Double diffusive mixing (thermohaline convection) 1. Semiconvection ( ⇋ diffusive convection) 2. saltfingering ( ⇋ 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 )

‘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 Saltfingering semiconvection

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 convection layered convection: 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 ➙ ➙

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 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

ST Development from Kato oscillations

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 x microscopic diffusivity mixing in saltfingering case (‘thermohaline’) is limited by small scale of the process