2.1. TWA operation

The TWA used in these experiments is constructed from a subset of the modules used in the low-power helicon experiments at DIII-D . The design of the antenna is detailed in Pinsker et al. (Reference Pinsker2018). A summary of the operational principles follows. The set of modules are aligned along the direction of the static (axial) magnetic field. Each individual module has its current-carrying element aligned perpendicular to the static magnetic field. The coordinate system shown in figure 1 corresponds to the array being aligned in the $\hat {z}$ direction ( $\boldsymbol{B_0}=B_0\hat {z}$ ) and the current elements in the $\hat {y}$ direction ( $\boldsymbol{J}=J_{\mathrm{mod}}\hat {y}$ ). Power is fed into the antenna structure at one end, a fraction of the power is radiated to the plasma and residual power is mutually coupled to the subsequent module. The inner modules are each fed through the mutual inductive coupling to the previous ‘upstream’ module. The last module has a feed through so that any uncoupled power can be dissipated in an external 50 Ohm dummy load. The antenna launches waves directionally, with the sign of the axial component of the launched wavevector determined by the selection of which end of the antenna is fed. The direction of the wave launch follows the direction of the power flow along the antenna array.

The feed for one end can be seen in figure 2, but is identical on the other end of the antenna array. The antenna array is symmetric, so by exchanging the feed end and dummy load, power can be coupled into the opposite end of the antenna. This exchange results in a change of the sign of the principal imposed parallel wavenumber and change in the directionality. The design is such that the input impedance of the antenna structure is largely determined by the mutual coupling between antenna modules rather than the plasma loading of any one individual module. This results in small changes in the reflected power for variations in the plasma at the antenna face. That is not to say that antenna loading is unaffected by changes in the plasma. Changes in the absolute density and density profile can affect the antenna loading, as will be shown in § 6. The loading is characterized by the rate of decay in the power measured along the array. A series of RF pick-up loops are integrated into each module to measure the power at any given module. The phase jump between modules shifts as the frequency is varied within the pass-band of the antenna. Figure 5 shows a model of the antenna frequency $f$ response where the phase jump ranges from 0 to $\pi$ as the applied frequency is varied from the lower to upper edges of the passband (Van Compernolle et al. Reference Van Compernolle2021). The phase jump, module size and separation determine the imposed parallel wavenumber spectrum the antenna excites. Therefore, a change in the drive frequency of the antenna results in a change in the principal imposed parallel wavenumber.

Figure 5. Model of antenna transmission, reflection coefficients and phase shift between modules over the passband of the antenna array.

3.1. Density profiles

The accessible range of densities sets a region in which we expect to measure fast waves launched from the antenna. Having conducted these experiments throughout the entirety of the LAPD discharge, we sample density during the ‘afterglow’ portion of the discharge, i.e. after the discharge power is turned off and the plasma density decays in time. Figure 6(a) shows the time history of an LAPD discharge. The DC power to the discharge is switched off at t = 10 ms, so that the subsequent portion of the discharge is the ‘afterglow’, during which the density drops monotonically. The specific case shown is with the nominal $B_0$ = 0.2 T static magnetic field. The fast-wave cutoff density and the confluence density are shown for $f_0$ = 476 MHz, $n_\parallel$ = 3.0. We can see that there is a limited time during the discharge where the line-averaged density measurement suggests that the plasma could support fast-wave propagation. Selecting a time during this window and using the spatially resolved density measurement, an example of which is shown in figure 6(b), we can characterize the spatial extent of the region that supports fast-wave propagation. Figure 6(b) also shows that the density across the machine is mostly flat, with the density rising from plasma edges but essentially flat across the central region $-10 \leqslant x \leqslant 20$  cm. These density profiles allow us to construct a picture of where we might expect fast-wave propagation in our plasma across the field configurations, densities and launch frequency. Figure 7 shows the accessible region of densities as the area enclosed in the red contours. The solid red lines show the region in which the density was calculated using a swept Langmuir probe for density and temperature and calibrated against the microwave interferometer. After 10 ms we no longer had reliable temperature data from the probes, so a flattened out temperature profile was assumed to calculate the density from the ion saturation current and microwave interferometer, this region is denoted by the dotted red line contours. The two rows of figure 7 are for two static magnetic field cases ( $B_0=0.1$ T and $B_0=0.2$ T). In the higher-field case we note that there is a more extended region, both in discharge time and spatially, in which fast waves can propagate (as indicated in the dispersion relation shown in figure 4 a). As we increase the launch frequency (and thereby the magnitude of the imposed $n_\parallel$ ) the accessible region of densities shifts to higher densities. Therefore, we expect fast-wave propagation later in the discharge (that is, at lower plasma density) at lower launch frequencies. The region outside the contours at late discharge times does not support fast-wave propagation because the density is below the fast-wave cutoff density everywhere in the plasma at that time. The slow branch can propagate at those low densities, so wave power measured at these times is likely to be due to direct excitation of the slow branch.

Figure 6. Density measurements in LAPD with $B_0=0.2$ T; (a) is the line-averaged density measured by an interferometer, (b) is a density profile across the radial axis of the machine at a selected time during the discharge ( $t=17$ ms) as measured by a Langmuir probe. Accessible region of densities shown for $f_0=476$  MHz and $n_\parallel =3.0$ .

Figure 7. Measurements of density shown at two magnetic field configurations: $B_0$ = 0.1 T (a–c); $B_0$ = 0.2 T (d–f). The color intensity of the image corresponds to the absolute density measurement from a Langmuir probe. Solid lines denote time where temperature profiles were included in density calculation from swept probes (0–10 ms). Overplotted is a contour of the accessible region of densities, i.e. densities between the fast-wave cutoff and confluence densities for a given combination of $B_0$ and $n_\parallel$ .

3.2. Wave amplitude measurements

Next, we can overlay the accessible region of densities onto plots of the measured wave magnetic fields. We compare the regions in which we measure wave power with the expected domain of fast-wave propagation. Within the overplotted contour both branches may propagate, so power measured therein may result from either the fast or slow branch excited by the antenna. Figure 8 shows measurements of the wave amplitudes for combinations of magnetic field and $n_\parallel$ . During early discharge times ( $t_{\mathrm{dis}}\leqslant 10$ ms), when the discharge is powered, we see significant wave power on the plasma edge. During this time in the discharge the edge density can fluctuate drastically due to drift waves. The simple picture of an time-average density profile does not entirely describe the wave propagation region during those times. Panels (d) and (e) in figure 8 show regions within the contours with significant measured wave power. It is clear from all the panels of figure 8 that there is significant wave power at late times in the discharge after the density has dropped below the fast-wave cutoff density across the plasma column. This suggests that, at these later times (lower densities), the antenna couples power directly to the slow mode. We will later, in § 6, show that at these late times the magnitude of the antenna loading is smaller when only slow wave coupling is present. While measurements of wave power at densities well below the fast-wave cutoff density suggests coupling to the slow mode, we can use characteristics of the two modes to further investigate the wave mode present.

Figure 8. Measurements of the wave magnetic field from $\dot {B}$ probes. Overplotted is the contour of the region of accessible densities for the given combination of $B_0$ and $n_\parallel$ .

4.1. Wave front measurements

Measurements of the wave fronts in the XZ plane give us information about both the parallel and perpendicular structure of the propagating waves. A $\dot {B}$ probe is moved from shot to shot across the XZ plane, to each position within the data run plane. We then record 20 plasma shots at each position. These measurements were made over an area that spanned 20 cm in the z-direction and 40 cm in the x-direction with 1 cm resolution in each direction. The extent of the measurement domain restricted us from making reliable direct measurements in the change in the imposed parallel wavenumber spectrum. In order to recover the phase information of the wave fronts at each position we calculate the cross-spectrum between the antenna forward power and the respective component of the magnetic field measurement. This is mathematically equivalent to measuring the correlation between the two signals and maps out the phase and amplitude of the moving probe relative to the fixed reference phase of the antenna forward power. The spatial dependence of the relative phase results from the spatial structure of the wave, i.e. the wavenumber spectrum. From here we can work to identify the two modes. Figure 9 shows wave fronts measured for the nominal 476 MHz, 0.2 T case. The selected times shown are indicative of three distinct cases from the density contours in § 3: within the contour when the density is high enough to support fast-wave propagation, shortly after the density has dropped below the cutoff and late after the density is too low to support fast-wave propagation. Recall that the antenna is at x = −30 cm and z = 0 cm in our coordinate system. Waves launched from the antenna propagate away from the antenna structure, so the group velocity of the waves must be in the positive $\hat {x}$ and positive $\hat {z}$ direction pointing towards the bottom right of corner in each panel. For the high-density case we see a long-wavelength mode that spans the range $-25$  cm $\leqslant x \leqslant -12.5$  cm. The radial ( $\hat {x}$ -direction) component of the phase velocity of this mode is in the positive x-direction, suggesting that the mode is a forward-propagating mode or fast-wave-like. We can contrast this with the late discharge time measurement that show wave fronts with an oppositely oriented radial phase velocity. This mode also has a much shorter perpendicular length scale. The combination of low-density propagation, backward phase velocity and short wavelength all support that at late discharge times we are measuring a propagating slow wave. Hence, for a given static magnetic field and $n_\parallel$ , at low enough densities the antenna directly excites the slow mode. A visual interpretation of the intermediate case is not as straightforward. A combination of modes exist in this case as well as a clearer dependence on the density profile for the wavelength of the slow branch. Section 4.2 will show calculations of the perpendicular wavenumber compared with measurements of the $\hat {x}$ -component wavenumber from the XZ planes show in this section. From this we can see a clear delineation of the modes from the magnitude and sign of the measured $k_x$ .

Figure 9. Cross-spectrum measurements of the y-component of the wave magnetic field in the XZ plane for the 476 MHz, 0.2 T case. Discharge time is used as a proxy for density here; increasing discharge time corresponds to lower densities. (a) High-density case in the early afterglow t = 17 ms. (b) Intermediate discharge time, t = 23 ms. (c) Late afterglow, t = 29 ms.

Figure 10. Cross-spectrum measurements of the y-component of the wave magnetic field in the XZ plane for the 466 MHz, 0.2 T case. (a) High-density case in the early afterglow t = 17 ms. (b) Intermediate discharge time, t = 23 ms. (c) Late afterglow, t = 29 ms.

Figure 10 shows the lower frequency case. At lower drive frequency on the antenna we expect a lower launched $n_\parallel$ ; for 466 MHz we estimate $n_\parallel \approx 2.3$ from the measured phase jump between adjacent modules. With a lower launch $n_\parallel$ the fast-wave cutoff density is reduced, so we expect that fast-wave propagation should occur at later discharge times. This is seen in figure 10 where panels (a) and (b) clearly both show the long-wavelength forward wave mode. In panel (c) we see a combination of modes as even at the late discharge time the density is still high enough to support fast-wave propagation.

4.2. Comparison with the cold-plasma dispersion relationship

A comparison with the cold-plasma dispersion relationship can be made for each case. Using the static magnetic field, predicted imposed parallel wavelength and line-averaged density, we compute the perpendicular wavenumber ( $k_\perp$ ) for both cold-plasma branches as a function of density. From the wavefront images in § 4.1 we take the spatial Fourier transform to extract the $\hat {x}$ -component of the wavenumber spectrum as a function of discharge time. Matching the discharge time to the line-averaged density, we can see significant overlap in the predicted spectrum with the measured spectrum from the experiments. Figure 11 shows this for the nominal 476 MHz, $B_0$ = 0.2 T case. We can see that there are clearly two branches present in the measurement, as expected from the calculation using the cold-plasma dispersion relationship. At earlier discharge times we see that the plasma supports both the fast and slow branches for the given set of antenna and plasma parameters. As the discharge progresses and the density drops, the fast-wave branch decreases in wavenumber (increases in wavelength), as seen both in the expected $k_\perp$ and in the measurements. Throughout the discharge we see an oppositely signed $k_x$ that follows the expected $k_\perp$ for the slow branch. Figure 12 shows this comparison for four cases: (a) 466 MHz and $B_0$ = 0.1 T, (b) 486 MHz and $B_0$ = 0.1 T, (c) 466 MHz and $B_0$ = 0.2 T, (d) 486 MHz and $B_0$ = 0.2 T. Again, we see good agreement between the expected $k_\perp$ and measured $k_x$ for each case. Each case demonstrates two branches with oppositely signed $k_x$ which have correspondingly separated length scales. However, since we are using the line-averaged density to compare with our measured spectrum we note that there should be significant spread in the measured spectrum due to the fact that our plasma is not uniform in density across the whole column. The non-uniform density gives rise to a range of wavelengths. Additionally, this does not capture the spatial profiles of the waves. For example, figure 12(a) shows wave power at discharge times before the line-averaged density would suggest the plasma supports wave propagation. Further investigation of this case shows that the waves are only in the edge of the plasma, where the density is low enough to support wave propagation. At later times where the line-averaged density suggests waves may propagate we see that the waves are measured in the bulk of the plasma rather than only in the edge. Although there may be some discrepancy between the measured onset of fast-wave propagation and what is predicted using the line-averaged density, the trend in relation to the change in the imposed parallel wavenumber matches well. This shows that we are able to vary the imposed parallel wavenumber by changing the antenna drive frequency, as intended. Further, it demonstrates that the antenna does launch fast waves as intended when the plasma supports fast-wave propagation. However, it also shows that there is direct coupling to the slow mode for plasma conditions in which the density is too low for fast-wave propagation. It is important to point out that, although figures 11 and 12 show little to no power in the wavenumber spectrum at the early discharge times, there were in fact significant levels of wave power measured. During the powered portion of the discharge, $0\leqslant t_{\mathrm{dis}} \leqslant 10$ ms, wave power was measured in the plasma edge but we were unable to discern the structure using the methods described here. This is most likely due to drift waves in the plasma edge during the powered phase of the discharge, during which the strong drift mode activity creates a time-varying coupling and propagation region for the waves in the edge. This incoherent shot-to-shot variation prohibits us from gathering meaningful phase information. We also note that wave power was measured in the case 466 MHz, $B_0$ = 0.1 T case during $20\leqslant t_{\mathrm{dis}} \leqslant 40$ ms, well into the afterglow of the discharge where drift modes are less prevalent. We are not able to measure the $k_x$ spectrum here, possibly due to the fact that the plasma density is well above the confluence density for the principal imposed $n_\parallel$ so waves would propagate in and mode convert to propagate back out as a slow wave. This process could lead to interference and a complicated wave front pattern that is not discernible in the measurements.

Figure 11. Comparison of measured $k_x$ spectrum with calculated cold-plasma dispersion $k_\perp$ for both cold-plasma branches at 476 MHz and $B_0$ = 0.2 T.

Figure 12. Comparison of measured $k_x$ spectrum with calculated cold-plasma dispersion $k_\perp$ for both cold-plasma branches for: (a) 466 MHz, 0.1 T; (b) 486 MHz, 0.1 T; (c) 466 MHz, 0.2 T; (d) 486 MHz, 0.2 T.

Figure 13. Comparison of wave magnetic fields as measured by $\dot {B}$ probes when exchanging antenna feed side; the left column shows the POS side launch feed case and the right column shows the NEG side launch feed case.

7.1. Full-wave axisymmetric cold-plasma model

To gain an understanding of the modes present at a given density we aimed to create a simple model of our experiment. While the previous sections have shown, through comparison with the cold-plasma dispersion relationship, the presence of each mode, this additional step to demonstrate the measured wave fields are in fact the corresponding branch ensures that geometric effects or reflections are not responsible for the resulting wave patterns. Within the model we can see that the measured fields are a result of the directly launched wave from the antenna structure.

A finite element method (FEM) solver, COMSOL Multiphysics (AB, 2024), was used to solve the wave equation in a cold plasma with density profiles similar to our experiment. The model uses a cylindrical LAPD-like plasma in a two-dimensional axisymmetric geometry. Cylindrical coordinates are used: $\vec{r}=(r,\phi,z)$ . We assume symmetry about the center axis of the LAPD vessel such that the quantities have the following dependence:

(7.1) \begin{equation} \boldsymbol{E}(r,\phi ,z) = \boldsymbol{E}(r,z)e^{im\phi } ,\end{equation}

where $m$ is the azimuthal mode number. The FEM solver then solves the following equation for the wave electric field:

(7.2) \begin{equation} \nabla \times \nabla \times \boldsymbol{E} - \frac {\omega ^2}{c^2}\left({\epsilon _{\mathrm{ r}}}-\frac {i{\sigma }}{\omega \epsilon _0}\right)\boldsymbol{E}= -i\omega \mu _0\boldsymbol{J}_{\mathrm{ext}}. \end{equation}

Although the antenna’s finite vertical extent breaks the axisymmetry we can use the model to understand how the azimuthal modes contribute to excite the fast and slow branches. To model the antenna, we used a series of current-carrying rings set at the antenna radius. Phasing between the rings, $\delta _{\mathrm{n}}$ , can be changed to set the imposed parallel wavelength or $n_\parallel$ . Additionally, a decay factor, $\beta$ , represents the decay in module power along the current-carrying rings that can be set to account for antenna loading, this being equivalent to the $\epsilon _p$ quantity from § 6. The total external current is

(7.3) \begin{equation} \boldsymbol{J}_{\mathrm{ext}}(r,\phi ,z) = \sum _{n=0}^{N}\delta (r-a_{\mathrm{ant}})e^{(-((z-z_n)/w_{\mathrm{mod}})^2)}e^{-(\beta +i\delta _{\mathrm{ n}})n}e^{im\phi }\hat {\phi } ,\end{equation}

where n is the module number, N is the total number of modules, $\delta (r)$ is the Dirac delta function, $a_{\mathrm{ant}}$ is the antenna position radius, $z_{\mathrm{n}}$ is the module position along the array and $w_{\mathrm{ mod}}$ is the field parallel length scale of the current-carrying element in the module. Density profiles in the model take the form

(7.4) \begin{equation} n_{\mathrm{e}}(r) = n_0(1-\tanh {((r/a_{\mathrm{plasma}})^4)}). \end{equation}

The parameter $n_0$ is used to adjust the peak density as measured in the experiment and $a_{plasma}$ is used to adjust the profile width. This profile is essentially flat around the center, but captures the density gradient in the edge. An artificial conductivity $\sigma$ is imposed at the edges of the simulation domain to damp the waves as there was little evidence of significant end reflections in our experimental data set. For all of the following figures, an m-number of $m=6$ was chosen. While we do not expect the antenna to excite a single m-number, a selection $m=6$ is justified in that the antenna’s vertical extent subtends $\sim 1/6$ of the plasma circumference. From here we will discuss the comparison of the model results with the experiment.

Figure 15. Full domain of COMSOL model run using nominal 476 MHz, $B_0$ = 0.2 T case, $m=6$ . Densities are representative of timings in figure 9: $n_0=1.5\times 10^{17}$ (m $^{-3}$ ) (top), $n_0=5.5\times 10^{17}$ (m $^{-3}$ ) (bottom).

Figure 15 shows the wave fields within the entire simulation domain for two density cases: $n_0=1.5\times 10^{17}$ (m $^{-3}$ ) and $n_0=5.5\times 10^{17}$ (m $^{-3}$ ) in the nominal antenna configuration of $f_0=476$  MHz, $B_0 = 0.2$ T. The antenna is centered at $z=0$  cm and $r=30$  cm. We can immediately see that waves are launched with a preferential direction from the antenna. Using the nominal 476 MHz, $B_0$ = 0.2 T case, figure 16 shows the wave fields in the region where we took measurements during the experiments. The three panels show wave fronts for density profiles that are representative of times during the discharge shown in figure 9. We can see in Figure 16(a) that the long-wavelength forward-propagating mode is present as it was in the experimental data. At a lower density/later discharge time, figure 16(b), the wave front pattern become more complicated as a combination of modes is present. At low density/late discharge time, figure 16(c), the long-wavelength forward-propagating mode is no longer present, only a shorter backward-propagating mode can be seen. This agrees well with what was measured experimentally.

Figure 16. The COMSOL model run using nominal 476 MHz, $B_0$ = 0.2 T case, $m=6$ . Densities are representative of timings in corresponding panels of figure 9: (a) $n_0=5.5\times 10^{17}$ (m $^{-3}$ ), (b) $n_0=2.5\times 10^{17}$ (m $^{-3}$ ), (c) $n_0=1.5\times 10^{17}$ (m $^{-3}$ ).

Figure 17. The COMSOL model run using 466 MHz, $B_0$ = 0.2T case, $m=6$ . Densities are representative of timings in corresponding panels of figure 10: (a) $n_0=5.5\times 10^{17}$ (m $^{-3}$ ), (b) $n_0=2.5\times 10^{17}$ (m $^{-3}$ ), (c) $n_0=1.5\times 10^{17}$ (m $^{-3}$ ).

Figure 17 shows the same section of the simulation domain with the parameters adjusted to reflect the antenna being run at 466 MHz. This involves adjusting the phasing between the current rings to excite an $n_\parallel \approx 2.3$ . At this lower antenna frequency and imposed $n_\parallel$ we see that the perpendicular wavelength in the high-density case, figure 17(a), is shorter than the 476 MHz case, as expected from the dispersion relationship and in agreement with what was measured in the experiment. In the intermediate density case, the long-wavelength forward-propagating mode is still present but with a longer perpendicular wavelength as expected from the dispersion relationship. Lastly, the low-density case shows only the slow mode in the core with some evidence of an edge mode present. Again these panels agree well with what was measured in the experiment.

It is important to note that the model uses only one m-number to generate the solutions. The total fields from the launched m-spectrum are not accurately represented by a single m-number; but we aim to show that the fast mode is present at densities above the fast-wave cutoff density and only the slow mode is present below it. Additionally, we aim to show that the orientation of the phase fronts flip accordingly, demonstrating fast- vs slow-wave propagation. In that regard the single m-number solutions qualitatively agree with the experimental measurements.

Figure 18. Comparison of directionality in wave magnetic fields from COMSOL model. Phasing in model set to launch in + $\hat {z}$ -direction.

Finally, we use the model to demonstrate the directional behavior of the antenna and compare with the experimental data. Figure 18 shows the model when the antenna currents are set to launch in the + $\hat {z}$ -direction. We see that the wave power is concentrated on the + $\hat {z}$ -side of the antenna, and there is a $\sim 10\times$ separation in the field amplitude measured on the intended launch side compared with the opposite side of the antenna. This is similar to what was measured in the experiment.

7.2. RFPisa

Further comparison with modeling was conducted using the RFPisa code. This code can capture more completely the three-dimensional structure of the wave fields given the ideal antenna spectrum. Utilizing the plasma and antenna parameters from the experiment, we worked to simulate the LAPD case in RFPisa. In particular, we ensured that the quantity of modes included in the simulation was large enough to reproduce correctly the expected ideal antenna spectrum. The spatially resolved density measurements from the experiment were used to set the profile in the highest-density case, then scaled accordingly to represent the low-density cases.

7.2.1. Description of the RFPisa code

RFPisa is a one-dimensional-plasma, full-wave, hot-plasma simulation code entirely developed at TAE Technologies for the cylindrical geometry and applied to a diversity of problems (Ceccherini et al. Reference Ceccherini, Galeotti, Barnes and Dettrick2024). It is originally based on the reduced finite Larmor radius equations which include the so-called Swanson–Colestock–Kashuba approximation (Swanson Reference Swanson1981; Colestock & Kashuba Reference Colestock and Kashuba1983) and that were previously derived by M. Brambilla for toroidal geometry (Brambilla Reference Brambilla1989, Reference Brambilla1999a ,Reference Brambilla b , Reference Brambilla2012). The general vector wave equation that RFPisa solves is given by

(7.5) \begin{equation} \nabla \times \nabla \times \boldsymbol{E}(\boldsymbol{r},t) + \frac {1}{c^2} \frac {\partial ^2 \boldsymbol{E}(\boldsymbol{r},t)}{\partial t^2} = -\frac {4 \pi }{c^2} \frac {\partial }{\partial t} [ \boldsymbol{J}_{\mathrm{ant}}(\boldsymbol{r},t) + \boldsymbol{J}_{\mathrm{ pla}}(\boldsymbol{E}(\boldsymbol{r},t))], \end{equation}

where $\boldsymbol{E}({\boldsymbol{r}}, t)$ is the perturbation electric field, $\boldsymbol{J}_{\mathrm{ant}}$ is the antenna current density and $\boldsymbol{J}_{\mathrm{pla}} (\boldsymbol{E}(\boldsymbol{r},t))$ is the plasma current density obtained from the linearized Vlasov equations. For the cylindrical geometry a substantial simplification is achieved if azimuthal symmetry and infinite length in the axial direction can be assumed. It is worth noting that these assumptions are well matched to the LAPD geometry. Once these assumptions are taken into account, a Fourier decomposition in the azimuthal and axial directions can be applied to the perturbation electric field

(7.6) \begin{equation} \boldsymbol{E}(\boldsymbol{r},t) = \sum _{m,n} \boldsymbol{E}^{(m,n)}(r)e^{i(m\theta + n k_{\parallel , \mathrm{min}} z - \omega t ) } ,\end{equation}

where $k_{\parallel , \mathrm{ min}}$ is the lowest possible value for the parallel wavevector. The wave equation (7.5) is reduced for each mode $(m, n)$ to

(7.7) \begin{equation} \nabla \times \nabla \times \boldsymbol{E}^{(m,n)}(r)= \frac {\omega ^2}{c^2} \bigg [ \boldsymbol{E}^{(m,n)} (r)+ \frac {4\pi i}{\omega } \bigg ( \boldsymbol{J}_i^{(m,n),}(r) +\boldsymbol{J}_e^{(m,n)} (r)+\boldsymbol{J}_{\mathrm{ ant}}^{(m,n)}(r) \bigg ) \bigg ] ,\end{equation}

where $\boldsymbol{J}_i^{(m,n)}$ , $\boldsymbol{J}_{\textrm{e}}^{(m,n)}$ and $\boldsymbol{J}_{\mathrm{ant}}^{(m,n)}$ represent the $(m,n)$ mode component of the electron and ion Larmor radius current (up to second order and including displacement current), and antenna current, respectively. Once the appropriate analytical expressions for the aforementioned terms are included in (7.7) the resulting equation can be reduced to a set of three coupled equations for complex electric field components. The set of coupled complex equations is discretized on a grid and solved numerically independently for each mode to obtain the corresponding electric field and hence, through Faraday’s law, the magnetic field.

Reiterating the solution of the coupled equations through a wide range of modes allows us to generate a database which can be sourced to create Fourier three-dimensional syntheses of both electric and magnetic fields. In fact, given any antenna configuration, it is possible to compute the weight corresponding to each mode $(m, n)$ and, through an appropriate summation, to calculate the total value of any field at any location of the cylindrical volume. An example of the full three-dimensional reconstruction of the fields in the cylindrical volume is shown in figure 19. Here, the antenna modules are overlaid onto the two-dimensional plane cuts of the tangential magnetic field. In principle, any region within the simulation domain can be examined. The following section will show the resulting wave fields in the measurement region of the experiment.

Figure 19. Two-dimensional plane cuts from three-dimensional reconstruction of experiment relevant plasma in RFPisa. Density profile used is representative of figure 6(b).

While the RFPisa code takes into account hot-plasma effects, it should be noted here that we expect little to no hot-plasma effects in the LAPD case. The low temperatures present in the LAPD make the wave dispersion well described by the cold-plasma model. The use of RFPisa here is then not to validate the code’s ability to capture the hot-plasma effects, but instead to validate the complicated multi-step process used to synthesize the full three-dimensional reconstruction of the fields. This is the first time RFPisa has been compared against experimental results, and therefore the validation shown is very important to the future applications of RFPisa.

7.2.2. Wave fields

The resulting fields calculated using the RFPisa code for the nominal $f_0=476$  MHz, $B_0=0.2$ T case are shown in figure 20. Figure 20 shows the region of the simulation where we have taken data in the experiment. Here, we can see that, in the highest-density case (figure 20 a), when we expect that fast waves can propagate in the plasma, we do indeed see a forward-propagating mode. The intermediate density case, figure 20 b), shows the mixture of the long-wavelength fast wave as well as the short-wavelength slow mode. At the late discharge time, figure 20 c), we again see only the slow wave as the backward-propagating mode. While the wave patterns do not exactly match the experiment, we do see the same behavior of the observed waves in each of the density cases. It is not entirely surprising that there is some disagreement between the RFPisa code and the experiment. In fact, the RFPisa code is given an ideal launch spectrum from the antenna. Moreover, particularities about the antenna geometry such as the Faraday screens on the modules as well as non-ideal fields excited in front of the modules are not included. Since the exact geometry of the antenna is not included in the model, we therefore do not expect an exact match in the measured fields. The comparison here does show, however, that inclusion of a more complete description of the ideal spectrum still shows the transition of fast to slow wave launching from the antenna. From this we have seen that we can use RFPisa to confidently identify the plasma conditions necessary for an antenna that excites the ideal spectrum expected from the TWA to launch fast waves in an LAPD plasma.

Figure 20. The RFPisa model run using 476 MHz, $B_0$ = 0.2 T case. Densities are representative of timings in corresponding panels of figure 9: (a) $n_0=5.5\times 10^{17}$ (m $^{-3}$ ), (b) $n_0=2.5\times 10^{17}$ (m $^{-3}$ ), (c) $n_0=1.5\times 10^{17}$ (m $^{-3}$ ).