磁化等离子体中Richtmyer-Meshkov不稳定性的理论研究
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摘要
当激波入射到受扰动的流体界面或者受扰动的流体界面被突然加速时,通常会有Richtmyer-Meshkov(RM)不稳定性的发生。RM不稳定性在天体物理中的超新星爆发过程以及惯性约束核聚变中,都扮演着重要的角色。当星球内核塌缩所产生的向外传播的激波穿越氘氚分界面时,会有RM不稳定性的发生。在惯性约束核聚变中,由激波引起的靶丸上缺陷的增长,将破坏靶丸的对称性,成为聚变燃料在内爆末期达到点火条件的主要障碍。对于RM不稳定性的理论描述,存在着几种理论,包括Richtmyer脉冲模型、线性可压缩理论、非线性理论以及烧蚀RM理论等。本文,我们讨论磁化等离子体中的RM不稳定性,具体内容包括:
1、我们讨论了切向磁场与剪切速度对RM不稳定性的影响。通过求解磁流体力学方程组,我们解析地得到了描述界面扰动幅度的表达式以及不稳定性发生的判据。结果表明,当(1-A_t~2)δ_u~2＞4v_a~2时,界面扰动幅度随时间指数增长。其中A_t是Atwood数,δ_u是界面两侧流体的切向相对速度,v_a是平均阿尔芬速度。增长率随着切向相对速度的增加而变大,随着磁场的增强而减小。因此,磁场对RM不稳定性有着致稳作用,剪切速度对RM不稳定性有着退稳作用。当二者都存在时,二者之间存在着相互竞争;系统是否稳定,取决于二者究竟谁占主导地位。
2、由粘性磁流体力学方程组出发,我们解析地得到了激波斜入射情况下粘性磁流体中描述RM不稳定性界面扰动幅度的表达式以及不稳定性发生的条件。结果表明,切向磁场的存在会提高不稳定性发生条件的阈值。当磁场足够强使得△v＜2v_a时,界面扰动幅度将以衰减的形式发展,不稳定性不会发生。因此,切向磁场对RM不稳定性有着抑制作用。同时,结果显示,不稳定性发生的条件与粘滞不相关,即△v＞2v_a时,不管粘滞系数取何值,都会有不稳定性的发生。但是粘滞的存在会减缓不稳定性的增长速度。在这种意义上说,粘滞在一定程度上也抑制着RM不稳定性。
3、我们研究了被突然加速的置于均匀切向外磁场中的N层流体系统中的RM不稳定性。通过分析描述界面扰动速度的二阶线性方程,我们得到了可以求解增长率与初始条件的特征方程,进而我们可以得到描述界面扰动幅度的表达式。于是,我们可以通过将流体划分为这样一个多层密度均匀的不连续系统,来分析任意密度分布的流体系统中的RM不稳定性的演化。作为例子,我们讨论了流体层数N=2和N=3这两种情况下的RM不稳定性。结果表明,切向磁场的存在会导致界面扰动幅度以振荡的形式发展,而不是线性或指数增长。这说明,切向磁场对RM不稳定性有着抑制作用。我们着重讨论了N=3情况下的“夹心饼干”模型(ρ_1=ρ_3=ρ＜ρ_2)。结果显示,由于切向磁场的存在,界面扰动幅度将以一个比较高的频率振荡,同时这些振荡又以一个比较慢的拍频周期变化。当中间层厚度比扰动波长小或与之相当(kt＜1或kt～1)时,中间层厚度对界面扰动幅度有较大的影响,且这时两个界面的扰动都将互相影响;若中间层厚度远大于界面扰动波长(kt＞＞1),界面扰动幅度都将独立发展而不受另外一个界面的影响。
The Richtmyer-Meshkov(RM) instability arises when an incident shock wave impacts a corrugated interface of a stratified fluid or a stratified system is impulsively accelerated.The RM instability plays an important role in applications ranging from the supernova in astrophysics to combustion in inertial confinement fusion(ICF).It is believed that the RM instability occurs when the outward propagating shock wave generated by the collapsing core of a dying star travels through the helium-hydrogen interface. In ICF,the shock inducing growth of the defects will destroy the symmetry of the shell which encapsulates the deuterium-tritium fuel and be a major obstacle to achieve required conditions for the fuel ignition at the end of the implosion.There have existed several theories to investigated RM instability,including the Richtmyer's impulsive model,linear compressible theory,nonlinear theory and ablative RM theory etc.. In this dissertation,we discuss the RM instability in magnetized plasmas.The main content are as follows.
First,the effects of transverse magnetic field and shear flow on the RM instability is investigated.An explicit expression describing the behavior of the interface perturbation and criterion of the stability is analytically obtained by solving the linearized magnetohydrodynamics equations.It is shown that the perturbation develops exponentially with time rather when(1-A_t~2)δ_u~2＞4v_a~2,where A_t is Atwood number,v_a is the mean Alfv(?)n velocity andδ_u the relative shear velocity,respectively.The growth rate of the instability increases with the growth of the shear velocity and decreases with the growth of the strength of magnetic field.Thus,RM instability is suppressed by the transverse magnetic field and reinforce by the presence of shear flow.When both the transverse magnetic field and shear flow are present,there exists a competition between the effects of magnetic field and shear flow.Whether the system is stable or not is determined by the dominating one of the stabilizing mechanism of magnetic field and the destabilizing mechanism of shear flow.
Second,we have investigated the behavior of an obliquely impulsively-accelerated interface which separates two semi-infinite uniform viscous fluids.The fluids are immersed in magnetic fields which are parallel to the interface.We analytically obtained the expression of the interface amplitude and the criterion of the instability.It turns out that the presence of the magnetic field will elevate the threshold of the RM instability. When the magnetic field is strong enough thatΔv＜2v_a,the interface amplitude will damp and the instability will not arise.Thus the transverse magnetic field can suppress the RM instability.It is also shown that criterion is independent on the viscosity, which means the RM instability will always occur no matter how big the coefficient of kinematic viscosity is whenΔv＞2v_a.However,the presence of the viscosity will slow down the growth speed.Therefore,viscosity can somehow suppress but cannot eliminate the RM instability when the interface is obliquely accelerated.
Finally,we have investigated the behaviors of the interfaces of an N-layer stratified fluid,which is impulsively accelerated.The whole system is immersed in an ambient homogeneous transverse magnetic field.By analytically solving the initial value problem and the second-order linear differential equation describing the perturbed velocities, we can get the growth rates and initial conditions.Hence,we can get the expressions of the interface amplitudes.Thus we can treat an arbitrary density distribution problem with a transverse magnetic field in presence by stratifying the fluid into an N-layer discrete discontinuous fluid.As examples,we considered two special cases N=2 and N=3. It is shown when a transverse magnetic field is present,the interfaces will develop as oscillation rather than linearly or exponentially growth.We focused ourselves a lot on the "layer-cake" model(ρ_1=ρ_3=ρ＜ρ_2).Results show when the system is embedded by a transverse magnetic field,the interfaces oscillate with a high fr(?)quency and the oscillations repeat themselves periodically with a low frequency.We also discussed the influence of the thickness of the middle layer on the evolution of the interfaces.It is illustrated that thickness greatly affects the evolution when the thickness is comparable to or smaller than the perturbed wave length(kt＜1 or kt～1).However,when the middle layer is thick enough(kt＞＞1),the two interfaces will develop independently.
1、我们讨论了切向磁场与剪切速度对RM不稳定性的影响。通过求解磁流体力学方程组,我们解析地得到了描述界面扰动幅度的表达式以及不稳定性发生的判据。结果表明,当(1-A_t~2)δ_u~2＞4v_a~2时,界面扰动幅度随时间指数增长。其中A_t是Atwood数,δ_u是界面两侧流体的切向相对速度,v_a是平均阿尔芬速度。增长率随着切向相对速度的增加而变大,随着磁场的增强而减小。因此,磁场对RM不稳定性有着致稳作用,剪切速度对RM不稳定性有着退稳作用。当二者都存在时,二者之间存在着相互竞争;系统是否稳定,取决于二者究竟谁占主导地位。
2、由粘性磁流体力学方程组出发,我们解析地得到了激波斜入射情况下粘性磁流体中描述RM不稳定性界面扰动幅度的表达式以及不稳定性发生的条件。结果表明,切向磁场的存在会提高不稳定性发生条件的阈值。当磁场足够强使得△v＜2v_a时,界面扰动幅度将以衰减的形式发展,不稳定性不会发生。因此,切向磁场对RM不稳定性有着抑制作用。同时,结果显示,不稳定性发生的条件与粘滞不相关,即△v＞2v_a时,不管粘滞系数取何值,都会有不稳定性的发生。但是粘滞的存在会减缓不稳定性的增长速度。在这种意义上说,粘滞在一定程度上也抑制着RM不稳定性。
3、我们研究了被突然加速的置于均匀切向外磁场中的N层流体系统中的RM不稳定性。通过分析描述界面扰动速度的二阶线性方程,我们得到了可以求解增长率与初始条件的特征方程,进而我们可以得到描述界面扰动幅度的表达式。于是,我们可以通过将流体划分为这样一个多层密度均匀的不连续系统,来分析任意密度分布的流体系统中的RM不稳定性的演化。作为例子,我们讨论了流体层数N=2和N=3这两种情况下的RM不稳定性。结果表明,切向磁场的存在会导致界面扰动幅度以振荡的形式发展,而不是线性或指数增长。这说明,切向磁场对RM不稳定性有着抑制作用。我们着重讨论了N=3情况下的“夹心饼干”模型(ρ_1=ρ_3=ρ＜ρ_2)。结果显示,由于切向磁场的存在,界面扰动幅度将以一个比较高的频率振荡,同时这些振荡又以一个比较慢的拍频周期变化。当中间层厚度比扰动波长小或与之相当(kt＜1或kt～1)时,中间层厚度对界面扰动幅度有较大的影响,且这时两个界面的扰动都将互相影响;若中间层厚度远大于界面扰动波长(kt＞＞1),界面扰动幅度都将独立发展而不受另外一个界面的影响。
The Richtmyer-Meshkov(RM) instability arises when an incident shock wave impacts a corrugated interface of a stratified fluid or a stratified system is impulsively accelerated.The RM instability plays an important role in applications ranging from the supernova in astrophysics to combustion in inertial confinement fusion(ICF).It is believed that the RM instability occurs when the outward propagating shock wave generated by the collapsing core of a dying star travels through the helium-hydrogen interface. In ICF,the shock inducing growth of the defects will destroy the symmetry of the shell which encapsulates the deuterium-tritium fuel and be a major obstacle to achieve required conditions for the fuel ignition at the end of the implosion.There have existed several theories to investigated RM instability,including the Richtmyer's impulsive model,linear compressible theory,nonlinear theory and ablative RM theory etc.. In this dissertation,we discuss the RM instability in magnetized plasmas.The main content are as follows.
First,the effects of transverse magnetic field and shear flow on the RM instability is investigated.An explicit expression describing the behavior of the interface perturbation and criterion of the stability is analytically obtained by solving the linearized magnetohydrodynamics equations.It is shown that the perturbation develops exponentially with time rather when(1-A_t~2)δ_u~2＞4v_a~2,where A_t is Atwood number,v_a is the mean Alfv(?)n velocity andδ_u the relative shear velocity,respectively.The growth rate of the instability increases with the growth of the shear velocity and decreases with the growth of the strength of magnetic field.Thus,RM instability is suppressed by the transverse magnetic field and reinforce by the presence of shear flow.When both the transverse magnetic field and shear flow are present,there exists a competition between the effects of magnetic field and shear flow.Whether the system is stable or not is determined by the dominating one of the stabilizing mechanism of magnetic field and the destabilizing mechanism of shear flow.
Second,we have investigated the behavior of an obliquely impulsively-accelerated interface which separates two semi-infinite uniform viscous fluids.The fluids are immersed in magnetic fields which are parallel to the interface.We analytically obtained the expression of the interface amplitude and the criterion of the instability.It turns out that the presence of the magnetic field will elevate the threshold of the RM instability. When the magnetic field is strong enough thatΔv＜2v_a,the interface amplitude will damp and the instability will not arise.Thus the transverse magnetic field can suppress the RM instability.It is also shown that criterion is independent on the viscosity, which means the RM instability will always occur no matter how big the coefficient of kinematic viscosity is whenΔv＞2v_a.However,the presence of the viscosity will slow down the growth speed.Therefore,viscosity can somehow suppress but cannot eliminate the RM instability when the interface is obliquely accelerated.
Finally,we have investigated the behaviors of the interfaces of an N-layer stratified fluid,which is impulsively accelerated.The whole system is immersed in an ambient homogeneous transverse magnetic field.By analytically solving the initial value problem and the second-order linear differential equation describing the perturbed velocities, we can get the growth rates and initial conditions.Hence,we can get the expressions of the interface amplitudes.Thus we can treat an arbitrary density distribution problem with a transverse magnetic field in presence by stratifying the fluid into an N-layer discrete discontinuous fluid.As examples,we considered two special cases N=2 and N=3. It is shown when a transverse magnetic field is present,the interfaces will develop as oscillation rather than linearly or exponentially growth.We focused ourselves a lot on the "layer-cake" model(ρ_1=ρ_3=ρ＜ρ_2).Results show when the system is embedded by a transverse magnetic field,the interfaces oscillate with a high fr(?)quency and the oscillations repeat themselves periodically with a low frequency.We also discussed the influence of the thickness of the middle layer on the evolution of the interfaces.It is illustrated that thickness greatly affects the evolution when the thickness is comparable to or smaller than the perturbed wave length(kt＜1 or kt～1).However,when the middle layer is thick enough(kt＞＞1),the two interfaces will develop independently.
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