Some statistical features of particle dynamics in Tokamak ... · 5 Some statistical features of...

SOME STATISTICAL FEATURES OFPARTICLE DYNAMICS IN TOKAMAK PLASMA

IULIAN PETRISOR

Association Euratom-MEdC, RomaniaUniversity of Craiova, Department of Physics, 13 A.I. Cuza Street, 200585 Craiova, Dolj, Romania

E-mail: [email protected]

Received September 21, 2015

In order to study the transport of charged particles (ions, electrons) or dust par-ticles in tokamak plasma, we have used the numerical simulations method based on theTURBO code. The method can be applied for any value of the Kubo numbers K, Ks

specific for the analyzed problem. The particle’s transport depends on the level of tur-bulence and on the inhomogeneity of the magnetic field. The magnetic shear is shownto have a contribution in order to obtain the plasma stability. We present the specificautocorrelations for the electrostatic fluctuations implementing the TURBO code andwe have calculated and interpreted the radial and poloidal mean squared displacements,the kurtosis and the skewness.

Key words: plasma, turbulence, particle dynamics, transport.

PACS: 52.25.Fi, 52.35.Ra, 47.27.E-.

1. INTRODUCTION

To study the transport of ions, electrons or dust particles in tokamak plasma,we can use at least two alternative methods (the decorrelation trajectory method, seee.g. [1]-[13]) and numerical simulation [7], [14]. In the present paper we used onlythe numerical simulations method based on the TURBO code [15]. The TURBOcode is used for the study of particle transport in fields a priori fixed or in fields thatare generated from their prescribed statistical properties. The results for test-particlesimulations in numerically generated stochastic electrostatic field by TURBO can becompared with those corresponding to the traditional numerical simulations alreadydone in [7], [15]. If enough computer resources are available, the method can be ap-plied for any value of the Kubo numbersK,Ks etc. The particle transport depends onthe level of turbulence and on different kind of spatial variation of the magnetic field(inhomogeneity, magnetic shear etc.). We can determine the diffusion coefficientsand higher order moments for different levels of electrostatic turbulence and differ-ent kind of spatial variation of the magnetic field that are measured by dimensionlesscorresponding Kubo numbers. The magnetic shear is shown to have a contributionin order to obtain the plasma stability. The results can be used in the study of ITERtokamak. The paper is organized as follows. Sec. I represents the Introduction. In

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218 Iulian Petrisor 2

Sec. II we presented the magnetic field model with normal shear, the order of mag-nitude of the parameters and the significance of various Kubo numbers. In Sec. IIIwere presented the dimensionless guiding centre equations and calculated the meansquared displacements, skewness and kurtosis using a numerical approach based onTURBO code for various Kubo numbers specific to a realistic tokamak. The Sec. IVrepresents the conclusions of our work.

2. THE MODEL WITH NORMAL MAGNETIC SHEAR

We consider in slab sheared geometry an electrostatic turbulence representedby an electrostatic stochastic potential Φ(X,Z,t), where X = (X,Y ) are the Carte-sian coordinates in the plane perpendicular to the main magnetic field. The z com-ponent of the magnetic field Bz depends on the radial coordinate X and has the formBz=B0 (1 +X/R)−1ez . The global magnetic field is unperturbed and it depends onthe distance from the main symmetry axis Oz having the form

B =B0[b(X)ez +s(X)ey] (1)

where s(X) ≡ XL−1s with Ls the shear length and b(X) ≡ (1 +X/R)−1 with R

the major radius of the tokamak. The approximated guiding center trajectories aredetermined from

dX

dt' Ub+

E×B

B2(2)

where U is the parallel velocity, which we will approximate with the thermal oneVth. The magnitude of the magnetic field B is

B =B0

[b2 (X) +s2 (X)

]1/2 (3)

and the unit vector b is

b≡B

B=

[b(X)ez +s(X)ey]

[b2 (X) +s2 (X)]1/2≡ b′ (X)ez +s

′(X)ey (4)

where b′(X)≡ b(X)

[b2(X)+s2(X)]1/2and s

′(X)≡ s(X)

[b2(X)+s2(X)]1/2.

Using E =−∇Φ(X,Z,t) [i.e. neglecting the velocity generated by Ez=−∂Φ∂z when

compared with the thermal one] and the expressions (1-4) we get the following sys-tem of equations, if we neglect the quadratic terms in X like: s2 (X) , X2/R2 ands(X) ·X/R

dX

dt=− 1

B0b−1 (X)

∂Φ(X,Z,t)

∂Y

dY

dt=

1

B0b−1 (X)

∂Φ(X,Z,t)

∂X+s(X)Vth

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3 Some statistical features of particle dynamics in Tokamak plasma 219

dZ

dt=s(X)

B0

∂Φ(X,Z,t)

∂X+ b−1 (X)Vth (5)

The dimensionless quantities x, y,z, τ and ϕ are defined in terms of the dimensionalvariables by the following expressions

x =X

λ⊥; z =

Z

λ‖; τ =

t

τc; Φ(X, t) = εϕ

(X

λ⊥,Z

λ‖,t

τc

)(6)

Here, λ⊥ is the perpendicular correlation length, λ‖ is the parallel correlation lengthalong the main magnetic field, τc is the correlation time of the fluctuating electrostaticfield and ε is a dimensional quantity measuring the amplitude of the electrostatic fieldfluctuation. The correlation time τc is the maximum time interval over which the field(the electrostatic potential in our case) maintains a given structure. Introducing thedefinitions (6) in the system (5) we get

d(xλ⊥)

d(ττc)=− 1

B0b−1 (xλ⊥/R)

∂εϕ(

Xλ⊥, Zλ‖

, tτc

)∂ (yλ⊥)

d(yλ⊥)

d(ττc)=

1

B0b−1 (xλ⊥/R)

∂εϕ(

Xλ⊥, Zλ‖

, tτc

)∂ (xλ⊥)

+s(xλ⊥)Vth

d(zλ‖)

d(ττc)=s(xλ⊥)

B0

∂εϕ(

Xλ⊥, Zλ‖

, tτc

)∂ (xλ⊥)

+ b−1 (xλ⊥/R)Vth (7)

With the parameters defined above, the system of equations (7) becomesdx

dτ=−K (1 +αRx)

∂ϕ

∂y(8)

dy

dτ=K (1 +αRx)

∂ϕ

∂x+Ksx (9)

dz

dτ=Kzsx

∂ϕ

∂x+Kzb

−1 (αRx) (10)

We introduce the following parameters

K =ετcB0λ2

⊥∈ [0.1,100] (11)

that is the electrostatic Kubo number,

Ks =τcVthLs∈[10−1, 1

](12)

the shear Kubo number,

Kzs =ετc

B0λ‖Ls≡K

λ2⊥

λ‖Ls∈[10−8, 10−4

](13)

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the parallel shear Kubo number and

Kz =Vthτcλ‖

(14)

the parallel thermal Kubo number and the geometrical parameter αR which is a ratioof the stochastic perpendicular correlation length λ⊥ and the geometrical aspect ofthe device given by major radius R

αR = λ⊥/R (15)

We have considered here typical time and space scales observed by several plasmaturbulence diagnostics looking at the edge region of the tokamaks and in consequencewe introduced the specific Kubo numbers.

2.1. THE ORDER OF MAGNITUDE OF THE PARAMETERS

Typical time and space scales are about τc ' 10−5s for the correlation time andλ⊥ ' 10−2 m for the perpendicular correlation length as was observed by severalplasma turbulence diagnostics looking at the edge region of the tokamaks [16]. Typi-cal tokamak plasma are characterized by the following ranges for the Kubo numbers[1]:

K ∈ [0.1,100] , Kionz ∈

[10−2, 1

], Kelectron

z ∈ [1, 100] (16)For ions V ion

th ∈[105,106

]m/s and for electrons V el

th ∈[106,107

]m/s (see e.g.

[17]). Normally, the parallel correlation length is in the range λ‖ ∈ [10,100] m. Theshear length is of order Ls = 10 m and λ⊥/R ≡ c is of order 10−2 (the same valuelike in the paper [18]). With these values we get the following minima and maximafor the parameters(

Kionz

)min

=

(V ionth

)min

τc(λ‖)

max

=105 ·10−5

102= 10−2

(Kionz

)max

=

(V ionth

)max

τc(λ‖)

min

=106 ·10−5

10= 1

(Kionzs

)min

= (K)min

λ2⊥(

λ‖)

maxLs

= 10−1 10−4

102 ·10= 10−8

(Kionzs

)max

= (K)max

λ2⊥(

λ‖)

minLs

= 102 10−4

10 ·10= 10−4

(Kions

)min

=τc(V ionth

)min

Ls=

10−5 ·105

10= 10−1

(Kions

)max

=τc(V ionth

)max

Ls=

10−5 ·106

10= 1 (17)

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From these orders of magnitude we can state that the term containing Kzs can beneglected in the equation (10) of the system and also we can make the approximation

b−1 (αRx)' 1

The dimensionless system of equations becomes 2D, with the trivial third dimen-sionless equation

dz

dτ'Kz

and its solution

z 'Kzτ

or in dimensional case

Z ' Vtht

2.2. ABOUT THE SIGNIFICANCE OF KUBO NUMBERS

In the equation (11) the ratio

V =ε

B0λ⊥

is the drift perpendicular velocity V . Moving with this velocity the particle needs thetime of flight τfl in order to cover a distance λ⊥ perpendicular to the main magneticfield, i.e.

V =λ⊥τfl

and the expression (11) becomes

K =V τcλ⊥

In principle, the expression (14) gives the stochastic parallel Kubo number, which inour case contains the thermal velocity not a stochastic parallel velocity, V‖ (or U ).

In our analysis we have neglected the velocity generated byEz=−∂Φ∂z , because

the initial parallel kinetic energy is considered much greater than the electrostaticone. For the usual tokamak plasma conditions, the energy of the stochastic field is ofthe order of 10% of the thermal energy. If a stochastic parallel Kubo number K‖ isdefined by analogy with Kz (see 14) as

K‖ =U‖τc

λ‖

where U‖ is the parallel stochastic velocity generated by Ez=−∂Φ∂z , this it would be

of the order of [10−3,10−2] for ions and [0.1,10] for electrons [7]. This means that

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stochastic parallel motion is irrelevant for the diffusion of ions. Here, different casesare present when analyze the following equation

K′ ≡ K

Kz + 1

1) Kz� 1, i.e. K′ ≈K, the diffusion is not influenced by the parallel motion

2) Kz �max[K,1] the parallel motion is dominant and is equivalent with anabsence of the trapping.

3) 1 � Kz � K when both parallel motion and the trapping influence thediffusion. These cases are specific to particles that move with practically the samevelocity V‖ (or U ), i.e. the runaways electrons or ion beams. Because of the rangesspecified in the next comments, the parallel motion is important only for electronsbut the ion polarization drift can make it important also for ions.

3. THE NUMERICAL APPROACH

3.1. GUIDING CENTER AND AUTOCORRELATION OF THE POTENTIAL

We integrate numerically the equation (2)

dX

dt' Ub+

E×B

B2(18)

with time independent B and E. The magnetic field B is given by the explicit for-mula (1), but E is more complicated, generated from a dimensionless stochastic po-tential ϕ with known correlations, defined as

〈ϕ(0,0,0)ϕ(X1,X2,X3)〉= exp

[−X

21 +X2

3

2λ2⊥

]exp

[−X2

λ‖

](19)

where

λ‖ ≡Vthλ‖τc

Vthτc+λ‖

In the simulations, the coordinates are rotated to:X1 is the poloidal coordinate Y and X3 is the radial one X .X2 is the “temporal-toroidal” coordinate,

Z ≡X2 = Vtht

The Fourier transform definition is:

f̃(k) =1√2π

∫ ∞−∞

dxf(x)exp(ıkx) (20)

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Fig. 1 – The poloidal spectrum

and the corresponding Fourier transforms of the correlations are:

exp

[−|X2|λ‖

]→√

2

π

λ‖

k22λ‖

2+ 1

(21)

exp

[−X

21 +X2

3

2λ2⊥

]→ λ2

⊥ exp

[−λ2⊥2

(k21 +k2

3)

](22)

The electric potential is initialized as

ϕ̃(k1,k2,k3) = εexp(ıθk1k2k3)

√√√√ 1

k22λ‖

2+ 1

exp

[−λ2⊥2

(k21 +k2

3)

](23)

θk1k2k3 is a set of random phases and E0 is a parameter necessary to ensure thedimension of the potential, i.e.

ε≡ E0λ⊥λ‖1/2(

2

π

)1/4

(24)

Due to the nature of the parameters, the trajectories are expected to extend to adistance of

• about 4 to 5 λ⊥ in the radial direction (subdiffusion, almost trapping)

• more than 10 λ⊥ in the poloidal direction (superdiffusion, almost ballistic)

• between 1 and 10 λ‖ in the toroidal direction

These observations, together with the form of the spectrum, translate into con-straints on the number of modes for each direction.

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Fig. 2 – The radial spectrum

Because on the toroidal direction the spectrum is proportional to 1/k22 , many

modes are needed to achieve some scale separation (the amplitude of the highestmode must be small in some way); this is the main reason for discussing the exactmemory requirements.

3.2. NUMERICAL TOOLS

The TURBO code is designed to solve numerically the equations for an incom-pressible fluid in a three dimensional geometry with periodic boundary conditions inthe three directions. This code is written in FORTRAN, is a spectral code and alsohas been optimized for parallel computing. Our numerical code for tracking particletrajectories was developed in a modular fashion as an extension of TURBO solver[19]-[21]. Large amounts of memory are needed in order to keep the complex elec-tric field, because a long tailed distribution is used in the toroidal direction. Sincethe fields are initialized in Fourier space, the large amount of memory is requiredbecause scale separation between the lowest and highest wave numbers is harder toachieve. Because the code is designed as a pseudo-spectral solver for fluid equations,parallelization requires that fields are divided evenly between individual processors.For this set of simulations, the fields were divided along the radial direction, so thatparticles would not “pass between processors” very often. Because of the particularnature of the system involved, it is more efficient to run several simulations in parallelrather than running one big simulation on more processors. Lastly, the time step forthe individual particle trajectories is computed so that the local integration errors arebelow some predefined value. Because of this criterion, the trajectories for larger Krequire a smaller time step, hence longer simulations are required. For every run, inthe case of low level of electrostatic turbulence, i.e. for small Kubo number (K = 0.2)

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Fig. 3 – The toroidal spectrum

around 20 CPU hours computation time were used; in the case of relatively high levelof electrostatic turbulence, i.e. for large Kubo number (K = 5.0), around 1100 CPUhours computation time were used. Initially, we generated the prescribed turbulenceand in such turbulence background we have “introduced” around 150 particles thatwere tracked in order to evaluate the mean squared displacements.

3.3. MEAN SQUARED RADIAL AND POLOIDAL DISPLACEMENTS

By using TURBO code we have calculated the mean square displacements, i.e.practically the diffusion coefficients of the particles for different values of the shearKubo number Ks and from low (K = 0.2) to high (K = 5) electrostatic turbulencegiven by the electrostatic Kubo number K, as can be seen from Figs. 4. It is obviousthat the radial diffusion decreases if the shear Kubo number increases and the orderof magnitude of the diffusion coefficient is greater the greater level of turbulence.

From Figs. 5 it can be observed that the poloidal diffusion is practically notinfluenced by the shear Kubo number but the higher level of turbulence gives largerpoloidal diffusion. The geometrical parameter is in the range

[10−4, 1

]for all the

runs. The Kubo numberKzs is between 10−8 and 10−6 andKz is constant and equalto 10−1. These plots show very nicely how, for a fixed electrostatic Kubo number,the radial transport is reduced by increasing the shear Kubo number. The reversehappens for the poloidal transport, which is increased with the shear Kubo number.The poloidal diffusion is of one order of magnitude greater than the radial one. Weobserve also that in the early stages in the plots some irregularities appear beforethe normal diffusion is reached; this thing is relatively specific to the transition tothe asymptotic regime of diffusion. For times comparable to the typical trappingtimes a regime of anomalous diffusion can be expected. It may be caused by the

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0 2000 4000 6000 80000

500

1000

1500

2000K = 0.2

t

<((

x(t)

−x(

0))

2>

0 2000 4000 6000 80000

1000

2000

3000

4000

5000

6000K = 0.5

t

<((

x(t)

−x(

0))

2>

0 2000 4000 6000 80000

1000

2000

3000

4000

5000

6000

7000

8000K = 1.0

t

<((

x(t)

−x(

0))

2>

0 2000 4000 6000 80000

1

2

3

4

5

6

7x 10

4 K = 5.0

t

<((

x(t)

−x(

0))

2>

Ks=0.1

Ks=0.5

Ks=1.0

Ks=2.0

Fig. 4 – Radial dimensionless MSD for different values of shear Kubo numbers and for different valuesof electrostatic turbulence

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0 2000 4000 6000 80000

1

2

3

4

5

6

7x 10

6 K = 0.2

t

<((

y(t

)−y(0

))2>

Ks=0.1

Ks=0.5

Ks=1.0

Ks=2.0

0 2000 4000 6000 80000

0.5

1

1.5

2

2.5

3

3.5x 10

7 K = 0.5

t

<((

y(t

)−y(0

))2>

Ks=0.1

Ks=0.5

Ks=1.0

Ks=2.0

0 2000 4000 6000 80000

1

2

3

4

5

6x 10

7 K = 1.0

t

<((

y(t

)−y(0

))2>

Ks=0.1

Ks=0.5

Ks=1.0

Ks=2.0

0 2000 4000 6000 80000

1

2

3

4

5

6x 10

8 K = 5.0

t

<((

y(t

)−y(0

))2>

Ks=0.1

Ks=0.5

Ks=1.0

Ks=2.0

Fig. 5 – Poloidal dimensionless MSD for different values of Kubo number and shear Kubo numbers

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trapping effect (see e.g. in [22]). Comparing the resulted MSD with those specific in[2] we observed that, using the TURBO code, the results are in agreement. This isan important conclusion because the fields are obtained with TURBO code and notimplemented relatively artificially from outside the system.

3.4. SKEWNESS AND KURTOSIS

Using TURBO code we have calculated the higher order moments (practicallythe skewness (S) and the kurtosis (Kx) in the radial and (Ky) poloidal directions)for different values of the shear Kubo number Ks and for relatively low (K = 0.2)and high (K = 5) electrostatic turbulence given by the electrostatic Kubo number K,as can be seen from Figs. (6)-(9).

From Fig. (6) we observe that only for Ks = 0.5 and K = 0.2,0.5 and 1,the third order radial moment (practically the radial skewness) is a negative quan-tity corresponding to a longer left tail of the distribution (left-skewed or left-taileddistribution).

From Fig. (7) we observe that for all the levels of electrostatic turbulencegiven by the Kubo numbers K = 0.2,0.5,1 and 5 and only for Ks = 2 the third or-der poloidal moment (practically the poloidal skewness) is also a negative quantitycorresponding to a longer left tail of the distribution (left-skewed or left-tailed dis-tribution). It is obvious that the radial diffusion decreases if the shear Kubo numberincreases and the order of magnitude of the diffusion coefficient is greater the greaterlevel of turbulence is.

From Figs. 8 and 9 it can be observed that the radial and poloidal kurtosisare positive. The radial kurtosis have practically the same slope for any shear Kubonumbers and for the relatively small electrostatic turbulence given by K = 0.2,0.5and 1.

All the poloidal kurtosis have the same slope for any values of the shear Kubonumbers. As a conclusion we state that the radial diffusion is influenced by themagnetic shear in the sense of the decreasing for the increase ofKS . The electrostaticturbulence level increase the diffusion as expected.

4. CONCLUSIONS

Various regimes of prescribed turbulence are able to be generated with TURBOcode like an input for the movement of charged particles in such prescribed turbu-lence. The magnetic field is supposed to be inhomogeneous and our work could berelated to the study of specific models for particles (electrons, ions and dust) andenergy transport in SOL (from tokamaks or solar wind). In our complete studieswe intend to investigate the effect of a given turbulence in the transport of differ-ent charged particles. We must to express the kurtosis Ki (i = x,y) as function of

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Fig. 6 – Radial dimensionless skewness (Sx) for different values of shear Kubo numbers and for variouselectrostatic turbulence.

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Fig. 7 – Poloidal dimensionless skewness for different values of shear Kubo numbers and variouselectrostatic turbulence.

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Fig. 8 – Radial dimensionless kurtosis for different values of shear Kubo numbers and for variouselectrostatic turbulence.

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Fig. 9 – Poloidal dimensionless kurtosis for different values of shear Kubo numbers and for variouselectrostatic turbulence.

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the skewness Si, i.e. Ki(S) in order to verify if the dependence is quadratic, i.e.Ki(S) =AS2

i +BSi+C. Previous measurements in tokamak show a change of thesign of S from + to − when crossing the LCFS (last closed flux surface) from SOL(scrape off layer) to the plasma edge [23]. The existence of nonzero skewness isequivalent to an asymmetry of the fluctuation from the average. Intermittent plasmaobjects (IPO) are created at or near LCFS and the existence of positive or negativeIPO moving in opposite directions is given by the skewness. The existence of large SandKi implies a fluctuation depart from Gaussianity in SOL. Gaussian means S = 0.If positive IPO move into the SOL with

vr =Eθ×B

B2

implies that the plasma edge is diluted. The distribution function that admits S > 0and/or < 0 is the beta distribution. A question is very normal: What are the depen-dences of S and Ki on Kubo numbers?

We need for calculateKi, Si

orK (r) , S (r) , r =

(x2 +y2

)1/2If negative IPO exist this means that the impurities are moving into the plasma

core. As a conclusion we state that the radial diffusion is influenced by the mag-netic shear in the sense of the decreasing for the increase of KS . The electrostaticturbulence level increases the diffusion as expected. Various regimes of frozen (pre-scribed) turbulence are able to be generated with TURBO code like an input for themovement of charged particles in such prescribed turbulence. The electromagneticstochastic fields are supposed to be anisotropic, and this kind of work could be eas-ily related to the specific models for particles and energy transport in SOL (fromtokamaks). In our complete studies we intend to investigate the effect of a given tur-bulence in the transport of different charged particles. A full analysis is in progress.

Acknowledgements. This work has been carried out within the framework of the EUROfusionConsortium and has received funding from the Euratom research and training programme 2014-2018under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflectthose of the European Commission.

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