The spectral universality in seemingly case-dependent fluid turbulence, which reflects self-similar dynamics in the inertial range of an energy cascade, is one of the most surprising findings in modern physics1,2,3. The presence or absence of universality in collisionless magnetized plasma turbulence exemplified by space, astrophysical, and magnetically confined fusion plasmas has been explored out of scientific interest. This question not only comes out of pure scientific curiosity but is also strongly associated with important practical problems, so-called turbulent heating in the solar wind4,5, solar corona6,7, accretion disks8,9,10, and energy transport in magnetically confined fusion plasmas11,12,13. These problems raise the question: at which scales are ions accelerated and thermalized? Some examples of turbulent heating include the coronal heating problem and the nonadiabatic temperature evolution of the solar wind6. One of the major models for explaining the heating mechanisms of the high-temperature corona/solar wind is the wave turbulence heating model (one is the microflare/nanoflare theory). The energy flux of the Alfvén waves/kinetic Alfvén waves propagating from the solar photosphere and chromosphere into the corona is considered to be sufficient to heat the corona temperature up to the observed temperature of ~100 eV. However, this model encounters a complicated problem of what mechanisms (exemplified by ion cyclotron resonance, Alfvén resonance, linear phase mixing, and shock dissipation, among many others) are responsible for dissipating (partitioning) the wave energy of the Alfvén wave/kinetic Alfvén wave turbulence. The entropy cascade attributed to nonlinear phase mixing is a compelling hypothesis4 for the mechanism. This hypothesis motivates us to experimentally study the validity of the entropy cascade via nonlinear phase mixing.

According to Boltzmann’s H-theorem, entropy production, in other words, ion heating, is realized only by collisions in weakly collisional (collisionless) plasmas at the microscopic scale14 much smaller than the ion gyro-scales (the inner scale) with a significant ∂f/∂v or ∂2f/∂v2, where f and v are the ion velocity distribution function and the velocity vector of ions, respectively. However, macroscopic scales are typical energy inputs to the system (the outer scale), which are considerably separated from the dissipation scales (the inner scale). This fact strongly indicates that the injected energy is nonlinearly transferred as free energy (≈ negative signed entropy) from the outer scale to the inner scale in collisionless magnetized plasmas. This process is called an entropy cascade14,15,16. The gyrokinetic theory15,17,18 predicts that the nonlinear phase-mixing19 is responsible for the transfer of entropy in the inertial subrange14,20, and has been confirmed in a number of gyrokinetic numerical simulations regardless of two-dimensional (2D) electrostatic16 or three-dimensional (3D) electrostatic21 or electromagnetic turbulence4,22,23. In addition, in a real plasma, an indication of a free energy cascade was observed in electromagnetic turbulence in the Earth’s magnetosheath as fine-scale structures in the ion velocity distribution functions24. Regardless of the fact that the original concept of the entropy cascade by the nonlinear phase mixing considers only 2D electrostatic fluctuations14,19, signatures of the entropy cascade have been witnessed in magnetized plasma turbulence in various situations. This broad applicability of the entropy cascade picture to a wide range of collisionless magnetized plasma turbulence is attributed to the fact that turbulence of collisionless magnetized plasmas universally consists of highly anisotropic fluctuations with frequencies much lower than the ion cyclotron frequency in addition to the intrinsic electrostatic nature of perpendicular fluctuations.

Nonlinear phase mixing is a phase-randomizing mechanism of electrostatic fluctuations of quasi-2D magnetized plasmas by a finite-Larmor radius effect19. It couples ion dynamics in the position and the velocity spaces, resulting in entropy cascades in the phase space14,25.

Although the visualization of the entropy distribution in the phase space of ions has been implemented in some simulations25,26, no measurements have been conducted in real plasmas, including space and laboratory plasma experiments.

The gyrokinetic theory for treating gyroscale plasma phenomena considers kinetics of ‘charged rings’ of gyrating ions, whose statistics are represented by ring-averaged ion velocity distribution functions at a fixed guiding center. The gyrokinetic system is invariant under the scaling transformation; (g, ϕ, r, v) → (g, μ2ϕ, μr, μv), where g, ϕ, r, v, and μ are fluctuation components of the ring-averaged ion velocity distribution function, the electrostatic potential, the space coordinate, the perpendicular velocity of ions, and a scaling factor, respectively20. Note that g(R, v) is the gyro average of δf = f (r, v, t) − F0(r, v), which is the perturbed distribution function, where F0 is the background equilibrium distribution function that is steady on the timescale of turbulent fluctuations (a Maxwellian is assumed in the standard gyrokinetic theory), and R is the fixed guiding center position.

The dual self-similarity hypothesis for the position and velocity spaces based on the above scale invariance predicts the power laws of spectra of the free energy \({W}_{g1}=\iint \frac{g{({{{{{\bf{R}}}}}},{v}_{\perp })}^{2}}{2}d{v}_{\perp }d{{{{{\bf{R}}}}}}\) and energy \({E}_{g}=\frac{1}{2}\int \tilde{\phi }{({{{{{\bf{r}}}}}})}^{2}d{{{{{\bf{r}}}}}}\) with the use of g(R, v) for the 2D electrostatic turbulence. The dual self-similar dynamics in the phase-space are expected to be accompanied by a dual cascade of two kinds of turbulence energy in 2D electrostatic turbulence: the forward cascade of free energy and an inverse cascade of electrostatic energy25. This dual self-similarity hypothesis extends the renowned Kolmogorov self-similarity hypothesis in fluid turbulence27 to the phase space of ions in gyrokinetic turbulence. It argues that the turbulent behavior of ions at the sub-Larmor scales is independent of scale in both their position and velocity spaces. These are strongly coupled in the phase space by nonlinear phase mixing.

Therefore, in this experiment, we employed diagnostic instruments named Ring-averaged ion velocity distribution function probes (RIDFPs)28 to measure g(R, v) for evaluation of the theoretically predicted power laws. Although our previous paper29 showed agreement between measurements and the theoretical prediction on the power spectrum of Eg as a function of the fluctuation wavenumber perpendicular to the background magnetic field k, we still face a lack of information on the velocity space to conclude the validity of the dual-cascade theory of gyrokinetic turbulence.

Here, we show the direct visualization of the entropy distribution of ions in the phase space of laboratory-magnetized plasma turbulence. These results provide the evidence of the existence of dual self-similarity and the dual cascade of entropy and field energy in the phase space of turbulence.

Results and discussion

Experimental setup

In this experiment (“Methods”: Magnetized plasma experiment (MPX) device, ring-averaged ion velocity distribution function probes, other measurement tools, and the plasma parameters), we prepared four states of electrostatic 2D turbulence driven by resistive drift-waves having varied driving perpendicular wavenumbers kdriveρthi between ~0 and 10 with control of the density of the background neutral particles, where ρthi is the Larmor radius of ions at the thermal velocity vthi. While this control scheme of kdriveρthi is not a well-established technique, we prepared the turbulent states with the control of the neutral particle density with reference to the numerical work30, which describes the numerical investigation results of an energy partition of turbulence in a cylindrical plasma. We use Fourier–Hankel components of the free energy: \({W}_{g1}({k}_{\perp },p)\equiv p{\sum }_{|{{{{{{\bf{k}}}}}}}_{\perp }|={k}_{\perp }}|g({{{{{{\bf{k}}}}}}}_{\perp },p){|}^{2}=\pi {k}_{\perp }p|g({k}_{\perp },p){|}^{2}\), where g(k, p) is the Fourier–Hankel transform of g(R, v) concerning the guiding center position and the ion velocity20, and the parameter p indicates the reciprocal of the fluctuation scale of g(R, v) in the velocity direction corresponding to the wavenumber in the velocity space.

The gyrokinetic approximation requires ωΩci, where ω and Ωci are the wave frequencies and the ion cyclotron frequency, respectively. The higher the axial magnetic field strength is, the more improved the gyrokinetic condition. However, our experiment does not completely satisfy ωΩci (ω/2π ~ 0.1–4 kHz, Ωci/2π = 8.4 kHz), which is a consequence of the two following adverse impacts of the strong field on our plasma states: (i) the suppression of the excitation of the drift waves; in our plasmas, the high magnetic field stabilizes the drift waves because of a change in the radial profile of the density; and (ii) the reduction of the ion gyro-radius, which limits the performance of the diagnostic. In other words, the axial magnetic field strength is set at a weak value to fully use all the velocity channels of the RIDFPs by expanding the ion gyro radii. This improves the resolution of the RIDFP measurement in the velocity space. In the third paragraph from the end of this section, we explain the influence of this incomplete satisfaction of the gyrokinetic ordering on the interpretation of the experimental results.

Spectra of free energy in the kv space

Figure 1 shows time-averaged free energy spectra k|g(k, v)|2 as functions of kxρthi and v up to ~2vthi for (Fig. 1a) kdriveρthi ~ 1 (Wg1 injection at large scales) and (Fig. 1b) kdriveρthi ~ 10 (Wg1 injection at small scales), respectively. We regard kx ≈ k in evaluating k|g(k, v)|2 based on the assumption of isotropy and fast decay of |g(k, v)|2 for k. The factor k indicates the circumferential length in the k-space.

Fig. 1: Spectra of free energy in the wavenumber–velocity space.
figure 1

Time-averaged power spectra k|g(k, v)|2 measured by RIDFPs as functions of the wavenumber kx normalized by the ion Larmor radius ρthi at the thermal velocity vthi and the perpendicular ion velocity v up to ~2vthi for a driving perpendicular wavenumbers kdriveρthi ~ 1 and b kdriveρthi ~ 10, respectively.

k|g(k, v)|2 for (A) kdriveρthi ~ 1 exhibits, in the entire range of v, the large amplitude at kxρthi ~ 1−2 consistent with \(|{\tilde{\phi }}_{{{{{{\rm{f}}}}}}}(\,f,{k}_{y}){|}^{2}\) measured by a fine-scale Langmuir probe (FSLP) for kdriveρthi ~ 1 (Fig. 2a), which shows excitation of kxρthi ~ 1 components at f ~ 1.0−1.3 kHz. Accordingly, the low k components at kxρthi ~ 1–2 show variation in the v-direction at the scale of ~vthi in Fig. 1a. The amplitude of components for |kxρthi| > 2 decays in large kx with finer-scale variation in the v-direction. On the other hand, k|g(k, v)|2 for (B) kdriveρthi ~ 10 exhibits significant power at |kxρthi| > 10. These |kxρthi| > 10 components exhibit structures in the v-direction at sub-vthi scales. This indicates strong coupling between the position and velocity spaces. See the next subsection for a quantitative assessment of scales in v space using spectra.

Fig. 2: Variation in spectra of the invariants of gyro-kinetics.
figure 2

The free energy Wg1(k, p) and the electrostatic energy Eg(k) for varied driving perpendicular wavenumbers kdriveρthi between ~0 and 10, (the 1st column) kdriveρthi ~ 1, (the 2nd column) kdriveρthi ~ 3−4, (the 3rd column) kdriveρthi ~ 0−4, and (the 4th column) kdriveρthi ~ 10, respectively. The plots in the 1st row (ad) are the power spectra of potential fluctuation \({|{\tilde{\phi }}_{f}(f,{k}_{x})|}^{2}\) measured by FSLP. The plots shown in the 2nd row (eh) are Wg1(k, p) in the k- p-space, namely, position–velocity-space. The 3rd row (il) are (cross) 1D spectra Wg11D of the free energy as functions of kxρthi and (open square) p-spectra of Wg1(p), respectively. Eg(k) spectra are shown in the 4th row (mp). Respective kdriveρthi ranges are shown by purple color bars. The blue and red bars represent power laws predicted by the gyrokinetic theory20 for (blue) k < kdrive and (red) k > kdrive, respectively.

Comparison with theory

In the second–fourth rows of Fig. 2, we show variation in the spectra of the collisionless invariants of 2D electrostatic gyrokinetics, Wg1(k, p) and Eg(k), for varied kdriveρthi approximately between 0 and 10, (the first column) kdriveρthi ~ 1, (the second column) kdriveρthi ~ 3−4, (the third column) kdriveρthi ~ 0–4, and (the fourth column) kdriveρthi ~ 10. These values of kdriveρthi are evaluated from \({|{\tilde{\phi }}_{{{{{{\rm{f}}}}}}}(\,f,{k}_{y})|}^{2}\) measured by FSLP as shown in the first row of Fig. 2a–d. In the respective cases of different kdriveρthi, the following two distinct frequency bands can be identified in \({|{\tilde{\phi }}_{{{{{{\rm{f}}}}}}}(\,f,{k}_{y})|}^{2}\): one showing significant power with nonstatistical/deterministic distribution corresponding to the drive, for example, kyρthi ~ 1 component at f ~ 1.3 kHz in Fig. 2a (kyρthi ~ 3–4 component at f ~ 1.7 kHz in Fig. 2b, and kyρthi ~ 10 component at f ~ 3–4 kHz in Fig. 2d), and the other bands showing scattered, statistical distribution in the ky-direction corresponding to turbulent cascade. Nonlinear dynamics exhibited in these shots are local because no nonlocal resonant wave–wave interactions occurred except for the kdriveρthi ~ 0–4 state (the vertical array of Fig. 2c, g, k and o), which exhibits a continuous driving spectrum in the f-kyρthi diagram and is a turbulent state originating from a nonlinear coherent state of drift waves called solitary drift waves31.

The plots shown in the second row (Fig. 2e–h) are Wg1(k, p) (negative signed entropy) in the k-p space, namely, position–velocity space (see “Evaluation of 1D spectrum of free energy Wg11D(kx, p) and relationship with Wg1(k) and Wg1(p)” in Methods). In the following results of varied kdriveρthi, a position of the drive moves along the diagonal in the k-p space, as shown in Fig. 2e–h.

When Wg1 is injected at large scales (Fig. 2e), in which a significant power of Wg1 can be seen at the left bottom of the diagram at kρthi ~ pvthi ~ 1–4 corresponding to kdriveρthi, Wg1 broadly develops along the diagonal from the driving scale toward the upper-right direction, reaching the top right corner around kρthi ~ pvthi ~ 20. (A similar profile can be seen in Fig. 6 in the article by Tatsuno et al.26). According to the gyrokinetic Poisson Eq., the potential fluctuation \(\tilde{\phi }({{{{{{\bf{k}}}}}}}_{\perp })\) is generated solely by nonlinear phase mixing, that is, \(g({k}_{\perp },p={k}_{\perp })\) (Eq. (2.11) in the article by Plunk et al.20). This provides a Fjørtoft-type relationship Wg1(k, p = k) = kEg(k) (Eq. (7.14) in the article by Plunk et al.20). Therefore, diagonal transport of Wg1 in the k-p space is inevitable when a dual-cascade of Wg1 and Eg occurs.

This Wg1(k, p) spectrum exhibits the generation of Wg1 at finer scales than the drive. This indicates that the nonlinear phase mixing works, resulting in a forward cascade of Wg1. This cascading nature will be examined below by comparing their power laws obtained from the measurement and the theory.

As kdriveρthi increased to ≈10, as shown in Fig. 2f–h, the significant Wg1 part moved in the upper-right direction along the diagonal line toward higher kρthi ~ pvthi locations corresponding to the increase in kdriveρthi. In all the cases of kdriveρthi (Fig. 2e–h), the spread of Wg1(k, p) along with the diagonal directions (between the bottom left and the upper right) is obvious. In these cases, the Fjørtoft-type energetic transition25, namely, a transition involving only diagonal components, can be realized.

In the third row of Fig. 2i–l), (crosses) one-dimensional (1D) spectra Wg11D of the free energy as functions of kxρthi and (open squares) p-spectra of Wg1(p), respectively, are shown. Note that when Wg1(k) k−α, Wg11D(kx) kx−α (see “Evaluation of 1D spectrum of free energy Wg11D(kx, p) and relationship with Wg1(k) and Wg1(p)” in Methods). Respective kdriveρthi ranges are shown by purple bars. The blue and red bars represent the power laws predicted by the gyrokinetic theory for (blue) the inverse-cascade range k < kdrive and (red) the forward cascade range k > kdrive, respectively20. The respective power indices of Wg11D(kx) and Wg1(p) evaluated from the least-square fitting to the measured data points are indicated by numbers with the standard errors (SE). The noise levels of Wg11D(kx) and Wg1(p) estimated from a vacuum shot are included in Fig. 2i. Wg1(p) in Fig. 2i (injection at a large scale) shows power-law decay starting at kxρthi ~ kdriveρthi ~ 1 up to the detection limit of kxρthi ~ pvthi ~ 20, clearly indicating the existence of an inertial subrange of the entropy (free energy). The evaluated exponent for the measured Wg1(p), −1.0, approximately agrees with the theoretical value of −4/3 within 5 times SE. For smaller scales kxρthi > 3, Wg11D(kx) in Fig. 2i shows a similar decaying tendency to Wg1(p), which is a strong indication of the dual self-similarity dynamics, although the discrepancy of Wg11D(kx) from the theoretical exponent is larger than 25 times SE. Larger discrepancies between the theoretical exponents and the measured ones in Wg11D(kx) than those in Wg1(p) are seen in all cases of kdrive.

As kdriveρthi increases to 3–4 (Fig. 2j), Wg1(p) shows an inflection point at kxρthi ~ kdriveρthi ~ 3–4. Correspondingly, Wg11D(kx) follows this change in the spectrum. For an injection at small scales kdriveρthi ~ 10 (Fig. 2l), both Wg1(p) and Wg11D(kx) become independent of p and kx, indicating good agreement with the theory20. It should be noted that the decaying exponents of both Wg11D(kx) and Wg1(p) at kxρthi 1 exhibit similar values in the respective cases of kdriveρthi and similar variation for varied kdriveρthi in Fig. 2i–l. This indicates the formation of inertial subranges in both the position and velocity spaces and strong coupling between them. A similar result is obtained in the gyrokinetic simulation16,26.

In the case of kdriveρthi ~ 0–4 (Fig. 2k), despite the driving source in the target k-range (i.e., noninertial range), strong kp coupling occurs, indicating the universality of nonlinear phase mixing in magnetized plasma turbulence.

The abovementioned approximate agreement between the measurement and the theoretical prediction of Wg1(p) and Wg1(k) within ~1 − 25 times SE is supported by the Eg(k) spectra measured by FSLP shown in the fourth row of Fig. 2 (Fig. 2m–p). Note that when Eg(k) k−α, Eg11D(kx) kx−α (“Methods”: Evaluation of the 1D spectrum of the electrostatic field energy Eg1D(ky) and its relationship with Eg(k)). The blue and red bars represent the power laws predicted by kinetic theory for (blue) k < kdrive and (red) k > kdrive, respectively. The respective power indices of Eg11D(kx) evaluated from the least-square fitting to the measured data points are indicated aside by numbers with the standard errors. The noise levels of Eg11D(kx) estimated from a vacuum shot are included in Fig. 2m–p. The evaluated scaling for Eg11D(kx) in Fig. 2m is kx−3.0, close to the theoretical power law k−10/3. The theoretical prediction k−2 for the inertial range in the larger scale than the driving scale becomes dominant as kdriveρthi increases. The exponent evaluated for the measured Eg11D(kx) in Fig. 2p is kx−1.8. The observed behavior of these power laws in Wg1 and Eg is consistent with the theory based on the hypothesis of dual self-similarity and the dual cascade of the entropy and field energy of the gyrokinetic system20. Our result strongly supports the validity of the dual-self similar hypothesis of gyrokinetic turbulence.

Note that the nonlinear decorrelation time16,26 for these states is evaluated as τρ ~ 10 × 10−5 s; hence, the dimensionless parameter16,26, which represents the scale separation analogous to Reynolds number, D ~ 30–40. Accordingly, the cutoff wavenumber, above which the collisional dissipation dominates the cascade, kcutρthi ~ 2D3/5 ~ 20. This guarantees that the observed power-law spectral range corresponds to the inertial subrange.

In what follows, we discuss the applicability of the gyrokinetic theory and its impact on the interpretation of the experimental results. The theoretical aspects of the applicability of gyro kinetics to space and astrophysical plasmas are discussed in previous works15,18,23. In brief, strong magnetization, anisotropy, small amplitude, low fluctuation frequencies, equilibrium Maxwellian distribution, and nonrelativistic effects are assumed.

As described in Experimental setup, our experiment does not completely satisfy the strong magnetization condition ωΩci in consideration of the two adverse impacts of the strong field on our plasma states. One possible influence of this incomplete satisfaction of the gyrokinetic ordering is the possibility of stochastic ion heating32,33. According to ref. 34, the amplitude threshold for strong stochastic heating when β < 1 can be expressed in terms of the quantity εiqiδΦi/mv2, where β, qi, and δΦi are the thermal pressure normalized by the magnetic field pressure, electric charge of the ions, and root mean square amplitude of the electrostatic potential fluctuation at kρthi 1, respectively. In our case, εi 0.2; therefore, the influence of stochastic ion heating is considered to be small. Ion cyclotron resonance heating is negligible because ω Ωci.

One of the weakest points of the gyrokinetic theory is the use of a Maxwellian as F015. Therefore, highly nonequilibrium turbulent states are not subjects of gyrokinetic theory. Nevertheless, the gyrokinetic theory is in effect in a wide range of space and turbulent astrophysical plasmas, as shown in numerous numerical simulations and theoretical considerations. Although our measurement has not had sufficient accuracy to be able to discuss a degree of deviation of F0 from a Maxwellian distribution (Fig. 3d in “Methods” shows an example of the velocity distribution F0), we believe that F0 of our plasmas is not that different from a Maxwellian because of the following reasons: (i) no mechanisms that heat preferentially ions with specific velocity components exist; the collisional energy transfer caused by electron–ion collisions is the only possible heating source of ions; and (ii) ions are weak collisional, but not too weak. The mean-free path of ions for ion–ion collision is ~0.5 m < 1.6 m ~ L, which is the length of the plasma column. For the same reasons, pressure anisotropy is not considered.

Fig. 3: Experimental setup.
figure 3

a Schematic diagram of the MPX plasma device including diagnostic systems. b System diagram of RIDFP (Ring-averaged ion velocity distribution function probe). c Schematic of the orbit filters and collector of the incoming gyrating ions. d Examples of equilibrium ring-averaged ion velocity distribution functions F0 at a fixed guiding center position for the background magnetic field strength B0 = 0.022 T and B0 = 0.033 T, respectively. The red solid curves indicate Maxwellian having the temperature of 0.1 eV. Shrink of orbits of the ions (Larmor radii) for the B0 change from 0.022 T to 0.033 T can be clearly observed, indicating validity of RIDFP measurement.

The influence of the electron behavior on the entropy cascade in the ion phase space is considered negligible according to gyrokinetic numerical simulations4,16,23. The work done by Tatsuno et al. showed that no essential difference was observed between the cases of applying Boltzmann-response (3D) and no-response (2D) electrons in their simulations4. For more detailed electron energetics, please see, the work by Kawazura et al.23, which studied the partition of irreversible heating between ions and electrons in terms of a compressive drive using hybrid gyrokinetic simulations.


In laboratory experiments, we present the measurement results of the Gibbs entropy distribution in the phase space of ions in electrostatic gyrokinetic turbulence. We confirm good agreement between the measurement and the theoretical prediction with a dual self-similar hypothesis in the scaling laws of the spectra of the entropy and the electric field energy. This result indicates an interplay between the position and velocity spaces of ions by perpendicular nonlinear phase mixing accompanied by a cascade of entropy in the phase space. Excitation of spectral components of the free energy Wg1 with kp in the phase space gives evidence of nonlinear phase mixing at sub-Larmor scales. This observation corroborates the evidence of the existence of dual self-similarity and the dual-cascade of entropy and field energy.

An entropy cascade via nonlinear phase mixing is considered universal in magnetized plasma turbulence because it has been witnessed in various numerical simulations in different magnetized turbulence setups, including 2D16 and 3D electrostatic21, 3D electromagnetic turbulence, including the Alfvenic solar wind4,22, and a simulation targeting an Alfvenic hot accretion disk23. The essential physical components in entropy cascades via nonlinear phase mixing include the gyro motion of the particles at the sub-Larmor scales and the interaction between the particles and the electrostatic fluctuations in addition to the strong anisotropy (kk||). Our experiment prepares turbulent states with these essential components in a simplified setup. Therefore, although our turbulence setup is in a narrow range (electrostatic turbulence in a drift-kinetic regime), our result is crucial for verifying universality. We believe that verifying the entropy cascades via nonlinear phase mixing contributes to understanding turbulent heating in the solar corona, solar wind, accretion disks, and magnetically confined fusion plasmas.


Magnetized plasma experiment (MPX) device: experimental device

The experiment was conducted in the MPX device35 (Fig. 3a), which can prepare qualified plasma configurations for our purposes, namely, quasi-2D configurations with the magnetic field (the strength is denoted by B0), the density, and temperatures within the target ranges. The hot cathode generated plasmas with the introduction of argon gas as the working gas. g(R, v) was measured by two RIDFPs separated by 3 mm and 230 mm in the x- and the z-directions, respectively (Fig. 3a), to obtain perpendicular wavenumbers k of g(R, v) with the use of a two-point correlation technique36. RIDFPs were positioned at the center axis of the target plasmas to minimize the influence of steady E × B rotation of the plasmas. Figure 3 shows the equilibrium ring-averaged ion velocity distribution functions F0 for B0 = 0.022 T and B0 = 0.033 T, respectively. The red solid curves indicate Maxwellian with a temperature of 0.1 eV. It can be observed that the orbits of the ions (Larmor radii) shrink as B0 change from 0.022 T to 0.033 T, indicating the validity of the RIDFP measurement.

Ring-averaged ion velocity distribution function probes28

The RIDFP is a set of ion collectors that detect different velocity components and is immersed into magnetized plasmas. The RIDFP achieves momentum selection of incoming ions by selection of the ion Larmor radii by the orbit filter. To nullify the influence of the sheath potential surrounding the RIDFP on the incoming ions’ orbits, the RIDFP body’s electrostatic potential is automatically adjusted to coincide with the space potential of the target plasma with the use of an emissive probe and a voltage follower28.

For precise measurement of the fluctuation g(R, v) in time, the ground potential of the current detection circuits is adjusted to that of the RIDFP body in order to eliminate capacitive coupling between the ion collectors and the RIDFP body. The length of RIDFP L = 79 mm satisfies the condition of v||fci−1 > L, where v|| is the parallel component of the thermal velocity of ions. This inequation imposes a condition for the plasma (ions) residing in the magnetic flux tube, in which an RIDFP is allocated, not to being severed by the RIDFP.

In the experiments, the resolution of the RIDFP measurement in the p-space Δ(pvthi) can be evaluated as follows: Δpvupper_bound ~ Δp(2vthi) = j0 2.4, Δp 1.2/vthiΔ(pvthi) ~ 1.2, where vupper_bound and j0 are the upper bounds of the measurable ion velocity by RIDFP and the first zero of the Bessel function of the first kind, respectively. The maximum (pvthi) in the RIDFP measurement is obtained as: (pvthi)max ~ j0vthi/Δv ~ 20.

The other measurement tools and the plasma parameters

In addition to RIDFPs, we employed a few sets of Langmuir probes (LPs) for different purposes. An LP measured the plasma density in addition to a microwave interferometer37. The LP also measured the electron temperature and the space potential of the plasmas. A poloidal array of LPs named LPA diagnosed the macroscopic poloidal structures of fluctuations of ion saturation current and their floating potential. Another set of LPs named fine-scale LP (FSLP) was applied to measure the fluctuation of floating potential \({\tilde{\phi }}_{f}\approx \tilde{\phi }\) as functions of frequencies f and ky (kx), whose upper bound of measurable k is ~8 × 102 m−1.

The hot cathode generated plasmas with the introduction of argon gas as the working gas. The typical plasma density and temperature are ne ~ 0.5–5 × 1016 m−3 and Te ~ 1–5 eV, respectively, with a 15–20 cm diameter. Accordingly, the characteristic frequencies of the ion cyclotron fci, drift-wave fDW, ion–ion Coulomb collisions fii, and ion-neutral collisions fin have the following numbers:

fci := Ωci/2π ≈ 8.4 kHz > fDW ~ 1–4 kHz ~ fii ~ 1–5 kHz > fin ~ 0.1–0.5 kHz, where Ωci is the ion cyclotron angular frequency of singly ionized argon ion.

Evaluation of 1D spectrum of free energy W g1 1D(k x, p) and relationship with W g1(k ) and W g1(p)

From the RIDFP measurement, 1D spectrum of free energy \({{W}_{g1}}^{1D}({k}_{x},p)\equiv \pi p{\int }_{-\infty }^{\infty }|g({{{{{{\bf{k}}}}}}}_{\perp },p){|}^{2}d{k}_{y}=\pi p{\int }_{-\infty }^{\infty }|g({k}_{x},{k}_{y},p){|}^{2}d{k}_{y}\) is evaluated as follows, where: \(g({{{{{{\bf{k}}}}}}}_{\perp },p)\equiv \frac{1}{2\pi }{\int }_{{\mathbb{R}}}{d}^{2}{{{{{\bf{R}}}}}}{\int }_{0}^{{v}_{\perp }^{upper}}{v}_{\perp }d{v}_{\perp }{J}_{0}(p{v}_{\perp }){e}^{-i{{{{{{\bf{k}}}}}}}_{\perp }\cdot {{{{{\bf{R}}}}}}}g({{{{{\bf{R}}}}}},{v}_{\perp })\)

  1. (1)

    Application of the Hankel transform concerning ion velocity v to g(R0, v, t) and g(R0 + Δx, v, t) measured by RIDFP-1 and RIDFP-2 gives \(g({{{{{{\bf{R}}}}}}}_{0},p,t)\equiv \frac{1}{2\pi }{\int }_{0}^{{v}_{\perp }^{upper}}{v}_{\perp }d{v}_{\perp }{J}_{0}(p{v}_{\perp })g({{{{{{\bf{R}}}}}}}_{0},{v}_{\perp },t)\) and g(R0 + Δx, p, t), where R0 and R0 + Δx are the positions of the guiding center of RIDFP1 and RIDFP2, respectively.

  2. (2)

    Fourier transforms of g(R0, p, t) and g(R0 + Δx, p, t) about time t are applied to obtain g(R0, p, f, θ) and g(R0 + Δx, p, f, θ), where θ represents the phase of the component of the frequency f.

  3. (3)

    Wavenumbers kx for all p-components are calculated by applying the two-point correlation technique to g(R0, p, f, θ) and g(R0 + Δx, p, f, θ). Integration of those about f gives \({\int }_{-\infty }^{\infty }|g({k}_{x},{k}_{y},p){|}^{2}d{k}_{y}={{W}_{g1}}^{1D}({k}_{x},p)/(\pi p)\).

If turbulence is isotropic, \({{W}_{g1}}^{1D}({k}_{x},p)=\pi p{\int }_{-\infty }^{\infty }|g(\scriptstyle\sqrt{{k}_{x}^{2}+{k}_{y}^{2}},p){|}^{2}d{k}_{y}.\) Applying a variable change \(\tan \theta =\frac{{k}_{y}}{{k}_{x}}\) and a cutoff θcut for the upper bound of the integration, one can obtain

$${{{{{{{\boldsymbol{W}}}}}}}_{{{{{{\boldsymbol{g}}}}}}1}}^{1{{{{{\boldsymbol{D}}}}}}}({{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{x}}}}}}},{{{{{\boldsymbol{p}}}}}})=2p{{\int }_{0}^{{{{{{{\boldsymbol{\theta }}}}}}}_{{{{{{\boldsymbol{cut}}}}}}}}\left|{{{{{\boldsymbol{g}}}}}}\left({{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{x}}}}}}}\sqrt{1+\tan {{{{{\boldsymbol{\theta }}}}}}},{{{{{\boldsymbol{p}}}}}}\right)\right|}^{2}\frac{{{{{{\boldsymbol{d}}}}}}{{{{{\boldsymbol{\theta }}}}}}}{{\cos }^{2}{{{{{\boldsymbol{\theta }}}}}}}$$

When Wg1(k) = ∫k|g(k, p)|2dpk−α, Wg11D(kx) = ∫Wg11D(kx, p)dpkx−α.

$${{{{{{\boldsymbol{W}}}}}}}_{{{{{{\boldsymbol{g}}}}}}1}({{{{{{\boldsymbol{k}}}}}}}_{\perp },{{{{{\boldsymbol{p}}}}}})\equiv p\mathop{\sum}\limits_{|{{{{{{\bf{k}}}}}}}_{\perp }|={{{{{{\boldsymbol{k}}}}}}}_{\perp }}{|{{{{{\boldsymbol{g}}}}}}({{{{{{\bf{k}}}}}}}_{\perp },{{{{{\boldsymbol{p}}}}}})|}^{2}=\pi {{{{{{\boldsymbol{k}}}}}}}_{\perp }p{|{{{{{\boldsymbol{g}}}}}}({{{{{{\boldsymbol{k}}}}}}}_{\perp },{{{{{\boldsymbol{p}}}}}})|}^{2}.$$


$${{{{{{\boldsymbol{W}}}}}}}_{{{{{{\boldsymbol{g}}}}}}1}({{{{{\boldsymbol{p}}}}}}) \equiv \, \pi p{\int }_{0}^{\infty }\mathop{\sum}\limits_{|{{{{{{\bf{k}}}}}}}_{\perp }|={{{{{{\boldsymbol{k}}}}}}}_{\perp }}{|{{{{{\boldsymbol{g}}}}}}({{{{{{\bf{k}}}}}}}_{\perp },{{{{{\boldsymbol{p}}}}}})|}^{2}{{{{{\boldsymbol{d}}}}}}{{{{{{\boldsymbol{k}}}}}}}_{\perp }=\pi p\iint\limits_{{-\infty }}^{{\infty }}{|{{{{{\boldsymbol{g}}}}}}({{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{x}}}}}}},{{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{y}}}}}}},{{{{{\boldsymbol{p}}}}}})|}^{2}{{{{{\boldsymbol{d}}}}}}{{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{y}}}}}}}{{{{{\boldsymbol{d}}}}}}{{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{x}}}}}}}\\ =\, {\int }_{-\infty }^{\infty }{{{{{{{\boldsymbol{W}}}}}}}_{{{{{{\boldsymbol{g}}}}}}1}}^{1{{{{{\boldsymbol{D}}}}}}}({{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{x}}}}}}},{{{{{\boldsymbol{p}}}}}}){{{{{\boldsymbol{d}}}}}}{{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{x}}}}}}}$$

Evaluation of the 1D spectrum of the electrostatic field energy E g 1D(k y) and its relationship with E g(k )

From the FSLP measurement, the 1D spectrum of the electrostatic field energy \({{E}_{g}}^{1D}({k}_{y})\equiv {\int }_{-\infty }^{\infty }{|\tilde{\phi }({{{{{{\bf{k}}}}}}}_{\perp })|}^{2}d{k}_{x}\) is evaluated as follows:

If turbulence is isotropic, \({{E}_{g}}^{1D}({k}_{y})={\int }_{-\infty }^{\infty }{|\tilde{\phi }(\scriptstyle\sqrt{{k}_{x}^{2}+{k}_{y}^{2}})|}^{2}d{k}_{x}.\) Applying a variable change \(\tan \theta =\frac{{k}_{y}}{{k}_{x}}\) and a cutoff θcut for the upper bound of the integration, one can obtain

$${{{{{{{\boldsymbol{E}}}}}}}_{{{{{{\boldsymbol{g}}}}}}}}^{1{{{{{\boldsymbol{D}}}}}}}({{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{y}}}}}}})=2{\int }_{0}^{{{{{{{\boldsymbol{\theta }}}}}}}_{{{{{{\boldsymbol{cut}}}}}}}}{\left|\tilde{\phi }\left({{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{x}}}}}}}\sqrt{1+\tan {{{{{\boldsymbol{\theta }}}}}}}\right)\right|}^{2}\frac{{{{{{\boldsymbol{d}}}}}}{{{{{\boldsymbol{\theta }}}}}}}{{\cos }^{2}{{{{{\boldsymbol{\theta }}}}}}}.$$

When Eg(k) k−α, Eg1D(ky) = ∫Eg(k)dkxky−α.