物理现象．     

Theory of internal gravity wave saturation

Gravity wave saturation is an important process affecting the transport and deposition of momentum, heat, and constituents in the earth's atmosphere. This paper informally discusses several saturation mechanisms and their effects, including convection, Kelvin-Helmholtz instability, vortical mode instability, parametric subharmonic instability, and mean flow interaction. Convective saturation is emphasized. The parameterization of convective adjustment is discussed and a few remarks are made concerning the effects of turbulence localization on the convective saturation process. Several outstanding problems in saturation theory are identified that could be addressed with observational, numerical, and laboratory studies.     

地球远磁尾中的磁场重联（Ⅰ）流动撕裂模不稳定性

本文数值研究了地球远磁尾中流动撕裂模不稳定性所引起的磁场重联过程.结果表明,在短暂的指数增长之后,当磁岛宽度接近等离子体片厚度时,流动撕裂模不稳定性的增长率大大降低,最终磁岛宽度趋于饱和.磁岛的饱和宽度随着磁Reynolds数S的增大而减小,随着剪切流动层宽度δV的增大而增加.在S→∞的情况下,流动撕裂模将退化为流动颈缩模,磁场重联不再发生.当飞船通过由流动撕裂模不稳定性所形成的磁岛时,即可观测到磁场B_z分量由北转南或由南转北的现象.     

On the normal mode instability of modons and Wu-Verkley waves

Abstract

The normal mode instability of steady Wu-Verkley (1993) wave and modons by Verkley (1984, 1987, 1990) and Neven (1992) is considered. All these flows are solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere. A conservation law for infinitesimal perturbations to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (1953) average spectral number of the amplitude of an unstable mode must be equal to a specific number that depends on the degree of the solution in its inner and outer regions as well as on spectral distribution of the mode energy in these regions. Some properties of the conditions for different types of modons are discussed. The maximum growth (and decay) rate of the modes is estimated, and the orthogonality of the amplitude of each unstable, decaying, or non-stationary mode to the basic solution is shown in the energy inner product.

The new instability conditions confine the unstable disturbances of the WV wave and modon to a hypersurface in the perturbation space and allow interpretation of their energy structure. They are also useful both in estimating the maximum growth rate of unstable modes and in testing the numerical algorithms designed for the linear stability study.     

SNREI地球对表面负荷和引潮力的形变响应

基于PREM模型，利用非自转、球型分层、各向同性、理想弹性（SNREI）地球的形变理论，讨论了地球在不同驱动力作用下的形变特征．采用地球位移场方程的4阶Runge Kutta数值积分方法，解算了在表面负荷和日月引潮力作用下地球表面和内部形变和扰动位，并给出了地球表面的负荷Love数和体潮Love数．结果表明在固体内核中的形变很小，液核中低阶(n＜10)负荷位移随半径的变化非常复杂．当负荷阶数超过10时，地核中的形变和扰动位都很小，地球的响应主要表现为弹性地幔中的径向位移，且随深度增加急剧减弱，负荷阶数越高这种衰减的速度越快．SNREI地球的地表负荷Love数和体潮Love数与信号频率的依赖关系很弱．在计算体潮Love数的过程中，采用了SNREI地球的运动方程，同时考虑了由于地球自转和椭率引起的核幔边界附加压力，这一近似处理方法获得的结果能很好地符合地球表面重力潮汐实际观测结果．     

A numerical study of island wake generated by an elliptical tidal flow

An idealized numerical study of the influence of a tidal flow around an island has been undertaken with ROMS. The study focusses on coastal island wakes which are mainly controlled by elliptical tidal current flows on shallow shelves. This model is typical of some isolated continental shelf islands. The model is forced by a semi-diurnal barotropic inertia gravity wave imposed on the four open boundaries of a rectangular domain and its propagation results in an elliptical tidal flow within the domain in which the circular island lies. The influence of the surrounding island bathymetry and of the ellipse shape has been studied both in two and three dimensions. In the island vicinity, the residual circulation patterns over a tidal period show alongshore flow divergence along the major axis and convergence along the minor axis. A thin tidal ellipse (i.e. with a large ratio between major and minor axes) leads to strong eddy activity periods in the lee of the island during the flood and ebb phases, with eddy dissipation phases in between. By contrast, an almost round ellipse (axis ratio nearly 1) leads to vorticity filaments which continuously progress around the island without eddy shedding. The presence of a topographic slope in the vicinity of the island strengthens the eddy activity. This study suggests that the tidal current rotation favors the development of the eddy rotating in the same direction and weakens the development of the second eddy. In three dimensions with a surrounding bathymetry, an intense upwelling occurs in a large area in the lee of the island and the vertical velocities are stronger with thinner ellipses. With a flat bottom the vertical motions are almost fully generated by convergence and divergence of the secondary flow. With a varying bottom topography, the vertical motions come from a combination of this mechanism with convergence and divergence of the depth averaged flow.     

EXPERIMENTAL STUDY ON THE COHERENT STRUCTURE OF THE TURBULENT BOUNDARY LAYER IN OPEN-CHANNEL FLOW

1 INTRODUCTION The coherent structure of turbulence is the most important discovery of turbulence studies in the last several decades. The coherent structure is a joint spatial state (Liang, 1999). In this space, there exist all kinds of motions that are organized and related. Namely, strip structures, large vortical structures, and other organizing flows are present. In 1932, researchers found that turbulence wasn抰 a fully random phenomenon and came to realize that the repeated emergen…