AST242 LECTURE NOTES PART 5 Contents 1. Waves and ...

Euler’s equation (equation 122) becomesAST242 LECTURE NOTES PART 5 21ωu = 1 ρ 0k x p 1(127)ωv = 1 ρ 0k y p 1iωw = ρ 1 dp 0ρ 2 0 dz − 1 dp 1ρ 0 dzWe can use the first two expressions above to remove u, v from equation (126).(128) iωρ 1 + w dρ 0dz − i(k2 x + ky) 2 p 1ω + ρ dw0dz = 0We now relate pressure perturbations to density perturbations. We assume thatperturbations are locally adiabatic. This condition can be written( ) ( )∂ p p(129)+ (u · ∇) = 0∂t ρ γ ρ γExpanding this∂p(130)∂t − ∂ρc2 s∂t + (u · ∇)p − c2 s(u · ∇)ρ = 0where c 2 s = γp/ρ. To first order(131)or(132) iω(p 1 − c 2 sρ 1 ) + w∂p 1∂t − ∂ρ 1c2 s∂t + w dp 0dz − c2 sw dρ 0dz = 0( dp0dz − c2 s)dρ 0= 0.dzRemember that c 2 s depends on z! The right terms can be written in terms of theBrunt-Väisälä frequency (equation 118)(133) iω(p 1 − c 2 sρ 1 ) + w ρ 0c 2 sN 2Use hydrostatic equilibrium(134) g = − 1 ρ 0dp 0dzand the Brunt-Väisälä frequency(135) − dρ 0dz = N 2 ρ 0g+ gρ 0c 2 sg= 0.