Studies On Dynamics Of Turbulence With Shell Model And Stability Of Spiral Waves In Reaction Diffusion Systems

The studies on nonlinear dynamics compose of two parts.The first,dynam-ics and anomalous scaling of turbulence.The second,stability of spiral waves in Reaction-diffusion systems.The power law of the structure function is an important property of the turbulence.We consider temporal correlations of the turbulence's velocity here. For given temporal-difference structure function,we simply analyze the correla-tion of the velocity on the time serial.With numerical simulation on GOY shell model,we show the temporal scaling relation between the temporal-difference structure function G_n^p(Ï„)and time intervalÏ„with G_pⁿ(Ï„)～τ^(ηn)(P)for short-time interval.ηn(p)is trivial function of the order p and the exponents are equal to each other for different scales.This phenomenon implies the similarity of tem-poral structure on different scales.The relations between velocities is lost for long-time interval,we have G_n^p(Ï„)～k_n^-β(p)for fixedÏ„,and the exponentsβ(p) are typical anomalous function of the order p.With the increasing of the Reynolds number,the fluid transfers from fixed point to chaos state.During these processes,there is a phase transition between quasi-periodic state and intermittent chaos state with a critical valueδ₀.When we add a modulated periodic external force to the GOY model,the phase transi-tion can also be found with a critical valueδ_e.Due to coupling between the force and the intrinsic fluctuation of the velocity on shells of GOY model,the stability of the system has been changed,which results in the variation of the critical value.For proper intensity and period of the force,δ_e is unequal toδ₀.The crit-ical value is a nonlinear function of amplitude of the force,and the fluctuation of the velocity can resonate with the external force for certain period T_e.When we add feedback signal to the force-added shell,the stability of the system is also varied.If the feedback modulation and the intrinsic fluctuation are same (reversed)phase,the stability of the system can be increases(decreases).The phase differences are controlled by the feedback time. The temperature of the fluids is always considered as passive scalar for certain approximation.We introduced a shell-model of Kraichnan's passive scalar problem.Different to the original problem,the prescribed random velocity field is non-Gaussian andδcorrelated in time,which is inspired by She and Léveque. For comparison,we also give the passive scalar advected by the Gaussian random velocity field.The anomalous scaling exponents H(p)of passive scalar advected by these two kinds of random velocity above are determined for structure function up to p = 15 by Monte Carlo simulations of the random shell model with Gear methods to solve the stochastic differential equations.We observed that the H(p)'s advected by the non-Gaussian random velocity are not more anomalous than those advected by the Gaussian random velocity.Whether the advecting velocity is non-Gaussian or Gaussian,same scaling exponents of passive scalar are obtained with same molecular diffusivity.The spiral wave is a temporal and spatial self-organizing map.To understand the stability of its structure is very very important for us.We investigate breakup of spiral wave under no-flux,periodic and an equivalent boundary conditions respectively.When the parameterεis close to the critic value for Doppler-induced wave breakup,instability of the system caused by the boundary effect occurs in the last two cases.This effect results in the breakup of spiral wave near the boundary.With an order measure AOMSW defined by us,we quantify the degree of order of the system when boundary-induced breakup of spiral wave happens.Analyzing AOMSW and the outer diameter R of the spiral tip orbit, it is easy to find that this boundary effect is correlated with large values of R,especially under the equivalent boundary condition.But this correlation is nonlinear,so AOMSW sometimes oscillates with variation ofε.