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

More documents

Recommendations

Info

22 AST242 LECTURE NOTES PART 5we can remove vertical derivatives of ρ 0 and p 0 . Pulling together our equations(equation 128,127 and 133)(136)( N2iωρ 1 − wρ 0g + g )− ik 2 p 1c 2 ⊥s ω + ρ dw0dz= 0(137)(138)iωw + gρ 1+ 1 dp 1ρ 0 ρ 0 dziω(p 1 − c 2 sρ 1 ) + w ρ 0c 2 sN 2g= 0= 0The above involve vertical derivatives of w and p 1 but not ρ 1 so we can reduce theproblem to two coupled first order equations involving p 1 and w. These can bestudied by themselves with all variables depending on z.One can do a local analysis assuming that wavelengths are smaller than the atmospherescale height. In this case we can assume that ρ 1 , p 1 , w all depend on e ikzz andrequire that k z ≫ g/c 2 s. In this limit we find a local dispersion relation(139) ω 4 − (N 2 + k 2 c 2 s)ω 2 + N 2 k 2 ⊥c 2 s = 0with(140) k 2 = k 2 ⊥ + k 2 zThere are two separated regimes on a plot of ω 2 vs k⊥ 2 for allowed wave propagation(see Figure 4). There is a higher frequency set with ω 2 > N 2 which are known asp-waves and a lower frequency set which are known as g-waves. The acoustic orp-waves have phase velocities (ω/k) greater than the sound speed and the buoyancyor g-waves have phase velocities below the sound speed.In the limit of high ω, we find ω 2 ∼ N 2 + k 2 c 2 s.In the limit of high ω, k, we find ω 2 ∼ k 2 c 2 s and the waves are acoustic.In the limit of low ω, we find ω 2 ∼ N 2 k⊥ 2 c2 s.N 2 +k 2 c 2 sIn the limit of low ω, k, we find ω 2 ∼ k 2 c 2 s and the waves are acoustic.In the limit of low ω and high k we find ω 2 ∼ N 2 k⊥2 .k 2Some notes: Apparently low g waves are incompressible, (but why?) and thephase velocity is perpendicular to the group velocity (make this clear?). Why is thisimportant?If there are boundary conditions then there is a discrete set of modes. I have triedto be careful here not to use the word ‘mode’ instead of the word ‘wave’. Abovewe have a local dispersion relation. Prior to that we had coupled equations thatdepended on depth in the atmosphere. For a whole body one would not necessarilyuse a plane parallel approximation but instead consider the equations as a functionof radius, and if the system is rotating, as a function of latitude. Certain frequencies
AST242 LECTURE NOTES PART 5 23Figure 4. The grey regions show the allowed regions for wave propagationbased on the local dispersion relation for waves in a planeparallel atmosphere. Here N is the Brunt-Väisälä frequency and c sthe sound speed. Waves propagating with ω > N are p-waves andthose propagating with ω < N are g-waves.will resonate and these can be called modes. Observations of the spectrum of modescan be used to probe the structure of the object. For example thousands of modeshave been measured on the Sun. The study of these waves and modes is calledhelioseismology. The internal structures of the Sun, Earth and stars are tightlyconstrained by the properties of the modes of oscillation.7. AcknowledgementsInstability at an interface following Clarke and Carswell. Thermal instabilityfollowing Pringle and King. Convective instability following Clarke and Carswell.Waves in atmospheres following Pringle and King.
Page 1 and 2: AST242 LECTURE NOTES PART 5Contents
Page 3: AST242 LECTURE NOTES PART 5 3Follow
Page 9 and 10: AST242 LECTURE NOTES PART 5 9We req
Page 11 and 12: AST242 LECTURE NOTES PART 5 11Figur
Page 13 and 14: AST242 LECTURE NOTES PART 5 13We co
Page 15 and 16: AST242 LECTURE NOTES PART 5 15with
Page 17 and 18: has solution(106) Σ ∝ sinAST242
Page 19 and 20: AST242 LECTURE NOTES PART 5 19We ca
Page 21: Euler’s equation (equation 122) b

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

Create successful ePaper yourself

Delete template?

Save as template?