湍流廣泛存在於自然與工程流動中，其跨尺度耦合使瞬時流場呈現強烈波動且難 以預測。二維湍流與三維湍流具有根本上的差異，在理想二維不可壓縮流中，渦旋 拉伸因二維幾何限制而消失，使渦度擬能成為動能之外的另一個無黏守恆量，並形 成動能向大尺度傳輸與渦度擬能向小尺度傳輸的雙重級聯。過去研究多以能譜、平 均通量與結構函數研究渦度擬能的正向級聯，但這些全域統計量無法保留局部傳輸 事件的空間分布與組織方式。本論文以電磁力驅動薄層電解液產生準二維湍流，利 用粒子影像測速法重建速度場並計算渦度場，再透過空間粗粒化方法解析不同濾波 尺度下的局部渦度擬能傳輸。 在驅動尺度以下，平均局部渦度擬能傳輸為正，表示淨傳輸朝向較小尺度，但正 向與反向事件在所有量測尺度均同時存在，強烈正向傳輸主要出現在濾波渦度梯度 幅值與次濾波尺度渦度傳輸強度均較大，且兩者呈反向對齊的區域。拓樸條件統計 顯示，相較於渦旋區域，鞍點區域具有較高的平均傳輸與較大的傳輸波動，兩者的 主要差異集中於高渦度梯度與強次濾波傳輸事件，其中鞍點區域的傳輸與梯度排列 更偏向有利於正向傳輸的反向對齊。受 Kolmogorov refined similarity hypothesis 啟 發，以濾波尺度局部渦度擬能耗散進行條件化統計後發現，高耗散區域呈現較大的 正向平均傳輸、更寬的正負分布尾端及較高的正向事件機率。 本研究進一步建構一個無因次類熵產生的物理量，用以量化局部正向與反向傳輸 的機率不對稱，在高渦度擬能耗散區域，正負類熵產生事件的對數機率比在有限的 可解析範圍內接近斜率為一的漲落關係，且此行為在所有量測的驅動尺度以下範圍 均可辨識。 這些結果顯示，渦度擬能正向級聯是一個局部、間歇且具統計不對稱性的過程。 局部傳輸的大小與方向由濾波渦度梯度、次濾波尺度渦度傳輸及其相對對齊共同決 定，而流場拓樸與局部渦度擬能耗散進一步組織正向與反向事件的傳輸幅度與機率 不對稱。;Turbulence is widespread in natural and engineering flows， where interactions across scales generate strong fluctuations and make the instantaneous flow difficult to predict. Two-dimensional turbulence differs fundamentally from three-dimensional turbulence. In ideal incompressible two-dimensional flow， vortex stretching is absent， and enstrophy becomes an additional inviscid invariant alongside kinetic energy. These two conservation laws produce a dual cascade in which kinetic energy is transferred toward larger scales and enstrophy toward smaller scales. The direct enstrophy cascade is commonly characterized using energy spectra， mean fluxes， and structure functions， but these global statistics do not resolve the spatial organization of local transfer events.
This thesis investigates the local statistics of the direct enstrophy cascade in a quasi two-dimensional flow generated by electromagnetic forcing in a thin electrolyte layer. Particle image velocimetry was used to reconstruct the velocity field and calculate the vorticity field， and spatial coarse graining was applied to evaluate the local enstrophy transfer across filter scales.
Below the forcing scale， the mean local transfer was positive， indicating net transfer toward smaller scales， although forward and inverse events coexisted at every measured scale. Intense forward transfer occurred where the filtered vorticity gradient and subfilter vorticity transport were both strong and anti-aligned. Saddle regions exhibited larger mean transfer and stronger fluctuations than vortex regions. Their main difference was concentrated among events with large vorticity gradients and strong subfilter transport， for which saddle regions showed a stronger preference for anti-alignment that favors forward transfer. Motivated by the Kolmogorov refined similarity hypothesis， conditioning on the local enstrophy dissipation revealed that high enstrophy dissipation regions had a larger positive mean transfer， broader positive and negative tails， and a higher probability of forward events.
A dimensionless variable analogous to entropy production was further constructed to quantify the probability asymmetry between forward and inverse transfer events. In regions of high enstrophy dissipation， the logarithmic probability ratio of positive and negative events followed an approximately unit slope fluctuation relation over the resolved range. This behavior was observed at all measured filter scales below the forcing scale. These results show that the local enstrophy transfer is intermittent and statistically asymmetric whose magnitude， direction， and probability asymmetry are organized by the filtered vorticity gradient， subfilter vorticity transport， their relative alignment， flow topology， and local enstrophy dissipation.
