scholarly journals The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate

Fluids ◽  
2021 ◽  
Vol 6 (10) ◽  
pp. 348
Author(s):  
Thomas Meunier ◽  
J. H. LaCasce
Keyword(s):  
Growth Rate ◽  
Lyapunov Exponent ◽  
Velocity Structure ◽  
Initial Conditions ◽  
Mathematical Formulation ◽  
Finite Size ◽  
Finite Amplitude ◽  
Computational Method ◽  
Finite Time Lyapunov Exponent ◽  
Amplitude Growth

The finite size Lyapunov exponent (FSLE) has been used extensively since the late 1990s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian coherent structures in geophysical flows and two-dimensional turbulence. Historically, the FSLE was defined in terms of its computational method rather than via a mathematical formulation, and the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. Here, we propose an exact definition of the FSLE based on conditional averaging of the finite amplitude growth rate (FAGR) of the particle pair separation. With this new definition, we show that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the links between the FAGR and other relevant Lagrangian metrics, such as the finite time Lyapunov exponent and the second-order velocity structure function.

scholarly journals The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate

2021 ◽  
Author(s):  
Thomas Meunier ◽  
J.H. LaCasce
Keyword(s):  
Growth Rate ◽  
Lyapunov Exponent ◽  
Velocity Structure ◽  
Initial Conditions ◽  
Mathematical Formulation ◽  
Finite Size ◽  
Finite Amplitude ◽  
Computational Method ◽  
Finite Time Lyapunov Exponent ◽  
Amplitude Growth

The Finite size Lyapunov exponent (FSLE) has been used extensively since the late 1990’s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian coherent structures in geophysical flows and two-dimensional turbulence. Historically, the FSLE was defined in terms of its computational method rather than via a mathematical formulation, and the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. Here we propose an exact definition of the FSLE based on conditional averaging of the finite amplitude growth rate (FAGR) of the particle pair separation. With this new definition, we show that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the links between the FAGR and other relevant Lagrangian metrics, such as the finite time Lyapunov exponent and the second order velocity structure function.

Domains of Attraction for an Autoparametric Mass-Pendulum System: A Lyapunov Exponent Map Approach

2003 ◽  
Author(s):  
Khalid El-Rifai ◽  
George Haller ◽  
Anil K. Bajaj
Keyword(s):  
Lyapunov Exponent ◽  
Finite Time ◽  
Initial Conditions ◽  
Transient Behavior ◽  
System Response ◽  
Pendulum System ◽  
Domains Of Attraction ◽  
Nonlinear Dynamical ◽  
Finite Time Lyapunov Exponents ◽  
Finite Time Lyapunov Exponent

Many recent studies have been performed on resonantly excited mass-pendulum systems with autoparametric (internal) resonance capturing interesting local steady state phenomena. The objective of this work is to explore the transient behavior in such systems. The domains of attraction of the time-periodic system provide some help in understanding the transient dynamics, and these are sought using a recently developed algorithm that solves for the finite-time Lyapunov exponent over a grid of initial conditions. Though the use of finite-time Lyapunov exponents in nonlinear dynamical analyses is not novel, its application to multi-degree-offreedom forced nonlinear systems has not been reported in the literature. In addition to identifying regions of different final states, the technique used captures different levels of attraction within a domain. This sheds some light on the role played by other modes present in a multi-degree-of-freedom system in shaping the overall system response.

Seasonality of surface stirring by geostrophic flows in the Bay of Bengal

2020 ◽  
Author(s):  
Nihar Paul ◽  
Jai Sukhatme
Keyword(s):  
Lyapunov Exponent ◽  
Power Law ◽  
Bay Of Bengal ◽  
Numerical Models ◽  
Finite Size ◽  
Power Spectra ◽  
Relative Dispersion ◽  
Eddy Diffusivities ◽  
Finite Time Lyapunov Exponent ◽  
Passive Tracers

<p>Stirring of passive tracers in the Bay of Bengal driven by altimetry derived daily geostrophic surface currents, is studied on subseasonal timescales. To begin with, Hovmöller plots, wavenumber-frequency diagrams and power spectra confirm the multiscale nature of the flow. Advection of latitudinal and longitudinal bands highlights the chaotic nature of stirring in the Bay via repeated straining and filamentation of the tracer field. An immediate finding is that stirring is local, i.e. of the scale of the eddies, and does not span the entire basin. Further, stirring rates are enhanced along the coast of the Bay and are relatively higher in the pre- and post-monsoonal seasons. Indeed, Finite Time Lyapunov Exponent (FTLE) and Finite Size Lyapunov Exponent (FSLE) maps in all the seasons are patchy with minima scattered through the interior of the Bay. Further, these maps bring out a seasonal cycle wherein rapid stirring progressively moves from the northern to southern Bay during pre- and post-monsoonal periods, respectively. The non-uniform stirring of the Bay is reflected in long tailed probability density functions of FTLEs, that become more stretched for longer time intervals. Quantitatively, advection for a week shows the mean FTLE lies between 0.13±0.07 day<sup>-1</sup>, while extremes reach almost 0.6 day<sup>-1</sup> . Averaged over the Bay, Relative dispersion initially grows exponentially, followed by a power-law at scales between approximately 100 and 250 km, which finally transitions to an eddy-diffusive regime. Quantitatively, below 250 km, a scale dependent diffusion coefficient is extracted that behaves as a power-law with cluster size, while above 250 km, eddy-diffusivities range from 6 × 10<sup>3</sup> - 1.6 × 10<sup>4  </sup> m<sup>2</sup>s<sup>-1</sup> in different regions of the Bay. These estimates provide a useful guide for resolution dependent diffusivities in numerical models that hope to properly represent surface stirring in the Bay.</p>

Predictability of the Northeast Monsoon Forecasts by the Educational Global Climate Model

2015 ◽  
Vol 804 ◽  
pp. 243-246
Author(s):  
Sunisa Saiuparad ◽  
Dusadee Sukawat
Keyword(s):  
Lyapunov Exponent ◽  
Prediction Model ◽  
Climate Model ◽  
Global Climate ◽  
Global Climate Model ◽  
Weather Prediction ◽  
Finite Size ◽  
Largest Lyapunov Exponent ◽  
Northeast Monsoon ◽  
Finite Time Lyapunov Exponent

The predictability by an atmospheric prediction model is determined by the uncertainties in the initial condition and the imperfection of the model. It is difficult to provide accurate weather prediction and determines the predictability of a model. Atmospheric prediction model efficiency is obtained from the analysis of predictability measurement. Five existing predictability measurements; Lyapunov exponent, finite size Lyapunov exponent, finite time Lyapunov exponent, local Lyapunov exponent and largest Lyapunov exponent are used to measure predictability of the northeast monsoon (winter monsoon) by the Educational Global Climate Model (EdGCM) and to test sensitivity of the model to small initial perturbations. The EdGCM is run for 142-year predictions from the year 1958 to 2100. However, only the outputs of geopotential height at 500hPa of December from 2012 to 2100 are used for predictability measurement. The results show that the EdGCM predictability for the northeast monsoon forecast is about 120 years.

scholarly journals Nonlinear dynamics approach to the predictability of the Cane–Zebiak coupled ocean–atmosphere model

2014 ◽  
Vol 21 (1) ◽  
pp. 155-163 ◽  
Author(s):  
L. Siqueira ◽  
B. Kirtman
Keyword(s):  
Nonlinear Dynamics ◽  
Lyapunov Exponent ◽  
Finite Size ◽  
Coupled Model ◽  
Finite Amplitude ◽  
Maximum Lyapunov Exponent ◽  
Theoretical Concepts ◽  
Ocean Atmosphere ◽  
Atmosphere Model ◽  
Predictability Analysis

Abstract. The predictability of the Cane–Zebiak coupled ocean–atmosphere model is investigated using nonlinear dynamics analysis. Newer theoretical concepts are applied to the coupled model in order to help quantify maximal prediction horizons for finite amplitude perturbations on different scales. Predictability analysis based on the maximum Lyapunov exponent considers infinitesimal perturbations, which are associated with errors in the smallest fastest-evolving scales of motion. However, these errors become irrelevant for the predictability of larger scale motions. In this study we employed finite-size Lyapunov exponent analysis to assess the predictability of the Cane–Zebiak coupled ocean–atmosphere model as a function of scale. We demonstrate the existence of fast and slow timescales, as noted in earlier studies, and the expected enhanced predictability of the anomalies on large scales. The final results and conclusions clarify the applicability of these new methods to seasonal forecasting problems.

On Computation of the Motion of Elastic Rods

1974 ◽  
Vol 41 (3) ◽  
pp. 777-780 ◽  
Author(s):  
R. P. Nordgren
Keyword(s):  
Initial Conditions ◽  
Three Dimensional ◽  
Free Fall ◽  
Finite Amplitude ◽  
Computational Method ◽  
Elastic Rods ◽  
Explicit Method ◽  
Finite Difference Approximations ◽  
Method Of Solution ◽  
Rigid Plane

A computational method is developed for the finite-amplitude three-dimensional motion of inextensible elastic rods with equal principal stiffnesses. The method also applies to the two-dimensional motion of such rods with unequal principal stiffnesses. For these two classes of problems the equations of the classical theory of rods are reduced to a non-linear vector equation of motion together with the inextensibility condition and appropriate boundary and initial conditions. Consistent finite-difference approximations are introduced and a semi-explicit method of solution is devised. The approximate limitation for numerical stability of the method is shown to be the same as for the usual explicit method in linear beam dynamics. By way of example the method is applied to the free fall of a circular pipe through water onto a rigid plane from a suspended initial configuration.

scholarly journals Predicting Experimental Sepsis Survival with a Mathematical Model of Acute Inflammation

2021 ◽  
Vol 1 ◽  
Author(s):  
Jared Barber ◽  
Amy Carpenter ◽  
Allison Torsey ◽  
Tyler Borgard ◽  
Rami A. Namas ◽  
...  
Keyword(s):  
Immune Response ◽  
Growth Rate ◽  
Inflammatory Response ◽  
Inflammatory Responses ◽  
Initial Conditions ◽  
Mortality Data ◽  
Experimental Sepsis ◽  
Initial Inoculum ◽  
Molecular Pattern ◽  
Parameter Values

Sepsis is characterized by an overactive, dysregulated inflammatory response that drives organ dysfunction and often results in death. Mathematical modeling has emerged as an essential tool for understanding the underlying complex biological processes. A system of four ordinary differential equations (ODEs) was developed to simulate the dynamics of bacteria, the pro- and anti-inflammatory responses, and tissue damage (whose molecular correlate is damage-associated molecular pattern [DAMP] molecules and which integrates inputs from the other variables, feeds back to drive further inflammation, and serves as a proxy for whole-organism health status). The ODE model was calibrated to experimental data from E. coli infection in genetically identical rats and was validated with mortality data for these animals. The model demonstrated recovery, aseptic death, or septic death outcomes for a simulated infection while varying the initial inoculum, pathogen growth rate, strength of the local immune response, and activation of the pro-inflammatory response in the system. In general, more septic outcomes were encountered when the initial inoculum of bacteria was increased, the pathogen growth rate was increased, or the host immune response was decreased. The model demonstrated that small changes in parameter values, such as those governing the pathogen or the immune response, could explain the experimentally observed variability in mortality rates among septic rats. A local sensitivity analysis was conducted to understand the magnitude of such parameter effects on system dynamics. Despite successful predictions of mortality, simulated trajectories of bacteria, inflammatory responses, and damage were closely clustered during the initial stages of infection, suggesting that uncertainty in initial conditions could lead to difficulty in predicting outcomes of sepsis by using inflammation biomarker levels.

scholarly journals Growth Rate of Gravity Wave Amplitudes Observed in Sodium Lidar Density Profiles and Nightglow Image Data

2019 ◽  
Author(s):  
Fabio Vargas ◽  
Guotao Yang ◽  
Paulo Batista ◽  
Delano Gobbi
Keyword(s):  
Growth Rate ◽  
Gravity Wave ◽  
Gravity Waves ◽  
Growth Rates ◽  
Image Data ◽  
Dynamic Parameters ◽  
Density Profiles ◽  
Wave Dynamic ◽  
Lidar Observations ◽  
Amplitude Growth

Amplitude growth rates of monochromatic gravity waves were estimated and compared from multiple instrument measurements carried out in Brazil. Wave dynamic parameters were obtained from sodium density profiles from lidar observations carried out in Sao Jose dos Campos (23°S, 46°W), while all-sky images of multiple airglow layers provided amplitudes and parameters of waves over Cachoeira Paulista (23°S, 45°W). Growth rates of gravity wave amplitudes from lidar and airglow imager data were consistent with dissipative wave behavior. Only a small amount of the observed wave events presented freely propagating behavior. Part of the observed waves presented saturated amplitude. The general saturated/damped behavior is consistent with diffusive filtering processes imposing limits to amplitude growth rates of the observed gravity waves.

Exact Dynamics in a Model of the Bimolecular Reaction A+B→ 0

1992 ◽  
Vol 290 ◽  
Author(s):  
Eric Clément ◽  
Patrick Leroux-Hugon ◽  
Leonard M. Sander
Keyword(s):  
Exact Solution ◽  
Steady State ◽  
Initial Conditions ◽  
Finite Size ◽  
Bimolecular Reaction ◽  
Reaction Model

AbstractWe have previously given an exact solution [1] for the steady state of a model of the bimolecular reaction model A+B→ 0 due to Fichthorn et al. [2]. The dimensionality of the substrate plays a central role, and below d=2 segregation on macroscopic scales becomes important: above d=2 saturation sets in for finite size systems. Here we extend our treatment to give an exact account of the dynamics and show how various initial conditions develop into the segregated and saturated regimes. In certain conditions we find logarithmic relaxation which is related to the dimensionality.

The Richtmyer–Meshkov instability of a ‘V’ shaped air/ interface

2016 ◽  
Vol 802 ◽  
pp. 186-202 ◽  
Author(s):  
Xisheng Luo ◽  
Ping Dong ◽  
Ting Si ◽  
Zhigang Zhai
Keyword(s):  
Growth Rate ◽  
High Speed ◽  
Initial Conditions ◽  
Monotone Function ◽  
Flow Characteristics ◽  
Soap Film ◽  
Vertex Angle ◽  
Initial Growth ◽  
General Behaviour ◽  
Interface Width

The Richtmyer–Meshkov instability on a ‘V’ shaped air/SF$_{6}$ gaseous interface is experimentally studied in a shock tube. By the soap film technique, a discontinuous interface without supporting mesh is formed so that the initial conditions of the interface can be accurately controlled. Five ‘V’ shaped air/$\text{SF}_{6}$ interfaces with different vertex angles ($60^{\circ }$, $90^{\circ }$, $120^{\circ }$, $140^{\circ }$ and $160^{\circ }$) are created where the ratio of the initial interface amplitude to the wavelength varies to highlight the effects of initial condition on the flow characteristics. The wave patterns and interface morphologies are clearly identified in the high-speed schlieren sequences, which show that the interface deforms in a less pronounced manner with less vortices generated as the vertex angle increases. A regime change is observed in the interface width growth rate near a vertex angle of $160^{\circ }$, which provides an experimental evidence for the numerical results obtained by McFarland et al. (Phys. Scr. vol. T155, 2013, 014014). The growth rate of interface width in the linear phase is compared with the theoretical predictions from the classical impulsive model and a modified linear model, and the latter is proven to be effective for a moderate to large initial amplitude. It is found that the initial growth rate of the interface width is a non-monotone function of the initial vertex angle (amplitude–wavelength ratio), i.e. the interface width growth rate in the linear stage experiences an increase and then a decrease as the vertex angle increases. A similar conclusion was also reached by Dell et al. (Phys. Plasmas, vol. 22, 2015, 092711) numerically for a sinusoidal interface. Finally, the general behaviour of the interface width growth in the nonlinear stage can be well captured by the nonlinear model proposed by Dimonte & Ramaprabhu (Phys. Fluids, vol. 22, 2010, 014104).

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