Eddy, drift wave and zonal flow dynamics in a linear magnetized plasma

Turbulence and its structure formation are universal in neutral fluids and in plasmas. Turbulence annihilates global structures but can organize flows and eddies. The mutual-interactions between flow and the eddy give basic insights into the understanding of non-equilibrium and nonlinear interaction by turbulence. In fusion plasma, clarifying structure formation by Drift-wave turbulence, driven by density gradients in magnetized plasma, is an important issue. Here, a new mutual-interaction among eddy, drift wave and flow in magnetized plasma is discovered. A two-dimensional solitary eddy, which is a perturbation with circumnavigating motion localized radially and azimuthally, is transiently organized in a drift wave – zonal flow (azimuthally symmetric band-like shear flows) system. The excitation of the eddy is synchronized with zonal perturbation. The organization of the eddy has substantial impact on the acceleration of zonal flow.

Observations of the solitary eddy in nonlinear drift wave -zonal flow system. Figure 1(a) shows the spatio-temporal evolution of I is measured by an azimuthal probe array (r = 4 cm). A wave coherently propagates in the electron diamagnetic direction. The duration of the propagation is much longer than the period of the drift wave. This wave is called as a nonlinear drift wave. The mechanism to form triangular wave form of drift wave was explained in ref. 24. Although the nonlinear drift wave is stationary, its wave form slightly varies as it propagates. To observe details of temporal changes of azimuthal structure of the nonlinear drift wave, we reconstructed I is in the 'mode frame' of the nonlinear drift wave, as shown in Fig. 1(b). Here, the azimuthal phase velocity of nonlinear drift wave was subtracted; the azimuthal angle θ is transformed to θ′ = θ − v p t, where v p indicates the phase velocity of the nonlinear drift wave (2.4 × 10 3 π rad/s). The azimuthal structure of the nonlinear drift wave shows clear variation. In addition, a small sub-structure is found around θ′ = 0.7-1 rad/2π in Fig. 1(b). The sub-structure consists of two components with different phase velocities (or frequencies) in laboratory frame (i.e., including E × B velocity). One propagates at 4 × 10 3 π rad/s (pink colored arrows on Fig. 1(b)) and this component is called as 'splash' (I is,SP ) 25 . The splash has higher frequency than that of the fundamental drift waves and hence has a short life time. The other propagates at the velocity of 2.6 × 10 3 π rad/s (yellow colored arrow on Fig. 1(b)) and this component is density bump associated with 'solitary eddy' (discussed later). Figure 1(c) shows the temporal evolution of the sub-structure (slice at θ′ = 0.94/2π rad of Fig. 1(b)). To separate the splash and the density bump associated with solitary eddy from the sub-structure, high-pass (> 1 kHz) and low-pass (< 1 kHz) filters were applied to the signal in Fig. 1(c). The splash and the density bump were amplitude-modulated at ~0.4 kHz simultaneously. The temporal evolution of the floating potential fluctuation (0.1-1 kHz) at r = 4 cm of movable probe is shown in Fig. 1(d). A previous work indicated that the excitation of splash is synchronized with the periodic evolution of the zonal perturbation (~0.4 kHz) 25 . In this experiment, in addition to splash, the solitary eddy is observed to form and to interact with zonal perturbation. Behind the nonlinear drift wave, we can see a closed isoline of δV f , which is emphasized by a purple solid circle. This indicates the organization of a solitary (radially and azimuthally localized) eddy, since isolines of potential is equivalent to lines of flow in magnetized plasmas. The time-to-peak of the potential perturbation (solitary eddy) was shifted from that of the density bump. Figure 2(b) shows the reconstructed two-dimensional structure of the vorticity, ∇ × v ( ) z , and δV f at τ = 0, where filled contours denote ∇ × v ( ) z and contour lines denote δV f . The vorticity peak in the purple solid circle also indicates the organization of the solitary eddy.

Cross-section image of the solitary eddy and interactions with drift wave and zonal flow.
The spatio-temporal structure of the solitary eddy is closely related to that of the zonal flow as shown in Fig. 2(c-f). The organized solitary eddy shows spatial and temporal asymmetry with zonal flow. The temporal behavior of azimuthal zonal flow, V ZF , is evaluated from the radial electric field with θ τ V r ( , , ) f . The evaluated V ZF structure clearly demonstrates the formation of zonal flow in the region r = 2-6 cm as shown in Fig. 2(c). The positive/negative sign of V ZF denotes the electron/ion diamagnetic direction. Here, the azimuthal mean flow velocity is ~2.5 km/s at r = 3.75 cm in the electron diamagnetic direction. The solitary eddy is organized around an inner antinode of the zonal flow (hatched region of 2.5-4.5 cm). No eddy is observed in the outer antinode of the zonal flow. In this sense, the generation of eddy has a symmetry breaking with respect to the phase of zonal flow. Therefore, the eddy is not excited by the zonal flow shear. Note that the maximum velocity around the solitary eddy is ~1 km/s and is much faster than zonal flow (~0.1 km/s). The solitary eddy is organized quasi-periodically, synchronized with zonal flow, and it strengthens the electron diamagnetic flow. Figure 3(a) shows the temporal behavior of the Reynolds stress per mass density, volume-averaged around the azimuthal location of the solitary eddy, which is evaluated using π < > = θ r The dotted square indicates the excited time and radius of the solitary eddy (τ = − 0.7 ms-0.5 ms, r = 2.5-4.5 cm). The Reynolds stress is generated at the radial location where the eddy and zonal flow exist as shown in Fig. 3(b). The Reynolds force, which accelerates/decelerates azimuthal flow is calculated from the radial gradient of Reynolds stress, π −∂ < > θ r r / r r . The temporal behavior of Reynolds force is shown in Fig. 3(c), where a positive sign indicates the electron diamagnetic direction. Figure 3(e) shows the fluctuation component of Reynolds force associated with eddy (− 0.7 < τ < 0.5 ms) and drift wave (m = 1) volume-averaged over the radial location of the eddy (r = 2.5-4.5 cm). Here, the solid red line denotes the eddy-driven component and dotted line means the bandpass filtered (0.1-1 kHz) data. The black lines indicate the Reynolds force driven by drift wave (m = 1). Although the noise components are large, the filtered data indicates that the eddy accelerates the azimuthal flow toward electron diamagnetic direction which is synchronized with acceleration of the zonal flow as shown in Fig. 2(d). The drift wave also contributes to azimuthal flow which accelerates toward electron (or ion) diamagnetic direction at τ < 0 ms (or τ > 0 ms).
The effect of the solitary eddy on the zonal flow is substantially large. The dynamic behavior of zonal flow, explained above, cannot be explained by the Reynolds force driven by the drift wave. The phase of acceleration of the zonal flow is delayed from the Reynolds force driven by the drift wave as shown in Figs 2(d) and 3(e). In addition, the magnitude of Reynolds force driven by the eddy has the same order as that driven by the drift wave. Therefore, Reynolds force driven by the solitary eddy can, not only accelerates flow by the same order of magnitude as drift waves, but also governs the dynamics of zonal flow acceleration, as shown by the time-to-peak comparison in Fig. 3(e). The solitary eddy thus plays an important role on excitation and saturation process of zonal flow.
The energy balance among solitary eddy, zonal flow and drift wave is evaluated. For each entity, the energy per unit volume is calculated from squared flow velocity evaluated by floating potential. The energy density of solitary eddy and drift wave are ~10 −2 Jm −3 ; the energy of zonal flow is one order of magnitude lower than these of solitary eddy and drift wave. Taking into account that the eddy has small volume, the energy partition among drift wave, solitary eddy and zonal flow is approximately 10:1:1.

Discussion
Our result suggests a new channel of energy and momentum transfer from the drift wave to zonal flow via solitary eddy. In this channel, the energy density of the eddy is 1/10 of that of drift wave but the eddy efficiently transfer its momentum to the zonal flow and thus momentum transferred to the zonal flow from the eddy is the same as that from drift wave. Thus, the role of eddy on the momentum transport is significant. The experimental discovery of this new channel indicates the need for a novel picture of plasma turbulence, in which wave, flow and eddy coexist and interact with each other. The role of eddies in drift wave -zonal flow system has also been discussed theoretically 26 . The result presented here will open the way to enhance the zonal flow in fusion plasmas. This new picture is also beneficial in research of the non-equilibrium non-linear systems.

Methods
Large Mirror Device-Upgrade (LMD-U). The LMD-U 18 is a linear magnetized plasma device; The cylindrical plasma with the diameter of approximately 0.1 m and the axial length of 3.74 m is produced by the RF wave in a quartz tube and is radially bounded by the magnetic field. The vacuum chamber of the LMD-U is made of the stainless steel with the diameter of 0.445 m and the axial length of 3.74 m. Neutral gas (argon) is filled from the plasma source. The ionized vacuum indicator is mounted on the source to monitor the pressure of neutral gas. The mass flow controller controls the amount of neutral gas flow. Four turbo-molecular pumps are installed in nearly middle and end regions of the LMD-U. The total exhaust velocity is 1000 l/s. Those pumps evacuate the background neutral pressure (< 10 −3 Pa in this study) and reduce impurity gases ( . . e g N 2 and H 2 O gases in the atmosphere). The baffle plates with the inner diameter of 150 mm are installed at the source and end region to stabilize the gas flow and to improve the control. The coil system in the LMD-U produces a linear magnetic field configuration. Strength of the magnetic field is controlled by the coil current. The coil current is excited stationary during the discharge. The LMD-U plasma is produced by helicon wave heating. The helicon wave (m = 0,