2.1 The Large Plasma Device

The LAPD is a 20 m long, 1 m diameter basic plasma device at UCLA (Gekelman et al. Reference Gekelman2016). The LAPD has a variable magnetic field, from 250 G to 1.6 kG and can be varied axially. Probes inserted into the plasma can collect high-resolution, temporal information on density, temperature, potential and magnetic field fluctuations. In this study, the plasma was formed using an emissive, 72 cm diameter barium-oxide (BaO) cathode (mapped to 60 cm in a flat field) and a 72 cm diameter, 50 % transparent molybdenum anode that accelerate electrons across a configurable 40–70 V potential; voltages of $60$ and 63 V were used in this study. The source has since been upgraded to a lanthanum hexaboride (LaB6) cathode (Qian et al. Reference Qian2023) that enables access to higher-density, higher-temperature regimes, but all the data in this study are from plasma formed by a BaO cathode.

The flexible magnetic geometry of the LAPD was used to construct a variety of magnetic mirror configurations. The discharge current, fill pressure and other machine parameters were held constant. The typical plasma parameters observed in this study can be seen in table 2. Data at several mirror ratios and lengths were collected (see table 1) but emphasis is placed on the short cell because the highest mirror ratio possible ($M=2.68$) with a $500$ Gauss midplane field could be accessed and probes were able to be placed outside of the mirror cell. An overview of the axial magnetic field for the short mirror configurations and probe locations can be seen in figure 1. The 2- or 3-cell mirror configurations were also explored but are not examined in this study. All results presented below are from the short mirror-cell configuration unless otherwise specified.

Table 1. Magnetic mirror lengths and ratios. The lengths are measured where the curvature changes sign and the ratio is the maximum divided by the minimum. Approximately $3.5$ m must be added to the length if the good-curvature region is included. In the case of small asymmetries, the field strengths were averaged before calculation of the mirror ratio.

Table 2. The LAPD machine information and plasma parameters in the core and peak-fluctuation region ($x=x_\text {PF}$) at the midplane in this study. Dashed quantities are assumed to be identical to core quantities.

Figure 1. Cartoon of the LAPD and the coordinate system used. Only four of the eleven mirror configurations studied are plotted for clarity (mirrors of the same length have similar shapes and simply scale with mirror ratio). Diagnostic set varied by datarun; unlabelled diagnostics were used in both dataruns.

2.2 Diagnostics

All diagnostics were recorded with an effective sampling rate of 6.25 MHz (16-sample average at 100 Msps) and a spatial resolution of 0.5 cm. When necessary, averaging over time is done in the approximate steady-state period of the plasma discharge ($4.8$ to $11.2$ ms from the 1 kA trigger signal). Unless otherwise noted, all data presented will be from probes inside the mirror region ($z \approx 7$ m). An example of a raw ion saturation current – $I_\text {sat}=0.6 S n_e e \frac{\sqrt{T_e}}{m_i}$, where $S$ is effective area, $n_e$ electron density, $e$ elementary charge, $T_e$ electron temperature, and $m_i$ ion mass – signal and processing steps can be seen in figure 2. The raw signals are detrended by subtracting the mean across shots to obtain the fluctuations only. Fast Fourier transformations are then taken of these fluctuations for calculating power spectra and cross-correlated quantities. Frequencies above 200 kHz are dominated by electronics and instrumentation noise and thus are also ignored. Fluctuation power profiles can then be constructed.

Figure 2. Raw data and basic processing steps for LAPD probe diagnostics as demonstrated by an $I_\text {sat}$ trace from a DR1, $M=1$ mirror at 26 cm. Data are truncated from 4.8 to 11.2 ms ($a$) and detrended ($b$). Power spectral density is calculated ($c$), and a power profiles can be constructed ($d$). The shaded regions are excluded from this analysis.

The data presented were collected in two phases. The first phase (‘datarun’), DR1, collected Langmuir probe ($I_\text {sat}$ and Vf – floating potential) and magnetic fluctuation (‘Bdot’) (Everson et al. Reference Everson, Pribyl, Constantin, Zylstra, Schaeffer, Kugland and Niemann2009) traces. Fifty shots were taken at each position for every configuration. The second phase, DR2, was conducted with a similar set of diagnostics focused on temperature measurements (swept and triple probe) and the two-dimensional $x$–$y$ structure. Fifteen shots were taken at each position, except for Langmuir sweeps with 64 shots. When appropriate, all data for each position were shot averaged. The parameter ‘$I_\text {sat}$’ will be used interchangeably with ‘density’ and be presented with units of density (assuming a flat electron temperature $T_e = 4.5$ eV profile).

3.1 Profile modification

Because the field at the plasma source increases with $M$, the midplane plasma expands by a factor of $\sqrt {M}$. The physical locations of the peak-fluctuation region – $x_{\rm PF}$ (maximum gradient) – and the cathode radius $x_c$ can be seen in table 3. This expansion leads to broader plasma profiles and decreased core density but these profiles are similar in the core and at $x_{\rm PF}$ when magnetically mapped to the cathode radius $x_c$, as seen in figure 3. Dips between the core ($x/x_c=0)$ and the peak-fluctuation region ($x=x_{\rm PF})$ are seen, but fluctuation power from this region ($x/x_c = 0.5$ to $0.7$) is not significant (figure 7) so this region is not the focus of this study. The line-integrated density as measured by a 56 GHz heterodyne interferometer increases up to ${\sim }35\,\%$ from the $M=1$ case of ${\approx }8 \times 10^{13}\ {\rm cm}^{-2}$ (figure 4) but does not increase past a mirror ratio of 2.3. Discharge power increases only slightly ($3\,\%$) at higher mirror ratios, suggesting negligible impact on density. Langmuir sweeps and triple probe measurements of $T_e$ (DR2) show slightly (less than 25 %) depressed core and slightly elevated edge $T_e$ with increasing mirror ratio (figure 5) but otherwise remains unaffected. The temperature affects $I_\text {sat}$ measurements through the $\sqrt {T_e}$ term so small changes are insignificant. The low temperatures indicate that the plasma is collisional given the length scales of the system (as seen in table 2) and isotropic. Plasma potential decreases across the plasma (figure 6) when the mirror ratio exceeds 1.9. This drop in plasma potential may be caused by the grounding of the anode to the wall, which should begin at $M=1.93$ given the 72 cm anode and 100 cm vessel diameters. The reason for the local minimum in the $M=2.68$ is unknown. This potential profile creates a sheared $\boldsymbol {E} \boldsymbol {\times } \boldsymbol {B}$ velocity profile (figure 6) limited to $500\ {\rm m}\ {\rm s}^{-1}$ in the core and exceeding ${\sim }3\ {\rm km}\ {\rm s}^{-1}$ at the far edge. The flow does not exceed 4 % of the sound speed (table 2) in the core or gradient ($x=x_{\rm PF}$) region. The mirror ratio does not appear to significantly alter azimuthal flow. The floating potential (Vf) profile also exhibits similar behaviour to the plasma potential (figure 6), but is modified by the presence of primary electrons.

Table 3. The $x_c$ and $x_{\rm PF}$ locations for each mirror ratio when scaled by the expected magnetic expansion.

Figure 3. Midplane $I_\text {sat}$ profile, shot averaged and time averaged from 4.8 to 11.2 ms (we have assumed $T_e = 4.5$ eV based on triple probe and Langmuir sweep measurements). Effective area was calibrated using a nearby interferometer. Profile shape remains similar in the core and gradient region when mapped to the cathode radius $x_c$. The dips in profiles at higher $M$ below $x=x_\text {PF}$ are of unknown origin and are not the focus of this study. Shot-to-shot variation is less than 5 % for $x \leqslant 0.95 x_c$ and less than 9 % for $x \leqslant 1.4 x_c$ for all cases.

Figure 4. Line-integrated density as measured by a 56 GHz heterodyne interferometer as a function of mirror ratio, taken from four discharges for each mirror configuration. Density increases up to a mirror ratio of 2.3, where it appears to level off. The interferometer is located in the mirror-cell bad-curvature region at 9.59 m, 1.3 m closer to the cathode from the midplane.

Figure 5. The value of $T_e$ from Langmuir sweeps (DR2) at the midplane. Triple probe results are nearly identical. The increased temperatures directly at the plasma edge, indicated by dotted portions of the curves, are likely artefacts caused by sheath expansion in lower densities. Changes in mirror ratio lead to at most a 25 % change in $T_e$. The plasma is collisional and isotropic.

Figure 6. Plasma potential Vp ($a$) and derived $\boldsymbol {E}\boldsymbol {\times }\boldsymbol {B}$ velocity profiles ($b$) from Langmuir sweeps at the midplane. The range $x/x_c > 1.2$ has been excluded from the graph for greater clarity in the core and gradient region. The electric field was calculated by taking the gradient of the spline-smoothed plasma potential profile. The Mach number (in per cent) is calculated using the approximate sound speed evaluated at $x=x_{\rm PF}$ (table 2). The overall structure of the flows does not appreciably change when mirror ratio is varied.

3.2 Reduced particle flux

The density fluctuation power peaks at the steepest gradient region ($x_{\rm PF} = x/x_c \sim 0.88)$ as expected, as seen in figure 7. Here, $x_{\rm PF}$ occurs at nearly the same magnetically mapped coordinate for each mirror ratio. These density fluctuations are a large driver of changes in the cross-field particle flux (3.1). The Vf fluctuations also peak at the same location, but the total power across mirror ratios is similar and, relative to density fluctuations, much lower in the core. Core density fluctuations below 2 kHz are substantial in the core at lower mirror ratios, possibly caused by hollow profiles, non-uniform cathode emissivity or probe perturbations, but are outside the scope of this study.

Figure 7. The $I_\text {sat}$ ($a$) and $B_\perp$ ($b$) fluctuation power profiles for signals 2 kHz and up at $z=8.3$ m (midplane) and $z=7.7$ m, respectively. The lower-frequency components in $I_\text {sat}$ are associated with bulk profile evolution, dominate the core region and are not the focus of this study.

A spectral decomposition technique is used to calculate the time-averaged particle flux (Powers Reference Powers1974) as seen in figure 8

(3.1)\begin{equation} \varGamma_{\tilde{E} \times B} = \langle \tilde{n}\tilde{v}\rangle = \frac{2}{B} \int^\infty_0 k \left( \omega \right) \gamma_{n \phi} \left( \omega \right) \sin \left( \alpha_{n \phi} \right) \sqrt{ P_{\rm nn} \left( \omega \right) P_{\phi \phi} \left( \omega \right) } \,{\rm d}\omega , \end{equation}

where $k$ is the azimuthal wavenumber component, $\gamma$ is the coherency, $\alpha$ is the cross-phase, $P$ is the power spectrum and $\omega$ is the frequency. This method is more robust than the naive time integration of $n (t ) \tilde {E} ( t )$ because it accounts for the coherency of the density-potential fluctuations. This representation also enables inspection of each contributing term in the event of surprising or problematic results. A plot of the $I_\text {sat}$-Vf phase can be seen in figure 9. The flattened particle flux in the core is likely caused by primary electrons emitted by the cathode. These electrons have long mean free paths (greater than a few metres) and sample fluctuations along the length of the machine, mixing the phases of these fluctuations. Since the floating potential is set by the hotter electron population, the measured Vf fluctuations are no longer related to the local plasma potential fluctuations of a wave by bulk $T_e$ (Carter & Maggs Reference Carter and Maggs2009). These primary electrons have a significant effect in the core within the region mapped to the cathode $x \lesssim x_c$. The $I_\text {sat}$ fluctuations are not affected.

Figure 8. Cross-field, $\tilde {E_y} \times B$ fluctuation-based particle flux (calculated using (3.1)) with respect to mirror ratio. A monotonic decrease in particle flux is observed with increasing mirror ratio at the midplane. Particle flux is normalized by plasma circumference to the $M=1$ case to account for the geometry-induced decrease in particle flux caused by a larger-diameter plasma.

Figure 9. Phase ($a$) and coherency ($b$) of $I_\text {sat}$ current and Vf near $x_{\rm PF}$ at the midplane, smoothed. Positive phase means $I_\text {sat}$ leads Vf. Peaks in coherency occur between 3 and 5 kHz and at the drift-Alvén wave peaks between 12 and 25 kHz. These coherency peaks tend to have larger phase shifts than other nearby frequencies.

Azimuthal wavenumber is measured by two Vf probe tips $0.5$ cm apart. This wavenumber estimation technique yields good agreement with correlation-plane measurements (figure 17). Note that $\tilde {E}$ is not directly measured – it is instead calculated through the $k(\omega ) \sqrt {P_{\phi \phi } (\omega )}$ terms.

The $\tilde {E} \times B$ particle flux clearly decreases with mirror ratio; most of this decrease is attributed to the decrease in density fluctuation power. The particle flux for each mirror ratio was normalized to the $M=1$ case via the plasma circumference to compensate for the increased plasma surface area at the same magnetically mapped coordinate $x/x_c$. This particle flux is of the order of Bohm diffusion $D_B = \frac {1}{16} {T_e}/{B} \approx 6.25 \ \text {m}^{2}\ \text {s}^{-1}$, as observed in other transport studies (Maggs, Carter & Taylor Reference Maggs, Carter and Taylor2007).

The $T_e$ profiles and fluctuations may affect particle fluxes but measurements of both were not taken in the same datarun; nevertheless, a quantification of the effect of $T_e$ on particle flux is attempted. The $T_e$ fluctuations affect $I_\text {sat}$-based density measurements through the $T_e^{-1/2}$ term, and triple probe and Langmuir sweep $T_e$ measurements suggest that temperature gradients have a negligible impact. A naive incorporation of temperature fluctuation data from DR2 into particle fluxes from DR1 suggests that cross-field particle flux may be underestimated by up to $50\,\%$ via the $I_\text {sat}$ temperature term, but the trend and relative fluxes across mirror ratios remain unchanged. Such a naive incorporation should be treated with suspicion because of the sensitive nature of the flux with respect to the gradient and the differences in profiles between DR1 and DR2. These difference in profiles may be caused by the cathode condition, deposits on the anode or a different gas mix and are difficult to account for.

3.3 Drift waves

The $I_\text {sat}$ fluctuation power spectra in the region of peak power $x \sim x_\text {PF}$, also where the density gradient is strongest, can be seen in figure 10. Notably, the fluctuation peaks shift to higher frequencies and decrease in total fluctuation power. The shift in frequency may be the Doppler shift caused by the change in $\boldsymbol {E}\boldsymbol {\times }\boldsymbol {B}$ plasma rotation seen in figure 6 at the location $x/x_c \approx x_\text {PF}$. The shift in frequency is somewhat smaller than what would be expected from the field line-averaged increase in Alfvén speed at the longest possible wavelength. The phase angle of $I_\text {sat}$ and Vf provides insight into the nature of the driving instability. Including a non-zero resistivity $\eta$ in the drift wave leads to a small phase shift $\delta$ between the density and potential. This phase shift $\delta$ in a collisional plasmas is of the order of $\delta \approx \omega \nu _e / k_\parallel ^2 \bar {v}_e^2$ (Horton Reference Horton1999). Estimating this quantity using measured and typical values ($k_\parallel = 0.18\ {\rm rad}\ {\rm m}^{-1}$, $\bar {v}_e = 1300\ {\rm km}\ {\rm s}^{-1}$, $\nu _e = 3.7$ MHz, $\omega = 12$ kHz) yields a substantial phase shift of $\delta \approx 46^\circ$, which roughly agrees with the phase shifts in figure 9, although the implied increased phase shift at higher frequencies does not agree with measurements.

Figure 10. The $I_\text {sat}$ (density) fluctuation power averaged over a 1 cm region around $x_\text {PF}$ at the midplane. The fluctuation power is largely featureless below 2 kHz and beyond 40 kHz aside from electronics noise.

As seen in figure 9, the phase shift between $I_\text {sat}$ and Vf fluctuations is larger below 10 kHz, implying the presence of additional modes beyond or significant modification of resistive drift-wave fluctuations. The phase difference between two Vf probes, $3.83$ m apart, was used to calculate the parallel wavelength ${2 {\rm \pi}}/{\lambda } = k_\parallel = \phi _{\text {Vf1}, \text {Vf2}} / \Delta z$ assuming the wavelengths are greater than 7.66 m. The two probes mapped to the same field line only in the $M=1$ configuration, so parallel wavenumbers are available only for the flat case. Parallel wavenumbers are theoretically calculable from two-dimensional correlation planes but the coherency dropped dramatically when a mirror geometry was introduced.

A 34 m wavelength mode likely contributes to the measured $k_\parallel$ from $3$ to ${\gtrsim }10$ kHz (figure 11). Drift waves are long-wavelength modes so coherent density and potential fluctuations along the flux tube are expected. The coherency is a measure of similarity of the spectral content of two signals, in this case Vf probes 1 and 2. The coherency is defined as $\gamma = {|\langle P_{1,2} \rangle |}/{\langle |P_{1,1}|^2 \rangle \langle |P_{2,2}|^2 \rangle }$, where $P_{x,y}$ is the cross-spectrum between signals $x$ and $y$ and the angle brackets $\langle \rangle$ denote the mean over shots. The coherency between the two Vf probes drops off with increasing frequency, with a slight bump at around 12 kHz. There are several candidates for the driving mechanism of the 3–5 kHz mode, but the 12 kHz mode is most likely a drift-Alfvén wave.

Figure 11. Values of $k_\parallel$ ($a$) and coherency $\gamma$ ($b$) as a function of frequency. Only results from the $M=1$ case are available, but it is clear that there are long (${\gtrsim }34$ m) wavelength modes at 3 and 12 kHz. The probes used for calculating $k_\parallel$ were located at the midplane ($z=8.31$) and $z=12.14$ m, 3.83 m apart.

3.4 Turbulence modification

The wavenumber–power relation in figure 12 shows decreased fluctuation power when a mirror configuration is introduced. However, there is no discernible trend when the mirror ratio is increased further. The exponential nature of the curve also remains unchanged. The greatest decrease in fluctuation power occurred at low and high $k_y$ values, around $10$ and $70\ {\rm rad}\ {\rm m}^{-1}$. The shape of the power-$k_y$ curves follow an exponential distribution, and is inconsistent with a two-dimensional drift-wave turbulent cascade (Wakatani Hasegawa $k^{-3}$) (Wakatani & Hasegawa Reference Wakatani and Hasegawa1984). The steep dropoff in fluctuation power with $k_y$ suggests that higher-wavenumber fluctuations do not have a significant effect on transport.

Figure 12. Fluctuation power summed for each $k_y$ for frequencies up to 100 kHz, smoothed. The contribution to fluctuation power is negligible past 100 kHz. The fluctuation power decreases substantially when a mirror configuration is introduced, but no trend is seen otherwise and the $k_y$ spectra remain exponential. Note the logarithmic scale.

Previous simulations in a flat field (Friedman Reference Friedman2013) predicted frequency and wavenumber spectra that can be fit with many power laws or exponentials, but the data presented here (figures 10, 13, 12) appear to follow an exponential relationship within measurement variation.

Figure 13. The $B_\perp$ fluctuation power averaged at the core from 0 to 3 cm ($a$) and around the peak-fluctuation point $(x \sim x_{\rm PF})$ ($b$). Fluctuation power decreases across the board with mirror ratio except for core frequencies close to $\varOmega _{\rm ci}$. Peaks around 10–30 kHz at $x_{\rm PF}$ are consistent (region 2) with drift-Alfvén waves and the near-cyclotron frequency features in the core may be resonating Alfvén waves created by the magnetic mirror. Frequencies below 2 kHz are dominated by instrumentation noise and thus excluded.

Turbulence measurements can be directly compared with theoretical predictions and other devices, as summarized by Liewer (Reference Liewer1985). For saturated drift-wave turbulence, one expects the normalized fluctuation level $\tilde {n}/n \sim 1/\langle k_\perp \rangle L_n$, where $k_\perp$ is some typical wavenumber. The power-weighted $k_y$ (calculated from figure 12) was approximately $15\ {\rm rad}\ {\rm m}^{-1}$, which is an order of magnitude too small to satisfy this relationship. The $\tilde {n}/n$ scaling with $\rho _s / L_n$, however, is roughly consistent with drift-wave turbulence level saturation: the latter is ${\approx }3$ times larger. These comparisons suggest that the large, low-frequency fluctuations (${\sim }3$ kHz, which had even smaller $k_y$) may have a drift-wave turbulence component but are dominantly driven by other instabilities. No trend is seen in $\rho _s / L_n$ and $1/k_y L_n$ when the mirror ratio is varied.

Core fluctuations appear to decrease dramatically, as seen in the $I_\text {sat}$ fluctuation power (figure 7). The $I_\text {sat}$ decorrelation time increases from ${\sim }0.7$ ms for $M=1$ to ${\sim }2.5$ ms for $M=2.68$. At $x=x_{\rm PF}$, decorrelation times for all mirror ratios remained at $0.2$ ms.

3.5 Magnetic fluctuations

The perpendicular magnetic fluctuation ($B_\perp$) component of the drift-Alfvén wave can be seen in figure 13. These $B_\perp$ fluctuations are spatially and spectrally coincident with the electrostatic fluctuations (figure 10). Drift-Alfvén waves have been studied in the LAPD in the past (Maggs & Morales Reference Maggs and Morales1997; Vincena & Gekelman Reference Vincena and Gekelman2006); strong coupling is observed for $\beta _e > m_e / m_i$ (the electron-ion mass ratio), which is satisfied in this study. The Alfvén speed $\omega / k_\parallel = v_A = B/\sqrt {4 {\rm \pi}n M}$ (given $\omega \ll \varOmega _{\rm ci}$) when averaged over the entire column ranges from ${\sim }450$ to ${\sim }1600\ {\rm km}\ {\rm s}^{-1}$. A $k_\parallel$ corresponding to a wavelength $\lambda = 34$ m roughly falls within the bound established by the kinetic and inertial Alfvén wave dispersion relations at the frequency peaks observed at $x \sim x_{\rm PF}$ seen in figure 13. The lengthening of field lines caused by curvature accounts for at most $10\,\%$ of the change in frequency.

The spatial extent of the $B_\perp$ features identified in figure 13 are plotted in figure 14. Feature 1 at ${\approx }6$ kHz shows increased fluctuation amplitudes at $x=0$ for mirror ratios 1.9 and above, but for $M=1$ and $M=1.47$ there is no increase in fluctuation power. A similar feature, but at a much smaller level, is observed in the $I_\text {sat}$ fluctuation power in the core as well. This core feature may be caused by the hole in the core seen in the $I_\text {sat}$ profile (figure 3) driving low-amplitude waves or instabilities. Feature 2 in figure 14 is the magnetic component of the drift-Alfvén wave. The fluctuation power peaks at the gradient region and corresponds with the peak in density fluctuations (figure 7).

Figure 14. The $B_\perp$ fluctuation power profiles for the three regions shown in figure 13: region 1 (6 kHz) ($a$), region 2, where frequencies are taken from the peaks of the drift-Alfvén waves for each mirror ratio, ($b$) and region 3 (114 kHz) ($c$).

Feature 3 is particularly interesting because this the only fluctuating quantity to increase with mirror ratio, seen in figure 15. This feature may be broad evanescent Alfvénic fluctuations from the plasma source. These fluctuations have been observed in the LAPD in the source region alongside an Alfvén wave maser (Maggs, Morales & Carter Reference Maggs, Morales and Carter2005). Note that the Alfvén maser cannot enter the mirror cell at mirror ratios greater than 1.75 because the Alfvén maser resonates at 0.57 $f_{\rm ci}$ but the midplane is always at or near 500 G.

Figure 15. Summed fluctuation power of $B_\perp$ in the core ($x/x_c \leqslant 0.3$) as a function of mirror length and ratio. ($a$) The fluctuation power is normalized by the sum of the full-spectrum summed power. ($b$) The frequency of the power distribution >50 kHz weighted by the fluctuation power.

The sub-$2$ kHz modes in $B_\perp$ and the harmonics of these modes are nearly constant in power across the entire plasma; these features are likely perturbations from the magnet power supplies and thus ignored. The lack of radial, azimuthal and axial structure in these magnetic signals below 2 kHz and narrow bandwidth indicate a non-plasma origin. Significant radial and azimuthal structure in $B_\perp$ fluctuation power starts to appear in frequencies larger than 4 kHz.