Three-wave coupling observed between a shear Alfvén wave and a kink-unstable magnetic flux rope

Both the kink oscillation of magnetic flux ropes and the shear Alfvén wave are fundamental waves in magnetized plasmas existing at low frequencies as compared with the ion cyclotron frequency. Despite the fundamental nature of both waves, scant attention has been given to the potential nonlinear interaction of these modes. The present study aims to shed light on this interaction and emphasize its importance to both natural and laboratory plasmas.

A shear Alfvén wave is a fundamental mode of oscillation in magnetized plasmas,¹ characterized by magnetic field fluctuations perpendicular to the background magnetic field $B0$⁠, which propagates primarily parallel to this background field. The kinetic regime of the shear wave^2–5 emphasizes perpendicular propagation and dissipation effects characterized by several length scales, such as the ion sound gyroradius $ρs=(Te/mi)1/2$⁠, where T_e is the electron temperature in energy units and m_i is the ion mass. Shear Alfvén waves play a crucial role in various plasma phenomena, including particle acceleration and magnetic field restructuring in both laboratory and astrophysical plasmas.

A kink-unstable magnetic flux rope is a structure in a current-carrying plasma where magnetic field lines twist around a central axis. When the twist exceeds a critical threshold (the Kruskal–Shafranov limit⁶), the rope becomes kink-unstable, distorting and potentially causing explosive magnetic re-configurations. This phenomenon is significant in astrophysical contexts, particularly in understanding solar eruptions and coronal mass ejections. In tokamak plasmas, they can lead to undesirable disruptions in the operation of the plasma.

In the context of solar physics, there exist solar coronal loops, which are structures in the Sun's corona—essentially loop-shaped bundles of magnetic field lines filled with plasma. These loops, visible in the extreme ultraviolet and x-ray portions of the electromagnetic spectrum, are prominent features in the solar corona, varying in size from small loops close to the solar surface to large structures that extend hundreds of thousands of kilometers into the corona.⁷ Kinking oscillations of these filamentary structures were predicted in the 1970s^8,9 and finally measured in the late 1990s.^10,11 An extensive review on kink oscillations in coronal loops is provided by Nakariakov et al.¹² As more sensitive instrumentation has come online, the existence and behavior of shear Alfvén waves in the solar environment have begun to be understood,¹³ with perturbations on the solar surface being one driver of Alfvén waves.^14,15 Given the existence of kink oscillations and shear Alfvén waves at or near the sun, the potential interaction between the standing wave kink oscillations of coronal loops and the Alfvén waves merits investigation. In particular, nonlinear wave–wave interactions between these modes may lead to the generation of smaller perpendicular length scales where dissipation is more efficient, thus contributing to the high temperatures observed in the solar corona. The generation of such smaller scale structures is the subject of the present work.

Another area within plasma physics in which kink oscillations and Alfvén waves co-exist and are of major importance is that of magnetically confined fusion research. The growth of the kink instability, for example, limits the maximum toroidal current that can safely flow in tokamak plasmas.¹⁶ Additionally, the seminal work by Kadomtsev¹⁷ on sawtooth crashes in tokamaks attributed these to the nonlinear evolution of resistive kink mode instabilities. Further, tokamaks exhibit a variety of Alfvénic modes, including shear wave eigenmodes that can drive the expulsion of fast ions from the core plasma.¹⁸ Early heating experiments using shear Alfvén waves through antennas^19,20 highlighted the potential for these waves to excite discrete kink modes. However, the possibility of nonlinear interactions between kink modes and Alfvén waves was not explored in that work.

Basic laboratory studies have focused on investigating the fundamental properties of both the kink oscillations and the kinetic Alfvén waves. Between the two, the kinetic Alfvén wave has received a considerable amount of attention, which is too lengthy to be summarized here; however, an excellent review has been provided by Gekelman et al.²¹ Fundamental theoretical predictions⁴ and experimental verification of kinetic Alfvén wave properties have been documented in the laboratory.^22,23

Basic studies for the kink instability in basic plasma experiments have included configurations ranging from the Madison rotating wall experiment,²⁴ to the Caltech plasma jet experiments,²⁵ to the former Los Alamos Reconnection Scaling eXperiment (RSX).²⁶ Recent advancements in laboratory experiments have begun to see the co-existence of kink oscillations and Alfvénic oscillations. The PHASMA experiment²⁷ demonstrated that the kink instability could spontaneously excite Alfvénic oscillations, further corroborating the significance of these interactions. There have also been several experiments on the Large Plasma Device (LAPD) at UCLA²⁸ that studied the dynamics of single²⁹ and multiple flux ropes^30–32 generated by currents emitted from lanthanum hexaboride cathode discharges in a 20-m-long linear plasma device. Building on this foundation of flux rope studies, and a 30-year history of Alfvén wave experiments (dating back to Gekelman et al.³³), our current investigation in LAPD focuses on the nonlinear interaction between kink oscillations and antenna-launched kinetic Alfvén waves within a flux rope.

The remainder of this manuscript is organized as follows: Sec. II details the experimental setup employed in this study. Section III concentrates on the data and results obtained from the experiments. Finally, Sec. IV offers concluding remarks and outlines potential directions for future research.

This section presents details of the experimental setup and various plasma parameters. Unless explicitly stated otherwise, the reader may refer back to these parameters as applying to the core plasma throughout the manuscript. The experiments are conducted in the Large Plasma Device (LAPD). A full description of the device is available in the article by Gekelman et al.²⁸ The LAPD and the details of the experimental setup are depicted schematically in Fig. 1. The device comprises four main components: the vacuum vessel, the magnets, and two hot-cathode plasma sources. The cylindrical vacuum vessel has an overall length of 24.4 m, and the two plasma sources (at opposite axial ends of the device) are separated by a distance of 17.9 m within a 1-m-diameter chamber; this chamber is surrounded by electromagnets that operate at constant current and provide the solenoidal magnetic field of $B0$ = 0.075 T $ẑ$⁠, with B₀ being the magnitude of this field and $ẑ$ being the unit vector along the positive z direction. The radial center r = 0 and the axial location of the cathode at the right of Fig. 1 define the origin of the right-handed Cartesian coordinate system, with the unit vector $x̂$ pointing out of the page and the unit vector $ŷ$ in the vertical direction.

Both sources are planar, thermionic-emitting, cathode–anode discharges. Each source has its own anode and its own independent, transistor-switched capacitor bank. Both cathodes, however, are of a different construction. The cathode to the left of Fig. 1 (referred to as the “main” cathode) is a heated sheet of nickel, coated with a circular pattern of (primarily) barium oxide of radius 0.38 m to reduce the work function of the nickel where the coating exists. (Note: After this experiment was completed, the main source was replaced by a large, LaB₆ cathode.³⁴) This cathode provides an ambient plasma with a density $n=2×1018$ m⁻³, T_e = 2 eV, at a radius of 0.35 m (full width at half maximum). The anode for this cathode is separated by a fixed axial distance of 0.5 m and introduces no net current into the remainder of the device. The cathode to the right of Fig. 1 is made of a square tile of amorphous lanthanum-hexaboride (LaB₆) with a length of 0.2 m on a side. The emissivity of this cathode is on the order of ten times that of the main cathode and is suitable for producing smaller, but denser core plasmas. The anode for the LaB₆ cathode can be placed at several axial locations, but is here placed at z = 11.7 m. This current-carrying section of the device forms the primary region of the experiment. Since the LaB₆ has a square cross section, a 0.3 × 0.3 m² graphite plate with a hole of radius R = 0.076 m removed from the center is inserted at z = 2.78 m in order to produce a magnetic flux rope also of circular cross section by preventing LaB₆-emitted electrons from passing outside of the hole. The equilibrium axial current density of this flux rope is $jz0$ =  $5.0×104$ A/m², the electron density $ne=1.35×1019$ m⁻³, electron temperature T_e = 4 eV, and ion temperature $Ti≲1$ eV. Electron density is calibrated with a 96-GHz microwave interferometer, and plasma profiles and the electron temperature are obtained through swept Langmuir probe measurements.

An Alfvén wave antenna is placed along the magnetic axis at z = 10.1 m in order to drive waves with a chosen frequency and polarization. This antenna is well documented from previous experiments.³⁵ It is constructed from a pair of independent, coaxially fed magnetic dipoles. The dipoles are orthogonal to each other, with one dipole (diameter 0.08 m) in the xz-plane and the other (diameter 0.09 m) in the yz-plane, so that the antenna inductively drives a magnetic field perturbation that is primarily perpendicular to the background field. Parallel currents in the plasma are carried by electrons, and these currents (to which the antenna fields couple) close in the perpendicular direction via ion polarization. Each dipole is driven by an independent, resonant LC circuit. The inductance is fixed by the geometry, and the capacitors in the circuits as well as the amplitudes of the drivers are adjusted to produce Alfvén waves that propagate with circular polarization and an azimuthal mode number m = −1. The choice of m number here is somewhat arbitrary. During initial experiments, $m=+1$ was also explored, as was a separate antenna that drove an m = 0 mode. Results from the initial experiments appeared to have very similar outcomes apart from the three-wave azimuthal wave number matching criterion (explained in more detail later) adapting for each chosen Alfvén wave driver. The m = −1 driver was arbitrarily chosen to focus on one azimuthal mode in detail within experimental time constraints. The frequency of the wave is here chosen to be 87 kHz, which is 0.3 f_ci to avoid coupling to the compressional (fast wave) branch, which is cut off in the perpendicular direction (evanescent) for frequencies below the ion cyclotron frequency under present experimental conditions.³⁶ The antenna launches shear Alfvén waves in both axial directions; however, probe measurements are made in the 87% of the interaction region where the Alfvén waves propagate in the $−ẑ$ direction. Further details of the antenna wave pattern will be given in Sec. III C.

The main diagnostics are a pair of differential magnetic induction probes, with axial locations shown in Fig. 1. These probes are sensitive to the perturbed magnetic field vector, $δB(t)$⁠, and each probe measures all three mutually orthogonal components of $∂tδB(t)$⁠. These probes are differentially wound with a total of ten turns for each field component. The loops for each turn have a radius $r≈1.8×10−3$ m. The wires of each pair are coated with a flexible ceramic sheath to withstand the heat load in the plasma and are fed along the length of a non-magnetic, stainless steel probe shaft as twisted pairs to vacuum feedthroughs and connected outside to a 150-MHz differential amplifier. There are six inputs to the amplifier (two for each of the five turns in the oppositely wound differential set per component). The output of the amplifier thus combines the magnetic signals while subtracting any unwanted electrostatic pickup. The time series data for each probe are digitized at 12.5 MHz and recorded with 16-bit resolution. Ensembles of the time series from between 8 and 128 (as noted later) plasma discharges are stored to improve the signal-to-noise ratio. After storage, each probe can be automatically moved (using computer-controlled stepper motors) to a new spatial location, or another parameter can be changed as desired, and the process repeated. The recorded voltages for each component i of the perturbed field are proportional to $∂tδBi$⁠, and the signals are numerically integrated to produce the perturbed magnetic field vector $δB(t)$⁠. Other quantities, such as the perturbed axial current density $δjz$⁠, will be presented where data allowing computation of the necessary spatial derivatives are measured.

Several features become apparent from this scan. As in previous experiments,²⁹ the kink oscillation is destabilized when $αKS>1$⁠. Although a clear demarcation for onset is not obvious on the logarithmic color scale, the power in the kink oscillation rises above 1% of the saturated power level for $αKS≳1$⁠. A broadening of the background power spectra along with the development of harmonics of the kink oscillation frequency also occurs as $αKS$ is increased. The key point here, however, is that when the kink oscillation is present, sidebands about the Alfvén driver frequency $fA$ appear that are separated from the driver by multiples of the kink oscillation frequency f_k. The slight upshift in frequency for the kink oscillation for increasing $αKS$ is reflected in the equivalent frequency shifts of the sideband frequencies away from that of the broadcast Alfvén wave. To the authors' knowledge, this is the first observation of Alfvén wave sideband generation mediated by a kink oscillation. To simplify the presentation of these new results, this analysis focuses on the two sidebands at frequencies $f±=fA±fk$⁠. This frequency spacing suggests a three-wave interaction—the investigation of this hypothesis forms the bulk of the remainder of the manuscript.

A final piece of information can be gleaned from the data in Fig. 2: the amplitude dependence of the main sidebands on the main kink and launched Alfvén wave. It is not entirely clear from the colorbar used for the logarithmic plot for Fig. 2, but the amplitude of the kink wave, when present, is one to two orders of magnitude larger than that of the launched Alfvén wave. One might expect, therefore, the dominant driver of any interaction to be the kink oscillation. Indeed, as the amplitude of the Alfvén wave was varied from zero to the maximum valued used for this figure, there was a linear dependence of the main sideband amplitudes on the driver Alfvén wave amplitude. Additionally, the amplitude of the kink oscillation (and its harmonics) could not be seen to depend on the launched Alfvén wave amplitude to within noise levels. There is, unfortunately, no experimental “knob” to vary the kink wave amplitude—to a large degree, it is either stable or unstable; however, the kink oscillation exhibits here a consistent variation in amplitude as $αKS$ is varied over the experimental range $1.2≤αKS≤2.3$⁠. Note that using values $αKS>1$ ensures that the kink mode is, indeed, destabilized. The observed amplitude variation is utilized to plot the main sideband mode amplitudes as functions of the kink oscillation. The results of this analysis are presented in Fig. 3. Two sets of data are presented here on a log-linear scale. The blue dots connected by solid lines are mode amplitudes measured for the lower sidebands (at $f−=fA−fk$⁠); $αKS$ (except for the final point on the line) is increasing from left to right. For generality, the kink oscillation amplitudes have been normalized to the maximum value for all $αKS$⁠. Additionally, the sidemode amplitudes have been normalized by the amplitude of the mode at $fA$ for each value of $αKS$—this ensures that (when all plotted together) any launched wave amplitude variations are normalized out, as the plasma conditions are evolving somewhat for the lowest values of $αKS$⁠, which can affect the Alfvén wave coupling. The same analysis is presented for the upper sideband (at $f+=fA+fk$⁠), with the lines and symbols colored in orange. For both sidebands, two key features are revealed by this figure: first, the sideband amplitudes initially grow exponentially with kink mode amplitude; second, the sideband amplitudes saturate as the normalized kink oscillation amplitude reaches approximately 0.5. The least squares, exponential fitting function $y=A exp(γx)$ is plotted for both sidemodes, with $A=2.0×10−2$ for both modes and $γ−=4.8$ for $f−$ and $γ+=4.4$ for $f+$⁠. The exponential growth of the sidebands with the kink (pump) oscillation amplitude suggests a nonlinear instability being responsible for their appearance, but requires further study.

To proceed, a single value of $αKS=1.6$ is selected so that the kink oscillation (observed at f_k = 9.2 kHz) is fully destabilized, producing clear Alfvén sidebands, while minimizing the appearance of kink harmonics. The Alfvén driver frequency is adjusted to $fA=87.0$ kHz to minimize overlap with any kink frequency harmonics, and the background magnetic field is adjusted to 0.075 T to slightly improve LaB₆ emission and to help achieve the value of $αKS=1.6$⁠. Magnetic field data $δBy$ are then acquired with the reference probe for a 1.3-ms window both before and during the Alfvén wave broadcast, and the resulting power spectra are displayed in Fig. 4. The blue trace (before the Alfvén waves are launched) and red trace (during the Alfvén waves) are substantially similar below 60 kHz with small baseline differences due to the time offset between the two cases. During the broadcast of the Alfvén waves, the appearance of the sidebands about the launched frequency is evident. The upper sideband is peaked at $f+$ = 96.1 kHz. The lower sideband is peaked at $f−$ = 77.8 kHz. The expected values from three-wave coupling are $f+=fA+fk$ = 96.2 kHz and $f−=fA−fk$ = 77.8 kHz. Thus, the three-wave frequency matching criteria are well satisfied, given the 0.38 kHz binning uncertainty from the Fast Fourier Transform (FFT). It is important to point out that this measurement is at a single spatial location. The spatial morphology of the wave will be examined in Sec. III B in the context of extracting the perpendicular wavenumbers for the four frequency peaks of interest as a next step in establishing the required three-wave wavenumber matching conditions: $k±=kA±kk$⁠.

The two circled points in Fig. 5 labeled as (a) and (b) correspond to the sidebands of interest. To within the modest fundamental bandwidth of the FFT used to compute b², point (a) is located at $f1=78$ kHz and $f2=9$ kHz; the third frequency must, therefore, be at $f3=f1+f2=87$ kHz. This indicates that the lower sideband at $f−=78$ kHz results, in part, from a phase-coherent or quadratic nonlinear interaction between the modes at $f2=fk$ and $f3=fA$⁠. The extent of the nonlinearity is measured by $b2=0.88$—or that 88% of the power at $f−$ is due to this interaction. Note that differences of 1 kHz from values here and elsewhere in the manuscript are due to slightly different lengths in the time series used in the Fourier transforms. Similarly, mode (b) has $f1=87$ kHz, $f2=9$ kHz, and $f3=f1+f2=96$ kHz. So that $f3=f+$⁠, with $b2=0.76$⁠, or 76% of the power in the upper sideband is due to a quadratic nonlinear interaction between f_k and $fA$⁠. In this manuscript, we cannot yet offer an explanation for what this interaction is, only that there is strong evidence to support its existence. The third point, labeled (c), is the first harmonic of the kink wave, produced by the fundamental kink frequency interacting with itself and the power fraction in this harmonic is $b2=0.87$ due to such an interaction.

To measure the mode structures perpendicular to the imposed magnetic field $B0ẑ$⁠, the magnetic field probe at z = 7.58 m is attached to a computer-controlled probe drive capable of positioning the probe in the xy-plane. This is facilitated through the use of a novel, deferentially pumped, ball-in-socket motion feedthrough.²² The probe is translated to the first position in a 35 × 35 square grid where 21 realizations of the experiment are performed and the probe signals are digitized and stored. The probe then moves to the next location, and the process is repeated until data from all spatial locations are acquired. In post processing, the individual frequencies (f_k, $fA$⁠, $f±$⁠) are isolated using narrow-pass filters in the frequency domain. The resulting spatial patterns of the perpendicular $(δBxx̂,δByŷ)$ magnetic field vectors for f_k and $fA$ are shown in Figs. 6(a) and 6(b), respectively. The particular phase of each wave is chosen arbitrarily. As functions of time, the kink wave rotates in the right hand sense with respect to the background magnetic field, and the Alfvén wave rotates in the opposite (left-handed) sense. The direction of rotation is determined after transforming back into the time domain and double-checked using the raw data in the time domain where the rotation of the kink mode is evident. The direction of rotation for the kink mode also agrees with the direction of rotation of the flux rope as observed using a fast, framing camera. The camera is set to measure visible light at 100 kHz, primarily of neutrals excited by the current of the flux rope. The camera is located at x = 0 m, $y=−0.4$ m, and z = −2 m.

In Fig. 6, the vector patterns show that both waves comprise two, oppositely flowing parallel electron current channels. Accordingly, the axial current density is computed from the measured magnetic fields using the low-frequency approximation of Ampère's law: $jz=(∂xBy−∂yBx)/μ0$⁠, and displayed in Figs. 6(c) and 6(d). Given the cylindrical nature of the experiment and the wave patterns, it is natural to think of assigning azimuthal mode numbers to these waves. Generally, the mode number m would be one minus either the complete number of positive or negative current channels in the scalar axial current pattern for the wave. This will be replaced by a quantified definition later; however, this simple definition is useful here to predict what is expected for a three-wave interaction in terms of the mode numbers. As in the three-wave frequency matching condition for the sidebands: $f±=fA±fk$⁠, waves with a dependence on $eimθ$ will yield $m±=mA±mk$⁠. Given the number of current channels and the respective senses of rotation in the θ direction, $mk=+1$ and $mA=−1$⁠. The lower sideband is, therefore, expected at $m−=−2$ and the upper sideband at $m+=0$⁠.

Figure 7 displays the perpendicular vector components of the wave magnetic fields for the two main sidebands. Immediately beneath each vector plot is the corresponding axial component of the current density derived from the magnetic field patterns. Parts (a) and (c) are for the lower sideband at frequency $f−$ and clearly show the expected m = −2 pattern. The sign of m is determined as in Fig. 6. Similarly, parts (b) and (d) are for the sideband at $f+$⁠, and again the predicted spatial pattern (here m = 0) is observed. It remains, however, desirable to quantify the observed m numbers. Additionally, there is a qualitative shift toward higher wavenumbers from the parent modes to the sidebands, and a quantified measure of the radial wavenumbers is still sought to evaluate the radial component of the three-wave matching conditions.

In practice, measurement resolution only permits an upper bound on the infinite series of Eq. (4). Additionally, the integrals for the coefficients are approximated using a five-point Newton–Cotes integration scheme after interpolation of the integrand onto an $(r,θ)$ grid. It is important to note that $jz(r,θ)$ must contain both an amplitude and phase to encode the correct signs for the m numbers. This can be achieved using the complex conjugate of the temporal Fourier transform for the signal.

The results of the decompositions of the four frequencies of interest are presented in Fig. 8. This figure displays the two-dimensional power spectra, $|cmj|2$⁠, obtained for the complex-valued c_mj via Eq. (5). Additionally, a summary of the decomposition analysis is provided in Table I. In general, the magnitude of the azimuthal wavenumbers m is well isolated and in agreement with the previous qualitative analysis of current channel counting. The signs for the nonzero m values are also in agreement with the direction of rotation for the patterns that could not be shown in the static images from Figs. 6 and 7. The quantitative analysis confirms the qualitative mode number matching: $m±=mA±mk$⁠.

The qualitative observation of the sideband patterns being shifted to primarily higher perpendicular wavenumbers is also established and quantified. In Fig. 8 and Table I, the perpendicular wavenumbers have been scaled by the ion sound gyroradius $ρs=cs/ωci=0.54$ cm, which is the appropriate cross field length scale for the kinetic Alfvén wave when $Ti/Te≪1$⁠.⁴ Here $cs=Te/mi$ is the ion sound speed and $ωci=eB0/mi$ is the ion cyclotron frequency, with e being the magnitude of the electron charge. From Table I, for the kink wave $k⊥,kρs=0.12±0.05$ and for the launched Alfvén wave, $k⊥,Aρs=0.23±0.05$⁠. Both sidebands are measured to have $k⊥,±ρs=0.37±0.05$⁠. Because these are not plane waves and the wavenumbers from the decomposition are all positive-definite, a naive expectation that $k⊥,±$ =  $k⊥,A±k⊥,k$ cannot be satisfied. Indeed, the above perpendicular three-wave-matching criterion is often demonstrated by considering the multiplication of two complex exponentials, whereas here the sidebands arise from the product of Bessel functions, which have no simple addition and subtraction rules. Consistent with the observations, however, is that $k⊥,±$ =  $k⊥,A+k⊥,k$⁠. This scaled sum is $k⊥,±ρs=0.35±0.07$ vs the measured $k⊥,±ρs=0.37±0.05$⁠.

Measurement of the parallel wavenumbers is straightforward. Magnetic field data are collected from the moving xy-moving probe at z = 7.58 m and the reference probe at z = 7.9 m. This is done for only the ensemble of 21 plasma discharges when the moving probe and the reference probe have the same xy-location: $(x,y)=(0.05,0.0)$ m. A spectral cross correlation is performed that yields the axial phase difference as a function of frequency $Δϕ(ω)$⁠. The axial wavenumbers are simply $k∥(ω)=Δϕ(ω)/Δz$⁠. The axial probe separation of one machine access port (⁠ $Δz=0.32$ m) minimizes the phase shifts from perpendicular phase differences at the expense of greater phase uncertainty for the long wavelength kink mode.

Table II summarizes the parallel wavenumber measurements for the modes of interest. Wave vectors measured for the launched Alfvén wave $k∥,A$ and the sidebands $k∥,±$ are all negative, indicating propagation away from the antenna for all three waves (toward negative z). For the kink wave, at frequency f_k, the phase difference cannot be distinguished from zero to within experimental error. A zero phase difference is consistent with a standing wave. In this case, the parallel wavenumbers of $k∥,k=±π/L$ (L = 11.7 m, the LaB₆ cathode–anode spacing) are assumed for the kink mode, and the relatively small error with this assumption comes from the small uncertainty in determining L. The assumption for the parallel kink wavenumbers provides an excellent match for the three-wave parallel wavenumber matching conditions. Note that although the standing wave provides both positive and negative values for the kink wavenumber, only the co-propagating component of the kink oscillation $(k∥,k=−π/L)$ satisfies the parallel three-wave matching conditions: $k∥,±=k∥,A±k∥,k$ that agree with the data. From Table II, the measured, launched Alfvén wavenumber $k∥,A=(−2.42±0.06)$ m⁻¹ and the upper sideband $k∥$ matching condition $k∥,+=k∥,A−π/L$ yield $(−2.69±0.06)$ m⁻¹ against the measured value of $(−2.69±0.07)$ m⁻¹. Similarly, the lower sideband satisfies, $k∥,−=k∥,A−(−π/L)=k∥,A+π/L=(−2.15±0.06)$ m⁻¹, with the directly measured value being $(−2.14±0.05)$ m⁻¹. The conclusion is that the parallel wavenumber three-wave matching conditions are well satisfied using the launched Alfvén wave and the assumption of the co-propagating component of the half-wavelength standing kink mode. That the sidebands also fall on an Alfvén wave dispersion curve is explored next.

Figure 9 displays, in two panels, plots of the dimensionless dispersion relations [Eq. (7)] along with the measurements of the parallel and perpendicular wavenumbers. Panel (a) shows (in green) the dispersion curve for the launched Alfvén wave at angular frequency $ωA$ and the curve (in red) for that of the upper sideband at $ω+=ωA+ωk$⁠, with ω_k being the angular kink oscillation frequency. The measured parallel and perpendicular wavenumber points are plotted (with errorbars) using the same color convention as for the dispersion curves. Both points fall distinctly on their respective dispersion curves. This should not be surprising for the launched wave; however, this demonstrates that the upper sideband is also a kinetic shear Alfvén wave. Additionally, the wave vectors for the launched wave (green) and upper sideband (red) have been drawn with the addition of the kink wave vector (black) drawn starting at the end of the launched wave vector. This clearly demonstrates that (in this vector plane) the upper sideband wave vector is the sum of the launched wave vector and the kink wave vector: $k+=kA+kk$⁠. The important assumptions are, again, that $k∥,k=−(π/L)ẑ$⁠, and that in our positive, definite decomposition of the perpendicular wave vectors that $k⊥,±$ =  $k⊥,A+k⊥,k$⁠. For the case presented here, however, both assumptions are highly consistent with the experimental measurements.

Similarly, panel (b) of Fig. 9 displays here in green the dispersion curve for the launched Alfvén wave along with the upper curve for the lower sideband in blue, at angular wave frequency $ω−=ωA−ωk$⁠. Again, the measured parallel and perpendicular wavenumber points are plotted (with errorbars) using the same color convention as for the dispersion curves. Both points clearly fall on their respective dispersion curves, well within errorbars. Like the upper sideband, this demonstrates that the lower sideband is also a kinetic shear Alfvén wave. The wave vectors, once more, are drawn with their respective color-matching vectors: the launched wave (green) and lower sideband (blue) have been drawn with the addition of the kink wave vector (black) drawn starting at the end of the launched wave vector. Again, with previous assumptions in mind, the lower sideband is produced from an interaction of the launched wave and the co-propagating kink wave.

With this total $k$ matching and dispersion analysis, it is apparent that the reason why only the co-propagating component of the standing kink wave interacts with the launched Alfvén wave is that the counter-propagating component interaction would attempt to excite sidebands that do not fall on dispersion curves for propagating Alfvén waves given the measured sideband frequencies appropriate for their respective three-wave frequency matching criterion.

The key experimental findings of this manuscript are enumerated as follows:

1. The Large Plasma Device (LAPD) is used to generate spatially co-existing kink-unstable flux ropes and kinetic shear Alfvén waves. Due to the nature of the kink instability and the present capability of the Alfvén wave drivers, the magnitude of the magnetic perturbation of the kink oscillation was an order of magnitude larger than that of the shear wave. Thus, the kink wave may be taken as the driver of any nonlinear processes.

2. The current density in the flux rope was varied in order to verify (in accord with previous observations^26,29) that the instability onset was one-half the classic Kruskal–Shafranov limit.

3. The perturbed magnetic field power spectra show that, in the presence of the large-amplitude kink wave, the driver Alfvén wave develops sidebands separated by integer multiples of the kink wave frequency.

4. The perturbed axial current density is used to quantitatively measure (for all four modes under consideration) the parallel wave vectors, perpendicular wave vectors, and azimuthal mode numbers, with the exception of the standing kink mode but assuming $k∥=±π/L$⁠. All of these (along with the frequency spacing) were consistent with a three-wave coupling process between the kink and launched Alfvén waves and the main sidebands on either side of the Alfvén wave in the power spectrum. Notably, only the co-propagating component of the kink wave appears to interact with the Alfvén wave to produce the two, co-propagating sidebands—the counter-propagating kink wave component being unable to link to propagating solutions of the shear Alfvén wave dispersion relation for the required sideband frequencies.

5. Bispectral analysis of the magnetic field time series data measured during the interaction indicates a high degree of a quadratic nonlinear (or phase-coherent) interaction between the kink and Alfvén modes as the driver for the production of the sidebands.

6. The amplitudes of the sideband Alfvén waves scale exponentially with the amplitude of the kink, pump wave until the pump wave saturates at approximately one-half its maximum observed value.

Together, these measurements demonstrate that the sidebands are generated via a three-wave interaction between the kink oscillation and the launched wave. Additionally, some questions remain unanswered and the research suggests directions for future experiments.

First, the theoretical mechanism behind the observed interactions remains to be discovered. Understanding this mechanism could provide deeper insights into the fundamental processes governing wave interactions in plasma.

Additionally, the parameter scan used to destabilize the kink mode naturally involves changes to the density and temperature as the magnetic flux rope develops. Despite these changes, the three-wave coupling appears to persist throughout this evolution. A study focusing on how the coupling varies as parallel and perpendicular wavenumbers change based on these parametric adjustments could offer valuable information.

The flux rope is a fully three-dimensional structure, being line-tied at the LaB₆ source and free at the anode due to the imperfect plasma conductivity and sheath at the anode. Extending the present analysis over the entire length of the flux rope, including axial parameter variation, may yield further insights into the coupling process.

Kink oscillations in flux ropes are observed to be quite common from the solar surface into the solar corona. With increasing sophistication in detectors and techniques, shear Alfvén waves also appear to exist in these same regions. We conjecture that the wave–wave interactions observed in this experiment between the kink wave and the kinetic shear Alfvén wave, leading to shorter perpendicular wavelengths and either direct particle heating (or through a two-step process involving phase mixing), could be an important part of the energy budget in the solar coronal heating problem.

Finally, these experiments were tailored to have single-frequency kink and Alfvén parent waves. An outstanding question remains: what does the net power spectrum look like for a broad spectrum of mutually interacting waves? Or, more simply, what would the net frequency spectrum look like for a single-frequency kink oscillation interacting with a broadband Alfvén wave disturbance? Answering such questions may shed light on the observed turbulent spectra where both modes are present.

S.V. would like to thank Tim DeHaas and Troy A. Carter for illuminating discussions. The authors would also like to thank Zoltan Lucky, Marvin Drandell, and Tai Ly for their expert technical support. The work was performed at the Basic Plasma Science Facility, which is a DOE Office of Science, FES collaborative user facility and is funded by DOE (No. DE-FC02-07ER54918) and the National Science Foundation (No. NSF-PHY 1036140).

The authors have no conflicts to disclose.

S. Vincena: Conceptualization (lead); Formal analysis (lead); Investigation (equal); Resources (equal); Writing – original draft (lead); Writing – review & editing (equal). S. K. P. Tripathi: Formal analysis (supporting); Investigation (equal); Resources (equal); Writing – review & editing (equal). W. Gekelman: Investigation (equal); Resources (lead); Writing – original draft (supporting); Writing – review & editing (equal). P. Pribyl: Investigation (equal); Resources (equal).

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