Plasma turbulence has been shown to play a critical role in many astrophysical and space environments. In the solar corona and solar wind, this turbulence involves the nonlinear interaction of kinetic Alfvén waves. In the Earth's magnetosphere, the turbulence is dominated by inertial Alfvén wave collisions. Observations of these wave–wave interactions in space and in laboratory plasma environments have shown that, in addition to the nonlinear cascade of energy to small scales, the interaction also produces nonlinear beat waves that have a frequency defined by $f3\xb1=|f1\xb1f2|$. Although the temporal behavior of the beat wave has been well documented, this paper presents the first detailed analysis of the spatial structure of the nonlinearly generated beat wave.

## I. INTRODUCTION

Plasma turbulence is important for our understanding of the dynamics of various space and astrophysical plasma environments. One such environment is the interstellar medium (ISM), which is a partially to fully ionized collisionless plasma, depending on the region of space being observed. Inside this region, there exist density fluctuations with length scales on the order of ∼10^{9 }m.^{1} These density fluctuations cause radio waves to scatter, giving rise to interstellar scintillation of distant stars and pulsars.^{2} Observations of this radio scattering has shown that the 3-D density fluctuation power spectrum has the form of $Cn2k\u2212\alpha $, where *k* is the wavenumber, *C _{n}* is the normalized constant describing the turbulence intensity, and

*α*is roughly 11/3 over 10

^{8 }<

*k*< 10

^{14 }cm.

^{3,4}Since the density spectrum spans at least five orders of magnitude and the power spectrum follows closely the Kolmogorov law

^{5}for a neutral fluid, this suggests that the density fluctuations must be due to turbulence. In most environments, turbulence dissipates its energy in the form of heat. In the ISM, dissipation time scales of the turbulent interactions are much shorter than other time scales, so this makes turbulent dissipation an effective mechanism for heating.

^{6}

Radio propagation measurements, such as angular broadening and pulsar dispersion, have been used to study turbulent structures in the ISM.^{7–11} These techniques have shown that the (a) irregularities in the ISM are anisotropic; (b) from the power law, *C _{n}* can vary greatly particularly in the medium density ionized region; and (c) turbulent structures in the magnetic field are detectable on the order of

*δB*/

*B*

_{0}≤

*δn*/

*n*

_{0}. Although radio measurements have found numerous results—here we have only listed three—we still do not have enough detailed measurements of the fluctuations to enable us to develop a predictive theory for turbulence and its dissipation in the ISM. This lack of data is mostly due to the fact that radio waves in space have very long wavelengths, so in order to achieve sufficient resolution to study the small-scale turbulent structures, a very large telescope or array of smaller telescopes is required.

Another environment in which turbulent processes play a key role is the solar corona. The corona is the extended outer atmosphere of the sun that starts around 2000 km above the photosphere and continuously extends outward, eventually becoming the solar wind. It is a collisionless plasma with a temperature millions of degrees higher than the photosphere, which averages a temperature of around 6000 K. Because of the extreme temperatures and the relatively low density (∼10^{10 }cm^{−3}), the heating is thought to be due to the turbulent cascade and dissipation of energy, involving nonlinear interactions among Alfvén waves.^{12,13} These waves propagate outward with velocities of around 20 km/s and periods of 100–500 s through the corona. The magnetic and kinetic energy of these Alfvén waves^{14} is sufficient to provide the energy needed to accelerate the solar wind.^{15}

Turbulent fluctuations in the solar wind have been observed *in situ* for several decades.^{16–20} This data have helped the scientific community to understand some of the properties of the solar wind, such as (a) differential ion temperatures; (b) anisotropic heating of the plasma, leading to a temperature in the perpendicular direction slightly greater than in the parallel direction; and (c) fluctuations in the magnetic field and velocity of the solar wind that are mostly perpendicular to the interplanetary magnetic field.^{21} However, this data are limited by the single-point nature of spacecraft measurements and, therefore, provides limited information about the 3-D structure of the turbulent fluctuations.

Turbulence has been shown to play a critical role in many other astrophysical and space environments, such as heat transport in galaxy clusters,^{22,23} transport of energy and mass into the Earth's magnetosphere,^{24,25} and regulation of star formation.^{9,26,27} Although these seem to be strikingly different environments, the turbulence in these plasmas is predicted to be dominated by the nonlinear interaction between counterpropagating Alfvén waves, denoted *Alfvén wave collisions*.^{28} A key aspect of this nonlinear interaction between counterpropagating Alfvén waves is the formation of quasimodes in the plasma.

When two counterpropagating Alfvén waves interact nonlinearly, they generate quasimodes with $f3\xb1=|f1\xb1f2|$, as detailed in Sec. III. These so-called beat modes have been observed in a variety of different astrophysical plasma environments^{29–32} and in laboratory plasma experiments.^{33–36} Although the presence of these modes is well documented, there are relatively few analyses examining the 3-D structure of these nonlinearly generated beat modes.

The experiments reported here represent the first controlled study of the spatial and temporal structures of the nonlinear beat wave arising from the interaction of two counterpropagating Alfvén waves in both the kinetic and inertial Alfvén wave regime. In Sec. II, we describe the experimental setup and parameters, with emphasis on the two antennas used to produce the inertial and kinetic Alfvén waves. In Sec. III, we outline the theory underlying the nonlinear generation of beat waves and explain the connection to our previous work on the secular transfer of energy to small scales in Alfvén wave collisions. In Sec. IV, we discuss the experimental results including a comparison of the measured beat wave signal to theoretical predictions.

## II. EXPERIMENTAL SETUP

The experiments were conducted in the Large Plasma Device (LaPD) at the Basic Plasma Physics Research Facility at UCLA.^{37} The LaPD produces a 15.5 m long, 40–70 cm diameter plasma column by using an indirectly heated barium-oxide coated cathode. The discharge typically lasts Δ*t* = 10–15 ms with a repetition rate of 1 Hz. The experiment took place in a 50% ionized^{38} hydrogen plasma. From a swept Langmuir probe, in conjunction with a microwave interferometer, the density and temperature in the measurement region were determined for each experiment. The resulting parameters, along with the background axial magnetic field (*B*_{0}), are shown in Table I. The parameters indicate that the first experiment will produce Alfvén waves in the kinetic regime,^{39,40} *v _{te}* >

*v*, and the second will produce Alfvén waves in the inertial regime,

_{A}^{39,41,42}

*v*<

_{te}*v*. The ion temperature in the LaPD is typically on the order of 1 eV,

_{A}^{43}although it was not directly measured in these experiments.

Expt. . | N [cm_{e}^{−3}]
. | T [eV]
. _{e} | B_{0} [G]
. | V [cm/s]
. _{A} | ρ [cm]
. _{s} | δ [cm]
. _{e} | v/_{te}v
. _{A} |
---|---|---|---|---|---|---|---|

1 | 1.0 × 10^{12} | 5 | 800 | 1.75 × 10^{8} | 0.29 | 0.53 | 1.52 |

2 | 1.25 × 10^{12} | 7 | 1800 | 3.20 × 10^{8} | 0.15 | 0.47 | 0.89 |

Expt. . | N [cm_{e}^{−3}]
. | T [eV]
. _{e} | B_{0} [G]
. | V [cm/s]
. _{A} | ρ [cm]
. _{s} | δ [cm]
. _{e} | v/_{te}v
. _{A} |
---|---|---|---|---|---|---|---|

1 | 1.0 × 10^{12} | 5 | 800 | 1.75 × 10^{8} | 0.29 | 0.53 | 1.52 |

2 | 1.25 × 10^{12} | 7 | 1800 | 3.20 × 10^{8} | 0.15 | 0.47 | 0.89 |

Two antennas were employed for these experiments, as shown schematically in Fig. 1.^{44} With *z* = 0 m at the cathode, the Iowa Arbitrary Spatial Waveform (ASW) antenna^{42,45} was placed at *z* = 15 m. This antenna, shown in Fig. 2(a), has a dimension of 30.5 cm × 30.5 cm and consists of 48 vertical copper mesh grids. Each element is driven by a separate amplifier that allows the current to be adjusted to a maximum value with a multiplicative factor between −1 and +1. The plane of the mesh grid is oriented perpendicular to the axial magnetic field of the LaPD. By varying the amplitude of each grid element, we are able to create an arbitrary spatial waveform across the array in the $x\u0302$ direction with effectively no variation in the $y\u0302$ direction. Since Alfvén waves have $\delta B\u2225=0$ and $\u2207\xb7B=0$, then $k\u22a5\u22a5\delta B\u22a5$. For the first experiment (see Table I), the ASW antenna generated a kinetic Alfvén wave with a frequency of *f*_{1} = 270 kHz and a perpendicular wavevector *k _{x}*

_{1}

*ρ*≃ ± 0.18; for the second experiment, the ASW antenna generated an inertial Alfvén wave with a frequency

_{s}*f*

_{1}= 525 kHz and a perpendicular wavevector

*k*

_{x}_{1}

*δ*≃ ± 0.55. Both waves propagate anti-parallel to the background axial magnetic field,

_{e}*B*

_{0}(blue wave in Fig. 1).

The second antenna used in these experiments was the UCLA Loop antenna,^{46} which was placed at *z* = 4.2 m. This antenna, shown in Fig. 3(a), consists of two electrically isolated, perpendicularly oriented rectangular loops of dimensions 21.5 cm × 29.5 cm. Varying the relative phase of the driving signal on each loop, a large amplitude Alfvén wave can be produced with a magnetic field predominantly in the $x\u0302$ direction and a perpendicular wavevector in the $y\u0302$ direction. For the first experiment, the Loop antenna generated an Alfvén wave with a frequency of *f*_{2} = 60 kHz and a perpendicular wavevector *k _{y}*

_{2}

*ρ*≃ ± 0.05, which propagates parallel to

_{s}*B*

_{0}(red wave in Fig. 1). For the second experiment, the Loop antenna generated a wave with a frequency of

*f*

_{2}= 116 kHz and a perpendicular wavevector

*k*

_{y}_{2}

*δ*≃ ± 0.15.

_{e}The perpendicular components of the magnetic field were measured using two Elsässer probes^{47} placed at *z* = 6.4 m and *z* = 14 m. Measurements were conducted over a perpendicular plane of size 30 cm × 30 cm at a spacing of Δ = 0.75 cm. With a sampling rate of 12.5 MHz, a time series was recorded at each spatial position in the plasma with a specified starting time during the plasma pulse, or shot. Since the shot-to-shot variation in the LaPD is modest, averaging over 10 shots per spatial position was sufficient to achieve an RMS noise level of ∼0.25 mG.

The procedure for measuring the magnetic field of the two counterpropagating waves follows for experiment 2. At *t* = 8.45 ms, where *t* = 0 is at the start of the plasma discharge, the Loop antenna launches a wavepacket consisting of 30 wave periods, which lasts for around 260 *μ*s. At t = 8.57 ms, the ASW antenna launches a wavepacket consisting of 20 wave periods, or 38 *μ*s. Since the Loop antenna launches waves both towards the ASW antenna and the cathode, this time delay allows for ample time for the combined direct and reflected waves to reach a steady state before the ASW antenna launches its wave. The magnetic field is measured 100 *μ*s before and after the Loop antenna is turned on and off. This allows us to measure not only the entire interval when both antennas are turned on but also the background noise in the plasma. A similar procedure was used to measure the perpendicular magnetic field components for experiment 1.^{48} Note that each experiment actually consisted of three data runs: (1) the ASW antenna only on, (2) Loop antenna only on, (3) both antennas on simultaneously.

The measured perpendicular magnetic fields for experiment 1 for the ASW antenna at *t* = 8.24 ms and the Loop antenna at *t* = 8.32 ms are shown in Figs. 2(b) and 3(b), respectively. For experiment 2, both antennas were measured at *t* = 8.58 ms and are shown in Figs. 2(c) and 3(c). All the figures include a linear interpolation to fill the locations in the plot between the actual measurements. Since the Elsässer probe employs a “B-dot” coil to measure the magnetic field, the measured signals are integrated in time in order to determine *δ***B**_{⊥}. The colormaps show *δB _{y}* for the ASW antenna and

*δB*for the Loop antenna. The vectors in each figure indicate the total vector $\delta B\u22a5$. For experiment 1, the figures indicate that the ASW antenna has approximately a 30 mG peak-to-peak amplitude in

_{x}*δB*with almost no

_{y}*δB*contribution. On the other hand, the Loop antenna has a dominant

_{x}*δB*component with a peak-to peak value of around 3500 mG and a small, but not insignificant,

_{x}*δB*component, ∼400 mG. For experiment 2, the figures show that the ASW antenna has approximately a 16 mG peak-to-peak amplitude in

_{y}*δB*, and the Loop antenna has a peak-to peak value of around 2500 mG in

_{y}*δB*and a small

_{x}*δB*component of ∼300 mG.

_{y}## III. THEORY

Early research on incompressible magnetohydrodynamic (MHD) turbulence in the 1960s^{49,50} emphasized the wave-like nature of turbulent plasma motions, suggesting that nonlinear interactions between counterpropagating Alfvén waves—or Alfvén wave collisions—govern the turbulent cascade of energy from large to small scales. This nonlinear energy transfer is mediated by a mode generated by the nonlinear beat interaction between the two counterpropagating Alfvén waves, motivating this experimental investigation.

To understand the nonlinear physics underlying the beat wave phenomenon measured in the experiments reported here, we begin with the equations of ideal, incompressible MHD

where **v** is the fluid velocity, **B** is the magnetic field, *ρ*_{0} is the mass density, *P* is the total pressure (thermal plus magnetic), and *μ*_{0} is the permeability of free space. When the magnetic field is expressed in velocity units as $B/\mu 0\rho 0$, the equations have a symmetric form. By adding and subtracting Eqs. (1) and (2), the equations of incompressible MHD can be written in the symmetric Elsässer form^{51}

Here, the magnetic field is decomposed into its equilibrium and fluctuating parts $B=B0+\delta B$, the Alfvén velocity due to the equilibrium magnetic field $B0=B0z\u0302$ is given by $vA=B0/4\pi \rho 0$, and $z\xb1=v\xb1\delta B/\mu 0\rho 0$ are the Elsässer fields. The Elsässer variable **z**^{+} corresponds to a wave traveling anti-parallel to the background magnetic field **B**_{0}, and **z**^{–} corresponds to a wave traveling parallel to the field. The second term on the left-hand side of Eq. (4) represents the linear propagation of the Elsässer variables at the Alfvén speed along the equilibrium magnetic field. The first term on the right-hand side represents the nonlinear interaction between counterpropagating waves, so this term determines the nonlinear energy transfer in plasma turbulence via Alfvén wave collisions. The second term on the right-hand side is a nonlinear term that enforces incompressibility.^{28,52}

For sufficiently small amplitude Alfvén waves, the nonlinear terms on the right-hand side of Eq. (4) are small compared to the linear term on the left-hand side, so perturbation theory can be used to obtain an analytical solution for the evolution of Alfvén wave collisions in the weakly nonlinear limit. Following significant previous studies on weak incompressible MHD turbulence,^{3,53,54} the nonlinear energy transfer in Alfvén wave collisions has recently been solved analytically in the weakly nonlinear limit,^{28} confirmed numerically with gyrokinetic numerical simulations,^{55} and verified experimentally in the laboratory,^{44,48,56} establishing Alfvén wave collisions as the fundamental building block of astrophysical plasma turbulence.

### A. General nonlinear interaction at the lowest order

The analytical calculation of Alfvén wave collisions by Howes and Nielson^{28} solved an idealized problem involving the nonlinear interaction between two counterpropagating Alfvén waves with $k1=k\u22a5x\u0302\u2212k\u2225z\u0302$ and $k2=k\u22a5y\u0302+k\u2225z\u0302$, where *k*_{⊥} and *k*_{∥} are positive constants. In order to interpret the results of the nonlinear beat interaction experiments presented in this paper, here we generalize the results for two counterpropagating Alfvén waves with $k1=k\u22a51x\u0302\u2212k\u22251z\u0302$ and $k2=k\u22a52y\u0302+k\u22252z\u0302$, where *k*_{⊥}_{1} ≠ *k*_{⊥}_{2} and *k*_{∥}_{1} ≠ *k*_{∥}_{2}. The linear frequency for an Alfvén wave is $\omega =|k\u2225|vA$, so the linear frequencies for these two initial waves are *ω*_{1} = *k*_{∥}_{1}*v _{A}* and

*ω*

_{2}=

*k*

_{∥}

_{2}

*v*. Note that we have adopted the convention that the wave frequency

_{A}*ω*≥ 0, so that the sign of

*k*

_{∥}determines the direction of propagation of the wave along the equilibrium magnetic field.

In the weakly nonlinear limit relevant to these experiments, the lowest-order nonlinear solution for this general case results from a three-wave interaction that yields Fourier modes $k3\xb1$ that satisfy the wavevector matching condition

and have corresponding beat frequencies

It is important to note that the modes $k3\xb1$, arising from the nonlinear beat interaction, do not necessarily satisfy the linear Alfvén wave dispersion relation, $\omega 3\xb1=|k\u22253\xb1|vA$. Note that, expressed as a linear (rather than angular) frequency, Eq. (6) yields $f3\xb1=|f1\xb1f2|$.

The key to reconciling the measurements of nonlinear beat waves reported here with our previous results confirming the nonlinear energy transfer to small scales due to Alfvén wave collisions^{44,48} is to recognize that the Loop-antenna Alfvén wave used in these experiments (see Fig. 3) can be decomposed into Fourier plane-wave modes that have different parallel wavenumbers, with two distinct modes yielding dominant contributions. The first dominant contribution has a parallel wavenumber that is determined by the driving frequency and the Alfvén wave speed, given by *k*_{∥}_{2} = *ω*_{2}/*v _{A}*. But, in addition, because the parallel wavelength of the Loop-antenna wave is much longer than the length over which the two counterpropagating Alfvén waves interact in the LaPD chamber, the Loop-antenna wave appears to have a significant plane-wave component with

*k*

_{∥}

_{2}= 0, as explained in detail in Howes

*et al.*

^{56}

The nature of the nonlinear interaction depends sensitively on the parallel component of the plane-wave mode constituent under consideration, **k**_{2}. If *k*_{∥}_{2} ≠ 0, then the lowest-order, three-wave nonlinear interaction is nonresonant. In this case, although oscillating beat waves with $\omega 3\xb1=|\omega 1\xb1\omega 2|$ are generated, there is no secular transfer of energy into these modes. If *k*_{∥}_{2} = 0, a resonant three-wave interaction can mediate the secular transfer of energy from the primary counterpropagating Alfvén waves to Alfvén waves with smaller perpendicular wavelengths, thus manifesting a nonlinear cascade of energy from large to small scales.

### B. Non-resonant beat wave interaction

In the case that *k*_{∥}_{2} ≠ 0, let us demonstrate the non-resonant nature of the nonlinear interaction. To make direct connection with the experiments described in Sec. II, we take *ω*_{1} > *ω*_{2} and *k*_{∥}_{1} > *k*_{∥}_{2}, where the ASW Alfvén wave is signified by the subscript 1 and the Loop Alfvén wave is signified by the subscript 2.

For the summed beat wave interaction, the $z\u0302$-component of the wavevector matching condition (Eq. (5)) yields $k\u22253+=\u2212(k\u22251\u2212k\u22252)$ and the frequency matching condition yields $\omega 3+=\omega 1+\omega 2=(k\u22251+k\u22252)vA$. The linear wave frequency for this nonlinearly generated beat mode is given by $\omega 3,lin+=|k\u22253+|vA=(k\u22251\u2212k\u22252)vA$. The fact that the nonlinearly generated beat mode frequency is not a natural (linear) frequency of the system, $\omega 3+\u2260\omega 3,lin+$, means that this interaction is non-resonant.

Similarly, for the differenced beat wave interaction, we obtain $k\u22253\u2212=\u2212(k\u22251+k\u22252)$ and $\omega 3+=\omega 1\u2212\omega 2=(k\u22251\u2212k\u22252)vA$. The linear wave frequency for this nonlinearly generated beat mode is given by $\omega 3,lin\u2212=|k\u22253\u2212|vA=(k\u22251+k\u22252)vA$, so we find that this interaction is also non-resonant, $\omega 3\u2212\u2260\omega 3,lin\u2212$.

So, what does it mean that a nonlinear interaction is non-resonant? The nonlinearly driven beat modes, which here are plane-wave modes with wavevectors $k3\xb1$, oscillate at their respective beat mode frequencies, $\omega 3\xb1$, but these modes do not gain energy secularly in time. A secular transfer of energy via nonlinear interactions can occur *only* if one of the natural oscillation frequencies of the plasma for a particular plane-wave mode **k**_{3} (i.e., a linear frequency of that **k**_{3} mode) matches the frequency at which that nonlinear beat mode is driven. Otherwise, the linear plasma response will eventually slip out of phase with respect to the forcing by the nonlinearly generated beat wave mode, ultimately causing the nonlinear forcing to remove any energy previously transferred into that mode. A simple physical analogy is pushing a child on a swing. If you push at a frequency different from the natural frequency of the swing, eventually you will slip out of phase, and find yourself pushing as the swing comes toward you, thus slowing down the swing.

The non-resonant nature of the nonlinear interaction is important in determining the long-time behavior of the system. But, in these experiments, the propagation time for waves along the plasma chamber is on the order of a single wave period, so the non-resonant beat wave contributes significantly to the nonlinear behavior measured in the experiment. The beat wave physics examined here is part of the nonlinear evolution that arises from the *k*_{∥}_{2} ≠ 0 component of the Loop Alfvén wave, complementing the resonant nonlinear energy transfer by the *k*_{∥}_{2} = 0 component that has been previously investigated, ^{28,44,48,55,56} as discussed in Subsection III C.

### C. Resonant nonlinear energy transfer

As described in detail in Howes *et al*.,^{56} the section of the Loop Alfvén wave that is sampled by the counterpropagating ASW Alfvén wave before it is measured has a significant *k*_{∥}_{2} = 0 component. The corresponding frequency of this component, as experienced by the ASW Alfvén wave, is *ω*_{2} = 0. Even though it may seem that the *k*_{∥}_{2} = 0 component of an Alfvén wave cannot propagate in fact, it can. This apparent contradiction arises from the limitation of considering only plane-waves of infinite extent. An Alfvén wave packet of finite extent indeed can support the propagation of a *k*_{∥}_{2} = 0 component. Physically, this *k*_{∥}_{2} = 0 component of the perturbed magnetic field *δ***B** corresponds to a shear in the magnetic field along which the counterpropagating waves attempt to propagate, leading to a shearing of that wave.

For the nonlinear beat interaction with the *k*_{∥}_{2} = 0 component, we obtain $k\u22253\xb1=\u2212k\u22251$ and $\omega 3\xb1=\omega 1$, so the nonlinearly generated modes satisfy the linear Alfvén wave dispersion relation $\omega =|k\u2225|vA$, leading to a resonant three-wave interaction. Note that, although the parallel component of the wavevector and the frequency are unchanged from the ASW Alfvén wave **k**_{1}, the perpendicular component indeed does change, $k\u22a53\xb1=k\u22a51\xb1k\u22a52$, indicating the transfer of energy to smaller perpendicular length scales.

This case of resonant three-wave interactions was the focus of our previous experimental investigations.^{44,48} It is these resonant three-wave interactions that dominate the secular transfer of energy to smaller perpendicular scales in plasma turbulence, so the confirmation of this nonlinear energy transfer in the laboratory was the primary motivation for our previous studies. But for the short times corresponding to the interaction between counterpropagating Alfvén waves in the experiments, the full nonlinear evolution of the ASW Alfvén wave involves significant contributions from interactions with both the *k*_{∥2} = 0 and *k*_{∥2} ≠ 0 components. The resonant three-wave interaction with the *k*_{∥2} = 0 component has been the subject of our previous investigations.^{28,44,48,55,56} The investigation of the non-resonant beat wave interactions with the *k*_{∥2} ≠ 0 component completes the full description of the nonlinear evolution of the ASW Alfvén wave, forming the subject of the present work.

One final point helps to connect these two complementary investigations. In the case that the two initial counterpropagating Alfvén waves have equal and opposite parallel components, *k*_{∥1} = *k*_{∥2}, the non-resonant beat interaction generates a mode with $k\u22253+=0$. In the absence of a pre-existing *k*_{∥} = 0 component of the initial Alfvén waves, this nonlinear beat interaction self-consistently generates the *k*_{∥} = 0 mode required to mediate the resonant transfer of energy to smaller scales, only in this instance the secular transfer of energy is accomplished by a resonant four-wave interaction.^{28,53} Therefore, the nonlinear beat wave physics explored here indeed plays a key role in the nonlinear transfer of energy to small scales when a *k*_{∥} = 0 component is not already present in the initial counterpropagating Alfvén wavepackets.

## IV. EXPERIMENTAL RESULTS AND DISCUSSION

The ASW antenna wave period is a fraction of the period of the Loop antenna wave. As such, the *k*_{∥2} = 0 component of the perturbed magnetic field due to the Loop antenna Alfvén wave can be thought of as causing a shear in the axial magnetic field.^{56} This leads to a distortion in the ASW Alfvén wave as it propagates along this sheared magnetic field, and is the physical process underlying the nonlinear transfer of energy to Alfvén waves with smaller perpendicular scales. In addition, the Loop antenna Alfvén wave also has a *k*_{∥2} ≠ 0 component that interacts nonlinearly with the ASW Alfvén wave to generate a nonlinear beat waves. Theory predicts that the beat waves will have two characteristic features:

The beat waves will have frequencies that correspond to the algebraic sum and difference of the frequencies of the two parent waves, $f3\xb1=|f1\xb1f2|$, as described in Sec. III.

The spatial structure of the beat wave can be roughly predicted by examining the form of the nonlinear term in Eq. (4), $z1+\xb7\u2207z2\u2212$, where $z1+$ is the ASW-antenna Alfvén wave and $z2\u2212$ is the Loop-antenna Alfvén wave. Physically, this means the nonlinearly generated beat wave signal should peak at spatial locations, where the gradient of the Loop Alfvén wave ∇

*δB*_{⊥2}aligns with the ASW Alfvén wave $\delta B\u22a51$.

As discussed previously, the ASW antenna produces an Alfvén wave almost exclusively in the $y\u0302$ direction. Although both the $x\u0302$ and $y\u0302$ components of the magnetic field for the waves were measured, for this paper, only the interactions corresponding to the $y\u0302$-component will be discussed.

### A. Frequency selection

The first theoretically predicted property of the nonlinear beat waves is that their frequencies will correspond to $f3\xb1=|f1\xb1f2|$. For experiment 1 in the kinetic Alfvén wave regime, the frequencies of the ASW and Loop antenna were *f*_{1} = 270 kHz and *f*_{2} = 60 kHz, respectively, leading to the two possible beat frequencies, $f3\u2212=210$ kHz and $f3+=330$ kHz. The antenna frequencies were intentionally chosen so that both beat frequencies $f3\xb1$ lie between the harmonics of the Loop Alfvén wave (*n*_{2} = 3, *n*_{2} = 4, *n*_{2} = 5, and *n*_{2} = 6) and the fundamental frequency of the ASW kinetic Alfvén wave (*n*_{1} = 1), allowing for clear observation of the two beat waves in frequency space. Fig. 4(a) shows the transform to frequency space of *δB _{y}*(

*t*) signal for each of the three data runs that comprise the experiment: (1) when the ASW-antenna Alfvén wave was launched and the Loop antenna was turned off (green line); (2) when the Loop-antenna Alfvén wave was launched and the ASW antenna was turned off (red line); and (3) when both antennas launched Alfven waves simultaneously (blue line). As can be seen in the figure, two peaks corresponding to the nonlinear beat waves at $f3\u2212=210\u2009kHz$ and $f3+=330\u2009kHz$ appear only when both antennas are on, demonstrating definitively that these modes arise through the nonlinear interaction between the two counterpropagating Alfvén waves.

For experiment 2 in the inertial Alfvén wave regime, we see a similar result. For this experiment, we chose an ASW inertial Alfvén wave frequency of *f*_{1} = 525 kHz and a Loop Alfvén wave frequency of *f*_{2} = 116 kHz. The theoretically predicted beat wave frequencies for this experiment are $f3\u2212=409\u2009kHz$ and $f3+=641\u2009kHz$. Fig. 4(b) shows the same three frequency plots as in the kinetic regime. The harmonics of the Loop Alfvén wave (*n*_{2} = 2, *n*_{2} = 3, *n*_{2} = 4, and *n*_{2} = 5) and the fundamental frequency of the ASW inertial Alfvén wave (*n*_{1} = 1) are clearly visible. In between, clear peaks at $f3\u2212=409\u2009kHz$ and $f3+=641\u2009kHz$ are seen only for the run when both antennas are operating (blue line), again demonstrating the nonlinear nature of the two beat wave modes measured here.

### B. Spatial localization

The spatial structure of the nonlinearly generated beat waves can be roughly predicted by examining the form of the nonlinear term in Eq. (4), $z1+\xb7\u2207z2\u2212$, where $z1+$ is the ASW-antenna Alfvén wave and $z2\u2212$ is the Loop-antenna Alfvén wave. Note that there is also a contribution from the nonlinear term in the complementary equation, $z2\u2212\xb7\u2207z1+$, but this contribution is subdominant due to the fact that the Loop Alfvén wave amplitude is much larger than that of the ASW Alfvén wave.

For the experiments performed in the kinetic regime, we observe from Fig. 3(b) that the gradient of the Loop-antenna perpendicular magnetic field pattern $\delta B\u22a52$ is the largest between −5.0 < *y* < 0.0 cm, where it points in $+y\u0302$-direction, and between 5.0 < *y* < 10.0 cm, where it points in $\u2212y\u0302$-direction. In Fig. 5, we plot a colormap of the *δB _{y}*(

*x*,

*y*) component after narrow band filtering the signal in frequency space around (a) $f3\u2212=210\u2009kHz$ and (b) $f3+=330\u2009kHz$. These beat wave spatial structures indeed peak at the locations corresponding to the maximum gradients of the Loop antenna Alfvén wave. The nonlinear term also yields a phase shift of

*π*/2, as is easily seen when Fourier transforming the nonlinear term to obtain a signal proportional to $iz\u03021+(k1)\xb7k\u22a52z\u03022\u2212(k2)$. Since the sign of the gradient in the two regions differs, this leads to a phase shift in the beat wave pattern of +

*π*/2 in the lower region (since

*i*=

*e*

^{iπ}^{∕}

^{2}) and a shift of −

*π*/2 in the upper region (since $\u2212i=e\u2212i\pi /2$). Thus, the upper and lower regions of the beat wave spatial structure should be

*π*out of phase with each other, as evident in Fig. 5. Close examination of the phase of the beat wave structures indeed confirms that the upper or lower regions in Fig. 5 are +

*π*/2 and −

*π*/2 out of phase with respect to the ASW antenna structure in Fig. 2(b).

Performing the same analysis for experiment 2 in the inertial Alfvén wave regime achieves similar results, as shown in Fig. 6. Again, the form of the nonlinear term in Eq. (4) enables a simple prediction of the spatial structure of the nonlinear beat waves from the equations of incompressible MHD theory.

In addition to being able to predict the spatial structure of the beat modes, the nonlinear term in Eq. (4) suggests that that the amplitude of the interaction should scale linearly with $z1+\xb7z2\u2212$. In Fig. 7, we plot the average power for both the observed modes from Fig. 4(a) along the x = −2.0 cm line. The figure shows that the amplitude does scale linearly with the product of the two pump wave amplitudes. However, it is also evident from the figure that the two plots have different slopes. This indicates that the two modes have different excitation efficiencies, which explains the observed asymmetry between the two modes. Similar results were observed for the beat modes produced by the nonlinear interaction of counterpropagating inertial Alfvén waves.

## V. CONCLUSIONS

In this paper, we present the first experimental measurements of the frequency and spatial location of the nonlinear beat waves arising from the interaction of two counterpropagating Alfvén waves. We have confirmed that the nonlinear interaction does produce modes with beat frequencies, as demonstrated in Fig. 4. By comparing the contour plots in Figs. 2 and 3 with those in Figs. 5 and 6, it is clear that the spatial location of the nonlinear beat wave appears at the location where both the magnitude of ASW antenna and the gradient of the Loop antenna are maximum, as predicted in Eq. (4). Our success in predicting the spatial pattern of the beat wave modes, based on the form of the nonlinear term in the incompressible MHD equations, supports the use of idealized fluid models for understanding the nonlinear evolution of weakly collisional plasmas^{28,55,57} which are relevant to many space and astrophysical plasma environments.

## ACKNOWLEDGMENTS

The experiments presented here were conducted at the Basic Plasma Science Facility at UCLA, which is funded in part by the U.S. Department of Energy and the NSF. Funding for this project was provided by the National Science Foundation (NSF) (Grant Nos. ATM 03-17310, PHY-10033446, and AGS-1054061).