In Eq. (8) the first-order quantities are v_(s)\mathbf{v}_{s} and E\mathbf{E}. The first-order magnetic field B^((1))\mathbf{B}^{(1)} would only appear in this equation of motion if v_(s)^((0))\mathbf{v}_{s}^{(0)} were finite (see Sec.2-8\mathrm{Sec} .2-8 ). One notes that elements of the plasma fluid, as modeled by Eq. (8), oscillate like jelly about fixed positions in space under the influence of the wave’s electromagnetic field. Hidden in this fluid picture is the underlying 在公式 (8) 中,一阶量为 v_(s)\mathbf{v}_{s} 和 E\mathbf{E} 。只有当 v_(s)^((0))\mathbf{v}_{s}^{(0)} 有限时,一阶磁场 B^((1))\mathbf{B}^{(1)} 才会出现在这个运动方程中(见 Sec.2-8\mathrm{Sec} .2-8 )。我们注意到,等离子体流体中的元素,如公式 (8) 所示,在波的电磁场影响下,像果冻一样围绕空间中的固定位置摆动。隐藏在这幅流体图中的是
structure of the collisionless plasma with particles free-streaming along their zero-order trajectories, only slightly perturbed in this motion by the presence of a wave. 无碰撞等离子体的结构,粒子沿其零阶轨迹自由流动,这种运动只受到波的轻微扰动。
Recalling that B_(0)= widehat(z)B_(0)\mathbf{B}_{0}=\widehat{\mathbf{z}} B_{0}, we denote (cf. Probs. 5 and 6) 回顾 B_(0)= widehat(z)B_(0)\mathbf{B}_{0}=\widehat{\mathbf{z}} B_{0} ,我们表示(参见 Probs.)
v^(+-)=(1)/(2)(v_(x)+-iv_(y))quad" and "quadE^(+-)=(1)/(2)(E_(x)+-iE_(y))v^{ \pm}=\frac{1}{2}\left(v_{x} \pm i v_{y}\right) \quad \text { and } \quad E^{ \pm}=\frac{1}{2}\left(E_{x} \pm i E_{y}\right)
where Omega_(s)\Omega_{s} (or omega_(cs)\omega_{c s}, which many authors use) is the algebraic cyclotron or gyrofrequency for particles of type ss : 其中, Omega_(s)\Omega_{s} (或 omega_(cs)\omega_{c s} ,许多作者使用)是 ss 类型粒子的代数回旋频率或陀螺频率:
Then using Eqs. (9)-(14) to go from v^(+-),E^(+-)v^{ \pm}, E^{ \pm}back to v_(x),v_(y),E_(x)v_{x}, v_{y}, E_{x}, and E_(y)E_{y}, one may find 然后,利用公式 (9)-(14) 从 v^(+-),E^(+-)v^{ \pm}, E^{ \pm} 回到 v_(x),v_(y),E_(x)v_{x}, v_{y}, E_{x} 和 E_(y)E_{y} ,可以发现
leading to the cold-plasma dielectric tensor 导致冷等离子体介电张量
epsilon*E=([S,-iD,0],[iD,S,0],[0,0,P])([E_(x)],[E_(y)],[E_(z)])\epsilon \cdot \mathbf{E}=\left(\begin{array}{ccc}
S & -i D & 0 \\
i D & S & 0 \\
0 & 0 & P
\end{array}\right)\left(\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z}
\end{array}\right)
in which the quantities S\boldsymbol{S} (for sum), D\boldsymbol{D} (for difference), and P\boldsymbol{P} (for plasma) are defined: 中定义了 S\boldsymbol{S} (表示和)、 D\boldsymbol{D} (表示差)和 P\boldsymbol{P} (表示等离子体):
{:[S=(1)/(2)(R+L)","quad D=(1)/(2)(R-L)","],[R-=1+sum_(s)chi_(s)^(-)=1-sum_(s)(omega_(rho s)^(2))/(omega(omega+Omega_(s)))","],[L-=1+sum_(s)chi_(s)^(+)=1-sum_(s)(omega_(rho s)^(2))/(omega(omega-Omega_(s)))","],[P-=1-sum_(s)(omega_(ps)^(2))/(omega^(2))]:}\begin{gathered}
S=\frac{1}{2}(R+L), \quad D=\frac{1}{2}(R-L), \\
R \equiv 1+\sum_{s} \chi_{s}^{-}=1-\sum_{s} \frac{\omega_{\rho s}^{2}}{\omega\left(\omega+\Omega_{s}\right)}, \\
L \equiv 1+\sum_{s} \chi_{s}^{+}=1-\sum_{s} \frac{\omega_{\rho s}^{2}}{\omega\left(\omega-\Omega_{s}\right)}, \\
P \equiv 1-\sum_{s} \frac{\omega_{p s}^{2}}{\omega^{2}}
\end{gathered}
It should be noted that Eqs. (20)-(22) are ambiguous at omega=0\omega=0 and omega\omega=+-Omega_(s)= \pm \Omega_{\mathrm{s}}. The correct treatment of these singularities, still within the context of cold-plasma susceptibilities, is given in Eqs. (3-23)-(3-25). 需要注意的是,公式 (20)-(22) 在 omega=0\omega=0 和 omega\omega=+-Omega_(s)= \pm \Omega_{\mathrm{s}} 处是模糊的。对这些奇点的正确处理,仍然是在冷等离子体感性的背景下,在公式 (3-23)-(3-25) 中给出。
A more formal justification for the cold-plasma approximation will be obtained in Secs. 10-710-7 and 11-511-5 where Eqs. (20)-(22) can be deduced from the kinetic theory hot-plasma result by making use of expansions valid for low temperatures. Also see Prob. 10-12. It is at this point that the jellylike oscillations of the cold-plasma fiuid will be reconciled with the underlying structure of almost unperturbed free-streaming particles. 在 10-710-7 和 11-511-5 两节中,我们将得到冷等离子体近似的更正式的理由,其中公式(20)-(22)可以利用对低温有效的展开式从动力学理论热等离子体结果中推导出来。另见问题 10-12。在这一点上,冷等离子体流体的果冻状振荡将与几乎不受扰动的自由流粒子的基本结构相协调。
1-3 The Dispersion Relation 1-3 分散关系
Having obtained the dielectric tensor epsilon\epsilon, we can solve Maxwell’s equations for plane waves. We have 在得到介电张量 epsilon\epsilon 之后,我们就可以求解平面波的麦克斯韦方程了。我们有
grad xxE=-(1)/(c)(del(B))/(del t).\nabla \times \mathbf{E}=-\frac{1}{c} \frac{\partial \mathrm{~B}}{\partial t} .
After Fourier analysis in time and space, Eqs. (23) and (24) combine to give the homogeneous-plasma wave equation 在对时间和空间进行傅立叶分析后,公式 (23) 和 (24) 结合起来就得到了均质等离子体波方程
It is convenient to introduce the dimensionless vector n\mathbf{n} which has the direction of the propagation vector kk and has the magnitude of the refractive index 我们可以方便地引入无量纲矢量 n\mathbf{n} ,它具有传播矢量 kk 的方向,并且具有折射率的大小
The magnitude n=|n|n=|n| is the ratio of the velocity of light to the wave phase velocity. The wave normal surface is the locus of the tip of the vector n^(-1)-=n//n^(2)n^{-1} \equiv n / n^{2}. 幅值 n=|n|n=|n| 是光速与波相位速度之比。波法线面是矢量 n^(-1)-=n//n^(2)n^{-1} \equiv n / n^{2} 尖端的位置。
Using nn, the wave equation (25) may be written simply 使用 nn ,波方程 (25) 可以简单地写成
If we use theta\theta to denote the angle between B_(0)= hat(z)B_(0)\mathbf{B}_{0}=\hat{\mathbf{z}} B_{0} and nn, and if we assume nn to be in the x,zx, z plane, Eq. (27) becomes 如果我们用 theta\theta 表示 B_(0)= hat(z)B_(0)\mathbf{B}_{0}=\hat{\mathbf{z}} B_{0} 和 nn 之间的夹角,并假设 nn 位于 x,zx, z 平面内,公式 (27) 将变为
([S-n^(2)cos^(2)theta,-iD,n^(2)cos theta sin theta],[iD,S-n^(2),0],[n^(2)cos theta sin theta,0,P-n^(2)sin^(2)theta])([E_(x)],[E_(y)],[E_(z)])=0\left(\begin{array}{ccc}
S-n^{2} \cos ^{2} \theta & -i D & n^{2} \cos \theta \sin \theta \\
i D & S-n^{2} & 0 \\
n^{2} \cos \theta \sin \theta & 0 & P-n^{2} \sin ^{2} \theta
\end{array}\right)\left(\begin{array}{l}
E_{x} \\
E_{y} \\
E_{z}
\end{array}\right)=0
The condition for a nontrivial solution of the vector wave equation (28) is that the determinant of the 3xx33 \times 3 matrix be zero. This condition gives the dispersion relation, that is, a scalar relation that determines omega\omega as a function of k,omega=omega(k)\mathbf{k}, \omega=\omega(\mathbf{k}). As there are no source terms in Eq. (28), the root or roots of the dispersion relation describe natural modes of oscillation of the system. The dispersion relation clearly provides the equation for the wave normal surface, and for waves in a cold plasma this equation was obtained by E\mathbf{E}. Åström (1950), A. G. Sitenko and K. N. Stepanov (1956), and W. P. Allis (1959), 矢量波方程 (28) 的非微分解的条件是 3xx33 \times 3 矩阵的行列式为零。这一条件给出了频散关系,即决定 omega\omega 作为 k,omega=omega(k)\mathbf{k}, \omega=\omega(\mathbf{k}) 函数的标量关系。由于公式 (28) 中没有源项,频散关系的一个或多个根描述了系统的自然振荡模式。频散关系显然提供了波法线面的方程,对于冷等离子体中的波,该方程由 E\mathbf{E} 得到。Åström (1950)、A. G. Sitenko 和 K. N. Stepanov (1956) 以及 W. P. Allis (1959)、
The dispersion relations for propagation at theta=0\theta=0 and theta=pi//2\theta=\pi / 2 are quickly obtained from Eq. (36). 根据公式 (36) 可以快速得到 theta=0\theta=0 和 theta=pi//2\theta=\pi / 2 处传播的色散关系。
Before proceeding to a general discussion of the dispersion relation, it will be useful to derive equations relating the phases and magnitudes of the velocity and field components. We recall that the oscillating field quantities were assumed to vary as exp(ik*r-i omega t)\exp (i \mathbf{k} \cdot \mathbf{r}-i \omega t). It is clear from equations such as (5), (18), and (28) that the components of the field amplitudes such as j_(s)(k,omega)j_{s}(k, \omega) and E(k,omega)E(k, \omega) are then complex numbers. But say that we want to work with 在对频散关系进行一般性讨论之前,我们有必要先推导出速度分量和场分量的相位和大小关系式。我们记得,振荡场量被假定为 exp(ik*r-i omega t)\exp (i \mathbf{k} \cdot \mathbf{r}-i \omega t) 变化。从 (5)、(18) 和 (28) 等式中可以清楚地看出,场振幅的分量,如 j_(s)(k,omega)j_{s}(k, \omega) 和 E(k,omega)E(k, \omega) 都是复数。但是,假设我们想用
real quantities, such as E_(x)(r,t)∼cos(k*r-omega t)E_{x}(\mathbf{r}, t) \sim \cos (\mathbf{k} \cdot \mathbf{r}-\omega t). Rather than inverting E(k,omega)\mathbf{E}(\mathbf{k}, \omega) through Fourier analysis to find E(r,t)\mathbf{E}(\mathbf{r}, \boldsymbol{t}), a shortcut representationwhen dealing with single values for kk and omega\omega-is to write simply 实量,如 E_(x)(r,t)∼cos(k*r-omega t)E_{x}(\mathbf{r}, t) \sim \cos (\mathbf{k} \cdot \mathbf{r}-\omega t) 。在处理 kk 和 omega\omega 的单个值时,与其通过傅立叶分析反演 E(k,omega)\mathbf{E}(\mathbf{k}, \omega) 以求得 E(r,t)\mathbf{E}(\mathbf{r}, \boldsymbol{t}) ,不如简单写成
where “Re” denotes “real part of.” In the same vein, the symbol “Im” will denote “imaginary part of.” [The formal Fourier transform and inversion relations are given in Eqs. (3-3), (3-4) and (4-58), (4-64).] Separating E(k,omega)\mathbf{E}(\mathbf{k}, \omega) into its real and imaginary parts, E(k,omega)=E_(r)+iE_(i)\mathbf{E}(\mathbf{k}, \omega)=\mathbf{E}_{r}+i \mathbf{E}_{i}, one obtains 其中,"Re "表示 "实部"。同样,符号 "Im "表示 "虚部"。[正式的傅里叶变换和反演关系见式 (3-3)、(3-4) 和 (4-58)、(4-64)。将 E(k,omega)\mathbf{E}(\mathbf{k}, \omega) 分解为实部和虚部 E(k,omega)=E_(r)+iE_(i)\mathbf{E}(\mathbf{k}, \omega)=\mathbf{E}_{r}+i \mathbf{E}_{i} 可以得到
Applying the same shortcut representation to a formula such as Eq. (5), j_(s)\mathrm{j}_{s}=-(i omega//4pi)chi_(s)*E=-(i \omega / 4 \pi) \chi_{s} \cdot \mathbf{E}, one would find 对公式 (5) j_(s)\mathrm{j}_{s}=-(i omega//4pi)chi_(s)*E=-(i \omega / 4 \pi) \chi_{s} \cdot \mathbf{E} 应用同样的快捷表示法,可以发现
Turning now to the question of wave polarization, we suppress the k*r\mathbf{k} \cdot \mathbf{r} dependence and note that, for omega > 0\omega>0, the case for pure right-hand circular polarization is given by A_(x)=a cos(-omega t)=a(Ree^(-i omega t))A_{x}=a \cos (-\omega t)=a\left(\operatorname{Re} e^{-i \omega t}\right) and A_(y)=-a sinA_{y}=-a \sin(-omega t)=a(Re ie^(-i omega t))(-\omega t)=a\left(\operatorname{Re} i e^{-i \omega t}\right), so that in our notation of complex amplitudes iA_(x)//i A_{x} /A_(y)=1A_{y}=1. Similarly, for left-hand circular polarization, iA_(x)//A_(y)=-1i A_{x} / A_{y}=-1. Polarization is defined here, using positive values of omega\omega, with respect to the zz direction, the direction of the static magnetic field. (In optics and quantum mechanics, the usual convention defines the polarization with respect to the wave propagation vector, k. See Prob. 3.) 现在来谈谈波的极化问题,我们抑制了 k*r\mathbf{k} \cdot \mathbf{r} 的依赖性,并注意到,对于 omega > 0\omega>0 ,纯右旋圆极化的情况由 A_(x)=a cos(-omega t)=a(Ree^(-i omega t))A_{x}=a \cos (-\omega t)=a\left(\operatorname{Re} e^{-i \omega t}\right) 和 A_(y)=-a sinA_{y}=-a \sin(-omega t)=a(Re ie^(-i omega t))(-\omega t)=a\left(\operatorname{Re} i e^{-i \omega t}\right) 给出,因此在我们的复振幅符号中 iA_(x)//i A_{x} /A_(y)=1A_{y}=1 。同样,对于左旋圆极化, iA_(x)//A_(y)=-1i A_{x} / A_{y}=-1 。这里使用 omega\omega 的正值来定义极化,相对于 zz 方向,即静态磁场的方向。(在光学和量子力学中,通常定义极化的方向是波的传播矢量 k。)
The polarization of the transverse electric fields may be taken from the middle line of Eq. (28): 横向电场的极化可从公式 (28) 的中间一行得出:
Making use of the definitions in Eq. (19) for the case of theta=0\theta=0 with n^(2)n^{2}=R=R, Eq. (42) becomes iE_(x)//E_(y)=1i E_{x} / E_{y}=1, while for the case of theta=0\theta=0 with n^(2)n^{2}=L=L, Eq. (39) becomes iE_(x)//E_(y)=-1i E_{x} / E_{y}=-1. We thus verify that the polarization is circular with a right-hand or left-hand sense according to n^(2)=Rn^{2}=R or n^(2)n^{2}=L=L, respectively. 利用公式 (19) 中的定义,在 theta=0\theta=0 与 n^(2)n^{2}=R=R 的情况下,公式 (42) 变为 iE_(x)//E_(y)=1i E_{x} / E_{y}=1 ;而在 theta=0\theta=0 与 n^(2)n^{2}=L=L 的情况下,公式 (39) 变为 iE_(x)//E_(y)=-1i E_{x} / E_{y}=-1 。因此,我们可以根据 n^(2)=Rn^{2}=R 或 n^(2)n^{2}=L=L 分别验证极化是右手或左手意义上的圆形。
A similar relation may be obtained for the macroscopic fluid velocities. Using Eqs. (9), (13) and again, chi_(s)^(+-)E^(+-)=(4pi i//omega)n_(s)q_(s)v_(s)^(+-)\chi_{s}^{ \pm} E^{ \pm}=(4 \pi i / \omega) n_{s} q_{s} v_{s}^{ \pm}, one may find 宏观流体速度也有类似的关系。利用公式 (9)、(13) 和 chi_(s)^(+-)E^(+-)=(4pi i//omega)n_(s)q_(s)v_(s)^(+-)\chi_{s}^{ \pm} E^{ \pm}=(4 \pi i / \omega) n_{s} q_{s} v_{s}^{ \pm} 可以发现
where iE_(x)//E_(y)i E_{x} / E_{y} has been evaluated by Eq. (42). As in Eq. (42) for the electric fields, we see in Eq. (43) that the motion is exactly circular and that the sense of rotation is right-handed or left-handed when n^(2)=Rn^{2}=R and n^(2)=Ln^{2}=L, respectively. 其中 iE_(x)//E_(y)i E_{x} / E_{y} 已由式 (42) 求得。与公式 (42) 中的电场一样,我们在公式 (43) 中看到,运动是完全圆的,当 n^(2)=Rn^{2}=R 和 n^(2)=Ln^{2}=L 时,旋转感分别为右旋或左旋。
In the presence of just a uniform static zero-order magnetic field, B_(0)\mathbf{B}_{0}= hat(z)B_(0)=\hat{\mathbf{z}} B_{0}, and no E\mathbf{E} field, the zero-order motion of charged particles will be helical, the particles spiraling around the magnetic lines of force. From the single-particle equation of motion (7), one readily sees that positive ions will rotate around B_(0)\mathbf{B}_{0} in a left-handed sense. These directions are consistent with the resonant denominators in Eqs. (20) and (21): for omegaB_(0) > 0\omega B_{0}>0, electrons are resonant for omega+Omega_(e)rarr0\omega+\Omega_{e} \rightarrow 0, ions for omega-Omega_(i)rarr0\omega-\Omega_{i} \rightarrow 0. 如果只有均匀的静态零阶磁场 B_(0)\mathbf{B}_{0}= hat(z)B_(0)=\hat{\mathbf{z}} B_{0} ,而没有 E\mathbf{E} 场,带电粒子的零阶运动将是螺旋式的,粒子围绕磁力线螺旋运动。根据单粒子运动方程 (7),我们很容易发现正离子将以左旋方式围绕 B_(0)\mathbf{B}_{0} 旋转。这些方向与公式 (20) 和 (21) 中的共振分母一致:对于 omegaB_(0) > 0\omega B_{0}>0 ,电子与 omega+Omega_(e)rarr0\omega+\Omega_{e} \rightarrow 0 共振,离子与 omega-Omega_(i)rarr0\omega-\Omega_{i} \rightarrow 0 共振。
Further consideration is given to the topic of wave polarization in Probs. 4, 5,6 , and 7 . 在问题 4、5、6 和 7 中,我们将进一步讨论波的极化问题。4、5、6 和 7。
1-5 Cutoff and Resonance 1-5 截止和共振
For certain values of the parameters, n^(2)n^{2} goes to zero or to infinity. W. P. Allis (1959) terms the former case a cutoff and the latter case a resonance. Cutoff occurs, according to Eqs. (29)-(32), when 对于某些参数值, n^(2)n^{2} 会趋于零或无穷大。W. P. Allis (1959) 将前一种情况称为截止,将后一种情况称为共振。根据公式 (29)-(32) ,截止发生在以下情况
P=0quad" or "quad R=0quad" or "quad L=0P=0 \quad \text { or } \quad R=0 \quad \text { or } \quad L=0
For real values of theta\boldsymbol{\theta}, Eq. (35) shows that F^(2)\boldsymbol{F}^{\mathbf{2}} is positive so that n\boldsymbol{n} in Eq. (34) is either pure real or pure imaginary. In going through cutoff n^(2)n^{2} goes through zero, and the transition is made from a region of possible propagation to a region of evanescence.* It will be shown in Chap. 13 that reflection occurs in this circumstance. It will also be shown there that reflection may occur, however, when only a single component of kk passes through zero (which is a less stringent condition than cutoff), while the other two components of kk are fixed by periodicity or boundary conditions. 对于 theta\boldsymbol{\theta} 的实值,公式 (35) 表明 F^(2)\boldsymbol{F}^{\mathbf{2}} 为正值,因此公式 (34) 中的 n\boldsymbol{n} 要么是纯实值,要么是纯虚值。在通过截止时, n^(2)n^{2} 归零,从可能传播区域过渡到衰减区域*。在第 13 章中,我们还将看到,当 kk 中只有一个分量通过零点(这是一个比截止点更宽松的条件),而 kk 的其他两个分量被周期性或边界条件固定时,反射也可能发生。
Resonance occurs for propagation at the angle theta\theta that satisfies the criterion 当传播角度 theta\theta 满足以下标准时,就会发生共振
*The term “evanescence” is used to describe the spatial decay of a wave, where the decay occurs for electromagnetic or kinematic reasons. By contrast, spatial attenuation of a wave can also occur because of absorption processes, and spatial growth because of instability mechanisms. In the latter cases, the divergence of the power flow is nonzero. *衰减 "一词用于描述波的空间衰减,这种衰减是由于电磁或运动学原因造成的。相比之下,波的空间衰减也可能是由于吸收过程,而空间增长则是由于不稳定机制。在后一种情况下,功率流的发散不为零。
tan^(2)theta=-P//S\tan ^{2} \theta=-P / S
In the transition region between propagation and evanescence that occurs where n^(2)n^{2} goes through oo\infty, absorption and/or reflection may occur. This transition will also be discussed in Chap. 13. 在 n^(2)n^{2} 穿过 oo\infty 的传播和衰减之间的过渡区域,可能会发生吸收和/或反射。第 13 章还将讨论这一过渡。
At theta=0\theta=0, resonance occurs for S=(1)/(2)(R+L)rarr+-ooS=\frac{1}{2}(R+L) \rightarrow \pm \infty, and it may be seen from Eqs. (20) and (21) that R rarr+-ooR \rightarrow \pm \infty corresponds to electron cyclotron resonance for a positive omega\omega, and L rarr+-ooL \rightarrow \pm \infty corresponds to an ion cyclotron resonance for a positive omega\omega. (In Prob. 1 it is shown that RR and LL do not diverge at omega=0\omega=0.) At theta=pi//2\theta=\pi / 2, resonance occurs for S=0S=0, which is the condition for the hybrid resonances discussed in the next chapter. Allis terms the resonances at theta=0\theta=0 and theta=pi//2\theta=\pi / 2 principal resonances. 在 theta=0\theta=0 时, S=(1)/(2)(R+L)rarr+-ooS=\frac{1}{2}(R+L) \rightarrow \pm \infty 发生共振,从公式 (20) 和 (21) 可以看出, R rarr+-ooR \rightarrow \pm \infty 对应于正 omega\omega 的电子回旋共振,而 L rarr+-ooL \rightarrow \pm \infty 对应于正 omega\omega 的离子回旋共振。(概率 1 中显示, RR 和 LL 在 omega=0\omega=0 处不发散)。在 theta=pi//2\theta=\pi / 2 处, S=0S=0 发生共振,这就是下一章讨论的混合共振的条件。Allis 将 theta=0\theta=0 和 theta=pi//2\theta=\pi / 2 处的共振称为主共振。
Equation (42) shows a principal resonance also at theta=0\theta=0 and P=0P=0. At this double limit, all the coefficients of Eq. (29) go to zero. The value of nn at the double limit depends on the path of approach in the theta,P\theta, P plane. Resonance results from certain avenues of approach, but for the same branch of Eq. (29) different avenues can give finite values for nn or even n=0n=0 (cutoff). 公式 (42) 显示,在 theta=0\theta=0 和 P=0P=0 处也会产生主共振。在这个双重极限处,公式 (29) 的所有系数都归零。双极限处的 nn 值取决于 theta,P\theta, P 平面上的接近路径。某些路径会产生共振,但对于公式 (29) 的同一分支,不同的路径会产生 nn 甚至 n=0n=0 的有限值(截止值)。
1-6 Wave Normal Surfaces 1-6 波浪法线表面
We now have the pieces of information most needed to discuss normal surfaces for waves propagating through a magnetized uniform cold plasma. As described in the introductory section for this chapter, the wave normal surface is the locus of the phase-velocity vector, v_("phase ")=(omega//k) hat(k)\mathbf{v}_{\text {phase }}=(\omega / k) \hat{\mathbf{k}}, where hat(k)=k//k\hat{\mathbf{k}}=\mathbf{k} / k. The wave normal surfaces are figures of revolution about the B_(0)\mathbf{B}_{0} or hat(z)\hat{\mathbf{z}} axis, and their cross section is a two-dimensional polar plot of omega//k\omega / k vs theta\theta. With k\mathbf{k} in the x,zx, z plane as in Eq. (28), this cross section may be equally well represented as the plot, in Cartesian coordinates, of omegak_(z)//k^(2)\omega k_{z} / k^{2} vs omegak_(x)//k^(2)\omega k_{x} / k^{2}. In either case, one must keep in mind that omega\omega is the solution of the dispersion relation, Eq. (29), omega=omega(k,theta)\omega=\omega(k, \theta) or omega=omega(k_(x),k_(z))\omega=\omega\left(k_{x}, k_{z}\right). 现在,我们已经掌握了讨论波在磁化均匀冷等离子体中传播的法线表面时最需要的信息。正如本章导言部分所述,波法线表面是相速度矢量 v_("phase ")=(omega//k) hat(k)\mathbf{v}_{\text {phase }}=(\omega / k) \hat{\mathbf{k}} 的位置,其中 hat(k)=k//k\hat{\mathbf{k}}=\mathbf{k} / k 。波法线表面是绕 B_(0)\mathbf{B}_{0} 或 hat(z)\hat{\mathbf{z}} 轴旋转的图形,其横截面是 omega//k\omega / k 与 theta\theta 的二维极坐标图。当 k\mathbf{k} 位于 x,zx, z 平面时,如公式 (28) 所示,该横截面同样可以表示为直角坐标下的 omegak_(z)//k^(2)\omega k_{z} / k^{2} vs omegak_(x)//k^(2)\omega k_{x} / k^{2} 图。无论哪种情况,我们都必须牢记 omega\omega 是色散关系式 (29) omega=omega(k,theta)\omega=\omega(k, \theta) 或 omega=omega(k_(x),k_(z))\omega=\omega\left(k_{x}, k_{z}\right) 的解。
The equation for the wave normal surface is easily obtained from Eq. (29), solving for the dimensionless wave phase velocity u=omega//kc=1//nu=\omega / k c=1 / n : 根据公式 (29),求解无量纲波相位速度 u=omega//kc=1//nu=\omega / k c=1 / n 即可轻松得到波法线面方程:
Cu^(4)-Bu^(2)+A=0,C u^{4}-B u^{2}+A=0,
with A,BA, B, and CC given in Eqs. (30)-(32). The properties of the solutions to Eq. (46) are discussed in detail in the following four sections, but it is immediately obvious that if u(theta)u(\theta) is a solution, so are u(-theta),u(pi-theta)u(-\theta), u(\pi-\theta), and u(theta-pi)u(\theta-\pi). The proof is simply that A,BA, B, and CC are functions only of sin^(2)theta\sin ^{2} \theta and cos^(2)theta\cos ^{2} \theta. 其中 A,BA, B 和 CC 在式 (30)-(32) 中给出。下面四节将详细讨论式 (46) 的解的性质,但显而易见的是,如果 u(theta)u(\theta) 是一个解,那么 u(-theta),u(pi-theta)u(-\theta), u(\pi-\theta) 和 u(theta-pi)u(\theta-\pi) 也是一个解。只需证明 A,BA, B , 和 CC 仅是 sin^(2)theta\sin ^{2} \theta 和 cos^(2)theta\cos ^{2} \theta 的函数即可。
Another point of interest is the number of independent parameters in Eq. (46). In Eqs. (30)-(32), A,BA, B, and CC are expressed in terms of P,R,LP, R, L, and S=(R+L)//2S=(R+L) / 2. But for a plasma containing electrons and a single species of ions, the number of free parameters is, in fact, only two. In an explicit example, we assume charge neutrality, Zn_(i)=n_(e)Z n_{i}=n_{e}, and take 另一个值得关注的问题是公式 (46) 中独立参数的数量。在式 (30)-(32) 中, A,BA, B 和 CC 是用 P,R,LP, R, L 和 S=(R+L)//2S=(R+L) / 2 来表示的。但对于包含电子和单一离子的等离子体,自由参数的数量实际上只有两个。在一个明确的例子中,我们假设电荷中性,即 Zn_(i)=n_(e)Z n_{i}=n_{e} ,并取
The formulation of R,LR, L, and PP in terms of just alpha\alpha and beta\beta (the mass ratio, mu\mu, is not considered a free parameter) will be used for the CMA diagram described in the following section and sketched in Figs. 2-1 and 2-2. alpha\alpha and beta\beta form, in fact, the abscissa and ordinate for this diagram. Each added ion species that brings in a new charge-to-mass ratio would add another free parameter to set (48) and another dimension to the CMA diagram, perhaps proportional to the density or plasma fraction for the new species. R,LR, L 和 PP 的表述方式只涉及 alpha\alpha 和 beta\beta (质量比 mu\mu 不视为自由参数),将用于下一节描述的 CMA 图,并在图 2-1 和图 2-2 中绘制。 alpha\alpha 和 beta\beta 实际上构成了该图的横座标和纵座标。每增加一个离子种类,带来一个新的电荷质量比,都会为 (48) 设定增加一个自由参数,并为 CMA 图增加一个维度,或许与新种类的密度或等离子体分数成正比。 gamma\gamma and beta\beta in set (48) comprise an alternative pair of independent parameters, with the advantage that only one member of the pair depends on the frequency omega\omega. 集合 (48) 中的 gamma\gamma 和 beta\beta 是另一对独立参数,其优点是这对参数中只有一个参数取决于频率 omega\omega 。
Figures 1-1 to 1-31-3 present some representative wave normal surfaces corresponding, as will be seen in the next chapter, to the Alfvén modes, the ion cyclotron and fast wave, and the whistler mode. Equations (30) and (45) indicate that more curious shapes for the wave normal surfaces will occur near P=0,S=0P=0, S=0, or S rarr+-ooS \rightarrow \pm \infty, and two sets of such surfaces are illustrated in Figs. 1-4 and 1-5. Problem 13 suggests the drawing of additional wave normal surfaces. 图 1-1 至 1-31-3 展示了一些具有代表性的波法线表面,下一章将介绍阿尔弗韦恩模式、离子回旋和快波以及惠斯勒模式。方程 (30) 和 (45) 表明,在 P=0,S=0P=0, S=0 或 S rarr+-ooS \rightarrow \pm \infty 附近会出现更奇特的波法线表面形状,图 1-4 和图 1-5 展示了两组这样的表面。问题 13 建议绘制更多的波法线表面。
P. C. Clemmow 和 R. F. Mullaly (1955) 提出了一个绘图,W. P. Allis (1959) 以修改的形式提出了一个图表,它非常清楚地说明了冷等离子体中波的分类。双组分等离子体的典型 CMA 图如图 2-1 和 2-2 所示。我们考虑一个坐标系,其中不同方向的尺度长度与等离子体的参数成正比,例如电子密度、静电磁场强度、离子种类的百分比组成和波频率。由这些坐标确定的空间我们称为参数 space。对于 CMA 图,在参数空间中绘制某些表面,这些 Surface 将此空间划分为多个体积块。(我们将参数空间中的这些体积称为有界体积,因为形成它们的表面是有界表面,但我们并不意味着它们的所有尺寸都是有限的。在没有边界表面干预的情况下,有界体积在参数空间中拉伸到无穷大。对于双组分等离子体,
The bounding surfaces are the surfaces for cutoff and for the principal resonances. In Sec. 1-5 these were found to be the P=0,R=0P=0, R=0, and L=0L=0 surfaces for cutoff, P=0,R rarr ooP=0, R \rightarrow \infty, and L rarr ooL \rightarrow \infty for resonance at theta=0\theta=0, and the surface S=0S=0 for resonance at theta=pi//2\theta=\pi / 2. 边界面是截止面和主共振面。在第 1-5 章中,我们发现这些表面是用于截止的 P=0,R=0P=0, R=0 和 L=0L=0 面,用于 theta=0\theta=0 处共振的 P=0,R rarr ooP=0, R \rightarrow \infty 和 L rarr ooL \rightarrow \infty 面,以及用于 theta=pi//2\theta=\pi / 2 处共振的 S=0S=0 面。
In this section and in the one that follows, we shall show that inside each bounded volume in parameter space the shapes or topological genera of the wave normal surfaces are unchanged. Referring to the CMA diagram, Fig. 2-1, the wave normal surfaces are sketched inside each bounded volume and each sketch remains topologically correct throughout the bounded volume. On the other hand, a geometrically correct representation of a wave normal 在本节和接下来的章节中,我们将说明在参数空间的每个有界体积内,波法线表面的形状或拓扑属概是不变的。参照 CMA 图(图 2-1),波法线面在每个有界体积内都有草图,而且每个草图在整个有界体积内都保持拓扑正确。另一方面,波法线面的几何正确表示方法是
在列出了我们的断言 1 到 5 之后,我们可以讨论波法线表面的形状或拓扑属。波法向表面是通过绘制无量纲相速度来形成的u=omega//kc=1//nu=\omega / k c=1 / n对theta\theta、方程。(46)-(48).我们首先考虑以下可能性:PP和SS是同一星座。然后,通过断言2,u2, u不能在有界卷内变为零,并且通过断言1,u1, u无法转到oo\infty在有界体积内。因此,通过断言 3,如果uu在任何地方都是实数,那么对于所有theta\theta,并且波法线表面在拓扑上必须等同于球体。 在列出了我们的断言 1 到 5 之后,我们可以讨论波法线表面的形状或拓扑属。波法向表面是通过绘制无量纲相速度来形成的 u=omega//kc=1//nu=\omega / k c=1 / n 对 theta\theta 、方程。(46)-(48)。我们首先考虑以下可能性: PP 和 SS 是同一星座。然后,通过断言 2,u2, u 不能在有界卷内变为零,并且通过断言 1,u1, u 无法转到 oo\infty 在有界体积内。因此,通过断言 3,如果 uu 在任何地方都是实数,那么对于所有 theta\theta ,并且波法线表面在拓扑上必须等同于球体。
另一种可能性是PP和SS是相反的星座。然后通过断言 2 有一个u^(2)u^{2}在角度处变为 0theta_("res ")\theta_{\text {res }}和pi\pi - theta_("res ")\theta_{\text {res }}在有界体积内的每个点。在u=0u=0,波法线表面看起来像两个相同的同轴圆锥体与顶点相遇的关节。通过断言 2,uu可以在没有其他角度的情况下变为 0,并且通过断言1,u1, u无法转到oo\infty.因此uu除了圆锥的顶点之外,在任何地方都是有限的,并且通过断言3,u3, u要么在圆锥体内为实,在圆锥体外为虚,要么在圆锥体内为虚,在外为实。这两种可能性产生的波法向表面在拓扑学上等同于 lemniscoids(旋转的 lemniscates),在第一种情况下类似于哑铃,在第二种情况下类似于轮子。波法线表面将在参数空间中有界体积的整个内部保留哑铃 lemniscoid 或 wheel lemniscoid 的拓扑属。 (球体的拓扑属、哑铃 lemniscoid 和 wheel lemniscoid 彼此不同,可以通过相对于原点周围的球体反转表面来更容易地可视化。这种转变承载着 要么在圆锥体内为实,在圆锥体外为虚,要么在圆锥体内为虚,在外为实。这两种可能性产生的波法向表面在拓扑学上等同于 lemniscoids(旋转的 lemniscates),在第一种情况下类似于哑铃,在第二种情况下类似于轮子。波法线表面将在参数空间界中有体积的整个内部保留哑铃lemniscoid 或 wheel lemniscoid 的拓扑属。 (球体的拓扑属、哑铃 lemniscoid 和 wheel lemniscoid 彼此不同,可以通过相对于原点周围的球体反转表面来更容易地可视化。这种转变承载着uu into nn. These three surfaces are then transformed into a sphere, a hyperboloid of two sheets, and a hyperboloid of one sheet, respectively.) 另一种可能性是 PP 和 SS 是相反的星座。然后通过断言 2 有一个 u^(2)u^{2} 在角度处变为 0 theta_("res ")\theta_{\text {res }} 和 pi\pi -。 theta_("res ")\theta_{\text {res }} 在有界体积内的每个点。在 u=0u=0 ,波法线表面看起来像两个相同的同轴圆锥体与顶点相遇的关节。通过断言 2, uu 可以在没有其他角度的情况下变为 0,并且通过断言 1,u1, u 无法转到 oo\infty .因此 uu 除了圆锥的顶点之外,在任何地方都是有限的,并且通过断言 3,u3, u 要么在圆锥体内为实,在圆锥体外为虚,要么在圆锥体内为虚,在外为实。这两种可能性产生的波法向表面在拓扑学上等同于 lemniscoids(旋转的lelemniscates),在第一种情况下类似于哑铃,在第二种情况下类似于轮子。波法线表面将在参数空间中有体积的整个内部保留哑铃 lemniscoid 或 wheel lemniscoid 的拓扑属。(球体的拓扑属、哑铃 lemniscoid 和轮子 lemniscoid 彼此不同,可以通过相对于原点周围的球体反转表面来更容易地可视化。这种转变承载着 uu 到 nn 。然后将这三个表面分别转化为一个球体、一个由两个薄片组成的双曲面和一个由一个薄片组成的双曲面)。
Finally, we must consider the second branch of u^(2)u^{2} for the case where PP and SS are of opposite signs. We want to demonstrate that if uu is real, then this branch gives a wave normal surface in the form of a topological sphere. We know from assertion 1 that uu cannot go to oo\infty inside the bounded volume, and so we have only to show that it cannot go to 0 either. If the second branch of u^(2)u^{2} goes to zero inside the bounding volume, it must do so at tan^(2)theta\tan ^{2} \theta=-P//S=-P / S,由方程 (45) 计算。如果我们考虑方程(46)在谐振角附近的系数,我们可以使用 计算。如果我们考虑方程(46)在谐振角附近的系数,我们可以使用计算。如果我们考虑方程(46)在谐振角附近的系数,我们可以使用sin^(2)theta≃-Pcos^(2)theta//S\sin ^{2} \theta \simeq-P \cos ^{2} \theta / S与方程(33)一起近似于谐振角附近的方程(46), 最后,我们必须考虑 u^(2)u^{2} 的第二个分支,即 PP 和 SS 符号相反的情况。我们要证明,如果 uu 是实数,那么这个分支给出的波法线面是拓扑球的形式。我们从断言 1 中知道, uu 在有界体积内不会变为 oo\infty ,因此我们只需证明它也不会变为 0。如果 u^(2)u^{2} 的第二个分支在有界卷内归零,它必须在 tan^(2)theta\tan ^{2} \theta=-P//S=-P / S 处归零,由方程 (45)计算。计算。如果我们考虑方程(46)在谐振角附近的系数,我们可以使用 sin^(2)theta≃-Pcos^(2)theta//S\sin ^{2} \theta \simeq-P \cos ^{2} \theta / S 与方程(33)一起近似于谐振角附近的方程(46),
PRLu^(4)-(P)/(S)(D^(2)cos^(2)theta+S^(2))u^(2)+(∼0)=0P R L u^{4}-\frac{P}{S}\left(D^{2} \cos ^{2} \theta+S^{2}\right) u^{2}+(\sim 0)=0
完整过渡。将参数空间中的边界面作为截止面和主共振面。如果一个模式在特定的边界表面上没有遇到 cutoff 或 principal resony,它将以完整的过渡穿过这个表面。 将参数空间中的边界面作为截止面和主共振面。如果一个模式在特定的边界表面上没有遇到截止 或 principal resony,它将以完整的过渡穿过这个表面。