Electrostatic Waves in Unmagnetized and Magnetized Plasmas

1. Complex Harmonic Wave Representations

Small-amplitude perturbations in plasma fluid quantities (density, velocity, electric and magnetic fields) are expanded as superpositions of plane waves:

$$\psi(\vec{r}, t) = \psi_0 + \psi_1 \exp\left[ i(\vec{k}\cdot\vec{r} - \omega t) \right]$$

where $\psi_0$ is the unperturbed background equilibrium state, $\psi_1$ is the first-order perturbation amplitude ($|\psi_1| \ll |\psi_0|$), $\vec{k}$ is the wavevector, and $\omega$ is the angular frequency. Under this harmonic convention, differential operators transform into algebraic multipliers:

$$\nabla \to i\vec{k}, \quad \frac{\partial}{\partial t} \to -i\omega$$

2. Phase Velocity vs Group Velocity

The speed at which surfaces of constant wave phase propagate is the phase velocity $\vec{v}_{ph}$:

$$\vec{v}_{ph} = \frac{\omega}{k} \hat{k}$$

The speed and direction at which a modulated wave packet—and consequently physical energy and information—transmits through the plasma is the group velocity $\vec{v}_g$:

$$\vec{v}_g = \nabla_k \omega = \frac{\partial \omega}{\partial k} \hat{k}$$

In dispersive plasma media where $\omega(k)$ is non-linear, $v_{ph} \ne v_g$.

1. The Linearization Procedure

For electrostatic waves, the magnetic field perturbation vanishes ($\vec{B}_1 = 0$), so the electric field is purely curl-free and derived from an electrostatic potential: $\vec{E}_1 = -\nabla \phi_1 = -i\vec{k}\phi_1$.

Each fluid variable is decomposed into equilibrium plus first-order perturbation:

$$n_\alpha = n_0 + n_{\alpha 1}, \quad \vec{u}_\alpha = 0 + \vec{u}_{\alpha 1}, \quad \vec{E} = 0 + \vec{E}_1$$

Neglecting second-order nonlinear terms ($n_1 \vec{u}_1 \approx 0$ and $(\vec{u}_1\cdot\nabla)\vec{u}_1 \approx 0$):

• Linearized Continuity: $$-i\omega n_{\alpha 1} + i n_0 \vec{k}\cdot\vec{u}_{\alpha 1} = 0 \implies n_{\alpha 1} = n_0 \frac{\vec{k}\cdot\vec{u}_{\alpha 1}}{\omega}$$
• Linearized Momentum: $$-i\omega m_\alpha n_0 \vec{u}_{\alpha 1} = q_\alpha n_0 \vec{E}_1 - \gamma_\alpha k_B T_\alpha (i\vec{k} n_{\alpha 1})$$
• Poisson's Equation: $$i\vec{k}\cdot\vec{E}_1 = \frac{e(n_{i1} - n_{e1})}{\varepsilon_0}$$

1. The Cold Plasma Limit

In the cold plasma limit ($T_e = 0$), electron pressure vanishes. Because the oscillation frequency is high, massive ions cannot respond ($n_{i1} = 0, \vec{u}_{i1} = 0$). The linearized electron momentum equation reduces to:

$$-i\omega m_e \vec{u}_{e1} = -e \vec{E}_1 \implies \vec{u}_{e1} = \frac{e}{i\omega m_e}\vec{E}_1$$

Substitute $\vec{u}_{e1}$ into continuity:

$$n_{e1} = n_0 \frac{\vec{k}\cdot\vec{u}_{e1}}{\omega} = \frac{n_0 e}{i\omega^2 m_e}\vec{k}\cdot\vec{E}_1$$

Substitute $n_{e1}$ into Poisson's equation $i\vec{k}\cdot\vec{E}_1 = -\frac{e n_{e1}}{\varepsilon_0}$:

$$i\vec{k}\cdot\vec{E}_1 = -\frac{e}{\varepsilon_0}\left( \frac{n_0 e}{i\omega^2 m_e}\vec{k}\cdot\vec{E}_1 \right) = \frac{n_0 e^2}{\varepsilon_0 m_e \omega^2} (i\vec{k}\cdot\vec{E}_1)$$

For a non-trivial wave solution ($\vec{k}\cdot\vec{E}_1 \ne 0$):

$$1 - \frac{n_0 e^2}{\varepsilon_0 m_e \omega^2} = 0 \implies \omega^2 = \frac{n_0 e^2}{\varepsilon_0 m_e} \equiv \omega_{pe}^2$$

In a cold plasma, electron oscillations occur at the single constant frequency $\omega = \omega_{pe}$, independent of wavevector $k$. Consequently:

$$v_g = \frac{d\omega}{dk} = 0$$

Cold electron plasma oscillations do not propagate; they are purely local stationary oscillations!

1. Thermal Pressure Correction

When electron temperature is non-zero ($T_e > 0$), thermal pressure $\nabla P_{e1} = \gamma_e k_B T_e \nabla n_{e1}$ provides an additional restoring force. For one-dimensional high-frequency compressions along $\vec{k}$, there is only one translational degree of freedom ($d=1$), so the adiabatic index is $\gamma_e = (d+2)/d = 3$.

The linearized electron momentum equation becomes:

$$-i\omega m_e n_0 u_{e1} = -e n_0 E_1 - 3 k_B T_e (i k n_{e1})$$

Using continuity $n_{e1} = n_0 \frac{k u_{e1}}{\omega}$:

$$-i\omega m_e u_{e1} = -e E_1 - 3 k_B T_e i k \left( \frac{k u_{e1}}{\omega} \right) \implies u_{e1}\left( -i\omega + \frac{3 k_B T_e i k^2}{\omega m_e} \right) = -\frac{e}{m_e} E_1$$ $$u_{e1} = \frac{e E_1}{i m_e \omega} \left( 1 - \frac{3 k^2 v_{\text{th},e}^2}{\omega^2} \right)^{-1}$$

where $v_{\text{th},e} = \sqrt{k_B T_e / m_e}$ is the electron thermal speed.

2. The Bohm-Gross Dispersion Relation

Substituting $n_{e1}$ into Poisson's equation $i k E_1 = -\frac{e n_{e1}}{\varepsilon_0} = -\frac{e n_0 k u_{e1}}{\varepsilon_0 \omega}$:

$$1 - \frac{\omega_{pe}^2}{\omega^2 - 3 k^2 v_{\text{th},e}^2} = 0 \implies \omega^2 = \omega_{pe}^2 + 3 k^2 v_{\text{th},e}^2$$

This is the famous Bohm-Gross dispersion relation for warm electron plasma waves (Langmuir waves).

Differentiating with respect to $k$ yields a non-zero group velocity:

$$2\omega \frac{d\omega}{dk} = 6 k v_{\text{th},e}^2 \implies v_g = \frac{3 k v_{\text{th},e}^2}{\omega} = \frac{3 v_{\text{th},e}^2}{v_{ph}}$$

Thermal pressure allows electron plasma waves to propagate and carry energy across the plasma!

1. Low-Frequency Dynamics

At low frequencies ($\omega \ll \omega_{pe}$), electrons move so rapidly compared to the wave that they remain in continuous thermodynamic equilibrium, establishing a Boltzmann distribution in the wave potential $\phi_1$:

$$n_{e1} = n_0 \frac{e \phi_1}{k_B T_e}$$

The heavy ions, however, are accelerated dynamically by the wave electric field $E_1 = -ik\phi_1$. Linearized ion continuity and momentum equations with ion temperature $T_i$:

$$-i\omega n_{i1} + i n_0 k u_{i1} = 0 \implies n_{i1} = n_0 \frac{k u_{i1}}{\omega}$$ $$-i\omega M_i n_0 u_{i1} = e n_0 (-i k \phi_1) - \gamma_i k_B T_i (i k n_{i1})$$

Solving for ion density perturbation $n_{i1}$:

$$n_{i1} = \frac{n_0 e k^2}{M_i (\omega^2 - \gamma_i k^2 v_{\text{th},i}^2)} \phi_1$$

2. The Ion Acoustic Dispersion Relation

Substituting $n_{e1}$ and $n_{i1}$ into Poisson's equation $k^2 \phi_1 = \frac{e(n_{i1} - n_{e1})}{\varepsilon_0}$:

$$k^2 \phi_1 = \frac{e}{\varepsilon_0}\left[ \frac{n_0 e k^2}{M_i (\omega^2 - \gamma_i k^2 v_{\text{th},i}^2)}\phi_1 - \frac{n_0 e \phi_1}{k_B T_e} \right]$$

Dividing by $\phi_1$ and using $\lambda_{De}^2 = \frac{\varepsilon_0 k_B T_e}{n_0 e^2}$ and $\omega_{pi}^2 = \frac{n_0 e^2}{\varepsilon_0 M_i}$:

$$k^2 + \frac{1}{\lambda_{De}^2} = \frac{\omega_{pi}^2 k^2}{\omega^2 - \gamma_i k^2 v_{\text{th},i}^2}$$

For cold ions ($T_i \ll T_e$):

$$\omega^2 = \frac{k^2 \omega_{pi}^2 \lambda_{De}^2}{1 + k^2 \lambda_{De}^2}$$

Noting that $\omega_{pi} \lambda_{De} = \sqrt{\frac{n_0 e^2}{\varepsilon_0 M_i}}\sqrt{\frac{\varepsilon_0 k_B T_e}{n_0 e^2}} = \sqrt{\frac{k_B T_e}{M_i}} \equiv c_s$, where $c_s$ is the ion sound speed:

$$\omega^2 = \frac{k^2 c_s^2}{1 + k^2 \lambda_{De}^2}$$

Including finite ion temperature ($T_i > 0$ with 1D adiabatic compression $\gamma_i = 3$):

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

1. Limiting Regimes of Ion Acoustic Waves

The ion acoustic dispersion relation exhibits two fundamentally distinct physical behaviors depending on wavelength relative to the Debye length:

1. Long-Wavelength Acoustic Limit ($k \lambda_{De} \ll 1$, $\lambda \gg \lambda_{De}$): $$\omega \approx k c_s, \quad v_{ph} = v_g = c_s = \text{const}$$ The wave is non-dispersive and behaves identically to an ordinary acoustic sound wave in neutral gas. In this limit, electrons perfectly shield ion charge fluctuations, maintaining quasi-neutrality ($n_{e1} \approx n_{i1}$).
2. Short-Wavelength Ion Plasma Limit ($k \lambda_{De} \gg 1$, $\lambda \ll \lambda_{De}$): $$\omega \approx \frac{k c_s}{k \lambda_{De}} = \frac{c_s}{\lambda_{De}} = \omega_{pi}$$ At wavelengths shorter than the Debye length, electron Debye shielding breaks down completely. The electrons can no longer cluster to screen the ions, and the wave degenerates into constant-frequency ion plasma oscillations at $\omega = \omega_{pi}$, with zero group velocity ($v_g \to 0$).

2. Fundamental Comparison: Electron Waves vs Ion Waves

Feature	Electron Plasma Wave (Langmuir)	Ion Acoustic Wave (Sound)
Restoring Force	Electric field + Electron thermal pressure	Electron thermal pressure ($T_e$)
Inertia	Electron mass $m_e$	Ion mass $M_i$
Characteristic Frequency	High ($\omega \ge \omega_{pe} \sim 10^{11}\text{ rad/s}$)	Low ($\omega \le \omega_{pi} \sim 10^9\text{ rad/s}$)
Ion Motion	Stationary neutralizing background	Oscillating fluid elements

1. Waves Propagating Perpendicular to B0: Upper Hybrid Oscillations

When a static magnetic field $\vec{B}_0 = B_0 \hat{z}$ is present, consider electrostatic electron waves propagating perpendicular to the field ($\vec{k} = k \hat{x} \perp \vec{B}_0$).

The electrons experience two restoring forces simultaneously:

• The electrostatic space-charge restoring force ($-\omega_{pe}^2$).
• The magnetic Lorentz force restoring gyration ($-\omega_{ce}^2$).

Solving the linearized electron equations of motion yields the Upper Hybrid frequency $\omega_{UH}$:

$$\omega^2 = \omega_{UH}^2 = \omega_{pe}^2 + \omega_{ce}^2$$

Including electron thermal pressure: $\omega^2 = \omega_{UH}^2 + 3 k^2 v_{\text{th},e}^2$.

2. Lower Hybrid Oscillations

When both electron and ion motions are included for perpendicular propagation, an intermediate resonance occurs between the electron and ion cyclotron frequencies, termed the Lower Hybrid frequency $\omega_{LH}$:

$$\frac{1}{\omega_{LH}^2} = \frac{1}{\omega_{pi}^2 + \omega_{ci}^2} + \frac{1}{\omega_{ce}\omega_{ci}} \implies \omega_{LH} \approx \sqrt{\omega_{ce}\omega_{ci}}$$

3. Electrostatic Ion Cyclotron (EIC) Waves

For waves propagating at an oblique angle nearly perpendicular to $\vec{B}_0$ ($k_\perp \gg k_\parallel$) with frequencies near the ion cyclotron frequency $\omega \sim \omega_{ci}$:

$$\omega^2 = \omega_{ci}^2 + k_\perp^2 c_s^2$$

These electrostatic ion cyclotron (EIC) waves are driven by parallel electron currents and play a major role in auroral ion heating and tokamak scrape-off layers.