Diffusion Theory of charged Particles in turbulent Magnetoplasma

Plasma Physics and Controlled Fusion, Vol 29. KO. 7, pp 8 2 5 to 856, 1987 Printed in Great Britain

0741-3335 8733 00+ .00 i 1987 I O P Publishing Ltd. and Pergamoii Journals Ltd

DIFFUSION OF CHARGED PARTICLES IN TURBULENT MAGNETOPLASMAS

J. H. MISGUICH,” R. BALESCU,? H. L. PECSELI,: T. MIKKELSEN,; S. E. LARSEN; and QIL XIAOMING §

* Association EURATOM-C.E.A. sur la Fusion, Centre #Etudes Nucleaires de CADARACHE, 13108 Saint-Paul-les-Durance Cedex. France, t Association EURATOM-Etat Belge. Faculte des Sciences. C.P. 23 I . Campus Plaine. Universite Libre de Bruxelles. Belgique. Association EURATOM-Rim Kational Laboratory, P.O. Box 49, DK-4000 Roskilde, Denmark and S: Southwestern Institute of Phqsics, Leshan,

Sichuan, People’s Republic of China

(Receiced 27 June 1986: arid in reciseciforni 15 Jnnuar:, 1987)

Abstract-Diffusion of charged particles in a turbulent, strongly magnetized plasma is considered. The analysis deals with two-dimensional electrostatic fluctuations in the plane perpendicular to an externally imposed homogeneous magnetic field. In the analysis particles are transported in this plane by the gyrocenter drift motion. A nonlinear description is given for absohte diffusion, which yields a Bohm scaling in the case of frozen turbulence or for large amplitudes of electrostatic fluctuations. Particular attention is given to the description of relatire diffusion of two charged particles. This process is described by a generalization of nonlinear Brownian motion, including a first stage of very slow initial relative diffusion. followed by a stage of rapid separation, until a final stage is reached where the particles become uncorrelated and classical Brownian-like diffusion is reached asymptotically. The stage of exponential growth (Lvhich has been measured in fluid turbulence) corresponds to the “ciunip effeci” in plasmas: it is a consequence of nonvanishing statistical correlations between particle trajectories. For a drift-wave turbulent spectrum we obtained an analytical expression for the Lyapunov exponent for the exponential particle separation. a typical feature of chaotic phenomena. The analysis applies to rather general power-spectra for the turbulent electric field fluctuations. Some special cases can be solved analytically. Particular attention is given to a spectral wavenumber subrange, characterized by a spectral index -3, because of its general occurrence: such a subrange is observed for the enstrophp-cascade subrange in ideally two- dimensional fluid turbulence and also in Tokamak plasma drift-wave turbulence. Finally, some practical implications of the results are outlined.

1. I N T R O D U C T I O N ONE OF the most important features of a turbulent medium (fluid or plasma) is its ability to disperse particles at a rate which is many orders of magnitude larger than the one characterizing diffusion by molecular collisions. Two basically different aspects of this problem can be distinguished. One is associated with the mean square displacement of one test particle with respect to its point of release. The other problem concerns the relative displacement of m’o particles. The mean square particle concentration for a cloud of simultaneously released particles can subsequently be calculated by classical methods when the two-particle probability distribution is known (BATCHELOR, 1952).

In the present study we consider the dispersion of charged test particles in a turbulent, strongly magnetized plasma. We consider electrostatic fluctuations in the plane perpendicular to the B-field. As a consequence the cross-field transport of charged particles is due to the randomly varying gyrocenter drift with a local velocity given by v = cE A BIB2, where E = -V@ is the fluctuating electric field and B is an externally imposed homogeneous magnetic field. Finite Larmor radius corrections can, at least to lowest order, be incorporated in such a description, as shown by e.g.

825

826 J. H. M~SGUICH et al.

PECSELI and MIKKELSEN (1985); this small effect will be ignored here and consequently v represents the velocity across the B-field lines for both ions and electrons. With the conditions outlined here our analysis thus deals with two-dimensional diffusion of test particles in a fluctuating plasma, specified through the properties of the randomly varying electric field. The problem thus bears a strong similarity with particle dispersion in two-dimensional incompressible turbulent flows, where the electrostatic potential 0 plays the role of the stream function, and - V2@ of the vor- ticity. Seen in this light, it is realized that for instance the statistical description of the time evolution of a cloud, or just a pair, of test particles applied in meteorological investigations is analogous to the study of transport in turbulent plasmas. Such a bridge between two different domains of turbulence theory appears to be most fruitful.

1 .1 . “Clump eifect” in fluid and in plasma turbulence It is well known that the growth of an initially small cloud of pollen in water, or

of pollutant in the atmosphere, does not only depend on the absolute diffusion of its elements, but also on the relirtice dif;fsion process. The physical reason why ele- mentary objects do not diffuse independently of each other can be understood from the fact that the force exerted on these objects by the turbulent medium has a nonvanishing correlation length: neighbouring objects experience correlated forces and consequently move along correlated trajectories. The effect of trajectory correlations generally result in a strong decrease of the relative separation process-as compared with Brownian-like uncorrelated diffusion-at least for neighbouring orbits which are separated by less than a correlation length. Particles propagate along similar paths, their relative diffusion is reduced: particles are said to “stick together”.

The problem of relative dispersion of test particles in turbulent fluids was investi- gated by ROBERTS (1961) using an Eulerian direct interaction approximation. How- ever, this approximation is not invariant with respect to random Gallilean transfor- mations. The Lagrangian direct interaction approximation described by KRAICHKAN (1966) will remedy this shortcoming on the expense of a complication of the theory. The analysis of e.g. SMITH (1959); SMITH and HAY (1961); CSANADY (1973) and SAWFORD (1 982) bring out the underlying physical mechanism more clearly.

The physical mechanism characterizing this relative diffusion problem can briefly be outlined as follows: fluctuations having a scale size larger than the actual mean square particle separation ( r 2 ( t ) ) + give rise to a simultaneous displacement of both particles, i.e. a propagation along correlated trajectories in space. Only turbulent fluctuations having length-scales smaller than the distance between the particles contribute to their relatice displacement. However, as the particles separate in space, fluctuations with a length-scale which originally gave rise to a bulk displacement of both particles, may at a later instant be smaller than ( r 2 ( r ) ) * and consequently contribute to the expansion instead. In particular it can be argued on intuitive grounds that, as the particles separate, a larger and larger fraction of the spectral energy will be available for their relative diffusion, which consequently proceeds at an increasing rate. Relative diffusion is weak in a first stage, and then becomes an accelerating process. Obviously the spectral shape of the turbulence plays an important role in determining the actual time variation of ( r 2 ( t ) ) * .

This is a very general feature of random media. It is remarkable to note that such a picture already results from the ideas developed for fluid turbulence studies by

Diffusion of charged particles in turbulent magnetoplasmas 827

RICHARDSON (1926), see also CSANADY (1973). This picture is also the basic explanation of the “clump effect” introduced in plasma turbulence theory by DUPREE (1 972) and by KADOMTSEV and POGUTSE (1971). More recently this clump effect has been observed to occur in the relative diffusion of balloons in the atmospheric turbulence (MOREL and LARCHEVEQUE, 1974). It may even be found in the old RICHARDSON data (1926) on atmospheric dispersion (see also LARCHEVEQUE and LESIEUR, 1981). In other words, clumps of balloons have been observed in the atmospheric turbulence: their size extend up to several lo3 km and their life-time last up to 10 days. In the present work, the life-time for clumps of ions for drift-wave turbulence in the experimental conditions of a Tokamak plasma is found to be of 20.14 ms which is a rather long time for microscopic processes.

The time evolution of the separation ( r2( t))* between initially close particles involves various successive time-regimes. One generally observes an initial stage of very slow relative motion, followed by a rapid exponential grobvth of the separation. For times longer than the clump life-time, of course, the trajectories of such initially close particles become decorrelated. Asymptotically the relative diffusion process reduces to the sum of independent absolute diffusion processes. The equation describing the evolution through these various stages has been described for the first time for plasma turbulence by MISGUICH and BALESCU (1982) who treated the case of electrons in an unmagnetized turbulent plasma. The exponential growth has been shown to be a particular example of a Suzuki scaling regime (SUZUKI, 1978 and 1 9 8 4 ~ ) for non-linear Langevin equations. This stage is associated with a positive Ljwpunoc e.uponenf: in that case the chaotic behaviour is due to the effect of reson- ance broadening which results in an overlapping of the successive resonances with the different Fourier components of the turbulent electric field.

In the present work we describe the various stages of relative diffusion for test particles in a turbulent spectrum of electrostatic fluctuations perpendicular to an external homogeneous magnetic field.

1.2. Relative dflusion in magnetized and unmagnetized plasma It is important to emphasize the basic difference between relative diffusion in an

unmagnetized plasma, as discussed by DUPREE (1972) and MISGUICH and BALESCU (1982), and the problem considered in the present work. If we imagine that the turbulent fluctuations are quenched by some device, then all particle motion would be stopped in our problem, i.e. the particles remain on the B-field line they occupied when the turbulence was arrested. The plasma would be confined in the perpendicular direction. Any relative motion could only be due to particle collisions, which are ignored in the present analysis. In an unmagnetized plasma on the other hand, two particles continue to separate along straight line orbits with their relative velocity, even after turning off the fluctuations. The analysis of turbulent transport of charged particles is consequently very different in the two cases. This difference becomes obvious when the results of the present paper are compared with those of e.g. DUPREE (1972), MISCUICH and BALESCU (1982) and SUZUKI (19844.

This important difference between magnetized and unmagnetized plasma turbulence occurs even for uncorrelated absolute diffusion which appears to have very different time dependence in the two kinds of systems. This can be understood analytically in a very simple way. It comes from the fact that in the unmagnetized

828 J. H. MISCUICH et ai.

case there exists in the corresponding Langevin equation a stoachastic force, the electric field: dvidt z E. A diffusion process occurs in the velocity space, with a velocity dffusion coefjcient which asymptotically goes to a constant. This velocity diffusion ( v 2 ( t ) ) % Dot simply yields (x2( t ) ) % Dbt3, i.e. a spatial diffusion which is cubic in time. On the other hand, in a strong magnetic field, the particle motion perpendicular to the B-field is described by the turbulent drift motion of its guiding center. In the corresponding Langevin equation dx,/dt z E A B the stochastic variable is thus the velocity and a Brownian-like diffusion process is found to occur in physical space. As a consequence, uncorrelated diffusion in a strongly magnetized plasma appears to be lineur in time in the perpendicular direction: ( xZ( t ) ) z Dt with a spatial diffusion coefjcient D. In the parallel direction, in turn, turbulent spatial diffusion would remain cubic in time, like in the unmagnetized plasma.

1.3. Non-linear analysis For magnetized plasmas, the first studies describing relative diffusion were derived

by taking the moments of approximate kinetic equations which take clump effects into account. The equations obtained describe the motion perpendicular and parallel to the B-field, see DIAMOND (1979) or HIRSHMAN and DIAMOND (1979). However, only the linear approximation of the diffusion equation were treated explicitly here. This is also the case in fluid turbulence (LARCHEVEQUE and LESIEUR, 1981). A generalization to the study of relative diffusion in sheared magnetic fields has been considered by HIRSHMAN and DIAMOVD (1979) in the framework of the drift kinetic equation, but only within the linear approximation of the equation of motion. More recently applications to Tokamak turbulence have been presented by TERRY and DIAMOND (1984).

Here we focus on the basic equation of motion in order to allow for a test of the nonlinear approximations involved in the analysis. By using a very simple stochastic method on the corresponding Langevin equation, we explicitly solve the nonlinear equations which describe the time-dependence of absolute and relative diffusion. We prove that a linear approximation yields indeed an exponential solution which describes the clump effect, but with a much slower increase than the exact solution: a linear approximation thus seriously overestimates the characteristic clump life-time. We derive and solve here the complete non-linear equation which describes the transitions through different time regimes for absolute and for relative diffusion.

The same method has been used by MISGUICH and BALESCU (1982) for an unmagnetized plasma. Time-dependence of relative diffusion in this case is described by a non-linear third-order equation which generalizes the linear approximation used by DUPREE (1972). The complete equation describes three successive regimes of relative diffusion, in f3, e' and t3. This method (MISGUICH and BALESCU, 1982) treated the problem of the nonlinear equation of motion for test particles in a prescribed turbulent spectrum. Compared to the full self-consistent problem of plasma turbulence (see e.g. BALESCU and MISGUICH, 1984) this represents a reduction which allows one to derive explicit results for a highly nonlinear problem in the framework of well- defined approximations. The validity of the latter can then be tested a posteriori by experimental measurements and, or numerical simulations.


1.4. Diffusion of guiding centers For magnetized plasmas, the Langevin method has been developed in a prelim-

inary work of Qru XIAOMIKG and BALESCU (1982) in the usual case of isotropic, homogeneous and essentially two-dimensional electrostatic turbulence in the plane perpendicular to a constant magnetic field. They obtained a linearized equation for transverse relative diffusion and, by taking trajectory correlations into account, they observed an important decrease of relative diffusion for neighbouring particles. An analytical expression is derived for the characteristic time of relative diffusion over a characteristic wavelength (the clump life-time), which is found to depend logarith- mically on the initial distance.

On the other hand, the same problem of relative diffusion of guiding centers was considered by PECSELI et al. (1982) and PECSELI (1982), by a very different method. These authors used a method originating in the studies of turbulent diffusion of pol- lutants in the environment (SMITH and HAY, 1961, 1968) and further developed by MIKKELSEN (1982). In this description (see e.g. CSANADY, 1972), the velocity autocorrelation function of the exact diffusion equation is modelled by an exponential decay with an appropriate time-scale.

The aim of the present work is to unify the advantages of these two kinds of approaches: the first method is extended in order to include the non-linearity of the diffusion equation, the second method is extended in order to obtain a solvable model valid in the different time-regimes. Moreover we want to take full account of the detailed form of the nonlinearity of the relative diffusion equation: it appears as a,filter effect in the wavenumber space, which selects at each time the spectral energy of structures smaller than the actual size of the cloud. Hence, as the time goes on, a larger and larger part of the energy spectrum contributes to the relative diffusion process. As explained in Section 1.1, this effect is responsible for the rapid acceleration of the process, hence for the clump effect. In order to describe this effect accurately, we thus need an explicit expression of the spectral energy in wavenumber space, as discussed in Section 4.

The organization of our paper is as follows. In Section 2 we outline the basic nonlinear effects in the analysis of absolute diffusion of guiding centers across the magnetic field. From the basic guiding center equation of motion, which is the Langevin equation for our system, we derive the exact diffusion equation by standard methods. The result, which has the form of a Green-Kubo integrand (see e.g. BALESCU, 1975), is known as the Taylor theorem in fluid turbulence studies. By applying simple standard approximations of the Gaussian-type (Corrsin’s factorization and Wein- stock’s second cumulant approximation) we derive a non-linear equation for the time evolution of the absolute diffusion in terms of the turbulent spectral energy in (k , CO) space. In the case of high amplitude of turbulence a Bohm scaling is obtained for the diffusion coefficient. This result is compared with similar ones obtained previously by DUPREE (1967) and for equilibrium plasmas by TAYLOR and MCNAMARA (1971) and by MONTGOMERY (1975, 1976). An explicit expression of this Bohm-like diffusion coefficient is given in Appendix A for the general case of an energy spectrum involving k-’ and k3 power-law spectral subranges, as observed in e.g. Tokamak drift-wave turbulence (TFR GROUP and TRUC, 1984) and discussed also by TCHEN et al. (1980) or PECSELI et al. (1983).

In Section 3 we use the same method to derive our basic equations for the time

830 J. H. MISCUICH ef al.

evolution of the mean square particle separation for an a priori given turbulent spectrum. The Jilter effects appears in a natural way as a consequence of nonlinear effects kept in the analysis. These results are applied in Section 4 to the general k-' and power-law spectrum, resulting in the nonlinear equation (46).

A t this stage the problem can be separated into two coupled equations. (i) The first one, equation (47), describes the nonlinear evolution of the relative

separation in terms of a time-like variable, and involves the clump effect. This equation is solved numerically for various shapes of the energy spectrum, with particular attention to the TFR drift-wave turbulence spectrum. An important difference is found with the solution of the simple linear equation previously used in the literature. The Lyapunov exponent which characterizes the exponential separation of close particles is given analytically.

(ii) The second equation (equation 62a or 67) describes the exact time-dependence of this time-like variable which actually depends on the modelization of the correlation time which characterizes the decay of the autocorrelation of relative velocities, as explained in Section 1.3. Two different models are analysed in Section 5 which allows us to complete the analysis of relative diffusion, and yields an explicit evalua- tion of the clump effect.

Finally, in Section 6 we outline some practical applications of our results with reference to a situation where a cloud of contaminants is released in a turbulent plasma. Extended versions of our analysis are available in report form (MIKKELSEN, 1982; MISGUICH et al., 1985).

2. A B S O L U T E D I F F U S I O N : N O N L I N E A R E F F E C T S We consider a system of charged particles in a constant and uniform magnetic

field B in presence of an electrostatic turbulent field E(x, r ) which is space- and time- dependent. In the limit of a strong magnetic field, the motion of the particles in the plane perpendicular to the B-field lines can be described by the drift velocity of the guiding centers. The basic equation of motion for the study of absolute and relative diffusion is:

where a small correction due to finite Larmor radii is ignored. In this limit, v represents the velocity of both ions and electrons. As a special case, we consider electrostatic fluctuations perpendicular to the magnetic field, characterized by frequencies much below the ion gyrofrequency and spatial scales much larger than the ion Larmor radius; these conditions can be satisfied in strong B fields, for instance in present Tokamak plasmas. As noted previously by MISGUICH and BALESCU (1982) this equation of motion has the form of a non-linear Langecin equation and its formal solution is:

where the non-linearity is due to the space dependence of the fluctuating field, here in


Fourier transform. The position of a test particle at a time t is obtained formally by integrating (1) along the Lagrangian orbit:

x(t) = x(0) + dt’v(t’). (3)

Introduction dx(t) = x(t) - x(O), we usually derive the following diffusion equation for the average value of the square displacement:

d dt

dT(V(t).v(t - 7 ) ) D( t ) (4)

where D ( t ) is the running diffusion coefficient, and the brackets ( ) indicate an average over an ensemble of realizations of the turbulent electric field E. Here we consider a homogeneous and stationary turbulence, hence the average field vanishes (E(x(t), f ) ) = 0, and so does the average displacement: (dx(t)) = 0. The equation (4) yields the Green-Kubo classical expression of the (asymptotic) diffusion coefficient:

D = 2 dz(v(r).v(t - 7 ) ) .

Introducing in equation (4) the normalized Lagrangian velocity autocorrelation

we have:

d dt - (d.x2(t)) = 2 ( u 2 ( t ) ) ( 7 )

In fluid turbulence this equation is known as Taylor’s theorem (TAYLOR, 1922; CSANADY, 1973). For time stationary turbulence R(7, t ) = R(7), and equation (7 ) is rewritten as:

where the quantity ( u 2 ) can be obtained for incompressible flows by a simple Eulerian measurement. In terms of a Lagrangian frequency spectrum @(w) defined by

(v(t).v(t - 7 ) ) = (9)

we have from (8):

832 J. H. MISGLICH et al.

as described for instance by LUMLEY and PANOFSKY (1964). The coefficient of @(CO) in the integrand acts as a time varying frequency filter which accentuates smaller and smaller frequencies as t ---f a. The problem with these very general results is that neither the correlation function R(z) nor the corresponding Lagrangian spectrum @(U) are known, nor easily accessible for measurements.

In order to express the result in terms of an Eulerian wave-number spectrum (which is measurable at least in principle) we evaluate the Lagrangian velocity autocorrelation:

where b B/B and nY. is the volume of the homogeneous three-dimensional homogeneous system under consideration. To proceed we make two approximations:

(i) We factorize the ensemble average in the integrand as the product of two averages. In atmospheric turbulence this approximation is known as the CORSSIN (1959) hypothesis.

(ii) Third and higher order terms in the cumulant expansion of the average of the exponential are neglected (WEINSTOCK, 1969).

The restrictions implied by this Gaussian-like approximation are well-known in plasma turbulence (DUPREE, 1966; WEIKSTOCK, 1969; MISGUICH and BALESCU, 1975; BALESCU and MISGCICH, 1984) and will not be discussed here. Another approach to this problem was described by e.g. P~CSELI and MIKKELSEN (1985).

Introducing Ax(t, z) = x(t) - x ( t - z) we thus obtain

Since Ax(t, r ) =

the second cumulant becomes:

dt’v(t’) = 1; d&(t - 6) we have as before ( A x ( t , t ) ) = 0, while i‘. P r

k . ( A x ( t , z)Ax(t, t ) ) . k = 2 k . J o dz’[“‘’ dQ(v(t - t’)v(t - T’ - Q ) ) . k . (13)

With r(t) E 3(Ax2(z, 7)) = f(sx’(z)) we finally obtain from (4):

where we explicitly made use of the homogeneity and isotropy of the problem in the plane perpendicular to the B field, and introduced:


which reduces to 26(k )Yk, when the electrostatic field fluctuations only involve k, perpendicular to the B field. The equation (14) should be solved with initial conditions r(0) = r'(0) = 0, where the prime denotes differentiation with respect to T .

The simplest case is obtained when the correlation time of the turbulence is very large compared to the time-scale of the exponential (MONTGOMERY, 1976). This corresponds to low frequencies, o r rapid diffusion processes i.e. a large amplitude of turbulence. This limit can be modelled by a frozen flow where the velocity field is a function of the spatial variable only, in each realization of the ensemble. In this limit we obtain from (14):

giving a first integral in the form:

Although the limiting case of a frozen turbulence simplifies the calculation con- siderably it is appropriate to note that there are certain anomalies associated with this case in a more general analysis (KRAICHNAN, 1970).

The diffusion limit is obtained for z + x as T(z) --+ +DT with

This result, which holds for guiding center diffusion in a very low frequency given turbulence, gives rise to a Bohm-type, 1 / B scaling with the magnetic field.

A very similar result D = +DE was obtained by TAYLOR and MCNAMARA (1971) who proved the Bohm scaling to be the general rule for a two-dimensional guiding center plasma moving in its self-consistent turbulent electrostatic field. From the work of MOKTGOMERY (1975) it is easily seen that self-consistency introduces a simple multiplicative factor in Equation (1 4): the frequency dependence for thermal fluctuations is proved to introduce in that case a damping in the form

which reduces the diffusion coefficient by a factor 2. An alternative approximate analysis has been first given by DUPREE (1967). A

result D D = D B , v h is found when the exact non-linear dependence in r is treated in an approximate way: assuming in equation (14) that r = )DDt also holds for short times gives:

834 J. H. MISGUICH et al.

where we used the superscript D for Dupree. Integrating (20) we obtain:

Again for large times the diffusion limit results with

or

where the dependence on DD on the right hand side appears as a renormalization. For k2D much larger than frequencies characterizing Y,,,, i.e. for low frequencies or large amplitudes, one obtains DD = D/$ where D is given by (18). This deviation is due to Dupree’s assumption of a short time diffusion process.

For a general solution of the non-linear equation (23) the frequency dependence of Y,,, has to be specified. In particular, the quasi-linear result is obtained by ignoring the renormalization terms, i.e. assuming k f D D to be very small, which gives

For the simple case where the frequency spectrum only involves eigenmodes w = ok, we have

and, from the position of the poles in the complex plane, the quasi-linear result is obtained as:

DQL = 2 dk,Y,(r = 0) Re s We thus find that there is no unique analysis for single particle diffusion in the

literature, according to the treatment of the nonlinearity (nonlinear or quasilinear diffusion), of the approximate description of the initial nondiffusive regime (which reduces DB by a factor of $), and of the self-consistency (which reduces DE by a


factor 2). We note, however, that in any case the diffusion limit depends on an integral of the entire wavenumber spectrum, weighted with some function of k. This remark holds also for other approaches, mentioned previously (PECSELI and MIKKELSEN, 1985).

However, often most of the energy is contained in the largest scales which can be accommodated in the physical system. For these, the assumption of homogeneity and isotropy could break down and the theoretical results for such systems may become inconsistent with the initial assumptions, since the diffusion coefficient in this case will be dominated by the advective contributions from the largest scales. In Tokamak plasmas, however, the isotropy of turbulence in the plane perpendicular to B is well proved by the measurements (TFR GROUP and TRUC, 1984). For relatice diffusion on the other hand, the physical arguments developed in the introduction indicate that at least in a certain time interval, small scales dominate the time variation of the mean square particle separation. In this sense the latter problem is more tractable for a theoretical analysis. Nevertheless, the results of this section will serve as useful guidelines for the analysis of two-particle relative diffusion, to be presented in the following Section.

3. R E L A T I V E D I F F U S I O N : B A S I C R E S U L T S Investigating the relative diffusion between two charged test particles, we formu-

late the problem the following way: the two particles are released at t = 0 with initial separation r(0) which is the same in each realization of the ensemble. The orientation of the vector r(0) is, however, random i.e. (r(0)) = 0. With x1 and x2 being the particle positions, we introduce the barycentric and separation coordinates as R = +(xl + x2) and r = x1 - x2, respectively. The relative velocity is then g r = v 1 - v2 X I - Xz, i.e. using (1):

C E{x2(t), t)] A b = -AE(t) B A b

Following the procedure outlined in Section 2 we readily obtain

dT(g(t).g(t - 5 ) ) dt

which is the two-particle generalization of equation (4). Introducing the normalized Lagrangian autocorrelation function for the relative velocities R l , 2 (g(t) g(t - 7));

(g2(t)) we have the equivalent of (7)

____- d'r2(r ) ) - 2(g2(t)) sf drR,,,(T, t) dt 0

i.e. a two-particle Tuylor's theorem. The important difference is that the time separation of the two particles is a nonstationary process: even if the turbulence is stationary R1,2 explicitly depends on both 7 and t. The expression corresponding to (1 1) can be expressed as

836 J . H. MISCUICH et al.

(g(t).g(t - T ) ) = -(:I2 $ i d k

[R(f)-R(f-7)] - b A EkcnE.k(t - T ) A b sin {$k.r(t)) sin { tk . r ( t - T ) ) ) . (30)

Using simple trigonometric relations we obtain

k . ( t ) . E-k_(f - T ) cos { t k . Ar(t, T ) } i 2c2 87c3 ( g ( t ) . g ( t - T ) ) - ~ dk(eLk L R ( ' ) "]E B2 $/

[l - {cos (k.r(t)) + sin (k.r(t)) tan (+k.Ar(t, T ) ) } ] ) (31)

where we introduced Ar(t, T ) = r(t) - r(t - T ) . By comparison with (11) we easily realize that the curly bracket in (31) describes the important effect of trajectory correlations between particles i.e. (x1x2) , In particular we have

(t).E-k (t)[1 - cos (k.r(t))])

We identify the cos (k r(t)) term as the one accounting for trajectory correlations and clump effects, similarly to what occurs in unmagnetized plasmas (DUPREE, 1972; MISGLICH and BALESCU, 1982). It should be noted that a simple physical explanation of this term has been given for fluid turbulence by CSANADY (1971).

Following the procedure outlined in Section 2 we split the average in (32) as the product of averages. With the second cumulant approximation also used in Section 2 we obtain

where Yk(0)-see (1 5) with 7 = 0--is independent of t for stationary turbulence. Introducing the Lagrangian time-scale for relative motion

we rewrite (29) as

Together with the initial condition for r(0) this equation forms the basis for the discussion in the following section. The effect of trajectory correlations between particles is responsible for the second term in the bracket, i.e. for the nonlinearity of this diffusion equation. Introducing the two-dimensional spectrum S,_ defined from (15) by Y,(O) = ~ ( c / B ) ~ S ( ~ )Ski, in o rder that 1 d3k,4p, = ( c / B ) ~ d2k,Skl, we find


where we made explicit use of the isotropy and homogeneity of the turbulence. In (36) we see that the bracket acts as a filter on the wavenumber spectrum, suppressing scales larger than the instantaneous mean square particle separation ,/m, in full agreement with the intuitive arguments given in the introduction. Also, a possible deviation from local homogeneity and isotropy for the largest scale will be of no consequence for the time evolutiofl in (36), at least as long as the particle separation is not too large. As apparent in (31) and (32) a similar non-linear effect will enter through tR( t ) . However, as noted already by CSANADY (1971), the time evolution of ( r 2 ( t ) ) is rather insensitive to the exact form of R I 2 ( ~ , t ) in (34). In particular we will demonstrate in the following that t,(f) is simple to model by physical arguments, which consequently complete the equation (36); by this way we avoid a description of the exact and subtle time dependence of the velocity autocorrelation. The spectrum S, also has to be specified. It is assumed to be a priori given just like in Section 2. In agreement with experiments, we assume Sk- to be composed of possibly several subranges having a power law dependence in wavenumbers.

4. D I S C U S S I O N O F T W O - P A R T I C L E R E L A T I V E D I F F U S I O N In this section we will discuss the solution of equation (36) for various models for

tR( t ) and S,(O). Some simple cases can be solved directly. First we consider a short time regime

With this modelling we may solve (36) analytically for 1 > 2 > 3 with the result

where c = -a{(3 - r)l(l - x)}r{(3 - c()/2} and q = 1/(3 - E) where I' here denotes the gamma function.

A k-' spectrum gives rise to a diverging energy integral, implying that this form of spectrum can only be considered as a subrange. For a k-3 spectrum on the other hand, the divergence for k + 0 is too strong to be quenched by the Gaussian filter function in (36). This case was solved by additional approximations by MIKKELSEN (1982). PECSELI et al. (1983). However, this particular spectrum deserves special attention since i t appears to be a generic form for two-dimensional turbulence. In fluid turbulence it characterizes the enstrophy cascade subrange (KRAICHNAN, 1967) and it is also found as a subrange in electrostatic, resistive drift-wave turbulence where the ion dynamics is essentially two-dimensional (TCHEN et al., 1980). I t is interesting to note that, in the latter case, a k-3 subrange can be obtained by simple dimensional arguments (CHEN, 1965), although this approach may be questioned (PECSELI et al., 1983). For the ion dynamics in drift-wave turbulence we may consider the fluctuations essentially two-dimensional and neglect B-parallel electric fields. Propagation along the magnetic field is, however, important for the electron dynamics.

838 J . H. MISGUICH et al.

In order to make our analysis reasonably general we will model here the turbulent spectrum involved in equation (36), according to the theoretical results of TCHEN et al. (1 980) which have received considerable experimental support (MIKKELSEN and PECSELI, 1978; P~CSELI, 1982; PECSELI et al., 1983). We thus consider, in agreement also with experimental observation in the TFR Tokamak (TRF GROUP and TRLC, 1984), a spectral energy involving essentially two subranges in k-’ and k3, respectively. The spectral energy is simply given, in terms of the spectrum S,, defined previously, by 8, = $ V k S , in such a way that the total energy in the system is

where the spatial integration runs over the volume nY. of the system. For simplicity we omit the index 1. here and in the following. We thus consider

and zero elsewhere. Here 6’ characterizes the energy density of the fluctuating field at the intermediate wavenumber k,, while k,,, and k,,, denote the spectral range, including the case k,,, -+ x. The abrupt change in spectral index at k = k , in (39) is of course an artifact of the modelling, but it is without serious consequence in the analysis that follows. We are aware that a power law wavenumber spectrum cannot prevail for k,,, -+ a. For very large wavenumbers we expect an exponential fall-off (PECSELI et al., 1983). The upper limit cutoff at k,,, serves as a substitute for this subrange. For many purposes we may, however, let k,,, -+ x. Theoretical and experimental results of e.g. KATOU (1982) and OKABAYASHI and ARUNASALAM (1 977) indicate that we may safely assume the small-scale motion to be locally homogeneous and isotropic in two dimensions perpendicular to the externally imposed homogeneous magnetic field. Such an isotropy of the turbulence spectrum is also observed in measurements in the T F R Tokamak.

The spectrum (39) is characterized by the k-’ and k-3 subranges, of respective widths p k,,,/k, < 1 and 11 = k,/k,,, < 1. The simplest case is that of a k-3 spectrum ( p = 1) of infinite width (v = 0). We define an energy parameter

6 = 6’(l - In p 2 - \ i 2 ) (40)

and a mean square electric field such that E’ (8nli”)E = (B/2ck,)’6. The mean square turbulent drift velocity ud = cE/B allows us to introduce the characteristic frequency

which is related to the energy parameter by

6 = (2k,cE/B)’ E 4Q2.


An average wavevector k , is introduced as

For the present spectrum we find

(43)

The analysis absolute diffusion under the influence of the spectrum (39) is presented in Appendix A. As in Section 2, we have explicitly considered the case of large amplitude (or low frequency or Jkozen turbulence). In this case the time evolution of the running diffusion coefficient is obtained as the numerical solution of a first-order differential equation. The solution asymptotically approaches a Bohm-like value for the diffusion coefficient. An analytical result is derived for this asymptotic diffusion coefficient. It is found to depend on the width of the two spectral subranges of the spectral energy (39). For more details we refer to MISGUICH et ul. (1985).

By virtue of the properties of the model spectrum in equation (39), we may rewrite the relation (36) for the problem of relutiw diffusion as

The integrals can be expressed in terms of the exponential integral functions (ABRAMOWITZ and STEGUN, 1970) and (45) is rewritten as

1 - In$ + E , ( Z ) - El (p2Z) - E2(Z) - 1’’ i where E,,(Z) = JF d t e-”]tfl. In particular E 2 ( Z ) = e-’ - ZE,(Z) , and


where %? = 0.577 2 1 6 . . . is the Euler constant. We note that the expansions of E, and E, are non-analytic at the origin; nevertheless ZE,(Z) --f 0 for Z -+ 0.

Formally, we can remove the Lagrangian time-scale for relative motion t,(t) from (46) by introducing the time-like variable 8(t) = !2’ lh dt’t,(t’) and obtain

with the initial condition at Q ( r = 0) = 0

Z(Q = 0) = $kir’(O) Z,. (48)

The last term in (47) accounts for the “clump effects” i.e. the trajectory correlations which are responsible for the deviation from a simple linear relation between Z(8) and 8. For the moment we ignore the transformation from 19 to t since it does not enter the relation (47) explicitly, and consider first the general variation of Z = Z(0) as given by (47).

4.1. Approximate analytical results

1 ‘k, and 1 ik,,, giving the corresponding scales for 2: The model spectrum (39) is characterized by three length-scales, namely I/k,,,,

The small cloud regime 8 < Q1 only exists for sufficiently small initial distances (smaller than, roughly, the smallest wavelength), i.e. Z, < 3v’. This can only be satisfied with a finite k-3 subrange i.e. v > 0.

More generally, the initial regime is defined by 8 < d,, i.e. as long as the distance remains smaller than-roughly-the intermediate lengthscale: this is the relevant regime for the clump effects which can be described as follows. Considering the initial evolution of neighbouring particles, we may approximate (47) for 8 < e2, i.e. Z < j b y

(50)

where we introduced 8’ = O/(l - In pz - v’) for simplicity. In the particular simple case of an infinitely wide k-3 subrange v = 0, this equation reduces to

where p ( p ) 2 - V - pz with the solution

Diffusion of charged particles in turbulent magnetoplasmas 84 1

In the limit 8' -+ x we find that Z,, -+ eP(p). This saturated value is, however, outside the region of validity for (51) which requires Z,,, < 3. We now define an average clump life time BCl as the time at which the r.m.s. particle separation becomes comparable to the characteristic length-scale 1 'k*. Considering one Cartesian component on the mean square separation. ( r , ' ) = 4 ( r 2 ) , we may for instance define QCl by ( r ; ( O c , ) ) = kG2, corresponding to

z(e,,) = 4. ( 5 3 )

Assuming that the solution Z,, holds at least approximately up to the extreme end 8 = O2 of the initial regime (this will be checked numerically in Section 4.2), we find (for 11 = 0)

This important quantity, which is indeed positive for Z, < 3, is shown in Fig. 1 as a function of log Z, in the case of a k-3 spectrum extending to infinity (11 = 0. p = 1). Note that an additional k-' subrange (i.e. ,D < 1) actually enhances the clump lifetime for the same energy.

We emphasize that the approximate initial solution (52) differs significantly from the one obtained by the simple linearized version of equation (51)

dZ,;dO' = ( 2 - V - p2)Z, ( 5 5 )

which gives a simple and much slower exponential growth in contrast with the one given by (52) . Such a linearized approximation, often used in the literature, would actually exaggerate the growth of the corresponding clump life-time and would give 8i(Z,) = (2 - $? - p2)-l In (+Z,), which is also drawn in Fig. 1.

7

6

5

- L

3 U a

10 - 8 - 6 - L - 2 0 log zo

FIG. 1.-Dependence of the clump life-time O,, on the initial distance Z,, derived from the analytical solution (equation 54) in the initial regime ( p = 1 , v = 0) and compared with the

dependence Qfi(Z,) given by the linearized equation ( 5 5 ) .

842 J. H. MISCUICH et al.

The above results emphasize the case v = 0. When this condition is relaxed in order to describe a finite-width k-3 subrange, a new smallest wavelength appears in the problem, namely l/kmax corresponding to (49) in terms of the normalized quantity Z. This allows the appearance of a new “small clump” regime, which is a “scaling regime” in the sense of SUZUKI (1978) as explained by MISGUICH and BALESCU (1982) or SUZUKI (1984~7, b). For 0 < O , , i.e. Z < )v2, we may expand the r.h.s. of (50) to obtain

~ (56) dZ, - 1 - p 2 - In v 2 d0 1 - l n p 2 - v 2 ZS

-

where the logarithmic term has been exactly cancelled. The solution describes the relative diffusion of an initially small clump since it holds for Z, < )v’:

Zs = Z, e2xo. (57)

The quantity defined in (43) plays the role of a positive Lyapunov exponent in the plane perpendicular to the B field, which is also the phase space of the corresponding Hamiltonian system. The solution (57) remains valid up to times

o1 =-1n(’j 1

2%

Although (58) strictly speaking requires a finite width of the k-3 subrange i.e. 1’ > 0, this solution for small values of \J is, however, not very different from the first growing stage of the initial solution Z,, (equation 52) which has been derived for 1 1 = 0.

This series of analytical solutions in the various time-regimes has to be completed by the asymptotic solution which holds for 0 > 0 3 . In this limit we obtain

= 1 dZ, d0 (59)

i.e. Z,, = Z, + 0 z 0. This result simply demonstrates that for very large times the two particles diffuse independently of each other, i.e. all collective trajectory correlations are lost when the distance exceeds the largest turbulent length-scale. The relation (59) is independent of the specific values of p and v.

4.2. Numerical solutions Solutions of the non-linear relative diffusion equation (47) are obtained numeri-

cally. Figure 2 shows the evolution of Z(0) for two initial conditions Z, in a k-3 spectrum extending to infinity (p = 1 , $1 = 0). An important conclusion from this figure is that the analytic expression for the initial evolution Z,, as given by (52) is an excellent approximation up to 2 z 3. This implies that the expression (54) for the clump life-time e,,, shown in Fig. 1, is actually very accurate. Although frequently used in the literature, the simple solution Z , of the linearized equation (55) is found to be inadequate for representing the initial growing stage of the relative diffusion for small initial distances.


3 " " ' 2 -

-7 -6 -5 - L - 3 - 2 -1 0 1 2 3 L log e

FIG. 2.-Numerical solution Z(0) of equation (47) for two different initial conditions 2,. Dotted line shows the initial regime Z,,,(O). interrupted line shows the linear solution Z,(O), while the dashed line represents the uncorrelated reference Zc. The spectrum parameters

are v = 0 and i( = 1.

In order to emphasize the non-linear effect of trajectory correlations we represent by a dashed line in Fig. 2 the evolution Z , which would result if the two particles diffused independently from the very beginning. This result is obtained by neglecting the entire nonlinear contribution 9JZ) in equation (47). As expected we find that the numerical solution of the complete equation (47) asymptotically joins in a smooth way to the uncorrelated solution 2, for large values of 8.

In Figs. 3 and 4 we present the numerical solutions of (47) for a full spectrum with k-' and k-3 subranges. The importance of a k-' subrange in the energy spectrum is seen in Fig. 3 which presents the diffusion curves for a wide k-3 spectrum with and without a k-' subrange (i.e. ,/,i = 0.6, v = 1 and p = 1, v = 0, respectively). It is found that by introducing a k-' subrange in the spectrum, we also enhance the clump effect. Similarly, it has been shown by MISGUICH et a/. (1985) that by introducing a finite width of the k-3 subrange (v > 0), we also increase the clump effect as com-

-L

-5 -6

-7 -6 -5 - 4 -3 -2 -1 0 1 2 3 log e

FIG. 3.-Enhancement of the clump effect on Z(0) In a k+ spectrum ( v = 0) when the width of the k-' subrange increases ( p = 1. 0.6 and 0.1).

844 J. H. MISGUICH et ul.

31 I I I I I I I I I n

2

1

ol ,o - 2

-3 -L

-5

-6 -7 -6 -5 - 4 -3 -2 -1 0 1 2 3

log e

FIG. 4.-Numerical solutions Z(0) for the spectral parameters v = 0.3125 and p = 0.4 corresponding to recent TFR (1984) results with k, L 12.5 cm-’. The case v = 0 and p = 1

is included for reference, along with the initial approximation Z,,,,,

pared to the case of an infinitely wide spectrum (v = 0). In other words, both realistic properties of the observed spectrum in Tokamak plasmas for instance (,U < 1 and v > 0) actually result in an enhancement of the clump effect as compared to the simple model spectrum (v = 0, ,U = 1).

In Fig. 4 we present results for a full spectrum where ,U = 0.4 and \) = 0.3125. This particular spectrum is chosen as an adequate representation of recent measurements in the TFR Tokamak as presented by the TFR GROLP (1984). In this case of a relatively narrow spectrum, the clump effect is particularly important. The clump effect is increased by an order of magnitude as compared to the case of a k3 spectrum extending to infinitely small wavelengths. Explicit values are given in Section 5.

5. M O D E L L I N G O F T H E L A G R A N G I A N T I M E SCALE FOR RELATIVE D I F F U S I O N

In the previous section, the evolution of the mean square particle separation was described in terms of a time-like variable 0, defined in equation (47). In the present section we shall demonstrate that a convincing model for the Lagrangian time-scale for relative diffusion can be obtained by physical arguments, which consequently provides the relation between Q and time. Rather general discussions are presented by MISGUICH et al. (1985). Here we concentrate on two different models.

5.1. Constant Lagrangian memory model The normalized Lagrangian autocorrelation function for the relative velocities

describes the memory-loss of the relative velocity g(t’) at successive times. Thus R,,,(T, t ) is sharply peaked around T = 0. We assume that this function can be written as a dimensionless function of just one dimensionless time variable. To be explicit we let this function be an exponential although it is not essential for the argument, i.e.

R,,,(z, t ) = e - m (60)

with R,,,(O, t ) = 1. The function tR( t ) can then readily be obtained as


where we note that r ( t ) z ,( t) / tR(t) > 1 for all t , i.e. the microscopic correlation time z, is always larger than the Lagrangian time-scale tR in the present model. The relation (61) can also be formulated as

We thus find that t , is always smaller than, or at most equal to, the real time t, in this model. The time-like variable 8 in Section 4 can be expressed as

In order to proceed we have to specify z,(t). The simplest, but still realistic, model turns out to be a constant value for 7,. Then (61a) reproduces the main features argued by CSANADY (1973) and MIKKELSEN (1982), namely a linear regime tR( t ) z t for t 6 z,, and an asymptotic saturated stage tR ( t ) z z, for t 9 7, . From (62a) we now obtain

(63)

Obviously T , is not just an arbitrary constant. We know that the relative diffusion coefficient asymptotically, for t + x, reaches twice the absolute diffusion coefficient, D. This implies

2

z, = 46) D where we used (59) with (63) in the limit of large t .

5.2. Size-scaled memory model In this section we consider a second model

giving

t R ( t ) = .rc(t) erf ~ (2;)

846

and

J. H. MISCUICH er al.

where erf ( ) denotes the error-function. The numerical coefficient n/4 is chosen for convenience, to normalize the total integral of this function to that of other models. To estimate ~ , ( t ) we use dimensional reasoning to obtain

where I is a length-scale and I; a velocity characterizing the problem. It is reasonable to assume on physical grounds that the memory of an expanding cloud of test particles actually grows, similarly to turbulent vortices, with the size of the cloud, and finally saturates when this size reaches some characteristic wavelength ic to be specified later on. The only obvious choice for I’(t) is ( r 2 ( t ) ) itself before the memory saturation ( t < tSaJ and if afterwards. The characteristic velocity v is the r.m.s. velocity associated with that part of the spectrum which contributes to the relative particle separation, i.e. v2(t) = (g2 ( t ) ) . Consequently

or, in terms of the dimensionless variable Z

Using (29) and (34) we rewrite (69b) as

2Q-2 d In 2’ T f ( f ) = ~

d0

This result holds for times smaller than the saturation time t,,, at which (r2(t,,,)) = i:, afterwards the quantity I has to be limited by the characteristic scale size 2, which remains to be specified later on.

An important feature of the present model is the coupling between T , and the time evolution of Z: the memory is size-dependent, as described by the set of equations (47), (67) and (69c). To describe the time-variation of z, we first note that according to the definition (69) it has a non-vanishing initial value given by

Diffusion of charged particles in turbulent magnetoplasmas

TABLE 1 .-VALUES OF i f c (0 ) I N THE SIZE-SCALED MODEL FOR VARIOUS COMBINATIONS OF THE SPECTRUM

= 1 .oo 0.60 0.40 0.10 0.06

v = o 0.41 0.52 0.61 0.84 0.9 1

I = 0.001 0.19 0.26 0.3 1 0.44 0.47 0.0 1 0.23 0.32 0.38 0.52 0.57 0.04 0.28 0.38 0.44 0.61 0.67 0.08 0.31 0.42 0.49 0.68 0.74 0.10 0.33 0.44 0.51 0.7 1 0.77 0.20 0.39 0.51 0.59 0.81 0.88 0.3125 0.44 0.57 0.66 0.91 0.99

847

PAKAMETERS /.I AND V

These values represent the minimum necessary value of the absolute diffusion coefficient D in order that the size-scaled model of Section 5.2 be different from the constant Lagrangian memory model of Section 5.1., i.e. in the domain of high amplitudes of turbulence.

From the short time evolution equations (50) and (51) we determine the denominator as function of p and v, i.e.

1 - l n p 2 (for v = 0)

2 Q2 (2 -

t,2(0) = - - p 2 ) - In Z,

and

2 1 - l n p 2 - v 2 1 Q2 1 - p 2 - In v2

-- - xQ2

(for v 2 > 22,). 2 r,(O) = -

In Table 1 we present the normalized values @2t,(O) +?,(O) for different combinations of (p, v) for log Z, = -6 obtained using (71a) and (71b). We observe that .r,(O) is of the order of l /Q for a wide range of spectral parameters. Asymptotically for t + x we know that dZidQ -, 1, see (59). In this limit we identify dZidt with i k i D , corresponding to ( r 2 ( t ) ) = 2Dt, see Section 5.1, where D again denotes the absolute, or single particle, diffusion coefficient discussed in Section 2 and Appendix A. The saturation value for the microscopic Lagrangian correlation time, in the size scaled memory model, is consequently given by tSat = $kiDQ-2; in dimensionless units, b $k:D/Q, this means

It corresponds to the scale size

Physically E., corresponds to the integral length scale, i.e. for scales larger than this all trajectory correlations are effectively lost. Using this identity together with (69c) we obtain an implicit equation for tSat: it is the time at which the time-dependent correlation time .I.,(t) reaches its limit value 2 d :

Tt,(t,,,) = J2de/dlnz = 2 0 . (74)

848 J. H. MISCUICH et al.

1 .o

0.5

0 * ~.r = 0.1 D 4.21

-7 -6 -5 - 4 -3 - 2 -1 0 1 2 3 log

FIG. 5.-Time variation of the correlation time 7, versus reduced time t , = 2Dt in the size- scaled memory model of Section 5.2.

Examples of the time-variation found, in this model for the correlation time, are given in Fig. 5; to be specific we have fixed the absolute diffusion coefficient to its Bohm-like value D&, v) given in (A.5) for p = 1 and 0.1. The dimensionless correlation time Z, = QT, is plotted as function of the reduced time t , = 2DRt: from its initial value (71) it grows according to (69) up to its saturation value (72).

This condition (74) also determines the specificity of the model as compared with the constant Lagrangian memory model of Section 5.1. A growing correlation time is found in this model only when this limiting value is larger than the initial value Z,,, 25> 2 Q O ) , which is given in Table 1. This necessarily implies that the model discussed in this section is specific only for high amplitudes of turbulence such that the absolute diffusion coefficient b is larger than the values of Table 1. When this requirement is not fulfilled, the model described in Section 5.1 will suffice.

Considering for instance the TFR spectrum ( p = 0.4 and v = 0.3125) we see in Table 1 that the present size-scaled model differs from the constant Lagrangian memory model only at high turbulence levels such that d 20.66 [which is almost half the Bohm value (A.7) for such a spectrum]. Since the experimental value of d, inferred from global energy balance,* is about d = 0.02 in such conditions, we conclude that the constant Lagrangian memory model is sufficient in that weak turbulence case.

This spectrum is explicitly taken into account in Fig. 6 where the solution of the coupled equations (47) and (63) is shown as a function of the dimensionless variable 7 = at for various values of D including the value estimated in the TFR experimental conditions. In this latter case trajectory correlations can be seen up to a time F Z lo3, i.e. up to ~ 0 . 1 4 ms which is a rather long time for microscopic processes. Clump effects are thus far from negligible in experimental Tokamak conditions. The characteristic clump life-time (at which Z = 3) is found from Fig. 6 to be 7 % 240, i.e. t z 33 ,us for this value (Z , = of the initial distance. Using 7 = Q/(26), the

*The experimental conditions are (TFR GROUP and TRUC, 1984): k , = 12.5 cm-', B = 4 Tesla, = 5 1013 ~ m - ~ , T = 1 keV, the turbulent energy density = 2.5 lo-' of the thermal energy, hence

E = 2.36 v cm-', ud = 5.3 lo5 cm ss' and the macroscopic drift time T 2niR = 0.95 p s ,


6

4

2 - " N O - E?

-2

- 4

-6 -6 - 4 -2 0 2 L

log T

e =l.LLS(Bohm)

10.02 (exp.) D =0.2

I3 z0.002

FIG. 6.-Numerical solution for Z($ for the TFR parameters, see Fig. 4. Note that the abscissa is f i t here while i t is the time-like variable 0 in Fig. 4. With the value d = 0.02 inferred from TFR experiments. the clump effect is seen to last up to 7 5 lo3.

order of magnitude of this result is confirmed by the analytical approximation (equation 54) which holds, however, only for 1' = 0. Nevertheless, this formula clearly indicates the dependence of the clump life-time on the initial distance Z,. The dependence on Z, is further illustrated in Fig. 7 which gives the time-behaviour of Z for various initial values Z,. Clearly smaller clouds have a longer clump life-time.

In order to illustrate the difference between the two models summarized in Section 5.1 and 5.2, we present a comparison in Fig. 8 where the value (A.5)

I 1 1 + v 2

is used. This Bohm-like value of the absolute diffusion coefficient is obtained in Appendix A by using the spectrum given in (39) with (40) in the analytical result

6

L

2

8 -2

- L

-6

-8 -6 - L - 2 0 2 L

log T

FIG. 7.-Time evolution of Z(i) for parameters as in Figs. 4 and 6, for varying initial conditions Z , and d = 0.02: the smaller cloud has the longer life-time.


31 I I 1 I I I I I I

2

1

0

r. -1

m -2

-3

- L

-5 -6

I

n

N

-

-7 -6 -5 -L -3 -2 -1 0 1 2 3 log t o

FIG. 8.-Relative diffusion of a small cloud ( Z , = in an infinitely wide k3 spec_trum ( v = 0 and p = 1) shown as a function of the energy scaled time-variable t , = 2Dt. For small values of the intensity of the turbulence (d = and lo-') the results in this representation are almost not sensitive to the exact value of the absolute diffusion coefficient.

(18). The comparison between the two models is most conveniently expressed in terms of the reduced time variable

tD E 2DF= 3k;Dt (76)

The advantages of this scaling becomes clear when we note that, in terms of this energy-scaled time variable, the asymptotic relative diffusion becomes Z,, = t D and is thus independent of the intensity of the turbulence. Moreover, we find as shown in Fig. 8 that for weak turbulence (D 5 0.1) the relative diffusion curve is rather insensitive to the exact value of the absolute diffusion coefficient D.

Referring to the numerical solutions presented in Fig. 9 we note that the size- scaled Lagrangian memory model predicts a slower increase in relative diffusion, i.e.

3 2

1

- 0 - -1

E? -2

a 4-

N

- -3

- 4 -5 -6

-7 -6 -5 - L -3 -2 -1 0 1 2 3 log tD

FIG. 9.-Time evolution of Z(t,) for the two Lagrangian memory models denoted by A (Section 5.1) and B (Section 5.2) in the case of a wide k-3 spectrum with (p = 0.1, I' = 0) and without (p = 1, L' = 0) a k-' subrange. The amplitude of turbulence is such as to yield the corresponding Bohm-like value (AS) of the absolute diffusion coefficient with these parameters. The lower value of the microscopic correlation time in model B at short times is

responsible for the observed decrease of relative diffusion.

Diffusion of charged particles in turbulent magnetoplasmas

- r. 1 N

ZO’

85 1

! -.-,_ - - -.-A - - -

distant particles // ! / I

z* I zo/eQ’t‘ j zn zo &QiW I

/ close particles initially close particles I d m p regime for

/’ /

an enhanced clump effect. The difference between the two models is however not particularly dramatic, even at the highest value of the absolute diffusion coefficient

Hence, we may apply the simple constant memory model of Section 5.1 for an overview of the time evolution of ( r 2 ( t ) ) , as summarized in Fig. 10. We distinguish four regimes in a ( Z , t ) plane. The path describing the time evolution of 2 = Z(t) in this plane is essentially determined by the initial value Z,, for given parameters characterizing the spectrum. For initially close particles, i.e. Z, < $e-(4R*~1’, we find Z = Z, e(nf)’ in region I for small times t < 27,. For later times we have Z = Z, exp (4!227,t) in the clump regime I1 characterized by an exponential separation of nearby trajectories (Lyapunov exponent x > 0). Finally Z enters region IV for independent diffusion of the two particles. In this region the distance between particles is much larger than the largest scalelength associated with the turbulent flow, while the time after release is larger than the corresponding time scale. For initially distant particles on the other hand, 2 evolves through region 111 into the region IV characterizing Brownian relative diffusion.

6. C O N C L U S I O N In this work we presented a simple nonlinear theory for absolute and relative

diffusion of test particles in a two-dimensional electrostatic turbulence, where the propagation across an externally imposed homogeneous strong magnetic field is described by the E A B gyrocenter velocity.

For this case we derive diffusion equations in the form of Green-Kubo integrals of the velocity autocorrelation functions. The approximation used in order to derive soluble equations are the Corrsin factorization approximation and the Weinstock second cumulant approximation.

Absolute diffusion is obtained as the asymptotic stage of evolution of the nonlinear equation (14). In the particular case of low frequency (or large amplitude or frozen turbulence), we explicitly derive an absolute diffusion coefficient which has a Bohm scaling in 1IB. The turbulent power spectrum of the electric field fluctations enters

852 J. H. MISGCICH e t al.

the analysis as an U priori given quantity. Particular attention is given to a model for locally homogeneous and isotropic spectrum characterizing drift-wave turbulence as it occurs in present day Tokamak devices. This spectrum, taken from the TFR Tokamak experiment, involves k-' and k-3 subranges in the spectral energy. An explicit value is then obtained for the Bohm-like diffusion coefficient which appears to depend on the form parameters characterizing the widths of these subranges of the spectral energy.

The same method has been generalized to the study of relative diffusion. Nonlinear dynamic equations are derived to describe the time evolution of the mean square separation between two released particles. The nonlinear equation (36) has been completed by two different models for the relative Lagrangian correlation time, resulting in the explicit coupled equations (47) and (63) or (67). These two models apply to weak and strong turbulence, respectively. The advantage of using scaled variables in the analysis was demonstrated. Different time regimes were identified for the time- variation of the relative particle diffusion. The results emphasize in particular the importance of the nonlinear effect of trajectory correlations (or the clump effect) for the exponential separation of nearby particles. In the case of the TFR experimental parameters, it is found that the clump life-time reaches 0.7 ms, which is rather long for a microscopic effect.

We are not aware of any existing experimental results where our analysis can be applied. However, we believe that at least the two types of experiments to be discussed in the following can serve as future test of the theory.

6.1. Barium cloud releases in the ionosphere Artificially released Barium clouds have been used for many years in the investi-

gation of the properties of the ionosphere in several altitude ranges. A large fraction of the released material is ionized almost instantaneously by the u.v.-radiation from the sun. The position and size of both the ionized and neutral cloud components can be detected optically. A d.c.-electric field is usually present in the ionosphere, giving rise to an E A B drift of the ionized cloud, ultimately separating it from the neutral remnant.

Actually, the main purpose of these Barium releases is simply to estimate the electric fields by following the drift of the ionized material, i.e. the average motion of the cloud. For this reason the Barium density is kept as low as possible (the lower limit being determined by the optical detection system) in order to ensure that the electric field, determined by this method, is characteristic for an ionosphere un- perturbed by the presence of the cloud.

Although the main scope of these investigations is to follow the center of mass motion, the expunsion of the ionized cloud is also discussed by e.g. LLOYD and GOLOMB (1967). The scale sizes for the cloud, both in the directions along and perpendicular to the Earth's magnetic field, can be rather accurately determined as a function of time after release by using the star background as a frame of reference. The cloud expands very rapidly along B as expected, actually in good agreement with a theory for a simple ambipolar diffusion coefficient. However, the diffusion coefficient perpendicular to the B-field lines is much larger than expected from such a simple theory, and observed values exceed the simple Bohm diffusion coefficient (derived from a spectrum of thermal fluctuations) by almost an order of magnitude.


It is thus likely that the perpendicular expansion rate is controlled by plasma turbulence.

The reported value of the effective cloud radius perpendicular to B scales very accurately as 4, corresponding to a classical Brownian relative diffusion. An exponential regime may be present initially. However, during this time, unfortunately, the ionized and neutral clouds overlap and no unambiguous detection of the growth rate is possible. We hope that future experiments with Barium cloud releases, where particular attention would be given to the expansion across magnetic field lines, will provide evidence also for the initial evolution. It is worth emphasizing that such results may actually provide a useful information on the power spectrum of the fluctuating part of the electric field.

6.2. Pellet refueling of hot laboratory plasmas The possibility of refuelling a fusion reactor by injecting solid deuterium pellets

has been considered for some time; the work in this field is reviewed by e.g. CHANG et al. (1980). As such a solid pellet (typically at 24°K) enters the plasma, a shielding blanket of a dense neutral gas develops. Subsequently, the outer layer of this gas becomes ionized by the bombardment of the hot plasma. Thus as the pellet propa- gates through the magnetized plasma together with its shielding neutral cloud, a trace of ionized cold pellet material is left behind. This additional component of the plasma will expand along the magnetic field, and eventually be distributed on the entire magnetic surface. This, however, requires that the plasma ring is traversed many times in the toroidal direction. Turbulent electric field fluctuations perpendicular to the magnetic field may greatly enhance the distribution of this plasma across the surface. Actually, it is well-known that turbulent electrostatic field fluctuations are observed in Tokamaks, especially at the edge of the torus.

It is still an open question whether the expansion of the trail left behind the pellet is experimentally observable. An optical detection seems most promising. In order to take advantage from this method, i t will be necessary to add some tracer material to the pellet: Neon seems to be a good candidate since it is easy to handle and not naturally present in a Tokamak plasma.

The condition that the pellet material does not introduce any major perturbation in the plasma, may not be satisfied in present day experiments since the number of atoms in the pellet is comparable to the total number of ions in the entire Tokamak plasma, and a strong cooling process is observed. An increased injection velocity will, however, reduce the density in the trace behind the pellet, thus making the experiment feasible. The question of how the plasma originating from the pellet is distributed in the torus is necessarily of importance for the design of an optimum pellet injection scheme. The same argument obviously applies to any fragment of foreign material entering the plasma.

In addition to experimental results, we expect that a numerical simulation of the problem may provide an important insight into the accuracy of the approximations involved in our theoretical analysis.

Acknon,iedgemerits-We thank N. AUBY for assistance in solving some of these equations numerically. We are indebted to Mrs M. LAKCHEVEQUE, P. TERRY, M. PETTINI and A. VLLPIANI for several important discussions and to Dr .I. OLIVAIN, A. TRUC and H. DRAWN for useful informations about TFR measurements. The kind interest of C. M. TCHEN is gratefully acknowledged.

854 J. H. MISGUICH et a1

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A P P E N D I X A In Section 2 we have derived the absolute diffusion equation (14). along with the Bohm-like diffusion

coefficient (18) which holds in the case of high amplitude or frozen turbulence. By using the same tech- niques as in Section 4, we apply here these results to the specific case of the TFR energy spectrum (39) with k1 and k-3 subranges.

With (39) the evolution equation (14) takes the form:

in the case of high amplitude or frozen turbulence s ~ ( T ) z s k ( ~ = 0). Introducing the dimensionless quantities T ( T ) = #:r(s), ? = QT and X = (k,:k,)', we obtain

This can be written in terms of the exponential integral functions introduced in Section 4

with the initial conditions r(0) = t'(0) = 0. By using the recurrence formula

equation (A.2) can be written

which exactly reduces to the first order equation

in which the nonlinear contributions are given by the functional

The reduction from (A.3) to (A.4) is the mathematical explanation for the appearance in frozen turbulence of a Bohm scaling of the diffusion coefficient (see MONTGOMERY, 1975). Asymptotically indeed, . F ( f ) goes to zero and the asymptotic diffusion coefficient is simply

/ 1 1 f Y4

This is the dimensionless form of the diffusion coefficient (18) which has a Bohm scaling in E'B:

E E

b, - C S I -+ B - n - ~-

We note that its value (A.5) explicitly depends on the width parameters p and v of the two subranges in the energy spectrum (39). In the simple case of a k - 3 spectrum extending to infinitely small wavelengths, we have

856 J. H. MISGLICH et al.

1 &(p = 1, v = 0) = -

vh while for the TFR experimental spectrum

D E ( p = 0.4, t' = 0.3125) = 1.45. (A.7)

The nonlinear evolution equation (A.4) has been solved numerically. The running diffuusion coefjcient B( i ) reaches a plateau value in time, which depends on p and 1' as described by (AS). This plateau corresponds to the fact that the velocity autocorrelation function rapidly decreases to zero. Thus, after an initial regime, the mean square displacement r grows linearly with time, as expected for a classical Brownian-like diffusion process.

We note that both (i) the plateau value representing the Bohm-like diffusion coefficient (AS), as well as (ii) the Lagrangian correlation time characterizing the duration of the initial regime, are increasing functions of the width (1:~) of the k-' subrange in the energy spectrum. This dependence can be understood as due to the appearance, for increasing l /p , of longer and longer wavelengths in the turbulent spectrum of given energy, which results in an enhancement of the absolute diffusion.

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Diffusion Theory of charged Particles in turbulent Magnetoplasma