2.1. Equilibrium

The scale of inhomogeneity of the plasma parameters in JET is much greater than SW wavelength. Therefore, the ray-tracing method (or approach) can be used for such simulations. Since during propagation, the SW ‘feels’ the plasma density and magnetic field locally, the equilibrium code EFIT (Lao et al. Reference Lao, John, Stambaugh, Kellman and Pfeiffer1985) was used in order to reconstruct a JET equilibrium with one-zero divertor. The flux surface label is $\psi =1$ at the last closed magnetic surface (LCMS) and $\psi =0$ at the magnetic axis (figure 1).

Figure 1. The magnetic surface label $\psi$ distribution in the minor cross-section of the JET tokamak. ‘Fat’ equilibrium.

Figure 2. The magnetic flux label $\psi$ as a function of big radius R along the line that passes through the magnetic axis.

The realistic shape of the JET vacuum vessel was taken into account while constructing a mathematical model of the plasma density distribution in the peripheral plasma. Two distinguished magnetic configurations were used in these calculations. One of them will be referenced as ‘fat’ and the other as ‘thin’ (see figure 2).

The plasma density and temperature profiles were set as

(2.1) \begin{align} n\!\left(\psi \right)&=\left\{\begin{array}{l} n_{0}\frac{\exp \left(\xi _{n}\right)-\exp \left(A_{n}\xi _{n}\psi \right)}{\exp \left(\xi _{n}\right)-1},\quad \psi \leq 1\\ n_{b}+\left(n_{x}-n_{b}\right)\exp \left[\left(1-\psi \right)/\delta \right]\!,\quad \psi \gt 1 \end{array}\right.\!,\nonumber\\T\left(\psi \right)&=\left\{\begin{array}{l} T_{0}\frac{\exp \left(\xi _{T}\right)-\exp \left(A_{T}\xi _{T}\psi \right)}{\exp \left(\xi _{T}\right)-1},\quad \psi \leq 1\\ T_{b}+\left(T_{x}-T_{b}\right)\exp \left[\left(1-\psi \right)/\delta \right]\!,\quad \psi \gt 1 \end{array}\right.\!, \end{align}

where $n_{0},\,T_{0}$ are density and temperature at the axis, $n_{b},\,T_{b}$ are density and temperature at the plasma boundary, $A_{n},\,A_{T}$ are smoothing coefficients, $n_{x},\,T_{x}$ are density and temperature at the LCMS, $\xi _{n},\,\xi _{T}$ are variable parameters ( $\xi _{n}=4$ for density and $\xi _{T}=-2$ for temperature were chosen) and $\delta$ is the e-folding length outside the LCMS.

2.2. Ray tracing

Propagation and absorption of the SW in the JET plasma were studied using the ray-tracing code SLOWAR-T (SLOw WAve Rays in Tokamak). The SLOWAR-T code is the modified version of the code SLOWAR, described in Grekov (Reference Grekov, Albert, Turkin and Volkova2024). The verification of the dispersion relation conservation shows that it fulfills the precision of the Runge–Kutta solver, which was used in the code. The code solves the equations

(2.2) \begin{equation} \frac{\textrm{d}\vec{r}}{\textrm{d}t}=\frac{\partial D_{R}}{\partial \vec{k}}\Big/\frac{\partial D_{R}}{\partial \omega }\!,\frac{\textrm{d}\vec{k}}{\textrm{d}t}=-\frac{\partial D_{R}}{\partial \vec{r}}\Big/\frac{\partial D_{R}}{\partial \omega }\!, \end{equation}

where $\vec{k}$ is the wave vector, ω is the wave frequency, t is time, $\vec{r}=(X,\enspace Y,\enspace Z)$ are the Cartesian coordinates of the ray and $D(\omega ,\vec{r},\vec{k})=0$ is the dispersion equation

(2.3) \begin{equation} D=D_{R}+iD_{I}=\varepsilon _{1}N_{\bot }^{4}+\left[\left(\varepsilon _{3}+\varepsilon _{1}\right)\left(N_{||}^{2}-\varepsilon _{1}\right)+\varepsilon _{2}^{2}\right]N_{\bot }^{2}+\varepsilon _{3}\left[\left(N_{||}^{2}-\varepsilon _{1}\right)^{2}-\varepsilon _{2}^{2}\right]=0. \end{equation}

Here, $k_{||}=\vec{k}\cdot \vec{B}/B , \vec{B}$ is the tokamak magnetic field, $N_{||}=ck_{||}/\omega , k_{\bot }=(k^{2}-k_{||}^{2})^{1/2}$ and $N_{\bot }=ck_{\bot }/\omega$ . Cylindrical coordinates $(R,\enspace \phi ,\enspace Z)$ and quasi-toroidal coordinates $(r,\enspace \vartheta ,\enspace \phi )$ were also used in this investigation. Here, $R=\sqrt{X^{2}+Y^{2}}$ is the big radius of the torus, $\phi =\textit{arctg}(Y/X)$ is the toroidal angle, $\vartheta$ is the poloidal angle and $r$ is the minor radius of the torus.

The calculation of the dielectric tensor $\hat{\varepsilon }$ is based on the warm plasma model and resolves only electron Landau damping and fundamental ion cyclotron resonance. For ${\left(\omega -\omega _{c\alpha }\right)^{2}})/({2k_{||}^{2}v_{T\alpha }^{2}})\gt \gt 1$ the correction for the real part of ε ₁ is negligible. For ${\left((\omega -\omega _{c\alpha }\right)^{2}})/({2k_{||}^{2}v_{T\alpha }^{2}})\sim 1$ , Re(ε ₁ ) ∼ Im(ε ₁ ), D _I ∼ D _R and the ray tracing is not applicable. The proper operator in the code terminates the calculation. Components of the tensor are defined as

(2.4) \begin{equation} \begin{array}{l} \varepsilon _{1}=1-\sum _{\alpha }\frac{\omega _{p\alpha }^{2}}{\omega ^{2}-\omega _{c\alpha }^{2}}\\ \quad+\sum _{\alpha }i\sqrt{\frac{\pi }{8}}\frac{\omega _{p\alpha }^{2}}{\left| k_{||}\right| v_{T\alpha }\omega }\exp \left[-\frac{\left(\omega -\omega _{c\alpha }\right)^{2}}{2k_{||}^{2}v_{T\alpha }^{2}}\right] \end{array}, \end{equation}

(2.5) \begin{equation} \varepsilon _{2}=\sum _{\alpha }\frac{\omega }{\omega _{c\alpha }}\frac{\omega _{p\alpha }^{2}}{\omega ^{2}-\omega _{c\alpha }^{2}} ,\end{equation}

(2.6) \begin{equation} \varepsilon _{3}=1+\frac{\omega _{pe}^{2}}{k_{||}^{2}v_\textrm{Te}^{2}}\left[1+i\sqrt{\pi }z_{e}W\!\left(z_{e}\right)\right]+i\varepsilon_\textrm{3coll}, \end{equation}

where m_b and e_b are mass and charge of b particles, b=α, e  $\omega _{p\alpha }=(4\pi e^{2}n_{\alpha }/m_{\alpha })^{1/2}$ is the plasma frequency of α ions, $\omega _{c\alpha }=eB/cm_{\alpha }$ is the α ion cyclotron frequency, $v_{T\alpha }=\sqrt{T_{\alpha }/m}_{\alpha }$ is the α ion thermal velocity, $\omega _{pe}=(4\pi e^{2}n_{e}/m_{e})^{1/2}$ is the plasma frequency of electrons, $v_\textrm{Te}=\sqrt{T_{e}/m}_{e}$ is the thermal velocity of electrons, $z_{e}=\omega /\sqrt{2}| k_{||}| v_\textrm{Te} , W(z_{e})$ is the plasma dispersion function, $\varepsilon_\textrm{3coll}=\omega _{pe}^{2}\nu_\textrm{eff}/\omega ^{3}$ , $\nu_\textrm{eff}$ is the effective collisional frequency of electrons where $\nu _{eff}=({4}/{3})\sqrt{{2\pi }/{m_{e}}}({e^{4}Z^{2} \wedge n_{e}})/({T_{e}^{3/2}})$ and Λ is the Coulomb logarithm.

Seeing $| \varepsilon _{3}| \gt \gt | \varepsilon _{1}| ,\,| \varepsilon _{2}|$ , the solutions of equation (2.3) are

(2.7) \begin{equation} N_{\bot S}^{2}=-\varepsilon _{3}\left(N_{||}^{2}-\varepsilon _{1}\right)/\varepsilon _{1}, \end{equation}

for the SW and

(2.8) \begin{equation} N_{\bot F}^{2}=\left[\left(N_{||}^{2}-\varepsilon _{1}\right)^{2}-\varepsilon _{2}^{2}\right]/\left(\varepsilon _{1}-N_{||}^{2}\right) ,\end{equation}

for the FW. Equations (2.7) and (2.8) are correct outside the conversion region, where $\varepsilon _{1}\approx N_{||}^{2}$ . Strictly speaking, ray tracing is not applicable in the regions of ray reflection and mode conversion regions. In the region of reflection of the SW from Maxwell’s equations for the $E_{||}$ field of the SW the equation $({\textrm{d}^{2}E_{||}}/{\textrm{d}\xi ^{2}})+\left({\textrm{d}k_{\bot S}^{2}}/{\textrm{d}\xi }\right)_{\xi {ref}}\left(\xi -\xi_\textrm{ref}\right)E_{||}=0$ is valid. Here, $\xi$ is the coordinate in the direction of the gradient of $(N_{||}^{2}-\varepsilon _{1})$ . This equation is transformed to the Airy equation $({\textrm{d}^{2}E_{||}}/{\textrm{d}\varsigma ^{2}})+\varsigma E_{||}=0$ , where $\varsigma =\left(\xi -\xi_\textrm{ref}\right)\left(({\textrm{d}k_{\bot S}^{2}}/{\textrm{d}\xi })\right)_{\xi \textrm{ref}}^{\frac{1}{3}}$ . It follows that the reflection occurs in the region $\varsigma \sim 1$ , or $\left| \xi -\xi _{ref}\right| \approx {\unicode[Arial]{x0394}} r\sim \left(({\textrm{d}k_{\bot S}^{2}}/{\textrm{d}\xi })\right)_{\xi ref}^{-\frac{1}{3}}$ . For the selected modeling parameters, ${\unicode[Arial]{x0394}} r\sim 1\textrm{cm}$ . Beyond this region, the Airy function connects to the ray-tracing solutions. Thus, using the ray-tracing model only leads to inaccuracy in determining the phase only, which is of the order of $\pi$ . Such a value of imprecision is insignificant for the present consideration. As $\varepsilon _{3}\lt 0$ at $n_{e}\geq 10^{7}\textrm{cm}^{-3}$ , the SW propagates in peripheral plasma only if the conditions $0\lt \varepsilon _{1}\lt N_{||}^{2}$ are fulfilled. Both of these inequalities are satisfied in hydrogen plasmas with ion cyclotron resonance frequency (ICRF) inverted minority heating (MH) or mode conversion heating (MC) regimes (Start et al. Reference Start1999; Lamalle et al. Reference Lamalle2006; Mayoral et al. Reference Mayoral2006). A key feature of these so-called inverted scenarios is that the minority ion species have a smaller charge-to-mass ratio than the majority ion species, i.e. Z _min /A _min $\lt$ Z _maj /A _maj. The wave frequency and the toroidal magnetic field are chosen so that the cyclotron resonance of the majority ions is located outside the plasma on the outer side of the torus in these experiments.

The power associated with a certain ray $Q$ is defined as $Q=Q_{0}e^{-{\unicode[Arial]{x0393}} }$ , where $Q_{0}$ is the initial RF power at the ray start location and ${\unicode{x1D6E4}} =\int _{0}^{t}\,\gamma \,\textrm{d}t , \gamma =-({D_{I}})/({\partial D_{R}/\partial \omega })$ . When calculating Q along the ray, electron Landau damping $p_{e}$ , ion cyclotron damping $p_{i\alpha }$ (α = 1, 2, 3) and collisional damping $p_{col}$ were taken into account. The power dissipated along the ray $P_\textrm{tot}$ was defined as $P_{tot}=Q_{0}(1-e^{-})$ . It can be represented as $P_{tot}=\int _{0}^{s}(p_\textrm{col}+p_{e}+p_{i1}+p_{i2}+p_{i3})ds$ , where $s=\int _{0}^{t}| \vec{v}_{g}| \textrm{d}t$ is the ray length. In what follows we assume Q ₀ = 1.

2.3. Initial conditions

To define $\vec{k}_{0}$ for SW ray tracing, the distribution of the wave electric field E _ϕ at the ICRF antenna must expanded into a Fourier series of poloidal (wavenumber m ₀) and toroidal (wavenumber l ₀) angles. It allows us to determine $N_{\phi 0}$ as $N_{\phi 0}=(cl_{0})/(\omega R_{0})$ . The value of $N_{z0}$ was defined as $N_{z0}=({cm_{0}}/{\omega \overline{a}})\cos \left[\textit{arctg}\left(Z_{0}/R_{0}\right)\right]$ , where $| m_{0}| \leq 50 , | Z_{0}| \leq 50cm$ , and $\overline{a}\approx 100cm$ is a mean minor radius. Accounting for $N^{2}=N_{R}^{2}+N_{\phi }^{2}+N_{z}^{2}$ and

(2.9) \begin{equation} N_{||}=N_{R}b_{R}+N_{\phi }b_{\phi }+N_{z}b_{z}, \end{equation}

where $(b_{R},\,b_{\phi },\,b_{z})$ are the components of the unit vector, which is parallel to the tokamak magnetic field, the initial value $N_{R0}$ was determined from the dispersion equation (2.3). Four roots of (2.3) were found by iterations, taking $\mathrm{Re}[W(z_{e})]$ in $\varepsilon _{3}$ and assuming $N_{R0}=0$ in $k_{||}$ at the first step. At subsequent iterations, the value $N_{R0}$ from the previous iteration was substituted into $\varepsilon _{3}$ . Two of the roots correspond to the FW. They are complex for the chosen value of $\psi _{0}$ . The other two roots are real in the inverted scenarios and correspond to the SW . Let us call the smaller one the first root and the larger one the second root.

3.1. Intrinsic features of the SW. Pure hydrogen plasma

The following set of parameters has been used for consideration in this subsection: B ₀ = 39 kG, n _e0 = 1 × 10¹³ cm⁻³, n _x = 4 × 10¹¹ cm⁻³, n _eb = 1 × 10¹¹ cm⁻³, n _H /n _e = 1.0, $T_{e0}=3\,\textrm{keV}$ , T _x = 60 eV, $T_{eb}=10\,\textrm{e}V , f=\omega /2\pi =32\,\textrm{MHz}$ , the ‘fat’ equilibrium. The ray starts from a fixed point R ₀ = 389 cm, Z ₀ = 40 cm which in Z direction is very close to the magnetic axis.

As can be seen from figure 3(a), the SW propagates predominantly in the toroidal direction. The JET tokamak antenna system consists of two systems spaced at a toroidal angle of $\pi$ (Kaye et al. Reference Kaye, Brown, Bhatnagar, Crawley, Jacquinot, Lobel, Plancouline, Rebut, Wade and Walker1994). Therefore, the RF power emitted as SWs is uniformly distributed in the toroidal direction. In this case, the angular displacement in the poloidal direction is relatively small, of the order of $\pi$ /4 (see figure 3 b). This is the essential difference between SWs in the ion cyclotron frequency range and lower hybrid SWs. The latter, when propagating in tokamaks, perform many reversals in the poloidal direction (Baranov & Fedorov Reference Baranov and Fedorov1980). Also, figure 3(b) shows that SWs propagate along the periphery of the plasma and can penetrate into the divertor region.

Figure 3. The SW rays in the equatorial (a) and minor (b) cross-sections of the torus for l ₀ = −40, m ₀ = 2, second root – red lines and for l ₀ = +40, m ₀ = 2, first root – blue lines. ‘Fat’ equilibrium.

The SWs moving in the $co-\vec{B}$ direction ( $N_{||}\gt 0$ , red lines in figures 3 and 4) propagate in the direction of increasing plasma density (figures 4 a and 4 b). The SWs have anomalous dispersion. The group velocity $v_{g\nabla \psi }\lt 0$ together with $N_{\nabla \psi }\gt 0$ at the initial part of the ray $s\lt 550\,\textrm{cm}$ . The maximum density at which $\varepsilon _{1}=N_{||}^{2}$ or $N_{\nabla \psi }=0$ is more than 3 × 10 ¹² cm⁻³. Since for these waves the maximum value of $N_{||}$ is less than 14, their Landau damping is negligible. The collisional damping (figure 4 d) is also rather small.

Figure 4. The flux label (a), the plasma density (b), the refractive index parallel to the magnetic field (c) and the total absorbed power (d) vs. the ray length for l ₀ = −40 – red lines and for l ₀ = +40 – blue lines. ‘Fat’ equilibrium.

The SWs moving in the $\textit{counter}-\vec{B}$ direction ( $N_{||}\lt 0$ , blue lines in figures 3 and 4) propagate in the direction of the wall at first. After reflection from the wall, they slowly penetrate towards higher-density plasma. The maximum achieved density of 2.5 × 10¹² cm⁻³ is comparable to the former case. The absolute value of $N_{||}$ is strongly increasing at the final section of the ray. This leads to complete absorption of the SW due to Landau damping in the divertor region.

3.2. Influence of plasma density changes

In this subsection, we will study how the variation of the plasma density profile affects the propagation of the SWs. For this purpose, all parameters were fixed as in the previous subsection. At first, the parameter $\xi _{n}$ in equation (2.1) was reduced from 4 to 0. In this case, the density profile changes from very flat to parabolic.

As can be seen from figure 5(a), as $\xi _{n}$ decreases, the minimum value of $\psi$ decreases, too. This means that the SW penetrates closer and closer to the magnetic axis. However, the maximum value of the plasma density on the ray, which increases with decreasing ψ, still declines (figure 5 b). The path in the poloidal section of the torus, taken by the SW along the periphery of the plasma, increases significantly (figure 5 c). The number of revolutions of the SW around the torus also increases.

Figure 5. The flux label (a) and the plasma density (b) vs. the ray length for $\xi _{n}=4$ (red), $\xi _{n}=2$ (green) and $\xi _{n}=0$ (magenta), l ₀ = −40. ‘Fat’ equilibrium. The SW rays in the minor cross-section (c) for the same set of parameters.

Now the influence of the plasma density value on the magnetic axis on the wave propagation will be studied. The parameter $\xi _{n}$ is equal to 4, as before. As can be seen from figures 6(a) and 6(b), a trend similar to figure 5 is observed again. The decrease in plasma density in the center also leads to an increase in the penetration depth of the wave. And the maximum density value achievable by the SW also decreases.

Figure 6. The flux label (a) and the plasma density (b) vs. the ray length for n _e0 = 3 × 10¹³ cm⁻³ (blue), n _e0 = 1 × 10¹³ cm⁻³ (red) and n _e0 = 3 × 10¹² cm⁻³ (green), l ₀ = −40. ‘Fat’ equilibrium.

3.3. Effect of the initial poloidal wavenumber m₀

As can be seen from figure 16 of the paper by Kaye et al. (Reference Kaye, Brown, Bhatnagar, Crawley, Jacquinot, Lobel, Plancouline, Rebut, Wade and Walker1994), the JET tokamak antenna is shielded by a screen consisting of 46 conductors. The complex structure of the electric field penetrating through such a screen can be seen in figure 2 of Lyssoivan et al. (Reference Lyssoivan2012). When such a field is expanded in a series by poloidal wavenumbers, harmonics with an initial $| m_{0}|$ of the order of 40 appear. The influence of such m ₀ is shown in figure 7 for l ₀ = +40.

Figure 7. The refractive index parallel to the magnetic field (a) and the total absorbed power (b) vs. the ray length for m ₀ = 2 – red, m ₀ = −30 – blue and m ₀ = +30 – magenta. The SW rays in the minor cross-section (c) for the same set of parameters. ‘Fat’ equilibrium, l ₀ = −40.

As shown in figure 7(a), the initial m ₀ has a strong effect on the initial $N_{||}$ . At m ₀ = 50, the initial $N_{||}$ becomes even larger than shown in the figure. At m ₀ = −50, the SW is not emitted because the initial $N_{||}$ is too small. As can be seen from figure 7(b), the collisional damping of the SW is small, of the order of several percent, in all cases. The increase in absorption for the case of m ₀ = 30 is associated with the large values of $N_{||}$ at the initial section of the trajectory. This leads to Landau damping of the SW on electrons. As follows from figure 7(c), the larger the initial value of m ₀, the longer the path along the plasma periphery the SW travels before hitting the wall.

3.4. The SW in the ‘fat’ and ‘thin’ magnetic configurations

As can be seen from figure 2, in the ‘thin’ magnetic configuration, the region of low-density plasma between the separatrix and the wall, where the propagation of a SW is possible, is much wider than in the ‘fat’ configuration. The length of the ray in the ‘thin’ case is more than twice the length of the ray in the ‘fat’ case (figure 8 a). The first turn of the SW around the torus passes through low-density plasma. Then the plasma density along the ray path increases and reaches one third of the density on the magnetic axis (figure 8 b).

Figure 8. The SW rays in the equatorial cross-section (a), the plasma density (b), the refractive index parallel to the magnetic field (c) and the total absorbed power (d) vs. the ray length. ‘Thin’ equilibrium – red, ‘Fat’ equilibrium – blue.

Increasing $| N_{||}|$ along the ray path (figure 8 c) leads to rapid absorption of the SW due to Landau damping in both cases. In the ‘thin’ case, this is preceded by the absorption of a small, approximately 20 %, part of the RF power due to collisional damping (figure 8 d). In contrast to the ‘fat’ case, in the ‘thin’ case the propagation and absorption of the SW weakly depends on the initial value of m ₀.