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Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics

Kip S. Thorne, Roger D. Blandford

Chapter 21

Waves in Cold Plasmas: Two-Fluid Formalism - all with Video Answers

Educators

Chapter Questions

Problem 1

We developed a one-fluid (MHD) description of plasma in Chap. 19, and in Chap. 20 we showed how to describe the orbits of individual charged particles in a magnetic field that varies slowly compared with the particles' orbital periods and radii. In this chapter, we describe the plasma as two or more cold fluids. We relate these three approaches in this exercise.
(a) Generalize Eq. (21.13b) to a single fluid as:
$$
\rho \mathbf{a}=\rho \mathbf{g}-\nabla \cdot \mathbf{P}+\rho_e \mathbf{E}+\mathbf{j} \times \mathbf{B},
$$
where $\mathbf{a}$ is the fluid acceleration, and $\mathbf{g}$ is the acceleration of gravity, and write the pressure tensor as $\mathbf{P}=P_{\perp} \mathbf{g}+\left(P_{\|}-P_{\perp}\right) \mathbf{B} \otimes \mathrm{B} / B^2$, where we suppress the subscript $s .{ }^1$ Show that the component of current density perpendicular to the local magnetic field is
$$
\mathbf{j}_{\perp}=\frac{\mathbf{B} \times \boldsymbol{\nabla} \cdot \mathbf{P}}{B^2}+\rho_e \frac{\mathbf{E} \times \mathbf{B}}{B^2}+\rho \frac{(\mathbf{g}-\mathbf{a}) \times \mathbf{B}}{B^2} .
$$
(b) Use vector identities to rewrite Eq. (21.16a) in the form:
$$
\mathbf{j}_{\perp}=P_{\|} \frac{\mathbf{B} \times(\mathbf{B} \cdot \boldsymbol{\nabla}) \mathbf{B}}{B^4}+P_{\perp} \frac{\mathbf{B} \times \boldsymbol{\nabla} \boldsymbol{B}}{B^3}-(\boldsymbol{\nabla} \times \mathbf{M})_{\perp}+\rho_e \frac{\mathbf{E} \times \mathbf{B}}{B^2}+\rho \frac{(\mathbf{g}-\mathbf{a}) \times \mathbf{B}}{B^2},
$$
where
$$
\mathrm{M}=P_{\perp} \frac{\mathrm{B}}{B^3}
$$
is the magnetization.
(c) Identify the first term of Eq. (21.16b) with the curvature drift (20.49) and the second term with the gradient drift (20.51).
(d) Using a diagram, explain how the magnetization (20.54)-the magnetic moment per unit volume-can contribute to the current density. In particular, consider what might happen at the walls of a cavity containing plasma. ${ }^2$ Argue that there should also be a local magnetization current parallel to the magnetic field, even when there is no net drift of the particles.
(e) Associate the final two terms of Eq. (21.16b) with the "E $\times \mathbf{B}^"$ drift (20.47) and the gravitational drift (20.48). Explain the presence of the acceleration in the gravitational drift.
(f) Discuss how to combine these contributions to rederive the standard formulation of MHD, and specify some circumstances under which MHD might be a poor approximation.

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01:27

Problem 2

Derive Eqs. (21.27) for the phase and group velocities of electromagnetic modes in a plasma.

Urvashi Arora
Urvashi Arora
Numerade Educator

Problem 3

Verify Eq. (21.28).

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01:27

Problem 4

Consider a transverse electromagnetic wave mode propagating in an unmagnetized, partially ionized gas in which the electron-neutral collision frequency is $v_e$. Include the effects of collisions in the electron equation of motion (21.17), by introducing a term $-n_e m_e v_e \mathbf{u}_e$ on the right-hand side. Ignore ion motion and electron-ion and electron-electron collisions.

Derive the dispersion relation when $\omega \gg v_e$, and show by explicit calculation that the rate of loss of energy per unit volume ( $-\nabla \cdot \mathbf{F}_{\mathrm{EM}}$, where $\mathbf{F}_{\mathrm{EM}}$ is the Poynting flux) is balanced by the Ohmic heating of the plasma. [Hint: It may be easiest to regard $\omega$

Urvashi Arora
Urvashi Arora
Numerade Educator
02:30

Problem 5

Ion-acoustic waves can propagate in an unmagnetized plasma when the electron temperature $T_e$ greatly exceeds the ion temperature $T_p$. In this limit, the electron density $n_e$ can be approximated by $n_e=n_0 \exp \left[e \Phi /\left(k_B T_e\right)\right]$ where $n_0$ is the mean electron density, and $\Phi$ is the electrostatic potential.
(a) Show that for ion-acoustic waves that propagate in the $z$ direction, the nonlinear equations of continuity, the motion for the ion (proton) fluid, and Poisson's equation for the potential take the form
$$
\begin{aligned}
\frac{\partial n}{\partial t}+\frac{\partial(n u)}{\partial z} & =0, \\
\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial z} & =-\frac{e}{m_p} \frac{\partial \Phi}{\partial z}, \\
\frac{\partial^2 \Phi}{\partial z^2} & =-\frac{e}{\epsilon_0}\left(n-n_0 e^{e \Phi /\left(k_B T_e\right)}\right) .
\end{aligned}
$$

Here $n$ is the proton density, and $u$ is the proton fluid velocity (which points in the $z$ direction).
(b) Linearize Eqs. (21.35), and show that the dispersion relation for small-amplitude ion-acoustic modes is
$$
\omega=\omega_{p p}\left(1+\frac{1}{\lambda_D^2 k^2}\right)^{-1 / 2}=\left(\frac{k_B T_e / m_p}{1+\lambda_D^2 k^2}\right)^{1 / 2} k,
$$
where $\lambda_D$ is the Debye length.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator

Problem 6

In this exercise we explore nonlinear effects in ion-acoustic waves (Ex. 21.5), and show that they give rise to solitons that obey the same KdV equation as governs solitonic water waves (Sec. 16.3). This version of the solitons is only mildly nonlinear. In Sec. 23.6 we will generalize to strong nonlinearity.
(a) Introduce a bookkeeping expansion parameter $\varepsilon$ whose numerical value is unity, ${ }^5$ and expand the ion density, ion velocity, and potential in the forms
$$
\begin{aligned}
n & =n_0\left(1+\varepsilon n_1+\varepsilon^2 n_2+\ldots\right), \\
u & =\left(k_B T_e / m_p\right)^{1 / 2}\left(\varepsilon u_1+\varepsilon^2 u_2+\ldots\right), \\
\Phi & =\left(k_B T_e / e\right)\left(\varepsilon \Phi_1+\varepsilon^2 \Phi_2+\ldots\right) .
\end{aligned}
$$

Here $n_1, u_1$, and $\Phi_1$ are small compared to unity, and the factors of $\varepsilon$ tell us that, as the wave amplitude is decreased, these quantities scale proportionally to one another, while $n_2, u_2$, and $\Phi_2$ scale proportionally to the squares of $n_1, u_1$, and $\Phi_1$, respectively. Change independent variables from $(t, z)$ to $(\tau, \eta)$, where
$$
\begin{aligned}
\eta & =\sqrt{2} \varepsilon^{1 / 2} \lambda_D^{-1}\left[z-\left(k_B T_e / m_p\right)^{1 / 2} t\right], \\
\tau & =\sqrt{2} \varepsilon^{3 / 2} \omega_{p p} t .
\end{aligned}
$$

Explain, now or at the end, the chosen powers $\varepsilon^{1 / 2}$ and $\varepsilon^{3 / 2}$. By substituting Eqs. (21.37) and (21.38) into the nonlinear equations (21.35), equating terms of the same order in $\varepsilon$, and then setting $\varepsilon=1$ (bookkeeping parameter!), show that $n_1, u_1$, and $\Phi_1$ each satisfy the $\mathrm{KdV}$ equation (16.32):
$$
\frac{\partial \zeta}{\partial \tau}+\zeta \frac{\partial \zeta}{\partial \eta}+\frac{\partial^3 \zeta}{\partial \eta^3}=0
$$
(b) In Sec. 16.3 we discussed the exact, single-soliton solution (16.33) to this KdV equation. Show that for an ion-acoustic soliton, this solution propagates with the physical speed $\left(1+n_{1 o}\right)\left(k_B T_e / m_p\right)^{1 / 2}$ (where $n_{1 o}$ is the value of $n_1$ at the peak of the soliton), which is greater the larger is the wave's amplitude.

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Problem 7

Derive $\mathrm{Eq}$. (21.52) for Faraday rotation.

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Problem 8

A narrow bundle of magnetic field lines with cross sectional area $A$, together with the plasma attached to them, can be thought of as like a stretched string. When such a string is plucked, waves travel down it with phase speed $\sqrt{T / \Lambda}$, where $T$ is the string's tension, and $\Lambda$ is its mass per unit length (Sec. 12.3.3). The plasma analog is Alfvén waves propagating parallel to the plasma-laden magnetic field.
(a) Analyzed nonrelativistically, the tension for our bundle of field lines is $T=$ $\left[B^2 /\left(2 \mu_0\right)\right] A$ and the mass per unit length is $\Lambda=\rho A$, so we expect a phase velocity $\sqrt{T / \Lambda}=\sqrt{B^2 /\left(2 \mu_0 \rho\right)}$, which is $1 / \sqrt{2}$ of the correct result. Where is the error? [Hint: In addition to the restoring force on bent field lines, due to tension along the field, there is also the curvature force (B $\cdot \boldsymbol{\nabla}) \mathbf{B} / \mu_0 ;$ Eq. (19.15).]
(b) In special relativity, the plasma-laden magnetic field has a tensorial inertial mass per unit volume that is discussed in Ex. 2.27. Explain why, when the field lines (which point in the $z$ direction) are plucked so they vibrate in the $x$ direction, the inertial mass per unit length that resists this motion is $\Lambda=\left(T^{00}+T^{x x}\right) A=$ $\left[\rho+B^2 /\left(\mu_0 c^2\right)\right] A$. (In the first expression for $\Lambda, T^{00}$ is the mass-energy density of plasma and magnetic field, $T^{x x}$ is the magnetic pressure along the $x$ direction, and the speed of light is set to unity as in Chap. 2; in the second expression, the speed of light has been restored to the equation using dimensional arguments.) Show that the magnetic contribution to this inertial mass gives the relativistic correction $1 / \sqrt{1+a^2 / c^2}$ to the Alfvén waves' phase speed, Eq. (21,56).

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Problem 9

Derive Eq. (21.65).

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09:54

Problem 10

A radio pulsar emits regular pulses at $1-s$ intervals, which propagate to Earth through the ionized interstellar plasma with electron density $n_e \simeq 3 \times 10^4 \mathrm{~m}^{-3}$. The pulses observed at $f=100 \mathrm{MHz}$ are believed to be emitted at the same time as those observed at much higher frequency, but they arrive with a delay of $100 \mathrm{~ms}$.
(a) Explain briefly why pulses travel at the group velocity instead of the phase velocity, and show that the expected time delay of the $f=100-\mathrm{MHz}$ pulses relative to the high-frequency pulses is given by
$$
\Delta t=\frac{e^2}{8 \pi^2 m_e \epsilon_0 f^2 c} \int n_e d x
$$
where the integral is along the waves' propagation path. Hence compute the distance to the pulsar.
(b) Now suppose that the pulses are linearly polarized and that their propagation is accurately described by the quasi-longitudinal approximation. Show that the plane of polarization will be Faraday rotated through an angle
$$
\Delta \chi=\frac{e \Delta t}{m_e}\left\langle B_1\right\rangle,
$$
where $\left\langle B_{\|}\right\rangle=\int n_e \mathbf{B} \cdot d \mathbf{x} / \int n_e d x$. The plane of polarization of the pulses emitted at $100 \mathrm{MHz}$ is believed to be the same as the emission plane for higher frequencies, but when the pulses arrive at Earth, the $100-\mathrm{MHz}$ polarization plane is observed to be rotated through 3 radians relative to that at high frequencies. Calculate the mean parallel component of the interstellar magnetic field.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:10

Problem 11

Consider a wave mode propagating through a plasma-for example, the ionospherein which the direction of the background magnetic field is slowly changing. We have just demonstrated that so long as B is not almost perpendicular to $\mathbf{k}$, we can use the quasi-longitudinal approximation, the difference in phase velocity between the two eigenmodes is $\alpha \mathbf{B} \cdot \mathbf{k}$, and the integral for the magnitude of the rotation of the plane of polarization is $\propto \int n_e \mathbf{B} \cdot d \mathbf{x}$.

Now, suppose that the parallel component of the magnetic field changes sign. It has been implicitly assumed that the faster eigenmode, which is circularly polarized in the quasi-longitudinal approximation, becomes the slower eigenmode (and vice versa) when the field is reversed, and the Faraday rotation is undone. However, if we track the modes using the full dispersion relation, we find that the faster quasi-longitudinal eigenmode remains the faster eigenmode in the quasi-perpendicular regime, and it becomes the faster eigenmode with opposite sense of circular polarization in the field-reversed quasi-longitudinal regime. Now, let there be a second field reversal where an analogous transition occurs. Following this logic, the net rotation should be $\propto \int n_e|\mathbf{B} \cdot d \mathbf{x}|$. What is going on?

Narayan Hari
Narayan Hari
Numerade Educator
09:40

Problem 12

The free electron density in the night-time ionosphere increases exponentially from $10^9 \mathrm{~m}^{-3}$ to $10^{11} \mathrm{~m}^{-3}$ as the altitude increases from 100 to $200 \mathrm{~km}$, and the density diminishes above this height. Use Snell's law [Eq. (7.49)] to calculate the maximum range of $10-\mathrm{MHz}$ waves transmitted from Earth's surface, assuming a single ionospheric reflection.

Ameer Said
Ameer Said
Numerade Educator
22:13

Problem 13

Verify that the group velocity of a wave mode is perpendicular to the refractive-index surface (Fig. 21.6b).

Prachita Kush
Prachita Kush
Numerade Educator
01:27

Problem 14

For each of the following modes studied earlier in this chapter, identify in the CMA diagram the phase speed, as a function of frequency $\omega$, and verify that the turning on and cutting off of the modes, and the relative speeds of the modes, are in accord with the CMA diagram's wave-normal curves.
(a) Electromagnetic modes in an unmagnetized plasma.
(b) Left and right modes for parallel propagation in a magnetized plasma.
(c) Ordinary and extraordinary modes for perpendicular propagation in a magnetized plasma.

Urvashi Arora
Urvashi Arora
Numerade Educator

Problem 15

Verify Eq. (21.76).

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01:49

Problem 16

In a very strong magnetic field, we can consider electrons as constrained to move in 1 dimension along the direction of the magnetic field. Consider a beam of relativistic protons propagating with density $n_b$ and speed $u_b \sim c$ through a cold electron-proton plasma along B. Generalize the dispersion relation (21.74) for modes with $\mathbf{k} \|$ B.

Anand Jangid
Anand Jangid
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Problem 17

Another type of wave mode that can be found from a fluid description of a plasma (but requires a kinetic treatment to understand completely) is a drift wave. Just as the two-stream instability provides a mechanism for plasmas to erase nonuniformity in velocity space, so drift waves can rapidly remove spatial irregularities.

The limiting case that we consider here is a modification of an ion-acoustic mode in a strongly magnetized plasma with a density gradient. Suppose that the magnetic field is uniform and points in the $\mathbf{e}_{\mathbf{z}}$ direction. Let there be a gradient in the equilibrium density of both the electrons and the protons: $n_0=n_0(x)$. In the spirit of our description of ion-acoustic modes in an unmagnetized, homogeneous plasma [cf. Eq. (21.33)], treat the proton fluid as cold, but allow the electrons to be warm and isothermal with temperature $T_e$. We seek modes of frequency $\omega$ propagating perpendicular to the density gradient $\left[\right.$ i.e., with $\left.\mathbf{k}=\left(0, k_y, k_z\right)\right]$.
(a) Consider the equilibrium of the warm electron fluid, and show that there must be a fluid drift velocity along the direction $\mathbf{e}_y$ of magnitude
$$
V_{d e}=-\frac{V_{i a}^2}{\omega_{c i}} \frac{1}{n_0} \frac{d n_0}{d x}
$$
where $V_{i a}=\left(k_B T_e / m_p\right)^{1 / 2}$ is the ion-acoustic speed. Explain in physical terms the origin of this drift and why we can ignore the equilibrium drift motion for the ions (protons).
(b) We limit our attention to low-frequency electrostatic modes that have phase velocities below the Alfvén speed. Under these circumstances, perturbations to the magnetic field can be ignored, and the electric field can be written as $\mathbf{E}=-\nabla \Phi$. Write down the three components of the linearized proton equation of motion in terms of the perturbation to the proton density $n$, the proton fluid velocity $\mathbf{u}$, and the electrostatic potential $\Phi$.
(c) Write down the linearized equation of proton continuity, including the gradient in $n_0$, and combine with the equation of motion to obtain an equation for the fractional proton density perturbation at low frequencies:
$$
\frac{\delta n}{n_0}=\left(\frac{\left(\omega_{c p}^2 k_z^2-\omega^2 k^2\right) V_{i a}^2+\omega_{c p}^2 \omega k_y V_{d e}}{\omega^2\left(\omega_{c p}^2-\omega^2\right)}\right)\left(\frac{e \Phi}{k_B T_e}\right) .
$$
(d) Argue that the fractional electron-density perturbation follows a linearized Boltzmann distribution, so that
$$
\frac{\delta n_e}{n_0}=\frac{e \Phi}{k_B T_e}
$$
(e) Use both the proton- and the electron-density perturbations in Poisson's equation to obtain the electrostatic drift wave dispersion relation in the low-frequency $\left(\omega \ll \omega_{c p}\right)$, long-wavelength $\left(k \lambda_D \ll 1\right)$ limit:
$$
\omega=\frac{k_y V_{d e}}{2} \pm \frac{1}{2}\left(k_y^2 V_{d e}^2+4 k_z^2 V_{i a}^2\right)^{1 / 2} .
$$

Describe the physical character of the mode in the additional limit $k_2 \rightarrow 0$. A proper justification of this procedure requires a kinetic treatment, which also shows that, under some circumstances, drift waves can be unstable and grow exponentially.

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