A Concept of Cross-Ferroic Plasma Turbulence

The variety of scalar and vector fields in laboratory and nature plasmas is formed by plasma turbulence. Drift-wave fluctuations, driven by density gradients in magnetized plasmas, are known to relax the density gradient while they can generate flows. On the other hand, the sheared flow in the direction of magnetic fields causes Kelvin-Helmholtz type instabilities, which mix particle and momentum. These different types of fluctuations coexist in laboratory and nature, so that the multiple mechanisms for structural formation exist in extremely non-equilibrium plasmas. Here we report the discovery of a new order in plasma turbulence, in which chained structure formation is realized by cross-interaction between inhomogeneities of scalar and vector fields. The concept of cross-ferroic turbulence is developed, and the causal relation in the multiple mechanisms behind structural formation is identified, by measuring the relaxation rate and dissipation power caused by the complex turbulence-driven flux.

corresponds to the electron diamagnetic direction, is defined as right-hand direction of z. A helicon plasma source is located at one side of a vessel, where a double loop antenna on a quartz tube (radius 5 cm, located at z = 0 m) is installed. Argon gas at 0.1 Pa is fed in the quartz tube and is ionized by 2.7 kW of 7 MHz rf waves from an antenna on the tube. On the other side of the vessel, the plasma is terminated by an endplate (stainless steel). This boundary condition (source at one end and metal plate at the other end) induces a plasma flow and weak gradient of electron pressure in the direction of the magnetic field, and determines the axial mode structure of the drift wave. In the region of z < 2.5 m, the cross-field transport by turbulence is dominant loss channel of plasma, and the plasma weakly varies along the field line. Typical plasma parameters are as follows: plasma radius a ~ 5 cm, plasma length L ~ 4 m, electron density 〈 n e 〉 ~ 1 × 10 19 m −3 , electron temperature 〈 T e 〉 ~ 3 eV, and ion temperature 〈 T i 〉 ~ 0.3 eV, where 〈 〉 denotes long time average. A 4-tips probe is used to measure scalar-fields (n e , T e ) and flow vector-fields (V r , V θ , V z ) (see 'Methods'). The fluctuation component of a general quantity A is defined as The axial location of the probe can be translocated to z = 0.625, 1.125, 1.625, 2.625, 3.125, 3.375 m in a shot by shot manner. Figure 1(a) illustrates the density profile and time-averaged plasma flow (arrows) on the r-z plane. Figure 1(b,c) show radial profiles of density and axial flow at two axial locations, z A = 0.625 m and z B = 3.375 m. A striking observation is that the particle flux is inward (up-gradient) and the axial flow is reversed in some area (e.g., near center at z = z A ).
These complex behaviors of flow are induced by fluctuations. The spatiotemporal structure of fluctuations in the PANTA plasma is measured by a 64-channel probe array 12 . Figure 2 indicates that the frequencies of the strongest coherent peaks in the power spectrum of ∼ V r at r = 4 cm are 2.8 kHz (which satisfies the dispersion relation of drift waves) and 1.2 kHz (a mediator of the streamer 12 ). In contrast, at r = 2 cm, the peak of spectrum appears at 6.5 kHz, and the higher frequency broadband modes are enhanced compared to that at r = 4 cm. The peak at 6.5 kHz is considered to be a D' Angelo mode (the details are discussed later).  The momentum and particle transport driven by fluctuations are shown in Fig. 3 (at z = z A ). Here Reynolds stress is calculated as 13 , where ≡ n n e e . The first, second and third terms are shown in Fig. 3(a). The total Reynolds stress has a dip around r = 2.5-3 cm. Thus the opposite of the divergence of Reynolds stress, − (r −1 ∂ (rΠ rz )/∂ r), i.e. the driving force of the flow, is negative in the region of r = 3-4 cm, where the inversion of axial flow is observed. The four terms in the force balance equation, , (where p is total pressure, m i is the ion mass, ν in is the ion-neutral collision frequency) are shown in Fig. 3(b). The inertia term V z ∂ V z /∂ z is negligibly small compared to the other terms in the region of measurement ( Fig. 1(a)). The parallel pressure gradient term is also small at this location (z = z A ). Hence the dominant drive of the flow is the Reynolds stress term, which balances with neutral drag to determine V z . We substitute a neutral drag coefficient ν in = 40 kHz, which is assumed to be constant over the plasma column and is evaluated based on neutral pressure measurement with an ionization gauge 14 . The V z estimated by force balance (dotted line in Fig. 3(c)) is consistent with the observed steep gradient and inversion of flow near the center (r = 2 ~ 3 cm). This inversion of axial flow in Fig. 1(c) is caused by the Reynolds stress of drift-wave fluctuations. Drift-waves are excited by radial density gradient and drive an outward particle flux in the outer portion of plasma (r > 2.5 cm) as shown in Fig. 3(d). The outward flux relaxes the density gradient. On the other hand, an inward particle flux is generated at r = 2 cm by D' Angelo mode, which is excited by the steep gradient of velocity near the center. The inward flux steepens the density profile in the central region. This is a cross-ferroic structure formation via plasma turbulence, in the sense that different fields (density and velocity) and corresponding fluxes (particle and momentum) cross-interact.
The force balance is also evaluated at z = 1.123 m. The Reynolds force plays a dominant role again in the flow-drive terms. In the region (z < 2 m), the turbulence driven-radial flux is dominant over the loss along the field line as the relaxation channel of particle/mass. The calculation is compared with observations as shown in Fig. 4. There is small discrepancy (off-set) in the magnitude between the observation and calculation. This originates from uncertainty of the estimation of density profile of neutral particle in the model. Although there is uncertainty in the evaluation of neutral drag term, the calculated flow is consistent with the observation qualitatively in that: 1) Strong axial flow and shear layer is formed in the central region.
2) The dip of the axial flow is located at around r = 3 cm. We thus stress that the picture of structure formation by turbulence holds in wider region along the field line.
This chained structure formation can be understood quantitatively by analyzing the stability condition of D' Angelo 6 . The instability condition is obtained via a fluid model as Here, c s is the ion sound speed, ρ s is the ion sound Larmor radius, ω *e is the drift wave frequency, 15 . Figure 5 shows the stability diagram for experimental parameters, ρ = θ k 1 s . D' Angelo mode can be unstable in inner (2-3 cm) regions close to the source region (z = 0.625, 1.125 m). When the velocity profile is monotonic and the gradient is weak (as is the case at z = 3.375 m in Fig. 1(c)), the D' Angelo mode is linearly stable. When the drift-wave and D' Angelo mode fluctuations coexist, the interference of particle and momentum fluxes takes place. The result of quasi-linear theory for particle and momentum fluxes is:    shear dominates the fluctuations. In turn, while releasing free energy stored in ∇ n e , the drift waves exert Reynolds stress to drive macroscopic flows. The momentum flow by ∇ n e -driven drift-wave turbulence has been discussed in the literature 13,16,17 . However, the present observation indicates, for the first time, interferences between fluxes of particle and axial momentum. We found a chained interaction in the observed turbulence. That is, the drift wave induces momentum flux, which reverses the flow direction. The enhanced velocity gradient excites the D' Angelo mode, which is stable without flow reversal (for the present parameters). D' Angelo mode causes an up-hill particle flux near the axis.

Discussion
The primary origin in this chained structural formation is examined via the rate of dissipations 18,19 . Each energy relaxation rate W j associated with the j-th turbulence-driven flow V j (j = r, θ, z), is evaluated from the volume integral of W j = F j V j , where F is the force density, F = (− ∂ p e /∂ r, − n i m i ∂ (rΠ rθ )/r∂ r, − n i m i ∂ (rΠ rz )/r∂ r). The energy relaxation rates (W r , W θ , W z ) are calculated as (76 W, 0.16 W, 11 W), respectively. These values are obtained by integrating the power density over a volume (0.625 m < z < 3.375 m, 2 cm < r < 6 cm, 0 < θ < 2π ). The dissipation power of fluctuation energy, W turb, is evaluated by the volume integral of τ n m V /2 i i 2 corr . Here the correlation time of turbulence is estimated as τ corr ~ 1 ms from Fig. 2, and W turb is estimated as 4.1 W. From these evaluations, we find that The power W r is dominant and thus the main relaxation channel of turbulent processes is the decay of density profile via radial flow.
In conclusion, in cross-ferroic turbulence, we analyzed the turbulent structural formation of scalar and vector fields of extremely nonequilibrium magnetized plasma. The results show elements in cross-ferroic turbulence: (i) Scalar-fields and vector-fields spontaneously produce global-scale orders. (ii) Both fields interfere with each other, yielding a chained structure in a selected region of plasma. (iii) The dominant cause can be detected by evaluating the non-linear relaxation rate. Namely, the drift-wave fluctuations drive momentum transport, which generates the inversion and strong shear of axial flow in the central region. The enhanced gradient of flow velocity, in turn, drives the D' Angelo mode, which causes particle pinch and collimates the plasma density in the central region.

4-tips probe measurement.
Two tips are aligned in the axial direction, to measure the axial velocity, 〈 V z 〉 and Ṽ z , with Mach probe technique. The other two are coordinated in the azimuthal direction for radial velocity measurement (〈 V r 〉 and Ṽ ) r , through the difference of floating potentials, φ f . The axial flow velocity is evaluated as V z = c s (2T e /T i ) 1/2 (I u − I d )/(I u + I d ) 20 , where I u and I d denote ion saturation currents in the upstream and downstream side, respectively. An alternative model to evaluate the flow velocity yields identical results. The upstream tip and downstream tip are exchanged in a shot-by-shot manner to balance the small individual variability of tips. The error bars in the Reynolds force term in Fig. 3 are originated from the asymmetry of the tips. Double-probe method is also applied by using a pair of tips, where 〈 T e 〉 and 〈 n e 〉 are evaluated by using similar discharges. The ion temperature of 〈 T i 〉 ~ 0.3 eV is measured by use of an ion sensitive probe. The long time-averaged potential 〈 φ〉 is estimated from the floating potential 〈 φ f 〉 , while considering the impact of 〈 T e 〉 . Relative density fluctuations are provided by relative ion saturation current fluctuations, as =   n n I I / / e e is is . The temperature fluctuation amplitude  T T ( / ) e e of the drift-wave is evaluated as ~10% of  n n / e e and φ  e T / f e by using a triple-probe method in PANTA 21 . In this study, the temperature fluctuation is not monitored but taken into account in the measurement error for evaluating the level of density and flow velocity fluctuations. The measurement errors by temperature fluctuation are estimated to be at most 5% for  n e and 50% for φ  f , respectively. The probe head is movable (r > 2 cm), without disturbing plasma turbulence.