10. Electrostatic Waves
10. Electrostatic Waves¶
Learning Objectives¶
- Understand the general framework for electrostatic wave dispersion relations
- Derive the Bohm-Gross dispersion for Langmuir waves in warm plasmas
- Analyze ion acoustic waves and their damping conditions
- Compute upper hybrid and lower hybrid resonances in magnetized plasmas
- Learn about Bernstein modes and their role in wave heating
- Solve electrostatic dispersion relations numerically for various plasma parameters
Introduction¶
In the previous lessons, we studied MHD and electromagnetic waves, which involve magnetic field perturbations. Now we focus on electrostatic waves, where:
$$\mathbf{B}_1 = 0, \quad \mathbf{E}_1 = -\nabla\phi_1$$
These waves involve only electric field fluctuations and can be derived from the scalar potential. They arise from charge separation and are governed by Poisson's equation:
$$\nabla \cdot \mathbf{E}_1 = \frac{\rho_1}{\epsilon_0}$$
Electrostatic waves are important for: - Plasma heating and current drive (electron cyclotron, lower hybrid) - Diagnostics (Thomson scattering, collective modes) - Understanding instabilities (two-stream, drift waves) - Wave-particle interactions (Landau damping, quasilinear theory)
The key tool is the dielectric function $\epsilon(\mathbf{k}, \omega)$, which encodes the plasma's response to electromagnetic perturbations.
1. General Electrostatic Dispersion Framework¶
1.1 Dielectric Function and Susceptibility¶
For a perturbation $\propto e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)}$, Poisson's equation becomes:
$$\mathbf{k} \cdot \mathbf{E}_1 = \frac{\rho_1}{\epsilon_0}$$
The induced charge density $\rho_1$ depends on the electric field through the plasma response. Define the dielectric function $\epsilon(\mathbf{k}, \omega)$ such that:
$$\mathbf{k} \cdot \mathbf{D} = \mathbf{k} \cdot (\epsilon_0 \epsilon \mathbf{E}_1) = 0$$
For electrostatic waves, this gives:
$$\boxed{\epsilon(\mathbf{k}, \omega) = 0}$$
This is the electrostatic dispersion relation.
The dielectric function is related to the susceptibility $\chi_s$ of each species:
$$\epsilon = 1 + \sum_s \chi_s(\mathbf{k}, \omega)$$
The susceptibility describes how species $s$ responds to the electric field:
$$\chi_s = \frac{n_{1s}/n_0}{\epsilon_0 E_1 / (e_s n_0)}$$
From the linearized Vlasov equation, the susceptibility for species $s$ is:
$$\chi_s(\mathbf{k}, \omega) = -\frac{\omega_{ps}^2}{k^2} \int \frac{\mathbf{k} \cdot \partial f_0/\partial \mathbf{v}}{\omega - \mathbf{k}\cdot\mathbf{v}} d^3v$$
where $\omega_{ps}^2 = n_0 e_s^2 / (\epsilon_0 m_s)$ is the plasma frequency.
1.2 Cold Plasma Limit¶
For a cold plasma ($T = 0$, $f_0 = n_0 \delta(\mathbf{v})$), the integral simplifies. Particles respond to the wave electric field via:
$$m \frac{d\mathbf{v}_1}{dt} = e \mathbf{E}_1$$
For harmonic perturbations $\propto e^{-i\omega t}$: $$-i\omega m \mathbf{v}_1 = e \mathbf{E}_1$$
The density perturbation from continuity is: $$-i\omega n_1 + n_0 i\mathbf{k}\cdot\mathbf{v}_1 = 0$$
Combining: $$n_1 = \frac{n_0 \mathbf{k}\cdot\mathbf{v}_1}{\omega} = \frac{n_0 e \mathbf{k}\cdot\mathbf{E}_1}{m\omega^2}$$
The susceptibility is: $$\chi_s = \frac{n_1 e_s}{\epsilon_0 E_1 n_0} = -\frac{\omega_{ps}^2}{\omega^2}$$
The dielectric function for multiple cold species:
$$\epsilon = 1 - \sum_s \frac{\omega_{ps}^2}{\omega^2}$$
1.3 Warm Plasma: Kinetic Effects¶
For a warm plasma with thermal velocity $v_{th}$, kinetic effects become important when:
$$\frac{\omega}{k} \sim v_{th}$$
The phase velocity $v_\phi = \omega/k$ is comparable to particle velocities. This leads to: - Landau damping: resonant wave-particle interaction - Thermal dispersion: wave frequency depends on $k v_{th}$
For a Maxwellian distribution: $$f_0(\mathbf{v}) = n_0 \left(\frac{m}{2\pi k_B T}\right)^{3/2} \exp\left(-\frac{m v^2}{2k_B T}\right)$$
the 1D susceptibility (for $\mathbf{k} = k \hat{z}$) involves the plasma dispersion function $Z(\zeta)$:
$$\chi_s = -\frac{\omega_{ps}^2}{k^2 v_{th,s}^2} \left[1 + \zeta_s Z(\zeta_s)\right]$$
where $\zeta_s = \omega/(k v_{th,s})$ and:
$$Z(\zeta) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} \frac{e^{-x^2}}{x - \zeta} dx$$
For large $|\zeta|$ (phase velocity $\gg v_{th}$): $$Z(\zeta) \approx -\frac{1}{\zeta}\left(1 + \frac{1}{2\zeta^2} + \frac{3}{4\zeta^4} + \cdots\right)$$
This gives thermal corrections to the cold plasma result.
2. Langmuir Waves (Electron Plasma Oscillations)¶
2.1 Cold Plasma: Pure Oscillation¶
In an unmagnetized plasma with stationary ions, the cold plasma dispersion is:
$$\epsilon = 1 - \frac{\omega_{pe}^2}{\omega^2} = 0$$
This gives: $$\boxed{\omega = \omega_{pe}}$$
These are Langmuir waves (or electron plasma oscillations), first observed by Langmuir and Tonks in 1929.
Properties: - Dispersionless: $\omega$ independent of $k$ - No propagation: group velocity $v_g = d\omega/dk = 0$ - Standing oscillation: electrons oscillate collectively - High frequency: $\omega_{pe} \sim 10^{9}-10^{11}$ rad/s for typical plasmas
Physical picture:
Time t=0: Time t=T/4: Time t=T/2:
- + - + - - - - + + + - + - + -
Ions (stationary) Electrons Back to equilibrium
displaced
2.2 Warm Plasma: Bohm-Gross Dispersion¶
Including electron thermal motion, the dispersion becomes:
$$1 - \frac{\omega_{pe}^2}{k^2 v_{th,e}^2}[1 + \zeta_e Z(\zeta_e)] = 0$$
For $\omega \gg k v_{th,e}$ (weak thermal effects), expand to first order:
$$1 - \frac{\omega_{pe}^2}{\omega^2}\left(1 + \frac{3k^2 v_{th,e}^2}{\omega^2}\right) = 0$$
Solving for $\omega^2$:
$$\boxed{\omega^2 = \omega_{pe}^2 + 3k^2 v_{th,e}^2}$$
This is the Bohm-Gross dispersion relation (1949), where $v_{th,e} = \sqrt{k_B T_e / m_e}$.
Properties: - Dispersive: $\omega$ depends on $k$ - Phase velocity: $v_\phi = \omega/k = \sqrt{\omega_{pe}^2/k^2 + 3v_{th,e}^2}$ - Group velocity: $v_g = d\omega/dk = 3k v_{th,e}^2/\omega$ - Cutoff: minimum frequency $\omega_{pe}$ at $k = 0$
The thermal correction is important when: $$k\lambda_D \sim 1$$
where $\lambda_D = v_{th,e}/\omega_{pe}$ is the Debye length.
2.3 Landau Damping of Langmuir Waves¶
For $\omega/k \sim v_{th,e}$, the imaginary part of $Z(\zeta)$ gives damping:
$$\text{Im}\,\omega = -\sqrt{\frac{\pi}{8}} \frac{\omega_{pe}}{k^3 \lambda_D^3} \exp\left(-\frac{\omega_{pe}^2}{2k^2 v_{th,e}^2} - \frac{3}{2}\right)$$
The damping rate decreases exponentially for $k\lambda_D \ll 1$.
Physical picture: particles moving at $v \approx \omega/k$ can exchange energy with the wave resonantly. If there are more slow particles than fast particles (Maxwellian has negative slope at high $v$), the wave loses energy.
3. Ion Acoustic Waves¶
3.1 Derivation: Two-Fluid Model¶
Consider low-frequency oscillations where: - Electrons respond adiabatically (fast, isothermal) - Ions respond dynamically (slow, inertial)
Electron response (Boltzmann relation): $$n_e = n_0 \exp\left(\frac{e\phi}{k_B T_e}\right) \approx n_0\left(1 + \frac{e\phi}{k_B T_e}\right)$$
Ion continuity and momentum: $$\frac{\partial n_i}{\partial t} + \nabla\cdot(n_i \mathbf{v}_i) = 0$$
$$m_i n_i \frac{\partial \mathbf{v}_i}{\partial t} = -e n_i \nabla\phi - \nabla(n_i k_B T_i)$$
Poisson equation: $$\epsilon_0 \nabla^2 \phi = e(n_e - n_i)$$
Linearizing with $n_e = n_0 + n_{e1}$, $n_i = n_0 + n_{i1}$, $\phi = \phi_1$, and assuming $e^{i(kx - \omega t)}$:
$$-\omega^2 m_i n_{i1} = -k^2 e n_{i1} \phi_1 - k^2 n_{i1} k_B T_i$$
$$-k^2 \epsilon_0 \phi_1 = e\left(\frac{e n_0 \phi_1}{k_B T_e} - n_{i1}\right)$$
Combining: $$-k^2 \epsilon_0 \phi_1 = e\left(\frac{e n_0 \phi_1}{k_B T_e} - \frac{k^2 e \phi_1 (n_0 + k_B T_i k^2/\omega^2 m_i)}{\omega^2 m_i - k^2 k_B T_i}\right)$$
After algebra, the dispersion relation is:
$$\omega^2 = \frac{k^2 k_B T_e}{m_i (1 + k^2 \lambda_D^2)} \left(\frac{1 + 3T_i/T_e}{1 + k^2\lambda_D^2}\right)$$
For $k\lambda_D \ll 1$:
$$\boxed{\omega \approx k c_s}$$
where the ion acoustic speed is:
$$\boxed{c_s = \sqrt{\frac{k_B T_e}{m_i}}}$$
assuming $T_i \ll T_e$.
3.2 Physical Picture¶
Ion acoustic waves are analogous to sound waves in a neutral gas, but: - Restoring force: electron pressure (not ion pressure) - Inertia: ion mass - Electrons act as a thermodynamic reservoir
Ion density: High Low High Low
| | | |
v ^ v ^
+++ --- +++ ---
Electron cloud: (responds adiabatically)
Wave propagates at c_s ~ √(Te/mi)
Typical speeds: - $T_e = 10$ eV, hydrogen: $c_s \approx 50$ km/s - $T_e = 1$ keV, deuterium: $c_s \approx 200$ km/s
Much slower than electron thermal velocity ($v_{th,e} \sim 10^3$ km/s), but faster than ion thermal velocity ($v_{th,i} \sim 10$ km/s).
3.3 Ion Landau Damping¶
For $\omega/k \sim v_{th,i}$, ions moving near the wave phase velocity can damp the wave:
$$\gamma \approx -\sqrt{\frac{\pi}{8}} \frac{c_s}{k\lambda_D} \left(\frac{T_i}{T_e}\right)^{3/2} \exp\left(-\frac{1}{2k^2\lambda_D^2} - \frac{3}{2}\right)$$
The condition for weak damping is:
$$T_e \gg T_i$$
If $T_e \sim T_i$, the waves are heavily damped and don't propagate effectively.
This is observed in lab plasmas: - Hot electrons, cold ions: strong ion acoustic waves - Comparable temperatures: waves damped out
3.4 Role in Turbulence¶
Ion acoustic turbulence is ubiquitous in plasmas: - Driven by current (ion acoustic instability) - Nonlinear interactions create spectrum of modes - Leads to anomalous resistivity, heating
Observed in: - Ionosphere (radar scatter) - Solar wind - Tokamak edge plasmas
4. Waves in Magnetized Plasmas¶
4.1 Upper Hybrid Resonance¶
In a magnetized plasma, electrons gyrate at $\omega_{ce}$ while also oscillating at $\omega_{pe}$. For waves propagating perpendicular to $\mathbf{B}$ ($\mathbf{k} \perp \mathbf{B}$), the two motions couple.
The dispersion relation for electrostatic waves across $\mathbf{B}$ is:
$$\epsilon_{\perp} = 1 - \frac{\omega_{pe}^2}{\omega^2 - \omega_{ce}^2} = 0$$
This gives:
$$\boxed{\omega_{UH}^2 = \omega_{pe}^2 + \omega_{ce}^2}$$
This is the upper hybrid frequency.
Physical picture: - Electrons oscillate at $\omega_{pe}$ - Magnetic field adds restoring force from $\mathbf{v} \times \mathbf{B}$ - Effective frequency is geometric mean of the two
The upper hybrid layer (where $\omega = \omega_{UH}$) is used for: - Plasma heating (upper hybrid resonance heating) - Current drive - Diagnostics (reflectometry)
4.2 Lower Hybrid Resonance¶
For lower frequencies involving ion motion, the lower hybrid frequency arises from coupling of ion cyclotron and ion plasma oscillations:
$$\frac{1}{\omega_{LH}^2} = \frac{1}{\omega_{ci}\omega_{ce}} + \frac{1}{\omega_{pi}^2 + \omega_{ci}^2}$$
For $\omega_{pi}^2 \gg \omega_{ci}^2$:
$$\boxed{\omega_{LH} \approx \sqrt{\frac{\omega_{ci}\omega_{ce}}{1 + \omega_{pe}^2/\omega_{ce}^2}} \approx \sqrt{\omega_{ci}\omega_{ce}}}$$
(assuming $\omega_{pe}^2 \ll \omega_{ce}^2$).
Typical values: - $B = 5$ T, $n = 10^{20}$ m$^{-3}$, hydrogen - $f_{ce} \approx 140$ GHz, $f_{ci} \approx 76$ MHz - $f_{LH} \approx 3.3$ GHz
Lower hybrid waves are used extensively for: - Current drive: can drive currents efficiently via Landau damping - Heating (though less efficient than ECRH or ICRH) - Plasma startup in tokamaks
The waves can propagate at specific angles to $\mathbf{B}$, allowing localized deposition.
4.3 Warm Plasma Corrections¶
Including finite temperature, the cutoff and resonances shift:
$$\omega_{UH}^2 = \omega_{pe}^2 + \omega_{ce}^2 + 3k^2 v_{th,e}^2$$
For lower hybrid: $$\omega_{LH}^2 \approx \omega_{ci}\omega_{ce} \left(1 + \frac{\omega_{pe}^2}{\omega_{ce}^2}\right)^{-1}\left(1 + k^2\lambda_D^2\right)$$
The thermal corrections become important when $k\lambda_D \sim 1$ or $k\rho_L \sim 1$ (where $\rho_L$ is Larmor radius).
5. Bernstein Modes¶
5.1 Electron Bernstein Modes¶
At harmonics of the electron cyclotron frequency, electrostatic modes can propagate even when electromagnetic waves are cutoff. These are Bernstein modes.
The dispersion relation involves Bessel functions. Near the $n$-th harmonic:
$$\omega \approx n\omega_{ce}\left(1 + \frac{k^2 v_{th,e}^2}{2\omega_{ce}^2}\right)$$
These modes exist for $\omega \approx n\omega_{ce}$, $n = 1, 2, 3, \ldots$
Properties: - Electrostatic (no $\mathbf{B}_1$) - Can propagate where EM waves cannot - Strong interaction with resonant electrons - Used for heating and current drive
The hot plasma dielectric for $\mathbf{k} \perp \mathbf{B}$ is:
$$\epsilon_{\perp} = 1 + \frac{\omega_{pe}^2}{k^2 v_{th,e}^2} \sum_{n=-\infty}^{\infty} \frac{I_n(\lambda) e^{-\lambda}}{\omega/\omega_{ce} - n} \left(\frac{\omega}{k v_{th,e}}\right)Z\left(\frac{\omega - n\omega_{ce}}{k v_{th,e}}\right)$$
where $\lambda = k^2 \rho_L^2$ and $I_n$ is the modified Bessel function.
5.2 Ion Bernstein Modes¶
Similarly, ion Bernstein modes exist near harmonics of $\omega_{ci}$:
$$\omega \approx n\omega_{ci}\left(1 + \frac{k^2 v_{th,i}^2}{2\omega_{ci}^2}\right)$$
These are important for: - Ion heating in dense plasmas - Mode conversion heating (EM wave converts to electrostatic Bernstein mode) - Understanding kinetic Alfvén waves at small scales
5.3 Mode Conversion¶
In inhomogeneous plasmas, electromagnetic waves can mode convert to electrostatic waves at specific locations:
EM wave (O-mode or X-mode)
|
v
Cutoff/Resonance layer
|
v
Bernstein wave (electrostatic)
|
v
Absorption via Landau/cyclotron damping
This is used in mode conversion heating schemes, avoiding cutoff issues with direct EM wave injection.
6. Python Implementation¶
6.1 Langmuir Wave Dispersion¶
import numpy as np
import matplotlib.pyplot as plt
from scipy.special import wofz
def plasma_dispersion_Z(zeta):
"""
Plasma dispersion function Z(ζ).
Z(ζ) = (1/√π) ∫ exp(-x²)/(x - ζ) dx
Uses Faddeeva function: Z(ζ) = i√π w(ζ) where w(z) = exp(-z²)erfc(-iz)
"""
return 1j * np.sqrt(np.pi) * wofz(zeta)
def langmuir_dispersion_cold(k, omega_pe):
"""
Cold plasma dispersion: ω = ω_pe (independent of k).
"""
return omega_pe * np.ones_like(k)
def langmuir_dispersion_warm(k, omega_pe, v_th):
"""
Warm plasma dispersion (Bohm-Gross): ω² = ω_pe² + 3k²v_th².
"""
return np.sqrt(omega_pe**2 + 3 * k**2 * v_th**2)
def langmuir_dispersion_kinetic(k, omega_pe, v_th, num_points=100):
"""
Solve full kinetic dispersion using plasma dispersion function.
ε = 1 - (ω_pe²/k²v_th²)[1 + ζZ(ζ)] = 0
where ζ = ω/(kv_th)
"""
omega_solutions = []
for k_val in k:
if k_val == 0:
omega_solutions.append(omega_pe)
continue
# Search for real part of ω
omega_range = np.linspace(omega_pe, omega_pe * 1.5, num_points)
epsilon = []
for omega in omega_range:
zeta = omega / (k_val * v_th)
Z_val = plasma_dispersion_Z(zeta)
eps = 1 - (omega_pe**2 / (k_val**2 * v_th**2)) * (1 + zeta * Z_val)
epsilon.append(np.real(eps))
epsilon = np.array(epsilon)
# Find zero crossing
sign_changes = np.where(np.diff(np.sign(epsilon)))[0]
if len(sign_changes) > 0:
idx = sign_changes[0]
omega_solutions.append(omega_range[idx])
else:
omega_solutions.append(omega_range[-1])
return np.array(omega_solutions)
# Parameters
n = 1e19 # m^-3
T_e = 10 # eV
m_e = 9.109e-31 # kg
e = 1.602e-19 # C
epsilon_0 = 8.854e-12 # F/m
k_B = 1.381e-23 # J/K
omega_pe = np.sqrt(n * e**2 / (epsilon_0 * m_e))
v_th = np.sqrt(2 * k_B * T_e * e / m_e)
lambda_D = v_th / omega_pe
print(f"Plasma frequency: f_pe = {omega_pe / (2*np.pi) / 1e9:.2f} GHz")
print(f"Thermal velocity: v_th = {v_th / 1e6:.2f} Mm/s")
print(f"Debye length: λ_D = {lambda_D * 1e6:.2f} μm")
# Wavenumber array
k = np.linspace(0.01, 2, 100) / lambda_D # Normalize to 1/λ_D
# Compute dispersions
omega_cold = langmuir_dispersion_cold(k, omega_pe)
omega_warm = langmuir_dispersion_warm(k, omega_pe, v_th)
omega_kinetic = langmuir_dispersion_kinetic(k, omega_pe, v_th)
# Plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
# Dispersion relation
ax1.plot(k * lambda_D, omega_cold / omega_pe, 'k--', linewidth=2, label='Cold plasma')
ax1.plot(k * lambda_D, omega_warm / omega_pe, 'b-', linewidth=2, label='Bohm-Gross')
ax1.plot(k * lambda_D, omega_kinetic / omega_pe, 'r:', linewidth=2, label='Kinetic (full)')
ax1.set_xlabel('$k\\lambda_D$', fontsize=13)
ax1.set_ylabel('$\\omega / \\omega_{pe}$', fontsize=13)
ax1.set_title('Langmuir Wave Dispersion', fontsize=14)
ax1.legend(fontsize=11)
ax1.grid(True, alpha=0.3)
ax1.set_ylim([0.95, 1.4])
# Phase and group velocity
v_phase_warm = omega_warm / k
v_group_warm = 3 * k * v_th**2 / omega_warm
ax2.plot(k * lambda_D, v_phase_warm / v_th, 'b-', linewidth=2, label='$v_\\phi$ (Bohm-Gross)')
ax2.plot(k * lambda_D, v_group_warm / v_th, 'r--', linewidth=2, label='$v_g$ (Bohm-Gross)')
ax2.axhline(1, color='k', linestyle=':', alpha=0.5, label='$v_{th}$')
ax2.set_xlabel('$k\\lambda_D$', fontsize=13)
ax2.set_ylabel('Velocity / $v_{th}$', fontsize=13)
ax2.set_title('Phase and Group Velocity', fontsize=14)
ax2.legend(fontsize=11)
ax2.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('langmuir_dispersion.png', dpi=150, bbox_inches='tight')
plt.show()
6.2 Ion Acoustic Wave Dispersion¶
def ion_acoustic_dispersion(k, n, T_e, T_i, m_i, Z=1):
"""
Ion acoustic wave dispersion with damping.
Parameters:
-----------
k : array
Wavenumber (m^-1)
n : float
Density (m^-3)
T_e : float
Electron temperature (eV)
T_i : float
Ion temperature (eV)
m_i : float
Ion mass (kg)
Z : int
Ion charge number
Returns:
--------
omega_real : array
Real frequency
gamma : array
Damping rate
"""
T_e_J = T_e * e
T_i_J = T_i * e
# Ion acoustic speed
c_s = np.sqrt(k_B * T_e_J / m_i)
# Debye length
v_th_e = np.sqrt(2 * k_B * T_e_J / m_e)
omega_pe = np.sqrt(n * e**2 / (epsilon_0 * m_e))
lambda_D = v_th_e / omega_pe
# Real frequency
omega_real = k * c_s / np.sqrt(1 + k**2 * lambda_D**2)
# Damping rate (approximate, for T_e >> T_i)
v_th_i = np.sqrt(2 * k_B * T_i_J / m_i)
zeta_i = omega_real / (k * v_th_i)
# Landau damping (asymptotic form for large ζ)
gamma = -np.sqrt(np.pi/8) * (omega_real / (k * lambda_D)) * \
(T_i / T_e)**(3/2) * np.exp(-1/(2*k**2*lambda_D**2) - 3/2)
return omega_real, gamma
# Parameters
n = 1e19 # m^-3
T_e = 10 # eV
m_i = 1.673e-27 # kg (proton)
T_i_array = np.array([0.1, 0.5, 1.0, 5.0]) # eV
# Compute Debye length
v_th_e = np.sqrt(2 * k_B * T_e * e / m_e)
omega_pe = np.sqrt(n * e**2 / (epsilon_0 * m_e))
lambda_D = v_th_e / omega_pe
c_s = np.sqrt(k_B * T_e * e / m_i)
print(f"Ion acoustic speed: c_s = {c_s / 1e3:.1f} km/s")
k = np.linspace(0.01, 3, 200) / lambda_D
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
colors = ['blue', 'green', 'orange', 'red']
for i, T_i in enumerate(T_i_array):
omega_real, gamma = ion_acoustic_dispersion(k, n, T_e, T_i, m_i)
ax1.plot(k * lambda_D, omega_real * lambda_D / c_s, color=colors[i],
linewidth=2, label=f'$T_i = {T_i}$ eV')
ax2.semilogy(k * lambda_D, np.abs(gamma) / omega_real, color=colors[i],
linewidth=2, label=f'$T_i = {T_i}$ eV')
# Add linear dispersion line
k_linear = k[k * lambda_D < 0.5]
ax1.plot(k_linear * lambda_D, k_linear * lambda_D, 'k--', alpha=0.5,
label='$\\omega = kc_s$')
ax1.set_xlabel('$k\\lambda_D$', fontsize=13)
ax1.set_ylabel('$\\omega\\lambda_D / c_s$', fontsize=13)
ax1.set_title('Ion Acoustic Dispersion', fontsize=14)
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)
ax2.set_xlabel('$k\\lambda_D$', fontsize=13)
ax2.set_ylabel('$|\\gamma| / \\omega$', fontsize=13)
ax2.set_title('Landau Damping Rate', fontsize=14)
ax2.legend(fontsize=10)
ax2.grid(True, alpha=0.3)
ax2.set_ylim([1e-6, 1])
plt.tight_layout()
plt.savefig('ion_acoustic_dispersion.png', dpi=150, bbox_inches='tight')
plt.show()
6.3 Upper and Lower Hybrid Frequencies¶
def hybrid_frequencies(n, B, Z=1, A=1):
"""
Compute upper and lower hybrid frequencies.
Parameters:
-----------
n : array
Density (m^-3)
B : array
Magnetic field (T)
Z : int
Ion charge
A : float
Ion mass number
Returns:
--------
dict with frequencies
"""
m_i = A * 1.673e-27
omega_pe = np.sqrt(n * e**2 / (epsilon_0 * m_e))
omega_pi = np.sqrt(n * Z**2 * e**2 / (epsilon_0 * m_i))
omega_ce = e * B / m_e
omega_ci = Z * e * B / m_i
omega_UH = np.sqrt(omega_pe**2 + omega_ce**2)
# Lower hybrid
term1 = 1 / (omega_ci * omega_ce)
term2 = 1 / (omega_pi**2 + omega_ci**2)
omega_LH = 1 / np.sqrt(term1 + term2)
return {
'omega_pe': omega_pe,
'omega_pi': omega_pi,
'omega_ce': omega_ce,
'omega_ci': omega_ci,
'omega_UH': omega_UH,
'omega_LH': omega_LH
}
# Parameter ranges
B_array = np.linspace(0.5, 10, 100) # T
n_ref = 1e20 # m^-3
freq = hybrid_frequencies(n_ref, B_array, Z=1, A=2) # Deuterium
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
# Upper hybrid
ax1.plot(B_array, freq['omega_pe'] / (2*np.pi) / 1e9, 'b--',
linewidth=2, label='$f_{pe}$')
ax1.plot(B_array, freq['omega_ce'] / (2*np.pi) / 1e9, 'r--',
linewidth=2, label='$f_{ce}$')
ax1.plot(B_array, freq['omega_UH'] / (2*np.pi) / 1e9, 'k-',
linewidth=2.5, label='$f_{UH}$')
ax1.set_xlabel('Magnetic Field (T)', fontsize=13)
ax1.set_ylabel('Frequency (GHz)', fontsize=13)
ax1.set_title(f'Upper Hybrid Frequency ($n = {n_ref:.0e}$ m$^{{-3}}$)', fontsize=14)
ax1.legend(fontsize=11)
ax1.grid(True, alpha=0.3)
# Lower hybrid
ax2.plot(B_array, freq['omega_ci'] / (2*np.pi) / 1e6, 'r--',
linewidth=2, label='$f_{ci}$')
ax2.plot(B_array, freq['omega_LH'] / (2*np.pi) / 1e6, 'k-',
linewidth=2.5, label='$f_{LH}$')
ax2.set_xlabel('Magnetic Field (T)', fontsize=13)
ax2.set_ylabel('Frequency (MHz)', fontsize=13)
ax2.set_title(f'Lower Hybrid Frequency ($n = {n_ref:.0e}$ m$^{{-3}}$)', fontsize=14)
ax2.legend(fontsize=11)
ax2.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('hybrid_frequencies.png', dpi=150, bbox_inches='tight')
plt.show()
# Print typical ITER values
B_ITER = 5.3 # T
n_ITER = 1e20 # m^-3
freq_ITER = hybrid_frequencies(n_ITER, B_ITER, Z=1, A=2)
print(f"\nITER-like parameters (B = {B_ITER} T, n = {n_ITER:.0e} m^-3):")
print(f" f_pe = {freq_ITER['omega_pe'][0] / (2*np.pi) / 1e9:.1f} GHz")
print(f" f_ce = {freq_ITER['omega_ce'][0] / (2*np.pi) / 1e9:.1f} GHz")
print(f" f_UH = {freq_ITER['omega_UH'][0] / (2*np.pi) / 1e9:.1f} GHz")
print(f" f_ci = {freq_ITER['omega_ci'][0] / (2*np.pi) / 1e6:.1f} MHz")
print(f" f_LH = {freq_ITER['omega_LH'][0] / (2*np.pi) / 1e6:.1f} MHz")
6.4 Bernstein Mode Dispersion¶
from scipy.special import iv # Modified Bessel function
def bernstein_mode_approximate(n_harmonic, k_perp, rho_L, omega_c):
"""
Approximate Bernstein mode dispersion near n-th harmonic.
ω ≈ nω_c(1 + k²v_th²/(2ω_c²))
"""
lambda_val = (k_perp * rho_L)**2
omega = n_harmonic * omega_c * (1 + lambda_val / 2)
return omega
# Parameters
B = 2.0 # T
T_e = 5e3 # eV (5 keV)
n = 5e19 # m^-3
omega_ce = e * B / m_e
v_th_e = np.sqrt(2 * k_B * T_e * e / m_e)
rho_L = v_th_e / omega_ce
print(f"Electron cyclotron frequency: f_ce = {omega_ce / (2*np.pi) / 1e9:.2f} GHz")
print(f"Larmor radius: ρ_L = {rho_L * 1e3:.2f} mm")
# Wavenumber
k_perp = np.linspace(0, 100, 500) / rho_L
fig, ax = plt.subplots(figsize=(10, 7))
harmonics = [1, 2, 3, 4, 5]
colors = plt.cm.viridis(np.linspace(0, 1, len(harmonics)))
for i, n_harm in enumerate(harmonics):
omega = bernstein_mode_approximate(n_harm, k_perp, rho_L, omega_ce)
ax.plot(k_perp * rho_L, omega / omega_ce, color=colors[i],
linewidth=2, label=f'n = {n_harm}')
# Mark the harmonic
ax.axhline(n_harm, color=colors[i], linestyle='--', alpha=0.3)
ax.set_xlabel('$k_\\perp \\rho_L$', fontsize=13)
ax.set_ylabel('$\\omega / \\omega_{ce}$', fontsize=13)
ax.set_title('Electron Bernstein Mode Dispersion', fontsize=14)
ax.legend(fontsize=11, loc='upper left')
ax.grid(True, alpha=0.3)
ax.set_xlim([0, 10])
ax.set_ylim([0, 6])
plt.tight_layout()
plt.savefig('bernstein_modes.png', dpi=150, bbox_inches='tight')
plt.show()
Summary¶
Electrostatic waves are fundamental modes in plasmas, characterized by:
General framework: - Dispersion relation: $\epsilon(\mathbf{k}, \omega) = 0$ - Dielectric function from susceptibilities: $\epsilon = 1 + \sum_s \chi_s$ - Cold plasma: algebraic dispersion - Warm plasma: kinetic effects via plasma dispersion function $Z(\zeta)$
Langmuir waves: - Cold plasma: $\omega = \omega_{pe}$ (no propagation) - Warm plasma (Bohm-Gross): $\omega^2 = \omega_{pe}^2 + 3k^2v_{th}^2$ - Group velocity: $v_g = 3kv_{th}^2/\omega$ - Landau damping when $\omega/k \sim v_{th}$
Ion acoustic waves: - Dispersion: $\omega = kc_s$ where $c_s = \sqrt{k_B T_e/m_i}$ - Electron pressure restoring force, ion inertia - Weak damping requires $T_e \gg T_i$ - Important for turbulence and transport
Magnetized plasma resonances: - Upper hybrid: $\omega_{UH}^2 = \omega_{pe}^2 + \omega_{ce}^2$ - Lower hybrid: $\omega_{LH} \approx \sqrt{\omega_{ci}\omega_{ce}}$ - Used for plasma heating and current drive
Bernstein modes: - Electrostatic modes near cyclotron harmonics: $\omega \approx n\omega_c$ - Exist where EM waves are cutoff - Mode conversion allows heating in overdense plasmas
Applications include: - Plasma diagnostics (Thomson scattering, reflectometry) - Heating and current drive (ECRH, LHCD) - Understanding instabilities and turbulence - Astrophysical phenomena (solar radio bursts, pulsar emissions)
Practice Problems¶
Problem 1: Langmuir Wave Properties¶
A plasma has $n = 5 \times 10^{18}$ m$^{-3}$ and $T_e = 5$ eV.
(a) Calculate the plasma frequency $f_{pe}$ and Debye length $\lambda_D$.
(b) For a Langmuir wave with $k = 0.1/\lambda_D$, calculate the frequency using the Bohm-Gross relation.
(c) Compute the phase velocity $v_\phi$ and group velocity $v_g$. How do they compare to $v_{th,e}$?
(d) At what wavenumber does the thermal correction to $\omega$ become 10% of $\omega_{pe}$?
Problem 2: Ion Acoustic Wave Damping¶
An argon plasma ($A = 40$) has $n = 10^{19}$ m$^{-3}$, $T_e = 20$ eV, $T_i = 2$ eV.
(a) Calculate the ion acoustic speed $c_s$.
(b) For $k\lambda_D = 0.3$, find the real frequency $\omega$ and the Landau damping rate $\gamma$.
(c) Calculate the damping length $L_d = v_g/|\gamma|$ where $v_g = d\omega/dk$.
(d) If the ion temperature increases to $T_i = 10$ eV, how does the damping change? Is the wave still observable?
Problem 3: Hybrid Resonances¶
A tokamak has $B = 3$ T and a density that varies from $n = 10^{20}$ m$^{-3}$ (core) to $n = 10^{18}$ m$^{-3}$ (edge).
(a) Calculate $f_{UH}$ at the core and edge.
(b) A heating system operates at 110 GHz. Where is the upper hybrid resonance layer located?
(c) Calculate $f_{LH}$ at the core and edge (assume deuterium).
(d) If a lower hybrid system operates at 5 GHz, sketch where the resonance layer is located as a function of radius.
Problem 4: Bernstein Wave Heating¶
Consider a dense plasma where $\omega_{pe} > \omega_{ce}$, making the plasma overdense for electron cyclotron waves.
(a) For $B = 1.5$ T, $T_e = 3$ keV, find the maximum density for which $\omega_{pe} < \omega_{ce}$.
(b) If $n = 5 \times 10^{20}$ m$^{-3}$ (overdense), calculate $\omega_{UH}/\omega_{ce}$.
(c) An O-mode wave at $\omega = \omega_{ce}$ cannot penetrate. Explain how mode conversion to a Bernstein wave at the upper hybrid layer can allow heating.
(d) Estimate the perpendicular wavenumber $k_\perp \rho_L$ of the Bernstein wave needed for efficient electron heating at the first harmonic.
Problem 5: Dispersion Relation Analysis¶
The electrostatic dispersion relation for an electron-ion plasma is:
$$1 - \frac{\omega_{pe}^2}{k^2 v_{th,e}^2}[1 + \zeta_e Z(\zeta_e)] - \frac{\omega_{pi}^2}{k^2 v_{th,i}^2}[1 + \zeta_i Z(\zeta_i)] = 0$$
where $\zeta_s = \omega/(k v_{th,s})$.
(a) In the limit $\omega \gg k v_{th,e}$, expand to show this gives the Bohm-Gross relation.
(b) In the limit $\omega \ll k v_{th,e}$ but $\omega \sim k v_{th,i}$, show this gives ion acoustic waves with electron Boltzmann response.
(c) For $m_i/m_e = 1836$, $T_e = T_i = 1$ keV, numerically solve for $\omega(k)$ over the range $0.01 < k\lambda_D < 10$. Identify the Langmuir and ion acoustic branches.
(d) At what value of $k\lambda_D$ do the two branches have comparable frequencies? What physics occurs in this regime?
Previous: 9. Collisional Kinetics Next: 11. Electromagnetic Waves