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Lecture 8

Lecture 8. Double-diffusive convection Discovered in oceanographic context Nowadays many applications Here: two diffusivities Thermal diffusivity acts destabilizing In astrophysics: semiconvection. D ifferent diffusivities crucial. k S =1.3 x 10 -5 cm 2 s -1 (sea water)

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Lecture 8

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  1. Lecture 8 • Double-diffusive convection • Discovered in oceanographic context • Nowadays many applications • Here: two diffusivities • Thermal diffusivity acts destabilizing • In astrophysics: semiconvection

  2. Different diffusivities crucial • kS=1.3 x 10-5 cm2 s-1 (sea water) • kC=1.5 x 10-3 cm2 s-1 • Le = kT/kC= Sc/Pr=Schmidt/Prandtl • is large • temperature in equilibrium, salinity not • Diffusion can destabilize the system! • not with single diffusing component

  3. Two active scalars • C for concentration of salinity • avoid S, which we used for entropy • both aT>0 and aC>0 • Opposing trends: • if T increases, r decreases • if C increases, r increases  Rayleigh-Benard-like problem

  4. Origin in oceanography • Stommel et al (1954) • Stern (1960)

  5. Case 1: more salt on top dT/dz>0 dC/dz>0 dr/dz<0 displacement hot salty blob lighter than surroundings blob heavier than surroundings fresh cold unstable stable unstable (top-heavy) but density stably stratified

  6. Case 2: salt stabilizes unstable dT/dz dT/dz<0 dC/dz<0 dr/dz<0 displacement cold fresh blob heavier than surroundings blob lighter than surroundings salty hot “overstable” unstable Stable Stabilizing! again density stably stratified

  7. Overstability? Originated from stellar stability  (Gough 2003)  Hopf bifurcation

  8. Boussinesq equations

  9. Recall lecture 3, eq.(10) and (11) linearized, & double-curl Proceed analogously where

  10. Reduce to single equation apply to both sides of and use on rhs

  11. ….to obtain Do simplest case: stress-free boundaries assume that principle of exchange of stabilities applicable

  12. …works only for nonoscillatory onset But we see that that instability is possible if remember: Either negative T gradient big enough (i.e. 1st term dominant):  similar to Rayleigh-Benard convection but could be stabilized by negative C gradient or C gradient is positive and big enough (2nd term dominant): salt fingers

  13. Dispersion relation substitute Buoyancy frequencies  Cubic equation Onset, nonoscillatory (zero frequency) Solve numerically also: decaying, oscillatory modes (pair of finite frequencies)

  14. Salt fingers

  15. Dispersion relation: opposite case growing oscillatory modes (again pair of frequencies)

  16. The tworegimes

  17. Staircase formation

  18. Staircase formation Empirical fact, not from linear theory (nor weakly nonlinear theory)

  19. Many applications! Publications in double-diffusive • Oceanography • Geology (magma chambers) • Astrophysics • Engineering (metallurgy) Yet, not very highly thought of back in the 1950s

  20. Astrophysical application: m-gradient Kato (1966)

  21. In astrophysics: semi-convection  Lots of current research! Layered convection (Zaussinger & Spruit 2013)

  22. Backup slides

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  28. “Salt fingers” in magma chamber

  29. Comparison: the 2 regimes

  30. Overstability?

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