Part 4, Chapter 3

Ion Acoustic Waves

Sound waves in plasma: two-fluid derivation and ion Landau damping

3.1 Physical Picture

Ion acoustic waves (IAWs) are the plasma analogue of ordinary sound waves. Unlike Langmuir waves, which involve electron oscillations against a stationary ion background, ion acoustic waves are low-frequency oscillations in which both ions and electrons participate. The ions provide the inertia while the electrons, being much lighter, respond nearly instantaneously and provide the restoring pressure force through their thermal motion.

The key requirement for weakly damped ion acoustic waves is that the electron temperature greatly exceeds the ion temperature: $T_e \gg T_i$. When this condition is violated, strong ion Landau damping occurs because the ion thermal speed approaches the wave phase velocity.

The characteristic speed of ion acoustic waves is the ion sound speed:

$$c_s = \sqrt{\frac{T_e + 3T_i}{m_i}}$$

3.2 Two-Fluid Derivation

We treat both electrons and ions as separate fluids. The electrons are assumed to be in Boltzmann equilibrium due to their small inertia at these low frequencies ($\omega \ll \omega_{pe}$):

$$n_e = n_0 \exp\!\left(\frac{e\phi}{T_e}\right) \approx n_0\left(1 + \frac{e\phi}{T_e}\right)$$

The ion fluid equations are the continuity and momentum equations:

$$\frac{\partial n_i}{\partial t} + \nabla \cdot (n_i \mathbf{v}_i) = 0$$$$m_i n_i \frac{\partial \mathbf{v}_i}{\partial t} = e n_i \mathbf{E} - \nabla p_i$$

Combined with Poisson's equation $\nabla^2 \phi = -e(n_i - n_e)/\epsilon_0$, we linearize all quantities and assume plane-wave perturbations $\propto e^{i(kx-\omega t)}$. Using the adiabatic ion pressure $p_{i1} = 3T_i n_{i1}$ and the Boltzmann electron response, we obtain:

$$-\omega^2 n_{i1} = -k^2 \frac{e n_0}{m_i}\phi_1 - \frac{3k^2 T_i}{m_i}n_{i1}$$

Using quasi-neutrality in Poisson's equation, $n_{i1} \approx n_0 e\phi_1/T_e \cdot 1/(1+k^2\lambda_D^2)$, we arrive at the ion acoustic dispersion relation:

$$\boxed{\omega^2 = \frac{k^2 c_s^2}{1 + k^2\lambda_D^2}}$$

In the long-wavelength limit $k\lambda_D \ll 1$, this reduces to$\omega = k c_s$ -- a non-dispersive acoustic wave. At short wavelengths $k\lambda_D \gg 1$, the frequency saturates at the ion plasma frequency $\omega_{pi}$.

3.3 Ion Landau Damping

The phase velocity of ion acoustic waves is $v_\phi = \omega/k \approx c_s$. The ion thermal speed is $v_{ti} = \sqrt{T_i/m_i}$. The ratio of these speeds determines the strength of ion Landau damping:

$$\frac{v_\phi}{v_{ti}} \approx \sqrt{\frac{T_e}{T_i}}$$

When $T_e \gg T_i$, the phase velocity lies far out on the tail of the ion distribution and damping is weak. When $T_i \sim T_e$, the phase velocity is comparable to $v_{ti}$, placing it near the peak of the ion distribution where many particles can resonate with the wave. The kinetic damping rate is:

$$\frac{\gamma}{|\omega|} \approx -\sqrt{\frac{\pi}{8}}\left(\frac{T_e}{T_i}\right)^{3/2}\exp\!\left(-\frac{T_e}{2T_i}\right)$$

This exponential dependence means that for $T_e/T_i = 10$, the damping is negligible, but for $T_e/T_i = 1$, ion acoustic waves are so heavily damped that they cannot propagate even one wavelength. This is why ion acoustic waves are prominent in laser-produced plasmas (where $T_e \gg T_i$) but rarely observed in thermalized fusion plasmas.

3.4 Computational Exploration

The code below plots the ion acoustic dispersion relation comparing cold ions ($T_i = 0$) with warm ions ($T_i = 0.1\,T_e$ and $T_i = T_e$), and shows the damping rate as a function of the temperature ratio.

Ion Acoustic Waves Simulation

Python

script.py51 lines

import numpy as np
import matplotlib.pyplot as plt

# Normalized units: T_e = 1, m_i = 1, lambda_D = 1
T_e = 1.0
m_i = 1.0
lambda_D = 1.0

k = np.linspace(0.01, 5.0, 500)

# Ion acoustic dispersion for different Ti/Te
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

for Ti_ratio, color, label in [(0.0, 'orange', '$T_i = 0$'),
                                 (0.1, 'cyan', '$T_i = 0.1\,T_e$'),
                                 (1.0, 'red', '$T_i = T_e$')]:
    T_i = Ti_ratio * T_e
    cs2 = (T_e + 3 * T_i) / m_i
    omega = np.sqrt(k**2 * cs2 / (1 + k**2 * lambda_D**2))
    ax1.plot(k * lambda_D, omega / np.sqrt(T_e / m_i), color=color, linewidth=2, label=label)

# Plot the acoustic limit omega = k*cs for cold ions
cs_cold = np.sqrt(T_e / m_i)
ax1.plot(k * lambda_D, k * lambda_D, '--', color='gray', linewidth=1, label=r'$\omega = k c_s$ (cold)')

ax1.set_xlabel(r'$k\lambda_D$', fontsize=13)
ax1.set_ylabel(r'$\omega / (T_e/m_i)^{1/2}$', fontsize=13)
ax1.set_title('Ion Acoustic Dispersion', fontsize=14)
ax1.set_xlim(0, 5)
ax1.set_ylim(0, 4)
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)

# Ion Landau damping rate vs Te/Ti
Te_Ti = np.linspace(1.0, 20.0, 200)
gamma_norm = np.sqrt(np.pi / 8) * Te_Ti**(1.5) * np.exp(-Te_Ti / 2.0)

ax2.semilogy(Te_Ti, gamma_norm, 'orange', linewidth=2)
ax2.axhline(y=1.0, color='red', linestyle='--', alpha=0.5, label='Critical damping')
ax2.set_xlabel(r'$T_e / T_i$', fontsize=13)
ax2.set_ylabel(r'$|\gamma/\omega|$', fontsize=13)
ax2.set_title('Ion Landau Damping Rate', fontsize=14)
ax2.set_xlim(1, 20)
ax2.legend(fontsize=10)
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('output.png', dpi=100, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Ion acoustic wave dispersion and damping plotted.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

3.5 Nonlinear Ion Acoustic Waves

At finite amplitude, ion acoustic waves exhibit nonlinear steepening: the wave crest moves faster than the trough because the local sound speed depends on density. This steepening is balanced by the dispersive term $k^2\lambda_D^2$ in the dispersion relation, leading to solitary wave solutions.

The nonlinear evolution is described by the Korteweg-de Vries (KdV) equation:

$$\frac{\partial \phi}{\partial t} + c_s \frac{\partial \phi}{\partial x} + \alpha \phi \frac{\partial \phi}{\partial x} + \beta \frac{\partial^3 \phi}{\partial x^3} = 0$$

which admits soliton solutions of the form:

$$\phi(x,t) = \phi_0 \,\text{sech}^2\!\left(\frac{x - Vt}{\Delta}\right)$$

where the soliton width $\Delta \propto 1/\sqrt{\phi_0}$ -- larger amplitude solitons are narrower and travel faster. Ion acoustic solitons have been observed in laboratory plasmas, the auroral ionosphere (by the FAST and Cluster satellites), and in laser-produced plasmas.

3.6 Physical Significance

Ion acoustic waves are central to many plasma phenomena:

Stimulated Brillouin scattering: In laser-plasma interactions, incident light can decay into scattered light plus an ion acoustic wave, a major energy loss mechanism in inertial confinement fusion.
Ion acoustic turbulence: Current-driven ion acoustic instabilities lead to anomalous resistivity in collisionless shocks and magnetic reconnection regions.
Ion acoustic solitons: Nonlinear steepening balanced by dispersion produces solitary structures described by the Korteweg-de Vries equation, observed in the auroral ionosphere and laboratory plasmas.
Plasma diagnostics: Ion acoustic wave scattering is used in Thomson scattering diagnostics to measure ion temperature and plasma flow velocity.

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