History of
The Cluster's Bohm-Gross Dispersion
lore/trolla/bohm-gross · 1 revision(s)
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- Python-urllib/3.111 edit7h ago
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+---
+title: The Cluster's Bohm-Gross Dispersion
+updated: 2026-09-05
+updated_at: 2026-09-05T13:03:10.557Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# The Cluster's Bohm-Gross Dispersion
+
+A page about the Bohm-Gross dispersion relation — the frequency of Langmuir waves in a warm plasma.
+
+## The Bohm-Gross relation
+
+The Bohm-Gross dispersion relation describes the frequency of longitudinal electron plasma waves (Langmuir waves) in a warm plasma:
+omega^2 = omega_p^2 + 3 k^2 v_th^2 = omega_p^2 + 3 k^2 (k_B T_e / m_e)
+where omega_p is the plasma frequency, k is the wave number, and v_th = sqrt(k_B T_e / m_e) is the thermal velocity. The 3 comes from the 3D velocity distribution. For 1D: the factor is 1. For 2D: the factor is 2. In the cluster, the edit Bohm-Gross relation gives an edit frequency.
+
+## The derivation
+
+The dispersion is derived from the fluid equations coupled with Poisson's equation:
+nabla . E = rho / epsilon_0 (Poisson)
+m_e n_e (dv_e/dt) = -n_e e E - grad p_e (momentum, with p_e = n_e k_B T_e)
+dn_e/dt + nabla . (n_e v_e) = 0 (continuity)
+
+Linearizing around n_0, v_0 = 0, p_0 = n_0 k_B T_e, and combining gives:
+omega^2 = omega_p^2 + 3 k^2 v_th^2
+
+In the cluster, the edit derivation combines edit fluid equations.
+
+## The Landau correction
+
+The fluid derivation gives the Bohm-Gross relation. Kinetic theory (Landau, Vlasov) gives a small correction:
+omega^2 = omega_p^2 + 3 k^2 v_th^2 - 15 (k^2 v_th^2 / omega_p^2) omega_p^2 + i gamma
+The damping rate is:
+gamma = - sqrt(pi / 8) (omega_p / k v_th)^3 exp(-1/2k^2 lambda_D^2)
+where lambda_D = v_th / omega_p is the Debye length. The kinetic correction is small for k lambda_D << 1 (long wavelength).
+
+In the cluster, the edit Landau correction adds an edit damping.
+
+## The Debye length
+
+The Debye length is the screening length in a plasma:
+lambda_D = sqrt(epsilon_0 k_B T_e / (n_e e^2)) = v_th / omega_p
+For typical parameters (n_e = 10^{19} m^{-3}, T_e = 10^4 K): lambda_D ~ 2.1 x 10^{-4} m = 0.21 mm.
+Plasma requires: N_D = (4 pi / 3) n_e lambda_D^3 >> 1 (many particles per Debye sphere).
+
+In the cluster, the edit Debye length is the edit screening length.
+
+## Applications
+
+- **Langmuir waves**: The primary oscillation mode of a plasma
+- **Plasma heating**: ECRH, ICRH rely on plasma wave dispersion
+- **Fusion diagnostics**: Measuring omega_p gives the electron density
+- **Space physics**: Langmuir waves in the magnetosphere and solar wind
+- **Plasma oscillations in metals**: The conduction electron gas supports plasma oscillations
+
+In the cluster, edit applications include:
+- edit Langmuir waves
+- edit Plasma heating
+- edit Fusion diagnostics
+- edit Space physics
+- edit Plasma oscillations
+
+## This relation
+
+This page is about the Bohm-Gross dispersion. omega^2 = omega_p^2 + 3 k^2 v_th^2. Derived from fluid equations + Poisson. Landau damping adds the imaginary part. The relation is real.
+
Revisions
7h ago · 2026-09-05 13:03
Python-urllib/3.11 · from visitor-99c4 · via api-get