Nonequilibrium initial conditions of a Brownian oscillator system observed by two-dimensional spectroscopy

2001 ◽  
Vol 115 (5) ◽  
pp. 2267-2281 ◽  
Author(s):  
Yoko Suzuki ◽  
Yoshitaka Tanimura
Keyword(s):  
Initial Conditions ◽  
Two Dimensional ◽  
Oscillator System ◽  
Brownian Oscillator

Waves and turbulence on a beta-plane

1975 ◽  
Vol 69 (3) ◽  
pp. 417-443 ◽  
Author(s):  
Peter B. Rhines
Keyword(s):  
Weak Interaction ◽  
Initial Conditions ◽  
Computer Experiments ◽  
Zonal Flow ◽  
Similarity Solutions ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Homogeneous Fluid ◽  
Restoring Force ◽  
Model Equations

Two-dimensional eddies in a homogeneous fluid at large Reynolds number, if closely packed, are known to evolve towards larger scales. In the presence of a restoring force, the geophysical beta-effect, this cascade produces a field of waves without loss of energy, and the turbulent migration of the dominant scale nearly ceases at a wavenumber kβ = (β/2U)½ independent of the initial conditions other than U, the r.m.s. particle speed, and β, the northward gradient of the Coriolis frequency.The conversion of turbulence into waves yields, in addition, more narrowly peaked wavenumber spectra and less fine-structure in the spatial maps, while smoothly distributing the energy about physical space.The theory is discussed, using known integral constraints and similarity solutions, model equations, weak-interaction wave theory (which provides the terminus for the cascade) and other linearized instability theory. Computer experiments with both finite-difference and spectral codes are reported. The central quantity is the cascade rate, defined as \[ T = 2\int_0^{\infty} kF(k)dk/U^3\langle k\rangle , \] where F is the nonlinear transfer spectrum and 〈k〉 the mean wavenumber of the energy spectrum. (In unforced inviscid flow T is simply U−1d〈k〉−1/dt, or the rate at which the dominant scale expands in time t.) T is shown to have a mean value of 3·0 × 10−2 for pure two-dimensional turbulence, but this decreases by a factor of five at the transition to wave motion. We infer from weak-interaction theory even smaller values for k [Lt ] kβ.After passing through a state of propagating waves, the homogeneous cascade tends towards a flow of alternating zonal jets which, we suggest, are almost perfectly steady. When the energy is intermittent in space, however, model equations show that the cascade is halted simply by the spreading of energy about space, and then the end state of a zonal flow is probably not achieved.The geophysical application is that the cascade of pure turbulence to large scales is defeated by wave propagation, helping to explain why the energy-containing eddies in the ocean and atmosphere, though significantly nonlinear, fail to reach the size of their respective domains, and are much smaller. For typical ocean flows, $k_{\beta}^{-1} = 70\,{\rm km} $, while for the atmosphere, $k_{\beta}^{-1} = 1000\,{\rm km}$. In addition the cascade generates, by itself, zonal flow (or more generally, flow along geostrophic contours).

scholarly journals R-Process Nucleosynthesis in MHD Jet Explosions of Core-Collapse Supernovae

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Motoaki Saruwatari ◽  
Masa-aki Hashimoto ◽  
Ryohei Fukuda ◽  
Shin-ichiro Fujimoto
Keyword(s):  
Magnetic Field ◽  
Heavy Element ◽  
Initial Conditions ◽  
Core Collapse ◽  
Combined Effects ◽  
Two Dimensional ◽  
The Third ◽  
R Process ◽  
Helium Star ◽  
Hydrodynamic Code

We investigate the r-process nucleosynthesis during the magnetohydrodynamical (MHD) explosion of a supernova in a helium star of 3.3 M⊙, where effects of neutrinos are taken into account using the leakage scheme in the two-dimensional (2D) hydrodynamic code. Jet-like explosion due to the combined effects of differential rotation and magnetic field is able to erode the lower electron fraction matter from the inner layers. We find that the ejected material of low electron fraction responsible for the r-process comes out from just outside the neutrino sphere deep inside the Fe-core. It is found that heavy element nucleosynthesis depends on the initial conditions of rotational and magnetic fields. In particular, the third peak of the distribution is significantly overproduced relative to the solar system abundances, which would indicate a possible r-process site owing to MHD jets in supernovae.

scholarly journals Phase Transition in Dynamical Systems: Defining Classes of Universality for Two-Dimensional Hamiltonian Mappings via Critical Exponents

2009 ◽  
Vol 2009 ◽  
pp. 1-22 ◽  
Author(s):  
Edson D. Leonel
Keyword(s):  
Phase Transition ◽  
Critical Exponents ◽  
Scaling Laws ◽  
Scaling Function ◽  
Initial Conditions ◽  
Invariant Tori ◽  
Dynamical Variable ◽  
Two Dimensional ◽  
Control Parameters ◽  
Average Value

A phase transition from integrability to nonintegrability in two-dimensional Hamiltonian mappings is described and characterized in terms of scaling arguments. The mappings considered produce a mixed structure in the phase space in the sense that, depending on the combination of the control parameters and initial conditions, KAM islands which are surrounded by chaotic seas that are limited by invariant tori are observed. Some dynamical properties for the largest component of the chaotic sea are obtained and described in terms of the control parameters. The average value and the deviation of the average value for chaotic components of a dynamical variable are described in terms of scaling laws, therefore critical exponents characterizing a scaling function that describes a phase transition are obtained and then classes of universality are characterized. The three models considered are: The Fermi-Ulam accelerator model, a periodically corrugate waveguide, and variant of the standard nontwist map.

ON THE JOINT APPLICATION OF THE COLLOCATION BOUNDARY ELEMENT METHOD AND THE FOURIER METHOD FOR SOLVING PROBLEMS OF HEAT CONDUCTION IN FINITE CYLINDERS WITH SMOOTH DIRECTRICES

2021 ◽  
pp. 15-38
Author(s):  
D.Y. Ivanov ◽  
Keyword(s):  
Fourier Series ◽  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Initial Conditions ◽  
Fourier Method ◽  
Time Interval ◽  
Cylindrical Domain ◽  
Two Dimensional ◽  
Element Method

Here we consider the initial-boundary value problems in a homogeneous cylindrical domain YI Ω ×+ ( Ω+ is an open two-dimensional bounded simply connected domain with a boundary 5 ∂Ω ∈C , 2 \ Ω≡ Ω − + R is the open exterior of the domain Ω+ , [0, ] YI ≡ Y is the height of the cylinder) on a time interval [0, ] TI ≡ T . The initial conditions and the boundary conditions on the bases of the cylinder are zero, and the boundary conditions on the lateral surface of the cylinder are given by the function 1 2 wx x yt ( , , ,) ( 1 2 (, ) x x ∈∂Ω , Y y ∈ I , T t I ∈ ). An approximate solution of such problems is obtained through the combined use of the Fourier method and the collocation boundary element method based on piecewise quadratic interpolation (PQI). The solution to the problem in the cylinder is expanded in a Fourier series in terms of eigenfunctions of the operator 2 By yy ≡ ∂ with the corresponding zero boundary conditions. The coefficients of such a Fourier series are solutions of problems for two-dimensional heat equations 2 2 t ∇ =∂ + u u ku . With a low smoothness of the functions w in the variable y, the weight of solutions at large values of k increases and the accuracy of solving the problem in the cylinder decreases. To maintain accuracy on a uniform grid, the step of discretization of the boundary function w with respect to the variable y is decreased by a factor of j. Here j is an averaged value of the quantity Y k π depending on the function w. In addition, the steps of discretization of functions ( ) 2 exp − τ k with respect to the variable τ in domains τ≤πT k are reduced by a factor of 2 2 k π . The steps in the remaining ranges of values τ and the steps by the other variables remain unchanged. The approximate solutions obtained on the basis of this procedure converge stably to exact solutions in the 2 ( ) LI I Y T × -norm with a cubic velocity uniformly with respect to sets of functions w, bounded by norm of functions with low smoothness in the variable y, uniformly along the length of the generatrix of the cylinder Y , and uniformly in the domain Ω . The latter is also associated with the use of PQI along the curve ∂Ω over the variable 2 2 ρ≡ − r d , which is carried out at small values of r ( d and r are the distances from the observed point of the domain Ω to the boundary ∂Ω and to the current point of integration along ∂Ω , respectively). The theoretical conclusions are confirmed by the results of the numerical solution of the problem in a circular cylinder, where the dependence of the boundary functions w on y is given by the normalized eigenfunctions of the differential operator By which vary in a sufficiently large range of values of k .

Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Penghe Ge ◽  
Hongjun Cao
Keyword(s):  
Neuron Model ◽  
Initial Conditions ◽  
Stability Conditions ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Fast Subsystem ◽  
One Dimensional ◽  
Sensitive Dependence ◽  
The Stability ◽  
Slow Subsystem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

Dripping instability of a two-dimensional liquid film under an inclined plate

2021 ◽  
Vol 932 ◽  
Author(s):  
Guangzhao Zhou ◽  
Andrea Prosperetti
Keyword(s):  
Liquid Film ◽  
Initial Conditions ◽  
Strong Dependence ◽  
Bond Number ◽  
Quasi Equilibrium ◽  
Two Dimensional ◽  
Inclined Plate ◽  
Maximum Value ◽  
Critical Curvature ◽  
Rayleigh Taylor Instability

It is known that the dripping of a liquid film on the underside of a plate can be suppressed by tilting the plate so as to cause a sufficiently strong flow. This paper uses two-dimensional numerical simulations in a closed-flow framework to study several aspects of this phenomenon. It is shown that, in quasi-equilibrium conditions, the onset of dripping is closely associated with the curvature of the wave crests approaching a well-defined maximum value. When dynamic effects become significant, this connection between curvature and dripping weakens, although the critical curvature remains a useful reference point as it is intimately related to the short length scales promoted by the Rayleigh–Taylor instability. In the absence of flow, when the film is on the underside of a horizontal plate, the concept of a limit curvature is relevant only for small liquid volumes close to a critical value. Otherwise, the drops that form have a smaller curvature and a large volume. The paper also illustrates the peculiarly strong dependence of the dripping transition on the initial conditions of the simulations. This feature prevents the development of phase maps dependent only on the governing parameters (Reynolds number, Bond number, etc.) similar to those available for film flow on the upper side of an inclined plate.

Richtmyer–Meshkov instability on two-dimensional multi-mode interfaces

2021 ◽  
Vol 928 ◽  
Author(s):  
Yu Liang ◽  
Lili Liu ◽  
Zhigang Zhai ◽  
Juchun Ding ◽  
Ting Si ◽  
...  
Keyword(s):  
Nonlinear Model ◽  
Initial Conditions ◽  
Superposition Principle ◽  
Mode Competition ◽  
Two Dimensional ◽  
Competition Effect ◽  
Density Ratios ◽  
Multi Mode ◽  
Amplitude Growth ◽  
Perturbation Growth

Shock-tube experiments on eight kinds of two-dimensional multi-mode air–SF $_6$ interface with controllable initial conditions are performed to examine the dependence of perturbation growth on initial spectra. We deduce and demonstrate experimentally that the amplitude development of each mode is influenced by the mode-competition effect from quasi-linear stages. It is confirmed that the mode-competition effect is closely related to initial spectra, including the wavenumber, the phase and the initial amplitude of constituent modes. By considering both the mode-competition effect and the high-order harmonics effect, a nonlinear model is established based on initial spectra to predict the amplitude growth of each individual mode. The nonlinear model is validated by the present experiments and data in the literature by considering diverse initial spectra, shock intensities and density ratios. Moreover, the nonlinear model is successfully extended based on the superposition principle to predict the growths of the total perturbation width and the bubble/spike width from quasi-linear to nonlinear stages.

scholarly journals Impact of initial conditions on the prediction of the spread of thermal pollution in rivers

2018 ◽  
Vol 40 ◽  
pp. 05048
Author(s):  
Monika B. Kalinowska ◽  
Pawel M. Rowiński ◽  
Artur Magnuszewski
Keyword(s):  
Open Channel ◽  
Flow Model ◽  
River Flow ◽  
Initial Conditions ◽  
Two Dimensional ◽  
River Temperature ◽  
Heated Water ◽  
Different Temperatures ◽  
Rivmix Model ◽  
Water Depths

The influence of initial conditions on the prediction of the increase of river temperature below the point of release of heated water for a designed power plant has been analysed in this study. The results for different assumed values of river flow and different temperatures of the discharged heated water have been presented. The results have been analysed taking into account existing legal frames. The two-dimensional inhouse RivMix model has been used to simulate the temperature distribution whereas the two-dimensional depth-averaged turbulent open channel flow model CCHE2D has been used to simulate the velocity fields and the water depths for the selected flows of the river.

STOCK-HARVEST DYNAMICS IN MULTIAGENT FISHERIES

2011 ◽  
Vol 21 (01) ◽  
pp. 363-372
Author(s):  
EN-GUO GU
Keyword(s):  
Initial Conditions ◽  
Profit Maximization ◽  
Fish Stock ◽  
Maximization Problem ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Fishery Resource ◽  
Global Dynamic ◽  
Existence And Stability ◽  
Fishery Model

In this work, we build a two-dimensional dynamical fishery model in which the total harvest is obtained by a multiagent game with best reply strategy and naive expectations, i.e. each agent decides the harvest quantity by solving a profit maximization problem. Special attention is paid to the global dynamic analysis in the light of feasible domains (initial conditions giving non-negative trajectories converging to an equilibrium), which is related to the crisis of extinction. We also study the existence and stability of non-negative equilibria for models through mathematical analysis and numerical simulations. We discover the increase in the margin price of fish stock may lead to instability of the fixed point and make the system sink into chaotic attractors. Thus the fishery resource may fluctuate in a stochastic form.

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