2 Full-wave modelling of ion cyclotron heating with the TWA

To demonstrate that the downshift in $k_\parallel$ can enhance ion heating and current drive efficiency, we simulate four different mounting positions of the TWA in a large DEMO-like plasma with a major radius of 9.38 m and an on-axis magnetic field of 5.1 T. The antennae are centred at the poloidal angles $0^\circ$, $23^\circ$, $45^\circ$ and $67^\circ$ (with respect to the centre of the vacuum vessel). These positions are referred to as positions 1 to 4, respectively, and are illustrated in figure 2. Note that in this work, the scrape-off layer is modelled as a vacuum, hence the discontinuity in the electric field across the last closed flux surface. With both toroidal field and plasma current oriented clockwise, the wave is upshifted below the equatorial plane and downshifted above the equatorial plane (assuming that $\phi$ runs anticlockwise and that $n_\phi > 0$). For equatorial launch, the total effect of the shift leads to a broadening of the poloidal wave spectrum, but the effect on power deposition is negligible (Zaar, Johnson & Vallejos Reference Zaar, Johnson and Vallejos2023). As the antenna is moved off the equatorial plane, the whole wave is increasingly downshifted.

Figure 2.

Magnitude of the left-hand polarised component $E_+$ for positions 1–4 shown in panels (a)–(d), respectively. Here, $T_0 = 20\,{\rm keV}$ and $n_\phi = 43$. Green curves denote deuterium cyclotron resonances ($n = 1$) and blue curves denote tritium/helium-3 cyclotron resonances ($n = 2/1$).

The effect of the downshifted antenna spectrum on the propagation and damping of the fast magnetosonic wave is quantified using the full-wave solver FEMIC (Vallejos et al. Reference Vallejos, Johnson, Ragona, Hellsten and Frassinetti2019). FEMIC has recently been updated (see Zaar et al. Reference Zaar, Johnson, Bilato and Vallejos2024) with an iterative method to account for the effect of the poloidal magnetic field on the parallel wavenumber

(2.1)\begin{equation} k_\parallel{=} \frac{B_\phi}{BR}\left( n_\phi + \frac{m_\theta}{q}\right), \end{equation}

where

(2.2)\begin{equation} q = \frac{L}{2{\rm \pi} R} \frac{B_\phi}{B_\theta}. \end{equation}

Here, $R$ is the major radius, $L$ is the circumference of the local flux surface, and $n_\phi$ and $m_\theta$ are toroidal and poloidal mode numbers, respectively. Additionally, $B_\phi$ and $B_\theta$ denote the toroidal and poloidal components of the magnetic field, and $B = (B_\phi ^2 + B_\theta ^2)^{1/2}$. The poloidal mode number $m_\theta$ is defined through the Fourier decomposition of the electric wave field, such that

(2.3)\begin{equation} \boldsymbol{E} = \sum_{m_\theta} \boldsymbol{\hat{E}}_{m_\theta} \mathrm{e}^{\mathrm{i} m_\theta\theta}, \end{equation}

where $\theta$ is an equal-arc-length angle.

To see how efficiently the ions are heated during the ramp-up phase of the plasma, we are considering core temperatures (equal for ions and electrons) between 10 and 30 keV. The plasma consists of equal parts deuterium and tritium with a small $^3$He minority (1 %) to improve the ion damping. The density is kept constant as the temperature is increased. To avoid excessively energetic ions, the frequency for positions 1 and 2 is tuned to place the cyclotron resonance slightly off-axis, near $\rho _\mathrm {pol} = 0.2$ in the equatorial plane, which corresponds to the antenna frequency $f = 50\,{\rm MHz}$. In this work, $\rho _\mathrm {pol}$ is used to denote the square root of the normalised poloidal flux. For positions 3 and 4, the antenna frequency is set such that power deposition is as central as possible. This corresponds to $f = 52\,{\rm MHz}$ and $f =55\,{\rm MHz}$, respectively. Power deposition profiles for $n_\phi = 43$ can be seen in figure 3 for positions 1–4. The power deposition profiles are indeed peaked around $\rho _\mathrm {pol} = 0.2$ for antenna positions 1–3, whereas it is peaked near $\rho _\mathrm {pol} = 0.4$ for position 4.

Figure 3.

Normalised flux surface average of the absorbed power density for $T_0 = 20\,{\rm keV}$ and $n_\phi = 43$. Inset shows the integrated power deposition.

To understand how the downshift affects the electron damping, the fraction of the power absorbed by the ions, $P_\mathrm{i}$, is plotted versus temperature in figure 4(a). Here, we set $n_\phi = 43$, which is one of the dominant toroidal mode numbers in a power spectrum tuned for low parallel wavenumbers. At $T_0 = 10\,{\rm keV}$, electron damping is weak and almost all power is absorbed by the ions for all antenna positions. The electron damping then increases with the temperature such that most of the wave energy is damped by the electrons for antenna positions 1 and 2. Electron damping is slightly weaker for position 2 than the equatorial launch due to the modest downshift in $k_\parallel$. In contrast, antenna positions 3 and in particular 4 largely retain their ability to deposit the majority of the wave energy on the ions all the way up to $T_0 = 30\,{\rm keV}$. In fact, for position 4, even at $T_0 = 20\,{\rm keV}$, approximately 94 % of the power is absorbed by the ions. This is attributed to the stronger downshift, as illustrated in table 1. Here, the fraction of absorbed power for the four antenna positions is summarised for $T_0 = 20\,{\rm keV}$ and $n_\phi = 43$ including and excluding the shift in the parallel wavenumber. The importance of the shift is similar for all probed temperatures, and it is evident that accounting for the effect of the poloidal magnetic field is essential for the modelling in this work.

Figure 4.

(a) Absorbed power partition and (b–d) power partition after collisional redistribution for (b) 50 MW, (c) 100 MW, and (d) 200 MW coupled power for $n_\phi = 43$.

Table 1.

Fraction of total coupled power absorbed by ions, $P_\mathrm{i}$, including and excluding the shift in $k_\parallel$ for $T_0 = 20\,{\rm keV}$ and $n_\phi = 43$.

As the energetic minority and tritium ions collide with the background plasma, the plasma is heated. To quantify the collisional power redistribution, electric fields and power deposition profiles from FEMIC are fed into the Foppler code. Foppler is a one-dimensional pitch angle averaged Fokker–Planck solver for ICRF heating that contains Doppler physics and an ad hoc model for the pitch angle distribution, based on the Dendy model (see Dendy et al. Reference Dendy, Hastie, McClements and Martin1995). The distribution function is assumed to be isotropic below the critical energy and becomes exponentially narrower around $\mu = E/B_\mathrm {res}$, which is the magnetic moment of an orbit with turning points on the unshifted cyclotron resonance. A detailed description of the Foppler code and the pitch angle distribution will be presented in a future publication by Bähner et al. Note that in this work, there is no self-consistent coupling between FEMIC and Foppler.

The collisional power redistribution is evaluated for a few different values of the total coupled power. As shown in figure 4(b–d), position 4 (and to some extent position 3) is able to deliver significantly more ion heating for all probed temperatures and power levels compared with equatorial launch. In particular, for 100 MW and 200 MW of coupled power, the ions are heated more than twice as much by an antenna located in position 4 than position 1. Also for 50 MW, the ions are heated twice as much for higher temperatures, and approximately 50 % more for $T_0 = 10\,{\rm keV}$ and 15 keV. Position 3 also delivers significantly enhanced ion heating, whereas the improvement with position 2 is more modest.

It is evident from figure 4 that absorbed power partition alone does not explain the enhanced ion heating for the downshifted waves. At $T_0 = 10\,{\rm keV}$, the absorbed power partition is fairly similar for all four launch positions, whereas the ion heating differs significantly. The main contributor to this discrepancy is the flux surface averaged power deposition, as higher absorbed power densities lead to more energetic tails in the distribution functions. Ions with energies above the critical energy are mainly slowed down by (and are thus heating) electrons (Stix Reference Stix1972).

Although the power deposition profiles are peaked near the same flux surface for positions 1–3, we observe in figure 3 that the maximum flux averaged absorbed power density becomes lower as we elevate the antenna. This is partially explained by the alignment of the resonance and the flux surfaces where the wave is absorbed. For position 1, we see in figure 2(a) that the resonance is somewhat tangential to the flux surface where the wave is damped. For position 3, however, we see in figure 2(c) that the resonance is completely perpendicular to the central flux surfaces. This spreads the wave front over a wider range of flux surfaces, lowering the power density. This effect is most prominent at lower temperatures. As the temperature increases, the resonance layers are broadened, widening the power deposition profile for position 1 as well.

At $T_0 \gtrsim 20\,{\rm keV}$, the enhanced ion heating is mainly explained by the more favourable absorbed power partition. Note that for geometrical reasons, the antenna in position 4 deposits its power around $\rho _\mathrm {pol} = 0.4$, rather than $\rho _\mathrm {pol} = 0.2$, further reducing the flux surface averaged power deposition compared with the other positions. Heating the plasma this far off-axis may not be desirable.

3 Current drive prospects

The downshift in the parallel wavenumber that permits dominant ion absorption may also be beneficial for FWCD. A higher parallel phase velocity $\omega /k_\parallel$ means that the fast magnetosonic wave resonates with electrons with higher parallel velocity, which in turn have a lower collisionality. However, Landau damping and transit-time magnetic pumping decay exponentially with the parallel phase velocity, as there are fewer electrons to accelerate. Hence, there is a trade-off between current drive efficiency and strong electron damping, which both are desirable for efficient current drive. In this section, we investigate the fast wave current drive prospects for the four mounting positions of the TWA antenna described in § 2.

3.1 Current drive model

In this work, we have implemented a local parametrised current drive (CD) efficiency $\eta = \eta (\psi,\theta )$ (see Ehst & Karney Reference Ehst and Karney1991) on the form

(3.1)\begin{equation} \eta = f_\mathrm{neo} \eta^* \eta_0, \end{equation}

where $f_\mathrm {neo}$ is the neoclassical correction factor for trapped electrons, $\eta ^*$ is a normalisation constant and $\eta _0$ is the dimensionless parametrised straight-field CD efficiency. Ehst & Karney (Reference Ehst and Karney1991) evaluate the radio frequency (RF) driven current assuming a ray tracing description of the wave. To adapt their model to a full-wave solution, the driven current density along the magnetic field, denoted by $J_{\parallel,\mathrm {CD}}$, is in this work implicitly expressed as

(3.2)\begin{equation} \frac{\langle J_{{\parallel},CD} B \rangle}{\langle B^2 \rangle} = \frac{\mathrm{d} V}{\mathrm{d} \psi} \left\langle \eta(\psi,\theta) Q_\mathrm{e}(\psi,\theta)\right\rangle, \end{equation}

where $V(\psi )$ is the volume contained by the flux surface $\psi$, $Q_\mathrm{e}(\psi,\theta )$ is the RF power density absorbed by the electrons and $\langle \cdot \rangle$ denotes flux surface average. The flux averaged toroidal RF-driven current density $J_\mathrm {CD}(\psi )$ can then be written as

(3.3)\begin{equation} J_\mathrm{CD}(\psi) = f_\mathrm{geom}(\psi) \left\langle \eta(\psi,\theta) Q_\mathrm{e}(\psi,\theta)\right\rangle, \end{equation}

where $f_\mathrm {geom}(\psi ) = F(\psi )\langle R^{-2} \rangle /\langle B \rangle \langle R^{-1} \rangle$ is a geometrical factor that depends on the magnetic field geometry and $F(\psi ) = RB_\phi$. The effect of the correction factors $f_\mathrm {geom}$ and $f_\mathrm {neo}$ on the current drive efficiency is illustrated in figure 5, where they are plotted as functions of $\rho _\mathrm {pol}$. We see that $f_\mathrm {geom}$ is near unity everywhere, and that $\langle f_\mathrm {neo} \rangle$ is significantly smaller than unity outside the very centre of the plasma due to the effect of trapped electrons. Here, $f_\mathrm {neo}$ has been evaluated for $m_\theta = 0$. It is apparent that off-axis current drive rapidly becomes less efficient as the toroidal mode number (and thus $k_\parallel$) increases.

Figure 5.

(a) Geometric and (b) neoclassical correction factors for current drive efficiency.

When accounting for parallel dispersion, FEMIC evaluates the absorbed power density as

(3.4)\begin{equation} Q_\mathrm{e}(\psi,\theta) = \frac{\omega \epsilon_0}{2} \mathrm{Im}{\left(\boldsymbol{E}^* \boldsymbol{\cdot} \sum_{m_\theta}\hat{\boldsymbol{\chi}}_\mathrm{e}(\psi,\theta,m_\theta) \boldsymbol{\cdot} \hat{\boldsymbol{E}}_{m_\theta} \mathrm{e}^{\mathrm{i} m_\theta\theta}\right)}, \end{equation}

where $\boldsymbol {\hat {\chi }}$ is the quasi-homogeneous susceptibility tensor in Fourier space (see Stix Reference Stix1992; Brambilla Reference Brambilla1998; Swanson Reference Swanson2003), $\boldsymbol {E} = \boldsymbol {E}(\psi,\theta )$ is the electric wave field and $m_\theta$ is the poloidal mode number. Here, caret denotes Fourier transform. To include the up- and downshift in the parallel wavenumber when evaluating the FWCD, we use a quasi-homogeneous current drive efficiency $\eta (\psi,\theta,m_\theta )$, where $k_\parallel$ is evaluated using (2.1), hence the dependency on $m_\theta$. The flux averaged toroidal RF-driven current density then becomes

(3.5)\begin{equation} J_\mathrm{CD} = \frac{\omega \varepsilon_0 f_\mathrm{geom}(\psi)}{2} \left\langle \mathrm{Im}{\left(\boldsymbol{E}^* \boldsymbol{\cdot} \sum_{m_\theta}\eta(\psi,\theta,m_\theta)\hat{\boldsymbol{\chi}}_\mathrm{e}(\psi,\theta,m_\theta) \boldsymbol{\cdot}\hat{\boldsymbol{E}}_{m_\theta} \mathrm{e}^{\mathrm{i} m_\theta \theta}\right)} \right\rangle. \end{equation}

3.2 Full-wave modelling results for FWCD

To maximise the single pass electron damping, the antenna frequency is increased to move the second harmonic tritium resonance as far away from the antenna as possible without having the second harmonic deuterium/alpha resonance enter the antenna box. In practice, alpha particles may resonate far away from the ‘cold’ resonance due to significant Doppler shift. Hence, care must be taken such that fast ions are not accelerated in front of the antenna (Hannan, Hellsten & Johnson Reference Hannan, Hellsten and Johnson2013). In this work, we set the antenna frequency to $f = 59\,{\rm MHz}$ for positions 1–3 and $f = 63\,{\rm MHz}$ for position 4. The resulting electric fields are visualised in figure 6. Note that the difference between the frequencies used for heating and current drive operation coincides with the bandwidth of the TWA antenna concept proposed by Ragona et al. (Reference Ragona, Messiaen, Ongena, Van Eester, Van Schoor, Bernard, Hillairet and Noterdaeme2020). As previously stated, increasing the frequency further is not desirable. However, it could be valuable to be able to position the minority resonance further on the low field side during the ramp-up phase.

Figure 6.

Electric field norm for positions 1–4 shown in panels (a–d), respectively. Here, $T_0 = 30\,{\rm keV}$ and $n_\phi = 53$. Green curves denote deuterium cyclotron resonances ($n = 1$ in panels a–c and $n = 2$ in panel d) and blue curves denote tritium cyclotron resonances ($n = 2$).

The RF-driven current is computed assuming a plasma consisting of deuterium and tritium in equal parts. Diluting the plasma with helium ashes does not strongly affect the wave propagation properties, but using the scaling relation from Kazakov et al. (Reference Kazakov, Van Eester, Wauters, Lerche and Ongena2014), we find that increasing $Z_\mathrm {eff}$ to 1.2 (corresponding to 10 % helium ash concentration) reduces $\eta$ by approximately 10 %. At $Z_\mathrm {eff} = 1.5$, the current drive efficiency is reduced by approximately 20 %.

Normalised current drive profiles for positions 1–4 evaluated for $n_\phi = 53$ are shown in figure 7. As expected, moving the antenna off the equatorial plane also moves the RF-driven current off-axis. For position 2, this effect is negligible, 80 % of the current is still driven inside $\rho _\mathrm {pol} = 0.5$, compared with $\rho _\mathrm {pol} = 0.45$ for equatorial launch. For position 4, however, the single-pass electron damping is low due to the downshift in $k_\parallel$, and we have significant current drive everywhere from $\rho _\mathrm {pol} = 0.25$ out to the last closed flux surface. Which current drive profile is the most appropriate for a given scenario depends on the bootstrap current and the availability of electron cyclotron heating.

Figure 7.

Normalised integrated current drive profiles for $n_\phi = 53$.

Both electron damping and current drive efficiency are sensitive to the parallel wavenumber. The electron power partition and total driven current are evaluated over a range of $n_\phi$ and shown in figure 8. It is evident that the equatorial plane launch is efficient for driving current at lower toroidal mode numbers. This is possible due to a large fraction of electron absorption combined with high current drive efficiency $\eta$ (due to low $k_\parallel$). Position 3 (and to a smaller extent, position 2) suffers from weak electron damping due to the downshift in $k_\parallel$. This is also reflected in the total integrated current in figure 8(b). Position 4 has significantly better electron damping than position 3 despite the larger downshift. This is attributed to the launch angle of the wave and higher antenna frequency $\omega$, which causes the wave to partially miss the tritium resonance, allowing more energy to be absorbed by the electrons. However, as the wave never reaches the innermost flux surfaces (see figure 6d), the current drive efficiency $\eta$ is limited by trapped electrons (see figure 5).

Figure 8.

(a) Electron power partition and (b) total driven current evaluated over a range of $n_\phi$. The grey dotted curve sketches the power spectrum for a relevant TWA antenna (in arbitrary units).

Moving to higher toroidal mode numbers, electron damping increases and a similar fraction of electron absorption $(P_\mathrm{e} > 90\,\%)$ is obtained for all antenna positions. The total driven current is then largely determined by the current drive efficiency, which is highest for antenna positions 3 and 4. It is noteworthy that for position 4, the total driven current does not change much across the most relevant parts of the toroidal spectrum, whereas the total driven current achieved using positions 2 and 3 is clearly peaked around $n_\phi = 43$ and $n_\phi = 53$, respectively. To maximise the RF-driven current for these positions, the antenna spectrum should be tuned so that it peaks around these values of $n_\phi$ as well.

The total driven current can be estimated by summing over all toroidal modes $n_\phi$ (neglecting coupling between toroidal modes), weighted by the normalised antenna power spectrum. For the spectrum shown in figure 8, we obtain approximately 35, 36, 28 and $27\,{\rm kA}\,{\rm MW}^{-1}$ for positions 1–4, respectively.