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

AST242 LECTURE NOTES PART 5 11Figure 2. Kelvin Helmholtz instability grows because of pressure differencescaused by the velocity perturbations. It’s not obvious here,but there must be a small lag between pressure and velocity differencesfor water waves to be excited by wind. If the velocity and pressure perturbationsare exactly in phase then there is no energy transfer betweenmedia.3.3. Kelvin-Helmholtz Instability. We consider the case when fluids are stableto the Rayleigh-Taylor instability but moving with respect to one another. Ourdispersion relation (equation 64) is a quadratic equation for ω. Grouping terms wecan write the dispersion relation as a polynomial of ω,(69) ω 2 (ρ + ρ ′ ) − ω2k(ρU + ρ ′ U ′ ) + k 2 (ρU 2 + ρ ′ U ′2 ) − kg(ρ − ρ ′ ) = 0The quadratic equation gives(70)ω =12(ρ + ρ ′ ) (2k(ρU + ρ′ U ′ )±√4k2 (ρU + ρ ′ U ′ ) 2 − 4(ρ + ρ ′ )(k 2 (ρU 2 + ρ ′ U ′2 ) − kg(ρ − ρ ′ ))).There is no real solution and instability occurs when(71) k > (ρ2 − ρ ′2 )gρρ ′ (U − U ′ ) 2 .Note ω is not necessarily small at the transition point. When gravity is unimportant,we can take g → 0, and then find that all wavelengths are unstable as long as U ′ ≠ U.When ρ > ρ ′ , then gravity stabilizes short k or long wavelengths. The larger k