Double diffusive mixing
1. Semiconvection
2. saltfingering
(thermohaline convection)
(⇋ diffusive convection)
(⇋ thermohaline mixing)
coincidences make these doable
Density
(
)
thermal diffusivity
(
), viscosity
solute (He) diffusivity
thermal overturning time
solute buoyancy frequency
(
)
astro:
Prandtl number
Lewis number
(elsewhere denoted )
Double - diffusive convection: (RT-) stable density gradient
Two cases:
‘saltfingering’, ‘thermohaline’
S destabilizes, T stabilizes
(
, incompressible approx.)
‘diffusive’, ‘semiconvection’
T destabilizes, S stabilizes
Saltfingering
semiconvection
Both can be studied
numerically, but only in a
limited parameter range
F. Zaussinger
W. Merryfield
Geophysical example: the East African volcanic lakes
Lake Kivu, (Ruanda ↔ DRC)
Lake Kivu
(Schmid et al 2010)
double-diffusive ‘staircases’
Linear stability (Kato 1966): predicts an oscillatory
form of instability
(‘overstability’)
cooling:
↑
displace up:
(in pressure
equilibrium)
↑
gravity,
temperature,
solute
⇓
downward
acceleration
↑
Why a layered state instead of Kato-oscillations?
- physics: energy argument
- applied math: ‘subcritical bifurcation’
energy argument
Energy needed to overturn (adiabatically)
a Ledoux-stable layer of thickness :
Per unit of mass:
vanishes as
: overturning in a stack of thin steps takes little energy.
sources:
- from Kato oscillation,
- from external noise
(internal gravity waves from a nearby convection zone)
Proctor 1981:
In the limit
a finite amplitude layered state exists
whenever the system in absence of the stabilizing solute is
convectively unstable.
conditions:
(i.e. astrophysical conditions)
‘weakly-nonlinear’ analysis of fluid instabilities
subcritical instability
(semiconvection)
supercritical instability
(e.g. ordinary convection)
onset of linear instability: Kato oscillations
layered convection:
← diffusion
convection
← diffusion
convection
← diffusion
diffusive interface
stable:
semiconvection: 2 separate problems.
1. fluxes of heat and solute for a given layer thickness
2. layer thickness and its evolution
1: can be done with a parameter study of single layers
2: layer formation depends on initial conditions,
evolution of thickness by merging: slow process,
computationally much more demanding than 1.
Calculations: a double-diffusive stack of thin layers
1. analytical model
2. num. sims.
layers thin: local problem
symmetries of the hydro equations: parameter space limited
5 parameters:
: Boussinesq approx.
Calculations: a double-diffusive stack of thin layers
1. analytical model
2. num. sims.
layers thin: local problem
symmetries of the hydro equations: parameter space limited
5 parameters:
: Boussinesq approx.
limit
: results independent of
Calculations: a double-diffusive stack of thin layers
1. analytical model
2. num. sims.
layers thin: local problem
symmetries of the hydro equations: parameter space limited
5 parameters:
: Boussinesq approx.
limit
: results independent of
a 3-parameter space covers all
➙
➙ fluxes:
+ scalings to astrophysical variables
Transport of S, T
by diffusion
Fit to laboratory convection expts
boundary layers
middle of stagnant zone
flow overturning
time
solute
temperature
- plume width
- solute contrast carried by plume is
limited by net buoyancy
Model
(cf. Linden & Shirtcliffe 1978)
Stagnant zone: transport of S, T by diffusion
Overturning zone:
- heat flux: fit to laboratory convection
- solute flux: width of plume
,
S-content given by buoyancy limit
- stationary: S, T fluxes continuous between
stagnant and overturning zone.
- limit
➙ fluxes (Nusselt numbers):
astro: heat flux
known, transform to
(
)
Heat flux held constant
Model predicts existence of a critical density ratio
(cf. analysis Proctor 1981, Linden & Shirtcliffe 1978)
Numerical
(F. Zaussinger & HS, A&A 2013)
Grid of 2-D simulations to cover the 3-parameter space
- single layer, free-slip top & bottom BC,
horizontally periodic, Boussinesq
- double layer simulations
- compressible comparison cases
Development from Kato oscillations
S
T
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Development of an interface
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Different initial conditions
Linear
Step
Model predicts existence of a critical density ratio
(cf. analysis Proctor 1981, Linden & Shirtcliffe 1978)
Wood, Garaud & Stellmach 2013:
Interpretation in terms of a turbulence model
Wood, Garaud & Stellmach 2013:
fitting formula to numerical results:
(not extrapolated to astrophysical conditions)
For astrophysical application:
independent of
valid in the range:
Semiconvective zone in a
MS star (Weiss):
Evolution of layer thickness
(can reach
?)
merging processes
Estimate using the value of
found
- merging involves redistribution of solute between neighboring layers
layer thickness cannot be discussed independent of system history
Conclusions
Semiconvection is a more astrophysically manageable process:
- thin layers ➙ local
- small Prandtl number limit simplifies the physics
- astrophysical case of known heat flux makes mixing rate
independent of layer thickness
- effective mixing rate only 100-1000 x microscopic diffusivity
mixing in saltfingering case (‘thermohaline’) is limited
by small scale of the process