Ringdown of inertial waves in a spheroidal shell of rotating fluid

The role of inertial waves in the dynamies of the Earth's fluid core has been investigated through laboratory experiments on a spheroidal shell of rotating fluid. In these experiments inertial waves of azimuthal wavenumber 1, Ekman number 0(10^?5), Rossby number 0(10^?1) were excited by precession of an inner spheroidal body. Proximity to resonance was achieved by adjusting the ratio of the frequency of precession of the inner body to the rotational speed of the container to be near the eigenfrequency of the inertial wave mode being studied. Once the system was near resonance the perturbation was stopped and ringdown records were obtained. Amplitude, eigenfrequency and decay rate were recovered simultaneously for the principal and neighbouring modes excited using an iterative linearized least squares procedure.The recovery of complex eigenfrequencies for non-axisymmetric inertial waves in this shell geometry has given experimental verification of their existence. For those waves of azimuthal wavenumber one, a significant nonlinear interaction among modes is inferred from the simultaneous recovery of neighbouring modes. Other non linear effects include a mean azimuthal flow which appears to be stable for the low spatial order modes studied. These results contrast with highly unstable mean flow found experimentally in similar experiments carried out in cylindrical geometry.     

Inertial waves in spherical shells at low Ekman numbers

Inertial waves as oscillatory motions in rotating fluids generate internal shear layers at critical latitudes. We investigated the nonlinear interaction of inertial waves for global flows (3D flows) in dependence on the Ekman number. When the value of the Ekman number decreases, the influence of the Ekman layers to the flow pattern increases. Critical latitudes, the attractor flow pattern and certainly internal shear layers are observable mainly at greater values of the Ekman number. Although, with decreasing the Ekman number smaller flow structures become visible, nonlinear interactions in shear layers drive an axisymmetric flow whose amplitude diverges at the limit of the vanishing Ekman number. We show that this conclusion is valid not only for zonal wind driven by inertial modes but also for similarly driven global flows.     

Can zonally symmetric inertial waves drive an oscillating zonal mean flow?

In the present paper zonal mean flow excitation by inertial waves is studied in analogy to mean flow excitation by gravity waves that plays an important role for the quasi-biennial oscillation in the equatorial atmosphere. In geophysical flows that are stratified and rotating, pure gravity and inertial waves correspond to the two limiting cases: gravity waves neglect rotation, inertial waves neglect stratification. The former are more relevant for fluids like the atmosphere, where stratification is dominant, the latter for the deep oceans or planet cores, where rotation dominates. In the present study a hierarchy of simple analytical and numerical models of zonally symmetric inertial wave-mean flow interactions is considered and the results are compared with data from a laboratory experiment. The main findings can be summarised as follows: (i) when the waves are decoupled from the mean flow they just drive a retrograde (eastward) zonal mean flow, independent of the sign of the meridional phase speed; (ii) when coupling is present and the zonal mean flow is assumed to be steady, the waves can drive vertically alternating jets, but still, in contrast to the gravity wave case, the structure is independent of the sign of the meridional phase speed; (iii) when coupling is present and time-dependent zonal mean flows are considered the waves can drive vertically and temporarily oscillating mean flows. The comparison with laboratory data from a rotating annulus experiment shows a qualitative agreement. It appears that the experiment captures the basic elements of the inertial wave mean flow coupling. The results might be relevant to understand how the Equatorial Deep Jets can be maintained against dissipation, a process currently discussed controversially.     

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.     

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.     

A local stability analysis of astronomical disks

Abstract

A fifth-order dispersion relation describing the local stability of a differentially rotating flow against small perturbations is derived. Finite viscosity and conductivity and both vertical (parallel to the rotation axis) and radial gradients in density, temperature and pressure are included. A general form is assumed for the equation of state, although this is not exploited in the paper. A number of special cases are studied: with negligible viscosity and conductivity, it is shown that modes can often be separated into two high frequency (modified acoustic), two intermediate frequency (combined inertial and internal waves) and a low frequency mode. In convectively unstable situations the intermediate frequency modes may be replaced by a damped/growing pair of instablities. Various criteria for mode excitation are given. It is shown that viscosity always inhibits instability at very short wavelengths, while non-zero conductivity may destabilize the flow. At intermediate wavelengths viscosity could also play a destabilizing role. A parameter study of the effects of fluctuations in the conductivity shows that it could cause mode excitation under certain circumstances.     

Seismic response control using electromagnetic inertial mass dampers

This paper presents a new type of electromagnetic damper with rotating inertial mass that has been developed to control the vibrations of structures subjected to earthquakes. The electromagnetic inertial mass damper (EIMD) consists of a ball screw that converts axial oscillation of the rod end into rotational motion of the internal flywheel and an electric generator that is turned by the rotation of the inner rod. The EIMD is able to generate a large inertial force created by the rotating flywheel and a variable damping force developed by the electric generator. Device performance tests of reduced‐scale and full‐scale EIMDs were undertaken to verify the basic characteristics of the damper and the validity of the derived theoretical formulae. Shaking table tests of a three‐story structure with EIMDs and earthquake response analyses of a building with EIMDs were conducted to demonstrate the seismic response control performance of the EIMD. The EIMD is able to reduce story drifts as well as accelerations and surpasses conventional types of dampers in reducing acceleration responses. Copyright © 2013 John Wiley & Sons, Ltd.     

Higher-mode force response in multi-story strongback-braced frames

Strongback-braced frames employ an essentially elastic steel truss, or strongback, that distributes demands more uniformly to delay or prevent story mechanisms. Because inertial forces are no longer limited by the formation of a story mechanism, strongback-braced frames can exhibit large elastic force demands, particularly in the higher modes. This paper characterizes the higher-mode force response of strongback-braced frames. Four-story archetypes were designed using nonlinear dynamic analyses to incorporate higher-mode force demands into the design process. The response of the archetypes was compared with that of reference buckling-restrained braced frames that were allowed to form story mechanisms. The force demands in the strongback were then described using equivalent-static forces to represent the inertial forces induced by the higher modes. Force demands in the strongback arise from a yielding first-mode ‘pivoting’ and elastic higher-mode ‘bending’ response. These higher-mode force demands are elastic, ill-constrained by the strength of the yield mechanism, and depend significantly on the choice of ground motion record used for the analysis. In remaining elastic in the higher modes, the strongback distributes demands more uniformly and mitigates the formation of story mechanisms. Consequently, design and analysis methods for strongback-braced frames need to include estimates for these near-elastic higher-mode force demands.     

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.     

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.     

A rotating laser doppler velocimeter and some new results on the spin-up experiment

Abstract

A laser Doppler velocimeter (LDV) has been successfully mounted on a high quality rotating turntable. The capability of this LDV is demonstrated by some detailed measurements of the relative flow during the spin-up of a homogeneous fluid in a cylinder. Local measurements in water of the zonal flow component of magnitude 0.1 cm/sec have been made with an error of about 0.003 cm/sec. The spatial resolution was about 0.1 cm and the temporal resolution about 0.5 Hz. Effects on the flow due to absorption of the low power laser beam (5 milliwatts) and to the low concentration (3 parts/million) of 0.5 micron diameter scattering particles were negligible. The results are compared with analytical theory and the agreement is good. For a Rossby number of 0.1, the weak inertial modes excited by the Ekman layer formation can be clearly seen and identified. The LDV offers great promise for checking numerical and analytical solutions against experiments. This is particularly true for contained flows where conventional probes often significantly disturb the flow.     

Wave patterns generated by disturbances travelling horizontally in rotating stratified fluids

The pattern and propagation of waves generated by steady or oscillatory disturbances travelling horizontally in a rotating, stratified fluid are studied following a technique developed by Lighthill. Both two‐ and three‐dimensional distrubances are investigated. The results show how rotation modifies internal wave patterns in a stratified fluid and how stratification modifies inertial wave patterns in a rotating fluid. The results are used to compute the effective diminution of Taylor column length due to the presence of density stratification. They also show that the appearance of wave crests upstream of a disturbance is possible only when the disturbance is unsteady and that observations of upstream blocking in a two‐dimensional stratified flow can be explained by the existence of a certain class of plane waves as modified by viscosity.     

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.     

Equation of a Barotropic Fluid on a Rotating Spherical Surface and its Inertial Manifold

The inertial manifold is a positive invariant set which exponentially attracts all the trajectories of a dissipative dynamic system. It was introduced for the purpose of studying the asymptotic behaviour of such systems. The initial infinitely dimensional dynamic system, generated by a partial differential evolution equation, can be projected on to it, in order to obtain the final system of ordinary differential equations (inertial equations). These equations then simulate the initial object. Although the inertial manifold is an object relatively simpler than the attractor (a very complicated set of non-integer dimension may be an attractor) it is more difficult to prove its existence than that of the attractor. The equation of a barotropic fluid on a rotating spherical surface is one of the examples of dissipative dynamic systems with an inertial manifold. This kindles the hope that also the equations of the dynamics of the real atmosphere will have an inertial manifold. The reduction of the sample system to this Lipschitz manifold of finite dimension thus justifies us in analysing the behaviour of the atmosphere on non-linear models of finite dimensions and few parameters, in a finite system of ordinary differential equations.     

Linear Magnetoconvection in Rotating Fluid Spheres Permeated by a Uniform Axial Magnetic Field

A linear analysis of thermally driven magnetoconvection is carried out with emphasis on its application to convection in the Earth's core. We consider a rotating and self-gravitating fluid sphere (or spherical shell) permeated by a uniform magnetic field parallel to the spin axis. In rapidly rotating cases, we find that five different convective modes appear as the uniform field is increased; namely, geostrophic, polar convective, magneto-geostrophic, fast magnetostrophic and slow magnetostrophic modes. The polar convective (P) and magneto-geostrophic (E) modes seem to be of geophysical interest. The P mode is characterized by such an axisymmetric meridional circulation that the fluid penetrates the equatorial plane, suggesting that generation of quadrapole from dipole fields could be explained by a linear process. The E mode is characterized by a few axially aligned columnar rolls which are almost two-dimensional due to a modified Proudman-Taylor theorem.     

Inertial waves in a fluid partially filling a cylindrical cavity during spin-up from rest

Abstract

Inertial waves are excited in a fluid contained in a slightly tilted rotating cylindrical cavity while the fluid is spinning up from rest. The surface of the fluid is free. Since the perturbation frequency is equal to the rotation speed resonance occurs at a critical height to radius aspect ratio of the fluid. Detailed study of a particular inertial wave shows that in solid body rotation this “eigenratio” agrees with predictions from linear inviscid theory to within 0.5%. Measured time dependence of the eigenratio during spin-up from rest is a function of the tilt amplitude and agrees favorably with predictions from a numerical study. Mean flow associated with the inertial wave becomes unstable during spin-up and in the steady state. A boundary for the unstable region is found experimentally.     

The bottom Ekman layer and the apparent violation of the maximum principle

The solution for the bottom Ekman layer has a somewhat counter intuitive character, which seems to violate the maximum principle: at a certain level the velocity within the Ekman layer is higher than the velocity in the geostrophic layer above. I explain this character by looking at an analogous problem in an inertial frame of reference and show that it is the result of observing the flow from a rotating frame of reference (i.e. within a system that is not in steady state). The flow in the bottom Ekman layer is a superposition of the flow that results from the force exerted on the fluid by the rotating Earth and of the flow that results from the pressure-gradient term. Therefore, at a certain level the speed is higher than the speed of the geostrophic layer above which results from the pressure gradient alone.     

On the stability of plane flow with horizontal shear to three-Dimensional nondivergent disturbances

Abstract

Arnold's (1965a) method is used to investigate the stability of a stationary, nonparallel, plane flow, with horizontal shear, to three-dimensional nondivergent disturbances in a Boussinesq fluid. It is shown that, if the fluid is statically stable, the Rayleigh condition is not sufficient to insure inertial stability to all disturbance modes. For channel flow it is possible to establish the sufficiency condition for stability to some of these modes.     

Inertial effects in an oceanic circulation model

Abstract

The flow in a mechanically driven thin barotropic rotating fluid system is analysed. The linear theory of Baker and Robinson (1969) is modified and extended into the non-linear regime.

An internal parameter, the “local Rossby number”, is indicative of the onset of nonlinear effects. If this parameter is 0(1) then inertial effects are as important as Coriolis accelerations in the interior of the transport-turning western boundary layer and both of its Ekman layers. The inertial effects in the Ekman layers, ignored in previous explorations of non-linear wind driven oceanic circulation, are retained here and calculated using an approximation of the Oseen type. The circulation problem is reduced to a system of scalar equations in only two independent variables; the system is valid for non-small local Rossby number provided only that the approximate total vorticity is positive.

To complete the solution for small Rossby number a boundary condition for the inertially induced transport is needed. It is found by examining the dynamics controlling this additional transport from the western boundary layer as the transport recirculates through the rest of the ocean basin. The strong constraint of total recirculation within the western boundary layer (zero net inertial transport) is derived.

The calculated primary inertial effects are in agreement with the observations of the laboratory model of Baker and Robinson (1969).

The analysis indicates the extent to which three-dimensional non-linear circulation can be reduced to a two dimensional problem.