Double diffusive mixing (thermohaline convection)

1. Semiconvection (⇋ diffusive convection)

2. saltfingering (⇋ thermohaline mixing)

coincidences make these doablecoincidences make these doable

Density ( )

thermal diffusivity ( ), viscosity

solute (He) diffusivity

thermal overturning time

solute buoyancy frequency ( )

astro:

Prandtl number Lewis number (elsewhere denoted )

‘saltfingering’, ‘thermohaline’S destabilizes, T stabilizes

‘diffusive’, ‘semiconvection’T destabilizes, S stabilizes

Double - diffusive convection: (RT-) stable density gradient

Two cases: ( , incompressible approx.)

Both can be studied numerically, but only in a limited parameter range

W. MerryfieldF. Zaussinger

Saltfingeringsemiconvection

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’)

↑displace up:(in pressure equilibrium)

cooling:

↑

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

← diffusion

← diffusion

← diffusion

convection

convection

layered convection:

diffusive interfacestable:

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 model2. 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 model2. 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 model2. 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

➙➙

Fit to laboratory convection expts

Transport of S, T by diffusion

boundary layers

flow overturning time

solute temperature - plume width

- solute contrast carried by plume is limited by net buoyancy

middle of stagnant zone

Model (cf. Linden & Shirtcliffe 1978)

Stagnant zone: transport of S, T by diffusionOverturning 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

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S T

Development from Kato oscillations

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Development of an interface

Different initial conditions

Step

Linear

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:

valid in the range:

independent of

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 limitedby small scale of the process