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Streaming Instability

Source lecture(s): pc368_lec11_streaming_ins

Intuition

A fast beam of electrons streaming through a background plasma can transfer energy to plasma waves when the beam speed exceeds the phase velocity of certain modes. This is the two-stream instability (often called the Buneman instability when \(v_0 \gg v_{te}\)). Instead of damping, the wave grows exponentially, creating holes and vortices in velocity space.

Formal Definition

The streaming instability is a kinetic instability that arises when a drifted Maxwellian beam co-exists with a background plasma, satisfying the resonance condition \(v_\phi \approx v_0\).

Mathematical Formulation

The dispersion relation for a cold beam of density \(n_b\) and drift \(v_0\) through a cold background plasma (\(n_0\)) is:

\[1 + \chi_0 + \chi_b = 0\]

with

\[\chi_{0,b} = -\frac{\omega_{p,0/b}^2}{(\omega - k v_{0/b})^2}\]

For \(m_b = m_e\), the electrostatic solution yields the Buneman instability growth rate:

\[\gamma_{\max} \approx \frac{(\pi/8)^{1/3}}{2^{7/6}} \frac{\omega_{pe}}{S^{1/3}}\]

where \(S = n_b / n_0\) is the density ratio.

Derivation

  1. Linearize the Vlasov–Poisson system for two cold species.
  2. Fourier transform: \((-i\omega + ikv) f_{1s} + (q_s/m_s) E \partial f_{0s}/\partial v = 0\).
  3. For cold beams \(f_{0s} = n_{0s} \delta(v - v_{0s})\), so \(\partial f_{0s}/\partial v\) is singular.
  4. Apply the Landau prescription to push the pole off the real axis.
  5. Solve the quartic polynomial in \(\omega\); unstable roots appear when \(v_0 > c_s\).

Worked Example

Threshold: For an electron beam (\(v_0 = 2\times 10^6\) m/s) through hydrogen plasma (\(n_0 = 10^{18}\) m\(^{-3}\)), determine if the Buneman instability is active. \(v_0 > c_s = \sqrt{k_B T_e/m_i}\)? With \(T_e = 1\) eV, \(c_s \sim 3\times 10^3\) m/s. Since \(v_0 \gg c_s\), the beam is super-sonic and the instability is strong.

Common Mistakes

  • Beam is the only driver. The background plasma provides the reactive inertia.
  • Ion acoustic instability is the same thing. It occurs for \(v_0 \sim c_s\) in two-species plasmas; Buneman is for \(v_0 \gg c_s\) and \(m_b \ll m_i\).
  • Landau damping always quenches instabilities. Near the resonance (\(\omega/k \approx v_0\)), the sign of \(\partial f_0/\partial v\) determines growth.

Quiz Questions

  1. Conceptual: Why does a beam–plasma system “prefer” to grow a wave rather than simply pass through undisturbed?
  2. Computational: Derive the threshold condition \(v_0 > c_s\) for Buneman.
  3. MCQ: Which feature distinguishes the streaming instability from pure Landau damping?
  4. A) Growth rate sign
  5. B) Frequency value
  6. C) Wavelength
  7. D) Magnetic field presence

Further Reading

  • O. Buneman, Dissipation of Currents in Ionized Media.