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physical series «Nuclei, Particles, Fields», issue ɍȾɄ 621.039.61

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N.A. Azarenkov, Zh.S. Kononenko Numerical study of first orbit losses...

NUMERICAL STUDY OF FIRST ORBIT LOSSES OF TRAPPED PARTICLES IN TOKAMAKS N.A. Azarenkov, Zh.S. Kononenko
V.N. Karazin Kharkiv National University Svobody sq. 4, 61077, Kharkiv, Ukraine e-mail: kononenko_zh@mail.ru Received 30 April, 2009 The analytical model of the 2D tokamak magnetic field for the test particle simulations is proposed. It is shown how to vary the model parameters to obtain the plasma shape with the different ellipticity and triangularity. The problems of the particle direct losses in tokamak are discussed. The loss cones of the particles are calculated for the different initial radial locations. The minimum trapped particle energy required to escape from the plasma due to the first orbit losses is found numerically for the different ion species. KEY WORDS: rotational transform, Poincare plot, trapped and passing particles, direct losses, loss cone. ɑɂɋɅȿɇɇɈȿ ɂɁɍɑȿɇɂȿ ɉɊəɆɕɏ ɉɈɌȿɊɖ ɁȺɉȿɊɌɕɏ ɑȺɋɌɂɐ ȼ ɌɈɄȺɆȺɄȺɏ ɇ.Ⱥ. Ⱥɡɚɪɟɧɤɨɜ, ɀ.ɋ. Ʉɨɧɨɧɟɧɤɨ ɏɚɪɶɤɨɜɫɤɢɣ ɧɚɰɢɨɧɚɥɶɧɵɣ ɭɧɢɜɟɪɫɢɬɟɬ ɢɦ. ȼ.ɇ. Ʉɚɪɚɡɢɧɚ, ɩɥ. ɋɜɨɛɨɞɵ, 4, 61077, ɝ. ɏɚɪɶɤɨɜ, ɍɤɪɚɢɧɚ ɉɪɟɞɥɨɠɟɧɚ ɚɧɚɥɢɬɢɱɟɫɤɚɹ ɦɨɞɟɥɶ 2D ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɬɨɤɚɦɚɤɨɜ ɞɥɹ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɦɟɬɨɞɨɦ ɬɟɫɬɨɜɵɯ ɱɚɫɬɢɰ. ɉɨɤɚɡɚɧɨ, ɤɚɤ ɩɨɫɪɟɞɫɬɜɨɦ ɢɡɦɟɧɟɧɢɹ ɦɨɞɟɥɶɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɞɨɛɢɬɶɫɹ ɫɟɱɟɧɢɹ ɩɥɚɡɦɵ ɫ ɪɚɡɧɵɦ ɡɧɚɱɟɧɢɟɦ ɷɥɥɢɩɬɢɱɧɨɫɬɢ ɢ ɬɪɟɭɝɨɥɶɧɨɫɬɢ. ɉɪɨɚɧɚɥɢɡɢɪɨɜɚɧɵ ɜɨɩɪɨɫɵ ɩɪɹɦɵɯ ɩɨɬɟɪɶ ɱɚɫɬɢɰ ɜ ɬɨɤɚɦɚɤɚɯ. ɉɨɫɬɪɨɟɧɵ ɤɨɧɭɫɵ ɩɨɬɟɪɶ ɱɚɫɬɢɰ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɧɚɱɚɥɶɧɵɯ ɪɚɞɢɚɥɶɧɵɯ ɩɨɥɨɠɟɧɢɣ. ɑɢɫɥɟɧɧɨ ɧɚɣɞɟɧɚ ɦɢɧɢɦɚɥɶɧɚɹ ɷɧɟɪɝɢɹ, ɩɪɢ ɤɨɬɨɪɨɣ ɜɨɡɦɨɠɧɚ ɩɪɹɦɚɹ ɩɨɬɟɪɹ ɡɚɩɟɪɬɵɯ ɱɚɫɬɢɰ ɢɡ ɩɥɚɡɦɵ, ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɫɨɪɬɨɜ ɢɨɧɨɜ ɩɥɚɡɦɵ. ɄɅɘɑȿȼɕȿ ɋɅɈȼȺ: ɭɝɨɥ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɨɬɨɛɪɚɠɟɧɢɟ ɉɭɚɧɤɚɪɟ, ɡɚɩɟɪɬɵɟ ɢ ɩɪɨɥɟɬɧɵɟ ɱɚɫɬɢɰɵ, ɩɪɹɦɵɟ ɩɨɬɟɪɢ, ɤɨɧɭɫ ɩɨɬɟɪɶ. ɑɂɋɅɈȼȿ ȾɈɋɅȱȾɀȿɇɇə ɉɊəɆɂɏ ȼɌɊȺɌ ɁȺɏɈɉɅȿɇɂɏ ɑȺɋɌɂɇɈɄ ɍ ɌɈɄȺɆȺɄȺɏ Ɇ.Ɉ. Ⱥɡɚɪɽɧɤɨɜ, ɀ.ɋ. Ʉɨɧɨɧɟɧɤɨ ɏɚɪɤɿɜɫɶɤɢɣ ɧɚɰɿɨɧɚɥɶɧɢɣ ɭɧɿɜɟɪɫɢɬɟɬ ɿɦ. ȼ.ɇ. Ʉɚɪɚɡɿɧɚ ɩɥ. ɋɜɨɛɨɞɢ, 4, 61077, ɦ. ɏɚɪɤɿɜ, ɍɤɪɚʀɧɚ Ɂɚɩɪɨɩɨɧɨɜɚɧɨ ɚɧɚɥɿɬɢɱɧɭ ɦɨɞɟɥɶ ɞɜɨɜɢɦɿɪɧɨɝɨ ɦɚɝɧɿɬɧɨɝɨ ɩɨɥɹ ɬɨɤɚɦɚɤɿɜ ɞɥɹ ɦɨɞɟɥɸɜɚɧɧɹ ɦɟɬɨɞɨɦ ɬɟɫɬɨɜɢɯ ɱɚɫɬɢɧɨɤ. ɉɨɤɚɡɚɧɨ, ɹɤ ɲɥɹɯɨɦ ɡɦɿɧɢ ɦɨɞɟɥɶɧɢɯ ɩɚɪɚɦɟɬɪɿɜ ɨɬɪɢɦɚɬɢ ɩɟɪɟɪɿɡ ɩɥɚɡɦɢ ɡ ɪɿɡɧɢɦ ɡɧɚɱɟɧɧɹɦ ɟɥɿɩɬɢɱɧɨɫɬɿ ɬɚ ɬɪɢɤɭɬɧɨɫɬɿ. ɉɪɨɚɧɚɥɿɡɨɜɚɧɨ ɩɢɬɚɧɧɹ ɩɪɹɦɢɯ ɜɬɪɚɬ ɱɚɫɬɢɧɨɤ ɭ ɬɨɤɚɦɚɤɚɯ. ɉɨɛɭɞɨɜɚɧɨ ɤɨɧɭɫɢ ɜɬɪɚɬ ɱɚɫɬɢɧɨɤ ɞɥɹ ɪɿɡɧɢɯ ɩɨɱɚɬɤɨɜɢɯ ɪɚɞɿɚɥɶɧɢɯ ɩɨɥɨɠɟɧɶ. Ⱦɥɹ ɪɿɡɧɢɯ ɫɨɪɬɿɜ ɿɨɧɿɜ ɩɥɚɡɦɢ ɱɢɫɥɨɜɢɦɢ ɦɟɬɨɞɚɦɢ ɡɧɚɣɞɟɧɨ ɦɿɧɿɦɚɥɶɧɭ ɟɧɟɪɝɿɸ, ɩɪɢ ɹɤɿɣ ɦɨɠɥɢɜɚ ɩɪɹɦɚ ɜɬɪɚɬɚ ɡɚɯɨɩɥɟɧɢɯ ɱɚɫɬɢɧɨɤ ɡ ɩɥɚɡɦɢ. ɄɅɘɑɈȼȱ ɋɅɈȼȺ: ɤɭɬ ɨɛɟɪɬɚɥɶɧɨɝɨ ɩɟɪɟɬɜɨɪɟɧɧɹ, ɜɿɞɨɛɪɚɠɟɧɧɹ ɉɭɚɧɤɚɪɟ, ɡɚɯɨɩɥɟɧɿ ɬɚ ɩɪɨɥɿɬɧɿ ɱɚɫɬɢɧɤɢ, ɩɪɹɦɿ ɜɬɪɚɬɢ, ɤɨɧɭɫ ɜɬɪɚɬ.

The test particle simulations play an important role in the understanding of the processes of particle transport in the present-day fusion devices such as tokamaks and stellarators. This method allows not only to verify the properties of the particle motion in the given magnetic field configuration [1] but also to estimate the radial dependence of the transport coefficients [2,3] as well as to calculate the heat loads on the different plasma-facing components [4]. Another important application of this technique is the study of different mechanisms of particle losses. In this paper we perform the numerical calculation of the particle orbits and direct losses of the hydrogen ions in a tokamak geometry. It is an important problem with respect to the fact of energetic tail production under the ICRF plasma heating in minority heating regime [5] which could decrease the plasma confinement. The purpose of the work is the study of the particle energies required to escape from the plasma for the different radial particle locations. TOKAMAK MAGNETIC FIELD MODEL In present-day tokamaks and stellarators the plasma is confined using the strong toroidal magnetic field. In order to compensate the radial drift and provide the confinement of particles the magnetic field should be made twisted. In stellarators the confining magnetic field is fully produced by the 3D external magnetic coils. In tokamaks the set of planar toroidal field (TF) coils is used to create the toroidal magnetic field. The rotational transform is produced due to the induced current flowing in the plasma. In contrast to the inherent 3D structure of the magnetic field in stellarators the tokamak geometry usually has the symmetry over the toroidal direction. Neglecting the effects with the toroidal ripple of the magnetic field due to the finite number of TF coils (for example, the number of TF coils for tokamak JET is 32 but could be switched to 16 to study the enhanced ripple losses [6]) the tokamak magnetic field can be considered as a 2D structure.

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«Journal of Kharkiv University», No.859, 2009 N.A. Azarenkov, Zh.S. Kononenko

We use the orthogonal curvilinear quasitoroidal coordinate system , where is the radial distance from the plasma center, and are the poloidal and toroidal angles, respectively (Fig. 1). This coordinates are linked to the cylindrical via the relations: , , , where is the major plasma radius. , , . We also The Lame coefficients for the coordinate system considered are the following: which will be useful for further calculations. introduce the quantity The stationary equilibrium magnetic field should satisfy the equations: (1) , . (2)

In the low case only the toroidal component of the plasma current is non-zero. Assuming the symmetry over the toroidal direction and using the explicit expression for (3) the standard radial dependence of the toroidal component of the magnetic field is obtained (4) is the magnetic field at the plasma axis. where Now we examine the expression for the divergence of the magnetic field . Under the assumptions made it is simplified to the condition . We now introduce two scalar functions and in the following way: , (7) . Then the equation (6) transforms to the condition: . The magnetic field is expressed in terms of the scalar functions as follows: . (9) (8) (6) (5)

FACTORIZATION OF THE SOLUTION Among all the functions and which satisfy the equation (8) we will consider only the solutions that could be written in the factorized form: , (10) . Using the separation of the variables one readily obtains , where , . Due to the finiteness of the solution at . Then the magnetic field model (9) transforms to the expression . , It should be emphasized that specifying any functions the form (12) automatically satisfies the equations (1) and (2). If we set field lines , and constant (11) we have the additional condition

(12) the magnetic field model in

CIRCULAR MAGNETIC SURFACES CASE the magnetic field will have zero radial component . From the equation of the magnetic (13)

it is immediately obtained that the magnetic surfaces will have the circular form,

.

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Numerical study of first orbit losses...

We will now connect the function with the rotational transform profile equation which characterizes the twisting of the magnetic field line . Let us denote

From (13) we obtain the (14)

. Then the magnetic field is written in such a simple form: . (15) is often called to be the rotational transform when using the magnetic field in a form (15). We will The function is the rotational transform profile only in the limit of the large show below that in a general case it is not true, and aspect ratio. The solution of equation (14) with the initial conditions , can be written in the following form: . The principle branch of the solution valid for can be found analytically . For we have . Adopting the definition of the rotational transform we obtain (18) is the exact rotational transform profile which is confirmed by the numerical integration of the field line Here equations. Finally, the more accurate expression for the magnetic field model with the circular magnetic surfaces and as the rotational transform profile has the form . (19) (17) (16)

To feel the difference between two models (15) and (19), let us to use the typical parameters of the JET tokamak: , , . The radial difference between the surfaces where is about 3 cm which could become important when the magnetic islands are presented in the plasma. and

Fig. 1. The quasitoroidal coordinate system

Fig. 2. Poincare plot of the D-shaped plasma surfaces

D-SHAPED MAGNETIC SURFACES Previously the strong plasma-wall interaction was prevented by using the limiters. In the such tokamaks the magnetic surfaces have almost circular form. But now with the development of divertor concept only a few tokamaks use the limiters. Most of present-day tokamaks operate with the non-circular plasma shapes. In this section we will show how to define the model parameters to obtain the magnetic surfaces with a different ellipticity and triangularity. The numerical integration of the magnetic field line equations , (20) . The is performed by the sixth order Runge-Kutta scheme (RK6). The initial point of each line is chosen , . Each magnetic field line is followed for 2000 toroidal initial radial grid is uniform, revolutions. The Poincare plot (the collection of intersection points of the magnetic field line with the chosen poloidal

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«Journal of Kharkiv University», No.859, 2009 N.A. Azarenkov, Zh.S. Kononenko

plane after each toroidal turn) for the surface is produced. Except the rational magnetic surfaces where is the ratio of two integer numbers, the magnetic field line in tokamaks is not the closed curve but fills up ergodically some surface which is called as the magnetic surface. The elliptic surfaces are obtained if we choose . For example, defining , , the near the axis and at the plasma edge. ellipticity of the magnetic surfaces is equal to The more general form of plasma surfaces is the D-shaped configuratuion with the non-zero triangularity . Such to . For example, if we choose surfaces can be obtained by adding the term proportional to the magnetic surfaces will have the shape shown in Fig. 2. In the considered case the at the center to at the edge. The triangularity increases from zero at the center ellipticity varies from at the plasma edge. The radial dependence of and could be controlled by the appropriate choice to . of the function SIMULATION RESULTS There are two groups of particles with the different type of orbits in a tokamak. Depending on the initial particle pitch-angle and the spatial location it could be either passing or trapped. The passing particles with the substantial parallel velocity rotate around the torus and form the drift surface. This surface is close to the magnetic surface but is somewhat radially shifted inward or outward depending on the sign of . Also the drift rotational transform which characterizes the twisting of the particle orbit differs from that of the magnetic field line. The trapped particles have low parallel velocity, at some point they are reflected from the high magnetic field side of tokamak. These particles do not complete the full turn around the torus, the poloidal projection of their orbits resembles the bananas. The banana trajectories of the hydrogen ions with the energy are shown in Fig. 3. It should be turn up radially inward while the particles with have the outward noted that the trapped particles with the banana width will be enough radial motion. With the increase of the energy higher than some critical energy large for the particles to be lost from the plasma volume. In contrast to ripple losses or heating induced losses this mechanism of particle losses is always presented in tokamaks. It is usually called as the direct or first-orbit losses mechanism. The separation between the trapped and passing particles in the velocity space occurs for some value of the pitchangle . In the limit of zero banana width it is given by the expression [7] , (21) is the inverse aspect ratio of the magnetic surface considered. The result (21) stands for the case of where . For finite particle energies the values of are slightly modified and particle location at the equatorial plane the separation cone in velocity space becomes asymmetric.

Fig. 3. The banana trajectories of the trapped particles of the hydrogen with the energy b) Outward banana trajectory a) Inward banana trajectory

:

The loss cones for the hydrogen ions for two radial surfaces are shown in Fig. 4 and Fig. 5. The minimum energy required for the trapped particle to escape from the plasma volume for different radial magnetic surfaces has been calculated for three ion species: hydrogen, deuterium and fully ionized carbon ions. The results are presented in Table 1. As expected, the critical energy strongly decreases with the approach of initial surface to the edge. This happens due to the radical dependence of the banana width on the particle energy [8] , (22)

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Numerical study of first orbit losses...

where

is the Larmor radius and

is the initial radial coordinate of the particle.

Fig. 4. Loss cone for

for hydrogen ions

Fig. 5. Loss cone for

for hydrogen ions

Among the considered species the deuterium has the lowest critical energy. It is twice smaller than the corresponding energy for the hydrogen. It means that with the increase of atomic mass number the energy decreases. The carbon ions have the same charge-to-mass ratio as deuterium ions. Their critical energy is three times larger than deuterium ones. This results in the following scaling for the critical energy as a function of ion charge state number and atomic mass (Table 2): , which is in agreement with the results of formula (22). Table 1. values for different radial locations and ion species Deuterium, Carbon C6+, Hydrogen, 0.6 0.7 0.8 0.9 -0.556 -0.592 -0.626 -0.656 -0.596 -0.621 -0.644 -0.664 112 57 23 5.2 to 4/3 1 56 28.5 11.4 2.6 for different ion species 3 6.4 7.2 8.1 335 171 68.5 15.5 (23)

Table 2. The ratio of 1 1/2 1/3

The obtained dependency implies that the tritium ions have the lowest critical energy and they will be lost stronger than the rest of the ions. The impurity ions have much higher critical energy than the hydrogen isotopes. It means the impurity should be heated selectively in order to extract them without the loss of plasma confinement. CONCLUSIONS The analytical model of 2D tokamak magnetic field has been developed which allows the simulations of particle motion in plasma with the non-circular shape. The first orbit losses of particles have been simulated. The radial dependence of the minimal energy required for the particle to be lost from the confinement volume has been calculated. The obtained loss cones in the velocity phase space can be used to estimate the particle flux induced by the external heating.
1. 2. 3. 4. 5. 6. 7. 8. REFERENCES D.V. Sivukhin Motion of charged particles in electromagnetic fields in the drift approximation // Reviews of Plasma Physics, edited by M.A. Leontovich (Consultants Bureau, New York, 1965), Vol. 1, p. 1. A.H. Boozer and G. Kuo-Petravic Monte Carlo evaluation of transport coefficients // Physics of Fluids. – 1981. – Vol. 24, ʋ5. – P.851-859. Zh.S. Kononenko and A.A. Shishkin Impurity ion dynamics near magnetic islands in the drift optimized stellarator configuration of the Wendelstein 7-X // Ukrainian Journal of Physics. – 2008. – Vol.53, ʋ5. – P.438-442. E. Strumberger Deposition patterns of fast ions on plasma facing components in W7-X // Nuclear Fusion. – 2000. – Vol. 40, ʋ 10. – P.1697-1713. M.-L. Mayoral et al Hydrogen plasmas with ICRF inverted minority and mode conversion heating regimes in the JET tokamak // Nuclear Fusion. – 2006. –Vol. 46, ʋ 7. – P.S550-S563. P.C. de Vries et al Effect of toroidal field ripple on plasma rotation in JET // Nuclear Fusion. – 2008. – Vol.48, ʋ3. – P.035007035012. J. Wesson Tokamaks. - Oxford, U.K.: Clarendon Press, 3rd edition, 2004. – 749 p. E.D. Volkov, V.A. Suprunenko and A.A. Shishkin Stellarator. – Kiev: Naukova dumka, 1983 – 310 p. (in Russian)