Impurity Perturbations by Sawtooth Crashes

In order to estimate the transport coefficient by the LSQ and GF methods, it is necessary to produce a substantial perturbation in the impurity fluxΓz and study the dynamic response of the impurity density profiles. Such perturbation can be generated externally by a laser blow-off [145], gas puff injection [146] or continuous harmonic source like ICRF [13]. The other option is to investigate perturbations in the intrinsic or injected impurity density generated by the sawtooth crashes [117] or a hybrid approach, where the external and internal perturbations are employed to improve accuracy over the whole plasma radius [147].

The discharge #32324 presented in Sec. 8.1.1 is well suited for the transport analysis during sawtooths cycles. The flattening by sawtooth crash and the following build-up phase, increasing the core W density by almost two orders of magnitude, generates sufficient variation in the impurity flux to estimate transport coefficients with reasonable

8.2. Impurity Perturbations by Sawtooth Crashes

accuracy. Additionally, a large number of the sawteeth provides the necessary statistics to assess the significance of the measured changes in the transport. On the other hand, this method cannot be used to infer the W transport coefficients outside of the sawtooth inversion radius (ρ_tor ≈0.26), because there the change of the flux is too small, while the uncertainty in the W density is significantly increased.

The interpretation of the inferred transport coefficients must be made with cau-tion because the assumpcau-tion of temporally constant transport coefficients applied in the transport analysis methods is not necessarily fulfilled. Since the gradients of Ti

and n_i are evolving during a sawtooth cycle, the transport coefficients will evolve as well. The impurity pinch (Eq. (5.25b)) will increase from zero after the crash when all the profiles are flattened to the largest value at the end of the sawtooth cycle when the profiles are the most peaked. As D can be interpreted as a²/τ, for the same characteristic length a, D is proportional to an inverted timescaleτ of the transport process. We can describe a sawtooth crash by two coupled recovery processes. The first applies to the background and the second to the impurity density. The effective diffusion coefficient of this coupled process is given by

D_eff⁻¹ ≈D⁻¹_W +D₀⁻¹, (8.3) where D₀ is diffusion coefficient of the background. If the tungsten diffusion coefficient D_W is significantly smaller than the background diffusion coefficientD0, i.e. D_W D0, the measured diffusion coefficient D_eff will correspond to the average of D_W during the investigated period. However, if D_W D₀, the tungsten density profile will be in the equilibrium with the background and the timescale of the tungsten evolution will depend solely on D0 and therefore D_eff will underestimate the real value ofD_W. Unfortunately, D₀ cannot be properly measured, due to the lack of core electron density measurements. We have to accept that the largest measured values of D_eff can underestimate the real D_W. On the other hand, the equilibrium value of the drift coefficient v/Dwill be unaffected.

8.2.1. Evaluation of the Transport Coefficient

The transport analysis was performed for each sawtooth within the constant phase of the current flattop in the discharge #32324. The beginning of the sawtooth cycle, influenced by the post-cursor was removed, and the quality of the data fits by the GF and the LSQ methods were carefully verified for each build-up phase. Fig. 8.3 shows the transport coefficients evaluated from the two sawtooth cycles presented in Fig. 8.2.

The radial profiles obtained from the GF and the LSQ methods are rather similar, demonstrating the consistency between the results obtained with these two methods.

8. Influence of Asymmetries on Radial Impurity Transport

Apparently, the diffusion coefficient for the sawtooth crash during the Q3 phase (cf.

Fig. 8.3a) is significantly larger than the coefficient estimated in the Q4 phase (Fig. 8.3d).

On the other hand, the drift coefficientsv/D in Fig. 8.3d,e have a similar shape and magnitude. This is consistent with the previous observation of the self-similar tungsten density profiles in both phases (cf. Fig. 8.1f).

The transport coefficients for each sawtooth cycle are depicted in Fig. 8.4. The values were evaluated close to the minima of the drift coefficient at ρ_tor=0.12, because there was also the largest change in the normalized particle flux and thus higher credibility of the results. The diffusion coefficient is significantly reduced from about D_W = 0.2–0.4 m²/s in the Q3 phase to D_W = 0.03–0.08 m²/s in the Q4 phase indicating a substantial impact of the poloidal asymmetry on the radial transport. Moreover, the LSQ method predicts slightly lower values ofD_W in the Q4 phase than GF method.

The drift coefficient v/D is in both phases between −20 and −40 m⁻¹ as is depicted in Fig. 8.4b. Also, in this case, a few points in the Q4 phase estimated by the LSQ method significantly deviate from others, which is a consequence of an underestimated D_W since D_W and v/D are correlated for small values of D_W. Otherwise, both methods typically match within the error bars.

8.2.2. Uncertainty Analysis

The estimation of the uncertainty in the transport parameters is not a trivial task due to the error propagation from the SXR tomography, theT_e,n_e profiles and the atomic data tonw and further propagation to the strongly regularized transport coefficients. The tungsten cooling factor L_W grows almost linearly with T_e in the relevant temperature range between 2.7–4 keV (cf. Fig. 8.2). SinceT_e was changing in the examined regions by about 30 %, the uncertainty inL_W should not significantly affect the W density varying by two orders of magnitude. Moreover, the transport coefficients are independent of the error in the scaling factor of the impurity density. The same argument is valid for n_e, changing by only 20 % during the sawtooth cycle. The main uncertainty in n_W thus comes from the SXR tomography. The radial resolution of the tomography is limited by the width of the LOS. A typical value for the LOS width in the plasma core is∆ρ_tor≈0.1 (cf. Fig. 4.1), resulting in the resolution of the radial gradients down to

∆ρ_tor ≈0.2, which is a rough guess of the maximal spatial resolution of the transport coefficients. The expected uncertainty in the W density estimated from all previous contributions is about 20 %.

The optimization problem solved by the LSQ and the GF method is in principle highly overdetermined because 12 free parameters of the transport coefficients are determined from n_r×n_t ∼ 30×200 measurements. However, the dependence on the temporal and spatial gradients and a correlation of v/D and D makes this problem also very