當磁場方向為南向的太陽風吹向地球時，會產生一個方向從黎明指向黃昏(dawn-to-dusk)的電場[1]，這個電場會和地球磁場作用並將在地球夜側磁尾(Magnetotail)附近的電漿粒子往地球方向飄移，當來到地球電漿層頂(plasmapause)時，因環境中的電漿密度增加[2]而屏蔽了dawn-to-dusk電場，使得電漿粒子因飄移效應的消失而留在電漿層頂，這些粒子稱為為注入粒子。注入粒子在這段過程因為所處環境的磁場變化導致垂直磁場方向的熱速度增加而改變原有的速度分布。本研究透過質點網格法(particle-in-cell)模擬注入粒子與電漿層頂外圍粒子混合時的情況，並藉由頻譜分析及觀察速度分布的變化來研究此現象。

首先，透過選取歸一化常數，帶入所使用的方程式中去求得用於電腦程式中的歸一化方程式。接著使用質點網格法將空間切割成網格狀並且藉由線性內插法將電場、磁場、電流密度、電荷密度記錄於格點。使用巨粒子(marcroparticle)模擬一團真實粒子，蛙跳法(leap-frog integration)則作為電磁場與粒子的更新方式，為了滿足蛙跳法的限制，更新粒子速度時採用Boris’s rotation算法[3]。

在產生粒子的位置和速度分布方面，使用程式語言Python的擴充程式庫NumPy中的Random Generator模組，這個模組採用了Permuted Congruential Generator (64-bit, PCG64)[4]作為亂數產生器。

本篇論文假設電漿層頂附近粒子的初始速度符合馬克示威-波茲曼分布(Maxwell–Boltzmann distribution)，同時假設注入粒子的初始速度符合馬克示威-波茲曼分布加上第一絕熱不變量在垂直磁場方向速度的加熱效應，背景磁場則假設符合地球磁場模型[5]，電漿環境滿足週期性邊界條件(periodic boundary condition)，時間尺度方面，選擇關注在電子的時間尺度上，而由於電子對於勞侖茲力（Lorentz force）的反應時間遠快於質子，因此將質子近似為靜止並均勻的分布於模擬環境中。

模擬了三個情境-無注入粒子、有和電漿層頂粒子溫度相同的注入粒子、有和電漿層頂粒子溫度不同的注入粒子。在能量方面，不管有無注入粒子，粒子總能量都占了系統總能量的99%以上，總電磁場能量則佔有不到1%。而在不到1%的總電磁場能量中，電場能量也是占有超過99%的比例，磁場能量則只占相當微小的比例。

在溫度方面，粒子總溫度在三種情境時大致都維持不變，但是對垂直和平行背景磁場方向上的溫度來說，當有注入粒子時，垂直方向的溫度會逐漸下降，平行方向的溫度則會逐漸上升直到平衡，無注入粒子時則兩方向的溫度都沒有變化。

透過頻譜分析發現，在有注入粒子時，0到0.02秒間在低頻(<200Hz)的電場能量普密度出現強度明顯高於其他頻率的快速上升，且平行於背景磁場方向的電場能量普密度上升幅度較大。我們推測這是造成粒子能量交換的主要原因。

另外我們也發現在不管有無注入粒子，分別在左旋極化波的終止頻率、右旋極化波的終止頻率、電子電漿頻率上出現了左旋極化波、右旋極化波和朗繆爾波。而在有注入粒子並且只有在溫度不同時，高頻(大於10個電子電漿頻率)範圍內出現了R、L、O、X波，且強度分布為X波> R、L波 > O波。

When the solar wind with the interplanetary magnetic field in the southward direction blows to the Earth, it produces a dawn-to-dusk electric field across the magnetosphere [1]. This electric field interacts with the Earth's magnetic field to produce an E×B drift effect. Plasma in the magnetotail is affected by this effect and drifts Earthward towards the magnetic equator. These plasma particles are called injected particles.

In this study, the particle-in-cell method was used to simulate the mixing of injected particles with particles at the region near the plasmapause. At the same time, linear interpolation, marcroparticle model, leap-frog scheme, and periodic boundary conditions are used in our model. The results are studied by spectral analysis and by observing the change of the velocity distribution.

There are three scenarios, without injected particles, with injected particles having the same temperature as the particles at the region near the plasmapause and with injected particles having different temperature from the particles at the region near the plasmapasue.

In terms of system energy, the particle energy accounts for more than 99% of the system’s energy and the electromagnetic energy accounts for less than 1%. And for the electromagnetic energy, the electric energy accounts for more than 99% of the electromagnetic energy and the magnetic energy only accounts for a very small proportion.

In terms of temperature, the total temperature of the particles remains roughly constant in the three scenarios, but for the temperature perpendicular and parallel to the background magnetic field, when there are injected particles, the temperature perpendicular to the background magnetic field gradually drops, and the temperature parallel to the background magnetic field gradually rises until balance. When there are no injected particles, the temperature in both directions does not change.

Through spectral analysis, it is found that when there are injected particles, the electric energy spectral density at low frequency (< 200Hz) has a rapid rise between 0 and 0.02 second that is significantly higher than those at other frequencies. And the electric energy spectral density in the direction parallel to the background magnetic field has a larger increase. It is speculated that this is the main reason for the energy exchange of particles.

In addition, we also found that regardless of the presence or absence of injected particles, there are left-handed polarized waves, right-handed polarized waves and Langmuir waves at the cut-off frequency of the left-handed polarized wave and the right-handed polarized wave, and at the electron plasma frequency. When there are injected particles having different temperature from the particles at the region near the plasmapasue , R, L, O, X waves appear in the high-frequency range (more than 10 times of the electron plasma frequency), their intensities differing as follows: X waves > R,L waves > O waves .

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