Small Amplitude Regime Research Articles

Interannual Variability in a Tropical Atmosphere–Ocean Model: Influence of the Basic State, Ocean Geometry and Nonlinearity

The behavior of a tropical coupled atmosphere/ocean model is analyzed for a range of different background states and ocean geometries. The model is essentially that of Cane and Zebiak for the tropical Pacific, except only temporally constant background states are considered here. For realistic background states and ocean geometry, the model solutions feature oscillations of period of 3–5 yr. By comparing the full model solution with a linearized version of the model, it is shown that the basic mechanism of the oscillation is contained within linear theory. A simple linear analog model is derived that describes the nature of the interannual variability in the coupled tropical atmosphere–ocean system. The analog model highlights the properties that produce coupled atmosphere–ocean instability in the eastern ocean basin, and the equatorial wave dynamics in the western ocean basin that are responsible for a delayed, negative feedback into this instability growth. The growth rate of the local instability c together with the magnitude b and lag of the wave-induced processes determine the nature of the interannual variability displayed in the coupled model. Specifically, these processes determine the growth rate of the coupled system and, when the solutions are oscillatory, the period of the oscillation. The terms b, c, and are set by the background state of the atmosphere and ocean, and the geometry of the ocean basin. The simple analog model is used to design and interpret a set of experiments using the full linear and nonlinear numerical models of the coupled atmosphere ocean system in the Pacific. In these experiments, we examine the effects of the assumed basic state and ocean geometry on the interannual variability of the coupled system. The simple model is shown to be a remarkably good proxy of the full linear and nonlinear numerical models. The limiting nonlinearity in the full numerical model is shown to be the dependence of the temperature of the upward water on the thermocline depth. However, we find the essential processes that describe the local instability growth rate and period of the interannual oscillations in the coupled system are linear. Nonlinearities primarily act as a bound on the amplitude of the final state oscillations, and decrease the period of the firm state oscillations by about 10 percent from that obtained in the small amplitude regime of the full coupled model and the linear analog model. The nonlinear analog model for the full numerical model is derived, and compared with that proposed by Suarez and Schopf. The numerical and analog models help to explain why organized, large amplitude, interannual variability is prominent in the tropical Pacific basin, and not in Atlantic and Indian basins.

Erratum: Dynamics of the nuclear one-body density: Small amplitude regime

Erratum: Dynamics of the nuclear one-body density: Small amplitude regime

Dynamics of the nuclear one-body density: Small amplitude regime.

We present a microscopic treatment for the small amplitude limit of the equations of motion for the nuclear one-body density. These were derived previously by means of projection techniques, and allow for the explicit separation of mean-field and collision effects which result from the dynamics of many-body correlations. The form of the nuclear response in the presence of collision effects is derived. An illustrative application to a soluble model is discussed.

Exact Soliton Solution for Superfluid Film Dynamics

Traveling wave solutions are examined for the nonlinear wave equation \begin{aligned} \text{i}\frac{\partial \psi}{\partial t}=-\frac{\partial^{2}\psi}{\partial x^{2}}+\left[1-\left(\frac{1+d}{1+d|\psi|^{2}}\right)^{2}\right]\psi, \end{aligned} which describes the dynamics of condensate wave function ψ( x , t ) in superfluid film of mean thickness d . An exact one-soliton solution is obtained analytically for arbitrary amplitude, and this suggests that the “quasi-solitons” found in the previous numerical work are stable at least in the asymptotic situation where quasi-solitons are essentially non-overlapping. It is shown explicitly that our solution reduces, in small amplitude regime, to the Korteweg-de Vries one-soliton solution.

Role of Trapped Particles and Waves in Plasma Solitons-Theory and Application

In this tutorial and review paper, we investigate the influence of trapped particles on the propagation of ion acoustic solitons and the role of trapped waves on the propagation of Langmuir solitons. The classical potential method allows us to construct finite amplitude soliton solutions, and trapping phenomena are found to retard the motion of solitons. Thus, ion acoustic solitons have minimum speed when they are based on an isothermal electron equation of state since this corresponds to "maximum electron trapping". In the small amplitude regime, deviations from the Boltzmann law lead to a new nonlinear term. Subsequently, the dynamics of the soliton is governed by a modified KdV-equation [e.g., (23)]. For Langmuir solitons an existence diagram is found and exhibits the comparison between experimentally observed localized structures and theory. The connection with the various small amplitude soliton solutions is also pointed out. Moreover, the inclusion of a pump and of dissipative terms in the coupled nonlinear Schrödinger-ion equation gives rise to a transient phenomenon called soliton flash, whose implication to the laboratory experiments is discussed. Finally, as an application in numerical analysis the propagation of solitons is followed by solving the KdV-equation in Fourier-space. This turns out to be an excellent tool to test stability and accuracy of numerical schemes. Our results reveal that the so-called aliasing interactions lead to erroneous solutions. The conservation of invariants does not guarantee the accuracy of a numerical algorithm, although it is needed for its stability.