在文獻 [1]中，Abbar博士提出了高傅立葉頻率的不穩定性會「泄漏」到較低
的頻率中，使得系統在小尺度到大尺度上普遍存在不穩定性。在本論文中，我
們擴展了對 [1]中所提出的效應的研究，並對其進行了深入探討。從非紊流系統
開始，其中不同的頻率並沒有耦合在一起，所以它們會依據自己所在的頻率獨
立演化，我們使用數值方法計算了這些非耦合系統，並通過 [2]的方法對其進
行了確認。然後，我們通過檢查計算的一致性確認了這種「泄漏」效應，並定
量地提供了紊流振幅與「泄漏」之間的關係。我們的結果表明：這種紊流耦合
系統的不穩定性增長率與非紊流系統相同。除了增長率之外，在「泄漏」發生
之前，系統在紊流耦合效應下的演化可以很好地由相應的非耦合系統和對應的
傅立葉頻率來預測。至於「泄漏」效應，雖然它僅在紊流耦合系統中出現。然
而，這只有在系統進入非線性階段時才會發生。因此，需要在非線性階段進行
完整的計算，以進一步探索「洩漏」效應是否能產生任何影響。

In [1], Dr. Abbar proposed that the instabilities at the higher Fourier modes would “leak” to lower modes thus making instabilities ubiquitous from small to large scales in the system. Here we extend the study of the system and the effect proposed in [1] and explore the system thoroughly. Starting from the non-turbulent system in which the modes are decoupled and evolve themselves independently, we calculate these non-coupled systems numerically and confirm them by following [2]. We verify this “leakage” effect by checking the consistency of the calculation, and we provide the relation between the amplitude of the turbulence and the “leakage” quantitatively. Our results show that the instability growth rate of this turbulentcoupled system is identical to the non-turbulent ones. Other than the growth rate, the evolution of the system under the turbulent-coupled effect could be well predicted by the non-coupled system with corresponding Fourier modes before the“leakage” occurs. For the “leakage” effect, it does emerge only in the turbulentcoupled system at later times. However, this happens only when the system has entered the non-linear regime. Thus, it requires a full calculation of the non-linear regime to further explore whether the “leakage” effect can produce any effect.

Contents
Abstract (Chinese) I
Acknowledgements (Chinese) II
Abstract III
Acknowledgements IV
Contents V
List of Figures VII
1 Introduction 1
2 Methodology 5
2.1 Density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Turbulent media fluctuation formulation . . . . . . . . . . . . . . . 8
2.4 Linear instability analysis . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Set up of the system . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.6 Linearize the equation of motion . . . . . . . . . . . . . . . . . . . . 10
2.7 Constant background medium . . . . . . . . . . . . . . . . . . . . . 11
2.8 Turbulent coupled equation . . . . . . . . . . . . . . . . . . . . . . 12
3 Results 15
3.1 Non-turbulent system . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.1 Instability map of the non-turbulent system . . . . . . . . . 15
3.1.2 Analysis of the instability map . . . . . . . . . . . . . . . . . 18
3.2 Turbulent coupled system . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Consistency of the turbulent-coupled system . . . . . . . . . 22
3.2.2 “leakage” effect in the turbulent system . . . . . . . . . . . . 29
3.2.3 The peak of the eigen-shape and the eigen-value . . . . . . . 36
3.3 Turbulent coupled system in the z evolution . . . . . . . . . . . . . 38
3.3.1 Consistency of the |Qk| evolution in z . . . . . . . . . . . . . 38
3.3.2 Qkevolution under constant µ[km−1] . . . . . . . . . . . . . 40
3.3.3 |Qk| evolution under µ(z) = µ0 × e−0.3z[km−1] . . . . . . . . 50
3.4 z evolution under non-coupled and coupled system . . . . . . . . . . 52
4 Discussion and conclusion 55
Bibliography 57

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