A laboratory demonstration of angular momentum mixing

Abstract

It is successfully demonstrated that substantial redistribution of the angular momentum within a completely liquid-filled cylinder in uniform rotation can be brought about by the induction of turbulent mixing through the resonant excitation of standing inertial waves. This means of mixing is accomplished without significant net circulation in the meridional plane, or strong boundary restraint.

Intense cyclonic vortices are created with an apparently high conversion of energy from the inertial wave excited. Visualizations and measurements of vortex strength and circulation distribution are presented and dimensional arguments are applied to interpret from the measurements the partition of the turbulence into relative velocity- and angular momentum-diffusing elements. This indicates tentatively the mechanism responsible; momentum advected by the inertial wave is irreversibly diffused by turbulence of smaller scale. Anisotropy with enhanced radial transport is an essential feature of the nett turbulence in such a mechanism. Similar combinations of large-scale waves and turbulence can be expected to occur in the geophysical situations to which the phenomenon of angular momentum mixing relates. The experiment does not, however, test the effectiveness of isotropic turbulence in the same rôle.     

On a general fluid dynamical theory of discrete unstable spiral modes in disk-shaped galaxies

Abstract

A general fluid dynamical theory of discrete unstable spiral modes in disk-shaped galaxies is described. This formulation of modes includes a radiation boundary condition and an exact numerical treatment of the Poisson equation. Thus, the modes are maintained by an outward transport of angular momentum, but they may be composed of both leading and trailing waves. A numerical scheme based on this formulation is described, and examples of modes obtained with this scheme are presented. These examples compare favorably with calculations based on the original asymptotic theory of Bertin, Lau, Lin and Mark. The implications of the present formulation of modes in galactic models support the hypothesis of a quasi-stationary spiral structure.     

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.     

Laboratory experiments on fronts

Abstract

Supercritically unstable density fronts near a vertical wall in a rotating, two-layer fluid were created on a laboratory turntable by withdrawing the outer wall of an annulus with a narrow gap, and allowing buoyant fluid from within the annulus to collapse toward a state of quasi-geostrophic balance. The resulting “coastal” current has a nearly uniform potential vorticity and is bounded by a front on which ageostrophic, wave-like disturbances grow. If the current width is comparable to the Rossby radius of deformation, the dominant length scale of disturbances is proportional to the width of the current. On the other hand, if the upper layer is much wider than the Rossby radius, then the observed length scale is a constant multiple of the Rossby radius. If the vertical boundary is omitted in the experiments, so that we are left with a circular anticyclonic vortex, the observed length scales and large-amplitude behaviour of disturbances are identical to those for the boundary currents, indicating that the wall has no significant influence on the flow.

At very large amplitude the growing waves lead to the formation of cyclone-anticyclone vortex pairs. For very wide currents, both the mean flow and the disturbances are first confined to a region within a few Rossby radii of the front. However, both the mean flow and the turbulent eddy motions slowly propagate into the previously stationary upper layer until, eventually, the whole of the upper layer is turbulent.     

A reduced model for nonlinear dispersive waves in a rotating environment

Abstract

The simplest model for geophysical flows is one layer of a constant density fluid with a free surface, where the fluid motions occur on a scale in which the Coriolis force is significant. In the linear shallow water limit, there are non-dispersive Kelvin waves, localized near a boundary or near the equator, and a large family of dispersive waves. We study weakly nonlinear and finite depth corrections to these waves, and derive a reduced system of equations governing the flow. For this system we find approximate solitary Kelvin waves, both for waves traveling along a boundary and along the equator. These waves induce jets perpendicular to their direction of propagation, which may have a role in mixing. We also derive an equivalent reduced system for the evolution of perturbations to a mean geostrophic flow.     

Understanding flow dynamics and density currents in a river-reservoir system under upstream reservoir releases

ABSTRACT

A three-dimensional flow and temperature model was applied for a 124 km river-reservoir system from Lewis Smith Dam tailrace to Bankhead Lock & Dam, Alabama. The model was calibrated against measured water levels, temperatures, velocities and flow rates from 4 May to 3 September 2011 under small constant release (2.83 m³/s) and large intermittent releases (~140 m³/s) from an upstream reservoir. Distributions of simulated flow and temperatures and particle tracking at various locations were analyzed which revealed the complex interactions of density currents, dynamic surface waves and solar heating. Flows in the surface and bottom layers moved in both upstream and downstream directions. If there was small constant release only from Smith Dam, simulated bottom temperatures at Cordova were on average 4.8°C higher than temperatures under actual releases. The momentum generated from large releases pushed bottom density currents downstream, but the released water took several days to reach Cordova.

Editor D. Koutsoyiannis; Associate editor B. Dewals     

An experimental study of forced barotropic rossby waves

Abstract

Barotropic Rossby waves are studied in a homogeneous fluid contained in a rotating cylindrical annulus with a radially sloping bottom boundary. The waves are forced by a simple source-sink distribution which can be rotated differentially relative to the annulus. When the speed of the source-sink distribution is close to the phase speed for a free Rossby wave of a given mode, resonant amplification occurs. The experimental results are in qualitative agreement with the predictions of a simple linear theory, but certain systematic differences between theory and experiment were observed.     

Alfén waves in a thermally stratified fluid

Abstract

The propagation of Alfvén waves along a uniform horizontal field in a highly conducting incompressible fluid, subject to the convective forces produced by a uniform vertical temperature gradient, is treated in a Boussinesq approximation. It is shown that there are exact solutions with large amplitude but restricted form. Their restricted form means that an arbitrary disturbing force produces other motions as well as Alfvén waves. An arbitrary initial disturbance of small amplitude produces waves whose state of polarization varies along the direction of propagation. For large amplitudes, however, any mixtures of polarization states causes scattering into new modes.     

Spontaneous Emission of Gravity Waves by Interacting Vortex Dipoles in a Stratified Fluid: Laboratory Experiments

Results from a new series of experiments on the geophysically important issue of spontaneous emission of internal gravity waves during unsteady interactions of vortical structures are presented. Vortex dipoles are a common element of a quasi-two-dimensional turbulent flow. Vortex dipoles perform translational motion and can collide with other vortices. During collision events the flow is unsteady and unbalanced and a further adjustment process associated with these events can therefore result in the spontaneous emission of gravity waves. Our laboratory experiments demonstrate that gravity waves are emitted when two translating vortex dipoles interact (collide) in a layered fluid, in accord with the current theoretical results. The emission was evident both in a two-layer system and in a fluid with a linear distribution of density with depth. The waves were generated during the period of deceleration of the secondary dipoles which constitute a vortex quadrupole emerging immediately after the collision of the primary dipoles.     

The existence of internal solitary waves in a two-fluid system near the KdV limit

Abstract

Two fluid layers of constant density lying one over the other on top of a rigid horizontal lower boundary with either a free upper surface or a rigid upper boundary can support solitary waves. The existence of a unique branch of such waves emanating from the horizontal flow at a critical speed U _? is demonstrated in both cases by use of the Nash—Moser implicit function theorem. These results complement the global results of Amick and Turner (1986) and are analogous to the work of Friedrichs and Hyers (1954) and Beale (1977) for surface waves. It is also noted that the most obvious variational principle which characterizes these waves as constrained extremals (Benjamin, 1984) is of indefinite type, having a Hessian with infinitely many positive and infinitely many negative eigenvalues.     

Internal gravity waves in a slowly varying,dissipative medium

Nonlinear internal gravity waves in a slightly dissipative, slightly compressible fluid are discussed for the case when the properties of the medium vary slowly on a scale determined by the local wave structure. A two‐timing technique is used to obtain transport equations which describe the changes in amplitude, phase and mean flow of a wave packet. Various solutions of these transport equations are discussed, with relevance to critical layer absorption.     

On the spin-up of a stably stratified,electrically conducting fluid. Part 1: Spin-up controlled by the hartmann layer

Abstract

The linear spin-up of a stably stratified, electrically conducting fluid within an electrically insulating cylindrical container in the presence of an applied axial magnetic field is analyzed for those cases in which electric currents generated within the steady Hartmann boundary layer control the fluid interior. It is shown how to obtain the known spin-up times for a homogeneous, nonconducting fluid (τ = E ^-½), a stably stratified, nonconducting fluid (τ = (σS/E, E ^?1) and a homogeneous conducting fluid (τ = α^?1 E ^-½) from the present formulation where τ = v/ωt, E = v/ωL ², σS = vN²/κω² and 2α² = σB²/pω. The problem is solved in the parameter range E?α²?1, α²/E?σS using the Laplace transform and two new spin-up times are obtained. Combined into one expression, they are τ = (1 + δ)α^?1E^-½ where δ = σμv. The spin-up mechanism is investigated and it is found that, in contrast to the homogeneous, conducting case, torsional Alfvén waves may be instrumental in the spin-up of a stratified conducting fluid. The effects of viscous and ohmic diffusion on the torsional Alfvén wave fronts are studied and the following regimes are identified: 0 < δ ?E/α², spin-up by meridional circulation of electric current with no Alfvén waves; E^-½/α ? δ ? 1, spin-up by meridional circulation of electric current with transient Alfvén waves; α/E^½ ? δ ? α²/E, spin-up by meridional circulation of current with weak Alfvén waves; 1 ? δ ? α/E^½, spin-up by strong Alfvén waves; α^½/E ? δ ? α²/E, spin-up by viscous diffusion with transient Alfvén waves; α/E ? δ < ∞, spin-up by viscous diffusion with no Alfvén waves.     

Why ripples become sweiis,and swells in shallow water decay rapidly

Abstract

In an ocean with a horizontal bottom where no wind is blowing it is shown that the spin (angular momentum) of the ocean is conserved. Thus, when energy is dissipated, at least one of three things will happen: i) Wave spectra may move towards lower frequencies. ii) The directional distribution may be changed towards long-crested waves. iii) Shear currents may be generated. By neglecting ii) and iii), the frequency shift of a spectrum is calculated due to molecular dissipation. When all energy transforming phenomena as e.g. wave breaking and turbulence generation are taken into account, the conservation of spin seems to be able to explain the frequency shift of wave spectra. In shallow water it is shown that there is energy transfer from the waves to shear currents.     

A Note on Standing Internal Inertial Gravity Waves of Finite Amplitude

The effects of finite amplitude are examined in two-dimensional, standing, internal gravity waves in a rectangular container which rotates about a vertical axis at frequency f/ 2. Expressions are given for the velocity components, density fluctuations and isopycnal displacements to second order in the wave steepness in fluids with buoyancy frequency, N , of general form, and the effect of finite amplitude on wave frequency is given in an expansion to third order. The first order solutions, and the solutions to second order in the absence of rotation, are shown to conserve energy during a wave cycle. Analytical solutions are found to second order for the first two modes in a deep fluid with N proportional to sech( az ), where z is the upward vertical coordinate and a is scaling factor. In the absence of rotation, results for the first mode in the latter stratification are found to be consistent with those for interfacial waves. An analytical solution to fourth order in a fluid with constant N is given and used to examine the effects of rotation on the development of static instability or of conditions in which shear instability may occur. As in progressive internal waves, an effect of rotation is to enhance the possibility of shear instability for waves with frequencies close to f . The analysis points to a significant difference between the dynamics of standing waves in containers of limited size and progressive internal waves in an unlimited fluid; the effect of boundaries on standing waves may inhibit the onset of instability. A possible application of the analysis is to transverse oscillations in long, narrow, steep-sided lakes such as Loch Ness, Scotland.     

Array measurements of the bottom boundary layer and the internal wave field on the continental slope

Abstract

An array of current meters was placed on the continental slope and rise for two months in the autumn of 1970. The bottom boundary layer was penetrated on the slope. On the smallest array scale, of the order of 1 kilometer, the array functioned as a directional internal wave antenna. Moving shoreward, the current spectra show strong suppression of the inertial peak and strong enhancement of the semidiurnal tide. The measured wave number spectra show that the tidal energy is almost completely baroclinic, and probably being generated in the region where the slope becomes “critical” for the tidal period. If this area is typical of worldwide conditions, a substantial fraction of the dissipation of surface tides takes place on the continental slopes by conversion to baroclinic waves. The bottom boundary layer has been modeled by an extension of the work of Ellison (1956) to a sloping boundary in a fluid of positive stability. An equivalent constant eddy coefficient has the value 3 cm²/sec as determined from the measurements.     

Turning surface behaviour for internal waves subject to general gravitational fields

Abstract

It is shown that the turning surfaces associated with internal waves in a uniformly rotating, density stratified, Boussinesq fluid in the presence of an arbitrary gravitational field are regular points for the governing eigenvalue differential equation. The results are illustrated for two particular examples that have geophysical and astrophysical significance, namely radially directed spherical gravity, and the gravitational field in a rapidly rotating cylinder.     

Hydromagnetic waves in a differentially rotating annulus. II. Resistive instabilities

Abstract

In part I of this study (Fearn, 1983b), instabilities of a conducting fluid driven by a toroidal magnetic field B were investigated. As well as confirming the results of a local stability analysis by Acheson (1983), a new resistive mode of instability was found. Here we investigate this mode in more detail and show that instability exists when B(s) has a zero at some radius s=s _c. Then (in the limit of small resistivity) the instability is concentrated in a critical layer centered on s _c . The importance of the region where B is small casts some doubt on the validity of the simplifications made to the momentum equation in I. Calculations were therefore repeated using the full momentum equation. These demonstrate that the neglect of viscous and inertial terms when the mean field is strong does not lead to spurious results even when there are regions where B is small.     

On the influence of the horizontal component of the earth's rotation on long period waves

Abstract

A general linearized wave equation for a stratified rotating fluid is derived and applied to obtain a dispersion relation for waves of short latitudinal extent in a thin shell of fluid. Long period wave solutions in three ocean models are compared: (1) for a stratified ocean with both components of the rotation vector; (2) for a stratified ocean without the horizontal component of rotation, and finally, (3) for a homogeneous ocean without horizontal rotation. The inclusion of the horizontal component of the Earth's rotation is found to have no noticeable effect on the dispersion relation of long period waves; its only influence is the introduction of a vertical phase shift in the motions. The origin of this phase shift is found in the tendency of the motions to satisfy the Taylor-Proudman theorem. The phase shift is of possible oceanographic relevance only for bottom-trapped buoyancy waves in a relatively weak stratification. The differences between the three ocean models are also discussed with the help of graphs of the numerically integrated dispersion relations. The relative influences of shell thinness and stratification in inhibiting the influence of the horizontal component of the earth's rotation are also briefly discussed.     

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.     

Time variations in parameters of split S waves from weak local earthquakes beneath eastern Hokkaido

Parameters of seismic waves from clusters of local weak earthquakes that occurred at the upper boundary of the seismofocal zone in depth intervals of 40–60 and 70–90 km along eastern Hokkaido are investigated for the period 1998–2003, including the strong (M = 8.0) Tokachi-Oki earthquake of September 26, 2003. Analysis of data indicates that the distribution of anisotropic properties along Hokkaido is inhomogeneous and parameters of split waves (the azimuth of the fast S wave and the time delay between split S waves) are sensitive to variations in the stress-strain state of the medium. Unstable behavior of split wave parameters and increased values of the ratio V _P /V _S for clusters of events in the areas of the Hidaka Mountains and Nemuro Peninsula imply that the medium is in a mechanically weakened state (in the regime of intense dilatant deformation). On the contrary, the regions beneath the Tokachi and Kushiro plains are more rigid and are characterized by lower V _P /V _S values and comparatively stable behavior of wave parameters. Anomalous parameters of split waves from events of clusters in areas of different stations correlate with each other and are related to occurrence times of large earthquakes around Hokkaido, which may point to a redistribution of stresses and strains and fluid migration in the subduction zone.