Neoclassical ion transport in the edge of axially-symmetric arbitrary cross-section tokamak with plasma subsonic toroidal flows

Abstract

General metrics of large aspect ratio tokamaks is used in the paper. General expressions for the neoclassical poloidal plasma rotation <img src="http:/img/fbpe/bjp/v32n1/03eq01.gif"> and radial ion heat flux gammaTi are obtained. Their dependence on the squared Mach number alpha = <img src="http:/img/fbpe/bjp/v32n1/03eq02.gif" align="absMiddle"> is analyzed (here Uzetai is the ion toroidal velocity and c s is the sound velocity, respectively). Some interesting peculiarities of this dependence are emphasized.

Neoclassical Ion Transport in the Edge of Axially-Symmetric Arbitrary Cross-Section Tokamak with Plasma Subsonic Toroidal Flows

J.H.F. Severo, V.S. Tsypin, I.C. Nascimento, R.M.O. Galvão,

M. Tendler¹, and A. N. Fagundes

Institute of Physics, University of São Paulo,

Rua do Matão, Travessa R, 187, 05508-900 São Paulo, Brasil

¹ The Alfvén Laboratory, EURATOM-Nuclear Fusion Research,

Royal Institute of Technology, 10044 Stockholm, Sweden

Received on 6 June, 2001 2

General metrics of large aspect ratio tokamaks is used in the paper. General expressions for the neoclassical poloidal plasma rotation and radial ion heat flux G_Ti are obtained. Their dependence on the squared Mach number a = is analyzed (here U_{z i} is the ion toroidal velocity and c_s is the sound velocity, respectively). Some interesting peculiarities of this dependence are emphasized.

I Introduction

The theoretical and experimental study of the neoclassical ion transport in edge tokamak plasmas are nowadays of renewed interest.^1-4 In particular, these studies are important in the so-called H-regimes in large toroidal facilities, characterized by sharp gradients in the radial profiles of macroscopic plasma quantities in the region of the transport barriers.^{1, 2} Plasmas of small tokamaks, which play an important role in investigating different physical phenomena in fusion research, are mainly in L-regime. Nevertheless, some problems regarding the neoclassical ion transport in these machines are not yet well understood and additional efforts are needed.^{3, 4} One of these problems is addressed in this paper, namely the neoclassical ion transport in the edge of an axisymmetric plasma column of arbitrary cross-section and with subsonic toroidal flows. These flows can be induced by parallel neutral beam injection or by radio frequency waves.

II Starting equations

The main neoclassical quantities observed experimentally are the ion radial heat flux and the ion poloidal velocity.^{1, 2} Until now there is no satisfactory theoretical explanation for the damping rate t_z of toroidal rotation, which is usually supposed to be approximately equal to the energy confinement time t_E, t_z» t_E.⁵

We follow Refs. 6 and 7, where the magnetic surface averaged ion radial heat flux for collisional plasmas was first obtained. We use the toroidal coordinates r,q,z, where r is the magnetic surface label, and q and z are poloidal and toroidal angles, respectively, assume axial symmetry, i.e., ¶/¶z = 0, large aspect ratio, and smooth profiles of the macroscopic plasma quantities. The magnetic surface average

of the radial contravariant component q_i^r of the ion heat flux (see Ref. [8]),

leads to

Here, T_i is the ion temperature, = T_i- áT_iñ is the oscillatory part (perturbation) of the ion temperature, p_i = nT_i, n = ánñ + is the plasma density and is the oscillatory part of it, w_ci = e_iB/cM_i is the ion cyclotron frequency, B and B^z are the magnitude and the z-contravariant component of the magnetic field B, respectively, and g¹¹ and g¹² are the contravariant components of the metric tensor , whose determinant is denoted g.

In obtaining Eq. (3) , we used also the expressions for the contravariant components of the magnetic filed

where c and f are poloidal and toroidal magnetic filed fluxes, respectively, and

The last equation follows from the condition

j

^r» 0, where

j

^r is the

r-contravariant component of the plasma current

j. The inequality

=

A- á

Añ

á

Añ was also employed, where

A stands for the plasma macroscopic quantities.

We see from Eq. (3) that, in order to find G_Ti, we need to calculate and . The equation for follows from the ion heat transport equation⁸

where

is the q-contravariant component of the ion velocity

V

_i, and

Perturbations of the electron temperature are neglected in Eq. (6) (see explanations after Eq. (18)). To find

one can employ the oscillatory part of the parallel component of the plasma one-fluid momentum equation

where

p =

p

_i +

p

_e =

n (

T

_i +

T

_e) is the plasma pressure and p

_|| is the parallel ion viscosity,

^9,10 and

Here we used also the well-known approximate expression (see, e.g., Ref. 3)

where p is the ion viscosity tensor. The velocity component

can be found from the poloidal average of the parallel component of the momentum equation (8)

The parallel ion viscosity p_||, entering Eqs. (8) and (9), is defined by^3,4,9,10

where

and

Thus, we have Eq. (3) to determine the radial heat flux G_Ti, and Eq. (9) to find the q-contravariant component of the plasma velocity . To calculate them we should solve Eqs. (6), (8), and (9).

III Solution of the perturbed equations

We find from Eq. (8)

and, consequently,

where a =

M

_i

/(

T

_i +

T

_e) is the squared Mach number,

=

(

T

_i +

T

_e) +

n

. To obtain Eq. (14), we used the covariant differentiation rules

and imposed that the metric tensor components

g

₁₂ =

g

₂₁ are periodical functions of the angle q.

Using Eqs. (6) and (7), we obtain the second order differential equation for the perturbed ion temperature

,

where

b = B²/B^q2 is the collisionality parameter, and l_i = is the ion mean free path. The order of the parameter B²/B^q2 for large aspect ratio tokamaks is approximately q²R², where q is the safety factor and R is the torus major radius. Thus we approximately have b = q²R²/ . For the collisional plasma the parameter b > 1. We consider the range 1 < b in this paper. In this case one can omit perturbations of the electron temperature in Eqs. (6) and (17). For the range 1 < b M_i/M_e, these perturbations should be taken into account (see, e.g., Ref. 3).

As far as the function f(q) and, consequently, are periodical in q and moreover, proportional to sinq for the circular cross-section tokamak³ and to sinq and sin2q for elliptical cross-section tokamak,⁴ the solution of Eq. (17) has the form

where

Comparison of Eq. (15) with Eqs. (17)-(20) shows that, to zero approximation, one can use

This expression can be substituted into Eqs. (3), (11), (12) and (18). The surface averaged parallel component of the momentum equation (9), taking into account Eq. (8), can be transformed into the form

Using also the identity

we get

Thus we can express every perturbed value via the oscillatory parts of the metric components g₂₂ and g₃₃.

IV Ion fluxes

Let us simplify expressions for perturbed values using Eqs. (21) and (24). The Fourier component of the perturbed ion temperature has the form

where

d

_s (

b) =

s

² + 2.17b

. The parallel viscosity p

_||, Eqs. (10)-(12), can be expressed in the form

Equation (22) can be transformed into

Substituting Eqs. (25) and (26) into Eq. (27), one finds the poloidal velocity

(to be compared with Ref. 12),

where

Equation (3) for the surface averaged ion heat flux can be rewritten as follows

Substitution of Eqs. (19), (25), (21), and (24) into Eq. (36) results in

where

V Estimates

Let us analyze the expressions for the ion poloidal velocity Eq. (28) and the ion heat flux Eq. (37) in a general case. When the squared Mach number a vanishes, a = 0, one obtains from Eq. (28)

which agrees with results of Refs. 11, 3, and 2. Equations (28) and (39) also confirm the Hazeltine theorem,¹³ which says that the so-called residual plasma poloidal rotation in tokamaks depends only on the ion temperature gradient and not on gradients of other macroscopic plasma parameters. Estimates show that the parameters A₂₃ and D₂₃ are negative, (A₂₃ < 0 and D₂₃ < 0). Hence, the parameter f₁ (a, b, A, D) is positive, f₁ (a, b, A, D) > 0, i.e., the denominator in Eqs. (28) and (37) is positive and has no roots as a function of a. A remarkable property of the poloidal velocity is the change of sign at a value a₀ of the parameter a. This results from the fact of taking into account inertial forces in the starting equations. Assuming that the poloidal velocity changes sign at a < 1 (see in detail Ref. 11), we find from Eq. (30)

From the approximate equation,

one finds the critical quantity a

_k,

corresponding to the maximum of the poloidal velocity

,

For a > a

_k, the poloidal velocity

decreases slowly with the growth of a.

Analysis of Eq. (37) shows that the magnetic surface averaged radial ion heat flux G_Ti is an increasing function of the parameter a. The factors that characterize the non-circularity of the plasma cross-section, such as ellipticity, triangularity, reduce the role of neoclassical effects in G_Ti for all values of the parameter a. These results coincide with the previous studies of this problem, fulfilled for elliptical and circular cross-section tokamaks,^3,4 and we demonstrate them here.

VI Elliptical tokamak

In the case of the elliptical tokamak we find from Eqs. (28)

where

U

_iq = r

,

U

_Ti = (1/

M

_iw

_ci) ¶

T

_i/¶r, r =

,

₁ and

₂ are the semiminor and semimajor axes of a tokamak cross-section, respectively,

d(b) = 1 + 2.2b. Obtaining Eq. (44), the parameter

was assumed to be small,

A

1.

Using Eq. (37), we derive the magnetic surface average of the radial ion heat flux in the Shafranov form,⁶

where

Equations (44) and (49) coincides with the proper equations of Ref. 4. As far as the coefficient in Eq. (49) is less than 1, < 1, the role of neoclassical effects in the radial ion heat flux decreases in noncircular cross-section tokamaks in comparison with circular cross-section ones.

The quantities G_u(a, b) and G_T(a, b) are plotted in Figs. 1 and 2, respectively. One can see that function G_u(a, b) (Fig. 1) changes sign at a₀» 2d(b)/b. For the collisional parameter b ~ , the quantity a is equal to a₀» 0.1. The maximum of function G_u(a,b) is achieved when b_m» 50, a_m» 1, and is G_u(a_m, b_m) » 3. It follows from Fig. 2 that the neoclassical contribution in the radial ion heat flux, as a function of a, is dropping with increasing b.

VII Conclusions

General expressions for neoclassical poloidal plasma rotation and radial ion heat flux G_Ti for an axially-symmetric arbitrary cross-section tokamak edge with plasma subsonic toroidal flows are obtained in the present paper. Their dependence on the squared Mach number a is analyzed. It is shown that there is a remarkable property of the poloidal velocity to change sign at a value a = a₀, which results from taking into account inertial forces in the starting equations. There also exists a critical value of a, a_k, which corresponds to the maximum of the poloidal velocity . For a > a_k, the poloidal velocity is a decreasing function of a. Analysis of the magnetic surface averaged radial ion heat flux G_Ti shows that this flux is an increasing function of the parameter a. The noncircular cross-section of a tokamak decreases the role of neoclassical effects in G_Ti for any value of the parameter a. These results confirm previous studies of this problem.

Acknowledgments

This work was supported by the Research Support Foundation of the State of São Paulo (FAPESP), National Council of Scientific and Technological Development (CNPq), and Excellence Research Programs (PRONEX) RMOG 50/70 grant from the Ministry of Science and Technology, Brazil.

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Publication Dates

• Publication in this collection
  26 Nov 2002
• Date of issue
  Mar 2002

History

• Received
  26 June 2001