Abstract
The turbulence of collisionless magnetized plasmas, as observed in space, astrophysical, and magnetically confined fusion plasmas, has attracted considerable interest for a longtime. The entropy cascade in collisionless magnetized plasmas is a theoretically proposed dynamics comparable to the Kolmogorov energy cascade in fluid turbulence. Here, we present evidence of an entropy cascade in laboratory plasmas by direct visualization of the entropy distribution in the phase space of turbulence in laboratory experiments. This measurement confirms the scaling laws predicted by the gyrokinetic theory with the dual selfsimilarity hypothesis, which reflects the interplay between the position and velocity of ions by perpendicular nonlinear phase mixing. This verification contributes to our understanding of turbulent heating in the solar corona, accretion disks, and magnetically confined fusion plasmas.
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Introduction
The spectral universality in seemingly casedependent fluid turbulence, which reflects selfsimilar dynamics in the inertial range of an energy cascade, is one of the most surprising findings in modern physics^{1,2,3}. 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, socalled turbulent heating in the solar wind^{4,5}, solar corona^{6,7}, accretion disks^{8,9,10}, and energy transport in magnetically confined fusion plasmas^{11,12,13}. 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 wind^{6}. One of the major models for explaining the heating mechanisms of the hightemperature 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 hypothesis^{4} for the mechanism. This hypothesis motivates us to experimentally study the validity of the entropy cascade via nonlinear phase mixing.
According to Boltzmann’s Htheorem, entropy production, in other words, ion heating, is realized only by collisions in weakly collisional (collisionless) plasmas at the microscopic scale^{14} much smaller than the ion gyroscales (the inner scale) with a significant ∂f/∂v or ∂^{2}f/∂v^{2}, 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 cascade^{14,15,16}. The gyrokinetic theory^{15,17,18} predicts that the nonlinear phasemixing^{19} is responsible for the transfer of entropy in the inertial subrange^{14,20}, and has been confirmed in a number of gyrokinetic numerical simulations regardless of twodimensional (2D) electrostatic^{16} or threedimensional (3D) electrostatic^{21} or electromagnetic turbulence^{4,22,23}. In addition, in a real plasma, an indication of a free energy cascade was observed in electromagnetic turbulence in the Earth’s magnetosheath as finescale structures in the ion velocity distribution functions^{24}. Regardless of the fact that the original concept of the entropy cascade by the nonlinear phase mixing considers only 2D electrostatic fluctuations^{14,19}, 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 phaserandomizing mechanism of electrostatic fluctuations of quasi2D magnetized plasmas by a finiteLarmor radius effect^{19}. It couples ion dynamics in the position and the velocity spaces, resulting in entropy cascades in the phase space^{14,25}.
Although the visualization of the entropy distribution in the phase space of ions has been implemented in some simulations^{25,26}, 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 ringaveraged ion velocity distribution functions at a fixed guiding center. The gyrokinetic system is invariant under the scaling transformation; (g, ϕ, r, v_{⊥}) → (g, μ^{2}ϕ, μr, μv_{⊥}), where g, ϕ, r, v_{⊥}, and μ are fluctuation components of the ringaveraged ion velocity distribution function, the electrostatic potential, the space coordinate, the perpendicular velocity of ions, and a scaling factor, respectively^{20}. Note that g(R, v_{⊥}) is the gyro average of δf = f (r, v, t) − F_{0}(r, v), which is the perturbed distribution function, where F_{0} 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 selfsimilarity 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 selfsimilar dynamics in the phasespace 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 energy^{25}. This dual selfsimilarity hypothesis extends the renowned Kolmogorov selfsimilarity hypothesis in fluid turbulence^{27} to the phase space of ions in gyrokinetic turbulence. It argues that the turbulent behavior of ions at the subLarmor 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 Ringaveraged ion velocity distribution function probes (RIDFPs)^{28} to measure g(R, v_{⊥}) for evaluation of the theoretically predicted power laws. Although our previous paper^{29} showed agreement between measurements and the theoretical prediction on the power spectrum of E_{g} 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 dualcascade theory of gyrokinetic turbulence.
Here, we show the direct visualization of the entropy distribution of ions in the phase space of laboratorymagnetized plasma turbulence. These results provide the evidence of the existence of dual selfsimilarity 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, ringaveraged ion velocity distribution function probes, other measurement tools, and the plasma parameters), we prepared four states of electrostatic 2D turbulence driven by resistive driftwaves having varied driving perpendicular wavenumbers k_{drive}ρ_{thi} between ~0 and 10 with control of the density of the background neutral particles, where ρ_{thi} is the Larmor radius of ions at the thermal velocity v_{thi}. While this control scheme of k_{drive}ρ_{thi} is not a wellestablished technique, we prepared the turbulent states with the control of the neutral particle density with reference to the numerical work^{30}, 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: \({W}_{g1}({k}_{\perp },p)\equiv p{\sum }_{{{{{{{\bf{k}}}}}}}_{\perp }={k}_{\perp }}g({{{{{{\bf{k}}}}}}}_{\perp },p){}^{2}=\pi {k}_{\perp }pg({k}_{\perp },p){}^{2}\), where g(k_{⊥}, p) is the Fourier–Hankel transform of g(R, v_{⊥}) concerning the guiding center position and the ion velocity^{20}, 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 ω ≪ Ω_{ci}, where ω and Ω_{ci} 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 ω ≪ Ω_{ci} (ω/2π ~ 0.1–4 kHz, Ω_{ci}/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 gyroradius, 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 k–v space
Figure 1 shows timeaveraged free energy spectra k_{⊥}g(k_{⊥}, v_{⊥})^{2} as functions of k_{x}ρ_{thi} and v_{⊥} up to ~2v_{thi} for (Fig. 1a) k_{drive}ρ_{thi} ~ 1 (W_{g1} injection at large scales) and (Fig. 1b) k_{drive}ρ_{thi} ~ 10 (W_{g1} injection at small scales), respectively. We regard k_{x} ≈ 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.
k_{⊥}g(k_{⊥}, v_{⊥})^{2} for (A) k_{drive}ρ_{thi} ~ 1 exhibits, in the entire range of v_{⊥}, the large amplitude at k_{x}ρ_{thi} ~ 1−2 consistent with \({\tilde{\phi }}_{{{{{{\rm{f}}}}}}}(\,f,{k}_{y}){}^{2}\) measured by a finescale Langmuir probe (FSLP) for k_{drive}ρ_{thi} ~ 1 (Fig. 2a), which shows excitation of k_{x}ρ_{thi} ~ 1 components at f ~ 1.0−1.3 kHz. Accordingly, the low k_{⊥} components at k_{x}ρ_{thi} ~ 1–2 show variation in the v_{⊥}direction at the scale of ~v_{thi} in Fig. 1a. The amplitude of components for k_{x}ρ_{thi} > 2 decays in large k_{x} with finerscale variation in the v_{⊥}direction. On the other hand, k_{⊥}g(k_{⊥}, v_{⊥})^{2} for (B) k_{drive}ρ_{thi} ~ 10 exhibits significant power at k_{x}ρ_{thi} > 10. These k_{x}ρ_{thi} > 10 components exhibit structures in the v_{⊥}direction at subv_{thi} 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.
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, W_{g1}(k_{⊥}, p) and E_{g}(k_{⊥}), for varied k_{drive}ρ_{thi} approximately between 0 and 10, (the first column) k_{drive}ρ_{thi} ~ 1, (the second column) k_{drive}ρ_{thi} ~ 3−4, (the third column) k_{drive}ρ_{thi} ~ 0–4, and (the fourth column) k_{drive}ρ_{thi} ~ 10. These values of k_{drive}ρ_{thi} 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 k_{drive}ρ_{thi}, 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, k_{y}ρ_{thi} ~ 1 component at f ~ 1.3 kHz in Fig. 2a (k_{y}ρ_{thi} ~ 3–4 component at f ~ 1.7 kHz in Fig. 2b, and k_{y}ρ_{thi} ~ 10 component at f ~ 3–4 kHz in Fig. 2d), and the other bands showing scattered, statistical distribution in the k_{y}direction corresponding to turbulent cascade. Nonlinear dynamics exhibited in these shots are local because no nonlocal resonant wave–wave interactions occurred except for the k_{drive}ρ_{thi} ~ 0–4 state (the vertical array of Fig. 2c, g, k and o), which exhibits a continuous driving spectrum in the fk_{y}ρ_{thi} diagram and is a turbulent state originating from a nonlinear coherent state of drift waves called solitary drift waves^{31}.
The plots shown in the second row (Fig. 2e–h) are W_{g1}(k_{⊥}, p) (negative signed entropy) in the k_{⊥}p space, namely, position–velocity space (see “Evaluation of 1D spectrum of free energy W_{g1}^{1D}(k_{x}, p) and relationship with W_{g1}(k_{⊥}) and W_{g1}(p)” in Methods). In the following results of varied k_{drive}ρ_{thi}, a position of the drive moves along the diagonal in the k_{⊥}p space, as shown in Fig. 2e–h.
When W_{g1} is injected at large scales (Fig. 2e), in which a significant power of W_{g1} can be seen at the left bottom of the diagram at k_{⊥}ρ_{thi} ~ pv_{thi} ~ 1–4 corresponding to k_{drive}ρ_{thi}, W_{g1} broadly develops along the diagonal from the driving scale toward the upperright direction, reaching the top right corner around k_{⊥}ρ_{thi} ~ pv_{thi} ~ 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 Fjørtofttype relationship W_{g1}(k_{⊥}, p = k_{⊥}) = k_{⊥}E_{g}(k_{⊥}) (Eq. (7.14) in the article by Plunk et al.^{20}). Therefore, diagonal transport of W_{g1} in the k_{⊥}p space is inevitable when a dualcascade of W_{g1} and E_{g} occurs.
This W_{g1}(k_{⊥}, p) spectrum exhibits the generation of W_{g1} at finer scales than the drive. This indicates that the nonlinear phase mixing works, resulting in a forward cascade of W_{g1}. This cascading nature will be examined below by comparing their power laws obtained from the measurement and the theory.
As k_{drive}ρ_{thi} increased to ≈10, as shown in Fig. 2f–h, the significant W_{g1} part moved in the upperright direction along the diagonal line toward higher k_{⊥}ρ_{thi} ~ pv_{thi} locations corresponding to the increase in k_{drive}ρ_{thi}. In all the cases of k_{drive}ρ_{thi} (Fig. 2e–h), the spread of W_{g1}(k_{⊥}, p) along with the diagonal directions (between the bottom left and the upper right) is obvious. In these cases, the Fjørtofttype energetic transition^{25}, namely, a transition involving only diagonal components, can be realized.
In the third row of Fig. 2i–l), (crosses) onedimensional (1D) spectra W_{g1}^{1D} of the free energy as functions of k_{x}ρ_{thi} and (open squares) pspectra of W_{g1}(p), respectively, are shown. Note that when W_{g1}(k_{⊥}) ∝ k_{⊥}^{−α}, W_{g1}^{1D}(k_{x}) ∝ k_{x}^{−α} (see “Evaluation of 1D spectrum of free energy W_{g1}^{1D}(k_{x}, p) and relationship with W_{g1}(k_{⊥}) and W_{g1}(p)” in Methods). Respective k_{drive}ρ_{thi} ranges are shown by purple bars. The blue and red bars represent the power laws predicted by the gyrokinetic theory for (blue) the inversecascade range k_{⊥} < k_{drive} and (red) the forward cascade range k_{⊥} > k_{drive}, respectively^{20}. The respective power indices of W_{g1}^{1D}(k_{x}) and W_{g1}(p) evaluated from the leastsquare fitting to the measured data points are indicated by numbers with the standard errors (SE). The noise levels of W_{g1}^{1D}(k_{x}) and W_{g1}(p) estimated from a vacuum shot are included in Fig. 2i. W_{g1}(p) in Fig. 2i (injection at a large scale) shows powerlaw decay starting at k_{x}ρ_{thi} ~ k_{drive}ρ_{thi} ~ 1 up to the detection limit of k_{x}ρ_{thi} ~ pv_{thi} ~ 20, clearly indicating the existence of an inertial subrange of the entropy (free energy). The evaluated exponent for the measured W_{g1}(p), −1.0, approximately agrees with the theoretical value of −4/3 within 5 times SE. For smaller scales k_{x}ρ_{thi} > 3, W_{g1}^{1D}(k_{x}) in Fig. 2i shows a similar decaying tendency to W_{g1}(p), which is a strong indication of the dual selfsimilarity dynamics, although the discrepancy of W_{g1}^{1D}(k_{x}) from the theoretical exponent is larger than 25 times SE. Larger discrepancies between the theoretical exponents and the measured ones in W_{g1}^{1D}(k_{x}) than those in W_{g1}(p) are seen in all cases of k_{drive}.
As k_{drive}ρ_{thi} increases to 3–4 (Fig. 2j), W_{g1}(p) shows an inflection point at k_{x}ρ_{thi} ~ k_{drive}ρ_{thi} ~ 3–4. Correspondingly, W_{g1}^{1D}(k_{x}) follows this change in the spectrum. For an injection at small scales k_{drive}ρ_{thi} ~ 10 (Fig. 2l), both W_{g1}(p) and W_{g1}^{1D}(k_{x}) become independent of p and k_{x}, indicating good agreement with the theory^{20}. It should be noted that the decaying exponents of both W_{g1}^{1D}(k_{x}) and W_{g1}(p) at k_{x}ρ_{thi} ≫ 1 exhibit similar values in the respective cases of k_{drive}ρ_{thi} and similar variation for varied k_{drive}ρ_{thi} 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 simulation^{16,26}.
In the case of k_{drive}ρ_{thi} ~ 0–4 (Fig. 2k), despite the driving source in the target krange (i.e., noninertial range), strong k–p 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 W_{g1}(p) and W_{g1}(k_{⊥}) within ~1 − 25 times SE is supported by the E_{g}(k_{⊥}) spectra measured by FSLP shown in the fourth row of Fig. 2 (Fig. 2m–p). Note that when E_{g}(k_{⊥}) ∝ k_{⊥}^{−α}, E_{g1}^{1D}(k_{x}) ∝ k_{x}^{−α} (“Methods”: Evaluation of the 1D spectrum of the electrostatic field energy E_{g}^{1D}(k_{y}) and its relationship with E_{g}(k_{⊥})). The blue and red bars represent the power laws predicted by kinetic theory for (blue) k_{⊥} < k_{drive} and (red) k_{⊥} > k_{drive}, respectively. The respective power indices of E_{g1}^{1D}(k_{x}) evaluated from the leastsquare fitting to the measured data points are indicated aside by numbers with the standard errors. The noise levels of E_{g1}^{1D}(k_{x}) estimated from a vacuum shot are included in Fig. 2m–p. The evaluated scaling for E_{g1}^{1D}(k_{x}) in Fig. 2m is k_{x}^{−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 k_{drive}ρ_{thi} increases. The exponent evaluated for the measured E_{g1}^{1D}(k_{x}) in Fig. 2p is k_{x}^{−1.8}. The observed behavior of these power laws in W_{g1} and E_{g} is consistent with the theory based on the hypothesis of dual selfsimilarity and the dual cascade of the entropy and field energy of the gyrokinetic system^{20}. Our result strongly supports the validity of the dualself similar hypothesis of gyrokinetic turbulence.
Note that the nonlinear decorrelation time^{16,26} for these states is evaluated as τ_{ρ} ~ 10 × 10^{−5} s; hence, the dimensionless parameter^{16,26}, which represents the scale separation analogous to Reynolds number, D ~ 30–40. Accordingly, the cutoff wavenumber, above which the collisional dissipation dominates the cascade, k_{cut}ρ_{thi} ~ 2D^{3/5} ~ 20. This guarantees that the observed powerlaw 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 works^{15,18,23}. 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 ω ≪ Ω_{ci} 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 heating^{32,33}. According to ref. ^{34}, the amplitude threshold for strong stochastic heating when β < 1 can be expressed in terms of the quantity ε_{i} ∼ q_{i}δΦ_{i}/mv^{2}, where β, q_{i}, and δΦ_{i} 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 k_{⊥}ρ_{thi} ∼ 1, respectively. In our case, ε_{i} ∼ 0.2; therefore, the influence of stochastic ion heating is considered to be small. Ion cyclotron resonance heating is negligible because ω ⋡ Ω_{ci}.
One of the weakest points of the gyrokinetic theory is the use of a Maxwellian as F_{0}^{15}. 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 F_{0} from a Maxwellian distribution (Fig. 3d in “Methods” shows an example of the velocity distribution F_{0}), we believe that F_{0} 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 meanfree 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.
The influence of the electron behavior on the entropy cascade in the ion phase space is considered negligible according to gyrokinetic numerical simulations^{4,16,23}. The work done by Tatsuno et al. showed that no essential difference was observed between the cases of applying Boltzmannresponse (3D) and noresponse (2D) electrons in their simulations^{4}. 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.
Conclusions
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 selfsimilar 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 W_{g1} with k_{⊥} ∼ p in the phase space gives evidence of nonlinear phase mixing at subLarmor scales. This observation corroborates the evidence of the existence of dual selfsimilarity and the dualcascade 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 2D^{16} and 3D electrostatic^{21}, 3D electromagnetic turbulence, including the Alfvenic solar wind^{4,22}, and a simulation targeting an Alfvenic hot accretion disk^{23}. The essential physical components in entropy cascades via nonlinear phase mixing include the gyro motion of the particles at the subLarmor scales and the interaction between the particles and the electrostatic fluctuations in addition to the strong anisotropy (k_{⊥} ≫ k_{}). 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 driftkinetic 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.
Methods
Magnetized plasma experiment (MPX) device: experimental device
The experiment was conducted in the MPX device^{35} (Fig. 3a), which can prepare qualified plasma configurations for our purposes, namely, quasi2D configurations with the magnetic field (the strength is denoted by B_{0}), 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 zdirections, respectively (Fig. 3a), to obtain perpendicular wavenumbers k_{⊥} of g(R, v_{⊥}) with the use of a twopoint correlation technique^{36}. 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 ringaveraged ion velocity distribution functions F_{0} for B_{0} = 0.022 T and B_{0} = 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 B_{0} change from 0.022 T to 0.033 T, indicating the validity of the RIDFP measurement.
Ringaveraged ion velocity distribution function probes^{28}
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 follower^{28}.
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_{}f_{ci}^{−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 pspace Δ(pv_{thi}) can be evaluated as follows: Δpv_{upper_bound} ~ Δp(2v_{thi}) = j_{0} ≒ 2.4, ∴Δp ≒ 1.2/v_{thi} ∴Δ(pv_{thi}) ~ 1.2, where v_{upper_bound} and j_{0} 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 (pv_{thi}) in the RIDFP measurement is obtained as: (pv_{thi})_{max} ~ j_{0}v_{thi}/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 interferometer^{37}. 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 finescale LP (FSLP) was applied to measure the fluctuation of floating potential \({\tilde{\phi }}_{f}\approx \tilde{\phi }\) as functions of frequencies f and k_{y} (k_{x}), whose upper bound of measurable k_{⊥} is ~8 × 10^{2 }m^{−1}.
The hot cathode generated plasmas with the introduction of argon gas as the working gas. The typical plasma density and temperature are n_{e} ~ 0.5–5 × 10^{16 }m^{−3} and T_{e} ~ 1–5 eV, respectively, with a 15–20 cm diameter. Accordingly, the characteristic frequencies of the ion cyclotron f_{ci}, driftwave f_{DW}, ion–ion Coulomb collisions f_{ii}, and ionneutral collisions f_{in} have the following numbers:
f_{ci} := Ω_{ci}/2π ≈ 8.4 kHz > f_{DW} ~ 1–4 kHz ~ f_{ii} ~ 1–5 kHz > f_{in} ~ 0.1–0.5 kHz, where Ω_{ci} 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)
Application of the Hankel transform concerning ion velocity v_{⊥} to g(R_{0}, v_{⊥}, t) and g(R_{0} + Δx, v_{⊥}, t) measured by RIDFP1 and RIDFP2 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(R_{0} + Δx, p, t), where R_{0} and R_{0} + Δx are the positions of the guiding center of RIDFP1 and RIDFP2, respectively.

(2)
Fourier transforms of g(R_{0}, p, t) and g(R_{0} + Δx, p, t) about time t are applied to obtain g(R_{0}, p, f, θ) and g(R_{0} + Δx, p, f, θ), where θ represents the phase of the component of the frequency f.

(3)
Wavenumbers k_{x} for all pcomponents are calculated by applying the twopoint correlation technique to g(R_{0}, p, f, θ) and g(R_{0} + 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 θ_{cut} for the upper bound of the integration, one can obtain
When W_{g1}(k_{⊥}) = ∫k_{⊥}g(k_{⊥}, p)^{2}dp ∝ k_{⊥}^{−α}, W_{g1}^{1D}(k_{x}) = ∫W_{g1}^{1D}(k_{x}, p)dp ∝ k_{x}^{−α}.
Then,
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 θ_{cut} for the upper bound of the integration, one can obtain
When E_{g}(k_{⊥}) ∝ k_{⊥}^{−α}, E_{g}^{1D}(k_{y}) = ∫E_{g}(k_{⊥})dk_{x} ∝ k_{y}^{−α}.
Data availability
The data supporting this study’s findings are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by GrantsinAid MOST 1112112M006025 from the Ministry of Science and Technology, Taiwan.
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E.K. designed the study and approaches. Y.L. designed the upgraded version of RIDFPs. Y.L. and E.K. performed the optimization experiments of the RIDFPs. E.K. and Y.L. developed the diagnostic tools. E.K. performed the experiments, measurements and analysis. E.K. and Y.L. wrote the paper. E.K. supervised the study.
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Kawamori, E., Lin, YT. Evidence of entropy cascade in collisionless magnetized plasma turbulence. Commun Phys 5, 338 (2022). https://doi.org/10.1038/s42005022011157
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DOI: https://doi.org/10.1038/s42005022011157
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