Diffusion of ions in an electrostatic stochastic field and a space-dependent unperturbed magnetic field

Marian NEGREA

Department of Physics Association Euratom-MEdC, Romania University of Craiova, A.I.Cuza str.13,200585 Craiova, Romania

Abstract We calculate the diffusion coefficients for ions moving in a prescribed electromagnetic field.The field is considered to be a superposition of an electrostatic stochastic field and a space-dependent and sheared magnetic field.We have considered as parameters involved in the calculation of the diffusion coefficients the shear ion Kubo number , the electrostatic Kubo number K, the parallel shear ion Kubo number , and the parallel thermal ion Kubo number . A geometrical parameter which is the measure of the product of the stochastic perpendicular correlation length and the gradient in the magnetic field strength (see definitions in the text) is found not to be important in our calculation. The results concerning the diffusion coefficients obtained in our model are in agreement with experimental data and with those corresponding to other models, and the neoclassical and anomalous values for the diffusion coefficients are obtained.

Keywords: magnetic field, turbulence, diffusion(Some figures may appear in colour only in the online journal)

1. Introduction

Transport of particles in fusion plasma is still a very important issue for researchers. The theoretical results obtained for the transport coefficients from classical transport theory (see e.g.[1]) are not in agreement with the experimental results. The agreement is improved if the geometry of the magnetic field is taken into account for the calculation of the transport coefficients and the neoclassical transport theory is built (see e.g.[2] and [3]). Even neoclassical theory cannot explain some aspects of the diffusion of particles in a chaotic (turbulent)plasma. In this kind of plasma, transport is called anomalous if turbulence dominates the classical and neoclassical contributions to transport. The approximation that treated trapping in a new manner compared with the papers of Isichenko M B [4, 5], the numerical treatment given in [6] and [7] and used also in our paper to calculate the diffusion coefficients is the decorrelation trajectory method(DCT).There exist also a large amount of experimental, theoretical, and numerical papers(see,for example,references[8-26])that are dedicated to the understanding and control of transport in magnetically confined plasmas. The DCT is presented in the following selection of papers [8, 9, 11, 14, 22, 23].

Results concerning the radial flux of particles obtained in reference [24] were used in order to calculate the effect of parallel fluctuations; in our paper only the perpendicular fluctuations (to the mean magnetic field) are taken into account. Separation of magnetic field lines, which is important for the calculation of diffusion coefficients, has been studied in many papers; see reference [25] and citations within. Particle behavior is a complex problem related to the confinement and transport of the bulk ions and electrons in plasma and to the plasma-wall interaction. This is a very important issue for the development of fusion reactors.

Our paper studied the problem of the diffusion of ions in a space-dependent magnetic field and a divergenceless electrostatic stochastic field. The shape of the diffusion coefficients of a particle moving in a magnetic field with shear combined with an electrostatic stochastic field has been studied in many papers (see e.g. [27]). Numerical studies (such as‘direct numerical simulations')are devoted to the transport of particles in such a combination of fields(see e.g.[28]).All these issues are presented in the very good book of Radu Balescu[29].Our theoretical results are in agreement with the experimental ones. In our paper the magnetic field is dependent on the radial coordinate and the electrostatic stochastic field influences the values of the diffusion coefficients. The movement of particles was analyzed also in a guiding center model with a stochastic anisotropic magnetic field [30]. The results concerning the diffusion coefficients are similar with those obtained in the present paper.The conclusion is that,at least from these two different physical models,practically we obtained the same diffusion coefficient behavior.In our paper we will use some results obtained in reference [8], and the paper is organized as follows. The magnetic field model and the approximate guiding center equations are established in section 2.In section 3,some details of the DCT approach are presented and the necessary Eulerian correlations are derived using a standard procedure of the DCT method.We have also defined here the parameters (specific to the ions) involved in the change of the diffusion coefficients, namely the ion shear Kubo number, the electrostatic ion Kubo number Kion,parallel shear ion Kubo number,the parallel thermal ion Kubo number Kionand the geometrical parameter αR, which is the measure of the product of the stochastic perpendicular correlation length and the gradient in the magnetic field strength. Here the specific parameters of the motion of ions are defined. Before the section 4 we introduce some comments on the issues related to Kubo numbers,trapping and the open trajectories. In section 4 we calculate the diffusion coefficients and averaged global velocities.We also calculate the trapping time for different values of the parameters. The conclusions are provided in section 5.

2. The magnetic field model and the approximate guiding center equations

In our paper we consider that particles are moving in an electrostatic stochastic electric field combined with a magnetic field that is unperturbed and has the form

The guiding center trajectories are determined from the approximate equations

where U is the parallel velocity, which we will approximate here with the thermal one,i.e.Vth.In order to make the system dimensionless we introduce the typical correlation lengths:λ⊥is the perpendicular correlation length and λ‖is the parallel correlation length along the main magnetic field. τcis the correlation time of the fluctuating electrostatic field and ε is a measure of the amplitude of the electrostatic field fluctuations.The correlation time τcis the maximum time interval over which the field (the electrostatic potential in our case) maintains a given structure and is about 10?5s, the perpendicular correlation length λ⊥is about 10?2m as was observed by several plasma turbulence diagnostics looking at the edge region of the tokamaks [31, 32] and λ‖is about 1-10 m.Using the results obtained in [8] and the dimensionless quantities x, y, z, τ and φ defined as

we obtain the following dimensionless Langevin system of equations [8]:

We write here the different Kubo numbers already defined in[8] entering the system of equations given in (4)-(6):

is the electrostatic Kubo number

is the shear Kubo number

is the parallel shear Kubo number and

is the parallel thermal Kubo number. The geometrical parameter αRis defined as the product of the stochastic perpendicular correlation length and the gradient in the magnetic field strength

Considering that the thermal velocity used in the former expressions corresponds to ions,the following corresponding Kubo numbers for ions can be defined:

3. Details of the DCT approach

From the orders of magnitude of the different Kubo numbers given in[8]we can suppose that the term containingcan be neglected in the equation (6). We also suppose that

where we can choose ( ) =z 0 0. An implicit dependence on time appears in the potential and the Langevin system of equations given in (4)-(6) becomes a 2D system

For this system the DCT method can be applied in order to calculate the diffusion coefficient and other quantities of interest. For the stochastic dimensionless electrostatic potential φ (x , z, τ)we can choose the following spatiotemporal autocorrelation

where

and

If we use solution (14), the function ( )τB becomes

where definition (13) was used.

The autocorrelation becomes

The statistical properties of all the fluctuating quantities are calculated by taking appropriate derivatives of equations(16)and (17). We introduce the notations for the ‘directly fluctuating velocities' as

From equation (19), we can deduce the Eulerian correlations between the directly fluctuating velocities Cij(i , j =x , y)and the mixed correlationsφCi( )=i x y, using the standard procedure given in [9-11]:

where the following general relations were used:

and also the antisymmetric tensor εij(ε12= ? ε21= 1,ε11=ε22= 0). Using these correlations we can develop the procedure of the DCT needed to calculate the components of the diffusion tensor components specific to our field. We introduce the following change of variable

With this change of variable system (15) becomes (in the following we will drop the subscript ‘ion' from τion)

where the following definitions are introduced:

and

The expression(29)and the definition given in(9)should not be confused. System (27) is studied using the DCT method.The total ensemble of the realizations of the stochastic system is a superposition of the subensembles S that are defined by the electrostatic potential φ and the ‘directly fluctuating velocities' videfined in (20) at time 0, i.e.

The probability distribution function of these initial values is defined as

The total Lagrangian correlation tensor components of the directly fluctuating velocities appearing in system (27) are

where

is the Lagrangian correlation tensor in a subensemble S. anddenotes the average in the subensemble. The last approximation in(33)is the essence of the DCT method.We have replaced the average velocity in a subensemble with the velocity obtained by using the solutions ( )τxSof the deterministic system given below in equations (37) and (38).

The Eulerian average directly fluctuating velocitiesin the subensemble S are calculated as

and the explicit expressions forare

Next,we define,in a subensemble S,a deterministic trajectory by the following equations of motion:

and

Using the equations (34), (35), (36), (40), (41) and (42) the terms are calculated as

and

The solution of deterministic system (37), (38) depends not only on the parameters that define the subensemble S i.e.but also on the Kubo numbers defined in equations (7), (8), (10).

The Lagrangian correlation tensor components are

and contribute to the calculation of the running diffusion tensor components as

4. DCT trajectories

In this section we present some trajectories and hodographs resulting from system (37), (38), for a subensemble defined asand vy(0 , 0) ==1and for different values of Kionand

In figures 1-3 we visualized trajectories and hodographs for the set of electrostatic Kubo numbers[0 .2, 0.5, 1, 5] for different values of the shear Kubo number=0.5(fgiure 1),=1(fgiure 2) and=5(fgiure 3). The shape is influenced by the electrostatic Kubo number only for a relatively high value of the level of turbulence as shown in the figures.For relatively small valuesof the level of turbulence,the shapes are almost the same except for the high value(see fgiures 1-3).

All these quantities are calculated for different values of the electrostatic Kubo numberdefined above in expression (28) and of the shear Kubo number

We have used in our calculations the electrostatic Kubo numbers

For each value of the electrostatic Kubo number we have considered the evolution of the quantities for different values of the shear Kubo numbers

Figure 4 presents the trajectories for a given subensemble,four fixed values for the electrostatic Kubo number Kion, a fixed value of the shear Kubo number=0.5and different values of the parameter: in the top left subplot= 10?3, in the top right subplot= 10?2, in the bottom left subplot= 10?1and in the bottom right subplot=1.The trajectories are closed for Kion= 5 and open in all the other cases,i.e.0.2,0.5,and 1 if＜1.For=1the trajectories corresponding to Kion= 5 are also open, so trapping is not present in this case.

4.1. Comments on Kubo numbers and trapping versus open trajectories

In the subensemble defnied as S:andthe shapes of the trajectories represented in figures 1-4 are influenced by the parameters already mentioned before.

Trapping (or the existence of closed trajectories) is present when particles are moving near the maxima or minima of the stochastic electrostatic potential, and is influenced by dimensionless quantities called Kubo numbers. In our paper there are several Kubo numbers defined in(7)-(10),(12)and(13) in the particular case of ions. The electrostatic Kubo number defined in(7)measures the stochastic character of the electrostatic field and represents the ratio between the distance Vthτccovered by a particle moving with the velocity Vthduring the correlation time τcand the perpendicular correlation length λ⊥,

5. Diffusion coefficients

In this section we present in figures 5-9 the behavior of the running poloidal and radial diffusion coefficients (Dyy, Dxx)and the averaged directly fluctuating velocities over the subensembles, i.e.In figures 5-9 the following labels are used for the radial running diffusion coefficients:=0.1(continuous red line),=1(continuous dotted red line),=2(dotted red line) and=5(continuous red x line) and the same labels in blue for the poloidal running diffusion coefficients.For Kion= 0.2 the smallest asymptotic value (≈0.6) of the radial diffusion coefficient Dxxis obtained for the greater value of the shear Kubo number, i.e.=5;the same behavior is obtained for the poloidal diffusion coefficient Dyybut the asymptotic value is different(≈0.75)(see figure 5).The averaged directly fluctuating velocitiesare represented also in figure 5;the poloidal one is positive for τ? ≥0 but the radial one is negative for=5and positive for the other values offor the entire time interval [ ]τ ∈ 0, 4.The asymptotic(i.e. for τ ≥ 4) values of the averaged velocities are all zero.Practically the same behavior is present for Kion= 0.5 but with other values for the asymptotic diffusion coefficients(see figure 6).For Kion= 1,the smallest asymptotic value(≈0.25)for the radial diffusion coefficient Dxxis obtained for the greater value of the shear Kubo number, i.e.=5;the same behavior is obtained for the poloidal diffusion coefficient Dyybut the asymptotic value is different (≈0.5) (see figure 7).

The averaged directly fulctuating velocitiesalso represented in figure 7 are positives for τ? ≥0 except for the radial one, which is negative for=5in the interval τ ∈ [1 , 2] . The asymptotic values are all zero.

For Kion= 5 the asymptotic radial diffusion coefficient values are diminished and they belong to the interval[0 .06, 0.075], but the small one is obtained for=0.1(see figure 8).For an increased value of the electrostatic Kubo number (Kion= 10) an inversion of the behavior of the diffusion coefficients is observed (see figure 9); a decreasing of the maximum of the radial diffusion coefficient is also observed. The dimensional expressions for the radial and poloidal diffusion coefficients are

and

respectively. If we considerandthe dimensional factor c is about

The expressions of DXXand DYYcan be defined also as functions ofas

and

where we remember that the correlation time is abouts. As a consequence,

and

Using the above relations and the results shown in the corresponding figures, we can state some observations. As stated in [33] for example, the neoclassical values for the diffusion coefficients are smaller than unity; e.g. they are of the order of 0.5 m2s?1,whereas in experiments they are much larger,with anomalous values that are found to be of the order of 4 m2s?1.

For Kion= 10 (see figure 9) the radial diffusion coefficient( )DXXmaxis in the range of the neoclassical values, i.e.for practically all values of. The same situation appears forfor practically all values of.For the asymptotic values of the radial diffusion coefficients the situation is the following:( )DXXasis in the range[0 .15, 0.35],and increases ifincreases to the range[0 .1, 5].(DXX)asis about 0.6 m2s?1practically for all.(DYY)asis in the range[0 .2, 1.2] and increases ifincreases to the range[ ]0.1, 5.

For Kion= 5 (see figure 8), =0.1 and 1, the radial diffusion coefficient has the maxima values 1.4 m2s?1for=1and 0.014 foris in the range[ ]0.75, 2 and increases ifincreases to the range[ ]0.1, 5.In figures 10-12 the labels used are as follows:Kion= 0.2 (red diamond), Kion= 0.5 (red squared), Kion=1(red triangle),Kion= 5(blue circle)and Kion= 10(blue x).We have calculated the influence ofon the radial and poloidal asymptotic diffusion coefficients for different levels of the electrostatic turbulence (see figure 10). It is obvious that an increase of the shear Kubo number produces a decrease of the asymptotic values for the radial and poloidal diffusion coefficients except for the level of electrostatic turbulence given here bythe asymptotic values increase slowly as a function ofbut the final values are smaller than those corresponding to Kion＜ 5. In figure 11 the moments τmaxcorresponding to the maxima achievements are represented as a function offor different Kion. For the radial diffusion coefficients (top of figure 11)the shapes are very similar to a hyperbolic decrease as a function of. The following order is obvious

The smallest value is( )τmax10and corresponds to Kion= 10.For the poloidal diffusion we represented in (bottom of figure 11) τmaxonly forThere are constant values for these moments; only their order of magnitude is different. We have also calculated the influence ofon the trapping time interval, i.e. onwhere τasis the moment corresponding to the beginning of the asymptotic regime and τmaxis the moment that corresponds to the maxima of the diffusion coefficients (see figure 12). We note that if trapping is present the diffusion coefficient shape begins with an increase(the ballistic regime)and continues with a subdiffusion regime up to a value corresponding to the beginning of the asymptotic regime.For the radial diffusion coefficient, we conclude that the trapping timeincreases for≤2for Kion∈{0 .2, 0.5, 1}and has relatively constant values for Kion∈{5 , 10}also for≤2. For≥2a relatively small region with a decrease of the time trapping is present for Kion= 5 followed by a decrease and an increase followed by a constant regime for Kion= 10. For Kion= 0.2 and 1 there exists a small increase, and for Kion= 0.5 a decrease is present. For the poloidal diffusion coefficient the shape of the trapping timerepresented in figure 12 only for a relatively large turbulence level is practically the same as for the corresponding radial situation.

6. Conclusions

In this paper,we have analyzed the diffusion of the ions using the Langevin equations corresponding to the guiding center and we applied the semi-analytical method of decorrelation trajectories (DCT). The latter can be considered as a generalization of the Corrsin approximation and takes into account the trapping effects (which necessarily exist in relatively strong turbulent plasmas).Using DCT,we have studied the transport of test particles (ions) by the electromagnetic drift that is produced by a stochastic electrostatic potential and by an inhomogeneous and sheared magnetic field. The DCT and the model used have given good qualitative results concerning the diffusion of ions. The radial and poloidal coefficients start with a linear part, indicating a ballistic regime,which is followed by a trapping regime.If trapping is present the diffusion coefficient has a shape that begins with an increase of the diffusion coefficient (the ballistic regime) and continues with a subdiffusion regime up to a value corresponding to the beginning of the asymptotic regime. We have calculated the influence ofon the radial and poloidal trapping time intervals for different values of Kion. After that the asymptotic value is reached (the trapping effects are visible in representations of trajectories). The diffusion coefficients increase with increasing levels of turbulence, i.e.with an increasing Kion. We have also represented the dependence of τmaxas a function ofand the trapping time interval as a function offor different levels of electrostatic turbulence. The maximum radial trapping time is reached for Kion= 0.5 for=2. The value=2 represents a critical value to which a corresponding critical value for the thermal ion velocity is obtained.For fixed values of the shear length and of the correlation time this critical value can be obtained. Forthe critical thermal velocity isThe results obtained in our model are in agreement with the experimental data: the neoclassical and anomalous values for the diffusion coefficients are obtained. Here the magnetic field model dependence on the radial coordinate and the electrostatic stochastic field influenced the values of the diffusion coefficients. Concerning the diffusion coefficients, similar results with those obtained in the present paper were found in [30].The conclusion is that from these two different physical models we obtained practically the same diffusion coefficient behavior. The magnetic shear, the inhomogeneity of the magnetic field and also the stochastic electrostatic field have the same influence on the ions'diffusion as has the stochastic magnetic drift.

In conclusion, we state that

(a) the diffusion present a pronounced trapping if Kion≥ 5;

(b) the maxima of the diffusion coefficients decreases if Kion≥ 5;

(c) the space dependence and the shear of the magnetic field modifies the diffusion coefficients.

Important results concerning the behavior of the magnetic field were obtained(see e.g.[34]and[35])analyzing the Grad-Shafranov equation. From here, the possibility of constructing different magnetic fields appears. The diffusion of stochastic isotropic and anisotropic magnetic field lines in turbulence with a magnetic average poloidal magnetic field component was studied in[36]and an extension of this paper in such a magnetic field would be of interest.It will be necessary that collisions are taken into consideration in such a study,but this issue is left for future work. Here, only the influence of the aforementioned parameters on motion were taken into account.

Acknowledgments

I would like to thank Dr. Iulian Petrisor for fruitful discussions.This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the EURATOM research and training program 2014-2018 under Grant Agreement No.633053.The views and opinions expressed herein do not necessarily reflect those of the European Commission.

ORCID iDs