Dispersion Relations
The relation between frequency and wavevector that classifies every linear wave and instability in a plasma.
Definition
A dispersion relation D(omega, k) = 0 links the angular frequency omega to the wavevector k for plane-wave perturbations of the form exp(i(k.x - omega t)). It is obtained by linearizing the governing equations, assuming this wave form, and demanding a nontrivial solution, which sets the determinant of the resulting linear system to zero. Every branch omega(k) is a distinct wave or instability.
Reading the Roots
Solving for complex omega = omega_r + i gamma reveals behavior directly. A real omega is an undamped propagating wave; gamma > 0 is exponential growth (instability); gamma < 0 is damping. The phase velocity is omega_r / k and the group velocity, which carries energy, is d omega_r / dk. Whether a mode is convective or absolute (growing at a fixed point) requires the Briggs-Bers analysis of the complex k-plane.
Examples Across Regimes
- Cold plasma waves: the dielectric-tensor determinant gives the O and X modes, R and L modes, and hybrid resonances.
- MHD: the fluid dispersion relation gives the Alfven and magnetosonic branches.
- Kinetic: the Vlasov dispersion relation contains the plasma dispersion function and yields Landau damping and kinetic instabilities.
Practical Use
Dispersion relations are the workhorse of heating and current-drive design: to deposit radio-frequency power one launches a wave that propagates to a resonance layer where D has a singularity or the group velocity vanishes. They also set instability thresholds by locating where gamma crosses zero.
For Kronos concepts, dispersion analysis governs both the D-T Hyperion breeder (Alfven-eigenmode and drift-wave stability) and the D-3He tandem-mirror burner, where wave accessibility to the high-field 26.49 T plug region constrains any radio-frequency scheme. These are modeling inputs; the machines are design and simulation studies.