Stratified atmosphere: dispersion, wave beams, and group vs phase velocity
Parameters
Dispersion Relations
ω = N · kₓ / |k| (where |k|=√(kₓ²+kᵤ²))
cₚ = ω/|k| (phase velocity ∥ k)
c_g = ∂ω/∂k (group velocity ⊥ phase!)
c_gx = N·kᵤ²/|k|³, c_gz = −N·kₓ·kᵤ/|k|³
ω—
|c_phase|—
|c_group|—
θ(k)—
T_period—
Physics
In a stably stratified fluid (dθ/dz > 0), displaced parcels oscillate at the buoyancy (Brunt-Väisälä) frequency N.
Key property: group velocity is perpendicular to phase velocity — energy propagates along beams angled at θ=arccos(ω/N) to the horizontal, while wave crests move perpendicular to that beam.
Colors show vertical displacement w(x,z,t). Arrows show group velocity direction (energy transport). Phase fronts move perpendicular to energy.