Higher order approximations in the theory of longitudinal plasma oscillations. II

1970 ◽  
Vol 4 (4) ◽  
pp. 819-824
Author(s):  
F. Einaudi ◽  
W. I. Axford
Keyword(s):  
Distribution Function ◽  
Time Scale ◽  
Asymptotic Series ◽  
Finite Amplitude ◽  
Landau Damping ◽  
Multiple Time ◽  
Complex Frequency ◽  
Previous Note ◽  
Non Linear ◽  
Independent Variable

In this note we comment and extend the results of a previous analysis in which the non-linear behaviour of one-dimensional electrostatic oscillations in a homogeneous, unbounded, collisionless and fully ionized plasma was considered. The evolution of a monochromatic wave of small, but finite amplitude is studied by expanding the dependent variables as well as the independent variable tin the form of asymptotic series; an ordering parameter e proportional to the initial amplitude of the electric field is introduced. The expansion of the independent variable in such a series allows us to eliminate secular terms from the part of the distribution function which does not depend on the free-streaming terms. This, in turn, allows us to determine corrections to the complex frequency a. Results of a previous note on non-linear Landau damping for an initially Maxwellian. distribution function are confirmed, but it is indicated that they apply to values of time up to a value τ1 rather than for all times. One can proceed to larger values of time in the manner of the multiple time-scale method. In particular it is found that the Landau damping is increased with respect to the linear value only initially during the first time scale.

Higher order approximations in the theory of longitudinal plasma oscillations

1967 ◽  
Vol 1 (4) ◽  
pp. 483-497 ◽  
Author(s):  
F. Einaudi ◽  
W. I. Axford
Keyword(s):  
Distribution Function ◽  
Electric Field ◽  
Linear Theory ◽  
Asymptotic Series ◽  
Finite Amplitude ◽  
Single Wave ◽  
Background Distribution ◽  
Damping Rate ◽  
Linear Behaviour ◽  
Longitudinal Plasma

The non-linear behaviour of one-dimensional electrostatic oscillations in a homogeneous, unbounded, collisionless and fully ionized plasma is considered for the case in which a single wave of small, but finite amplitude is excited initially. The Vlasov–Poisson equations are solved using the method of strained co-ordinates in which the independent variable t, the electric field and the distribution function are expanded in the form of asymptotic series, the terms of which are founded by an iterative procedure. An ordering parameter e is introduced, which is proportional to the initial amplitude of the electric field given by linear theory. Differential equations are derived which can be solved sequentially to obtain uniformly valid solutions to all orders in ε. Solutions are given to second order and applied to the case in which the background distribution function is Maxwellian. It is found that the changes in the real and imaginary part of the frequency are small in comparison to the values obtained in the linear theory; that the free-streaming terms decay exponentially in time with a damping rate proportional to ε2, in contrast with the linear theory where they are Un- damped; and that the analysis allows us to calculate the changes in the background distribution function for large time, resulting from particle-wave interactions.

Analysis of the Nonlinear Response of a Cantilever Beam Under Deterministic and Random Excitations

2003 ◽  
Author(s):  
F. Benedettini ◽  
D. Zulli ◽  
M. Vasta
Keyword(s):  
Differential Equations ◽  
Cantilever Beam ◽  
Time Scale ◽  
Input Power ◽  
Multiple Time ◽  
Second Order Statistics ◽  
Linear Behavior ◽  
Power Spectral ◽  
Non Linear ◽  
The Moment

Dynamical systems subject to random excitations exhibit non-linear behavior for sufficiently large motion. In the case of 1 d.o.f. models, the multiple time scale method has been extensively utilized in the framework of non-linear deterministic analysis to obtain two averaged first-order differential equations describing the slow time scale modulation of amplitude and phase response. In this paper a stochastic frequency-response relationship has been derived involving the response amplitude statistics and the input power spectral density. A low-intensity noise has been assumed to separate the strong mean motion from its weak fluctuations. The moment differential equations of phase and amplitude have been derived and a linearization technique applied to evaluate the second order statistics. The theory has been validated through digital simulations on a Duffing-Rayleigh oscillator and on a cantilever beam with tip force.

scholarly journals At which time scale does the complementary principle perform best on evaporation estimation?

2020 ◽  
Author(s):  
Liming Wang ◽  
Songjun Han ◽  
Fuqiang Tian
Keyword(s):  
Uncertainty Analysis ◽  
Eddy Covariance ◽  
Time Scales ◽  
Time Scale ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Energy Correction ◽  
Independent Variable ◽  
The Difference ◽  
Annual Time

Abstract. The complementary principle has been widely used to estimate evaporation under different conditions. However, it remains unclear that at which time scale the complementary principle performs best. In this study, evaporation estimation was assessed over 88 eddy covariance (EC) monitoring sites at multiple time scales (daily, weekly, monthly, and yearly) by using the sigmoid and polynomial generalized complementary functions. The results indicate that the generalized complementary functions exhibit the highest skill in estimating evaporation at the monthly scale. The uncertainty analysis shows that this conclusion is not affected by ecosystem types nor energy correction methods. Through comparisons at multiple time scales, we found that the slight difference between the two generalized complementary functions only exists when the independent variable (x) in the functions approaches 1. The difference results in different performance of the two models at daily and weekly scales. However, such difference vanishes at monthly and annual time scales as few high x occurrences. This study demonstrates the applicability of the generalized complementary functions across multiple time scales and provides a reference for choosing the suitable timestep for evaporation estimation in relevant studies.

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