Continuous Density Profile Research Articles

Models of vertical structure and the calibration of two-layer models

Methods of calibration layer models (choosing the adjustable parameters defifining the layer depths and density steps) are presented. The radii of deformation, the surface and bottom amplitudes of the vertical normal modes and the depth-averaged triple products of the modes are the important functionals of the stratification which determine the behavior of the system. We show how these can be matched to the equivalent two-layer model functions of the density step ε and upper-layer depth H 1 in three different physical situations: time-dependent wind forcing, bottom slope or friction influences, and nonlinear interactions. In each physical situation we illustrate two different choices of which characteristics of the behavior to match and make quantitative comparisons with a continuously stratified model with the stratification derived from oceanic data. These examples show that the optimal calibrations are very different in the three physical situations; i.e., any single choice of ε and H 1 will inevitably lead to serious errors in predicting the behavior in at least two of the three physical situations. This inaccuracy may lead to serious qualitative misbehavior of a two-layer model in a situation where two or more physical processes are competing (e.g., bottom topography and nonlinearity). We propose a two-mode model (of the same computational simplicity) which does not suffer from this problem, but is optimally calibrated in all three physical situations without changes in the parameters. In addition, it offers the advantages of being automatically calibrated by the specification of the mean continuous density profile and being readily applied to oceanic data of all kinds.

Dynamic Stabilization and the Rayleigh-Taylor Instability

Most problems in the dynamic stabilization of plasmas have been treated in a sharp boundary limit. The stability of an incompressible liquid and of a low-β plasma, having continuous density profiles, is considered here. The destabilizing force is gravity, and stability is sought by oscillating the system in a direction parallel to this force. In the case of a liquid, long wavelengths are stabilized but short wavelengths remain unstable. In the case of a plasma, it is found that modes which are unstable with dynamic stabilization alone or finite Larmor radius effects alone can be stabilized by the presence of both.

The ionosphere as a whispering gallery

The propagation of radio waves of very low frequency to great distances is conveniently treated by regarding the space between the earth and the ionosphere as a wave-guide. Several authors have found that the least attenuated modes are profoundly affected by the earth’s curvature. This effect is investigated for several models of the ionosphere. It is found, in particular, that for frequencies greater than about 30 kc/s some modes are possible for which the energy is concentrated in a region near the base of the ionosphere, and the field strength near the ground is small. It is useful to think of such modes as being composed of waves repeatedly reflected at the inside spherical surface of the ionosphere, the rays being chords of this sphere. By analogy with sound waves these modes are called ‘whispering gallery modes’. The theory uses wave admittance and reflexion coefficient variables because these satisfy differential equations which are convenient for integration using a digital computer. The curvature of the earth is allowed for by using the method of the modified refractive index, but the earth’s magnetic field is neglected. Formulae for the m ode condition and the excitation of the various modes by a transmitter are given and discussed. A new way of dealing with an ionosphere having a continuous electron density profile is presented. The results of some numerical calculations are given both for a sharply bounded homogeneous ionosphere and for an exponential profile of electron density.