在托卡馬克裡，為了解決長波長磁流體力學的不穩定性，我們發展了電磁迴旋動力模擬模型。我們從基礎的粒子模擬開始建立程式碼。首先，我們用基本的粒子網格法模擬朗道阻尼。然後，我們利用δf方法進一步發展程式碼並模擬電漿回聲，藉由這個方法可以減少統計雜訊的產生。

建立模擬的基礎後我們加入靜電迴旋動力模型的概念。利用絕熱電子的想法我們模擬了飄移波的不穩定性，在此模擬下頻率與密度梯度的相依性符合理論的預測；並且藉由朗道阻尼，不穩定性趨於飽和。為了追求電磁狀況下的粒子模擬，我們需要動力電子。因此我們模擬飄移波在磁鏡力下的運動狀況，得出在飽和狀況下須有較大的電場振幅，並且在相空間看到粒子的δf部分從束縛到穿透的加速過程。

最後我們更進一步在(x,p,t)座標下使用電磁迴旋動力理論，利用這個模型我們模擬了阿爾芬波在靜電極限的狀況也就是β_e等於零，即為粒子空間波，相依於離子與電子間質量比。但在(x,p,t)中，我們在數值上會遇到小項被大項掩蓋的問題，因此我們導出新方法在(x,v,t)座標。此方法中我們運用了一個荷姆赫茲方程式來解來自法拉第定律對時間微分的磁向量勢.

New electrostatic and electromagnetic gyrokinetic particle-in-cell (PIC) simulation methods are developed for future application of kinetic electron dynamics and long-wave-length magnetohydrodynamic instability in a fully three dimensional tokamak geometry. The PIC code is developed step by step. First, Landau damping of Langmuir wave is confirmed in “full- f” PIC. Second, by introducing the δf-method, Eulerian valuables of Vlasov equation is changed to Lagragian variables in the weight equations plus the equation of motion, the characteristics. Third, as one of the applications, simulation of plasma echo of the Langmuir wave by the δf-PIC is demonstrated by imposing two temporal impulsive electric fields.

Forth, benchmark computation of 1-D electrostatic drift-wave instability is performed. Variation of real frequency (and growth rates) as a function of density gradient matches with the linear theory of the drift-wave dispersion relation. Nonlinear saturation of the drift-wave due to Landau damping by kinetic electrons is demonstrated. Fifth, mirror force is given to simulate trapped electrons and ions. Electrostatic potential also saturates but with larger amplitude. Finally, in the new formulation of electromagnetic gyrokinetics, time derivative of Ampere’s law is incorporated to obtain inductive component in gyrokinetic equation (∂A/∂t) which originates from Faraday’s Law.

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