在行星際磁場為北向時，可以在磁層頂觀察到Kelvin-Helmholtz (K-H) 不穩定的發生。許多研究指出，K- H不穩定所產生的渦流為行星際空間與磁層頂間物質交換的主因。過去的理論解研究顯示，有一波長的擾動波可使K-H不穩定以最高的速率成長，稱為最不穩定波。這些研究假設擾動僅涵蓋一定的範圍，且不會向外擴張，但太空中的K-H不穩定發生時，擾動波厚度並不會固定不變。從過去的數值模擬研究也可發現，隨著K-H不穩定的成長，擾動厚度會不斷擴大，但同時成長率卻維持定值。本篇論文旨在探討固定背景流場下，K-H不穩定的擾動厚度改變時，成長率以及擾動波波長的變化情形，以及使K-H不穩定以等速率成長的機制。本論文中所研發的方法可在不限制邊界範圍下，找出K-H不穩定的成長率，並對應到不同的擾動邊界位置。我們的研究顯示，成長率會隨著擾動邊界遠離速度切中心而增加，並且存在一上限，我們稱為成長率臨界值。最不穩定波的K-H不穩定有最高的成長率臨界值，但其擾動邊界位置卻不是最遠的。當擾動波波長比最高成長率之波長還要長時，因其擾動層厚度較大，在能量與物質轉換的功能上有一定的影響力。因此在K-H不穩定事件中，不只要考慮成長率的高低，也要考慮擾動層厚度的大小。

Velocity shear can lead to Kelvin-Helmholtz (K-H) instability. During northward interplanetary magnetic field, K-H instability at the flank magnetopause plays an important role in mass and energy transfer between the solar wind and the magnetosphere. Previous theoretical studies consider the growth rate of K-H instability with fixed boundary conditions. Depending on the strength of the velocity shear, one can always find a wavelength, such that the corresponding surface wave shows highest growth rate, which is called the most unstable mode of K-H instability. However, the reason why the perturbations will grow simultaneously over the entire region without time delay and the reason why the perturbations should confine inside these boundaries have not been explained. Previous simulation studies of K-H instability show that the K-H instability with nearly constant growth rate can last for quite a long time before the nonlinear effects step in. The goal of this thesis work is to study the evolution of growth rates and wavelengths when the perturbation boundaries of the K-H instability expand. A new scheme is developed to find growth rates of K-H instability of the surface waves at different wavelengths without prescribing the locations of the perturbation boundaries. Instead, the locations of the perturbation boundaries become a part of the solutions in this thesis study. The results of this study indicate that the growth rate increases with increasing distance of the perturbation boundaries from the center of the velocity shear layer. However, the growth rates show upper-bound values, which are called the critical growth rates, in this thesis. It is shown that the most unstable mode has the highest critical growth rate, but the location of perturbation boundary increases with increasing wavelength. As a result, the waves with longer wavelength will eventually become the dominant modes in the development of the K-H instability.

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