紊流態是指流體或電漿中混沌且無序的流場狀態。儘管各物理系統的紊流態性質似乎不盡相同，但各流體系統中的紊流態存在著某種普遍性。這一普遍性是由Kolmogorov的普遍平衡理論所提出和驗證，這一理論成名於它對等向三維紊流能量譜所提出的-5⁄3能量串級尺度率。此一能量尺度率揭示了在紊流中能量由大尺度串跌至小尺度的現象。同樣地，類似的普遍性和尺度率關係在磁化電漿紊流態或稱迴旋動力電漿紊流中提出。此電漿紊流態電位震盪的時間和空間尺度皆小於離子迴旋半徑和頻率。此迴旋動力紊流態中的普遍性體現於離子於相空間的熵串級現象。這一相空間和實空間彼此統合的特性是迴旋動力紊流態中所獨有的，有別於一般的中性流體紊流。迴旋動力理論對迴旋動力紊流態中的能量譜和熵譜各自的尺度率提出預測。這些能量譜和熵譜是利用離子迴旋速度分布函數上的震盪 g(R,v_⊥ ) 所計算而得，其中 R 代表離子迴旋中心，v_⊥ 代表離子垂直於磁場方向的速度分量。本研究的目的在於透過直接量測實驗磁化電漿相空間中的熵，來驗證迴旋動力理論。
基於此目的，一種稱為離子迴旋速度分布函數探針的量測儀器被開發出來以量測 g(R,v_⊥ )。此探針利用篩選軌道的方式來量測速度分布函數。早期的探針訊號受到下列雜訊來源影響，1) 探針固有電容訊號，2) 類Langmuir探針訊號，3) 高能電子訊號，4) 類自由離子訊號。探針固有電容訊號藉由隔離電流-電壓轉換電路和數據收集系統來消除。類Langmuir探針訊號藉由在探針中加裝不銹鋼網來消除。高能電子訊號藉由探針外加裝不銹鋼網和鋁板來消除。類自由離子訊號藉由延長探針軌道長度來消除。藉由以上方法探針訊照比得到改善。改善後探針所量測的速度分布函數呈現 Maxwellian 的輪廓，其中所量測的離子溫度為 0.1 eV。此探針已被用於量測三種不同的電漿態，分別是 線性飄移波態 (linear drift-wave state, LDW)，非線性飄移波態 (non-linear drift-wave state, NDW) 和 飄移波紊流態 (drift wave turbulence state, DWT)。此實驗藉由量測相空間的熵 E ̂_g (k,p) 來檢視熵於實驗電漿相空間的傳播，其中熵 E ̂_g (k,p) 正比於 g(R,v_⊥ ) 於相空間的能量譜。在不同電漿態中不同的熵傳播模式已被觀測。在NDW態中，熵 E ̂_g (k,p) 於相空間呈現局部分布。在DWT態中，熵 E ̂_g (k,p) 在相空間中呈現散布的狀態，且延伸至實空間和速度空間尺度至 kρ_thi ~ 8 和 pv_thi ~ 6，其中 ρ_thi 為離子熱速度的迴旋半徑，v_thi 為離子熱速度。迴旋動力理論所預測的尺度率 E_ϕ (k) ~ k^(-10⁄3) 和 E_g (k) ~ k^(-4⁄3) 在DWT態中被觀測到，其中 E_ϕ (k) 為電漿電位的能量譜，E_g (k) 為熵在實空間的能量譜。這些量測結果部分驗證迴旋動力理論的預測，但在速度空間所量到的熵尺度率不符合理論預測 E_g (p) ~ p^(-4⁄3)。為了取得更確切的結論，進一步改善探針的訊照比是必須的。

Turbulence is referred to as chaotic and disordered flow/field of fluids and plasmas. Although properties of turbulence are seemingly situation-dependent, a certain kind of “universality” exists in various turbulence systems. This universality was proposed and in part revealed by Kolmogorov’s universal equilibrium theory famous for the -5⁄3 power law of turbulence energy spectrum in isotropic three-dimensional turbulence, which indicates an energy cascade process from large scales to small viscous scales. Similarly, a certain kind of universality was proposed for magnetized plasma turbulence called gyro-kinetic plasma turbulence, whose length and time scales of potential fluctuations are sub-ion gyro-radius and sub-ion gyro-frequency, respectively. The universality in gyro-kinetic turbulence proposed is entropy cascade in phase space of ions. This synthesis of configuration space and velocity space is remarkable difference from turbulence of neutral fluids. The theory predicts several power laws in power spectra of entropy and energy of gyro-kinetic turbulence. These power spectra are evaluated with the use of “fluctuations” of ion velocity distribution function at a fixed guiding center position g(R,v_⊥), where R and v_⊥ are the guiding center position and ion velocity component perpendicular to the background magnetic field, respectively. The purpose of this research is verification of the gyro-kinetic turbulence theory by direct measurement of entropy in phase space in laboratory magnetized plasmas.
To this end, we developed a measurement tool of g(R,v_⊥) called Ring-Ion Distribution Function Probe (RIDFP), which detects different velocity components of ions by selection of gyro-orbits. The early version of RIDFPs had serious problems in quality of signals. These stemmed from the following four spurious components: a) intrinsic capacitive noise, b) Langmuir probe-like signal, c) high energy electrons signal and d) free-stream like ions signal. Intrinsic capacitive noise is suppressed by isolating current-voltage converter circuits from the data acquisition system to prevent capacitive couplings. Langmuir probe-like signal is suppressed by installing mesh grids inside RIDFPs to capture electrons. High energy electron signal is suppressed by addition of entrance mesh grids and metal plates to prevent these electrons from entering into RIDFPs. Free-stream ion signal is suppressed by extension of the length of orbit filters. As a result of application of these measures, signal quality of RIDFPs are significantly improved. The resultant equilibrium distribution functions measured by RIDFPs display Maxwellian-like profiles with the ion temperature on the order of 0.1 [eV].
The developed RIDFPs have been applied to three states of magnetized plasmas, linear drift wave-excited (LDW) states, non-linear drift wave coherent state (NDW) and drift wave turbulence (DWT) states, respectively. Entropy transfer in MPX plasma was examined by measuring entropy in the phase space E ̂_g (k,p) (k : real space wavenumber, p : velocity space wavenumber) which is proportional to the power spectrum of g(R,v_⊥) in phase space (i.e. ∝ |g(k,p)|^2). A transition in E ̂_g (k,p) was observed between the three states. In the NDW states, E ̂_g (k,p) was localized in phase space. The localized position corresponds to the scale of the linear drift waves. In DWT states, E ̂_g (k,p) was broadened in the phase space, and extended to finer scales up to kρ_thi ~ 8 and pv_thi ~ 6, where ρ_thi and v_thi are the ion gyro-radius at thermal velocity and the thermal velocity, respectively. Power laws E_ϕ (k) ~ k^(-10⁄3),E_g (k) ~ k^(-4⁄3), where E_ϕ (k) and E_g (k) are power spectra of plasma potential and distribution functions, respectively, predicted by the gyro-kinetic theory were verified in the drift wave turbulence states. These results indicate validity of the hypothesis of entropy cascade in gyro-kinetic turbulence. Further improvement in signal-to-noise ratio is necessary to draw a firm conclusion. The measured power spectrum of entropy in velocity space, however, was much steeper than the predicted spectrum E_g (p) ~ p^(-4⁄3). Although yet clarified, one possible explanation for this discrepancy is a difference in the turbulence settings between the theory and our experiment, decaying turbulence and driven turbulence.

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