Stratified Liquid Research Articles

On the dynamical effects of a heterogeneous and compressible liquid core in the theory of Chandler wobble

SUMMARY The general 3-D scalar equations of motion of the liquid core (with respect to the radial components of displacements and cubic dilatation) are constructed as a superposition of the solutions of ordinary diVerential equations describing the dynamics of a stably stratified, heterogeneous, compressible and inviscid rotating fluid inside thin spherical layers (Molodensky & Sasao 1995). The estimation of dynamical eVects of a homogeneous and incompressible liquid core on the Chandler period (Groten, Lenhardt & Molodensky 1991) is generalized for the case of a heterogeneous, compressible, inviscid and neutrally stratified liquid core.

Linear theory of the propagation of internal wave beams in an arbitrarily stratified liquid

Beams of harmonic internal waves in a liquid with smoothly changing stratification are calculated in the Boussinesq approximation taking into account the effects of diffusion and viscosity. A procedure of local reduction of the beam in a medium with an arbitrary smooth stratification to the case of an exponentially stratified liquid is constructed. The coefficient of energy losses in the case of beam reflection on the critical level is calculated. Parameters of internal boundary flows with split scales of velocity and density that are formed by a wave beam on discontinuities of the buoyancy frequency and its higher derivatives are determined.

On the dynamical effects of an inhomogeneous liquid core in the theory of nutation

As shown in previous work, dynamical effects of a realistic model of a heterogeneous, compressible, stably stratified liquid core may be obtained by means of a simple analysis of the generalized two-dimensional Laplace tidal equation which describes tidal flows of an incompressible and non-gravitating fluid in a thin spherical layer with mobile boundaries. The solution was presented in the form of expansions in powers of a small parameter κ being the ratio of nutational motion frequency in space to the frequency of the Earth's diurnal rotation. Whereas in an earlier paper only first-order terms were taken into account, our present approach includes not only main second-order terms in the spherical harmonic expansions of the solutions, but also the terms of higher orders. These effects are calculated numerically for realistic models of the Earth's outer liquid core, solid inner core and anelastic mantle (PREM model). All tables are found in electronic version at http://www.tu-darmstadt.de/fb/vw/ipg/Welcome2.html

On the evolution of double-diffusive intrusions into a stably stratified liquid: a study of the layer merging process

The nonlinear evolution of double-diffusive instabilities into a laterally heated stably stratified motionless liquid is studied through direct numerical simulation in a two-dimensional set-up. In this paper, we consider liquids which are initially stratified through a constant salt gradient. The stages of evolution of the intrusions and their spatial scales correspond well with those observed in laboratory experiments. A central process in the evolution is that of layer merging. A particular case of layer merging is analysed in detail and a new physical description of this process is proposed, to which we refer as ‘layer sandwiching’.

On the evolution of double-diffusive intrusions into a stably stratified liquid: the physics of self-propagation

This is the second part of a study of the nonlinear evolution of double-diffusive instabilities into laterally heated stably stratified liquids. Flows developing in initially doubly stratified systems are considered, i.e. in addition to a stabilizing salinity distribution a destabilizing temperature distribution is present. Although the development of the intrusions is qualitatively similar to that in singly stratified liquids, important differences occur when the initial destabilizing temperature gradient becomes large. When the lateral heating is turned off, intrusions are still able to propagate. The main contribution of the paper is a detailed study of the physics of this self-propagation process.

Oscillatory convective instability in electrochemical systems

Abstract Oscillatory instability induced by buoyancy forces in an electrochemical cell with plane horizontal electrodes is considered. The space between the electrodes is filled with solution containing two types of single-charged cations and one type of single-charged anion. Under conditions of transport limitation of the electrochemical reaction rate, low-frequency convective–concentration oscillations may arise in the system, which are the internal standing waves in the concentration stratified liquid. The conditions for the development of the waves, their frequency and space period are found. A possibility for using internal standing waves for the construction of low-frequency electrochemical resonators is discussed.

Numerical investigation of the formation of three-dimensional structures in stably-stratified mixing layers

A temporally-growing mixing layer has been directly simulated with a pseudospectral technique, for initial bulk Richardson numbers from 0.0 to 0.2 and for Prandtl numbers from 0.00535 to 2.2. Several different initial conditions for the velocity fluctuations were imposed. For the two-dimensional (2-D) case only purely-deterministic conditions were used, whereas purely-deterministic, combined deterministic-random, or purely-random conditions were imposed in the three-dimensional (3-D) cases. The numerical procedure allowed fields with very different characteristic lengths to be resolved, with spectral accuracy maintained. The evolution of the velocity, active (temperature), and passive scalar fields were followed independently by adaptively redistributing collocation points in the regions of high shear and rapid scalar variations. The vertical boundary conditions were imposed at infinity to eliminate any boundary-layer effects and an exponential mapping was used to translate infinite physical space into finite computational space. The birth and time evolution of the longitudinal structures have been investigated. Variations in the initial modal forcing are reflected in different outcomes from the competition between core- and braid-centered instabilities in unstratified flow. For relatively strong fundamental forcing (compared to the 3-D forcing) the most unstable mode is braid-centered, whereas if the fundamental forcing is weak the most-unstable, core-centered mode determines the overall three-dimensionalization of the flow by generating large deformations of the main vortex cores. In subcritically-stratified air and water flow, instead, the braid-centered instability supersedes the core-centered instability, whatever the initial forcing. In stratified liquid sodium the shear-aligned convective instabilities observed in air and water are not excited. The conversion of potential into kinetic energy by convective overturning in the braid region does not occur because the high thermal conduction precludes the existence of the unstably-stratified regions necessary to drive the instabilities. The initial conditions, the Richardson, and the Prandtl numbers accordingly play a significant role in the free-shear-layer evolution and need to be explicitly considered for modeling purposes.

High-accuracy method for measuring profiles of buoyancy periods

We present a theoretical basis for a direct method of determining buoyancy period (frequency) as a function of depth by registering the displacements and oscillations of the hydrodynamic wake formed in a stationary liquid by freely ascending bubbles (the density marker method). We achieve an error no greater than 5% by comparing theoretical computations of parameters for internal waves generated by a free submerged thin wake having uniform transverse density distribution in an exponentially stratified liquid with measurements of internal waves in the liquid. We find that there is a power law for the decrease in amplitude of internal oscillations with time.

The conditions for the symmetric instability of the vortex motions of an ideal stratified liquid

The problem of the symmetric instability of the steady-state motions of an incompressible ideal liquid which is stratified with respect to its density is investigated in the case of two types of motion, axially symmetric and with translational symmetry. It is shown that the sufficient condition for stability obtained in [1] using a variational method (the direct Lyapunov method) for the motions under consideration is closely related to the extremal nature of their energy; stable motions are characterized by a conditional minimum of the energy. A minimum of the energy holds in the class of states for which a potential vortex, expressed in terms of the Lagrangian invariants, angular momentum and density, is represented by the same function of these invariants. Conditions for instability are formulated and estimates of the increase in the kinetic energy of perturbations are given.

A Hamiltonian approach to dissipative phenomena between the Earth's mantle and core, and effects on free nutations

SUMMARY The Hamiltonian formalism was recently applied by Getino (1995a,b) for the study of the rotation of a non-rigid earth with a heterogeneous and stratified liquid core. That earth model is generalized here by including the effect of the dissipation arising from the mantle-core interaction, using a model similar to that of Sasao, Okubo 8z Saito (1980), which includes both viscous and electromagnetic coupling. First, a solution for the free nutations is obtained following a classical approach, which in our opinion is more familiar to most of the readers than the Hamiltonian treatment. This solution provides a theoretical basis clear enough to study both the qualitative and quantitative effects of the dissipations considered in the hypotheses. The main qualitative features are, besides the delays, that the free core nutation (FCN) suffers an exponential damping, while the chandler wobble (CW) is not damped at first order, by the dissipation considered. The numerical values obtained for the complex compliances agree with the most recent experimental computations. Next, the problem is studied under a Hamiltonian formalism, and a solution equivalent to the above is obtained. Besides its interest from a theoretical point of view, this formalism is necessary in order to apply canonical perturbation methods in order to obtain analytical nutation series.

The structure of (linearly) stable double diffusive flow patterns in a laterally heated stratified liquid

The structure of (linearly) stable double diffusive flow patterns in a laterally heated stratified liquid

Interphase stability of stratified viscous liquids in Couette flow

An exact long-wavelength asymptotic solution is constructed for the problem of interphase stability in the Couette flow of a two-layer system of Newtonian liquids with different viscosities and with shallow depths. It is shown that the dispersion relation for the complex velocity of a wave of disturbances can be represented as an expansion in fractional powers of the wave number which are multiples of 1/3. New stability criteria, different from those obtained previously, are found.

The flow field of two stratified liquids in the presence of interfacial waves

The flow field of two stratified liquids in the presence of interfacial waves

The flow field of two stratified liquids in the presence of interfacial waves

Generated interfacial waves are under investigation in this paper. The flow field and the movement of the interface between two stratified newtonian fluids will be considered. Cylindrical geometry is examined. All results are given in an analytical form. With the help of an algebraic computation programm it is possible to reduce computation time for parameter studies. The dependence of the wavenumber of several dimensionless groups are studied intensively.

Experimental study of incompressible Richtmyer–Meshkov instability

The Richtmyer–Meshkov instability of a two-liquid system is investigated experimentally. These experiments utilize a novel technique that circumvents many of the experimental difficulties that have previously limited the study of Richtmyer–Meshkov instability. The instability is generated by vertically accelerating a tank containing two stratified liquids by bouncing it off of a fixed coil spring. A controlled two-dimensional sinusoidal initial shape is given to the interface by oscillating the container in the horizontal direction to produce standing waves. The motion of the interface is recorded during the experiments using standard video photography. Instability growth rates are measured and compared with existing linear theory. Disagreement between measured growth rates and the theory are accredited to the finite bounce length. When the linear stability theory is modified to account for an acceleration pulse of finite duration, much better agreement is attained. Late time growth curves of many different experiments seem to collapse to a single curve when correlated with the circulation deposited by the impulsive acceleration. A theory based on modeling the late time evolution of the instability using a row of vortices is developed. The growth curve given by this model has similar shape to those measured, but underestimates the late-time growth rate.

Comparison of the drag coefficients of bodies moving in liquids with various stratification profiles

For bodies moving in liquids with various stratification profiles, the relation between the drag coefficients considered as functions of the Froude number is investigated. The problems of stratified liquid dynamics have not previously been studied from this viewpoint, either experimentally or theoretically. On the range of the Froude numbers F1-1, the drag force coefficients obtained from bench measurements of the towing resistance to the uniform horizontal motion of models in two-layer and continuously stratified liquids are compared. The experimental data obtained in a thermocline are then compared with the results of [1,2] for linearly stratified and two-layer liquids.

Influence of a floating elastic plate on the surfaceeffects of internal waves generated by motion of a source in a non-homogeneous liquid

Perturbations of the surface of an exponentially stratified liquid of finite depth, free or covered by an elastic plate, are studied on the assumption that the perturbations are caused by internal waves generated by the steady motion of a constant -intensity source. The dependence of the spatial distribution of perturbations on the plate properties, velocity and the submersion depth of the source is considered.

The structure of (linearly) stable double diffusive flow patterns in a laterally heated stratified liquid

Layered double diffusive flow patterns in a laterally heated stably stratified liquid are considered in a configuration which allows for steady states to exist. For the heat/salt system, these flows are characterized by the thermal and solutal Rayleigh numbers RaT and RaS, or equivalently by RaT and the buoyancy ratio Rρ. The bifurcation structure of steady patterns with respect to RaT is computed for two cases: fixed RaS and fixed Rρ. For the first case, results in N. Tsitverblit and E. Kit [Phys. Fluids A 5, 1062 (1993)], are computed and extended, and it is shown that many of the previously found flow patterns are unstable; only in a small interval of RaT, multiple (linearly) stable steady states exist. For the second case, the physical relevance of the unstable steady states with respect to the evolution of the flow toward a stable steady state is demonstrated.

An interpretation of the core-mantle interaction problem

This paper deals with the canonical formulation of the rotation of a non-rigid earth with a heterogeneous stratified liquid core. The treatment of Kubo (1979) for a rigid core-rigid mantle earth model is extended by considering the core as liquid instead of rigid. We show that the tensional torque introduced by Kubo to explain the core-mantle interaction is due to the hypothesis of rigidity of the core, this interaction now being included in the new expression of the kinetic energy. More precisely, this interaction is due to a kind of variation or deformation of the inertia tensor of the core, which we have named inertial deformation, in correspondence with the inertial coupling of Sasao. Okubo & Saito (1980) and Kubo (1979) among other authors. The Hamiltonian obtained is compared with that of Sevilla & Romero (1987), showing the equivalence between them.

The reflection of beams of internal gravity waves at a flat rigid surface

The linear problem of the reflection of a beam of monochromatic internal waves at a rigid inclined wall in an exponentially stratified liquid with viscosity and diffusion is considered. In such a medium there is, as well as the reflected wave, a boundary-layer flow at the plane with split spatial length-scales for the velocity and density variation. Viscosity and diffusion restrict the limiting value of the geometrical compression coefficient of the beam. The solution has no singularities at critical angles of reflection. Calculations for the reflected beam and boundary flow are performed for an incident beam radiated by a point mass source.