Small-amplitude Disturbances Research Articles

A Conservation Law for Small-Amplitude Quasi-Geostrophic Disturbances on a Zonally Asymmetric Basic Flow

Abstract A quadratic conservation law is derived for small-amplitude quasi-geostrophic disturbances on a wavy basic state. The law may be useful for describing the three-dimensional propagation of disturbances on time-averaged flows. This parallels the use of the generalized Eliassen-Palm theorem in the description of waves propagating on zonally-averaged flows.

Coupled waves in stratified plasmas, treated by generalized polytropic equation of state

Applying a generalized polytropic law the author derives coupled wave equations governing small amplitude disturbances in horizontally stratified plasma. The methods of Clemmow and Heading and of Heading (1961) are used. With the usual choice of the polytropic indices the results referring to CGL plasmas and to MHD media can be found.

Embedded equations for the waves in the stratified CGL-plasma

This paper deals with wave equations describing small amplitude disturbances in an inhomogeneous CGL-plasma. The method developed by Heading is applied to find coupled equations nonsingular in the reflexion regions.

A Unified Three-Dimensional Instability Theory of the Onset of Blocking and Cyclogenesis

Abstract The instability characteristics of three-dimensional flow typical of the Northern Hemisphere average winter troposphere are examined in a two-layer spherical quasigeostrophic model. The properties of the four fastest-growing small-amplitude disturbances that develop on the basic state have been analyzed for three cases (case 1, 2a and 3) having increasingly larger static stability parameters. For case 1, the fastest-growing disturbance mode, which is propagating eastward rather rapidly, has a monopole cyclogenesis structure with maximum amplitudes slightly downstream of the jetstream maxima in the Pacific ocean and off the cast coast of North America. Comparisons of streamfunctions squared and momentum and heat fluxes with bandpass filtered observations for transient eddies, which pick out the developing storms, are quite reasonable considering that a two-layer model is used and the contributions from individual linear modes are considered. For case 2a, the fastest growing mode has large-scale hi...

Scale Selection and Energy Spectra of Disturbances in Southern Hemisphere Flows

The growth and nonlinear interaction of initially small-amplitude disturbances, having a complete spectrum of zonal wavenurnbers, with zonally averaged flows characteristic of January and May SouthernHemisphere situations, are studied in a multilevel primitive equation spectral model-incorporating spherical geometry and viscous dissipation. For January, the fastest growing scale has zonal wavenumberm = 9, in close agreement with linear instability theory. However, the contribution from m = 7 subsequently exceeds that of rn = 9 and is largely responsible for the general agreement between model andobserved eddy streamfunctions and fluxes. Although the individual wavenumber contributions, eg,m = 9 and m = 7 undergo vacillation cycles similar to ones found earlier with single zonal wavenumberdisturbances growing on the same zonal flows, the total eddy kinetic energy after the initial growth periodis relatively constant. This appears to be due to the interference between the larger number of waves. ForMay the fastest growing scale has m = 5 and initially grows on the polar jet. As the disturbance maturessecondary growth occurs on the subtropical jet and the disturbance achieves significant amplitude in theupper troposphere in the latitude band between 30 and 60. Unlike previous baroclinic instability results,the nonlinear model and observed eddy fluxes are also in close qualitative agreement. For January a very long-time viscous integration was carried out and the kinetic energy spectra werestudied at various stages. When zonal wavenumber m = 7 reaches its peak (day 23), the spectrum has anapproximate m-s power law. This. however, is not a statistical quasi-steady state and further integrationout to days 45-SO was needed to reach such a state. Then it was found that the spectra satisfy the m-3power law of classical two-dimensional and quasi-geostrophic turbulence theory.

Perturbed panel flutter - A simple model

The effects of small periodic disturbances on the response of a two-degree-of-freedom, nonconservative mechanical system are analyzed. The system is a simple model for panel flutter. The disturbance simulates the pressure fluctuations of a turbulent boundary layer on the panel. Asymptotic expansions of the solutions are obtained for small-amplitude disturbances. The qualitative features of the response depend on the prescribed variation of the disturbance frequency with the magnitude of the nonconservative applied force. The disturbance can induce a smooth transition to the fluttering states of the rods, or it may induce jump transitions. The results suggest a possible technique for delaying panel flutter by applying periodic forcing functions with appropriate frequencies.

Interaction of wave disturbances in an oscillating boundary layer

A theoretical analysis is made of the interaction of wave disturbances of small finite amplitude in a boundary layer in the case when the velocity distribution contains a periodic component that oscillates in time in accordance with a harmonic law. It is shown that it is in principle possible for there to be a four-wave synchronous (resonance) interaction in a cubic nonlinearity; equations are obtained for the amplitudes. Calculations made to test the effectiveness of the resonance phenomena have shown that the coupling coefficients are not sufficiently large for the superimposed oscillations to change significantly the nature of the interaction of the waves.

Response of Tradewind Cumuli to Large-Scale Processes

Abstract The two-dimensional slab-symmetric numerical cloud model used by Soong and Ogura (1973) for studying the evolution of an isolated cumulus cloud is extended to investigate the statistical properties of cumulus clouds which would be generated under a given large-scale forcing composed of the horizontal advection of temperature and water vapor mixing ratio, vertical velocity, sea surface temperature and radiative cooling. Random disturbances of small amplitude are introduced into the model at low levels to provide random motion for cloud formation. The model is applied to a case of suppressed weather conditions during BOMEX for the period 22–23 June 1969 when a nearly steady state prevailed. The composited temperature and mixing ratio profiles of these two days are used as initial conditions and the time-independent large-scale forcing terms estimated from the observations are applied to the model. The result of numerical integration shows that a number of small clouds start developing after 1 h. So...

On coupled waves in stratified CGL-plasma

Starting from the Chew-Goldbsrger-Low equations we derive wave-equations describing small amplitude disturbances in a horizontally stratified, continuously varying CGL-plasma. A set of equations of first-order matrix form is treated by the method of Clemmow and Heading.

Nonlinear forced oscillations in a closed tube: continuous solutions of a functional equation

The continuous small amplitude disturbances generated by the oscillations of a piston in a gas contained in a closed-ended tube are discussed. The coupled characteristic equations are integrated exactly for a model equation of state which approximates any stress–strain law with an error at O ([strain] 3 ). When the small amplitude limit is taken and the relative importance of amplitude dispersion and nonlinear interaction is assessed, the disturbances can be determined from solutions to a nonlinear functional equation. The motions are characterized by a similarity parameter, A , and a frequency parameter, ∆ . A simple algebraic scheme for constructing continuous periodic solutions of the functional equation is given, and the A–∆ plane is divided by a transition curve. The various resonances are then evident.

Partial equilibrium and relaxation in a plasma

This paper describes a study of the behavior of nitrogen and argon plasmas at about 4000°K temperature and 10 Pa pressure. Experiments show that such plasmas exhibit several possibilities of physical and chemical nonequilibrium. In the case of nitrogen, chemical relaxation phenomena which may be initiated by small-amplitude disturbances are first considered, in order to identify accurately the chemical kinetics of the plasma. Several reactive models were investigated; concentrations at chemical equilibrium and characteristic process times were computed in several cases. The influence of radiative processes, and that of various types of thermal freezing were investigated, leading to determination of the most probable reactive process. It is shown that full equilibrium is highly unlikely, whereas partial equilibrium is possible. Partial chemical equilibrium computations at one temperature were performed. Coupled relaxation processes at several temperatures were then investigated, and the laws governing the phenomena established. Two phenomena were investigated more specifically: • —relaxation behind a shock, • —relaxation after nitrogen injection into an argon plasma.

Initial value problem for Rayleigh–Taylor instability of viscous fluids

The initial value problem associated with the development of small amplitude disturbances in Rayleigh–Taylor unstable, viscous, incompressible fluids is studied. Solutions to the linearized equations of motion which satisfy general initial conditions are obtained in terms of Fourier–Laplace transforms of the hydrodynamic variables, without restriction on the density or viscosity of either fluid. When the two fluids have equal kinematic viscosities, these transforms can be inverted explicitly to express the fluid variables as integrals of Green’s functions multiplied by initial data. In addition to normal modes, a set of continuum modes, not treated explicitly in the literature, makes an important contribution to the development of the fluid motion.

Numerical calculation of the stability of parallel flows

Abstract : A numerical approach to solving the two-dimensional, incompressible Navier-Stokes equations is presented and is used to study the stability and evolution of disturbance in a boundary layer. Previous numerical approaches have used streamfunction-vorticity formulations that take advantage of the special properties of two-dimensional flow. The present method solves a finite-difference form of the momentum equations and may therefore be extended to three-dimensions more readily. Computations using this technique are compared with the result of linear stability theory for small amplitude disturbances and are found to give satisfactory agreement, while calculations at large amplitude support the conjecture that non-linear effects can be destabilizing. (Author)

On Itoh's finite amplitude stability theory for pipe flow

In two recent papers Itoh has developed a finite amplitude stability theory which indicates that nonlinearity increases the damping rate of a small but finite amplitude disturbance to flow in a circular pipe when the disturbance is concentrated near the axis of the pipe. For such a centre mode, which is the only mode considered by Itoh, Davey & Nguyen found, in an earlier paper, the opposite result that nonlinearity decreases the damping rate. We examine the reasons for this discrepancy and we explain the subtle difference between Itoh's method and the method of Reynolds & Potter, which was used by Davey & Nguyen.We suggest that for the centre mode of pipe flow neither Itoh's result nor Davey & Nguyen's result is a reliable guide to the true situation. However, for the wall mode of pipe flow, and especially for plane Couette flow, both methods give very similar results and we suggest that this similarity indicates that in these cases the damping rate is decreased by nonlinearity. For a particular problem we believe that it is only when the results of the two methods are very similar that either method is likely to be useful.

The Life Cycles of Some Nonlinear Baroclinic Waves

Abstract Some aspects of the nonlinear behavior of mid-latitude baroclinic waves are investigated by means of a series of integrations of the primitive equations with spherical geometry. Each integration has as initial conditions a balanced zonal flow perturbed by a small-amplitude disturbance of normal-mode form. Results are presented in detail for several zonal flows and perturbations which are confined initially to either zonal wavenumber 6 or zonal wavenumber 9. In each case a disturbance grows by baroclinic instability and develops a structure in some agreement with the usual synoptic picture of an occluding system. Its growth rate at low levels decreases more rapidly than that at higher levels, as found by Gall using a more severely truncated model, and upper-level amplitudes become larger relative to surface values than in the initial linear mode. This is more marked for wavenumber 6 than for wavenumber 9, and differences in linear structure are thus enhanced in the nonlinear regime. Barotropic pro...

Generalized Eliassen-Palm and Charney-Drazin Theorems for Waves oin Axismmetric Mean Flows in Compressible Atmospheres

The theorems, which exhibit the role of wave dissipation, excitation and transience in the forcing of mean flow changes of second order in wave amplitude by arbitrary, small-amplitude disturbances, are obtained 1) for the primitive equations in pressure coordinates on a sphere, and 2) in a more general form (applicable for instance to nonhydrostatic disturbances in tornadoes or hurricanes) establishing that no approximations beyond axisymmetry of the mean flow are necessary. It is shown how the results reduce to those found by Boyd (1976) for the case of sinusoidal, hydrostatic waves with exponentially growing or decaying amplitude, and it is explained why the approximation used by Boyd in the thermodynamic equation is not needed. The reduction to Boyd's results entails the use of a virial theorem. This theorem amounts to a generalization of the “equipartition” law derived in an earlier paper (Andrews and Mclntyre, 1976). That derivation appeared to rely on an assumption about relative phases of disturbance Fourier components; the present derivation shows that no such assumption is in fact necessary.

Stability of a liquid film with respect to initially finite three-dimensional disturbances

The nonlinear stability of a viscous liquid film flowing steadily down an inclined plane is studied in the phase plane. Explicit expressions of the critical points of the governing differential system are obtained. The nature of the critical points and the integral curves corresponding to various flow parameters, initial conditions, and disturbance characteristics are determined numerically. Based on the phase plane analysis and the numerical results, the following conclusions are reached: The film which is unstable according to linear theory may be stable with respect to a finite three-dimensional disturbance if the initial amplitude and the side-band width are sufficiently small. The particular values of the initial amplitude and the side-band width beyond which the film becomes unstable depend on the relevant flow parameters. The film which is stable according to linear theory is also shown to be stable with respect to three-dimensional small finite amplitude disturbances with finite bandwidths.

Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 1. An asymptotic theory valid for small amplitude disturbances

A strict distinction is made between the two fundamental assumptions in the Stuart-Watson theory of nonlinear stability, one of which is that the amplitude of disturbance is sufficiently small, while the other is that the damping or amplification rate for an infinitesimal disturbance is small. This distinction leads to classification of the nonlinear stability theory into two asymptotic theories: the theory based on the first assumption can be applied to subcritical flows with Reynolds numbers away from the neutral curve, even to flows with no neutral curve, such as plane Couette flow or pipe Poiseuille flow, while the theory based on the second assumption is available only for Reynolds numbers and wavenumbers in the neighbourhood of the neutral curve. In the theory based on the first assumption the concept of trajectories in phase space, together with the method of eigenfunction expansion, is introduced in order to display nonlinear behaviour of the disturbance amplitude and to provide the most rational definition of the Landau constant available for classification of the behaviour patterns.

The evolution of nonlinear standing waves in bounded media

The evolution of small amplitude disturbances in a bounded medium, under fixed and nearly fixed end conditions, is considered. The various physical effects accounted for are amplitude dispersion, frequency dispersion and dissipation due to both radiation of energy out of the medium and rate-dependence of the medium. In a nonlinear geometrical acoustics theory the transport equations which determine the signal carried by a component wave have the form of a simple wave equation, Korteweg-de Vries equation, damped simple wave equation and Burgers' equation.

Integration of Vlasov equation by vector method

To discuss the small amplitude disturbances propagating through a hot plasma in a uniform magnotic field, or to find the macroscopic quantities of a plasma, it. is necessary to solve the Vlasov equation. In this paper a general vector method is presented and it will give a solution of the linearized Vlasov equation with a collision term. The explicit expressions of the perturbed distribution functions for both anisotropic and isotropic equilibrium distributions are derived. The results can be easily applied to find the dispersion equation and the polarization properties of the waves.