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 Opposing trends:
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 hot salty
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
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   Hopf bifurcation
(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  Cubic equation Solve numerically
Buoyancy
frequencies
 Cubic equation
Solve numerically
Onset, nonoscillatory (zero frequency)
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 two regimes

17. Staircase formation

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

19. Many applications! Oceanography Geology (magma chambers) Astrophysics
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

23. x

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27. x

28. “Salt fingers” in magma chamber

29. Comparison: the 2 regimes

30. Overstability?