Showing papers on "Inertial wave published in 2006"

Wave turbulence in rapidly rotating flows

Abstract: An asymptotic quasi-normal Markovian (AQNM) model is developed in the limit of small Rossby number Ro and high Reynolds number, i.e. for rapidly rotating turbulent flow. Based on the 'slow' amplitudes of inertial waves, the kinetic equations are close to those that would be derived from Eulerian wave-turbulence theory. However, for their derivation we start from an EDQNM statistical closure model in which the velocity field is expanded in terms of the eigenmodes of the linear wave regime. Unlike most wave-turbulence studies, our model accounts for the detailed anisotropy as the angular dependence in Fourier space. Nonlinear equations at small Rossby number are derived for the set e, Z, h - energy, polarization anisotropy, helicity - of spectral quantities which characterize second-order two-point statistics in anisotropic turbulence, and which generate every quadratic moment of inertial wave amplitudes. In the simplest symmetry consistent with the background equations, i.e. axisymmetry without mirror symmetry, e, Z and h depend on both the wavevector modulus k and its orientation θ to the rotation axis. We put the emphasis on obtaining accurate numerical simulations of a generalized Lin equation for the angular-dependent energy spectrum e(k, θ , t ), in which the energy transfer reduces to integrals over surfaces given by the triadic resonant conditions of inertial waves. Starting from a pure three-dimensional isotropic state in which e depends only on k and Z = h = 0, the spectrum develops an inertial range in the usual fashion as well as angular anisotropy. After the development phase, we observe the following features: (a) A k^−3 power law for the spherically averaged energy spectrum. However, this is the average of power laws whose exponents vary with the direction of the wavevector from k^−2 for wavevectors near the plane perpendicular to the rotation axis, to k^−4 for parallel wavevectors. (b) The spectral evolution is self-similar. This excludes the possibility of a purely two-dimensional large-time limit. (c) The energy density is very large near the perpendicular wavevector plane, but this singularity is integrable. As a result, the total energy has contributions from all directions and is not dominated by this singular contribution. (d ) The kinetic energy decays as t^−0.8 , an exponent which is about half that one without rotation.

On the evolution of eddies in a rapidly rotating system

Abstract: The formation of columnar eddies in a rapidly rotating environment is often attributed to nonlinear processes, acting on the nonlinear time scale l/|u|. We argue that this is not the whole story, and that linear wave propagation can play an important role, at least on the short time scale of Ω -1 . In particular, we consider the initial value problem of a compact blob of vorticity (an eddy) sitting in a rapidly rotating environment. We show that, although the energy of the eddy disperses in all directions through inertial wave propagation, the axial components of its linear impulse and angular momentum disperse along the rotation axis only, remaining confined to the cylinder which circumscribes the initial vortex blob. This confinement has a crucial influence on the manner in which energy disperses from the eddy, with the energy density within the tangent cylinder remaining much higher than that outside (i.e. decaying as t-' inside the cylinder and t -3/2 outside). When the initial conditions consist of an array of vortex blobs the situation is more complicated, because the energy density within the tangent cylinder of any one blob is eventually swamped by the radiation released from all the other blobs. Nevertheless, we would expect that a turbulent flow which starts as a collection of blobs of vorticity will, for times of order Ω -1 , exhibit columnar vortices, albeit immersed in a random field of inertial waves. Laboratory experiments are described which do indeed show the emergence of columnar eddies through linear mechanisms, though these experiments are restricted to the case of inhomogeneous turbulence. Since the Rossby number in the experiments is of the order of unity, this suggests that linear effects can still influence and shape turbulence when nonlinear processes are also operating.

Three-dimensional Compressible Hydrodynamic Simulations of Vortices in Disks

Abstract: We carry out three-dimensional, high-resolution (up to 10242 × 256) hydrodynamic simulations of the evolution of vortices in vertically unstratified Keplerian disks using the shearing sheet approximation. The transient amplification of incompressible, linear amplitude leading waves (which has been proposed as a possible route to nonlinear hydrodynamic turbulence in disks) is used as one test of our algorithms; our methods accurately capture the predicted amplification, converge at second order, and are free from aliasing. Waves that are expected to reach nonlinear amplitude at peak amplification become unstable to Kelvin-Helmholtz modes when Wmax Ω (where Wmax is the local maximum of vorticity and Ω the angular velocity). We study the evolution of a power-law distribution of vorticity consistent with Kolmogorov turbulence; in two dimensions long-lived vortices emerge and decay slowly, similar to previous studies. In three dimensions, however, vortices are unstable to bending modes, leading to rapid decay. Only vortices with a length-to-width ratio smaller than 1 survive; in three dimensions the residual kinetic energy and shear stress is at least 1 order of magnitude smaller than in two dimensions. No evidence for sustained hydrodynamic turbulence and transport is observed in three dimensions. Instead, at late times the residual transport is determined by the amplitude of slowly decaying, large-scale vortices (with horizontal extent comparable to the scale height of the disk), with additional contributions from nearly incompressible inertial waves possible. Evaluating the role that large-scale vortices play in astrophysical accretion disks will require understanding the mechanisms that generate and destroy them.

Energy decay of rotating turbulence with confinement effects

Abstract: The energy decay of grid-generated turbulence in a rotating tank is experimentally investigated by means of particle image velocimetry. For times smaller than the Ekman time scale, a range of approximate self-similar decay is found, in the form u2(t)∝t−n, with the exponent n decreasing from 2 to values close to 1 as the rotation rate is increased. Even at very weak rotation rates, rotation is shown to have a strong indirect influence on the decay law, by making the integral length scale to quickly saturate to the experiment size through the propagation of inertial waves. The experimental decay exponents are found in good agreement with the predicted values from a phenomenological model based on the exponent of the energy spectrum, in which both the effects of the rotation and the confinement are taken into account.

On the evolution of eddies in a rapidly rotating system

Abstract: value problem of a compact blob of vorticity (an eddy) sitting in a rapidly rotating environment. We show that, although the energy of the eddy disperses in all directions through inertial wave propagation, the axial components of its linear impulse and angular momentum disperse along the rotation axis only, remaining confined to the cylinder which circumscribes the initial vortex blob. This confinement has a crucial influence on the manner in which energy disperses from the eddy, with the energy density within the tangent cylinder remaining much higher than that outside (i.e. decaying as t −1 inside the cylinder and t −3/2 outside). When the initial conditions consist of an array of vortex blobs the situation is more complicated, because the energy density within the tangent cylinder of any one blob is eventually swamped by the radiation released from all the other blobs. Nevertheless, we would expect that a turbulent flow which starts as a collection of blobs of vorticity will, for times of order Ω −1 , exhibit columnar vortices, albeit immersed in a random field of inertial waves. Laboratory experiments are described which do indeed show the emergence of columnar eddies through linear mechanisms, though these experiments are restricted to the case of inhomogeneous turbulence. Since the Rossby number in the experiments is of the order of unity, this suggests that linear effects can still influence and shape turbulence when nonlinear processes are also operating.

Three Dimensional Compressible Hydrodynamic Simulations of Vortices in Disks

Abstract: We carry out three-dimensional, high resolution (up to $1024^2\times 256$) hydrodynamic simulations of the evolution of vortices in vertically unstratified Keplerian disks using the shearing sheet approximation. The transient amplification of incompressible, linear amplitude leading waves (which has been proposed as a possible route to nonlinear hydrodynamical turbulence in disks) is used as one test of our algorithms; our methods accurately capture the predicted amplification, converges at second-order, and is free from aliasing. Waves expected to reach nonlinear amplitude at peak amplification become unstable to Kelvin-Helmholtz modes when $\mid W_{\rm max}\mid\gtrsim \Omega$ (where $W_{\rm max}$ is the local maximum of vorticity and $\Omega$ the angular velocity). We study the evolution of a power-law distribution of vorticity consistent with Kolmogorov turbulence; in two-dimensions long-lived vortices emerge and decay slowly, similar to previous studies. In three-dimensions, however, vortices are unstable to bending modes, leading to rapid decay. Only vortices with a length to width ratio smaller than one survive; in three-dimensions the residual kinetic energy and shear stress is at least one order of magnitude smaller than in two-dimensions. No evidence for sustained hydrodynamical turbulence and transport is observed in three-dimensions. Instead, at late times the residual transport is determined by the amplitude of slowly decaying, large-scale vortices (with horizontal extent comparable to the scale height of the disk), with additional contributions from nearly incompressible inertial waves possible. Evaluating the role that large-scale vortices play in astrophysical accretion disks will require understanding the mechanisms that generate and destroy them.

Magnetic field induced by elliptical instability in a rotating spheroid

Abstract: The tidal or the elliptical instability of the rotating fluid flows is generated by the resonant interaction of the inertial waves. In a slightly elliptically deformed rotating sphere, the most unstable linear mode is called the spin-over mode, and is a solid body rotation versus an axis aligned with the maximum strain direction. In the non-viscous case, this instability corresponds to the median moment of the inertial instability of the solid rotating bodies. This analogy is furthermore illustrated by an elliptical top experiment, which shows the expected inviscid heteroclinic behaviour. In geophysics, the elliptical instability may appear in the molten liquid cores of the rotating planets, which are slightly deformed by the tidal gravitational effects of the close bodies. It may then participate in the general outer core dynamics and possibly the geodynamo process. In this context, Kerswell and Malkus (Kerswell, R.R. and Malkus, W.V.R., Tidal instability as the source for Io's magnetic signature. Geophy...

Quasi-periodic Oscillations from Magnetorotational Turbulence

Abstract: Quasi-periodic oscillations (QPOs) in the X-ray light curves of accreting neutron star and black hole binaries have been widely interpreted as being due to standing wave modes in accretion disks. These disks are thought to be highly turbulent due to the magnetorotational instability (MRI). We study wave excitation by MRI turbulence in the shearing box geometry. We demonstrate that axisymmetric acoustic modes and radial epicyclic motions driven by MRI turbulence give rise to narrow, distinct peaks in the temporal power spectrum. Inertial waves, on the other hand, do not give rise to distinct peaks which rise significantly above the continuum noise spectrum set by MRI turbulence, even when the fluid motions are projected onto the eigenfunctions of the modes. This is a serious problem for QPO models based on inertial waves.

Inertial oscillations and related internal beat pulsations and surges in Lakes Michigan

Abstract: Records of water temperature, current, and wind, made during campaigns on Lakes Michigan 1963 and Ontario 1972, are searched for inertial responses to wind action: (i) circle-tracking currents; and (ii) isothermdepth undulations, i.e., long internal wave manifestations of forcing by wind. With stratification absent in winter, only response (i) is seen (and then only rarely) with periods very close to the local inertial period, Tin. After wholebasin stratification is complete, both responses (i) and (ii) occur frequently, as internal Kelvin waves (shore

Oscillatory jets and instabilities in a rotating cylinder

Abstract: The viscous flow inside a closed rotating cylinder of gas subject to periodic axial compression is investigated numerically. The numerical method is based on a spectral Galerkin expansion of the velocity field, assuming axisymmetry of the flow. If the forcing amplitude is weak and the angular forcing frequency is less than twice the rotation rate, inertial waves emanate from the corners, forming conical oscillatory jets which undergo reflections at the walls. Their thickness is O(E1∕3), or O(E1∕4) for particular forcing frequencies, where E is the Ekman number. For larger forcing amplitudes, the conical pattern breaks down. When the forcing frequency is resonant with a low-order inertial mode, the flow can undergo two types of parametric instabilities: a mode-triad resonance, and a subharmonic instability. The combination of both these mechanisms provides a possible route to quasiperiodicity of the flow.

Vertical structure of time-dependent currents in a mid-ocean ridge axial valley

Abstract: Velocity measurements with high vertical resolution, and a two-dimensional linear quasianalytic model for subinertial oscillatory flows, are used to analyze the vertical structure of flow in the axial valley at Endeavour Segment, Juan de Fuca Ridge. At a site away from hydrothermal vents, observed semidiurnal flows are independent of depth, rectilinear and parallel to the valley axis, while subinertial flows are intensified and re-aligned along-valley toward the bottom. This behavior is consistent with solutions from the model, which show attenuation of subinertial across-valley flow with depth. This cross-flow attenuation is most pronounced for valleys with widths less than the internal Rossby radius of deformation. Reduction of across-valley flow with depth results in a weakened Coriolis force that cannot fully balance the along-valley pressure-gradient force. The resulting force imbalance yields a directly accelerated bottom-intensified along-valley flow. The importance of this physical process in other submarine valleys depends on their geometry, stratification and latitude. If active, this mechanism provides a dynamic background environment for the axial valley to which hydrothermal venting would add complexity. The strong vertical shears and spiraling flows observed within the axial valley for diurnal tidal and lower-frequency flows have important implications in the transport of hydrothermal vent fluid and the dispersal of larvae of vent organisms by bottom currents.

Inertial oscillations and related internal beat pulsations and surges in Lakes Michigan and Ontario

Abstract: Records of water temperature, current, and wind, made during campaigns on Lakes Michigan 1963 and Ontario 1972, are searched for inertial responses to wind action: (i) circle-tracking currents; and (ii) isotherm depth undulations, i.e., long internal wave manifestations of forcing by wind. With stratification absent in winter, only response (i) is seen (and then only rarely) with periods very close to the local inertial period, Tin. After whole basin stratification is complete, both responses (i) and (ii) occur frequently, as internal Kelvin waves (shore trapped and not further treated here) and cross-basin Poincare internal seiche modes. Mode periods range between 1% and 15% less than Tin, depending on which of the here-modeled modes have responded and on differences between modes in their partitioning of kinetic and potential energy. When, in both lakes, short duration wind impulses were followed by a week or more of relative calm, Poincare mode combinations produced beat pulsations in both responses (i) and (ii), diagnostic features of which sometimes permitted participating modes to be identified. At a nearshore downwelled front in Lake Ontario, another type of poststorm adjustment was seen. Periodically released from the front, internal surges migrated across the basin through fields of inertially rotating, response (i) currents.

Characteristics and energy rays of equatorially trapped, zonally symmetric internal waves

Abstract: Characteristic curves of partial differential equations (PDEs) in general differ from short wave energy rays. We give conditions for linear, two dimensional second order PDEs that guarantee an exact correspondence between characteristics and energy rays. The findings are applied to time-harmonic, zonally-symmetric small-amplitude equatorial internal waves. It is shown that for a fairly general model of inertia-gravity waves, an exact characteristic-ray correspondence holds. When characteristics of internal waves, trapped in a meridional plane, are followed over several boundary reflections, a convergence towards a limit cycle (called equatorial wave attractor) can generally be found. The results on the characteristic-ray correspondence help to interpret physically equatorial wave attractors. Recent ideas on energy accumulation by near inertial waves, trapped on wave attractors at deep ocean sites are confirmed by our results, at least in the short wave (WKB) sense.

Second-order random internal and surface waves in a two-fluid system

Abstract: [1] Second-order solutions for internal and surface waves in a two-fluid system are derived using a perturbation technique. As expected, solutions of the second order are composed of the products of the first-order components. Super- and subharmonic transfer functions describing the relation between the first- and second-order wave amplitudes are introduced. For the self interaction of the first-order waves with a fixed wavenumber, there exist three combinations of linear waves. The associated superharmonic transfer function and the effects of second-order waves on wave profiles are examined. Furthermore, taking the density of the upper layer to be zero, present results include most existing theories for second-order surface waves as special cases.

An improved calculation of Coriolis terms on the C grid

Abstract: The central differencing grid with fully staggered velocity components (C grid) is widely used in primitive equation oceanographic models despite potential problems in simulating baroclinic inertia–gravity and Rossby waves that can arise due to the averaging of velocity components in the Coriolis terms. This note proposes a new averaging of the velocity components in order to calculate the Coriolis terms on the C grid. The averaging weights are calculated from the minimum of a suitably defined cost function that optimally minimizes the error in the inertial part of frequencies of inertia–gravity waves and maintains the second-order accuracy of the computations. The theoretical analysis of wave frequency diagrams shows that the new scheme results in more accurate frequencies of long inertia–gravity and Rossby waves, especially when the Rossby radius of deformation is not well resolved by the grid resolution.

Origin of Mass. Mass and Mass-Energy Equation from Classical-Mechanics Solution

Abstract: We establish the classical wave equation for a particle formed of a massless oscillatory elementary charge generally also traveling, and the resulting electromagnetic waves, of a generally Doppler-effected angular frequency ω, in the vacuum in three dimensions. We obtain from the solutions the total energy of the particle wave to be e = ¯cω, 2π¯c being a function expressed in wave-medium parameters and identifiable as the Planck constant. In respect to the train of the waves as a whole traveling at the finite velocity of light c, e = mc 2 represents thereby the translational kinetic energy of the wavetrain, m = ¯cω/c 2 being its inertial mass and thereby the inertial mass of the particle. Based on the solutions we also write down a set of semi-empirical equations for the particle’s de Broglie wave parameters. From the standpoint of overall modern experimental indications we comment on the origin of mass implied by the solution.

Driven inertial waves in spherical Couette flow

Abstract: Dynamo action in the cores of planets and stars gives rise to their magnetic fields. We explore spherical Couette flows in liquid sodium as a laboratory model of the Earth’s core. Below, we image various spatiotemporal magnetic-field modes present in spherical Couette flows. Our apparatus is comprised of two independently driven rotating spheres, radii 20 and 60 cm, with sodium filling the gap between them Fig. 1 . We apply a B0=50 G axial magnetic field from external magnets and image the resulting induced magnetic field. As the Lundquist number S= B0l * 2 0 −1/2 where l is a characteristic length scale, is the magnetic diffusivity, and is the density is small S=0.7 , the magnetic field is a passive probe of the internal flows. Data from an array of 21 Hall probes along a meridian M1. . .M21 , plus four more distributed along the equator E1. . .E4 , are used to characterize the induced magnetic field exiting the outer sphere. We have projected the resulting data onto a basis of spherical harmonics and used that projection to produce the magneticfield images shown Fig. 2 . Patterns of Coriolis-restored inertial waves are present when the Ekman number, E = / 2 ol where is the kinematic viscosity and o is the angular frequency of the outer sphere , is small, and they depend on the rotation rate ratio of the inner and outer spheres i / o. All data shown here are for fixed o =30 Hz E=1.2 10−8 . The spectrogram at the left of Fig. 2 shows the frequency content of a time series of a single equatorial probe. These wave modes are likely to occur in the Earth’s outer core, but would largely be masked from view by the mantle. FIG. 1. Experimental setup.

Inertial Motions during the Transient Adjustment of a Density Anomaly in the Equatorial Ocean with Application to the Western Pacific Warm Pool

Abstract: This paper is focused on the spontaneous transient adjustment of a buoyant lens of water with uniform density, initially at rest in the vicinity of the equator. For parameters typical of the western Pacific warm pool, the adjustment is shown to generate finite-amplitude wave motions with period ∼8 days, which are not covered by the standard theory of linear equatorial waves. This mechanism may be at the origin of inertial motions at the early stages of ENSO events in the western Pacific Ocean. The lens adjustment is studied within the 11/2-layer reduced-gravity approximation on the equatorial β plane, using the high-resolution finite-volume numerical methods that are specially designed to handle outcropping isopycnals. Under the reduced-gravity approximation, a buoyant region of light water with outcropping boundaries in the vicinity of the equator is described by two parameters: the meridional-to-zonal scale aspect ratio δ and the ratio γ of the Coriolis force to the pressure force on its meridi...

Wave Force on an Ocean Current

Abstract: Linear momentum of surface gravity waves changes with time during refraction by a horizontally variable current, as is predicted by ray theory; the momentum change per unit time requires a force by the current on the waves. According to Newton’s third law, the waves apply an equal but opposite force back on the current. The wave force of linear waves on the current is calculated for a steady horizontal shear current and it is found to be directly proportional to the wave momentum times the shear in the current. For a current like the Gulf Stream it is theoretically possible for the wave force on the current to be as large as the Coriolis force on the current to the depth of wave influence; the effect on equatorial surface currents is likely to be even more significant. Considering the reasonable conjecture that the orbital angular momentum of the waves cannot be exchanged with the current, the growth or decay of the wave amplitude in the shear current is computed as well. An exponential growth or...

Are geostrophic and quasi-geostrophic approximations valid in Venus’ differential super-rotation?

Abstract: Geostrophic dynamics in the horizontally differential super-rotation of Venus are examined using fA (the Coriolis parameter defined by the angular velocity of a basic flow in an inertial frame) and Γ (the differential rotation parameter defined by the latitudinal gradient of the angular velocity) under the conditions that vertical shear of the basic field is not considered and the intrinsic phase velocity has a magnitude comparable to that of an eddy horizontal flow. The geostrophic and quasi-geostrophic approximations are valid in the regions of weakly differential and rigid-body super-rotations; however, they are invalid in the regions of strongly differential super-rotation even when the Rossby number R O is sufficiently smaller than unity for synoptic eddies. In a general circulation model of a Venus-like atmosphere, the horizontal divergence that results from the strong differential cannot be ignored over a wide range within latitudes ±60° and below 60 km elevation because of large Γ/fA (≥1/2).

Near-inertial Current variability off the east coast of Korea

Abstract: in Korean) 118 Appendices I. Derivation of dispersion relation for near-inertial waves under background subinertial motions 119 II. Filling ADCP data gap caused by vertical migration of zooplankton 123 Funded programs 125 Acknowledgements 126 Curriculum Vitae 129

Laboratory Studies of Nonlinear and Breaking Surface Waves

Abstract: Author(s): Drazen, David | Abstract: A laboratory investigation of nonlinear and breaking surface waves is presented in two parts. The first focuses on the instability of progressive surface gravity waves incident on a vertical wall and the second on the measurement of the kinematics and dynamics of breaking progressive waves and the turbulence they generate.In Part I, Theoretical arguments suggest that progressive gravity waves incident on a vertical wall can produce periodic standing waves only if the incident wave steepness a k is quite small. Laboratory experiments are carried out in which an incident wavetrain of almost uniform amplitude meets a vertical barrier. When a k g 0.236, a growing instability is observed in which every third wave crest is steeper than its neighbours. The instability grows by a factor of about 2.2 for every three wave periods, almost independently of the incident wave steepness.In Part II, the measurement of the dissipation of wave energy by breaking over a significant range of parameter space allows the kinematics of breaking to be related to the underlying dynamics. Control volume analysis yields a measure of the change in energy flux across the volume and is related to the dissipation through the duration of active breaking. Assuming the plunging wave toe follows a ballistic trajectory, an inertial estimate of the dissipation is developed and found to predict the dissipation rate within an order of magnitude.Detailed measurements of the post-breaking velocity field using DPIV are conducted in the longitudinal and transverse planes. Statistical measures of the turbulence are presented. Separation of the surface-wave induced velocity from the full measured velocity helps isolate the effects of breaking, including the generation of coherent vorticity. Turbulent wave number spectra exhibit a deviation from the inertial subrange at high wave numbers, thought to be caused by an imbalance between the flux of energy from large scales and the dissipation at small scales. Measurements of terms in the turbulent kinetic energy density equation are presented. The relationship between the three-dimensional turbulent kinetic energy density and two-dimensional approximations are discussed. A comparison between various estimates of the rate of viscous dissipation is also given.

Inertial particle approximation to solutions of the Shallow Water Equations on the rotating spherical Earth

Abstract: The present study assesses the relationship between solutions of the mechanical problem of particle motion on the surface of a rotating sphere subject only to the gravitation force (called inertial particle motion) and the fluid dynamical problem there, described by the shallow water equations (SWE). Trajectories of fluid parcels advected by a time-dependent velocity field subject to the SWE on the sphere are computed numerically and compared to analytical formulae of inertial particle trajectories. In addition, the free surface height of an ensemble of noninteracting particles is estimated within the classical mechanics framework and compared to computed height of the SWE. The comparison between solutions of the two systems shows very good qualitative as well as quantitative agreement for times of several inertial periods in the following basic low-energy cases: inertial particle oscillations in mid-latitudes (corresponding to inertial waves in fluid dynamics) and inertial motion near the equator. Moreover, for realistic values of the reduced gravity ( gH of 1 to 100 m 2 s −2 ) and for time interval of 1–2 d the periods of the trajectories of fluid parcels nearly coincide with those of inertial particles. These results are obtained for a wide range of initial velocity fields and they imply that, at least for time intervals considered, the Coriolis force dominates the motion even after the pressure gradient forces become sufficiently large to affect the motion. They also highlight the fact that fluid parcels of non-linear inertialwaves are subject to the same westward drift as inertial particles and provide an explanation for existence of so called ‘inertial peak’ in the internal oceanic wave spectrum.

Numerical simulations of axisymmetric inertial waves in a rotating sphere by finite elements

Abstract: Axisymmetric inertial waves of a viscous fluid that fill a perturbed rotating spherical container are numerically simulated by finite elements. A laminar flow of an incompressible viscous fluid of Newtonian type is assumed in the numerical simulations. A monolithic computational code is employed, which is based on stabilized finite elements by means of a Streamline Upwind Petrov Galerkin (SUPG) and Pressure Stabilized Petrov Galerkin (PSPG) composed scheme. The Reynolds number is fixed as 50 000, while the ranges of the Rossby and Ekman numbers are 0.2 Ro 1 and 2 × 10-5 Ek 10-4 , respectively. Some flow visualizations are performed. The pressure coefficient spectrum at the centre of the sphere is plotted as a function of the frequency ratio and some resonant frequencies are identified. The position of these resonant frequencies are in good agreement with previous experimental and analytical ones in the inviscid limit. [To appear in International Journal of Computational Fluid Dynamics.Author Posting. (c) Taylor & Francis, 2007. ]

Scattering of barotropic Rossby waves by the Antarctic Circumpolar Current

Abstract: This study examines the interactions between barotropic Rossby waves and a zonal current, with particular reference to the Antarctic Circumpolar Current (ACC). In the high latitude of the Southern Ocean, the effect, which provides the restoring mechanism for Rossby waves, is relatively weak, resulting in barotropic waves that are extremely long, with periods of the order of 1 week or longer. We model the interactions between the current and the waves using the linearized potential vorticity (PV) equation with the inclusion of background PV terms corresponding to a barotropic zonal flow of finite width. An analytical solution is found for the simplest, piecewise-linear flow and numerical solutions obtained for more realistic, smoothly varying flows. The results show that, in general, the long waves are not appreciably modified or reflected by the shear flow, except in the case where the wave is incident at an oblique angle. Wave reflection is also more pronounced for shorter waves, such as Rossby waves having eastward group velocity or the case where the wave frequency is just below the cutoff frequency.

Quasi-Periodic Oscillations from Magnetorotational Turbulence

Abstract: Quasi-periodic oscillations (QPOs) in the X-ray lightcurves of accreting neutron star and black hole binaries have been widely interpreted as being due to standing wave modes in accretion disks. These disks are thought to be highly turbulent due to the magnetorotational instability (MRI). We study wave excitation by MRI turbulence in the shearing box geometry. We demonstrate that axisymmetric sound waves and radial epicyclic motions driven by MRI turbulence give rise to narrow, distinct peaks in the temporal power spectrum. Inertial waves, on the other hand, do not give rise to distinct peaks which rise significantly above the continuum noise spectrum set by MRI turbulence, even when the fluid motions are projected onto the eigenfunctions of the modes. This is a serious problem for QPO models based on inertial waves.

Inertial particle approximation to solutions of the Shallow Water Equations on the rotating spherical Earth

Abstract: The present study assesses the relationship between solutions of the mechanical problem of particle motion on the surface of a rotating sphere subject only to the gravitation force (called inertial particle motion) and the fluid dynamical problem there, described by the shallow water equations (SWE). Trajectories of fluid parcels advected by a time-dependent velocity field subject to the SWE on the sphere are computed numerically and compared to analytical formulae of inertial particle trajectories. In addition, the free surface height of an ensemble of noninteracting particles is estimated within the classical mechanics framework and compared to computed height of the SWE. The comparison between solutions of the two systems shows very good qualitative as well as quantitative agreement for times of several inertial periods in the following basic low-energy cases: inertial particle oscillations in mid-latitudes (corresponding to inertial waves in fluid dynamics) and inertial motion near the equator. Moreover, for realistic values of the reduced gravity (gH of 1 to 100 m 2 s −2 ) and for time interval of 1‐2 d the periods of the trajectories of fluid parcels nearly coincide with those of inertial particles. These results are obtained for a wide range of initial velocity fields and they imply that, at least for time intervals considered, the Coriolis force dominates the motion even after the pressure gradient forces become sufficiently large to affect the motion. They also highlight the fact that fluid parcels of non-linear inertial waves are subject to the same westward drift as inertial particles and provide an explanation for existence of so called ‘inertial peak’ in the internal oceanic wave spectrum.

Numerical simulations of axisymmetric inertial waves in a rotating sphere by finite elements

Abstract: Axisymmetric inertial waves of a viscous fluid that fill a perturbed rotating spherical container are numerically simulated by finite elements. A laminar flow of an incompressible viscous fluid of Newtonian type is assumed in the numerical simulations. A monolithic computational code is employed, which is based on stabilized finite elements by means of a Streamline Upwind Petrov Galerkin (SUPG) and Pressure Stabilized Petrov Galerkin (PSPG) composed scheme. The Reynolds number is fixed as 50,000, while the ranges of the Rossby and Ekman numbers are and respectively. Some flow visualizations are performed. The pressure coefficient spectrum at the centre of the sphere is plotted as a function of the frequency ratio and some resonant frequencies are identified. The position of these resonant frequencies are in good agreement with previous experimental and analytical ones in the inviscid limit.