Autocatalytic reaction—Diffusion model for intermittency in turbulence

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Chaos, Solitons & Fractals Vol. 6, pp. 367-371, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0960-0779195 $9.50 + .00

Autocatalytic Reaction-Diffusion Model for Intermittency in Turbulence A.S. M I K H A I L O V and

D.H. ZANETTE

Fritz tlaber Institute, Max Planck Society Faradayweg 4-6, Berlin 33, Germany Centro At6mico Bariloche, Instituto Balseiro and CONICET 8400 Bariloche, Rfo Negro, Argentina

A b s t r a c t - We present a model of a diffusing chemical species undergoing stochastic autocatalytic (birth-death) reactions. This system develops intermittent population distributions with strongly non-Gaussian statistics. The structure of such distributions is studied by means of numerical simulations, applying the cluster technique used to analyze fully developed hydrodynamical turbulence. We find that the population distributions of this simple birth-death model is strikingly similar to the vorticity distribution obtained in computer calculations on developed turbulence. INTRODUCTION Intermittency is a phenomenon present in the evolution of a wide class of stochastic dynamical systems. It occurs when the relevant fields of the system under study concentrate in small spatial and temporal regions, so that they take relatively high values in those regions and practically vanish elsewhere. In fully developed turbulence, for instance, experiments and numerical simulations show that vorticity and enstrophy (i.e., the rate of energy dissipation) are essentially different from zero only along the vortex lines [1]. These lines evolve in time and space, and can be created or annihilated. Their evolution dominates the behaviour of the whole system. In population dynamics, intermittency is observed when birth and death rates vary randomly in space and fluctuate in time [2]. In this case, a strongly nonuniform population distribution is established, characterized by the presence of narrow spikes that move randomly through the medium. Due to the occurrence of death and birth events, they can suddenly dissapear or spontaneously be created. In 1987, Zeldovich proposed a very simple model for an intermittent population [3]. It consists of a set of disconnected cells occupied by reproducing particles, evolving at discrete time steps. At a given time step the population of each site can be either duplicated or completely annihilated. Both events occur 367

A.S. MIKHAILOVand D.H. ZANETI'E

368

with the same probability

1/2.

This model represents the evolution of a nondiffusing chemical species

undergoing the pair of autocatalytic reactions A ~

2.4, A ~

0. If the initial population per cell equals

one, it is easily seen that, after t time steps, the population at a given site x will be

n(x,t)

= 2 t with

probability 2 -t, and zero with the complementary probability 1 - 2 -~. Although the mean population (n) remains constant, higher-order statistical moments grow with time as (n TM) = 2 (m-l)'.

(1)

This is a direct consequence of the fact that the population becomes concentrated in an increasingly small number of cells. The exponential dependence of the population moments in Eq.(1) on the order m is a typical feature of an intermittent distribution. Zeldovich's model is a particularly suitable starting point for numerical simulations of intermittent phenomena. In this paper we generalize this model by incorporating diffusion. This transport process is able to extend the spatial intermittency of the original model to the temporal domain. We also add a saturation process that insures that the total population in the numerical experiment - w h i c h necessarily involves a finite number of cells- will remain approximately constant, avoiding complete annihilation or population overflows. We study the geometrical features of the population distribution, and find a striking resemblance with analogous results of numerical calculations on fully developed turbulence. NUMERICAL

MODEL

The model used in our numerical simulations is a variant of the probabilistic cellular automata described in [4]. It consists of a one dimensional lattice of L =512 sites, with site spacing A x and periodic boundary conditions. The state of the system at time t is specified by a set of L integers representing the population at every node. The initial condition is

n(x,O) =

n(x, t)

1 for all x.

The evolution of the system is characterized by three parameters, ~'D, P0 and Pa. At each time step, a lattice site x is chosen at random.

Then, a decision is made whether all the particles at that site

will participate in diffusion or reaction. Diffusion occurs with a probability

UD, and

reaction has the

complementary probability 1 - YD. If diffusion is chosen, each particle at the selected node decides whether it will remain at its site, with probability P0, or move to one of the two nearest neighbours, with probabilities (1 -/)0)/2. This diffusion process is characterized by a diffusion length ~ =

Axe/Up(1 --/)0)/2(1 -

/]D).

On the other hand, when reaction is chosen, it is further decided whether the population will undergo annihilation or duplication. The annihilation event has probability Pa. In order to keep the total population approximately constant, the annihilation probability is assumed to depend on the total number

N(t) of particles

in the system as

p,(N)

a + (1 = 1 + (1 -

2a)N(t)/N(O) -

~

'

(2)

with 0 < a < 1/2. In our simulations a = 0.1. If the population becomes too large, the annihilation probability tends to one. On the contrary, it is also reduced. For

N(t)

is always given by 1 - p a .

N(t)

has significantly decreased, its annihilation probability

= N(0) we recover Zeldovich's model, as p~ = 1/2. The duplication probability

Reaction-diffusion m o d e l for intermittency in turbulence

369

1E+O06

]:

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X Figure 1: Intermittent population distribution for A = 1.

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6.0E+005

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,

26o

7

--~

300

400

X Figure 2: Intermittent population distributionfor A = 3 .

500

370

A.S.

ZANETTE

MIKHAILOV and D.H. 10

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<

n(x, t)

is greater than 50. Smaller dots show the sites where

50. Note that the threshold n = 50 corresponds to about 10 % of the total population

concentrated at a single site. The strong nonuniformity in the population distribution is apparent. CLUSTER ANALYSIS In order to analyze the statistics of geometrical properties in the population distributions we use a cluster method employed in studies of developed hydrodynamical turbulence [1]. We choose a certain population threshold h. Then, a cluster is defined as a connected region inside which the local population exceeds h. We introduce the following statistical properties of a generic population distribution: • The volume ratio

v(h)

is the fraction of sites with population higher than h, i.e., the fraction of

the lattice occupied by clusters. • The population ratio s(h) is the fraction of the total population concentrated in the clusters. • The number of clusters

c(h)

is the total number of spikes higher than h.

Since all these quantities depend on the threshold h, they can be plotted parametrically, as functions of each other. The properties of such plots might be used to characterize and compare intermittent distributions of various origins. Figures 3a and b show the dependence of the number of clusters and the population ratio on the volumen ratio, for different values of/~. These are average results over about 500 realizations, at t = 3 x 105. The results of the cluster analysis for our birth-death model can be compared with the data obtained from the numerical solution of Navier-Stokes equation for developed turbulence, on a 2563-site lattice [1]. In this system, the scalar field of the squared vorticity shows inttermitency properties. Figures 4a and

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Figure 4: Comparison between birth-death model (,) and turbulence data (o). b show the comparison between the birth-death model for one realization at ~ = 3 (black dots) and the turbulence problem (white dots, from [1]), where the relevant field is the squared vorticity distribution. As the turbulence simulations differ from ours in size and dimensionality, the comparison of the cluster number required to "extend" our data to three dimensions by taking the third power of both v(h) and c(h), and normalizing the result by a constant factor. CONCLUSIONS Strongly non-Gaussian intermittent distributions are typical of reproductive populations in fluctuating media. The statistical properties of these distributions can be studied using the techniques of cluster analysis employed in hydrodynamical turbulence problems. The cluster statistics of fully developed hydrodynamical turbulence is well reproduced by our simple autocatalytic reaction-diffusion model, under suitable choice of the relevant parameters. This qualitative connection between both systems suggests that some aspects of turbulence could be modeled through autocatalytic processes. Acknowledgement - D.H.Z. thanks Fundaci6n Antorchas, Argentina, for financial support. R E F E R E N C E S

1. T. Sanada, Phys. Rev. A46, 6480 (1991); Prog. Theor. Phys. 8T, 1323 (1992). 2. A.S. Mikhailov, Phys. Rep. 184, 307 (1989); Physica A 188, 367 (1992). 3. Ya.B. Zeldovich ctal., Soy. Phys. Usp. 30, 353 (1987). 4. D.H. Zanette, Phys. Rev. A46, 7573 (1992).