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Tungsten accumulation is one of the main challenges for successful operation of ITER and future reactors. For this reason, various thechniques have been developed recently in order to mitigate accumulation. One of such methods is the application of wave heating, in particular electron cyclotron resonance heating (ECRH) deposited close to the plasma center.

Recent 3D equilibrium calculations with safety factor $q\approx 1$ reveal that up to half of the ITER plasma can be helical [1]. Such helical cores with dominant mode numbers *m/n=1/1* are routinely observed between sawtooth crashes in ASDEX Upgrade discharges with central ECRH [2]. The long-standing mystery of these shots, which motivated present work, is deeply hollow tungsten density profile manifested in the inverted sawteeth on the sof X-ray signals.

In the present work it is shown that ECRH-generated hot electrons can be responsible for the tungsten hole. Such electrons 'run away' along RF-induced quasi-linear diffusion path in velocity space and form strongly anisotropic population with banana tips accumulated at cyclotron resonance position on the magnetic surface. Equilibrium distribution function of this population can be approximated as follows:

$f_{0e}^h\approx \frac{n_e^h}{1+\ln (\varepsilon_{max}/\varepsilon_{min})}\delta \left (\kappa^2-\kappa_{res}^2\right )\left (\varepsilon^{-3/2}-\varepsilon_{max}^{-3/2}\right )H(\varepsilon -\varepsilon_{min})H(\varepsilon_{max}-\varepsilon )$, (1)

where $\varepsilon (\kappa^2)$ is the energy (banana trapping parameter with $\kappa_{res}^2$ corresponding to banana tip at the cyclotron resonance), $\delta (H)$ is the Dirac (unit step) function, $\varepsilon_{min}$ is the runaway boundary given by $\varepsilon_{min}\approx T_e\left (\nu_c/\nu_{QL}\right )^{2/3}$ with $\nu_{c(QL)}$ thermal electron collision rate (quasi-linear diffusion rate), $\varepsilon_{max}$ is the high-energy cut-off due to relativistic effect [3], which is related with $\varepsilon_{min}$ by $\gamma_{max}=\gamma_{min}+\sqrt{\gamma_{min}^2-1}, \gamma_{min(max)}=1+\varepsilon_{min(max)}/m_ec^2$, and $n_e^h$ is the hot electron density given by $n_e^h\approx n_e\exp{\left [-(\nu_c/\nu_{QL})^{2/3}\right ]}$.

Internal kink induces *n=1* toroidal ripple in the magnetic field [4], which in turn modifies the second adiabatic invariant $J_b=\oint{v_\parallel dl}$. This invariant serves as a Hamiltonian for super-banana motion of hot electrons

$\frac{dr}{dt}=\frac{q}{2\omega_{ce}r\tau_b}\frac{\partial J_b}{\partial \zeta_0}\approx v_d\frac{\xi_0}{r}\sin{\zeta_0}\left [\frac{2E(\kappa )}{K(\kappa )}-1\right ]$, (2)

$\frac{d\zeta_0}{dt}=-\frac{q}{2\omega_{ce}r\tau_b}\frac{\partial J_b}{\partial r}\approx -\frac{v_d}{r}\left [\frac{2E(\kappa )}{K(\kappa )}-1\right ]+\frac{c}{r}\frac{E_r}{B_0}$, (3)

where $v_d\approx v^2/(2\omega_{ce}R_0)<0$ is the electron magnetic drift velocity, $\xi_0$ is the rigid kink displacement amplitude, $\zeta_0$ is the field line label defined by $\zeta =\zeta_0+q\theta $ with $\zeta (\theta )$ the toroidal (poloidal) angle, $K(E)$ is the elliptic integral of the first (second) kind, and $E_r$ is the radial electric field to be determined by ambipolarity condition $\Gamma_e^h(E_r)=\Gamma_i$, where $\Gamma_i$ is the non-ambipolar kink-induced flux of thermal ions [4] and $\Gamma_e^h$ is the flux of hot electrons, which can be calculated as follows.

Kink displacement induces perturbation of the hot electron distribution (1), which obeys banana drift kinetic equation

$\frac{d\zeta_0}{dt}\frac{\partial f_{1e}^h}{\partial \zeta_0}+\frac{dr}{dt}\frac{\partial f_{0e}^h}{\partial r}=\left \langle Q\left (f_{1e}^h\right )+C\left (f_{1e}^h\right )\right \rangle $, (4)

where $C(Q)$ is the collision (quasi-linear diffusion) operator, angular brackets denote bounce-average, and we have taken into account that for 'runaways' $\left \langle Q(f_{0e}^h)+C(f_{0e}^h)\right \rangle \approx \left \langle Q(f_{0e}^h)\right \rangle =0$, which yields Eq.(1). For the almost collisionless hot electrons, solution of Eq.(4) can be easily obtained replacing right-hand side by simple Krook operator with infinitesimal effective collision frequency, $\left \langle Q(f_{1e}^h)+C(f_{1e}^h)\right \rangle \approx -\nu_{eff}f_{1e}^h$, and taking the limit $\nu_{eff}\rightarrow 0$

$f_{1e}^h=-\lim_{\nu_{eff}\rightarrow 0}Re\left \{ \frac{1}{id\zeta_0/dt+\nu_{eff}}\frac{dr}{dt}\frac{\partial f_{0e}^h}{\partial r}\right \}=\pi \sin{\zeta_0}\left |\frac{\beta}{\alpha}\right |\delta \left (\kappa^2-\kappa_0^2\right )\xi_0\frac{\partial f_{0e}^h}{\partial r}$, (5)

where $\alpha =d[2E(\kappa )-K(\kappa )]/d\kappa^2|_{\kappa^2=\kappa_0^2}, \beta =2E(\kappa_0)/K(\kappa_0)-1$, and $\kappa_0^2(\varepsilon ,E_r)$ is the trapping parameter corresponding to resonance $d\zeta_0/dt =0$, which arises due to cancellation between reversed (i.e. co-current) magnetic precession of hot electrons and positive electric drift in Eq.(3). Note that such resonance is possible only with high-field side ECRH, which is consistent with experiment [2]. Equation (5) yields for the electron flux [5]

$\Gamma_e^h=\frac{1}{(2\pi )^2}\int_0^{2\pi}d\zeta_0\int_0^{2\pi}d\theta \int d^3v\frac{dr}{dt}f_e^h=$

$-\frac{\sqrt{\pi}}{2}\frac{cE_r}{B_0}\left (\frac{\xi_0}{r}\right )^2r\frac{dn_e^h}{dr}\left (1+\ln{\frac{\varepsilon_{max}}{\varepsilon_{min}}}\right )^{-1}\left \{1-\left [\frac{eR_0E_r}{|\beta (\kappa_{res})|\varepsilon_{max}}\right ]^{3/2}\right \}$. (6)

For parameters of the experiment [2], $\Gamma_e^h(E_r)$ at its maximum exceeds the non-resonant thermal ion flux by more than an order of magnitude. Ambipolarity condition then reduces to $\Gamma_e^h(E_r)=0$, which yields the stable 'electron root' $E_r=|\beta (\kappa_{res})|\varepsilon_{max}/eR_0\sim 30\div 50 kV/m$ (note that solution $E_r=0$ is unstable). Trace tungsten must obey Boltzmann relation $\nabla n_Z/n_Z\approx ZE_r/T_i>0$, which corresponds to the deep hole, consistent with experiment [2].

In summary, experimental results presented in Ref.[2] and theoretical results presented in this contribution and Ref.[5] imply that high-field side ECRH can be a viable option to prevent tungsten accumulation in the ITER discharges prone to spontaneous helical core formation [1], since recent modelling [6] suggests that helical core itself (i.e. without ECRH) *augments* impurity accumulation.

**References**

- A. Wingen
*et al.*, Nucl. Fusion**58**(3), 036004 (2018). - M. Sertoli
*et al.*, Nucl. Fusion**55**, 113029 (2015). - B. Hafizi and R.E. Aamodt, Phys. Fluids
**30**, 3059 (1987). - S.V. Putvinskij, Nucl. Fusion
**33**, 133 (1993). - V.S. Marchenko, Phys. Plasmas
**27**, 022516 (2020). - M. Ragunathan
*et al.*, Plasma Phys. Control. Fusion**59**, 124002 (2017).

Affiliation | Institute for Nuclear Research |
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Country or International Organization | Ukraine |