3.1 Coupled and decoupled solutions

In a homogeneous, magnetised plasma, the wave equation reads (Stix Reference Stix1992; Brambilla Reference Brambilla1998)

(3.1) $$\begin{eqnarray}\boldsymbol{k}\times (\boldsymbol{k}\times \boldsymbol{E})+k_{0}^{2}\unicode[STIX]{x1D646}\boldsymbol{\cdot }\boldsymbol{E}=0,\end{eqnarray}$$

where $\boldsymbol{k}$ is the wave vector, $\boldsymbol{E}$ the electric field associated with the wave, ${\it\omega}$ the angular frequency of the wave in vacuum, $k_{0}={\it\omega}/c$ the vacuum wavelength with $c$ the speed of light and $\unicode[STIX]{x1D646}$ the plasma dielectric tensor.

In the cold approximation and for a magnetic field $B_{0}$ in the $z$ -direction, the plasma dielectric tensor can be written as:

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D646}=\left(\begin{array}{@{}ccc@{}}S & -\text{i}D & 0\\ \text{i}D & S & 0\\ 0 & 0 & P\end{array}\right),\end{eqnarray}$$

using the Stix formulation; the elements $S$ , $D$ and $P$ are given in Stix (Reference Stix1992). The wave equation (3.1) can be rewritten as:

(3.3) $$\begin{eqnarray}\left(\begin{array}{@{}ccc@{}}k_{0}^{2}S-k_{\Vert }^{2} & -\text{i}k_{0}^{2}D & k_{\bot }k_{\Vert }\\ \text{i}k_{0}^{2}D & k_{0}^{2}S-k_{\Vert }^{2}-k_{\bot }^{2} & 0\\ k_{\bot }k_{\Vert } & 0 & k_{0}^{2}P-k_{\bot }^{2}\end{array}\right)\left(\begin{array}{@{}c@{}}E_{x}\\ E_{y}\\ E_{z}\end{array}\right)=0,\end{eqnarray}$$

where $k_{\Vert }$ and $k_{\bot }$ are respectively the parallel and perpendicular components of the wave vector with respect to $B_{0}$ . Equation (3.3) has non-trivial solutions if the determinant of the $3\times 3$ matrix is equal to zero, which leads to a fourth-order equation in $k_{\bot }$ :

(3.4) $$\begin{eqnarray}Ak_{\bot }^{4}+Bk_{\bot }^{2}+C=0.\end{eqnarray}$$

Here $A$ , $B$ and $C$ are respectively given by:

(3.5) $$\begin{eqnarray}\displaystyle & A=S, & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & B=k_{\Vert }^{2}[P+S]-k_{0}^{2}[S^{2}-D^{2}+PS] & \displaystyle\end{eqnarray}$$

and

(3.7) $$\begin{eqnarray}C=P[k_{0}^{2}[S^{2}-D^{2}]+k_{\Vert }^{4}-2k_{0}^{2}k_{\Vert }^{2}S].\end{eqnarray}$$

The full (or coupled) solutions for $k_{\bot }^{2}$ are:

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle k_{\bot ,1}^{2}=\frac{-B+\sqrt{(B^{2}-4AC)}}{2A} & \displaystyle\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle k_{\bot ,2}^{2}=\frac{-B-\sqrt{(B^{2}-4AC)}}{2A}. & \displaystyle\end{eqnarray}$$

Due to the difference between the ion and electron mass, $P$ is commonly much bigger than $S$ and $D$ . Exploiting this fact for the case where $B^{2}\gg 4AC$ yields the commonly known roots of the decoupled roots, a fast wave (FW) and a slow wave (SW):

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle k_{\bot ,FW}^{2}=\frac{[k_{0}^{2}S-k_{\Vert }^{2}]^{2}-[k_{0}^{2}D]^{2}}{k_{0}^{2}S-k_{\Vert }^{2}}=\frac{[k_{0}^{2}R-k_{\Vert }^{2}][k_{0}^{2}L-k_{\Vert }^{2}]}{k_{0}^{2}S-k_{\Vert }^{2}} & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle k_{\bot ,SW}^{2}=\frac{P}{S}[k_{0}^{2}S-k_{\Vert }^{2}]. & \displaystyle\end{eqnarray}$$

Equation (3.10) is the dispersion equation root for the decoupled fast wave; $R$ and $L$ can be found in Stix (Reference Stix1992). The fast wave has a cutoff at $k_{0}^{2}L=k_{\Vert }^{2}$ and at $k_{0}^{2}R=k_{\Vert }^{2}$ , the former being typical for the high density region and the latter for the low density region. Equation (3.11) characterises the slow wave. In view of the difference in size of $P$ w.r.t. $S$ and $D$ it can readily be seen that the wavelength of the slow wave is significantly shorter than that of the fast wave. Furthermore, the slow wave has a cutoff at $P=0$ and a resonance at $S=0$ , the latter being known as the lower hybrid resonance (LHR). Note that the decoupled slow wave cutoff location at $k_{0}^{2}S=k_{\Vert }^{2}$ coincides with a resonance for the decoupled fast wave. Plotting the roots of the full dispersion equation demonstrates that this point is actually a confluence. Although extremely useful in many cases, the decoupled roots do not uniformly reflect the actual physics.

3.2 Wave behaviour for typical tokamak parameters

The cold plasma description is adequate when doing wave studies in the edge of a tokamak, where temperatures are sufficiently low (normally a few tens of eV). Hence, they are used as standard when addressing topics of wave dynamics between the last closed flux surface of the tokamak and the launchers, be it that antenna design is routinely done assuming the density is low enough to neglect the presence of the plasma altogether at the location of the antenna straps inside the antenna box. Densities vary from nearly zero at the level of the antenna launcher to $10^{17}{-}10^{18}~\text{m}^{-3}$ at the plasma edge. Behind the last closed flux surface, the density typically decays exponentially with a decay length of a few cm. For typical tokamak edge conditions, the lower hybrid resonance layer, the low density fast wave cutoff and the slow wave cutoff are crossed at some point going from the antenna strap towards the central plasma. Depending on the exact parameters, these cutoffs and the resonance occur inside the antenna box or in front of it. Because in antenna codes the low density region is mostly not taken into account and replaced by a vacuum layer, the reality that the plasma impacts sensitively on the wave behaviour is typically sidestepped, which potentially has severe consequences on predictions of the coupled power and which does not allow adequate description of the near-field pattern that is crucial for describing the non-resonant acceleration of particles, thought to be the origin of sputtering. Theoretical and experimental verification of how the wave–plasma cross-talk actually happens seems crucial for optimising antenna designs. At the lower hybrid resonance, one of the roots of the dispersion equation (the slow wave in the decoupled formulation) changes its behaviour from propagative (for densities just below the LHR) to evanescent (for densities just above the LHR). In figure 2 the perpendicular wave vector $\boldsymbol{k}_{\bot }^{2}$ is plotted as a function of the density in the range ( $10^{13}{-}10^{20}~\text{m}^{-3}$ ). The calculations were done for parameters relevant for the edge region of tokamak plasmas: a magnetic field $B_{0}=2.0$ T, an antenna frequency $f=50$  MHz, the dominant parallel wave component $k_{\Vert }=5~\text{m}^{-1}$ of a typical dipole spectrum, a temperature $T_{e}=10$ eV and for a plasma composed of 95 % of deuterium and 5 % of hydrogen. Under these conditions the LHR (i.e. where $k_{\bot }^{2}$ goes to infinity) is located just below $10^{17}~\text{m}^{-3}$ , as can be seen on the figure. The cutoffs (i.e. where $k_{\bot }^{2}$ goes through zero) for slow and fast wave are also visible on the plot. The cutoffs are located around density values of the order of $5\times 10^{13}~\text{m}^{-3}$ and $1\times 10^{18}~\text{m}^{-3}$ , respectively. Figure 3 provides the corresponding Stix parameters, $S$ , $D$ and $P$ of the dielectric tensor. The $P$ value (responsible for the dielectric response in the direction parallel to the magnetic field) rises quickly with density, to values that are several orders of magnitude larger than the perpendicular ( $S$ and $D$ ) components, $10^{6}$ and above, compared to $10^{1}$ . The underlying reason is that both the cyclotron frequency and the square of the plasma frequency scale inversely with the particle mass, which means that the electron contribution dominates that of the ions in $P$ and that typically $P\gg S,D$ on account of the very different masses of ions and electrons (recall that ${\it\omega}\approx O({\it\Omega}_{i})\ll |{\it\Omega}|_{e}$ ). However, for low densities, and even up to the LHR layer, the $S$ , $D$ and $P$ , have values of order $1{-}10^{2}$ . This means – opposite to what is the case for typical core tokamak parameters – that the three directions are of comparable relative importance. For that reason the near-antenna field physics is very different from the core ICRF heating physics.

Figure 2.

$k_{\bot }^{2}$ values for full and decoupled solutions as a function of the plasma density and for typical tokamak parameters. The notation ‘log10’ denotes $\text{sign}(F)\log _{10}(|F|)$ when $|F|>1$ and 0 when $|F|<1$ . This notation is introduced to enable capturing of the very different length scales of the modes the cold plasma supports while allowing easy identification of regions of wave propagation and evanescence via the sign of $\log _{10}(k_{\bot }^{2})$ . In the low density region, both roots approach the vacuum limit, $k_{\bot }^{2}=k_{0}^{2}-k_{\Vert }^{2}$ , as expected, while at higher densities representative for the core, one roots is propagative (FW) and the other is deeply evanescent (SW).

Figure 3.

Stix parameters $S$ , $D$ and $P$ , as a function of the plasma density, and for typical tokamak parameters.

4.1 Test bed parameters

The maximal magnetic field in IShTAR is $B_{0}=0.24$ T, and presently in standard operation it is limited to $B_{0}=0.10$ T. As the magnetic field is one of the key ingredients setting apart a dielectric medium from a magnetised plasma, this restriction is potentially severe. Moreover – at least currently (Crombé et al. Reference Crombé2015; D’Inca et al. Reference D’Inca2015; Louche et al. Reference Louche2015) – the densities reached in IShTAR are smaller than those in the tokamak scrape-off layer. In figures 4 and 5, $k_{\bot }^{2}$ is plotted for the same conditions as figure 2, but now with the magnetic field is reduced to $B_{0}=0.10$ T. Figure 4 shows the solutions for a (95 % D, 5 % H) plasma and $k_{\Vert }=15~\text{m}^{-1}$ , whereas figure 5 is for a 100 % argon plasma and $k_{\Vert }=5~\text{m}^{-1}$ . An important thing to note regarding the figures is that the coupled and decoupled solutions are now different around the LHR, in fact the coupled solutions (which are the more correct ones) do not show the typical signature of a LHR as for the $B_{0}=2.0$ T case. Furthermore, in figure 4 it can be seen that a higher $k_{\Vert }$ value moves the interesting region for sheath studies to density values in the range $10^{17}{-}10^{20}~\text{m}^{-3}$ , which is outside the present operational domain. Argon is the main gas used for the initial operation phase of IShTAR because it is easier to handle than hydrogen. Operation with other gases, such as helium and hydrogen, are foreseen for future campaigns. The wave behaviour in an argon plasma situation is plotted in figure 5. The cutoff for the second root (FW type) is now moved to lower densities, to around $7\times 10^{17}~\text{m}^{-3}$ instead of $1\times 10^{18}~\text{m}^{-3}$ for the ‘tokamak’-case. For both roots (FW and SW type) a change from propagative to evanescent behaviour occurs at the same densities interval ( $1\times 10^{18}{-}5\times 10^{18}~\text{m}^{-3}$ ). The conclusion is clear: the test bed situation is very different from the tokamak edge with respect to wave propagation. However, understanding the interaction of the two modes is crucial and should experimentally be addressed in IShTAR, because it moreover aims at studying the consequences of this wave behaviour on the plasma density, the creation of static electric fields, material sputtering and formation of hot spots. A ‘fast wave antenna’ relies on running a current through a poloidally oriented strap to excite the fast wave, but such a current equally parasitically excites the slow wave in a magnetised plasma in which the poloidal component is non-zero. This can readily be checked by evaluation of the wave equation such that for a finite magnetic field, the poloidal electric field component at the antenna forces the radial component to be finite, and the latter in turn means that the parallel component is finite. The slow wave is propagative at densities between $S=0$ and $k_{0}^{2}S=k_{\Vert }^{2}$ and carries energy away from the antenna to the LHR. In Van Eester et al. (Reference Van Eester, Crombé and Kyrytsya2013a ,Reference Van Eester, Crombé and Kyrytsya b ) the interested reader can find examples of the slow wave exponentially decaying or adding a short wavelength propagation signature to the electric field structure close to the launcher. Such behaviour is at odds with the usual assumption that a fast wave antenna merely excites fast waves which need to tunnel through the evanescence layer to reach the $k_{0}^{2}R=k_{\Vert }^{2}$ cutoff and that the slow wave plays no role in the process.

Figure 4.

$k_{\bot }^{2}$ values for full and decoupled solutions as a function of the plasma density for similar parameters as in figure 2, but with a lower magnetic field value $B_{0}=0.1$ T and for $k_{\Vert }=15~\text{m}^{-1}$ .

Figure 5.

$k_{\bot }^{2}$ values for full and decoupled solutions as a function of the plasma density for similar parameters as in figure 4, but for a pure argon plasma (with partial ionisation  $8^{+}$ ).

4.2 Exploring the parameter domain

The reason for the difference in behaviour with respect to the tokamak case can be understood by scanning the frequency domain. In figure 6 four different operating frequencies are plotted in the range (15–60 MHz). For low frequencies (e.g. the 15 MHz case) the wave behaviour is similar to the standard tokamak scenario, with a slow and fast wave cutoff at clearly distinct densities and a typical LHR that only affects the SW, since the FW is still evanescent for that density. However for increasing frequencies (e.g. the 30 MHz case), the SW and FW cutoffs move closer to each other, and the FW becomes propagative at densities below the LHR (e.g. the 45 MHz case). At that moment (and this continues for even higher frequencies, e.g. the 60 MHz case) the effect of the resonance is visible on both waves as a sudden change from propagative to evanescent behaviour. Therefore, for studying the interaction characteristics of the RF waves and the plasma in the region between the antenna and the LCFS in a tokamak, adopting frequencies of ${\sim}$ 10 MHz must be envisaged, with the densities and magnetic fields that are obtainable in IShTAR. In addition, a lower operating frequency pushes the LHR towards lower densities – level $10^{16}~\text{m}^{-3}$ for 15 MHz instead of $10^{18}~\text{m}^{-3}$ for the 50 MHz example – which are easier to reach in the test bed.

Figure 6.

$k_{\bot }^{2}$ values for full and decoupled solutions as a function of the plasma density for similar parameters as in figure 5, but different frequencies: 15, 30, 45 and 60 MHz.

Since the Stix parameters also depend on the magnetic field, the role of the $B_{0}$ value was investigated, and is illustrated in figure 7 for various amplitudes: $B_{0}=0.05$  T, $B_{0}=0.15$ T, $B_{0}=0.30$ T and $B_{0}=1.0$ T. The calculations were performed for a fixed frequency of 50 MHz. An increasing magnetic field moves the fast wave cutoff towards higher densities. At $B_{0}=0.30$ T, the fast wave starts propagating only at around the LHR, and above that value the more standardly known separation between fast and slow waves is recovered. Presently IShTAR experiments are performed at $B_{0}=0.10$ T, as this creates the most homogeneous magnetic field configuration between the plasma source and the main vessel (Crombé et al. Reference Crombé2015; D’Inca et al. Reference D’Inca2015; Louche et al. Reference Louche2015). However, if the large magnetic field coils would be operated at maximal current of 8 kA instead of 2.4 kA, a field of $B_{0}=0.24$ T would be achievable, which could allow us to drive the antenna at higher frequencies while keeping the standard SW and FW behaviour similar to the tokamak conditions. A drawback of the higher frequency is that higher densities – outside the range of what presently can be reached in IShTAR – are needed to cross the LHR. The advantage would be that ASDEX generators can be used, which are available and can deliver high power to the antenna. At the moment only a low power (up to 1 kW) generator is available for the 10 MHz range of frequencies. As the usual collisionless damping mechanisms active in the core of tokamaks are absent, the waves inside IShTAR are poorly damped. Although standing wave patterns that will sensitively depend on the density profiles are bound to be set-up (making the analysis non-straightforward) these modest power levels are thought to be sufficient to initiate the study.

Figure 7.

$k_{\bot }^{2}$ values for full and decoupled solutions as a function of the plasma density for similar parameters as in figure 5, but different values of the $B$ -fields: 0.05, 0.15, 0.30 and 1.0 T.

Different gases can be used in the test facility. A dispersion equation study reveals that the choice of the working gas does not significantly modify the location of the relevant resonance and cutoffs. Hence, from the RF point of view, whatever gas is most easy to handle is suitable for the study.