Exponential Stratification Research Articles

Vortices and internal waves in a stratified fluid

Equations describing the evolution of potential vortices and internal waves in a stably stratified fluid which fills a half-space are derived in Euler-Lagrange variables. Asymptotic series in a small parameter are constructed which give approximate solutions of the non-linear problem. The equations of the linear approximation for a potential vortex and internal waves are independent, while the equations of the higher-order approximations describe the interaction between the potential vortices and the internal waves. It is shown that distributed sources (which can be interpreted as intense atmospheric rainfall) cause an exponential increase in the potential vorticity, which in turn may lead to a considerable increase in the amplitudes of the internal waves. Asymptotic forms for the field of the internal waves in different space-time regions are constructed for the case of an exponential stratification.

Intermodal interaction of internal solitary waves

Interaction of internal solitary waves of different modes in an incompressible fluid with an exponential stratification in density is considered. Two such waves are considered. These waves may be going in the same direction or in opposite directions. The analysis shows that the interaction produces two pairs of waves, whose mode indices are the difference and the sum of those of the original waves, respectively. Each pair consists of a right-going wave and a left-going one, and a part that vanishes eventually everywhere. Higher-order terms affecting interaction are briefly discussed. The interaction of more than two waves can be dealt with pair by pair.

Holographic array processing in a range-dependent ocean.

Imperfect knowledge of the salient characteristics of the propagation medium limits the performance of acoustic array processors at long ranges in the ocean. Holographic and phase conjugation techniques can be used to diminish the range-integrated effect of the medium and reconstruct the wave front in the vicinity of a scatterer or other signal source. Then, using a back-propagation technique that focuses sound at the position of the unknown source, the location of the source can be determine. In this paper, the WKB approximation is implemented for a range-dependent ocean with both N2 bilinear and exponential stratification of the sound-speed profile. The results of the analysis and simulation show the possibility of localizing a source at large distance with great accuracy.

On inherited fertility in biological systems: a model of correlated fluctuations in the stochastic branching process

A new evolutionary model with hereditary model considered as correlated fluctuations of fertility has been proposed. It has been demonstrated that the model allows the global statistical properties of the system to be evaluated, e.g. the ensemble average and the probability of extinction. The results obtained show the increase of instability of a population with the enhancement of inheritance efficiency. The existence of at least an exponential stratification in the population has also been shown. Possible applications of the present model are discussed.

A singly exponential stratification scheme for real semi-algebraic varieties and its applications

This paper describes an effective procedure for stratifying a real semi-algebraic set into cells of constant description size. The attractive feature of our method is that the number of cells produced is singly exponential in the number of input variables. This compares favorably with the doubly exponential size of Collins' decomposition. Unlike Collins' construction, however, our scheme does not produce a cell complex but only a smooth stratification. Nevertheless, we are able to apply our results in interesting ways to problems of point location and geometric optimization.

Effects of Hall currents and Coriolis forces on the instability of a self-gravitating plasma of variable density

The hydromagnetic instability of a self-gravitating, incompressible rotating plasma of variable density has been examined in the presence of Hall currents. The system is assumed to be permeated by a variable horizontal magnetic field. The solution of the relevant linearized perturbation equations has been obtained by the normal mode technique through a variational principle which is shown to characterize the problem. Proper solutions have been obtained for a semi-infinite plasma having exponential density stratification along the vertical. The dispersion relation has been derived and solved numerically for different values of the physical parameters involved. It is found that Hall currents and Coriolis forces have both destabilizing influence as the growth rate of the unstable modes is found to increase with the increase of both Hall currents and Coriolis forces.

Nonlinear, periodic, internal waves

It is shown how to calculate large-amplitude, periodic, internal waves in a channel filled with continuously stratified fluid by using a Stokes-type amplitude expansion along the channel and a modal expansion across the channel. We obtain explicit solutions for the case where the density increases exponentially with depth. It is found that periodic waves in exponentially stratified fluid are waves of depression: as the wave increases in amplitude the wave speed decreases and the mean density of the fluid at a given height decreases. The waves are limited in height by the formation of an eddy of fluid on the upper boundary above the trough of the wave. This is consistent with the description of waves of depression given by Amick and Toland (1984). There is no backflow before the eddy forms. By contrast, Amick and Toland (1984) have shown that solitary waves in the same system are waves of elevation, limited in height by a cusp in the streamlines at some interior point. A relationship is found between the mass flux and the mass displacement. The expressions necessary to calculate the potential and kinetic energies are given. A simple analytic solution for the internal wave is presented for a fluid with a weak exponential stratification, that is when the Boussinesq approximation is appropriate. For an Nth mode wave the limiting amplitude of the perturbation streamfunction is shown to be 1/Nπ. This corresponds to a maximum streamline displacement of 2z/Nπ, where z (=0.7391) is the solution to z = cos z.

Rayleigh–Taylor instability of a continuously stratified fluid under a general rotation field

An estimate of the upper bound of the growth rate of instability in a continuously unstable stratified fluid under a general rotation field is obtained. An example for exponential stratification is given. For magnitudes of the wavenumber k slightly above kc (where kc is the critical value for instability) it is shown that the angle of propagation for the perturbation associated to a maximum growth rate is θmax =0° (180°), where θmax is measured relative to a fixed horizontal component of rotation. For larger values of k bifurcation occurs with two maxima of the same magnitude appearing at angles different from θ=0° but symmetrically located. Increasing k leads to a unique maximum at θmax =90° (−90°). The bifurcation tends to disappear as the vertical component of rotation is increased.

Oscillations in a continuously stratified fluid in a moving vessel, and their control

Classical models of a heavy ideal fluid are used to study internal wave motions of a stable stratified fluid in a moving vessel, and methods of controlling these oscillations. The case of exponential stratification, which differs in a number of essential features from the case of discrete stratification studied earlier in /1–3/, is considered.

Polar Cap Accretion onto Magnetized Neutron Stars: An Analytic Solution

This abstract should be read in conjunction with the papers by Arons and by Klein and Arons in these proceedings. In the context of the accretion models described there, one can find an analytic solution for the flow down the polar field lines if a number of simplifying assumptions are made. These are (1) steady flow in the co–rotating frame; (2) radiation pressure large compared to gas pressure; (3) pure scattering for the Rosseland opacity, with the magnetic corrections set equal to constants instead of using the actual functions of temperature; (4) diffusion flux of radiative energy proportional to the gradient of the energy density alone, instead of the correct sum of terms proportional to the photon energy density and the number density gradients; and (5) below a radiative shock, subsonic flow in approximate hydrostatic equilibrium. We assumed dipole geometry, and also assume the mass flux is independent of distance from the magnetic axis. The essential trick is to use (1), (2) and (5) to write the advective contribution to the radiation transfer equation as Mg/area = rate at which gravity does work on a fluid element, and use (3) and (4) to write the nonlinear diffusion flux as the ratio of gradients in the energy density. Then the multidimensional diffusion equation can be cast in a separable, linear form by using the logarithmic radial gradient of the energy density as the basic variable (see also Kirk, J., 1985, Astron. and Astrophys., 142, 430). The result is exponential stratification of the energy density, velocity and mass density along B with scale height R*[L(EDeff)/4Lco-p]; the effective Eddington luminosity is discussed by Arons, these proceedings. This result can be understood as the result of almost exact balance between upward diffusion and downward advection of photons in the optically thick medium. The same fluid quantities are stratified in a Gaussian manner across B, with angular half width at half maximum Δθ = [L(EDeff)/Lcap](r/R*)3/2. These distributions agree well with more sophisticated computational results, during times when the flow is steady. When used as a basis for calculations of the radiative entropy, the calculated emergent spectra are not dissimilar to the spectra of high luminosity, accretion powered pulsars.

Effect of magnetic resistivity on the stability of a partially ionized compressible Hall plasma

The dynamic stability of a partially ionized, compressible Hall plasma of finite electrical conductivity has been investigated when the plasma is immersed in a uniform, horizontal magnetic field. Based on the variational principle, which is shown to characterize the problem, the solution has been obtained for a semi-infinite plasma confined between two planes and having an exponential density stratification along the vertical. It is found that the effect of neutral gas friction is stabilizing while magnetic resistivity, Hall currents and compressibility all have destabilizing influence.

The incidence of internal waves onto a thin submerged barrier

Abstract The problem of oblique incidence of internal ocean waves on a thin submerged ocean barrier is considered when the ocean has exponential density stratification. A Wiener-Hopf approach is used combined with numerical evaluation of series. Results for the reflected energy are obtained and reveal a complex dependence on incidence and barrier height. Application of this model to waves incident on the Mid-Atlantic ridge suggests that the ridge almosts isolates first mode energy on one side of the ocean from the other side. In certain circumstances there, is a surprising appearance of “barrier” waves. These waves are closely confined to the barrier and propagate along it.

Two-dimensional problem for internal waves generated by moving singular sources

When bodies move in a liquid with inhomogeneous density in a gravitational field waves are excited even at low velocities and in the absence of boundaries. They are the so-called internal waves (buoyancy waves), which play an important part in geophysical processes in the ocean and the atmosphere [1–4]. A method based on the replacement of the bodies by systems of point sources is now commonly used to calculate the fields of internal waves generated by moving bodies. However, even so the problems of the generation of waves by a point source and dipole are usually solved approximately or numerically [5–11]. In the present paper, we obtain exact results on the spectral distribution of the emitted waves and the total radiation energy per unit time for some of the simplest sources in the two-dimensional case for an incompressible fluid with exponential density stratification. The wave resistance is obtained simply by dividing the energy loss per unit time by the velocity of the source. In the final section, some results for the three-dimensional case are briefly formulated for comparison.

Internal solitary waves in a linearly stratified fluid

The mode shapes for internal solitary waves propagating in a linearly stratified fluid of finite depth are determined in the limit of weak stratification. The solutions obtained for both a fixed upper boundary and a free surface are compared with known results for exponential stratification. The profound changes which occur depending on whether the upper surface is fixed or free are derived from the strong influence of the free surface when the stratification is weak. Evidence is presented supporting the conjecture that the free surface can no longer effect qualitative differences in the modal structure when the fluid is heavily stratified.

On currents, internal and inertial waves in a stratified ocean due to variable winds. Part 2

The response of a continuously stratified viscous ocean to a variable wind and air pressure field has been studied during recent years by means of analytical methods. This requires the assumption of consntant eddy viscosity and exponential stratification. In shallow water constant eddy viscosity leads to irrealistically thick boundary layers compared to the water depth and, theorefore, eliminates internal waves. In the first part of a two-part study a theoretical model has been developed for arbitrary density stratification and eddy viscosity. The results of the integration are dealt with in this second part 2.

On currents, internal and inertial waves in a stratified ocean due to variable winds. Part 1

The response of a continuously stratified viscous ocean to a variable wind air pressure field has been studied during recent years by means of analytical methods. This requires the assumption of constant eddy viscosity and exponential stratification. In shallow water constant eddy viscosity leads to irrealistically thick boundary layers compared to the water depth and, therefore, eliminates internal waves. In this first part of a two-part study a theoretical model is developed for arbitrary density stratification and eddy viscosity. The results of the integration are dealt with in part 2.

Forced vertical oscillations in a viscous stratified fluid

AbstractThe linear fluid dynamics is considered when infinite vertical boundaries are set in oscillatory vertical motion. The case of exponential stratification with constant kinematic viscosity is explicitly studied. When the forcing frequency equals the Brunt–Vaisälä frequency for the fluid, the customary boundary layers are absent in the steady-state oscillation, however small be the kinematic viscosity; for a semi-infinite fluid the corresponding horizontal extent of the region influenced by the boundary motion is then of the order of the stratification length. The sign of the phase angle is everywhere dependent on whether the magnitude of the forcing frequency is greater than or less than that of the Brunt–Vaisäla frequency.