Particle simulations on propagation and resonance of lower hybrid wave launched by phased array antenna in linear devices

1.   Introduction

Radio-frequency (RF) waves are widely applied in magnetically confined plasmas for auxiliary heating, and current drive approaches [1–3]. Within the broad range of available wave frequencies, waves in the lower hybrid range of frequency, i.e. lower hybrid waves (LHWs), have extensive applications in many devices [4–8]. They involve helicon wave, lower hybrid fast and slow wave, applied for the current drive, plasma heating and plasma rotation in toroidal devices [9–11]. In addition, a recent study has demonstrated the effectiveness of stochastic ion heating of LHW in magnetic mirror devices, which opens a new aspect for ion heating studies in linear devices, such as tandem mirrors [12]. In addition, LHW is economical, which is very helpful for university research on fusion plasma heating.

In many magnetic confinement devices, the launching structure of LHW is generally a tunable phased waveguide or loop antenna array [13, 14]. This launching system is simple and flexible. It enables us to control the initial $k_{\Vert }$, i.e. the launching angle and polarization of the injected wave, which is necessary for wave propagation into the resonance range near the plasma core and for strong wave absorption under certain conditions. It is worth mentioning that an array antenna with fixed phasing produced by fixing the transmission line is often used in a linear device, which can keep the phasing constant with large reflection power. Meanwhile, with the development of decoupler technology and automatic matching technology, a tunable phased array antenna spectrum can easily be realized by controlling the output phase of RF sources [15–18]. However, to couple tremendous external energy into the plasmas under various conditions with high efficiency, it is important to understand wave coupling and plasma response. Although loop antenna sources with a tunable antenna wavenumber spectrum have been successfully achieved in the ion cyclotron range of frequency and a helicon plasma experiment [19, 20], the tunable phased array antenna has not been investigated in lower hybrid resonance heating (LHRH), in particular at simulation.

Nowadays, numerical simulation methods are adopted to study LHWs extensively in fusion plasma. Nevertheless, most methods developed only involve one or a couple of aspects in LHW physics. For example, ray-tracing methods, such as GENRAY and C3PO, are generally applied to study the propagation and linear absorption of LHWs [21, 22]. Full-wave codes, such as the semi-spectral solver TORLH, finite element solver LHEAF and ALOHA can be applied for coupling LHWs, and more finite studies of wave propagation and energy deposition from the vacuum region with waveguides to the plasma core, which should also have the capability to simulate the tunneling process of evanescent wave are described in this paper [23–25]. In addition, Fokker–Planck-based methods, such as LUKE and CQL3D, must be integrated to take the kinetic effect into consideration, which does not satisfy first principles [26, 27]. Recently, a series of research works on LHW current drive and heating, which were carried out based on the first-principles method without any additional modeling, were presented [28]. Specifically, a symplectic structure-preserving electromagnetic particle-in-cell (PIC) scheme is implemented in the SymPIC code (https://github.com/JianyuanXiao/SymPIC) [29]. These simulations use a fully kinetic model for ions and electrons, which guarantees first-principles approximations to the original system, and the preservation of non-canonical symplectic structure provides the ability for long-term simulation without inducing abominable numerical dissipation. Note that, currently, our model does not involve any collision term explicitly and we have also not included any special treatment of the wave absorption on the equation level. Energy transference between the wave and the plasma is effected through spontaneous wave-particle interaction or another kinetic collisionless process.

The SymPIC code has tremendous advantages. It has the ability to solve the antenna-plasma coupling problem including linear and nonlinear processes. Although this work basically only involves the linear coupling and heating process, with potential nonlinear physics due to high coupled power not being included here, we can still expect that when the coupled power reaches a certain level, the ponderomotive force and background plasma turbulence can be significant and thus result in various nonlinear processes. Nevertheless, the research interests of the previous works only focus on the physics of the injected LHW inside the plasma rather than the coupling and antenna-related physics. Here, we utilize this code to investigate the detailed coupling process of LHW in linear devices with nonuniform plasmas. Note that the 2D slab configuration is adopted in our simulation, and based on the device's known plasma parameters, our simulation results can provide practical guidance for antenna structure design.

In this work, different antenna power spectra excited by different phasings from 0 to $\pi$ for octuplet-loop antenna array are simulated, and good agreement of LHW propagation is found with previous work [30, 31]. As expected, the high phased antenna can launch a slow wave well, with it propagating into the LHR layer and heat plasma. However, there is an interesting discovery that in the low phased antenna simulation case, we observed that a large electric field is generated from the edge to the plasma core and penetrates the LHR layer. This electric field, in our opinion, behaves as a source that excites a fast wave in the inner region of the LHR layer, which consequently transforms to a slow wave through slow-fast wave mode conversion (MC) [32–34]. Moreover, the simulation results also show that the absorption power of plasma in a low phased antenna is higher than that in a high phased antenna in a few wave periods.

Based on all the observations, this work has been structured to study LHW propagation and plasma absorption through PIC simulations with a phased antenna array. Section 2 describes the experimental setup of a phased array antenna and a simulation setup in a linear device, discussion of LHWs in cold deuterium plasma, and the accessibility condition. Section 3 presents a detailed study and the simulation results of wave launching, propagation, coupling and absorption with phased antenna array for different phasing. Finally, section 4 summarizes the above results and conclusions.

2.   Description of a phased array antenna and simulation setup

2.1   Description of a phased array antenna

As is well known, the phased waveguide array or grill antenna has been widely used in launching high-power LHW into toroidal plasmas with a high magnetic field. However, for linear devices with a low magnetic field, the LHW should be excited by a phased multiple-loop array antenna, as shown in figure 1. In the experiment, we can fulfill the current feed of each antenna loop with a certain phase difference through a lossless phase shifter, impedance matcher and RF decoupler. The loop antenna scheme is used to excite the azimuthal mode number $m=0$ and the Fourier spectrum of the phased array antenna current density $J_{\varphi }\left(m=0,k_z,\mathrm{∆}\varphi \right),$ where $\mathrm{∆}\varphi$ is the phase difference of the adjacent antenna and $k_z$ is the axial wavenumber. The $J_{\varphi }\left(m=0,k_z,\mathrm{∆}\varphi \right)$ is defined as follows [19]:

Figure  1.  Octuple-loop antenna with phased loop current in $\mathrm{∆}\varphi .$

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where $W_A,$ $r_A$ and $n$ denote the width, radius and number of loops, respectively, and p is the pth loop. We defined another parameter $\left|J_{\varphi ,\; Norm}\right|$ as $\left|J_{\varphi }\right|$ divided by the maximum value $\left|J_{\mathit{\varphi },\max }\right|$ of $\left|J_{\varphi }\right|,$ that is, normalized the parameter to unity. The $\left|J_{\varphi ,\; Norm}\right|$ power spectra corresponding to different phases are shown in figure 6 for $W_A=6\mathrm{cm},$ $r_A=10\mathrm{cm},$ $d_A=10\mathrm{cm}$ and $n=8.$ We find that the dominant parallel wavenumber $\left|k_{\max }\right|$ corresponding to the maximum value $\left|J_{\varphi ,max}\right|$ of $\left|J_{\varphi }\right|$ can be determined by $d_A$ and $\mathrm{∆}\varphi$, which satisfies $\left|k_{z,\max }\right|=\mathrm{∆}\varphi /d_A.$

2.2   Simulation setup

The linear device is a cylinder rather than a slab, so the radial profile of the LHW electric field should be a Bessel function [35]. The geometry effect is important to the wave-particle resonance when $k_{\perp }r\gg 1$ is not well satisfied, particularly in the core. However, the 2D slab geometry simulations with greatly simplified computation are a good enough approximation for the cold plasma. In the paper, the simulation domain is in the x-z plane, where x denotes the radial direction and z denotes the axial direction for a linear device. The 2D slab geometry consists of the deuterium plasma with nonuniform density along the x-direction and a homogeneous magnetic field along the z-direction, as shown in figure 2. The phased array antenna is parallel placed along the z-direction with fixed phase difference $\mathrm{∆}\varphi$ for adjacent current, and the antenna current $J_{\varphi }$ is now reduced to two current sheets with opposite direction $\pm J_y$ on each side of the plasma in the x-z plane.

Figure  2.  Illustration of the wave launching by the phased array antenna for nonuniform deuterium plasma with a vacuum boundary. There are perfect electric conductor (PEC) layers at the boundary of the x-direction. Boundary conditions of the z-direction are periodic.

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Meanwhile, the antenna is placed in a vacuum with thick ${\delta }_{\mathrm{V}}$ adjacent to the plasma, which is based on the physical images of the LHW experiment in linear devices. The PEC condition is used to represent the conductor walls at the x boundaries and the periodic boundary condition is applied at the z boundaries. The external magnetic field $\mathit{B}=B_0\overrightarrow{{\mathbf{e}}_{\mathit{z}}},$ $B_0=\mathrm{2000}\mathrm{G},$ is uniform and the nonuniform deuterium plasma is cold: $m_{\mathrm{i}}=\mathrm{3672},m_{\mathrm{e}}=\mathrm{3}{\mathrm{.34}\times \mathrm{10}}^{-30}\mathrm{g},$ $T_{\mathrm{e}}=T_{\mathrm{i}}=0\mathrm{eV},$ and the radial distribution function of plasma density is as follows:

In our simulations, the grid size we set in the x-direction and z-direction are $\mathrm{∆}x=1\mathrm{mm}$ and $\mathrm{∆}z=4\mathrm{mm},$ respectively. The frequency is $f=50\mathrm{MHz},$ and is the same frequency as that in Porkolab's LHW experiments and theoretical calculations [30]. The time increment is $\mathrm{∆}t=$1.67 ns. The length and width of the simulation area are $L_z=\mathrm{2000}\mathrm{∆}z$ and $L_x=\mathrm{160}\mathrm{∆}x,$ respectively. The thickness of the vacuum is $\mathrm{30}\mathrm{∆}x.$ The choice of number of sampled particles per grid (NPG) is based on the consideration that balances the computational cost and numerical noise level, and NPG = 500 is chosen in our simulation.

2.3   LHWs in cold deuterium plasma and the accessibility condition

The essential features of LHW for cold plasmas in the cylindrical geometry of a linear device are approximated by a slab model so that the plasma dielectric tensor elements is given by [33],

in which the 'Stix symbols' $S,$ $D$ and $P$ for deuterium plasma are defined as,

Then, the cold-plasma wave dispersion relation is obtained:

where the parallel index of refraction $n_{\Vert }=k_{z}c/\omega ,$ the perpendicular index of refraction $n_{\perp }=\frac{k_{\perp }c}{\omega },$ $k_z=\vec{\mathit{k}}·\frac{\vec{\mathit{B}}}{\left|\vec{\mathit{B}}\right|},$ and $\overrightarrow{{\mathit{k}}_{\perp }}=\vec{\mathit{k}}-\frac{k_z\vec{\mathit{B}}}{\left|\vec{\mathit{B}}\right|}.$ Equation (7) indicates zero, one and two modes of wave polarization in terms of $k_{\perp }^2$ depending on which frequency $\omega$ and parallel wavenumber $k_z$ are chosen. Figure 3 exhibits $k_{\perp }$ as a function of density for three different parallel wavenumbers $k_z$ at a fixed frequency of $f=\mathrm{50}\mathrm{MHz},$ magnetic field of $B_0=2000\mathrm{G},$ and deuterium plasma. In addition, if the value of $k_{\perp }$ is positive, the wave is propagating and if it is negative, the wave is cut-off. It is important to note that there is a critical value of parallel wavenumber $k_z\equiv k_{z,\mathrm{C}}={\left(1-\frac{{\omega }^2}{{\omega }_{\mathrm{ce}}{\omega }_{\mathrm{ci}}}\right)}^{-\frac{1}{2}}\approx \mathrm{0.012}{\mathrm{cm}}^{-1}$ for which the slow wave just happens to be able to propagate from its cut-off to the resonance without mode converting to fast wave, as shown in figure 3(b) [33]. If $k_z=\mathrm{0.011}{\mathrm{m}}^{-1}<k_{z,\mathrm{C}}$ (figure 3(a)), there are two MC points between the slow wave and fast wave at the densities denoted by $N_{e,\mathrm{T1}}=\mathrm{1}{\mathrm{.2}\times \mathrm{10}}^9{\mathrm{cm}}^{-3}$ and $N_{e,\mathrm{T2}}=\mathrm{1.3}\times {\mathrm{10}}^{11}{\mathrm{cm}}^{-3}.$ The wave is evanescent in the region between the two MC points. If $k_z=0.03{\mathrm{cm}}^{-1}>\mathrm{}{k}_{z,\mathrm{C}}$ (figure 3(c)), the slow wave can propagate to its resonance at a density denoted by $N_{e,\mathrm{LH}}=\mathrm{1}{\mathrm{.6}\times \mathrm{10}}^{\mathrm{11}}{\mathrm{cm}}^{-3}$ without MC, which is launched at the cut-off density of $N_{e,S}=\mathrm{3}{\mathrm{.1}\times \mathrm{10}}^7{\mathrm{cm}}^{-3}.$ In addition, the fast wave can propagate to over-dense density straightaway when the fast wave is excited at a region larger than the fast-wave cut-off density of $N_{e,\mathrm{F}}=\mathrm{2}{\mathrm{.6}\times \mathrm{10}}^{\mathrm{10}}{\mathrm{cm}}^{-3}.$ The three parallel wavenumbers are adopted here only to illustrate the three cases of LHW propagation. Hence, we just need to compare the given $k_z$ with $k_{z,\mathrm{C}}$ to indicate the propagation behavior of waves. Furthermore, the polarization of the slow-wave electric field is calculated by equation (3): $\frac{\left|E_y\right|}{\left|E_x\right|}=\left|\frac{-D}{S-n^2}\right|\ll 1,$ and $\frac{\left|E_z\right|}{\left|E_x\right|}=\left|\frac{-n_z{n}_{\perp }}{P-n_{\perp }^2}\right|\ll 1$ for $N_{\mathrm{e}}>{2\times \mathrm{10}}^{\mathrm{10}}{\mathrm{cm}}^{-3}$ in this paper. This relation means that the dominating electric field component of the slow wave excited by the loop antenna is $E_x$ in the simulation study, and the results of the simulations are consistent with these rules. Thus, the LHW propagation results are given in terms of the $E_x$ component. Note that the above calculation is for monochromatic waves (single $k_z$); therefore, the analysis is straightforward. However, the real antenna spectrum should consist of a broad k spectrum; thus, the coupling is complicated. Consequently, this work is primarily concerned with the LHW performance excited by the phased antenna array with different phasings, which change the dominant parallel wavenumbers $k_{z,\max }$ of the realistic antenna wavenumber spectrum in a cylinder plasma.

Figure  3.  Evolution of the vertical wavenumber $k_{\perp }$ with range density for fixed $f=\mathrm{50}\mathrm{MHz}$ and $B_0=\mathrm{2000}\mathrm{G}.$ In addition, the parallel wavenumber is $k_z=\mathrm{0.011}{\mathrm{cm}}^{-1}(\mathrm{a});$ $k_z=\mathrm{0.012}{\mathrm{cm}}^{-1}(\mathrm{b});$ $k_z=\mathrm{0.03}{\mathrm{cm}}^{-1}(\mathrm{c}).$ Solid gray lines and red lines represent the propagation of slow and fast waves, respectively.

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3.   Benchmark of simulations

First, the case with phasing $\mathrm{∆}\varphi =\pi$ described here can well verify the correctness of our simulations. The LHW of a slow wave propagates symmetrically from the antenna to the LHR layer along the z-axis in figure 4(a). The dominant value of the parallel wavenumber of $E_x$ in this case is $k_{z,\max }=\mathrm{0.292}{\mathrm{cm}}^{-1},$ which is the parallel index of refraction $N_{z,\max }=\mathrm{27.9}\gg 1.$ Then, the perpendicular and parallel group velocities are in the following ratio [36]:

Figure  4.  Comparison of LHW propagations in plasma with (a) PIC simulation for the phasing $\mathrm{∆}\varphi =\pi$ at $t=134\mathrm{ns}$ and (b) the theoretical ray-tracing calculation with a source having a finite length source from $z=-110\mathrm{cm}$ to $z=-40\mathrm{cm}$ with electric field distribution of $E_x=\sin \left(k_z\left(\frac{z\left[\mathrm{cm}\right]+\mathrm{110}}{\mathrm{100}}\right)\right)$ at $x=5\mathrm{cm}.$

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The spectrum of single $k_z=k_{z,\max }=\mathrm{0.314}{\mathrm{cm}}^{-1}$ is excited by a source having a finite length from $z=-\mathrm{110}\mathrm{cm}$ to $z=-\mathrm{40}\mathrm{cm}$ with electric field distribution of $E_x=\sin \left(k_z\left(\frac{z\left[\mathrm{cm}\right]+\mathrm{110}}{\mathrm{100}}\right)\right)$ at $x=\mathrm{5}\mathrm{cm}.$ In addition, the slow-wave propagation by integration of equation (8) is shown in figure 4(b). The features of LHW propagation and a qualitative account of the accessibility conditions are demonstrated here, and these make it essential to use PIC simulations to obtain accurate electric field and plasma energy profiles.

The radial evolution of $E_x$ with fixed $k_z$ can be solved using the WKB method by Porkolab [13]:

where $\mathrm{g}\left(x\right)={\displaystyle {\int }_{x_0}^x{\left(-\frac{P\left(x\text{'}\right)}{S\left(x\text{'}\right)}\right)}^{\frac{1}{2}}}\mathrm{d}x\text{'},$ $A\left(x,x_0\right)={\left(\frac{S\left(x_0\right)P\left(x_0\right)}{P\left(x\right)S\left(x\right)}\right)}^{\frac{1}{4}}$ and $x_0$ is the starting point. The WKB method is valid when $k_{\perp }^{-1}\frac{\mathrm{d}{k}_{\perp }}{{\mathrm{d}}_x}\ll k_z.$ The radial evolution of $E_x$ for a slow wave at $k_z=\mathrm{0.292}{\mathrm{cm}}^{-1}$ calculated by equation (9) is presented in figure 5(a) as a red solid line. Through comparison in figure 5(a), the waveform matches well at $x=3.5-4.5\mathrm{cm}$ and the wavelength becomes less compressed in the simulation. The scatter diagram of $k_{\perp }$ calculated by a wavelength fitting of $E_x$ for simulations (the red scatters) and by dispersion relations as a function of plasma density is shown in figure 5(b). The radial grid size is $\mathrm{∆}x=1\mathrm{mm}$ and the maximum $k_{\perp }$ is ${\displaystyle \frac{\pi }{\mathrm{∆}x}}\approx \mathrm{31.4}{\mathrm{cm}}^{-1}$, which means that the LHR layers can be resolved well for cold-plasma wave dispersion, as shown in figure 3. From figures 5(a) and (b), it can be seen that the waveform and wavenumber show good agreement in the lower density side, and the mismatch just occurs near the LHR layer, which is very reasonable. Finally, the density displays little change in our simulations, as shown in figure 5(c), and the linear absorption physics is a good enough approximation for the parameters in this paper.

Figure  5.  (a) Comparison of the radial evolution of $E_x$ with theoretical calculation (the red solid line) at fixed $k_z=\mathrm{0.292}{\mathrm{cm}}^{-1}$ and the simulation results (the black scatters) at $t=\mathrm{134}\mathrm{ns};$ (b) comparison of $k_{\perp }$ with half-wave fitting calculation of the simulation results (the blue scatters) and theoretical calculation (the red scatters) as a function of plasma density; (c) comparison of the radial evolution of plasma density with $t=\mathrm{0}\mathrm{ns}$ (the blue solid line) and $t=\mathrm{134}\mathrm{ns}$ (the red solid line). Intersection of the red dotted lines denotes the LHW resonance point.

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4.   Results and discussion

4.1   Power spectrum of the antenna with different phasing

The plasma has different responses to the antenna with different configurations, so the comparative analysis of the parallel wavenumber power spectrum of the wave and power spectrum of the antenna is beneficial for a better understanding of the wave coupling process. The power spectrum of the wave $P\left(x,k_{\Vert }\right)$ is calculated from $E_x$ as follows [37]:

where $N$ is a normalization coefficient, $y_{\mathrm{s}}$ is the surface impedance, $N_{\mathrm{t}}$ is the number of time steps, $t_j$ is the corresponding time for each time step and ${\mathit{z}}_v$ is the discretized axial coordinate. Due to the finite length of the plasma domain $L_{\mathrm{p}}=800\mathrm{cm}$ and the grid size $\mathrm{∆}z=0.4\mathrm{cm},$ the minimum resolution of $k_z$ is $\mathrm{0.0039}{\mathrm{cm}}^{-1}$ and maximum of $k_z$ is $\mathrm{7.85}{\mathrm{cm}}^{-1}.$

The broadening of the normalized antenna power spectrum is the same for all $\mathrm{∆}\varphi$ for the vacuum case, except for the dominant value of the parallel wavenumber, which is linearly increased as the increase of $\mathrm{∆}\varphi ,$ as shown in figures 6(a)–(f) by the solid black line. Meanwhile, the solid red lines represent the power spectrum of $E_x$ inside the plasma, which indicates the absorption spectrum produced by the plasma. Note that, the power spectrum of $E_x$ only displays a slight difference with respect to different radial positions, which indicates that the inhomogeneous plasma density has little influence on the $E_x$ spectrum. Thus, only the power spectrum of $E_x$ at x = 3.6 cm is given here for simplicity. The normalized power spectrum of the injected wave in the plasma is consistent with the normalized antenna spectrum, except for the cases in figures 6(a), (b) and (d). For the case in figure 6(a), $\mathrm{∆}\varphi =0,$ although the dominant value of the parallel wavenumber is smaller than $k_{z,\mathrm{C}},$ i.e. away from the accessibility region, the simulation results show that the wave can still affect the plasma core and is significantly absorbed. The tunneling and MC effects play a key role in this situation, which we discuss in the next section. For the case of $\mathrm{∆}\varphi =\pi /8$ and $\mathrm{∆}\varphi =3\pi /8,$ the part of the power spectrum with $k_z<k_{z,\mathrm{C}},$ i.e. sidelobe, also shows dramatic excitation and absorption in figures 6(b) and (d). The excitation of the sidelobe is responsible for improving the heating efficiency in this specific case.

Figure  6.  Comparison of the normalized wavenumber power spectrum with the antenna (the solid black line) and $E_x$ (the solid red line) at x = 3.6 cm for different phasings.

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4.2   Propagation characteristics of the LHW with different phasings

The propagation images of LHW in plasma for $\mathrm{∆}\varphi =0$ at different times selected by a significant change in the state are described in figure 7. As expected, there is a perpendicular propagation wave packet, or in other words, a large electric field with a long extension in the perpendicular direction, from the antenna placed in the plasma margin to the plasma center, as shown in figure 7(a). Figure 7(b) shows that the field penetrates the LHR layer and forms a spot inside the LHR region. The corresponding spot has no spatial phasing, and it can be seen as an infinite k sum and be considered a source to excite LHW. As shown in figures 7(e)–(h), the new source evolves new propagation and absorption processes from the plasma core to the exterior. It can be seen that the wave deposition range is between the MC layer and the LHR layer.

Figure  7.  Propagations of LHW in plasma for $\mathrm{∆}\varphi =0$ at different times are selected by a significant change in state and marked in the upper right corner box from (a) to (h). Color map is for the $E_x$ component. Red dashed lines denote the theoretical resonance (LHR) layers of a slow wave, and the black dashed lines denote the theoretical MC layer.

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In addition, some relatively weak bright and dark stripes appear from the resonance layer to the plasma center in figure 7(h), which should be the evanescent wave. Figure 8 shows the radial evolution of $E_x$ at $z=0\mathrm{cm}$ by a solid black line and the exponential fitting between the resonance layer and plasma center indicating that the magnitude of $E_x$ meets the exponential attenuation in the red dashed line. The tunneling process of the evanescent wave generated proves the authenticity and superiority of the simulation code.

Figure  8.  Radial evolution of $E_x$ (the solid black line) at $z=0\mathrm{cm}$ and $t=134\mathrm{ns};$ blue dashed line is the LHR layer at $x=-1.6\mathrm{cm};$ red dashed line is the exponential fitting of $E_x$ amplitude with $x.$

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In order to understand the plasma absorption principle, a schematic diagram of LHW propagation is shown in figure 9. The antenna with a low phase excites an electric oscillation with a large radial characteristic scale, which can penetrate the plasma center through the tunneling effect and form a new source of excitation represented by the cyan ellipse labeled 'new source' in figure 9. It is worth mentioning that the tunneling effect is important because the radial wavelength is close to the evanescent region size for our simulation parameters and the tunneling effect will become weak for a higher $k_{\perp }$ value and larger global simulation domain. There should be only a fast wave being excited at the inner region of the LHR, where the plasma density is higher than the LHR density, as shown for the dispersion relation in figures 3(a)–(c). It is clear that the absorption of the fast wave propagating to high density excited by the new source in the plasma core is weak, as indicated by the black dotted arrow in the pink area of figure 9. Furthermore, the case of fast-wave propagation to low density needs to be discussed separately. For monochromatic waves of $k_z>k_{z,\mathrm{C}}$ in the wave packet represented by the cyan ellipse labeled 'Part 1' in figure 9, the decreasing density from the plasma core to the edge yielding $k_{\perp }$ is smaller. In addition, these fast waves gradually disappear as they approach the low-density cut-off layer, as shown in figures 7(d)–(h). For the component with $k_z\le k_{z,C}$ of the wave packet spot represented by the cyan ellipse labeled 'Part 2' in figure 9, it will reach the MC layer from the inner side and convert into slow-wave propagation back to the interior. Then, the slow wave propagates to the resonant layer and achieves resonance absorption. At the same time, resonance promotes the generation of the tunneling effect and improves wave absorption by plasma. Our phenomenological explanation based on the LHW dispersion relation well describes the whole propagation of LHW in the low-phase antenna launching case.

Figure  9.  Illustration of LHW propagation and mode conversion process excited by low-phase antenna.

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Next, the cases with phasing $\mathrm{∆}\varphi =\pi /8$ to $\mathrm{∆}\varphi =\pi /2$ are described in figure 10. The slow wave is no longer symmetrical like $\mathrm{∆}\varphi =0$ or $\mathrm{∆}\varphi =\pi ,$ and it propagates mainly to the right, which means that the phased antenna array can control the direction of the wave propagation. The $k_{z,\max }$ excited by the antenna is $\mathrm{0.039}{\mathrm{cm}}^{-1}$ at phasing $\mathrm{∆}\varphi =\pi /8$which satisfies $k_{z,\max }>k_{\Vert ,\mathrm{C}},$ thus, we can see the slow wave is partially propagated from the antenna to the LHR layer in figure 10(a). Because the power spectrum also contains the part of $k_{z,\max }<k_{\Vert ,\mathrm{C}},$ the mode conversion process from fast wave to slow wave also occurs like the case of $\mathrm{∆}\varphi =0.$ Meanwhile, the amplitudes of $E_x$ near the point ($z=\mathrm{40}\mathrm{cm},\left|x\right|=3\mathrm{cm}$) on the slow-wave trajectory, ($z=\mathrm{115}\mathrm{cm},\left|x\right|=\mathrm{1.6}\mathrm{cm}$) on the place where the slow-wave field and mode conversion field are superimposed, (z = -75 cm, |x| = 1.0 cm) on the place that only has fast-wave field and (z =-300 cm, |x| = 1.6 cm) on the place where mode conversion predominates are $\mathrm{210},\mathrm{300},\mathrm{60}$ and $\mathrm{230}$ V cm^-1, respectively. This electric field distribution is strong evidence for the existence of two propagation processes. For $\mathrm{∆}\varphi =\pi /4$ and $\mathrm{∆}\varphi =\pi /2,$ only the slow wave propagates and the amplitudes of $E_x$ are $210$ and $170\mathrm{V}{\mathrm{cm}}^{-1}$, respectively, as shown in figure 10(b). This is because the low $k_z$ part of the spectrum launched by the antenna (either the main lobe or the sidelobe) just does not excite, as shown in figure 6(c). For $\mathrm{∆}\varphi =3\pi /8,$ the slow wave and mode conversion both existed in figure 10(c) which is similar to $\mathrm{∆}\varphi =\pi /8$and the amplitudes of $E_x$ near the point ($z=\mathrm{10}\mathrm{cm},\left|x\right|=3\mathrm{cm}$) on the slow-wave trajectory, ($z=\mathrm{60}\mathrm{cm},\left|x\right|=\mathrm{1.6}\mathrm{cm}$) on the place where the slow-wave field and mode conversion field are superimposed, ($z=\mathrm{75}\mathrm{cm},$ $\left|x\right|=\mathrm{1}\mathrm{.0}\mathrm{cm}$) on the place that only has a fast wave and ($z=-\mathrm{320}\mathrm{cm},$ $\left|x\right|=\mathrm{1.6}\mathrm{cm}$) on the place where mode conversion predominates are $\mathrm{170},$ $\mathrm{210},\mathrm{50}$ and $\mathrm{150}\mathrm{V}{\mathrm{cm}}^{-1},$ respectively.

Figure  10.  Propagations of LHW in the $x-z$ plane for different phasings: (a) $\mathrm{∆}\varphi =\pi /8;$ (b) $\mathrm{∆}\varphi =\pi /4;$ (c) $\mathrm{∆}\varphi =3\pi /8;$ (d) $\mathrm{∆}\varphi =\pi /2$ at $t=134ns.$ Color map is for the $E_z$ component, and the red dashed lines denote the theoretical resonance layers of the slow wave.

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4.3   Particle inchoate response under injected waves with different antenna phasing

In order to guide the experiment better, the earlier plasma response by intense LHW needs to be considered here. Under the action of electrostatic oscillation force $E_{x0}q\cos (\omega \mathrm{t}),$ the influence of the magnetic field on ions can be ignored and the ions' main movement is along the $x$ direction with a trajectory approximately that of a straight line. The movement of electrons is mainly through electric drift movement in the electrostatic field and it is along the $y$ direction. Certainly, the electrons also have a motion component in the $x$ direction. When lower hybrid resonance conditions are achieved, both electrons and ions should be affected at the inchoate stage. The particle movement associated with the coherent wave and the drift motion near the LHW resonance layer leads to an increase in the initial plasma energy, and this effect at different phasings is studied.

Our simulation of the particle response results well agrees with above description. The plasma-averaged kinetic energy simulated by PIC also shows that $E_{ix}^k$ and $E_{\mathrm{e}y}^k$ are relatively strong and other kinetic energy components of particles are basically close to noise level. Note that the wave itself contains a certain amount of energy, $\mathit{W}=\frac{1}{16\pi }\left(B^{*}·B+E^{*}·\frac{\partial }{\partial \omega }\left(\omega {\mathit{ϵ}}_h\right)·E\right),$ where the second term in the bracket indicates the portion of the charged-particle kinetic energy that is associated with the coherent wave motion, i.e. part of the wave energy is intrinsically stored by particle, and as the wave penetrates to the interior, the portion of the electrostatic energy becomes large (the slow wave becomes more electrostatic). Therefore, the plasma energy increased notably along the wave trace. On the other hand, as the wave reached the resonance layer, the perpendicular phase velocity decreased dramatically. Therefore, the wave could interact with plasma, which produces an additional energy transfer channel. As can be seen from figures 11 and 12, the energy deposition of both $E_{ix}^k$ and $E_{\mathrm{e}y}^k$ are similar to the trajectories of $E_x$ and the kinetic energy increases more obviously near the slow-wave resonance layer at $t=134ns.$ In addition, $E_{ix}^k$ decreases from $4-6\mathrm{eV}$ to about 0.05 eV and $E_{\mathrm{e}y}^k$ decreases from $1-2\mathrm{eV}$ to 0.05 eV when the antenna phasing changes $\mathrm{∆}\varphi =0$ to $\mathrm{∆}\varphi =\pi .$

Figure  11.  Heating effect of the x-direction component of the ion kinetic energy $E_{ix}^k$ in $z-x$ plane at $t=\mathrm{134}\mathrm{}\mathrm{ns}$ for different phasings.

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Figure  12.  Heating effect of the y direction component of the electron kinetic energy $E_{\mathrm{e}y}^k$ in $z-x$ plane at $t=\mathrm{134}\mathrm{ns}$ for different phasings.

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Meanwhile, the time evolutions of plasma absorption energy $E_{\mathrm{abs}}$ for $E_{ix}^k$ and $E_{\mathrm{e}y}^k$ with different phasings are compared and analyzed. The $E_{\mathrm{abs}}$ increases linearly and the power absorption of ions is greater than that of electrons. The absorption powers of ions and electrons as a function of the antenna phasing are shown in figure 13. The $P_{\mathrm{abs}}$ for ions (electrons) decreases linearly from $\mathrm{30}\mathrm{W}(\mathrm{10}\mathrm{W})$ to $\mathrm{2}\mathrm{W}(\mathrm{0.5}\mathrm{W})$ with the antenna phasing increasing from $\mathrm{∆}\varphi =0$ to $\mathrm{∆}\varphi =\pi /4$ and decreasing slowly from $2\mathrm{W}(0.5\mathrm{W})$ to about $0.5W$ (0.06 W) with the antenna phasing increasing from $\mathrm{∆}\varphi =\pi /4$ to $\mathrm{∆}\varphi =\pi .$ The strong heating effect of low phased antenna is due to the existence of the mode conversion mechanism, which improves the plasma resonance absorption.

Figure  13.  Absorption power of ions (the red solid line) and electrons (the black solid line) as a function of $\mathrm{∆}\varphi .$

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5.   Conclusion

In this work, the 2D PIC simulation framework for LHWs launched by phased array antenna in nonuniform plasma is applied to a linear device. The reliability and accuracy of the program for cold-plasma wave dispersion relation and propagation can be well verified by theoretical calculation.

Simulation results show that the plasma absorption power spectrum is well consistent with the antenna power spectrum, which means we can use the antenna power spectrum to calculate the wave accessibility conditions and coupling effect. Meantime, in order to select a better heating method, the effect produced by the sidelobe of the antenna spectrum should also be considered in addition to the main lobe of the antenna spectrum.

Detailed analysis of wave trajectory over time for $\mathrm{∆}\varphi =0$ shows that the wave packet radially penetrates to the plasma center as a fast wave with high $k_z$ spectrum and gradually converts into a slow wave near the LHR layer. It has been suggested that mode conversion and tunneling effect can be used to enhance the efficiency of plasma heating in fusion plasmas [38]. The phenomenon reported here encourages us to develop such experiments in a linear device to systematically study the heating efficiency of the phased array antenna. As the antenna phasing gradually increases, the slow-wave propagation structure has a certain directivity. Meanwhile, the slow-wave propagation structure gradually strengthens and the mode conversion process gradually disappears. When the antenna phasing is $\mathrm{∆}\varphi =\pi ,$ the propagation direction is symmetric and there is only the slow-wave propagation process. The energy is deposited near the LHR layers as expected in this case. Since the electric field component $E_x$ excited in plasma is strongest, the kinetic energy components of $E_{ix}^k$ (ions mainly accelerated by $E_{x}q$) and $E_{\mathrm{e}y}^k$ (electrons mainly accelerated by $\frac{E_x\overrightarrow{e_x}\times B_z\overrightarrow{e_z}}{B^2}$) display obvious improvement, while the other kinetic energy components are still at the noise level.

Finally, the energy coupling effect of plasma gradually decreases until saturation with the antenna phasing increases. Based on the simulation results, it is better to use the antenna with low $\mathrm{∆}\varphi$ to achieve better coupling. However, this does not mean that we must use the low-phasing phased array antenna, because it can easily spark breakdown in high-power experiments. In addition, the high power can be fed into plasma quietly and reposefully for the slow wave excited by a large phasing antenna. If we need directional wave propagation, for example, heating the center plasma with the RF wave launched in the tandem mirror, the antenna phasing can be set up to $\mathrm{∆}\varphi =\pi /2$ or $-\pi /2$, which can realize remote plasma heating. In general, our studies here not only fill the void of PIC simulation for LHRH with phased array antenna coupling in the linear device, but provide strong guidance for linear device experiments in the future.