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

18 AST242 LECTURE NOTES PART 5heat exchange is slow. This is equivalent to moving the fluid element adiabatically.Because we consider adiabatic changes p ∝ ρ γ and( ) γ ( )p ∗(107)p = p′p = ρ∗p′ 1/γor ρ ∗ = ρρpThe pressure gradient for our atmosphere is dp . If we go up by δz from a particulardzlocation, then the ambient pressure and density are(108) p ′ = p + dpdz δzρ′ = ρ + dρdz δz.We insert p ′ into our previous equation finding( ) p′ 1/γρ ∗ = ρp( p + dp/dzδz= ρp(= ρ 1 + 1 )dpγp dz δz(109)= ρ + ρ dpγp dz δz.) 1/γThe difference in densities( ρ(110) ρ ∗ − ρ ′ dp=γp dz − dρ )δzdzThe system is stable if ρ ∗ > ρ ′ or(111)ρ dpγp dz > dρdzThe stability criterion is often called the Schwarzschild criterion but it is usuallyexpressed in terms of temperature.5.1. Schwarzschild criterion and the Brunt-Väisälä frequency. Using an idealgas p ∝ ρT orExpanding the derivative dρdz ,ρ ∝ p T .(112) ρ ′ = ρ + dρ [ ρdz δz = ρ + dpp dz − ρ T]dTδz.dz