磁化等离子体中热交换与旋转不稳定性的理论研究
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摘要
当等离子体中粒子的回旋半径远远小于平均碰撞自由程的时候,等离子体中的热流集中沿磁力线方向流动。这种各向异性的热传导在重力场中会引起一种磁流体力学不稳定性即所谓的磁热不稳定性(Magnetothermal Instability,MTI),这种流体不稳定性在天体环境中广泛存在,近年来受到越来越多的关注。另外一种空间等离子体中较为常见的流体不稳定性是由差分旋转效应引起的磁旋转不稳定性(Magnetorotational Instability,MRI)。MRI从上个世纪六十年代被发现以来,一直受到广泛的重视,其在诸多天体如吸积盘、原行星盘、恒星盘、超新星等等环境中起着重要作用。由于这两种不稳定性都是典型的磁流体力学不稳定性,用磁流体力学(Magnetohydrodynamic,MHD)方程组能够很好地描述,从而为理论分析带来很大便利。本文是基于前人的研究结果和方法对MTI和MRI进行研究,具体内容包括：
1、以非理想MHD方程组研究了粒子回旋时间远远小于平均碰撞时间的情况下,各向异性的电阻和粘滞耗散效应对集中沿磁力线方向输运的热流引起的MTI的影响。利用WKB近似与Boussinesq近似简化线性化扰动MHD方程组,得到了完整的色散关系,讨论了电阻和粘滞效应对MTI的影响,分别获得了电阻(粘滞)效应较小以及很大情况下MTI增长率的表达式。当温度沿重力方向逐渐减小时,扰动将被阻尼；当温度沿重力加速度方向逐渐增大时,扰动是磁热不稳定的,电阻和粘滞效应会减小MTI的增长率,使得非理想MHD情况下的增长率都小于理想MHD下的增长率。
2、基于理想MHD方程组,考察了任意磁场强度下的磁热不稳定性,探讨了密度梯度效应对MTI的影响,我们得到了关于轴向扰动速度的二阶常微分方程以及固定边界条件下MTI发生的临界条件和增长率的表达式。发现当温度沿重力加速度方向增大时,扰动是磁热不稳定的。MTI增长率起初随着密度梯度标长LD的增加而增大,当LD超过临界密度梯度标长LDc之后,增长率则开始随着LD增加而减小。最后,磁场对MTI起致稳作用,并能完全抑制不稳定性的发生。
3、基于包含静止尘埃颗粒效应的非理想MHD方程组,我们研究了各向同性的电阻与粘滞耗散效应以及尘埃颗粒效应对MRI的影响,采用WKB近似与局域近似,求得了完整的色散关系,详细讨论了两种极限情况：(?)Rd／(?)lnr>>Rd≥0以及Rm>>Rd,丨(?)Rd／(?)lnr丨>>Rd,这里,Rm是归一化的旋转频率,Rd是归一化的有效尘埃旋转频率。分别获得了这两种情况下的不稳定性判据,以及后一种情况下增长率的表达式。
4、利用包含静止尘埃颗粒效应的理想磁流体力学方程组考察了柱坐标系统下尘埃等离子体中的MTI与MRI,应用WKB近似和Boussinesq近似线性化方程组,得到了高-β极限下,MTI与MRI耦合下的色散关系。对于熵梯度模,密度不均匀性对MRI发生的临界条件有显著影响：当等离子体密度沿径向向外增大时,密度梯度效应起致稳作用；当等离子体密度沿径向向内增大时,密度梯度则起退稳作用。接下来我们讨论了MTI与MRI耦合的情况,得到了不稳定性判据以及各种情形下的增长率表达式,研究发现,尘埃颗粒对MTI起致稳作用；旋转效应则改变了MTI发生的临界条件,即使温度沿着重力加速度方向减小,只要rΩ2的值大于重力加速度9,依然会发生磁热不稳定性。
5、我们还考察了各向异性的电阻和粘滞耗散效应对MTI和MRI的影响,获得了完整的色散关系以及高-β极限下简化的色散关系与增长率的解析表达式,发现在现有模型下,各向异性的电阻和粘滞耗散对不稳定性发生的临界条件没有任何影响,不稳定性判据与理想磁流体情况下得到的结果完全一致：电阻与粘滞耗散都起着致稳作用,在其他参数固定的情况下,电阻项与粘滞项减小增长率γ的值,也就是说,尽管各向异性的耗散效应并没有改变不稳定性发生的临界条件,但对增长率的修正是显著的。
6、最后我们利用理想磁流体力学方程组研究了轴向磁场情况下,密度梯度效应和非轴对称效应对MRI的影响,发现密度梯度效应并不对MRI发生的判据产生影响。我们得到了增长率的表达式,重点分析了弱场近似下的增长率,考察了密度梯度效应对增长率的作用。结果显示,在κ2>0时,这里κ是周转频率,密度梯度标长较小的时候起退稳作用；密度梯度标长超过临界长度时对MRI有抑制作用。在其他参数不变的情况下,当密度梯度标长超过ge／κ2时,MRI完全被抑制。在κ2<0时,密度梯度起类似作用,但MRI不会完全被抑制,增长率最终趋于与LD无关的常数。在忽略密度梯度效应而关注非轴对称效应的情况下,我们求得了非轴对称效应修正下的色散关系以及弱场极限下的不稳定性判据。与密度梯度效应类似的是,此时的非轴对称效应对磁化等离子体中的旋转不稳定性发生的临界条件没有任何影响。
When the particle gyroradius is much less than the mean collisional free path, the heat is restricted to being transported primarily along the magnetic force lines. The anisotropic heat conduction will introduce a kind of convective magnetohydrodynamic (MHD) instability, which is referred to as the magnetothermal instability (MTI). MTI has attracted more and more attention due to its important application on astrophysical bodies. Another MHD instability considered in this dissertation is the so-called magne-torotational instability (MRI) which is induced by differential rotation. MRT was first discovered in 1959 and has significant application on many astrophysical bodies such as accretion disks, protoplanetary disks as well as core-collapse supernovae and so on. It is sufficient for us to use MHD equations to describe the two instabilities. The present dissertation is based on the previous method and results to investigate MTI and MRI. The main content are as follows.
1.When the gyro time is much less than the mean collisional time, we investigate the anisotropic resistivity and viscosity dissipative effects on the MTI due to the parallel heat transporting on the basis of the non-ideal MHD equations. The general dispersion relation is obtained by using the Wentzel-Kramers-Brillouin (WKB) approximation to simplify the linearized perturbed MHD equations in the Boussinesq limit. We discuss the effects on the MTI due to the resistivity and viscosity and obtained the growth rates for weak dissipation and strong dissipation, respectively. The perturbations are damped when the temperature decreases in the direction of gravity. When the temperature in-creases in the direction of gravity, the system is MTI unstable. Resistivity and viscosity are shown to reduce the value of MTI growth rate, which is then less than the growth rate obtained in the ideal MHD case.
2.The density gradient effect on the MTI in an arbitrary magnetic field is exam-ined by using ideal MHD equations. We obtained the second-order ordinary differen-tial equation (ODE) describing the axial perturbed velocity and derived the instability criterion and growth rate under the fixed boundary condition. It is shown that the per-turbation is MTI unstable when the temperature increases in the direction of gravity. The growth rate increases monotonically as Ld increases at first, where LD is the scale length of density gradient. When LD is greater than a critical value LDc, the growth rate starts to decrease as LD increases. Finally, magnetic fields show stabilizing effect on the MTI and can totally quench the instability.
3.The magnetorotational instability (MRI) in differential rotating dusty plasmas with dissipative effects is investigated by using local linear analysis on the basis of non-ideal MHD equations. We assume that the dust grains are heavy enough to be immobile. The general local dispersion relation is derived under the WKB approximation. Two limiting cases, (?)Rd/(?)lnr>>Rd≥0 and Rm>>Rd,丨(?)Rd/(?)lnr丨>>Rd, are discussed with respect to the dust-induced effect in detail, where Rm is the normalized rotation frequency and Rd is the normalized effective dusty rotation frequency. The instability criteria in the different limiting cases are presented and the growth rate of local MRI in the last case is demonstrated.
4. The local MTI and MRI in differential rotating plasmas are investigated in the ideal MHD framework which contains immobile dusty grains effects. By adopting the WKB and Boussinesq approximations, the general dispersion relation of MTI as-sociated with MRI is derived and presented in the high-βlimit. For entropy-gradient mode, density inhomogeneity has significant effect on the instability criterion. Density gradient shows stabilizing effect on the instability when increasing outward whereas has destabilizing effect when increasing inward. For the MTI combined with MRI, the instability criteria and growth rate show that dust has stabilizing effect on the MTI whereas the rotation changes the instability criterion. Even if the temperature decreases in the direction of gravity, the MTI will come into being provided that rΩ2 is greater than the gravity acceleration g.
5.Anisotropic resistivity and viscosity effect on the MTI and MRI is examined. We obtain the full general dispersion relation as well as the reduced dispersion rela-tion and growth rate in the high-βlimit. It is found that in our model, the anisotropic dissipative terms do not affect the instability criterion, which is identical with the one obtained in the ideal MHD case. On the other hand, both resistivity and viscosity re-duce the value ofγ. That is, they have stabilizing effect on the thermal convective and magnetorotational instability and modify the growth rate remarkably.
6.The density gradient and non-axisymmetric effects on the MRI are finally in-vestigated in the presence of axial magnetic field by using the ideal MHD equations. Our results show that density gradient has no effect on the instability criterion. We also obtain the instability growth rate and discuss the weak field case in detail. For positiveκ2, it is found that when the scale length of density gradient is small, density gradient shows destabilizing effect on the MRI. When the scale length exceeds a crit-ical value, density gradient has stabilizing effect on the instability. The MRI will be totally quenched by the density gradient when the latter exceeds ge/κ2. For negativeκ2, density gradient shows similar effect on the instability but the MRI can not be to-tally quenched and the growth rate goes to a constant independent of LD.Ignoring the density gradient, we then obtain the dispersion relation and instability criterion in weak fields limit with non-axisymmetric effect modifications. Similar to the density gradient, non-axisymmetric effect in this case do not change the MRI criterion.
1、以非理想MHD方程组研究了粒子回旋时间远远小于平均碰撞时间的情况下,各向异性的电阻和粘滞耗散效应对集中沿磁力线方向输运的热流引起的MTI的影响。利用WKB近似与Boussinesq近似简化线性化扰动MHD方程组,得到了完整的色散关系,讨论了电阻和粘滞效应对MTI的影响,分别获得了电阻(粘滞)效应较小以及很大情况下MTI增长率的表达式。当温度沿重力方向逐渐减小时,扰动将被阻尼；当温度沿重力加速度方向逐渐增大时,扰动是磁热不稳定的,电阻和粘滞效应会减小MTI的增长率,使得非理想MHD情况下的增长率都小于理想MHD下的增长率。
2、基于理想MHD方程组,考察了任意磁场强度下的磁热不稳定性,探讨了密度梯度效应对MTI的影响,我们得到了关于轴向扰动速度的二阶常微分方程以及固定边界条件下MTI发生的临界条件和增长率的表达式。发现当温度沿重力加速度方向增大时,扰动是磁热不稳定的。MTI增长率起初随着密度梯度标长LD的增加而增大,当LD超过临界密度梯度标长LDc之后,增长率则开始随着LD增加而减小。最后,磁场对MTI起致稳作用,并能完全抑制不稳定性的发生。
3、基于包含静止尘埃颗粒效应的非理想MHD方程组,我们研究了各向同性的电阻与粘滞耗散效应以及尘埃颗粒效应对MRI的影响,采用WKB近似与局域近似,求得了完整的色散关系,详细讨论了两种极限情况：(?)Rd／(?)lnr>>Rd≥0以及Rm>>Rd,丨(?)Rd／(?)lnr丨>>Rd,这里,Rm是归一化的旋转频率,Rd是归一化的有效尘埃旋转频率。分别获得了这两种情况下的不稳定性判据,以及后一种情况下增长率的表达式。
4、利用包含静止尘埃颗粒效应的理想磁流体力学方程组考察了柱坐标系统下尘埃等离子体中的MTI与MRI,应用WKB近似和Boussinesq近似线性化方程组,得到了高-β极限下,MTI与MRI耦合下的色散关系。对于熵梯度模,密度不均匀性对MRI发生的临界条件有显著影响：当等离子体密度沿径向向外增大时,密度梯度效应起致稳作用；当等离子体密度沿径向向内增大时,密度梯度则起退稳作用。接下来我们讨论了MTI与MRI耦合的情况,得到了不稳定性判据以及各种情形下的增长率表达式,研究发现,尘埃颗粒对MTI起致稳作用；旋转效应则改变了MTI发生的临界条件,即使温度沿着重力加速度方向减小,只要rΩ2的值大于重力加速度9,依然会发生磁热不稳定性。
5、我们还考察了各向异性的电阻和粘滞耗散效应对MTI和MRI的影响,获得了完整的色散关系以及高-β极限下简化的色散关系与增长率的解析表达式,发现在现有模型下,各向异性的电阻和粘滞耗散对不稳定性发生的临界条件没有任何影响,不稳定性判据与理想磁流体情况下得到的结果完全一致：电阻与粘滞耗散都起着致稳作用,在其他参数固定的情况下,电阻项与粘滞项减小增长率γ的值,也就是说,尽管各向异性的耗散效应并没有改变不稳定性发生的临界条件,但对增长率的修正是显著的。
6、最后我们利用理想磁流体力学方程组研究了轴向磁场情况下,密度梯度效应和非轴对称效应对MRI的影响,发现密度梯度效应并不对MRI发生的判据产生影响。我们得到了增长率的表达式,重点分析了弱场近似下的增长率,考察了密度梯度效应对增长率的作用。结果显示,在κ2>0时,这里κ是周转频率,密度梯度标长较小的时候起退稳作用；密度梯度标长超过临界长度时对MRI有抑制作用。在其他参数不变的情况下,当密度梯度标长超过ge／κ2时,MRI完全被抑制。在κ2<0时,密度梯度起类似作用,但MRI不会完全被抑制,增长率最终趋于与LD无关的常数。在忽略密度梯度效应而关注非轴对称效应的情况下,我们求得了非轴对称效应修正下的色散关系以及弱场极限下的不稳定性判据。与密度梯度效应类似的是,此时的非轴对称效应对磁化等离子体中的旋转不稳定性发生的临界条件没有任何影响。
When the particle gyroradius is much less than the mean collisional free path, the heat is restricted to being transported primarily along the magnetic force lines. The anisotropic heat conduction will introduce a kind of convective magnetohydrodynamic (MHD) instability, which is referred to as the magnetothermal instability (MTI). MTI has attracted more and more attention due to its important application on astrophysical bodies. Another MHD instability considered in this dissertation is the so-called magne-torotational instability (MRI) which is induced by differential rotation. MRT was first discovered in 1959 and has significant application on many astrophysical bodies such as accretion disks, protoplanetary disks as well as core-collapse supernovae and so on. It is sufficient for us to use MHD equations to describe the two instabilities. The present dissertation is based on the previous method and results to investigate MTI and MRI. The main content are as follows.
1.When the gyro time is much less than the mean collisional time, we investigate the anisotropic resistivity and viscosity dissipative effects on the MTI due to the parallel heat transporting on the basis of the non-ideal MHD equations. The general dispersion relation is obtained by using the Wentzel-Kramers-Brillouin (WKB) approximation to simplify the linearized perturbed MHD equations in the Boussinesq limit. We discuss the effects on the MTI due to the resistivity and viscosity and obtained the growth rates for weak dissipation and strong dissipation, respectively. The perturbations are damped when the temperature decreases in the direction of gravity. When the temperature in-creases in the direction of gravity, the system is MTI unstable. Resistivity and viscosity are shown to reduce the value of MTI growth rate, which is then less than the growth rate obtained in the ideal MHD case.
2.The density gradient effect on the MTI in an arbitrary magnetic field is exam-ined by using ideal MHD equations. We obtained the second-order ordinary differen-tial equation (ODE) describing the axial perturbed velocity and derived the instability criterion and growth rate under the fixed boundary condition. It is shown that the per-turbation is MTI unstable when the temperature increases in the direction of gravity. The growth rate increases monotonically as Ld increases at first, where LD is the scale length of density gradient. When LD is greater than a critical value LDc, the growth rate starts to decrease as LD increases. Finally, magnetic fields show stabilizing effect on the MTI and can totally quench the instability.
3.The magnetorotational instability (MRI) in differential rotating dusty plasmas with dissipative effects is investigated by using local linear analysis on the basis of non-ideal MHD equations. We assume that the dust grains are heavy enough to be immobile. The general local dispersion relation is derived under the WKB approximation. Two limiting cases, (?)Rd/(?)lnr>>Rd≥0 and Rm>>Rd,丨(?)Rd/(?)lnr丨>>Rd, are discussed with respect to the dust-induced effect in detail, where Rm is the normalized rotation frequency and Rd is the normalized effective dusty rotation frequency. The instability criteria in the different limiting cases are presented and the growth rate of local MRI in the last case is demonstrated.
4. The local MTI and MRI in differential rotating plasmas are investigated in the ideal MHD framework which contains immobile dusty grains effects. By adopting the WKB and Boussinesq approximations, the general dispersion relation of MTI as-sociated with MRI is derived and presented in the high-βlimit. For entropy-gradient mode, density inhomogeneity has significant effect on the instability criterion. Density gradient shows stabilizing effect on the instability when increasing outward whereas has destabilizing effect when increasing inward. For the MTI combined with MRI, the instability criteria and growth rate show that dust has stabilizing effect on the MTI whereas the rotation changes the instability criterion. Even if the temperature decreases in the direction of gravity, the MTI will come into being provided that rΩ2 is greater than the gravity acceleration g.
5.Anisotropic resistivity and viscosity effect on the MTI and MRI is examined. We obtain the full general dispersion relation as well as the reduced dispersion rela-tion and growth rate in the high-βlimit. It is found that in our model, the anisotropic dissipative terms do not affect the instability criterion, which is identical with the one obtained in the ideal MHD case. On the other hand, both resistivity and viscosity re-duce the value ofγ. That is, they have stabilizing effect on the thermal convective and magnetorotational instability and modify the growth rate remarkably.
6.The density gradient and non-axisymmetric effects on the MRI are finally in-vestigated in the presence of axial magnetic field by using the ideal MHD equations. Our results show that density gradient has no effect on the instability criterion. We also obtain the instability growth rate and discuss the weak field case in detail. For positiveκ2, it is found that when the scale length of density gradient is small, density gradient shows destabilizing effect on the MRI. When the scale length exceeds a crit-ical value, density gradient has stabilizing effect on the instability. The MRI will be totally quenched by the density gradient when the latter exceeds ge/κ2. For negativeκ2, density gradient shows similar effect on the instability but the MRI can not be to-tally quenched and the growth rate goes to a constant independent of LD.Ignoring the density gradient, we then obtain the dispersion relation and instability criterion in weak fields limit with non-axisymmetric effect modifications. Similar to the density gradient, non-axisymmetric effect in this case do not change the MRI criterion.
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