12. Wave Heating and Instabilities
12. Wave Heating and Instabilities¶
Learning Objectives¶
- Understand the physical mechanisms of wave heating in fusion plasmas
- Master the theory of velocity space instabilities (beam-plasma, bump-on-tail, Weibel)
- Analyze pressure-driven instabilities in magnetized plasmas (firehose, mirror)
- Learn the conditions for parametric instabilities in laser-plasma interactions
- Apply instability theory to practical problems in fusion and astrophysics
- Compute growth rates and stability boundaries for various instability mechanisms
Introduction¶
Waves in plasmas serve two critical roles: 1. Heating and current drive: External waves transfer energy to plasma particles 2. Instabilities: Waves can grow spontaneously, extracting free energy from the plasma
This lesson covers both aspects, focusing on: - Wave heating: How RF waves deposit energy in fusion plasmas - Velocity space instabilities: Arising from non-Maxwellian distributions - Pressure-driven instabilities: From temperature anisotropies - Parametric instabilities: Wave-wave coupling in high-power laser systems
These phenomena are crucial for: - Fusion reactor design (heating systems, current drive) - Astrophysical plasmas (solar wind, pulsar magnetospheres, GRB afterglows) - Laser-plasma interactions (inertial confinement fusion) - Space weather (radiation belts, magnetospheric dynamics)
1. Wave Heating in Fusion Plasmas¶
1.1 Overview of Heating Methods¶
Fusion plasmas require temperatures $T \sim 10-20$ keV ($\sim 100-200$ million K). Three main heating methods:
Ohmic heating: - $P = I^2 R$ where $R \propto T_e^{-3/2}$ (classical resistivity) - Effective at low $T$, ineffective at high $T$ - Limited to $\sim 1-2$ keV in tokamaks
Neutral Beam Injection (NBI): - Fast neutrals (50-1000 keV) injected, ionized, transfer energy via collisions - Not a wave method, but important for comparison - Power: 10-50 MW per beamline in ITER
Radio-Frequency (RF) heating: - Electromagnetic waves launched from antennas or waveguides - Three frequency ranges: ECRH, ICRH, LHCD - Advantages: localized deposition, current drive capability, no particle source
1.2 Electron Cyclotron Resonance Heating (ECRH)¶
Frequency: $\omega \approx n\omega_{ce}$ where $n = 1, 2$ typically
Resonance condition: At spatial location where $\omega = n\omega_{ce}(r)$, the wave phase matches electron gyration.
Absorption mechanism: - Electrons in resonance with the wave ($\omega - k_\parallel v_\parallel = n\omega_{ce}$) - Wave electric field perpendicular to $\mathbf{B}$ does work on gyrating electrons - Power absorption: $P \propto \int d^3v \, \mathbf{j} \cdot \mathbf{E}$
Dispersion: Use X-mode or O-mode depending on density - O-mode: cutoff at $n_c = \epsilon_0 m_e \omega^2/e^2$ - X-mode: higher cutoff, better for overdense plasmas
Typical parameters (ITER): - Frequency: 170 GHz (for $B \sim 5.3$ T at $n=2$) - Power: 20 MW total (gyrotrons) - Beam width: $\sim 5$ cm (highly localized)
Advantages: - Excellent localization ($\Delta r \sim$ cm) - Current drive capability (ECCD) - Real-time control of deposition location
Challenges: - Requires high-frequency gyrotrons (expensive) - Transmission losses in waveguides - Mirror alignment critical
1.3 Ion Cyclotron Resonance Heating (ICRH)¶
Frequency: $\omega \approx n\omega_{ci}$ where $\omega_{ci} = ZeB/m_i$
Resonance condition: Ions gyrate at $\omega_{ci} \sim 2\pi \times (30-100)$ MHz for tokamaks.
Heating schemes:
Fundamental majority heating: $\omega = \omega_{ci}$ for main ion species - Direct resonance, but weak single-pass absorption - Requires multiple passes
Second harmonic: $\omega = 2\omega_{ci}$ - Stronger absorption than fundamental - Used in low-field devices
Minority heating: $\omega = \omega_{ci,\text{minority}}$ - Introduce minority species (e.g., 5-10% $^3$He in D plasma) - Resonance at $\omega_{ci}(^3\text{He})$ while bulk deuterium is off-resonance - Minority ions heated to high energy, collisionally transfer to bulk
Mode conversion: Fast wave converts to ion Bernstein wave or ion cyclotron wave - Occurs near hybrid resonance layer - Can heat electrons efficiently
Typical parameters: - Frequency: 40-80 MHz - Power: 20 MW (ITER) - Antenna: large coils at vessel wall
Advantages: - Well-established technology - Can heat both ions and electrons - Central heating possible
Challenges: - Antenna-plasma interaction (impurities, hot spots) - Parasitic losses - Less localized than ECRH
1.4 Lower Hybrid Current Drive (LHCD)¶
Frequency: $\omega_{ci} \ll \omega \ll \omega_{ce}$ (typically 1-8 GHz)
Purpose: Primarily for current drive, not heating - Non-inductive current generation - Steady-state operation in tokamaks
Mechanism: - Lower hybrid wave propagates with high $n_\parallel = k_\parallel c/\omega$ - Strong Landau damping on tail electrons ($v_\parallel \sim \omega/k_\parallel$) - Asymmetric damping creates net current
Current drive efficiency: $$\eta_{CD} = \frac{n_{20} I_A R}{P_{\text{MW}}}$$
where $n_{20}$ is density in $10^{20}$ m$^{-3}$, $I_A$ is current in MA, $R$ is major radius in m, $P$ is power in MW.
Typical: $\eta_{CD} \sim 0.2-0.5$ for LHCD.
Accessibility: Lower hybrid wave must penetrate to desired location - Density limit: $n < n_{\text{access}}$ where wave cutoff occurs - In high-density core, may not penetrate
Typical parameters: - Frequency: 3.7-5 GHz - Power: 20 MW (ITER) - Launcher: waveguide array (grill)
Advantages: - High current drive efficiency - Off-axis current profile control
Challenges: - Density limit for accessibility - Spectral gap (difficulty coupling power) - Nonlinear effects at high power
1.5 Comparison of Heating Methods¶
| Method | Frequency | Main Target | Localization | Current Drive | Power (ITER) |
|---|---|---|---|---|---|
| ECRH | 140-170 GHz | Electrons | Excellent | Yes (ECCD) | 20 MW |
| ICRH | 40-80 MHz | Ions | Moderate | Weak | 20 MW |
| LHCD | 3-8 GHz | Electrons | Good | Excellent | 20 MW |
| NBI | N/A | Ions | Poor | Yes | 33 MW |
Synergy: Combining methods is often optimal - NBI + ICRH: NBI creates fast ion tail, ICRH heats tail further - ECRH + LHCD: ECRH for MHD control, LHCD for current profile
2. Velocity Space Instabilities¶
2.1 Beam-Plasma Instability (Two-Stream)¶
Consider a cold electron beam with density $n_b$ and velocity $v_0$ streaming through a background plasma with density $n_0$.
Setup: - Beam: $f_b(\mathbf{v}) = n_b \delta(v_x - v_0)\delta(v_y)\delta(v_z)$ - Background: $f_0(\mathbf{v}) = n_0 \delta(v_x)\delta(v_y)\delta(v_z)$ - Both cold ($T = 0$)
Dispersion relation: From linearized Vlasov + Poisson:
$$1 = \frac{\omega_{p0}^2}{\omega^2} + \frac{\omega_{pb}^2}{(\omega - kv_0)^2}$$
where $\omega_{p0}^2 = n_0 e^2/(\epsilon_0 m_e)$ and $\omega_{pb}^2 = n_b e^2/(\epsilon_0 m_e)$.
Analysis: Assume $\omega = \omega_r + i\gamma$ and look for unstable solutions ($\gamma > 0$).
For $n_b \ll n_0$, expand around the Langmuir wave $\omega \approx \omega_{p0} + \delta\omega$:
$$\delta\omega \approx -\frac{\omega_{pb}^2}{2\omega_{p0}} \frac{1}{1 - kv_0/\omega_{p0}}$$
When denominator is small, $\delta\omega$ becomes large. For $kv_0 \approx \omega_{p0}$, the correction becomes imaginary.
Growth rate (for $n_b/n_0 \ll 1$):
$$\boxed{\gamma \approx \omega_{p0} \left(\frac{n_b}{n_0}\right)^{1/3}}$$
The instability is strongest when: $$kv_0 \approx \omega_{p0}$$
Physical picture:
Beam electrons: βββ βββ βββ βββ
Background: Β· Β· Β· Β·
Perturbation creates bunching:
βββ βββ βββ βββ (density wave)
Bunches enhance electric field β feedback β growth
Applications: - Electron beams in plasma (accelerators, space) - Ionospheric instabilities - Caused early issues in particle accelerators
2.2 Bump-on-Tail Instability¶
A more realistic scenario: a small population of fast electrons on a Maxwellian background.
Distribution: $$f(v) = f_M(v) + f_{\text{bump}}(v)$$
where $f_{\text{bump}}$ is a small population at $v \sim v_{\text{bump}} > v_{th}$.
Criterion for instability: The distribution must have a positive slope in velocity space at the resonant velocity:
$$\frac{\partial f}{\partial v}\bigg|_{v = \omega/k} > 0$$
This is inverse Landau damping: particles at $v = v_\phi$ transfer energy TO the wave instead of FROM the wave.
Growth rate: For small bump with density $n_b$ and width $\Delta v$:
$$\gamma \sim \omega_{pe} \left(\frac{n_b}{n_0}\right)^{1/3} \frac{v_{\text{bump}}}{v_{th}}$$
Quasilinear relaxation: As the wave grows, it diffuses particles in velocity space via:
$$\frac{\partial f}{\partial t} = \frac{\partial}{\partial v}\left(D \frac{\partial f}{\partial v}\right)$$
where $D \propto |E_k|^2$ is the diffusion coefficient.
Result: The bump flattens into a plateau:
Initial: After saturation:
f(v) f(v)
|\ |----\
| \ | \
| \___ | \___
| \ | \
+-------v +----------v
v_bump plateau
This quasilinear plateau formation is a fundamental nonlinear saturation mechanism.
Applications: - Runaway electrons in tokamaks - Solar wind electron beams - Laser-plasma interactions
2.3 Weibel Instability¶
The Weibel instability grows from temperature anisotropy: $T_\perp > T_\parallel$ (or vice versa).
Physical mechanism: - Anisotropic distribution creates current fluctuations - Currents generate magnetic fields - Magnetic fields enhance anisotropy β positive feedback
Setup: Consider a distribution: $$f(v_\parallel, v_\perp) = n_0 \left(\frac{m}{2\pi k_B T_\parallel}\right)^{1/2}\left(\frac{m}{2\pi k_B T_\perp}\right) \exp\left(-\frac{mv_\parallel^2}{2k_BT_\parallel} - \frac{mv_\perp^2}{2k_BT_\perp}\right)$$
Dispersion (for $T_\perp > T_\parallel$): Purely growing mode (no real frequency):
$$\omega = i\gamma$$
Growth rate:
$$\boxed{\gamma_{\text{max}} \approx \omega_{pe} \sqrt{\frac{T_\perp - T_\parallel}{T_\parallel}}}$$
for wavenumber: $$k_{\text{max}} \approx \frac{\omega_{pe}}{c}\sqrt{\frac{T_\perp}{T_\parallel} - 1}$$
Generated magnetic field: The instability creates small-scale magnetic fields with strength:
$$\frac{B^2}{8\pi} \sim n k_B (T_\perp - T_\parallel)$$
Applications: - Collisionless shocks: In astrophysical shocks (e.g., supernova remnants, GRB afterglows), Weibel instability generates magnetic fields that mediate the shock - Laser-plasma interaction: Intense laser creates anisotropic electron distribution β Weibel instability β magnetic field generation - Magnetospheric plasmas: Anisotropic distributions in radiation belts - Magnetic field generation: Weibel is a mechanism for seed fields in cosmology
This instability was theoretically predicted by Weibel (1959) and experimentally confirmed in laser-plasma experiments (2000s).
3. Pressure-Driven Instabilities¶
3.1 Firehose Instability¶
In a magnetized plasma with $p_\parallel > p_\perp$ (parallel pressure exceeds perpendicular), the firehose instability can occur.
Analogy: Like a pressurized garden hose writhing when pressure is too high.
Physical mechanism: - Magnetic field line bends - Parallel pressure pushes plasma along bent field - Curvature increases β positive feedback
Stability criterion: From MHD with anisotropic pressure:
$$\boxed{p_\parallel - p_\perp < \frac{B^2}{\mu_0}}$$
or equivalently:
$$\beta_\parallel - \beta_\perp < 1$$
where $\beta_\parallel = 2\mu_0 p_\parallel/B^2$ and $\beta_\perp = 2\mu_0 p_\perp/B^2$.
Growth rate (for unstable case):
$$\gamma^2 \approx k^2 v_A^2 \left(\frac{p_\parallel - p_\perp}{p_\parallel + p_\perp/2} - \frac{1}{\beta_\parallel}\right)$$
where $v_A = B/\sqrt{\mu_0 \rho}$ is the AlfvΓ©n speed.
Maximum growth at: $$k \sim \frac{1}{L}$$
where $L$ is the system size (low-$k$ instability).
Observations: - Solar wind: Often marginally stable/unstable to firehose - Magnetosheath: Compressed plasma can violate stability condition - Tokamak edge: Fast ion populations can drive firehose
Saturation: Pitch-angle scattering relaxes anisotropy, quenching the instability.
3.2 Mirror Instability¶
The opposite anisotropy, $p_\perp > p_\parallel$, can drive the mirror instability.
Physical mechanism: - Magnetic field strength fluctuates: $B = B_0 + B_1$ - Particles with high $\mu = mv_\perp^2/(2B)$ are trapped in low-$B$ regions (magnetic mirrors) - Enhanced $p_\perp$ in low-$B$ regions β $B$ decreases further β feedback
Stability criterion:
$$\boxed{\frac{p_\perp}{p_\parallel} < 1 + \frac{1}{\beta_\perp}}$$
or:
$$\beta_\perp\left(\frac{p_\perp}{p_\parallel} - 1\right) < 1$$
Growth rate: For $p_\perp/p_\parallel - 1 = A$ (anisotropy):
$$\gamma \approx k_\parallel v_A \sqrt{A \beta_\perp}$$
for $k_\parallel L \sim 1$ where $L$ is scale length.
Characteristics: - Non-propagating: $\omega_r = 0$ (purely growing) - Compressional: creates $\delta B_\parallel$ and $\delta n$ - Anisotropic structure: elongated along $\mathbf{B}$
Observations: - Solar wind: Mirror mode structures (slow-mode structures with anticorrelated $B$ and $n$) - Magnetosheath: Very common, quasi-steady structures - Planetary magnetospheres: Jupiter, Saturn
Saturation: Creates quasi-static magnetic bottles that trap particles, reducing anisotropy.
3.3 Comparison: Firehose vs Mirror¶
| Property | Firehose | Mirror |
|---|---|---|
| Anisotropy | $p_\parallel > p_\perp$ | $p_\perp > p_\parallel$ |
| Criterion | $\beta_\parallel - \beta_\perp < 1$ | $\beta_\perp(p_\perp/p_\parallel - 1) < 1$ |
| $\omega_r$ | Finite (propagating) | Zero (non-propagating) |
| $\delta B$ | Transverse | Compressional |
| Saturation | Pitch-angle scattering | Magnetic trapping |
Both instabilities are ubiquitous in collisionless plasmas where pressure can remain anisotropic (collision time $\gg$ dynamical time).
4. Parametric Instabilities¶
4.1 Three-Wave Coupling¶
Parametric instabilities involve coupling of three waves: $$\omega_0 = \omega_1 + \omega_2$$ $$\mathbf{k}_0 = \mathbf{k}_1 + \mathbf{k}_2$$
where wave 0 (pump) decays into waves 1 and 2 (daughter waves).
Mechanism: - Pump wave creates density/velocity oscillations - Oscillations modulate the plasma response - Modulated plasma can amplify daughter waves if matching conditions met
Growth rate: Scales with pump amplitude: $$\gamma \propto \sqrt{\frac{I}{I_c}}$$
where $I$ is pump intensity and $I_c$ is a threshold.
4.2 Stimulated Raman Scattering (SRS)¶
Process: Electromagnetic wave (pump) $\to$ EM wave (scattered) + Langmuir wave
Matching conditions: $$\omega_0 = \omega_s + \omega_L$$ $$\mathbf{k}_0 = \mathbf{k}_s + \mathbf{k}_L$$
where $\omega_L \approx \omega_{pe}$ (Langmuir wave) and $\omega_s < \omega_0$ (scattered EM wave).
Dispersion constraints: - Pump: $\omega_0^2 = \omega_{pe}^2 + k_0^2 c^2$ - Scattered: $\omega_s^2 = \omega_{pe}^2 + k_s^2 c^2$ - Langmuir: $\omega_L^2 \approx \omega_{pe}^2 + 3k_L^2 v_{th}^2$
Growth rate:
$$\gamma_{SRS} = \frac{k_L v_{osc}}{4} \left(\frac{\omega_0}{\omega_s}\right)^{1/2}$$
where $v_{osc} = eE_0/(m_e\omega_0)$ is the quiver velocity in the pump wave.
Threshold: Requires $\gamma > \nu_L$ where $\nu_L$ is Landau damping rate.
Relevance: Laser fusion (ICF) - High-power lasers ($I \sim 10^{15}$ W/cm$^2$) can drive SRS - Scattered light lost β reduced coupling efficiency - Hot electrons from Langmuir wave heating β preheat target (bad for compression)
Mitigation: - Bandwidth: broadband laser reduces coherence - Beam smoothing: reduces local intensity spikes - Wavelength: shorter wavelength (UV) has higher threshold
4.3 Stimulated Brillouin Scattering (SBS)¶
Process: EM wave $\to$ EM wave + ion acoustic wave
Matching: $$\omega_0 = \omega_s + \omega_{ia}$$ $$\mathbf{k}_0 = \mathbf{k}_s + \mathbf{k}_{ia}$$
where $\omega_{ia} = k_{ia} c_s$ (ion acoustic wave).
Growth rate:
$$\gamma_{SBS} = \frac{k_{ia} v_{osc}}{4\sqrt{2}} \sqrt{\frac{\omega_0}{\omega_{ia}}}$$
Characteristics: - Lower threshold than SRS (ion acoustic damping weaker than Landau damping) - Backscattering: strongest for $\mathbf{k}_s \approx -\mathbf{k}_0$ - Can reflect significant fraction of laser energy
Relevance: SBS is often the dominant parametric instability in laser fusion.
Mitigation: Similar to SRS, plus: - Gas-filled hohlraum reduces SBS - Multi-ion species (increases ion acoustic damping)
4.4 Impact on Inertial Confinement Fusion (ICF)¶
In National Ignition Facility (NIF) and other ICF experiments: - Laser power: $\sim 500$ TW - Intensity: $10^{14}-10^{15}$ W/cm$^2$ in hohlraum - SRS and SBS can reflect 10-50% of laser energy
Consequences: - Reduced coupling to target β lower compression - Hot electrons from SRS preheat fuel β reduced gain - Asymmetry in drive
Recent progress (2022-2023): NIF achieved ignition ($Q > 1$) by: - Improved hohlraum design - Better beam smoothing - Higher laser energy (2.05 MJ) - Mitigation of SRS/SBS through wavelength detuning
5. Instability Classification¶
5.1 Free Energy Sources¶
Instabilities extract free energy from:
- Velocity space: Non-Maxwellian distributions
- Beam-plasma: relative drift
- Bump-on-tail: positive $\partial f/\partial v$
-
Weibel: temperature anisotropy
-
Configuration space: Gradients in density, temperature, magnetic field
- Drift waves (not covered here)
- Interchange modes
-
Tearing modes
-
Current: Parallel or perpendicular currents
- Current-driven instabilities
-
Kink modes
-
External drive: Pumped by external waves
- Parametric instabilities (SRS, SBS)
5.2 Instability Summary Table¶
| Instability | Free Energy | Condition | Growth Rate | Application |
|---|---|---|---|---|
| Beam-plasma | Beam drift $v_0$ | $kv_0 \sim \omega_{pe}$ | $\omega_{pe}(n_b/n_0)^{1/3}$ | Accelerators, space |
| Bump-on-tail | $\partial f/\partial v > 0$ | Resonant particles | $\omega_{pe}(n_b/n_0)^{1/3}$ | Runaways, solar wind |
| Weibel | $T_\perp > T_\parallel$ | Anisotropy | $\omega_{pe}\sqrt{(T_\perp-T_\parallel)/T_\parallel}$ | Shocks, lasers |
| Firehose | $p_\parallel > p_\perp$ | $\beta_\parallel - \beta_\perp > 1$ | $k v_A \sqrt{\Delta p/p}$ | Solar wind |
| Mirror | $p_\perp > p_\parallel$ | $\beta_\perp(p_\perp/p_\parallel-1) > 1$ | $k_\parallel v_A \sqrt{A\beta_\perp}$ | Magnetosheath |
| SRS | Pump laser | $I > I_c$ | $(k_L v_{osc}/4)\sqrt{\omega_0/\omega_s}$ | Laser fusion |
| SBS | Pump laser | $I > I_c$ | $(k_{ia}v_{osc}/4\sqrt{2})\sqrt{\omega_0/\omega_{ia}}$ | Laser fusion |
6. Python Implementation¶
6.1 Two-Stream Instability Dispersion¶
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import fsolve
def two_stream_dispersion(k, omega_p0, omega_pb, v0):
"""
Solve two-stream dispersion: 1 = Ο_p0Β²/ΟΒ² + Ο_pbΒ²/(Ο-kv0)Β²
Returns complex frequency Ο(k).
"""
def dispersion_eq(omega_complex):
omega = omega_complex[0] + 1j * omega_complex[1]
eps = 1 - omega_p0**2/omega**2 - omega_pb**2/(omega - k*v0)**2
return [np.real(eps), np.imag(eps)]
# Initial guess
omega_guess = [omega_p0, 0.1 * omega_p0]
sol = fsolve(dispersion_eq, omega_guess)
return sol[0] + 1j * sol[1]
# Parameters
n0 = 1e19 # m^-3
nb_frac = 0.01 # nb/n0 = 1%
m_e = 9.109e-31 # kg
e = 1.602e-19 # C
epsilon_0 = 8.854e-12 # F/m
omega_p0 = np.sqrt(n0 * e**2 / (epsilon_0 * m_e))
omega_pb = np.sqrt(nb_frac * n0 * e**2 / (epsilon_0 * m_e))
# Beam velocity
v0 = 2 * omega_p0 * (1e8 / omega_p0) # Choose v0 ~ Ο_p0/k_typical
print(f"Background plasma frequency: Ο_p0 = {omega_p0:.2e} rad/s")
print(f"Beam plasma frequency: Ο_pb = {omega_pb:.2e} rad/s")
print(f"Beam velocity: v0 = {v0:.2e} m/s")
# Wavenumber scan
k_array = np.linspace(0.5, 3, 100) * omega_p0 / v0
omega_real = []
omega_imag = []
for k in k_array:
omega = two_stream_dispersion(k, omega_p0, omega_pb, v0)
omega_real.append(np.real(omega))
omega_imag.append(np.imag(omega))
omega_real = np.array(omega_real)
omega_imag = np.array(omega_imag)
# Analytical approximation for small nb/n0
gamma_approx = omega_p0 * (nb_frac)**(1/3) * np.ones_like(k_array)
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
# Real frequency
ax1.plot(k_array * v0 / omega_p0, omega_real / omega_p0, 'b-',
linewidth=2, label='Numerical')
ax1.axhline(1, color='r', linestyle='--', label='$\\omega_{p0}$')
ax1.plot(k_array * v0 / omega_p0, k_array * v0 / omega_p0, 'g--',
label='$kv_0$')
ax1.set_xlabel('$kv_0 / \\omega_{p0}$', fontsize=13)
ax1.set_ylabel('$\\omega_r / \\omega_{p0}$', fontsize=13)
ax1.set_title('Two-Stream: Real Frequency', fontsize=14)
ax1.legend(fontsize=11)
ax1.grid(True, alpha=0.3)
# Growth rate
ax2.plot(k_array * v0 / omega_p0, omega_imag / omega_p0, 'b-',
linewidth=2, label='Numerical')
ax2.plot(k_array * v0 / omega_p0, gamma_approx / omega_p0, 'r--',
linewidth=1.5, label=f'Approx: $(n_b/n_0)^{{1/3}} = {nb_frac**(1/3):.3f}$')
ax2.set_xlabel('$kv_0 / \\omega_{p0}$', fontsize=13)
ax2.set_ylabel('$\\gamma / \\omega_{p0}$', fontsize=13)
ax2.set_title(f'Two-Stream: Growth Rate ($n_b/n_0 = {nb_frac}$)', fontsize=14)
ax2.legend(fontsize=11)
ax2.grid(True, alpha=0.3)
ax2.set_ylim([0, 0.5])
plt.tight_layout()
plt.savefig('two_stream_instability.png', dpi=150, bbox_inches='tight')
plt.show()
6.2 Weibel Instability Growth Rate¶
def weibel_growth_rate(T_perp, T_parallel, n, B0=0):
"""
Weibel instability growth rate.
Ξ³_max β Ο_pe β[(T_β₯ - T_β₯)/T_β₯]
"""
omega_pe = np.sqrt(n * e**2 / (epsilon_0 * m_e))
anisotropy = (T_perp - T_parallel) / T_parallel
if anisotropy > 0:
gamma_max = omega_pe * np.sqrt(anisotropy)
else:
gamma_max = 0
return gamma_max, omega_pe
# Parameters
n = 1e20 # m^-3
T_parallel = 1e3 # eV
T_perp_array = np.linspace(1e3, 10e3, 100) # eV
gamma_array = []
for T_perp in T_perp_array:
gamma, omega_pe = weibel_growth_rate(T_perp, T_parallel, n)
gamma_array.append(gamma)
gamma_array = np.array(gamma_array)
anisotropy_array = (T_perp_array - T_parallel) / T_parallel
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
# Growth rate vs anisotropy
ax1.plot(anisotropy_array, gamma_array / omega_pe, 'b-', linewidth=2)
ax1.set_xlabel('Anisotropy $(T_\\perp - T_\\parallel)/T_\\parallel$', fontsize=13)
ax1.set_ylabel('$\\gamma / \\omega_{pe}$', fontsize=13)
ax1.set_title('Weibel Instability Growth Rate', fontsize=14)
ax1.grid(True, alpha=0.3)
# Growth rate vs T_perp
ax2.plot(T_perp_array / 1e3, gamma_array / omega_pe, 'r-', linewidth=2)
ax2.axvline(T_parallel / 1e3, color='k', linestyle='--',
label=f'$T_\\parallel = {T_parallel/1e3:.0f}$ keV')
ax2.set_xlabel('$T_\\perp$ (keV)', fontsize=13)
ax2.set_ylabel('$\\gamma / \\omega_{pe}$', fontsize=13)
ax2.set_title(f'Growth Rate vs Perpendicular Temperature', fontsize=14)
ax2.legend(fontsize=11)
ax2.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('weibel_instability.png', dpi=150, bbox_inches='tight')
plt.show()
print(f"\nWeibel instability at T_β₯ = {T_perp_array[-1]/1e3:.0f} keV, T_β₯ = {T_parallel/1e3:.0f} keV:")
print(f" Anisotropy: {anisotropy_array[-1]:.1f}")
print(f" Ξ³/Ο_pe: {gamma_array[-1]/omega_pe:.2f}")
6.3 Firehose and Mirror Stability Boundaries¶
def firehose_criterion(beta_parallel, beta_perp):
"""
Firehose stability: Ξ²_β₯ - Ξ²_β₯ < 1
Returns True if stable.
"""
return (beta_parallel - beta_perp) < 1
def mirror_criterion(beta_perp, p_perp, p_parallel):
"""
Mirror stability: Ξ²_β₯(p_β₯/p_β₯ - 1) < 1
Returns True if stable.
"""
return beta_perp * (p_perp / p_parallel - 1) < 1
# Generate stability diagram
beta_perp_range = np.linspace(0, 5, 200)
beta_parallel_range = np.linspace(0, 5, 200)
Beta_perp, Beta_parallel = np.meshgrid(beta_perp_range, beta_parallel_range)
# Firehose boundary: Ξ²_β₯ - Ξ²_β₯ = 1
firehose_stable = Beta_parallel - Beta_perp < 1
# Mirror boundary: Ξ²_β₯(p_β₯/p_β₯ - 1) = 1
# β p_β₯/p_β₯ = 1 + 1/Ξ²_β₯
# Assume isotropic for simplicity in demo (real case needs p_ratio)
# For demo, use Ξ²_β₯(Ξ²_β₯/Ξ²_β₯ - 1) < 1 β Ξ²_β₯ < Ξ²_β₯ + 1
mirror_stable = Beta_parallel < Beta_perp + 1
# Combined stability region
stable = firehose_stable & mirror_stable
fig, ax = plt.subplots(figsize=(10, 8))
# Plot stability regions
ax.contourf(Beta_perp, Beta_parallel, stable.astype(int),
levels=[0, 0.5, 1], colors=['red', 'green'], alpha=0.3)
# Boundaries
beta_line = np.linspace(0, 5, 100)
ax.plot(beta_line, beta_line + 1, 'b-', linewidth=2,
label='Firehose boundary: $\\beta_\\parallel - \\beta_\\perp = 1$')
ax.plot(beta_line, beta_line - 1, 'r-', linewidth=2,
label='Mirror boundary (approx)')
# Diagonal
ax.plot(beta_line, beta_line, 'k--', alpha=0.5, label='$\\beta_\\parallel = \\beta_\\perp$')
ax.set_xlabel('$\\beta_\\perp$', fontsize=14)
ax.set_ylabel('$\\beta_\\parallel$', fontsize=14)
ax.set_title('Pressure Anisotropy Stability Diagram', fontsize=15)
ax.legend(fontsize=12)
ax.grid(True, alpha=0.3)
ax.set_xlim([0, 5])
ax.set_ylim([0, 5])
# Annotate regions
ax.text(1, 3.5, 'Firehose\nUnstable', fontsize=12, ha='center',
bbox=dict(boxstyle='round', facecolor='red', alpha=0.3))
ax.text(3.5, 1, 'Mirror\nUnstable', fontsize=12, ha='center',
bbox=dict(boxstyle='round', facecolor='orange', alpha=0.3))
ax.text(2, 2, 'Stable', fontsize=12, ha='center',
bbox=dict(boxstyle='round', facecolor='green', alpha=0.3))
plt.tight_layout()
plt.savefig('anisotropy_stability.png', dpi=150, bbox_inches='tight')
plt.show()
6.4 Parametric Instability Thresholds¶
def srs_growth_rate(I_laser, n, T_e, lambda_laser=1.053e-6):
"""
Stimulated Raman Scattering growth rate.
Parameters:
-----------
I_laser : float
Laser intensity (W/mΒ²)
n : float
Density (m^-3)
T_e : float
Electron temperature (eV)
lambda_laser : float
Laser wavelength (m)
Returns:
--------
gamma_SRS : float
Growth rate (s^-1)
"""
c = 3e8
omega_0 = 2 * np.pi * c / lambda_laser
omega_pe = np.sqrt(n * e**2 / (epsilon_0 * m_e))
# Quiver velocity
E_0 = np.sqrt(2 * I_laser / (c * epsilon_0))
v_osc = e * E_0 / (m_e * omega_0)
# Scattered wave frequency (backward scattering)
omega_s = omega_0 - omega_pe # Approximate
# Langmuir wavenumber
k_L = 2 * omega_0 / c # Backscatter
# Growth rate
gamma_SRS = (k_L * v_osc / 4) * np.sqrt(omega_0 / omega_s)
return gamma_SRS
# Laser parameters
lambda_laser = 351e-9 # m (UV, 3Ο Nd:glass)
I_array = np.logspace(13, 16, 100) # W/mΒ²
n = 0.1 * 1.1e21 # m^-3 (nc/10 where nc is critical density)
T_e = 3e3 # eV
gamma_array = []
for I in I_array:
gamma = srs_growth_rate(I, n, T_e, lambda_laser)
gamma_array.append(gamma)
gamma_array = np.array(gamma_array)
# Landau damping (approximate)
v_th = np.sqrt(2 * T_e * e / m_e)
omega_pe = np.sqrt(n * e**2 / (epsilon_0 * m_e))
k_L = 4 * np.pi / lambda_laser
zeta = omega_pe / (k_L * v_th)
gamma_Landau = omega_pe * np.sqrt(np.pi/8) * np.exp(-zeta**2/2) / (k_L**3 * (v_th/omega_pe)**3)
fig, ax = plt.subplots(figsize=(10, 6))
ax.loglog(I_array / 1e15, gamma_array / omega_pe, 'b-',
linewidth=2, label='SRS growth rate')
ax.axhline(gamma_Landau / omega_pe, color='r', linestyle='--',
linewidth=2, label=f'Landau damping: $\\gamma_L/\\omega_{{pe}} = {gamma_Landau/omega_pe:.2e}$')
# Threshold
I_threshold_idx = np.argmin(np.abs(gamma_array - gamma_Landau))
I_threshold = I_array[I_threshold_idx]
ax.axvline(I_threshold / 1e15, color='g', linestyle=':',
linewidth=2, label=f'Threshold: $I_{{th}} = {I_threshold/1e15:.2f}$ PW/cmΒ²')
ax.set_xlabel('Laser Intensity (PW/cmΒ²)', fontsize=13)
ax.set_ylabel('$\\gamma / \\omega_{pe}$', fontsize=13)
ax.set_title('Stimulated Raman Scattering Growth Rate', fontsize=14)
ax.legend(fontsize=11)
ax.grid(True, which='both', alpha=0.3)
plt.tight_layout()
plt.savefig('srs_threshold.png', dpi=150, bbox_inches='tight')
plt.show()
print(f"\nSRS parameters:")
print(f" Density: n = {n:.2e} m^-3 (n/n_c = {n/1.1e21:.2f})")
print(f" Temperature: T_e = {T_e/1e3:.0f} keV")
print(f" Threshold intensity: I_th = {I_threshold:.2e} W/mΒ² = {I_threshold/1e15:.2f} PW/cmΒ²")
Summary¶
Wave heating and instabilities are central to plasma physics:
Wave heating in fusion: - ECRH: $\omega \approx n\omega_{ce}$, 140-170 GHz, excellent localization, current drive - ICRH: $\omega \approx n\omega_{ci}$, 40-80 MHz, ion heating, minority schemes - LHCD: $\omega_{ci} \ll \omega \ll \omega_{ce}$, 3-8 GHz, efficient current drive - Synergistic use of multiple methods optimal for fusion reactors
Velocity space instabilities: - Beam-plasma: Cold beam on cold background, $\gamma \sim \omega_{pe}(n_b/n_0)^{1/3}$ - Bump-on-tail: Positive $\partial f/\partial v$ drives inverse Landau damping, quasilinear plateau - Weibel: Temperature anisotropy generates magnetic fields, $\gamma \sim \omega_{pe}\sqrt{\Delta T/T}$
Pressure-driven instabilities: - Firehose: $p_\parallel > p_\perp$, criterion $\beta_\parallel - \beta_\perp < 1$, bends field lines - Mirror: $p_\perp > p_\parallel$, criterion $\beta_\perp(p_\perp/p_\parallel - 1) < 1$, creates magnetic bottles - Both ubiquitous in collisionless plasmas (solar wind, magnetosphere)
Parametric instabilities: - SRS: EM $\to$ EM + Langmuir, hot electron generation, laser fusion issue - SBS: EM $\to$ EM + ion acoustic, backscattering, energy loss - Thresholds depend on pump intensity, damping rates - Major challenge in ICF, mitigation through bandwidth, smoothing
Applications span fusion energy, astrophysics, space physics, and high-energy-density physics. Understanding instabilities is essential for controlling and optimizing plasma performance.
Practice Problems¶
Problem 1: ECRH System Design¶
A tokamak has $B_0 = 2.5$ T on axis and a density profile $n(r) = n_0(1 - r^2/a^2)^2$ with $n_0 = 8 \times 10^{19}$ m$^{-3}$, $a = 0.5$ m.
(a) Calculate the electron cyclotron frequency $f_{ce}$ at the magnetic axis.
(b) For 2nd harmonic ECRH ($\omega = 2\omega_{ce}$), what gyrotron frequency is needed?
(c) Calculate the O-mode cutoff density at this frequency. Can the wave reach the core?
(d) If X-mode is used instead, where is the upper hybrid resonance layer located?
Problem 2: Two-Stream Instability¶
An electron beam with $n_b = 10^{17}$ m$^{-3}$, $v_0 = 10^7$ m/s propagates through a plasma with $n_0 = 10^{19}$ m$^{-3}$.
(a) Calculate the background plasma frequency $\omega_{p0}$.
(b) Estimate the growth rate $\gamma$ using $\gamma \approx \omega_{p0}(n_b/n_0)^{1/3}$.
(c) At what wavenumber $k$ is the instability resonant (i.e., $kv_0 \approx \omega_{p0}$)?
(d) How many $e$-folding times does it take for the wave amplitude to grow by a factor of 1000? If the beam transits the plasma in $L/v_0 = 1$ ΞΌs, is this sufficient for significant growth?
Problem 3: Weibel Instability in Laser Plasmas¶
A laser-heated plasma has $T_\perp = 500$ eV (heated by laser), $T_\parallel = 50$ eV (cold in the laser direction), and $n = 10^{21}$ m$^{-3}$.
(a) Calculate the anisotropy parameter $(T_\perp - T_\parallel)/T_\parallel$.
(b) Estimate the maximum Weibel growth rate $\gamma_{\text{max}}$.
(c) The generated magnetic field scales as $B^2/(8\pi) \sim nk_B(T_\perp - T_\parallel)$. Estimate the field strength in Tesla.
(d) Compare this field to the field required for electron gyroradius $\rho_L \sim 1/k_{\text{max}}$ where $k_{\text{max}}$ is the wavenumber of maximum growth. Are electrons magnetized by the self-generated field?
Problem 4: Solar Wind Anisotropy¶
Solar wind observations at 1 AU show $\beta_\parallel = 0.8$, $\beta_\perp = 1.5$.
(a) Check if the plasma is stable to the firehose instability.
(b) Check if the plasma is stable to the mirror instability.
(c) If unstable, estimate the growth rate for $v_A = 50$ km/s, $L = 10^6$ km.
(d) The observed anisotropy is quasi-steady, suggesting marginal stability. Propose a mechanism that maintains the plasma near the stability boundary.
Problem 5: Laser-Plasma Instabilities in ICF¶
A laser with intensity $I = 3 \times 10^{15}$ W/cm$^2$ and wavelength $\lambda = 351$ nm illuminates a plasma with $n = 0.1 n_c$ (where $n_c$ is critical density) and $T_e = 3$ keV.
(a) Calculate the critical density $n_c$ for this wavelength.
(b) Estimate the quiver velocity $v_{osc}$ of electrons in the laser field.
(c) Calculate the SRS growth rate using $\gamma_{SRS} \approx (k_L v_{osc}/4)\sqrt{\omega_0/\omega_s}$ where $k_L \approx 2\omega_0/c$ (backscatter).
(d) Compare to the Landau damping rate for Langmuir waves. Is SRS above threshold? What strategies could reduce SRS?
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