The Sequence-of-Bifurcations Approach towards Understanding Turbulent Fluid Flow

The sequence-of-bifurcations approach for the analysis of transitionsfrom simple to complex forms of fluid flow is described.Formulated for fluid systemswith external conditions that are homogeneous in two spatial dimensionsand in time,this approach determines bifurcations through theirsymmetry breaking properties.Solutions that are periodic in the homogeneous dimensionsand in time are generatedin this way. While the secondary solution induced by theinstability of the basichomogeneous state generically assumes the form of rolls orstripes, higher bifurcationsintroduce solutions exhibiting structures specific for thephysical system underconsideration. These structures often persist in the formof coherent structures inthe turbulent state of the system. Examples from the caseof thermal convection in alayer heated from below are used to illustrate thesequence-of-bifurcations approach.     

Point-to-curve raytracing by algebraic rasterization

Boundary-value raytracing problems can be concatenated to a smooth one-parameter family of problems, that can be solved by continuation. This has been the purpose of point-to-curve raytracing. A global approach, based on algorithms taken from computer graphics (algebraic rasterization of implicit curves), has several advantages. Subject to relatively mild assumptions-Lipschitz continuity of the emergence point as function of initial parameters-all solution branches are found, there are no problems with initialization, bifurcation, or closed loop solutions. The algebraic rasterization benefits to boundary value raytracing problems in a wide range of applications: shot-to-profile shooting, VSP raytracing, normal raytracing, and more. The algorithm is sufficiently robust to continue even beyond points where the Lipschitz continuity does not apply, such as faults.     

Nonlinear Dynamics in Atmospheric Chemistry Rate Equations

Nonlinear terms in the gas-phase rate equations of atmospheric trace constituents give rise to unexpected oscillations in the concentrations. For a simplified chemical scheme of the troposphere that contains only the generic reaction paths, the underlying dynamical structure is investigated. It is shown that the source strengths of CO and NO are the controlling parameters of the system. A linear stability analysis reveals that the steady state solutions lose stability due to the occurrence of two Hopf bifurcations. Furthermore, it appears that the dynamical behaviour of the oscillatory solutions is dominated by O₃ and CO. Based on the two-variable system (CO–O₃) it is shown that the oscillatory solution involves an autocatalytic ozone production phase which is followed by a phase in which CO is oxidised quickly. A simple expression is presented from which the period of the oscillation can be obtained. The implications for the present troposphere are unclear, since other hydrocarbons are present and transport is taking place. Nevertheless, the system nicely shows the general nonlinear mechanisms that operate in the tropospheric chemistry equations.     

A comparison of gravity wave phase velocities calculated from the impedance equation and pressure data

Abstract

Marked oscillations in wind speed, wind direction and pressure with periods of between 5 and 15 minutes were recorded on a number of occasions and it is assumed that they were produced by atmospheric gravity waves. Gossard and Munk (1954) have shown that the phase velocity of a gravity wave can be calculated using wind and pressure fluctuation data from a single station by means of the “impedance equation”. It is also possible to measure the phase velocity of a gravity wave using an array of microbarovariographs. In this investigation, the phase velocity of gravity waves calculated using these two different methods are compared and the feasibility of deriving wave characteristics from single point measurements is discussed. The results show that, while in a few isolated cases the two velocities agree reasonably well, the application of the impedance equation to data from a single station is limited due to the superposition of local wind systems upon the wave-induced perturbations.