速度切不穩定又稱作Kelvin-Helmholtz (K-H) 不穩定是自然界一種常見的流體能量與動量交換過程。自然界中所觀測到的K-H不穩定，擾動層厚度會不斷增加，但是早期理論研究卻假設擾動層的厚度是固定不變的，因此可以找到一個成長率最高的擾動波波長，此波長的波被視為該速度切條件下的最不穩定波。較新的理論研究(張益偉, 2016) 發現，給定速度切大小與擾動波的波長，隨著擾動層厚度增加，不穩定波的成長率也會隨之增加，直到達到該波長的最大擾動層厚度。最大擾動層厚度會隨著波長增加而增厚。本論文利用高階的二維磁流體數值模擬碼，模擬研究磁流體電漿中快波馬赫數小於1的K-H不穩定事件。我們成功找到了一種初始條件讓模擬結果可以驗證張益偉 (2016) 的理論解。我們比較數值模擬結果與理論解發現：K-H不穩定過程中的最不穩定波，並不見得是系統中最主要的擾動波。因為最不穩定波的有效擾動擾動層厚度比較薄，所以達到飽和的時間通常早於較長的擾動波。反之，較長擾動波的有效擾動層厚度比較厚，所以可以從背景電漿流中獲得較多的能量，故達到飽和的時間晚，且振幅較大。因此系統中的擾動波會往長波發展，使得擾動層厚度不斷增加。我們的模擬結果與理論解相比，在振幅與相位的空間分佈上都相當一致，但是模擬中的耗散項使得振幅的成長率略低於理論值。;Velocity shear instability, which is also called the Kelvin-Helmholtz (K-H) instability, provides an efficient mechanism for energy exchange and momentum exchange of different flows in nature. The thickness of the surface perturbation usually increases with time in the observed K-H instability in the natural environment. However, in early theoretical studies of K-H instability, the surface perturbation was assumed to be confined in a boundary layer. The halfwidth of the boundary layer is less than ten times the initial thickness of the velocity shear layer. As a result, one could always find a wave of a particular wavelength that has the highest growth rate in the confined boundary layer. This wave was considered to be the most unstable mode in the system. Recently, the theoretical study by Chang (2016) showed that the location of the maximum perturbation boundary varies with the tangential wavelength of the surface wave. The submagnetosonic K-H instabilities in a magnetohydrodynamic (MHD) plasma are studied by a higher-order two-dimensional MHD simulation in this thesis. We successfully find one type of initial conditions which allow broadband surface disturbances to grow simultaneously. As a result, our simulation results can verify the theoretical solutions obtained by Chang (2016). We analysis the amplitude distributions and the phase distributions of the broadband waves generated by the K-H instability and find the simulation results are in good agreement with the theoretical solutions. However, the growth rates obtained in our simulation are less than the corresponding growth rates found in the theoretical solutions due to the additional dissipation terms added in our simulation model. Our simulation results also show that the most unstable mode in the K-H instability will be the dominant mode only during the initial phase of the K-H instability. When the perturbed boundary grows beyond the outermost edge of the most unstable mode, the growth rate of the most unstable mode begins to decrease. Whereas, waves with longer wavelengths have a more extended outermost boundary for them to grow linearly. These waves can receive more energy from the flow field and continuously grow to higher amplitudes. Thus, these waves with longer wavelengths eventually become the dominant modes in the late phase of the K-H instability.
