Experiments and gyrokinetic simulations of the nonlinear interaction between spinning magnetized plasma pressure filaments

Filamentary plasma structures are characterized by pressure perturbations that are spatially localized in the plane perpendicular to a confining magnetic field and elongated along this field.^1,2 The cross field localization leads to the term “blobs” or “blob-filaments” in the literature. These structures can develop from linear instabilities, such as interchange-type modes, and emerge in the turbulent saturation phase.^3–5 They are most well studied in toroidal magnetized plasmas^6–8 but they can also appear in linear devices as well.^9–12 In the former case, charged particles within the filaments are subject to curvature and magnetic gradient drifts that cause charge separation and a vertical (polarization) electric field that drives the filaments radially outward via the $E×B$ drift. In the latter linear device case, filament polarization can occur via centrifugal forces from rigid body plasma rotation^13,14 or neutral flows arising from different heating rates in the plasma core and edge regions.¹⁵ If the filaments contain a substantial excess of particles and energy, this can be a significant component of the overall cross field thermal and particle fluxes.

A number of experiments have imaged the filaments using fast visual cameras and gas puff imaging (GPI).^8,16–18 With much improved resolution, the blob tracking and counting statistics of these structures have advanced.¹⁹ They have supported the conjecture that filamentary structures are a result of the nonlinear saturation of edge instabilities and evolve as propagating coherent structures that exhibit properties of intermittent turbulence. Numerical simulations have played an important role in confirming blob-filament theoretical and scaling studies using seeded blobs as initial conditions.²⁰ More recent work has focused on the blob-filament “birth zone” and physical processes leading to the observed blob spatiotemporal scales. A key ingredient appears to be sheared flows that act to breakup the radially extended streamers.²¹ 

One key aspect of blob-filament dynamics associated with the present study is the internal rotation or spin. In certain regimes, blob-filaments are not in thermal equilibrium with the background plasma, and “hot blobs” with an internal temperature profile, $Te(r)$⁠, may develop.^2,22–24 The elevated temperature of the filament induces a local increase in the floating potential, thus forming a monopolar electrostatic potential that leads to $E×B$ circulatory motion. This spin brakes the outward convective radial displacement of the blobs by reducing the internal charge polarization.^25,26 It can also stabilize certain internal instabilities, such as flute mode perturbations; however, other types of instabilities can occur driven by sheared rotation (Kelvin–Helmholtz); these can contribute to the blob dissipation.^27,28

The simulation models of filament dynamics can be broadly classified as two-fluid, multi-moment (based on drift-reduced Braginskii equations),^29–31 gyrofluid,^32,33 gyrokinetic continuum,³⁴ and gyrokinetic particle,³⁵ hybrid (particle ion, fluid electron),³⁶ and fully kinetic particle simulations.^37,38 These 2D and 3D models have mainly been carried out in slab, flux tube, and even more complex magnetic geometries while working under various assumptions and approximations. For example, 3D models include parallel electron dynamics and sheath boundary conditions, where a collisional Ohm's law is used to obtain the parallel electron current density.^27,28,39 These earlier simulation studies, including parallel electron dynamics, worked in the electrostatic limit, allowing for electric potential and density variation, and assumed electron–ion collisions dominated while neglecting electron inertia, viscosity, and electron-neutral collisions in the parallel momentum balance. Extensions have been made to incorporate electromagnetic perturbations,^40,41 a finite Larmor radius,⁴² neutral gas, and collisionality.^43,44 The two-fluid models have been applied to the study of multiple blob-filament interactions^31,45 and internal blob instabilities.^27,28,46,47

Large scale potential structures with blob-like properties are also observed in simple magnetized toroidal (SMT) devices.^7,48–51 Of particular interest in this work are the studies of blob-filament dynamics in linear magnetized plasma devices where these structures and intermittent dynamics are associated with instabilities in drift waves, flow shear, and rotational modes, where centrifugal forces can mimic magnetic curvature.^52–54 Blob-filament structures have been found in rapidly rotating plasma columns¹⁴ and in the shadow of a limiter placed inside the cylindrical column.¹⁰ The vast majority of experiments have focused on self-generated blob-filaments involving density striations, with the exception of pulsed ECH heating in a simple magnetized tori⁵⁵ and controlled electron temperature filaments formed using a thermalized electron beam source in a linear plasma device.⁵⁶ 

In our previous work on single and multiple magnetized electron temperature filament studies, using localized heat sources in the Large Plasma Device (LAPD),⁵⁷ we have characterized the dominant drift-Alfvén modes that are driven mainly by the cross field electron thermal gradient.⁵⁸ In the establishment of the single isolated temperature filament there is a transition from classical electron heat transport to anomalous cross field transport once the temperature gradient in the filament reaches a certain threshold.⁵⁹ When three filaments of this type are placed in close proximity (within a few electron collisionless skin depths), the classical transport phase is minimized and the system rapidly transitions to a turbulent phase. Three-dimensional effects are evident from analysis of the filament structure at different axial planes where the observed filament merging is highly nonuniform axially.⁶⁰ The system of bundled filaments exhibits intermittency in the fluctuations and transport dynamics originating from Lorentzian pulses associated with the saturation dynamics of the unstable drift-Alfvén modes.

In this work, we focus on the symmetry breaking of the temperature gradients through filament–filament interactions in close proximity and examine how the convective mode structure is altered. The local thermal gradients on each filament reorganize into a more global thermal gradient surrounding the filaments. Another feature uncovered in this study concerns the particle and energy transport between filaments which induces inter-filament polarization electric fields. These influence flows and filament rotation and are also linked to the filament merging dynamics.

The organization of this paper is as follows. Section II describes the experimental setup and parameters used to initialize multiple filamentary structures in close proximity. The results from Langmuir probe measurements are presented in Sec. III, and Sec. IV contains the analysis of gyrokinetic simulations of the filament–filament interaction. Finally, Sec. V gives the discussion and conclusions.

The fluctuation data used in this study were obtained from experiments conducted on the upgraded Large Plasma Device (LAPD) at the Basic Plasma Science Facility (BaPSF), University of California, Los Angeles (UCLA).⁵⁷ The basic setup of the magnetized plasma experiment is summarized in Fig. 1.

First, the background plasma conditions are presented. The linear device produces a magnetically confined cylindrical plasma column that is 18 m long and 60 cm in diameter. The cylindrical chamber is encircled by electromagnets that produce a uniform background magnetic field fixed at $B0=0.1$ T (1000 G) for these experiments. The He gas filling the chamber has a fill pressure of $1.33×10−4$ Torr (⁠ $1.77×10−5$ kPa). At one end of the chamber is the main plasma source in the form of a hot barium oxide (BaO) cathode and a mesh anode 50 cm away axially; a bias of $−65$ V is applied at the cathode with respect to the anode, and thermionic electrons accelerate into the main plasma chamber and collisionally ionize the He gas to form the plasma column. The main plasma discharge lasts for 12 ms and has a typical electron temperature of $Te∼3$ eV, a density of $n∼2×1012$ cm⁻³, and an ion temperature, $Ti$⁠, less than 1 eV. Once the main discharge bias is switched off, the plasma transitions to the afterglow phase, where the electron temperature rapidly cools to $Te∼0.25$ eV, whereas the density decays exponentially throughout the afterglow with an order of magnitude larger time constant. The neutral density during the afterglow phase is approximately $4−5×1012$ cm⁻³. The LAPD is a pulsed device and the process of making the plasma is repeated at a 1 Hz repetition rate; the plasma conditions are highly reproducible shot-to-shot.

Second, the localized heat sources are described. These consist of two or three 3-mm-diameter crystal cathodes of cerium hexaboride (CeB₆) mounted on probe shafts that are inserted into the plasma, as illustrated in Figs. 1(a) and 1(b). The crystals are supported by current carrying wires mounted on a ceramic base. CeB₆ has a low work function and high electron emissivity when it is heated to an operating temperature of around 1400°C; the heating is accomplished ohmically with around 10 W of power to the crystal. The axial location of the filaments is taken as $z0=0$ cm and the crystals can be arranged arbitrarily close to each other. Each crystal has independent heating and biasing circuitry, and the crystals are inserted from different sides of the machine. We did our best to match the emission power of each source; however, they are not perfectly symmetric. More details on the mounting and alignment can be found elsewhere.⁵⁸ 

A negative bias is applied between each crystal and the mesh anode starting at 2 ms into the afterglow phase; the time of application of the bias is taken as $t=0$ and the bias remains for 20 ms. When the bias is applied between the hot crystal cathodes and the mesh anode at the other end of the device, electrons are emitted from the CeB₆ crystals. The potential difference between the crystals and anode is kept below 20 V to ensure the thermionic electrons are below the ionization energy of helium (24.6 eV). The emitted electrons thermalize in the afterglow plasma after a few mean free paths, creating a heated region less than 1 m in axial extent and a few millimeters in diameter. Initially, the asymmetry in perpendicular and parallel thermal transport coefficients in the magnetized plasma causes the heated regions to rapidly form filamentary structures of elevated temperature that are less than 10 m in length with symmetric Gaussian-like transverse profiles about 1–2 cm in diameter. The peak temperature near the heat sources is 3–5 eV depending on the bias voltage and decreases continuously toward the end and edges of the filaments where the temperature equilibrates with the cold background plasma.

Third, the fluctuation measurements presented in this paper are all collected using small Langmuir probes inserted through the ports and biased to sample the temporal evolution of the current drawn from the plasma. Since the LAPD has a high repetition rate and plasma reproducibility, a probe located at an axial position (z) can be placed at a position in the transverse $(x,y)$ plane, collect several nearly identical shots, and then be moved to a new position in the plane to repeat the process; this allows the collection of large 2D data planes from an ensemble of plasma shots. The process of moving the probes and collecting the data are entirely automated with the probes mounted on probe drives capable of less than 1 mm accuracy. In these experiments measurement probes for collecting 2D planes of data are inserted at distances $z1=256$ cm and $z2=544$ cm from the crystal cathodes along with a reference probe that is manually inserted at $zR=320$ cm and is used for correlation analysis of the plasma shots. The probe at plane $z1$ is the one used for planes of $Isat$ as well as other quantities.

The Langmuir probes are used to make two types of measurements; ion saturation current and temporal Langmuir sweeps to obtain the electron temperature, plasma density, and space potential, $Vs$⁠, from a characteristic I–V curve. The $Isat$ measurements are decomposed into fluctuating and time-averaged components. The time resolution of the Langmuir probe is 1.3  $μ$s, which corresponds to a sampling rate of 781.25 kHz. For planes of $Isat$⁠, the time width averaged over is 5  $μ$s.

Figure 2 shows the two-filament case at a very early time (⁠ $t=0.07$ ms) just as the filaments are turning on. Panel (a) gives the $Isat$ for the far separation case (approximately 2 cm edge to edge, 2.5 cm center to center). The coordinates (0, 0) are defined to be the center of the bottom filament. The figure is presented in terms of $Isat$/ $Isat0$⁠, where $Isat0$ is defined as the $Isat$ of the background plasma at the given time, and calculated as the average of $Isat$ around the outer edge of the plot domain. This value dips below one in some places because the background edge is not uniform. Panel (b) shows $∇Isat$ for the far separation case and highlights the inhomogeneity in the gradient on the bottom filament along the top right corner, which will develop into a full “tail” region at later times. Panel (c) shows the close separation (1 cm edge to edge) case. In these early times, the filaments rapidly rotate around each other and merge into one large filament. The time, $t=0.07$ ms, is chosen to highlight the rotating filaments in this panel, with one larger filament on the left side and a smaller filament-blob on the lower right. In panel (d), the gradient map shows that the majority of the gradient is in a ring shape wrapped around the combining filament mixture but there is a small vertical division between the left and right merging filaments. It is worth noting that the (0, 0) position is defined based on the ultimate location of the bottom filament in later times (around 10 ms).

Figure 3 presents the evolution of the two-filament far case at 4 and 8 ms, with panel (a) depicting $Isat$ at 4 ms, panel (b) depicting $∇Isat$ at 4 ms, panel (c) depicting $Isat$ at 8 ms, and panel (d) depicting $∇Isat$ at 8 ms. Unlike the close case, the filaments do not merge and wrap into a single filament, but a tail still forms extending from the bottom filament in the counterclockwise direction. The gradient maps highlight the edges of the tail, and the 8 ms tail region is wider and extends diagonally upward rather than initially upward. This “tail region” is evidence for the interaction between the filaments and will be further explored using Figs. 4 and 6. We also remark here that a more distinct asymmetry in $Isat$ between the upper and lower filaments forms at 4 ms as compared to the earlier times shown in Fig. 2. We attribute this up–down asymmetry to be related to the slightly unequal heating rates on each filament. The advection of the temperature and density, discussed next, gives a clearer picture of the inter-filament transport.

Based on measurements from Langmuir I–V sweeps, the electron temperature, density, and space potential were constructed. Figure 4 shows $Te$⁠, $ne/nb$⁠, and $ΔVs$ at 8 ms for the far separation case, where $nb$ is the background density, defined using the average of the outside edge of the measurement area. The reference zero taken for $ΔVs$ is also defined as the average of the outside edge of the measurements. The small panels beside each of the data planes show the parameters along line cuts in $x=0$ and $y=0$ (the dotted lines). Panel (a) indicates the filaments have a significant temperature enhancement, which appears Gaussian-like and slightly asymmetric on the bottom filament due to the presence of the tail region. Panel (b) shows that the bottom filament has a density depletion while the top filament has a density enhancement. This is most likely caused by the transport of density along the filament tail toward the upper filament. The existence of a region of elevated density on the right side of the upper filament (the same side as the filament tail) further supports this conclusion. It is worth noting that the location of the apparent spot in density is not representative of the filament location. Overall, it is clear that the filament profile (and $Isat$⁠) is dominated by temperature, and the changes in density are small in comparison. This is expected, as the bias voltage for the crystals is kept below the ionization threshold of helium. Since there is no significant particle source, the spatial variation in density arises primarily due to redistribution through convective flows. Panel (c) illustrates the dip in potential along each of the filaments, in the tail region and between the filaments. It is worth noting that the temperature and potential are very similar in shape, though inverted in sign, and this relationship was used to calculate the potential based on the temperature maps in the simulations. All three of the quantities in this figure have been smoothed spatially due to the inherent noisiness in processing Langmuir sweeps. A moving filter of five data points using a Savitzky–Golay filter of polynomial order 3 is used to smooth the original data along the X and Y directions, and then the results of these two smoothings are averaged to produce the final plot.

Figure 5 gives the evolution of the two-filament close case at two later time steps in the time series. Panels (a) and (b) depict $Isat$ and $∇Isat$⁠, respectively, at 4 ms, and panels (c) and (d) show the same at 8 ms. By 4 ms, the rotating filaments have coalesced into one single larger filament centered at the location of the top crystal, with an extended ring outside the filament core. This is highlighted in the gradient map in panel (b), which indicates a strong gradient around the central filament core and a weaker gradient around the outside of the ring. In panels (c) and (d), the bottom filament has started to reemerge and the exterior ring has morphed into a wrapped tail extending out from the filament. Panel (d) illustrates the reemergence of the bottom filament in the gradient map, particularly the gradient along the outside filament tail. At later times (around 10 ms), the bottom filament continues to reemerge and rotate clockwise to eventually settle at the (0, 0) position. We believe the gradients referred to here are mainly in electron temperature since the far separation sweeps demonstrated spatial variation mainly in electron temperature compared with variation in the density, but we do not have Langmuir sweeps for this case to confirm this.

Next, we examine the mode structure starting with the far separation case. Figure 6 is taken at 8 ms with $Isat$ given in panel (a) along with the reconstructions of the $δIsat$ mode structure in panels (b) and (c). For this case the reference probe location was not in a position to make a proper cross correlation analysis. Instead, the crystal currents were used as a reference signal for correlating each shot's fluctuations into a coherent signal. Panel (b) is the reconstructed mode based on correlation with the top filament crystal current, and panel (c) shows the mode reconstructed by correlating with the bottom crystal current. The frequency range was selected based on the dominant mode frequency using wavelet transforms (not shown). The mode structure is illustrated using a two-ended log scale, which is linear near zero⁶¹ to highlight weak fluctuations. An azimuthal $m=1$ mode is dominant at both the top and bottom filaments, with mode structure extending out onto the tail of the bottom filament. It is interesting to note the partially formed but distorted mode structure on the top filament in panel (c) using the bottom crystal current, and the weak but present mode structure on the bottom filament in panel (b). This seems to suggest a middle ground between completely disconnected filaments and full coupling between the modes; some level of weak coupling is evident. This also demonstrates that the crystal currents are not perfectly coupled, as panels (b) and (c) highlight modes more clearly on the filament corresponding to the crystal used for correlation. This will be discussed further in a separate paper devoted to the two-filament case, where bicoherence analysis is used to quantify the degree of interaction.

Figure 7 illustrates the reconstructed mode structure from the two-filament close case. Panel (a) shows $Isat$ at 4 ms and panel (b) shows the reconstructed mode structure using Fourier transforms from $t=4–5$ ms and over a frequency range of $16–20$ kHz as determined by wavelet transforms to be the dominant mode. The dominant mode appears to have a $m=1$ mode structure centered around the main filament. However, there is also a part of the mode structure extending outward onto the “ring”/wrapped tail region and exterior gradient.

Next, we perform a similar analysis for the three-filament configuration illustrated in Fig. 8, using both $Isat$ and $∇Isat$ taken at 1 ms. Panels (a) and (b) show the far configuration (⁠ $∼2$ cm separation edge to edge) in $Isat$ and $∇Isat$⁠, respectively. Panels (c) and (d) correspond to the close configuration (⁠ $∼1$ cm separation) in $Isat$ and $∇Isat$⁠. As early as 1 ms, a significant tail was formed in the far case, extending counterclockwise outward from each of the filament centers. This is highlighted in the gradient map showing how they develop on both the inner and outer edges of the tails, with the strongest gradient on the opposite side of the tail region, just like in the two-filament case. In the close case, the filaments remain visibly distinct but with a significant region of elevated $Isat$ surrounding the filament group. The gradient map highlights the extent of the bottom filaments and shows the existence of a substantial gradient along the top edge of the top filament/ring, which drives significant mode activity as shown later in Fig. 12.

Figure 9 presents the evolution of the three-filament far case at 4 and 8 ms, with panels (a) and (b) showing $Isat$ and $∇Isat$ at 4 ms, and panels (c) and (d) showing $Isat$ and $∇Isat$ at 8 ms. The most notable feature of this figure is the “broadening” of the filaments over time. Compared to the 1 ms case in Fig. 8, there appears to be more of a gradient “closing the circle” around each of the filaments and less of an open-ended tail. The tail region also grows less pronounced as the filaments widen.

Figure 10 gives the evolution of the close case at 4 and 8 ms, with panels (a) and (b) showing $Isat$ and $∇Isat$ at 4 ms, and panels (c) and (d) are $Isat$ and $∇Isat$ at 8 ms. The strength of the exterior gradient extending on the right/top edge of the filaments is significant, as a strong mode pattern develops along this outer gradient. Although the individual filaments are still visible, they are strongly merged in this case, unlike the far case, and have weaker gradients surrounding them compared to the stronger gradient on the exterior edge.

Figure 11 depicts the reconstructed mode pattern using a manually placed reference probe. Panel (a) refers to $Isat$ at 1 ms, and panel (b) is the reconstructed $δIsat$ mode pattern over $t=1–2$ ms, filtered from 19 to 22 kHz. Each filament has the dominant $m=1$ mode as well as significant mode activity extending onto the filament tail. These modes are coherent despite being separate, unlike the two-filament far case because this is taken early in the time series before the modes have drifted out of phase with each other shot-to-shot. Not shown in this figure is a likely $m=2$ mode on the top filament (based on its frequency range of $∼40$ kHz) because it is not coherent with the reference probe. This $∼40$ kHz mode is the dominant mode for the top filament based on mode power—potentially due to the reduced tail region on the top filament or asymmetries in filament power.

Figure 12 illustrates the reconstructed mode pattern using a manually placed reference probe over two different frequency ranges to reconstruct the more complete mode pattern. The early mode structure is rather incoherent, so a time range of $4–5$ ms was chosen since the modes are more fully established by then. The dominant mode along the outer edge and bottom left/top filaments is shown in panel (b) from 13 to 16 kHz and reveals an extended mode structure, strongest along the outer edge of the filament and where the stronger gradient is present. The bottom left filament has dominant frequency activity in the $21–24$ kHz range and an $m=1$ mode is present, with traces of frequency activity in this range appearing to propagate along the filament tail.

The data from the multi-filament experiments was subsequently compared to three-dimensional gyrokinetic simulations. In this model, the electrons are treated as drift-kinetic particles and ions as gyrokinetic, which incorporates the gyro-averaged cross field $E×B$ drifts and exact parallel dynamics. The self-consistent evolution requires the solution of a gyrokinetic-Poisson equation to obtain the perpendicular and parallel electric fields, and the magnetic field perturbations arising from the parallel particle currents are computed via Ampère's law. The details of the simulation model is given elsewhere.⁶² This version of the model includes electron–ion collisions using a particle and energy conserving Monte Carlo collision operator.⁶³ Conducting boundary conditions are applied in the x–y plane while the axial boundary conditions are chosen to be periodic, which is the direction of the confining magnetic field, $Bo$⁠.

The simulation model allows for the use of realistic parameters when compared to the experiment. The 3D system has spatial scales of dimensions $Lx×Ly×Lz=256Δx×256Δy×16Δz$⁠, with grid spacing $Δx=Δy=0.0625δe$ and $Δz=69.5δe$⁠, where $δe≃0.53$ is the collisionless electron skin depth for an electron density of $1×1012$ cm⁻³. The simulations were carried out with a maximum of 80 particles per cell. The time step used in the simulation, normalized to the ion cyclotron frequency, is taken as $ωciΔt=1.0$ and corresponds to $0.2 μ$s, assuming a helium ion mass and 1 kG background magnetic field. All time steps quoted in the simulation results are normalized to the ion cyclotron frequency. The electron–ion collision frequency was chosen to be $νei/ωci=1$⁠. It should be emphasized here that the simulations are carried out on shorter time scales, comparable to the temporal evolution of the drift-Alfvén instability, which occurs on time scales of tens of microseconds. The larger scale convective flows that evolve on time scales of milliseconds is not captured in these results.

The simulation results for the two-filament interaction are presented in this section. The filament separations are chosen to be consistent with the experiment, namely, the far separation case with $d=2.5$ cm (⁠ $∼5δe$⁠) and the close separation case with $d=1$ cm (⁠ $∼2δe$⁠), with both distances being measured from filament center to center.

The electron temperature maps for both separations are displayed in Fig. 13, taken at the initial time and a later time level corresponding to the saturation of the unstable drift-Alfvén modes driven by the electron temperature gradient. For the far separation case, Fig. 13(b) illustrates the nonlinear evolution of the spinning filaments with spiral arms emanating from the maximum temperature gradient region where the mode amplitudes peak for each of the filaments. These arms extend outward and partially overlap at later times. In Fig. 13(c), the cross field electron temperature fluctuation map is shown at an earlier time, $t=80$⁠, corresponding to the pre-saturation of the instability and indicates the presence of several low mode number (⁠ $m=1,2,3$⁠) azimuthal perturbations. Saturation of the unstable modes occurs through a combination of mode coupling and background temperature gradient modification. We note here that the temperature fluctuation maps serve as a proxy for the drift-Alfvén eigenmode structure and relate directly to the $δIsat$⁠.

In the lower panels of Fig. 13, the close separation case reveals a spiral arm pattern emanating from the strongly overlapping eigenmodes on each filament. The mode pattern shown in the temperature fluctuation map, taken at saturation, indicates a dominant $m=1$ azimuthal mode with an $E×B$ circulation flow pattern running through and around each filament. The spiral arms extend over three times the distance of the initial size of the individual filaments. We note here that some smoothing is applied to the temperature maps using the Matlab function smoothn,⁶⁴ except for panels (b) and (e), which are unsmoothed to highlight the spiral arms extending outward from the filaments. Each of the panels is normalized to $Te_max_0$⁠, which is defined as the peak temperature at $t=0$⁠.

The three-filament configuration results are presented here with the separations also selected as far, with $d=3$ cm (⁠ $∼6δe$⁠), and a close separation distance $d=1$ cm (⁠ $∼2δe$⁠). Both distances are measured from filament center to center in the triangular pattern.

In the upper panels of Fig. 14, the mode patterns on each filament are similar to the two-filament case with broadening of the temperature peaks and spiral arms forming and inter-connecting. From the electron temperature fluctuation maps shown in Fig. 14(c), the azimuthal mode numbers, $m=1$ and $m=2$⁠, dominate and once again are associated with the unstable drift-Alfvén waves. The frequency of these modes in the simulations matches the range $18–21$ kHz that was measured in the experiment.

In the lower panels of Fig. 14, a global mode forms from the strongly overlapping eigenmodes and a spiral mode pattern evolves surrounding the triangular-shaped configuration. The electron temperature fluctuation map in Fig. 14(e) reveals a more dominant $m=3$ mode pattern, which is also consistent with the experiment observations in the $δIsat$ cross field planes. The same smoothing as Fig. 13 is applied here with all panels except for (b) and (e) smoothed. Although not shown here, for close separation, the initial shallow density gradients on each filament are quickly removed after about one rotation period of the spinning filaments, leading to a uniform density profile. In the far separation case, the initial weak density gradients are slightly perturbed by convective motion and spiral arms from neighboring filaments. As in the two-filament case, there is charge transfer between the filaments inducing a weak polarization electric field.

In summary, we have carried out experiments on small-scale filament–filament interactions using localized heat sources placed in close proximity. The electron beams from these sources rapidly thermalize and produce striations of elevated temperature (maximum of $∼1.5$ eV), which are embedded in a colder background plasma (∼0.25 eV). These seeded electron temperature filaments are of finite length along the confining magnetic field and isolated from the plasma boundaries. Our study focused on two-filament and three-filament interactions; both cases exhibited interesting nonlinear dynamical behavior.

In the first configuration consisting of two electron filaments, interactions between them began to occur at separations corresponding to about five times the size of individual filaments. Even at these distances there was a cross field flow of energy and charge between the filaments. This charge flow caused a deficit of density in one filament and an enhancement in the other, leading to a polarization electric field that formed between the filaments. This vertical electric field induced an $E×B$ rotation of the filaments about the center of mass. At the far separation, $d∼2.5$ cm $∼5δe$⁠, the observed rotation was small; however, for the closer distance, $d∼1$ cm $∼2δe$⁠, the rotation was significant and rapid, with merging time scales in the order of 0.5 ms. Gradient maps of the cross field $Isat$ planes have been particularly useful in tracking the rearrangement of temperature gradients, which ultimately controls the stability of the drift-Alfvén modes. In the far separation case, the steepest thermal gradient region is weakly perturbed and the interaction of the drift modes that develop surrounding the gradient region weakly overlap as the edges of the spiral arms make contact. This is confirmed with gyrokinetic simulations, indicating linear growth and saturation of drift-Alfvén modes on each filament and forming low azimuthal mode number temperature fluctuations. When the filaments are within approximately $2δe$⁠, the modes strongly overlap and the filaments merge, with the strongest thermal gradient forming outside of the two merged filaments, thus resulting in a more extended nonlinear eigenmode characterized by a rotating $m=1$ structure and spiral arms emanating from the merged region.

Another feature that is observed in this close case is the wrapping of one filament around the other (in counterclockwise direction, opposite the clockwise spinning of the merged filaments), thus forming a double layered thermal gradient structure. This reversed flow is indicative of a sign reversal of the radial component of the electric field, which is yet to be understood. To capture these dynamics in the simulations would require runs over much longer times since the convective dynamics occurs on millisecond time scales.

The second configuration that was investigated consisted of three filaments arranged in a triangular pattern, with equidistant filaments that varied from close (⁠ $d∼1$ cm $∼2δe$⁠) to far (⁠ $d∼3$ cm $∼6δe$⁠). In the latter case, energy and charge flows were observed between the filaments with tail-like structures emanating from each. The gradient maps revealed strong asymmetries, with gradients that were significantly reduced in the tails as the convective flows between filaments influenced their neighbors. The $m=1$ azimuthal modes from the drift-Alfvén instability dominated on each filament with propagating fluctuations following the tail-like structures in between the filaments. In the closer separation case, modes on each filament strongly overlapped leading to enhanced mixing of temperature and density. The thermal gradient on each filament is reduced and maximized in the region surrounding the tri-filament bundle. This outer temperature gradient excited a more global drift mode with a higher azimuthal mode number (⁠ $m=3$⁠). The nonlinear gyrokinetic simulations also demonstrated that a global mode structure with $m=3$ dominates and spiral arm like structures emanate from the strongly coupled filaments.

To conclude, we have characterized a small-scale (⁠ $∼δe$⁠, collisionless electron skin depth) filament–filament interaction in a magnetized plasma. These “seeded filaments” are created using low energy electron beam sources, forming long and thin enhanced temperature striations. Such structures may naturally be generated during the saturation of interchange-type instabilities. In toroidal magnetic geometry, filamentary structures (density and temperature) are present in the plasma edge and scrape-off layer (SOL) region. These blob-filament structures have internal polarization electric fields that are created through charged particle gradient-B and curvature drifts, causing the filaments to move radially outward. In the linear plasma device, this effect is absent; however, we found that charge and energy can flow between the filaments when they are sufficiently close with electron density enhancement and depletions, thus forming inter-filament electric fields. These electric fields, when crossed with the confining magnetic field, induces $E×B$ filament rotation about the center of mass and influences the merging dynamics. This work has emphasized the role of symmetry-breaking perturbations on the steep thermal gradients of individual interacting filaments and the redistribution of the thermal gradients to the region surrounding the filament bundle. The results of this study may be useful in the interpretation of interacting multi-filament structures present in magnetized plasma environments, both in space and in fusion laboratory experiments.

The authors acknowledge support from the Natural Sciences and Engineering Research Council of Canada (NSERC), the experiments were performed at the Basic Plasma Science Facility supported by DOE and NSF, with major facility instrumentation developed via an NSF Award No. AGS-9724366.

The authors have no conflicts to disclose.

R. D. Sydora: Conceptualization (lead); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). T. Simala-Grant: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). S. Karbashewski: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). F. Jimenez: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). B. Van Compernolle: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). M. J. Poulos: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.