Collisionless wave dissipation in a D-shaped tokamak with Soloviev type equilibrium

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Preprint submitted on 18 Oct 2004

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Collisionless wave dissipation in a D-shaped tokamak with Soloviev type equilibrium

Nikolay Grishanov, Abimael Loula, Alexandre Madureira, Joaquim Pereira Neto

To cite this version:

Nikolay Grishanov, Abimael Loula, Alexandre Madureira, Joaquim Pereira Neto. Collisionless wave dissipation in a D-shaped tokamak with Soloviev type equilibrium. 2004. �hal-00001704v2�

Collisionless wave dissipation in a D-shaped tokamak with Solov’ev type equilibrium

N.I. Grishanov¹, A.F.D. Loula¹, A.L. Madureira¹, J. Pereira Neto²

1Laboratório Nacional de Computação Científica, Petrópolis, RJ, BRASIL

2Universidade do Estado do Rio de Janeiro, Rio de Janeiro, RJ, BRASIL

Abstract:Parallel permittivity elements are derived for RF waves in an axisymmetric D-shaped tokamak with Solov’ev type equilibrium. Drift-kinetic equation is solved separately for untrapped (passing or circulating) and three groups of trapped particles as a boundary-value problem, accounting for the bounce resonances. A coordinate system with the "straight" magnetic field lines is used. Our dielectric permittivity elements are suitable to estimate the wave dissipation by electron Landau damping (e.g., during the plasma heating and current drive generation) in the frequency range of the Alfvén and fast magnetosonic waves, for both the large and low aspect ratio tokamaks with circular, elliptic and D-shaped magnetic surfaces. Dissipated wave power is expressed by the summation of terms including the imaginary parts of both the diagonal and non-diagonal elements of the parallel permittivity.

1. Introduction

Spherical Tokamaks (or Low Aspect Ratio Tokamaks) represent a promising route to magnetic thermonuclear fusion [1-6]. To achieve fusion conditions in these devices additional plasma heating must be employed. Effective schemes of heating and current drive in tokamaks can be realised using the collisionless dissipation of RF waves (e.g., Alfvén, Fast Magnetosonic and Lower Hybrid Waves) by electron Landau damping, transit time magnetic pumping (TTMP), cyclotron and bounce wave-particle interactions, etc.

Kinetic wave theory in any toroidal plasma should be based on the solution of Vlasov- Maxwell's equations [7-15]. However, this problem is not simple even in the scope of the linear theory since to solve the differential wave (or Maxwell's) equations one should use the complicated integral dielectric characteristics valid in the given frequency range for realistic two- or three-dimensional plasma models. The form of the dielectric (or wave conductivity) tensor components, H_ik, depends substantially on the geometry of an equilibrium magnetic field and, accordingly, on the chosen geometrical coordinates.

The specific toroidal effects arise since the parallel velocity of plasma particles in tokamaks is not constant, v_|| v˜hzconst. As a result,

- depending on the pitch angle the plasma particles should be split in the populations of the trapped(t) and untrapped (u) particles;

- trajectories of t- and u-particles are different;

- Vlasov equation should be resolved separately for each particle group;

- the t- and u-particles give different contributions to H_ik;

- the Cherenkov-resonance conditions of the t- and u-particles are different;

- wave dissipation by the u- and t-particles depends on the ratio of v_ph/v_T and Z/Z_b; - another interesting feature in tokamaks is the contribution of all spectrum of E-field to the m-th harmonic of the perturbed current density: ¦^rf´

´

´

/ ,

4Sij_i^m Z _mH_ik^mmE_k^m .

All these features of the Large Aspect Ratio Tokamaks (U/R0<<1) take place in the Low Aspect Ratio Toroidal Plasmas (U/R0<1) with circular [16-18], elliptic [19], and D- shaped magnetic surfaces [20]. In both the Large and Low Aspect Ratio Tokamaks with circular magnetic surfaces, the equilibrium magnetic field has only one minimum (or three extremums, with respect to the poloidal angle T). Accordingly, in such plasma models there

magnetic surfaces is the fact that the equilibrium magnetic field can have two local minimums (or five extremums, with respect to T ). As a result, together with untrapped and usual t- trapped particles, two additional groups of the so-called d-trapped (or double-trapped) particles can appear at such magnetic surfaces where the corresponding criterion [19] is satisfied: U^/R0h_T²b^2/a^{21 ,} here b/a is the elongation, and h_T is the poloidal component of the unit vector along the equilibrium magnetic field.

Many present-days tokamaks, mainly the spherical ones, have the D-shaped transverse cross-sections of the magnetic surfaces. In this paper, the parallel dielectric permittivity elements are derived for RF waves in an axisymmetric D-shaped toroidal plasma with a Solov’ev type equilibrium. A collisionless plasma model is considered. Drift-kinetic equation is solved separately for untrapped and three groups of trapped particles as a boundary-value problem, using an approach developed for low aspect ratio tokamaks with concentric circular [18], elliptic [19] and D-shaped [20] magnetic surfaces.

Fig. 1. D-shaped transverse magnetic surface cross-sections in a tokamak with Solov’ev type equilibrium

2. Plasma Model with Solov’ev Type Equilibrium

D-shaped transverse magnetic surface cross-sections corresponding to the Solov’ev type equilibrium [21, 22], see Fig.1, can be plotted by the following parametric equations for cylindrical (R,I^,Z)and quasi-toroidal coordinates (UTI^):

(1) T

U^cos 2 ₁

02 aR

R

R

_{I I}

T U G

T U D

cos 2

sin

2 1 0

1

aR R

Z aR

D D U G

1

2 2 2 0 2 2

2

2 ) (

4

aR

R R R

Z

^¸¸_¹

·

¨¨

©

§

₂

2 0

2 2

arctan

R R

R Z D T G

I I

where R0is the radius of the main magnetic axis;

R1is the major radius of the external magnetic surface;

a is the (minor) plasma radius;

b/a is the elongation;

d/a is the triangularity;

Uis the non-dimensional radius of a local magnetic surface (U^);

Tis the poloidal angleSTS

and the additional definitions are [22]

) ) (

( 2

) (

) (

) (

2 1 2 0

4 1 2 1 2 1

d R R

d R a

R a R

G ; ₂

1 2 0

2 2

) (

2 d R R

b

D ^. (2)

The force balance and Maxwell’s equations:

p uH c’

J , ’uH (4S/c)J, ’˜H 0, (3) can be reduced to the Grad-Shafranov equation [23, 24]

<

w w

<

w w

w

<

w w

<

w w

w p

p

I I R p

c RJ Z

R R

R R ₂ ²

2

4 4

1 S S

I , (4)

where J_I is the toroidal current density; c is the speed of light; p p(<) is the plasma pressure; < RA_I is the poloidal magnetic flux; i.e., H H_Ie_I ’u(A_Ie_I) and

RHI

I

I_{p p}(<) . The Solov’ev solution [21, 22] of the Grad-Shafranov equation has been found for the case

<

D S

D 2 1

1 2 0

0 aR

p p

p _, <

2

1 2 0

0 2 0 2

1 2 4

D D

S G

I aR H p

R

I_p , (5)

and can be written as

2 0^{22 1} ₂ ^{0 2}

2 2

2 2 1

0

1 2 ) 4

1 (

2 U

D S D D

D G D

S

»

¼

« º

¬

ª

< aR p

R R R

aR Z

p , (6)

where p₀ ^and H_0I are, respectively, the plasma pressure and toroidal magnetic field on the axis (R=R0 or U Accordingly, the non-dimensional radius U, see Eqs. (1), is introduced as

D D G S

D U D

1

2 2 0 2 2 2

2

0 1

2

2 ) (

4 2

1

aR

R R R

Z p

aR

< . (7)

As a result, the components of an equilibrium magnetic field can be written as

T U

T U T G

D U R 2 cos

cos 2

sin R 1

1 2

1 2 0

1 2

0 2

0

aR aR p

Z H_R R

w

<

w

T U

D GU E

D D G

I I

I

cos 2

1

) 1 1 ( 1

2 1 2

2 0 1

2 2 0

2

2 0 1

2 0 0 2 0

R aR H R

aR H p R R R H I

o p

<

(8)

T U G

T G T

DU U

D ^{R 2 cos}

cos ) R ( ) cos 1 ( 1

2 1

1 2

0

2 0 2

1 2 0

aR aR

p R

H_Z R

w

<

w

and module of an equilibrium magnetic field is

T U

T D U

U E T

U _I

I I

cos 2

1

) , 1 (

1 )

, (

2 0 1 2 2

0 0

2 2 2 0

R aR

g H

G H H H H H

o Z

R

(9)

where

2 2 2

0 1 2

2 0

1 2 0 1 2

2 2 0 2

0

2 1 cos 1 cos

2 cos 1

cos 2

1 cos

cos 2

1 sin ) ,

( »

¼

« º

¬

ª

¸¸¹

·

¨¨©

§

¸¸

¹

·

¨¨©

§ T

G T U

G T U

T U D

G T T U T

T

U R

aR R

aR R aR R

R

g aR ,

2 0 0 o

8

I

E S H

p is the E–value (central beta) on the main magnetic axis (R=R0 or U Now let us verify how many minimums can have H0UTin dependence of the poloidal angle T at the given by U magnetic surfaces. The corresponding plots are present in Fig. 2 for tokamaks with elliptic (b/a=2.5, d/a=0,U =0.5) and D-shaped (b/a=2.5, d/a=0.3, U =0.5) magnetic surfaces. As shown in Fig.2, plasma tends to have H0(UTconfigurations with one minimum. Configurations with two minimums of H0(UT(and, accordingly, additional groups of the trapped particles) can appear in elongated tokamaks with a high elongation b/a > 2 and high central EҠo.Of course, the preferable regimes-configurations are those with the one minimum of H0(UTsince the new-additional groups of the trapped particles can provoke the additional plasma-wave instabilities, modify the transport processes and so on.

Fig. 2. The dependence of the equilibrium magnetic field H0(UTon the poloidal angleTin tokamaks with elliptic (a) and D-shaped (b) magnetic surfaces.

Another important characteristic of tokamaks is a tokamak safety factor:

¸¸

¹

·

¨¨

©

§

3

G U

U U

U S

S G U D GU E

U U

1 2

0 1 1

2 0

1

2 0

1 2 2 0

2

2 0

1 2

, 4 2 , 4 2 2 1 2

) 1 1 ( 2

1 )

( R aR

aR aR

R aR R

aR R R

aR q q

o

o (10)

where

G E

D

2 0 0

2

1 1

) 0

( R R

q aR q

o

o is the safety factor at the main magnetic axis (R R₀ or U 0); and

³

3 ^K

M N M O N M

O

K _{0 2 2 2}

sin 1

sin ) 1

, ,

( d

(11) is the elliptic integral of the third kind. Note that ERand q_o are self-consistent, i.e.,

) (

) 1 (

2 0 2 0 2

2 2

1 2

G E D

R R q

R a

o

o . (12) The radial structure of q(U) in the low (a) and large (b) aspect ratio D-shaped tokamaks is present in Fig.3.

Fig. 3. Radial structure of q(U) in the low (a) and large (b) aspect ratio tokamaks:

a) Low Aspect Ratio Tokamak R1=0.7 m, a=0.5 m, b=1.25 m, d=0.15 m;

b) Large Aspect Ratio Tokamak R1=3.0 m, a=1.0 m, b=2.5 m, d=0.3 m.

3. Drift-Kinetic Equation

Using the conservation integrals v||²v_A² const and v_A² /2H₀ const , the new variables v and P in velocity space can be introduced instead of v|| and v_{A as}

v² v_||² v_A² ,

) , (

1

2 2

||

2

T P U

G v v

v

A A

. (13)

As a result, the drift-kinetic equation for the perturbed distribution functions,

( , , , ) ( , , , )exp( )

|| 1 U T P Z I

T

U v v f v i t in

f

s¦ s

A r (14)

in the zeroth order over the magnetization parameter, can be reduced to

F v E s v M f e R H

H ins v f

H Z H R

H s v f i

T s

s Z

R

s 2 ||

||

0

||

0

|| 2

w

¸ w

¹

¨ ·

©

§

w w w w

^I

T T

Z T (15)

where

^H₀ ^H_0I^G(U,T), R R₀² 2aR₁UcosT ,

T U D G

T E

T I 2 cos

1 ¹

2 2 0 1

0 R aR

aR H H Z

H_R R _Z ^o

w w w

w , (16)

¸¸

¹

·

¨¨

©

§ ₂

2 3

5 .

1 exp

T

T v

v v

F N

S ,

M v_T2 2T

.

By s r1 we distinguish the perturbed distribution functions fs , with positive and negative values of the parallel velocity (relative to H) v_|| sv 1-P ˜G (U,T).

4. Trapped and Untrapped Particles The parallel current density components can be calculated as

¦ ³ ³

r f

˜ ¹

0

) , ( / 1 0 3

||( , ) ( , ) ( , , , )

s

G s 0

dv d v f

v s H eG

j U T ^{j H0} S U T ^UT U T P P . (17)

Analysing the conditions

0 ) , ( - 1 )

,

||(P T sv P˜G U T

v (18) the phase volume of plasma particles should be split in the phase volumes of untrapped, t-trapped and d-trapped particles:

0dPdP_u S dT dS - for untrapped particles

P_u dPdP_t Tt dT dTt - for t-trapped particles

P_t dPd P_d T_t dT dT_d - ford-trapped particles

d

t P P

P d d T_d dT dT_t - ford-trapped particles

where the reflection points rT_t and rT_d for t-trapped and d-trapped particles can be defined by solving the equation ^G(U^,T⁾ P and the extremumsPuPt andPdare defined by

) ,

(U S

P_u G r , P_t G(U,0), P_d G(U,T_max) with the angles T_max satisfying the equation dg(U,T)/dT 0. In particular,

¸¸

¹

·

¨¨

©

§

¸¸

¹

·

¨¨

©

§

U G D

D U E

U P

2 0 1 2

2 0 2 2

2 0 1

2 1 1

1

2 1

R aR R

R aR

o

u ,

¸¸

¹

·

¨¨

©

§

¸¸

¹

·

¨¨

©

§

U G D

D U E

U P

2 0 2 1

2 0 2 2

2 0 1

2 1 1

1

2 1

R aR R

R aR

o

t . (19)

Using the boundary conditions

1) periodicity of fs overT for u-particles;̓

2) continuity of fs at the stop-points rT_t and rT_d for t-trapped and d-trapped particles, the perturbed distributions can be found in the form

¦^rf »

¼

« º

¬

ª

p u

u p s u

s inq

nq T p i f

f ( ) ( )

) (

2

, exp W T T T

S , (20)

¦^f

»»

¼ º

««

¬

ª

#

p td

d t

p s d

t

s inq

p T i f

f exp 2 ( ) ( )

, ,

,

, W T T T

S , (21)

wherep is the number of the bounce resonances;

³

T

G T K U

U P

K K T U

W

0

2 0

1 cos

1 2 ) , ( 1

) , ) (

(

R G aR

d

G (22)

is the new time-like variable to describe the bounce-periodic motion of u-,t- , d-particles; the corresponding bounce periods are: T_u 2W(S), T_t 4W(T_t)^, T_d 2(W(T_t)W(T_d)); and

¸¸

¹

·

¨¨

©

§

3

¸¸

¹

·

¨¨

©

§

3

U G

U U

U S

U G

U U

U T

S T T

1 2

0 1 1

2 0

1

1 2

0 1 1

2 0

1

2 , 4

2 , 4 2

2 , 4

2 , 4 ) 2

(

aR R

aR aR

R aR

aR R

aR aR

R aR

(23)

is the new poloidal angle for coordinates where the magnetic field lines are ‘straight’. The Fourier harmonics f_s^u_,p, fs^t_,p and f_s^d_,p can be derived after the corresponding bounce- averaging.

5. Parallel Permittivity Elements

To evaluate the parallel permittivity elements we use the Fourier expansions of the current density and electric field over T ^:

¦^rf

¸¸¹

·

¨¨©

§

m

m im

R j aR R

aR G

j 2 cos exp( )

1 2 cos

) 1 , (

) (

2 ||

0 1 2

0

|| 1

T G T

T U U T

U

T , (24)

¦^rf

^'

'

||

2 0

1

|| exp( ' )

2 cos 1

) , ( ) (

m

m im

E R

aR G

E T

G T U

T U

T . (25)

As a result,

¦

¦^{rf rf}

'

'

||

' ,

||, ' ,

||, ' ,

||, '

'

||

' ,

||

|| ( )

4

m

m m m

d m m

t m m

u m

m m m

m E E

i j

H H

H Z H

S (26)

and the contributions of u-,t-trapped and d-trapped particles to H_||^m,m' are

> @

¦ ³^f

f

¸¸

¹

·

¨¨

©

§

3

¸¸

¹

·

¨¨

©

§

p p p p

m p m p u

T

o p m

m u

u u i u W u d

nq p

A A T aR

R aR aR

R v aR

R aR R

q aR

R P

P S

U G

U U

S U S

G U Z U

H

0

3 2

2 '

1 2

0 1 1

2 0 2 1

2

2 0

1 2

0 2 1

2 0 2 '

,

||, 1 2 2 ( )

) ( 2

, 4 2 , 4 4 2

1 2 1 2

> @

¦ ³^f

¸¸

¹

·

¨¨

©

§

3

¸¸

¹

·

¨¨

©

§

1

3 2

' 2

1 2

0 1 1

2 0 2 1

2

2 0

1 2

0 1 2

2 0 2 '

,

||, 1 2 2 ( )

2 , 4

2 , 4 4 2

1 2 1 2

p t

u

p p p

m p m p t

T

o p m

m

t B B v i v W v d

p T aR

R aR aR

R v aR

R aR R

q aR

R _P

P

P S

U G

U U

S U S

G U Z U

H