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.     

Waves in a viscous fluid on a rotating sphere

Summary In this note the waves in a viscous fluid rotating on a sphere has been studied. The solution contains unknown constants which may be evaluated in particular cases applying the boundary conditions.     

On wave motions in a tropical spectral experiment

An attempt is made to use the barotropic vorticity equation in spectral form in order to study barotropic instability when the basic current has east-west quasi-stationary asymmetries on the scale of long waves. This is done by expressing the spectral equations in three different ways. In the first experiment a 9-component system is integrated and the long waves are allowed to propagate freely. In the second experiment the long waves are constrained to propagate slowly and in the third experiment, they are removed altogether.The motivation behind the present investigation is due to observations made in the motion fields of the tropics. These are characterized by quasi-stationary long waves and very energetically active and propagating short waves.The presence of quasi-stationary long waves seems to enhance the energy exchanges between the short waves and the mean zonal current and also allow for larger energy values for the short waves. Long term integration (90 days) shows a 6-day mode in thev time spectra at latitude 10°N and a 15-day mode in theu time spectra at individual grid points in the equatorial latitudes. Any possible connection between this peak and the observed peak of 15 days in the completely different physics of Kelvin waves is left as a conjecture.     

The catastrophe structure of thermohaline convection in a two-dimensional fluid model and a comparison with low-order box models

Abstract

We impose a surface forcing on the 2D, Boussinesq, thermohaline equations in a rectangular domain, in the form of equatorially symmetric cosine distributions of salinity flux and temperature. This system may be seen as an idealization of the ocean thermohaline circulation on the global scale over intervals of centuries or millenia. Multiple steady states are found numerically. They reflect the competition between the opposite signs of the temperature and salinity-driven equatorially symmetric circulations. There are also pole-to-pole, equatorially asymmetric circulations. In the control space of the temperature and salinity-flux forcing amplitudes, these equilibria form two cusp catastrophes, and transitions between stable equilibria occur through several distinct bifurcations. These catastrophes can be reproduced in simple box models connecting stirred reservoirs through capillary pipes. This steady-state analysis may provide a framework for a better understanding of climatic transitions between different stable regimes of the ocean-atmosphere system.     

The vortex created by mass transfer between layers of a rotating fluid

Abstract

A simple way to model stratification of the ocean or atmosphere is in terms of two superposed homogeneous layers of different density. Effects of cooling of the upper layer, such as that which occurs during bottom-water formation in the ocean, can be simulated by mass transfer from the upper layer to the lower layer. A model is constructed to see What effect such a mass transfer has on the flow when the mass transfer is confined to a limited region. The main effects are (i) doming of the interface, which maintains pressure gradients in balance with the velocity field, (ii) cyclonic rotation in the upper layer due to conservation of angular momentum of particles king drawn toward the sink, yet anticyclonic vorticity for those particles outside the mass transfer region due to shrinking of vortex lines drawn up over the dome. (iii) generally anticyclonic rotation in the lower layer due to particles tending to maintain their angular momentum while being pushed outwards, but some cyclonic rotation near the centre of mass transfer, due to momentum transfer from the upper layer. Similar effects to these are seen in the Greenland Sea where bottom water formation occurs. Results of the same sort are also found in a laboratory model of the process.     

Characteristics of surface ground motions induced by blasts in jointed rock mass

An in-situ experimental program in a jointed rock mass was designed and implemented to investigate the rock joint effects on stress wave propagation. Accelerometers were placed on rock surface along three lines at 0, 45 and 90° to the predominant rock joint strike direction. Eight blasts were detonated in a charge hole drilled in the rock mass. The equivalent TNT blast weight ranged from 2.5 to 50.0 kg and the loading density (charge weight divided by the charge chamber volume) varied from 1 to 20.0 kg/m³. A fully coupled detonation with a charge weight of 25.0 kg was also exploded to study the coupling effect. The recorded data are presented and analyzed in this paper. The effects of rock joints on characteristics of stress wave propagation such as peak value attenuation, spectrum, and spatial variations are discussed.     

On unsteady motion of a rotating fluid bounded by flat porous plates

Summary In this paper, an investigation is made of the unsteady flow generated in a viscous, incompressible and homogeneous fluid bounded by (i) an infinite horizontal porous plate atz=0, or (ii) two infinite horizontal porous plates atz=0 andz=D. The fluid together with the plate(s) is in a state of solid body rotation with a constant angular velocity about the z-axis normal to the plate(s), and additionally, the plate(s) performs non-torsional elliptic harmonic oscillations in its (their) own plane(s). A uniform suction or injection is introduced in the configurations through the porous plate(s) and its influence on the unsteady flow and the associated boundary layers is examined. The unsteady flow field as well as the associated boundary layers is obtained explicitly. In contrast to the unsteady rotating flow without suction, solutions of the present problem with suction exhibit no resonant phenomena. It is shown that the suction is responsible for making the boundary layers thinner and for the elimination of the resonant phenomena. It is confirmed that the velocity field and the associated multiple boundary layers are significantly modified by suction. Physical significances of the mathematical results are discussed. Several limiting cases of interest are recovered from this analysis. The initial value problem for both the configurations is exactly solved by the Heaviside operational calculus combined with the theory of residues.     

全球水质量迁移对海平面空间模式周年变化的影响

大气、陆地水和海洋之间的水质量迁移对海平面的影响一般假定为均匀薄层分布.但实际上,水质量负荷重新分布一方面会使地壳产生形变,另一方面会引起重力位势场变化(引力位和离心力位),这都会对海平面时空变化特征产生影响,两方面之和称为负荷自吸引效应(SAL).海洋模式模拟的时变洋底压力结果一般符合Boussinesq假设即体积守恒,忽略了大气、陆地水和海洋之间水质量交换的影响.本文基于海平面变化方程,联合陆地水模型、大气地表气压模型、海洋洋底压力模型和GRACE反演的冰川质量变化,详细讨论了2003-2010年SAL对海平面周年变化的影响.主要结论有:(1)SAL对全球海平面周年变化有显著影响,振幅在1.3~19 mm.其中近海岸和低纬度区域受影响较大.(2)在SAL引起的海平面周年振幅变化的因素中,陆地水储量变化因素最大,大气因素次之,非潮汐海洋影响最小.但非潮汐海洋对海平面周年相位空间变化的影响最为复杂.(3)通过与国际长期验潮站观测数据结果比较,在ECCO海洋模式估计的洋底压力结果中引入SAL,能多解释约5.3%观测信号方差.     

The mass transport velocity induced by free oscillations at a single frequency

Abstract

Second order effects due to the presence of a first order free oscillation at a single frequency in a variable depth rotating ocean are examined. It is found that the second order Lagrangian mean velocity (mass transport velocity) satisfies the linearized equations for unforced steady geostrophic motion. This implies that if the ocean basin is laterally bounded and contains no closed geostrophic contours, the second order Lagrangian mean velocity vanishes everywhere.     

Fluxes of fluid and heat from the oceanic crustal reservoir

Recent discoveries define a global scale fluid reservoir residing within the uppermost igneous oceanic crust, a region of seafloor that is both warm and may harbor a substantial biosphere. This hydrothermal fluid reservoir formed initially within volcanic rocks newly erupted at mid-ocean ridges, but extends to the vastly larger and older ridge flanks. Upper oceanic crust is porous and permeable due to the presence of lava drainbacks, fissuring, and inter-unit voids, and this porosity and permeability allows active fluid circulation to advect measurable quantities of lithospheric heat from the crust to an average age of 65 Myr. A compilation of crustal porosities shows that this fluid reservoir contains nearly 2% of the total volume of global seawater. Heat flow and sediment thickness data allow calculation of reservoir temperatures, predicting 40°C mean temperatures in Cretaceous crust. Utilizing these temperature estimates, heat flow measurements and models for the thermal structure and evolution of the oceanic lithosphere, we have computed mean hydrothermal fluxes into the deep ocean as a function of plate age. The total hydrothermal volume flux into the oceans approaches 20% of the total riverine input and may contribute to the global seawater mass balance.     

On the origin of jets in the ocean

We show a mechanism whereby the jets result during the development of β-plumes (i.e., low-frequency Rossby waves that establish gyre circulations) in a model of ocean-basin circulation. The energy originates in baroclinic meanders of circulation at the eastern boundary of the ocean. Eddies are intimately related and occur as a result of the instability of this process. This mechanism does not rely on the existence of the small-scale turbulence to establish zonal flows. Zonal jets can then be amplified by eddies arranged in certain order in the flow. The underlying dynamics include the propagation of linear and nonlinear basin scale Rossby waves. The related barotropic theory for these waves is developed here. We demonstrate the radiative development of jets and β-plumes in a laboratory experiment using a rotating fluid with a paraboloidal free surface. The dynamical fields are measured by the laboratory analog of the satellite altimetry.     

Travelling wave convection in a rotating layer

Abstract

Small amplitude two-dimensional Boussinesq convection in a plane layer with stress-free boundaries rotating uniformly about the vertical is studied. A horizontally unbounded layer is modelled by periodic boundary conditions. When the centrifugal force is balanced by an appropriate pressure gradient the resulting equations are translation invariant, and overstable convection can take the form of travelling waves. In the Prandtl number regime 0.53 < [sgrave] < 0.68 such solutions are preferred over the more usual standing waves. For [sgrave] < 0.53, travelling waves are stable provided the Taylor number is sufficiently large.     

On the completeness of inertial wave modes in rotating annular channels

An important unanswered mathematical question in the theory of rotating fluids has been the completeness of the inviscid eigenfunctions which are usually referred to as inertial waves or inertial modes. We provide for the first time a mathematical proof for the completeness of the inertial modes in a rotating annular channel by establishing the completeness relation, or Parseval’s equality, for any piecewise continuous, differentiable velocity of an incompressible fluid.     

Seismic wave propagation in a viscoelastic porous solid saturated by viscous liquid

A general solution is deduced of the differential equations describing the propagation of elastic waves in a dissipative liquid-filled viscoelastic porous solid. The velocities of three existing waves have been expressed in convenient form using the moduli of the solid phase and by introducing the frequency-dependent equivalent mass densities. The solution is then used to examine some of the phenomena which arise when each of the three-body waves, in turn, are incident on a traction-free plane boundary. Analytic expressions for the reflection coefficients are obtained. Numerical calculations have been made, for a particular model, in case of incidentP _I wave. Effect of viscoelasticity and viscosity on the reflection coefficients has also been exhibited.     

On the onset of thermal convection in slowly rotating fluid shells

Abstract

The problem of the removal of the degeneracy of the patterns of convective motion in a spherically symmetric fluid shell by the effects of rotation is considered. It is shown that the axisymmetric solution is preferred in sufficiently thick shells where the minimum Rayleigh number corresponds to degree l = 1 of the spherical harmonics. In all cases with l > 1 the solution described by sectional spherical harmonics Y^l _l (θ,φ) is preferred.     

Frontal upwelling in a rotating two-layer fluid

Abstract

This paper describes the source-sink driven flow in a two-layer fluid confined in a rotating annulus. Light fluid is injected at the inner wall, while denser fluid is withdrawn at the outer wall. The interface between the immiscible fluids intersects the bottom and thus produces a front. The net transport from the source to the sink is carried by Ekman layers at the bottom and at the interface, and by Stewartson layers at the side walls. A detached Stewartson layer arises at the front, leading to a pronounced upwelling circulation.     

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

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

Wave-induced steady streaming,mass transport and net sediment transport in rough turbulent ocean bottom boundary layers

The interaction between two important mechanisms which causes streaming has been investigated by numerical simulations of the seabed boundary layer beneath both sinusoidal waves and Stokes second order waves, as well as horizontally uniform bottom boundary layers with asymmetric forcing. These two mechanisms are streaming caused by turbulence asymmetry in successive wave half-cycles (beneath asymmetric forcing), and streaming caused by the presence of a vertical wave velocity within the seabed boundary layer as earlier explained by Longuet-Higgins. The effect of wave asymmetry, wave length to water depth ratio, and bottom roughness have been investigated for realistic physical situations. The streaming induced sediment dynamics near the ocean bottom has been investigated; both the resulting suspended load and bedload are presented. Finally, the mass transport (wave-averaged Lagrangian velocity) has been studied for a range of wave conditions. The streaming velocities beneath sinusoidal waves (Longuet-Higgins streaming) is always in the direction of wave propagation, while the streaming velocities in horizontally uniform boundary layers with asymmetric forcing are always negative. Thus the effect of asymmetry in second order Stokes waves is either to reduce the streaming velocity in the direction of wave propagation, or, for long waves relative to the water depth, to induce a streaming velocity against the direction of wave propagation. It appears that the Longuet-Higgins streaming decreases as the wave length increases for a given water depth, and the effect of wave asymmetry can dominate, leading to a steady streaming against the wave propagation. Furthermore, the asymmetry of second order Stokes waves reduces the mass transport (wave-averaged Lagrangian velocity) as compared with sinusoidal waves. The boundary layer streaming leads to a wave-averaged transport of suspended sediments and bedload in the direction of wave propagation.     

A numerical study of detached shear layers in a rotating sliced cylinder

Abstract

The low Rossby number flow in a rotating cylinder with an inclined bottom, of small slope, is examined when part of the lid of the container is rotating at a slightly different rate. The resulting flow is calculated numerically by solving the governing equations for the two-dimensional geostrophic motion which approximates the flow in most of the fluid including the inertially-modified E ^¼ -layers. The presence of ageostrophic regions, on the container walls and beneath the velocity discontinuity on the lid, is accounted for in the governing equations and their boundary conditions. This study supplements previous work on this configuration, in which the zero Rossby number flow was calculated and experimental results were presented, by enabling a direct comparison to be made between the results of the low Rossby number theory and the experiments. The numerical results for a range of Rossby and Ekman numbers compare well with those from the experiments despite a severe limitation on the size of the Rossby number arising from the analysis in the ageostrophic part of the detached shear layer.