The Kelvin-Helmholtz instability in weakly ionised flows . Downes 1 , 2 & A.C. Jones 1 T.P 1 School of Mathematical Sciences & National Centre for Plasma Science & Technology, Dublin City University 2 Dublin Institute for Advanced Studies 20th June 2012 T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 1 / 25
Collaborators Dr Stephen O’Sullivan (Dublin Institute of Technology) Dr Aoife Jones T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 2 / 25
Introduction Why weakly ionised? Certain regions of the ISM contain mostly neutral material Molecular clouds T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 3 / 25
Introduction Why weakly ionised? Certain regions of the ISM contain mostly neutral material Accretion disks around YSOs Weak ionisation (pretty much) implies multifluid effects at some length scale T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 4 / 25
Assumptions Assumptions The bulk flow velocity is the neutral velocity The majority of collisions experienced by each charged species occur with neutrals The charged species’ inertia is unimportant The charged species’ pressure gradient is unimportant We can derive a generalised Ohm’s law for this case. T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 5 / 25
Derivation Outline of derivation of Ohm’s law The momentum equations for the charged species are: α i ρ i ( E + v i × B ) + f i 1 = 0 (1) where i = 2 , . . . , N . Ignoring mass transfer between the charged species, we can say f ij = ρ i ρ j K ij v j − v i � � (2) T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 6 / 25
Derivation Outline of derivation of Ohm’s law Moving to the rest frame of the neutral fluid: − B E ′ + v ′ i × B ( α i ρ i v ′ � � 0 = α i ρ i i ) (3) β i T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 7 / 25
Derivation Outline of derivation of Ohm’s law After a little algebra: ⊥ + σ H ( E ′ × b ) J = σ � E ′ � + σ ⊥ E ′ (4) where b ≡ B B . Hence ( J · B ) B J × B B × ( J × B ) E ′ = r 0 + r 1 + r 2 (5) B 2 B B 2 T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 8 / 25
Derivation The induction equation Our induction equation then becomes ∂ B ( J · B ) B J × B B × ( J × B ) � � ∂ t + ∇ · { uB − Bu } = ∇ × r 0 + r 1 + r 2 B 2 B B 2 (6) T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 9 / 25
Derivation The equations ... ∂ρ i ∂ t + ∇ · ( ρ i u i ) 0, (1 ≤ i ≤ N ), = ∂ρ 1 u 1 � � ρ u 1 u 1 + a 2 ρ I J × B , + ∇ · = ∂ t ∂ B ( J · B ) B J × B B × ( J � ∂ t + ∇ · ( u 1 B − Bu 1 ) r 0 + r 1 + r 2 = ∇ × B 2 B B 2 α i ρ i ( E + u i × B ) − ρ i ρ 1 K i 1 ( u 1 − u i ) , 2 ≤ i ≤ N , = ∇ · B = 0 , ∇ × B J , = N � α i ρ i = 0 . i = 2 T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 10 / 25
Reflections Where’s the problem? We don’t like diffusive terms: For explicit algorithms they limit the time-step we can take with each iteration In extreme systems the Hall effect limits the time-step to zero. We don’t like implicit algorithms: Challenging to make multidimensional Challenging to parallelise T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 11 / 25
Reflections Where’s the problem? We don’t like diffusive terms: For explicit algorithms they limit the time-step we can take with each iteration In extreme systems the Hall effect limits the time-step to zero. We don’t like implicit algorithms: Challenging to make multidimensional Challenging to parallelise T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 11 / 25
Reflections Where’s the problem? “Diffusion” terms in our induction equation: ∂ B ( J · B ) B J × B B × ( J × B ) � � ∂ t + ∇ · { uB − Bu } = ∇ × r 0 + r 1 + r 2 B 2 B B 2 (7) Ambipolar diffusion causes a serious stable time-step problem. T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 12 / 25
Reflections Where’s the problem? “Diffusion” terms in our induction equation: ∂ B ( J · B ) B J × B B × ( J × B ) � � ∂ t + ∇ · { uB − Bu } = ∇ × r 0 + r 1 + r 2 B 2 B B 2 (8) Hall can be a very big problem. T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 13 / 25
Algorithm Outline of numerics Advance the entire system of equations using operator splitting (O’Sullivan & Downes 2006, 2007): Advance neutrals using Godunov-type method Apply “diffusion terms” using super-time-stepping and the Hall Diffusion Scheme Advance charged species densities assuming force balance Method of Dedner used to control ∇ · B Method is entirely explicit T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 14 / 25
Algorithm Outline of numerics Advance the entire system of equations using operator splitting (O’Sullivan & Downes 2006, 2007): Advance neutrals using Godunov-type method Apply “diffusion terms” using super-time-stepping and the Hall Diffusion Scheme Advance charged species densities assuming force balance Method of Dedner used to control ∇ · B Method is entirely explicit T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 14 / 25
Algorithm Scaling 2e+09 1024^3 sim with HYDRA Ideal scaling 1.8e+09 1.6e+09 1.4e+09 Zone updates/second 1.2e+09 1e+09 8e+08 6e+08 4e+08 2e+08 50000 100000 150000 200000 250000 Number of cores Strong scaling on the JUGENE BG/P system at Juelich ( 1024 3 ) T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 15 / 25
The KH instability Initial conditions - KH instability Isothermal, multifluid MHD: neutrals, electrons and ions. Computational domain in ( x , y ) of 32 L × L , resolution of 6400 × 200 Flow in the y direction, periodic boundaries at high and low y , gradient zero and high and low x Ambipolar dominated and Hall dominated flows (magnetic Reynolds numbers in the range of 28 – 280. T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 16 / 25
The KH instability Ambipolar dominated KH instability 1.0 0.48 1.0 0.14 0.8 0.8 0.32 0.11 0.6 0.6 y y 0.4 0.4 0.17 0.08 0.2 0.2 0.0 0.01 0.0 0.06 15 16 17 15 16 17 x x Magnitude (grey-scale) and vector field of the magnetic field for ideal (left panel) and ambipolar dominated (right panel) simulations at onset of saturation T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 17 / 25
The KH instability Ambipolar dominated KH instability 0 -5 2 ) log ( 1/2 ρ v x -10 -15 -20 -25 0 5 10 15 20 t / t s Transverse kinetic energy as a function of time (progressively thicker lines for high ambipolar resistivity) T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 18 / 25
The KH instability Ambipolar dominated KH instability 0.025 0.020 2 2 - 1/2 B 0 0.015 0.010 1/2 B 0.005 0.000 0 5 10 15 20 t / t s As previous slide, but for perturbed magnetic energy T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 19 / 25
The KH instability Hall dominated KH instability 0.025 0.020 2 2 - 1/2 B 0 0.015 0.010 1/2 B 0.005 0.000 0 5 10 15 20 t / t s Perturbed magnetic field evolution in Hall dominated, and ideal MHD simulations. T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 20 / 25
The KH instability Hall dominated KH instability 0.025 0.020 2 2 - 1/2 B 0 0.015 0.010 1/2 B 0.005 0.000 0 5 10 15 20 t / t s As previous slide, but decomposing magnetic energy into that in the xy -plane and that in the z direction. T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 21 / 25
The KH instability Hall dominated KH instability So now let’s boost the Hall resistivity even further ... T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 22 / 25
The KH instability Hall dominated KH instability 1.0 1.0 0.70 0.70 0.8 0.8 0.47 0.47 0.6 0.6 y y 0.4 0.4 0.24 0.24 0.2 0.2 0.0 0.0 0.00 0.00 15 16 17 15 16 17 x x 1.0 1.01 0.8 0.67 0.6 y 0.4 0.34 0.2 0.0 0.00 15 16 17 x Plots of the magnitude (grey-scale) and vector field of the neutral, ion and electron velocity fields T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 23 / 25
The KH instability Hall dominated KH instability 0.03 2 2 - 1/2 B 0 0.02 1/2 B 0.01 0.00 0 5 10 15 20 t / t s Perturbed magnetic field evolution in Hall dominated, and ideal MHD simulations. T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 24 / 25
Conclusions Conclusions Ambipolar diffusion dramatically reduces the magnetic energy generated, and marginally increases the peak transverse energy The Hall effect leads to a system which does not reach a quasi-steady state In extreme situations the Hall effect leads to strong dynamo action T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 25 / 25
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