# Ripple transport and neoclassical diffusion in IR-T1 tokamak

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## Abstract

In tokamaks, small variations in the magnetic field create ripple. The discontinuous nature of the magnetic field coils in an axisymmetric torus conduces to additional particle trapping, and it is responsible for an additional neoclassical diffusion. Ripples also reduce the particle removal efficiency and disturb plasma confinement and cause constraints in the design of magnet of fusion reactor. Therefore, it is quite important to include the ripple for the design of plasma edge components. Herein, several considerations are taken into account to calculate and evaluate the diffusion coefficient and ion heat conductivity in ripple transport and also to compare it with neoclassical mode.

## Keywords

Neoclassical diffusion Ripple magnetic field Ripple transport Diffusion coefficient Diffusivity Ion heat conductivity## Introduction

The axisymmetry of the magnetic field has been assumed as one of the substantial advantages of tokamaks. Apart from the simplicity of analysis, the radial motion of particles trapped in low collision frequencies is less than that of asymmetric fields [1].

It is expected that in an axisymmetric magnetic field configuration of a tokamak D-T reactor, fusion-produced alpha particles are well confined and thus provide for efficient internal plasma heating. The presence of magnetic perturbations, however, may greatly affect the retention of these particles and lead to their loss from the confinement region [2].

Small variations in the magnetic field are called magnetic ripple. In practice, the discrete nature of the magnetic field of the toroidal coils around the torous creates a ripple in the field intensity with a period equal to the distance between the centers of the coils. This perturbation is toroidal field ripple (TFR).

Ripple has negative effects on fusion plasma performance including the fusion energy reduction due to particle loss, the plasma beta reduction and the rotation break. Generally, various magnetic perturbations like toroidal field ripples can degrade fast ions confinement. In the outer region of tokamak plasmas due to loose winding of toroidal coils, TF ripples have strongest effect and can cause fast ions to loose very quickly which in turn can also damage the first wall of the tokamak [2, 3, 4, 5].

The ripple strength not only enhances the diffusion coefficient but also changes the energy dependence [6].

An ideal, axisymmetric tokamak has no ripple. In real-life machines, the finite widths of the toroidal field coils create a periodic ripple [7].

Deviation from the symmetry is generally considered to be fiddling. This effect will be more important probably in the next generation of machines, for two reasons. It is expected to reach higher temperatures which is accompanied by upsurging of asymmetry effect as the frequency of collision reduces. In some collisional regime for the present and next generation of tokamaks, the ripple diffusion changes inversely to frequency of collision [1].

In “Ripple transport” section, we discuss about the ripple transport and the ripple magnetic field in the toroidal coordinate and superbanana diffusion. In “A solution of the Fokker–Planck equation for trapped particles in the magnetic field ripples” and “Diffusion and heat transport” sections, the Fokker–Planck equation is presented for particles trapped in the ripple magnetic field and the equations for the particle flux and heat flux are obtained to calculate the diffusion coefficient and the ion heat conductivity and finally transport coefficients are compered in neoclassical and ripple mode for IR-T1 tokamak.

## Ripple transport

The ripple transport is due to the drift motion of particles distributed in the ripple of a magnetic field. The moderate number of toroidal field coils of a tokamak expunges the perfect axial symmetry of the system. The coils create a short wavelength in the magnetic field strength, namely “ripple,” as a field line is pursued around the torus.

*N*coils, the magnetic field can be presented by

*r*is the radial distance from the magnetic axis,

*R*is the radius of the magnetic axis and \( \varepsilon = r/R \) is the inverse aspect ratio.

*R*

_{outer}and

*R*

_{inner}are, respectively, the major radii of the outer and inner legs of the toroidal coils space [3].

For a particle to be trapped in the ripple wells it has to satisfy the condition that the parallel velocity sufficiently small that magnetic mirror reflection occurs in the ripple well. This is equivalent to the statement that the banana turning point positions of the particles lie inside the ripple well region. Particles are trapped into and re-trapped out of ripple well by collisional processes, in particular by pitch-angle scattering, which change the ratio between the parallel and perpendicular velocities. The typical residence time of a particle in ripple well is of the order \( \Delta t\sim\delta /\nu \), where \( \nu \) is the collisional deflection frequency. The fraction of particles which is ripple well trapped is of order \( \delta^{1/2} \).

*v*is the total particle velocity. Incorporating the ripple well trapped fraction, the drift velocity and the residence time, a diffusion coefficient for collisional ripple well transport can be estimated as:

*D*is radial diffusion coefficient for a circular plasma, averaged over the pitch angle. This implies a greater loss rate for ions than for electrons, and an ambipolar electric field will evolve to reduce the ion particle diffusion. Nevertheless, Eq. (2) remains an appropriate estimate. It should, however, be noted that its validity is restricted to collision frequencies \( \nu > v_{\text{d}} \delta /a \), where

*a*is plasma minor radius; otherwise, particles would drift out of the plasma before suffering a collision, producing a loss cone in velocity space rather than a diffusion. This restriction is particularly stringent as

*D*which is dominated by contributions from the more energetic particles, so that the inequality must hold for these particles. (Energies are approximately five times the plasma temperature.) The proper poloidal averaging procedure leads to a significant reduction of

*D*from that given in Eq. (2) [3].

*N*[3]. He showed that for values of \( \alpha = \varepsilon t/N\delta \) (\( t \) is the rotational transformation) in excess of unity the ripple does not outcome in the constitution of local magnetic mirrors and resultant trapping over the whole minor azimuth, and that, generally for \( \alpha \) of order unity, considerable diminution of the mirror depth arises at all angles. This can be seen from Eq. (1) by noting minimum in \( B_{l} \) (where \( l \) is arc length along a field line) arise only if \( \alpha \sin \theta < 1 \). Even when this retains, each ripple is an asymmetric mirror with mirror rotation \( R_{ \pm } (\theta ) \) where

Stringer’s estimation of the decrease in diffusion and ion heat flux is attained by including this decrease in the well depth from \( 2\delta \) to \( \Delta (\theta ) \). Though this effect is a highly important factor, it undermines the asymmetric distortion of the ripple wells. In the next part, here we offer the solution of the Fokker–Planck equation for the ripple-trapped distribution, and then we deduced the resultant transport coefficients.

*j*and \( \omega_{cj} \) is the gyrofrequency. Physically, it means that the effective collision frequency \( \nu_{\text{eff}} = \nu /\delta \) is less than the bounce time of a ripple-trapped particle \( \omega_{b} = \delta^{1/2} v_{\text{Th}} N/R \), but higher than the drift frequency circa of a complete superbanana orbit in the mirror azimuth of the torus [8, 9, 10, 11].

## A solution of the Fokker–Planck equation for trapped particles in the magnetic field ripples

*q*along the field. The gradient operators in Eq. (5) are taken into account at constant energy

And since \( \frac{\partial (\mu J)}{\partial \mu } = - \int_{{\phi_{1} }}^{{\phi_{2} }} {\frac{{{\text{d}}\phi \,}}{q}} (\mu B - q^{2} ). \)

And \( \frac{{q^{2} }}{\mu B} \sim O(\delta ). \)

## Diffusion and heat transport

*D*, when \( \nu > \delta \omega_{E} \) and \( \omega_{E} = E_{r} /rB \), the radial excursion of a ripple banana is limited by collisions and Eq. (2) is still true. When \( \omega_{E} (\varepsilon T/e)^{4} < \delta \omega_{E} < \nu \), the ripple diffusion is

The simple form for radial profiles of tokamaks can be considered as follows: \( n(r) = n_{0} (1 - x^{2} ), \)\( T_{i} = T_{i0} (1 - x^{2} ), \)\( \delta (r) = \delta_{a} x^{2} \)\( ,\,\,x = r/a \) [1].

The ratio of particle and heat transport for the ripple contribution can be written as follows: \( \chi_{iR} /D_{R} \sim2(m_{i} /m_{e} )^{1/2} (T_{i} /T_{e} )^{7/2} (1 + T_{i} /T_{e} )^{ - 1} \) while for toroidal contributions can be written as: \( \chi_{ib} /D_{b} \sim0.5\,(m_{i} /m_{e} )^{1/2} (T_{e} /T_{i} )^{7/2} (1 + T_{i} /T_{e} )^{ - 1} \). In the condition of present tokamaks where \( T_{i} /T_{e} \sim0.5 \), the \( \chi_{iR} \) is relatively less important than that of in the condition which is \( T_{i} = T_{e} \).

This difference in the results is due to the difference between values of \( T_{i} /T_{e} \) [14].

## Experimental setup and results

IR-T1 is an air-core transformer-type tokamak with a circular cross section, low beta and large aspect ratio, which has two stainless steels grounded fully, poloidal limiters and ohmic heating discharge system. The average pressure before discharge is in the range of 2.5–2.9 × 10^{−5} Torr. It contains magnetic, electric, rake Mach probes and Mirnov coils for plasma diagnostics. Also it contains toroidal coils, ohmic coils and central solenoid and vertical coils. In IR-T1, 16 toroidal field coils produce a magnetic field around the torus to confine the plasma.

Some of IR-T1 tokamak characteristics

Parameters | Value |
---|---|

Major radius ( | 0.45 m |

Minor radius ( | 0.125 m |

Material of first wall | Stainless steel |

Limiter type and diameter ( | Ring limiter and 0.250 |

Ripple at plasma edge and number of coils | 16 |

Toroidal magnetic field ( | 0.6–0.9 T |

Plasma current ( | Up to 40 kA |

Discharge duration | < 35 ms |

Energy confinement time | 1–3 ms |

Electron density | 0.7–1.5 × 10 |

| 2.6–8 V |

The ripple ratio \( \delta_{B} \) (magnitude of the magnetic field ripple) \( \delta_{B} = (B_{\hbox{max} } - B_{\hbox{min} } )/(B_{\hbox{max} } + B_{\hbox{min} } ) \) for IR-T1 with 16 coils is calculated as \( \delta_{B} \simeq 0.6 \). For TorSopra, \( \delta_{B} = 0.5 \), for JET, \( \delta_{B} = 0.08 \), for ASDEX, \( \delta_{B} = 0.6 \) and for JT060u \( \delta_{B} = 1 \) [17].

Losses due to particle and energy transport can create many limitations in the design of the reactor magnet. Of course the number of toroidal coils can affect the dissipation of energy. The amplitude of the ripples is clearly lower when the number of toroidal coils is higher. For example, in JET tokamak, by reducing the number of coils from 32 to 16, the ripples level increased from 1 to 10% [18, 19, 20, 21, 22, 23, 24].

## The variation in transport coefficients on IR-T1 tokamak

As we see in Figs. 5 and 6 in the IR-T1 tokamak conditions, particle diffusion coefficient and heat conductivity do not differ greatly. On both graphs, the neoclassical coefficient is decreasing by radius but the ripple distribution is increasing.

Also comparison of the plots shows that the thermal conductivity coefficient increases by increasing the radius, but this increase is faster than the diffusion coefficient with a factor about 4.1.

## Conclusion

In this work, we investigated transport coefficients (diffusivity and ion heat conductivity) considering the effects of ripples in the magnetic field. In addition to the neoclassical diffusion, we observed the ripple diffusion and compared the transport coefficients for these two modes. The ripple ratio is also calculated numerically, \( \delta_{B} \simeq 0.6 \), for IR-T1 tokamak magnetic field. The observations indicate that diffusivity and ion heat conductivity decrease, as we move away from the plasma center in neoclassical mode. But in ripple mode, transport coefficients increase with the radius, and the rate of growth of the conductivity coefficient is greater than the diffusion coefficient.

## Notes

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