The Kelvin-Helmholtz instability in weakly ionised flows . Downes 1 - - PowerPoint PPT Presentation

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The Kelvin-Helmholtz instability in weakly ionised flows . Downes 1 - - PowerPoint PPT Presentation

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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

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SLIDE 1

The Kelvin-Helmholtz instability in weakly ionised flows

T.P . Downes1,2 & A.C. Jones1

1School of Mathematical Sciences & National Centre for Plasma Science & Technology,

Dublin City University

2Dublin Institute for Advanced Studies

20th June 2012

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 1 / 25

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SLIDE 2

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

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SLIDE 3

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

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SLIDE 4

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

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SLIDE 5

Assumptions

Assumptions

The bulk flow velocity is the neutral velocity The majority of collisions experienced by each charged species

  • ccur 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

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SLIDE 6

Derivation

Outline of derivation of Ohm’s law

The momentum equations for the charged species are: αiρi (E + vi × B) + fi1 = 0 (1) where i = 2, . . . , N. Ignoring mass transfer between the charged species, we can say fij = ρiρjKij

  • vj − vi
  • (2)

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 6 / 25

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SLIDE 7

Derivation

Outline of derivation of Ohm’s law

Moving to the rest frame of the neutral fluid: 0 = αiρi

  • E′ + v′

i × B

  • − B

βi (αiρiv′

i)

(3)

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 7 / 25

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SLIDE 8

Derivation

Outline of derivation of Ohm’s law

After a little algebra: J = σE′

+ σ⊥E′ ⊥ + σH(E′ × b)

(4) where b ≡ B

B . Hence

E′ = r0 (J · B)B B2 + r1 J × B B + r2 B × (J × B) B2 (5)

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 8 / 25

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SLIDE 9

Derivation

The induction equation

Our induction equation then becomes ∂B ∂t + ∇ · {uB − Bu} = ∇ ×

  • r0

(J · B)B B2 + r1 J × B B + r2 B × (J × B) B2

  • (6)

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 9 / 25

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SLIDE 10

Derivation

The equations ...

∂ρi ∂t + ∇ · (ρiui) = 0, (1 ≤ i ≤ N), ∂ρ1u1 ∂t + ∇ ·

  • ρu1u1 + a2ρI
  • =

J × B, ∂B ∂t + ∇ · (u1B − Bu1) = ∇ ×

  • r0

(J · B)B B2 + r1 J × B B + r2 B × (J B2 αiρi (E + ui × B) = −ρiρ1Ki 1(u1 − ui), 2 ≤ i ≤ N, ∇ · B = 0, ∇ × B = J,

N

  • i=2

αiρi = 0.

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 10 / 25

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SLIDE 11

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

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SLIDE 12

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

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SLIDE 13

Reflections

Where’s the problem?

“Diffusion” terms in our induction equation: ∂B ∂t + ∇ · {uB − Bu} = ∇ ×

  • r0

(J · B)B B2 + r1 J × B B + r2 B × (J × B) B2

  • (7)

Ambipolar diffusion causes a serious stable time-step problem.

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 12 / 25

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SLIDE 14

Reflections

Where’s the problem?

“Diffusion” terms in our induction equation: ∂B ∂t + ∇ · {uB − Bu} = ∇ ×

  • r0

(J · B)B B2 + r1 J × B B + r2 B × (J × B) B2

  • (8)

Hall can be a very big problem.

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 13 / 25

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SLIDE 15

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

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SLIDE 16

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

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SLIDE 17

Algorithm

Scaling

2e+08 4e+08 6e+08 8e+08 1e+09 1.2e+09 1.4e+09 1.6e+09 1.8e+09 2e+09 50000 100000 150000 200000 250000 Zone updates/second Number of cores 1024^3 sim with HYDRA Ideal scaling

Strong scaling on the JUGENE BG/P system at Juelich (10243)

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 15 / 25

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SLIDE 18

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

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SLIDE 19

The KH instability

Ambipolar dominated KH instability

15 16 17 x 0.0 0.2 0.4 0.6 0.8 1.0 y 0.01 0.17 0.32 0.48 15 16 17 x 0.0 0.2 0.4 0.6 0.8 1.0 y 0.06 0.08 0.11 0.14

Magnitude (grey-scale) and vector field of the magnetic field for ideal (left panel) and ambipolar dominated (right panel) simulations at onset

  • f saturation

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 17 / 25

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SLIDE 20

The KH instability

Ambipolar dominated KH instability

5 10 15 20 t / ts

  • 25
  • 20
  • 15
  • 10
  • 5

log ( 1/2 ρ vx

2)

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

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SLIDE 21

The KH instability

Ambipolar dominated KH instability

5 10 15 20 t / ts 0.000 0.005 0.010 0.015 0.020 0.025 1/2 B

2 - 1/2 B0 2

As previous slide, but for perturbed magnetic energy

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 19 / 25

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SLIDE 22

The KH instability

Hall dominated KH instability

5 10 15 20 t / ts 0.000 0.005 0.010 0.015 0.020 0.025 1/2 B

2 - 1/2 B0 2

Perturbed magnetic field evolution in Hall dominated, and ideal MHD simulations.

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 20 / 25

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SLIDE 23

The KH instability

Hall dominated KH instability

5 10 15 20 t / ts 0.000 0.005 0.010 0.015 0.020 0.025 1/2 B

2 - 1/2 B0 2

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

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SLIDE 24

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

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SLIDE 25

The KH instability

Hall dominated KH instability

15 16 17 x 0.0 0.2 0.4 0.6 0.8 1.0 y 0.00 0.24 0.47 0.70 15 16 17 x 0.0 0.2 0.4 0.6 0.8 1.0 y 0.00 0.24 0.47 0.70 15 16 17 x 0.0 0.2 0.4 0.6 0.8 1.0 y 0.00 0.34 0.67 1.01

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

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SLIDE 26

The KH instability

Hall dominated KH instability

5 10 15 20 t / ts 0.00 0.01 0.02 0.03 1/2 B

2 - 1/2 B0 2

Perturbed magnetic field evolution in Hall dominated, and ideal MHD simulations.

T.P . Downes (DCU/DIAS) Multifluid KH instability 20th June 2012 24 / 25

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SLIDE 27

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

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