a study of the dynamics of the 3d kelvin-helmholtz...
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A study of the dynamics of the 3D Kelvin-Helmholtz instability
in a partially ionised plasmaAndrew Hillier, Naoki Nakamura, Shinsuke Takasao,
Harufumi Tamazawa & Akito Davis Kawamura
Kwasan Observatory, Kyoto University
In collaboration with the (PIP) coding team, KwasanObservatory
Partially (Weakly) ionized plasma in solar physics
The solar photosphere (turbulence & convection). Ionisation fraction
๐๐~10โ4 โ 10โ6
The solar chromosphere and prominences (waves, reconnection, instabilities and turbulence).
Ionisation fraction ๐๐~10โ1 โ 10โ3
How are neutrals coupled to magnetic fields
Perfect coupling๐๐๐/๐๐ = โ
no coupling๐๐๐/๐๐ =0
n
+
- โ
+
โก
โ Movement of magnetic fieldโก together with charged particles
โข Charged particles collide with neutrals
โฃ Neutrals move in the same direction as the charged particlesCollision
โข โฃ
weak coupling๐๐๐/๐๐ โช ๐๐ท๐๐
marginal coupling๐๐๐/๐๐~๐๐ท๐๐
strong coupling๐๐๐/๐๐ โซ ๐๐ท๐๐
โDynamic timeโ vs โCollision timeโ
Alfven waves
๐๐ด,๐๐ฟ
VS ๐๐๐/๐๐
TurbulenceDimensional arguments give:
๐
๐ฟ2
1/3
VS ๐๐๐/๐๐
(Like a partially ionized plasma Kolmogorov lengthscale)
Instabilities
๐ = ๐(๐,๐ถ๐, ๐๐ด, ๐ฟ,...) VS ๐๐๐/๐๐
L
Reconnection
๐๐ ๐ธ๐ถ~ ๐๐ด๐ฟ VS ๐๐๐/๐๐
โAnd the mountains flowed before the Lordโ
Judge 5.5
In solar physics (and astrophysics in general), we are interested in a huge range of temporal and spatial scales. We cannot only think of our โmountainsโ as โsolidโ or โfluidโ
c.f. Deborah number associated with polymeric fluids
๐ท๐ = ๐ก๐ ๐ธ๐ฟ๐ด๐
๐ก๐๐ต๐
The (PIP) codeNew code to study partially ionized plasma developed at Kyoto University
3D HD and 3D MHD conservative equations solved coupled by collisons (both fluids taken as ideal gases)
Schemes: Fourth-order central difference (eg Vogler et al 2005) and HLL type (e.g. Miyoshi & Kusano 2005)
MPI parallelized (no complications from collision terms)
๐๐๐๐๐ก
+ ๐ป โ ๐๐๐ฏ๐ = 0
๐
๐๐ก๐๐๐ฏ๐ + ๐ป โ ๐๐๐ฏ๐๐ฏ๐ + ๐๐๐
= โ๐๐๐๐๐๐ ๐ฏ๐ โ ๐ฏ๐๐๐๐๐๐ก
+ ๐ป โ ๐ฏ๐ ๐๐ + ๐๐ =
๐๐๐๐๐๐1
2(๐ฃ๐
2 โ ๐ฃ๐2) + 3๐ ๐(๐๐ โ ๐๐)
๐๐๐๐๐ก
+ ๐ป โ ๐๐๐ฏ๐ = 0
๐
๐๐ก๐๐๐ฏ๐ + ๐ป โ ๐๐๐ฏ๐๐ฏ๐ + ๐๐ +
๐2
8๐๐ โ
๐๐
4๐
= ๐๐๐๐๐๐ ๐ฏ๐ โ ๐ฏ๐๐๐๐๐๐ก
+ ๐ป โ ๐ฏ๐ ๐๐ + ๐๐ +๐ธ ร ๐ต
4๐
= ๐๐๐๐๐๐1
2(๐ฏ๐
2 โ ๐ฏ๐2) + 3๐ ๐(๐๐ โ ๐๐)
๐๐
๐๐ก+ ๐ป ร โ๐ฏ๐ ร ๐ + ฮท๐ป ร ๐ = 0
Solving for the collision terms๐๐๐ฏ
๐๐ก~ โ ๐๐๐๐ท
Two problems are associated with solving this equation:1) This equation is very stiff, meaning that the safety factor for the
CFL condition has to be annoyingly small2) When simulating a whole stellar atmosphere, where densities
can vary by many orders of magnitude, there will be regions with very high collision frequency that we need to make simulatable
But for strong coupling it is easy to show that ๐๐ท(๐ก) = ๐๐ท ๐ก0 exp(โ๐ผ๐๐๐๐๐ก)
Which allows for an analytic solution to be give for the integration at timescales below the dynamic timescale
The Magnetic Kelvin-Helmholtz instability in partially ionised plasma
Fundamental instability of astrophysical plasma.
Partially ionized effects have been investigated in molecular clouds (Jones & Downes 2012) and solar prominences (Diaz et al 2012)
๐๐๐๐ = 2
๐๐๐๐ = 1
๐ฝ = 0.3Magnetic field along line of sightX
๐๐ท๐ผ๐น๐น = 0.9๐ถ๐,๐
Ionisation fraction ๐๐ = 0.5
Most cases ๐๐๐ = 100
The simulation setup
All simulations on a 500x500 grid
Linear Regime
For strong coupling, the instability grows in the two fluids together. As coupling gets weaker a lag develops, and the fluids eventually develop their own growth rates (consistent with linear theory of Soler et al. 2012)
๐๐๐/๐๐ โซ ๐๐ท๐ผ๐น๐น๐ ๐๐๐/๐๐ โช ๐๐ท๐ผ๐น๐น
๐
The nonlinear PIP KH
Movie shows ๐๐ and ๐๐ with the arrows showing velocity vectors
for each fluid. Though the similar dynamics develop, the density distributions show stark differences in the centre of the vortex
๐๐ ๐๐๐๐๐/๐๐ = 100
Neutral depletion at vortex centre
Movie shows ๐๐
๐๐ with the arrows showing drift velocity ๐๐ท =๐ฏ๐ โ ๐ฏ๐. Strong regions of neutral depletion form in the vortex centre
Neutral depletion at vortex centre
๐๐
๐๐ (left) and vorticity (right). This highlights that though there are many vortices, only the long-lived vortex shows significant depletion
Ion-neutral centrifuge
Hydrodynamic Vortex MHD Vortex
However, coupling requires them to have the same temperature
Gas pressure gradientMagnetic pressure gradient
Neutral fluid can no longer be contained!
Summary
โข The (PIP) code is progressing well and ready for use in scientific research. Work is still in progress to improve its usage on supercomputers
โข Application to the Kelvin-Helmholtz instability shows that ion-neutral centrifuges are formed in the KH vortex (neutral dust can undergo a similar process c.f. Hendrix and Keppens 2014)
โข Inverse cascade (mainly a 2D effect?) can transport the non-ideal effects to be important at โidealโ length/timescales (as suggest by Jones and Downes 2012)