Numerical Simulations of the Wardle Instability Sam Falle, - PowerPoint PPT Presentation

Numerical Simulations of the Wardle Instability Sam Falle, Department of Applied Mathematics, University of Leeds. Tom Hartquist, Sven van Loo, School of Physics and Astronomy, University of Leeds.

Astrophysical Molecular Clouds Neutral density ∼ 10 6 particles cm − 3 Neutral Sound speed ∼ 0 . 2 km s − 1 Magnetic Field ∼ 10 − 3 G ⇒ Alfv´ en speed ∼ 2 km s − 1 Ionization fraction ∼ 10 − 4 − 10 − 9 ⇒ High Resistivity (Ambi-polar diffusion) ⇒ Shock structures dominated by resistivity Shock thickness ∼ 10 16 − 10 17 cm ≫ viscous shock thickness ( ≃ 10 13 cm) Magnetic Reynolds number ≃ 1000

Two-Fluid MHD (Draine 1980) Simplest model is a two fluid system. Neutral fluid No interaction with magnetic field Conducting fluid Tied to magnetic field (perfect conductor) The fluids interact via a drag term due to collisions. Fast heating and cooling ⇒ fluids behave isothermally: p n = a 2 p c = a 2 n ρ n , c ρ c where p n Neutral fluid pressure ρ n Neutral fluid density p c ρ c Conducting fluid pressure Conducting fluid density a n – Constant neutral fluid sound speed a c – Constant conducting fluid sound speed

Two Fluid Equations Neutral Fluid Conducting Fluid ∂ U n ∂t + ∂ F n ∂ U c ∂t + ∂ F c ∂x = S n ∂x = S c  ρ c   ρ n  ρ c v cx   U n = ρ n v nx U c =     ρ c v cy   ρ n v ny B y  ρ c v cx    ρ n v nx a 2 c ρ c + B 2 y / 2 + ρ c v 2  cx  a 2 n ρ n + ρ n v 2 F n = F c =  nx    ρ c v cx v cy − B x B y   ρ n v nx v ny v cx B y − v cy B x � 0   0 � S n = S c = − f   f 0 f = K m ρ n ρ c ( v c − v n )

Frozen Wavespeeds For short wavelengths, interaction force is negligible and two fluids are decoupled. Have ∂ U n ∂t + ∂ F n ∂ U c ∂t + ∂ F c ∂x = 0 , ∂x = 0 λ n 1 , 4 = v nx ∓ a n , λ n 2 , 3 = v nx Neutral Fluid λ c 1 , 4 = v cx ∓ c cf , λ c 2 , 3 = v cx ∓ c cs Conducting Fluid where the slow and fast speeds, c cs and c cf are given by cs,f = 1 � � 1 / 2 � 2 − 4 B 2 � c 2 a 2 c + B 2 /ρ c ∓ ( a 2 c + B 2 /ρ c ) x a 2 c /ρ c 2

Equilibrium Wavespeeds For long wavelengths have equilibrium S = 0 ⇒ v n = v c = v e i.e. a single fluid that obeys ideal MHD equations. Wavespeeds are λ e 1 , 5 = v ex ∓ c ef , λ e 2 , 4 = v ex ∓ c es , λ e 3 = v ex es,f = 1 � � 1 / 2 � 2 − 4 B 2 � c 2 a 2 c + B 2 /ρ e ∓ ( a 2 c + B 2 /ρ e ) x a 2 c /ρ e 2 ρ e = ρ n + ρ c , p e = ρ n a 2 n + ρ c a 2 c On the large scale have a shock of the equilibrium system. Shock structure governed by full equations.

C Shock Structure Oblique shock B x = 1 . 0 , B y = 0 . 6 , Alfv´ en Mach No = 12 . 9 , Mach No = 150 , Ionisation fraction = 10 − 4 . No ionisation and recombination.

Wardle Instability (Wardle 1990) Neutrals B Conducting fluid density at A larger than at B ⇒ Drag on charged fluid larger at A than at B ⇒ Increases buckling of field Neutrals A Magnetic field line en Mach number ( > 5 ) Unstable for sufficiently large Alfv´

Two Fluid Time Dependent Numerical Scheme Gasdynamic scheme for neutrals, MHD for conducting fluid. Add source terms. Subshocks captured in usual way. (e.g. Toth 1995a,b; Stone 1997; MacLow & Smith 1997) BUT Must have all Hall parameters β i = α i B ≫ 1 K i ρ n ρ n – neutral density α i – charge to mass ratio for fluid i K i – Momentum transfer coefficient for fluid i – true for ions and electrons, but not for grains If density of conducting fluid ≪ total density (ionisation fraction ≪ 1 ) ⇒ conducting fluid wavespeeds ≫ equilibrium wavespeeds ⇒ small timestep with explicit scheme

Non-linear Development for Perpendicular Shock (Stone 1997) en Mach No = 10, Ionisation fraction 10 − 3 Two fluid scheme (ZEUS), Alfv´

Multi-Fluid Some species with β i ≃ 1 Total density of charged species ≪ total density ⇒ neglect inertia of charged species Get single fluid with induction equation ∂ B = −∇ ∧ E ∂t = ∇ ∧ ( v ∧ B ) hyperbolic ( J · B ) − ∇ ∧ [ ν 0 B ] conduction parallel to field B 2 ( J ∧ B ) − ∇ ∧ [ ν 1 ] Hall effect B ( J ∧ B ) − ∇ ∧ [ ν 2 ∧ B ] ambipolar diffusion B 2

Resistivities Conductivities are σ 0 = 1 σ 1 = 1 α i ρ i β i σ 2 = − 1 α i ρ i � � � α i ρ i β i , i ) , (1 + β 2 (1 + β 2 B B B i ) i i i Resistivities are ν 0 = 1 σ 2 σ 1 ν 1 = − ν 2 = − ( σ 2 1 + σ 2 ( σ 2 1 + σ 2 σ 0 2 ) 2 ) Note | ν 1 | ≪ 1 if all β i ≫ 1 i.e. no Hall effect To compute these need charged species densities, ρ i .

Momentum equations for charged species reduce to β i B ( E + v i ∧ B ) + ( v 1 − v i ) = 0 i = 2 · · · N (Neglecting inertia and collisions between charged species) Also have � J = ∇ ∧ B = α i ρ i v i i These N equations determine E and the v i for i = 2 · · · N . Given the v i , determine the ρ i from the continuity equations

Subtleties Must include Lorentz force, J ∧ B as source term in momentum and energy equations to get correct relations across subshock. The obvious explicit scheme is unconditionally stable for pure Hall effect It is possible to get a stable explicit scheme for pure Hall with a more subtle differencing scheme for the Hall term (O’Sullivan & Downes 2006, Toth 2007). Could use implicit scheme for resistive terms (Falle 2003)

Even if we use an explicit scheme for field, the multi-fluid scheme is faster for low ionisation fraction, X i : Multi-fluid scheme Shock width L ∝ resistivity ν 2 ∝ 1 , mesh spacing ∆ x ∝ L X i Time step ∆ t ∝ ∆ x 2 ∝ L 2 X i ν 2 Flow time ∝ L ⇒ No of steps in a flow time ∝ L 1 ∆ t ∝ – independent of X i LX i Two-Fluid Scheme 1 Conducting fluid wavespeed c i ∝ X 1 / 2 i Time step ∆ t ∝ ∆ x ∝ LX 1 / 2 i c i ⇒ No of steps in a flow time ∝ L 1 ∆ t ∝ – increases as X i decreases X 1 / 2 i

Ionisation and Recombination Ionisation due to photoionisation or ionisation by cosmic rays ∝ ρ n Recombination ∝ ρ i ρ n ∝ X i ρ 2 n ⇒ In ionisation equilibrium X i ∝ 1 ρ n In fact timescale for ionisation equilibrium is < 0 . 01 shock flow time (Pineau des Forets, Flower, Hartquist, Dalgarno 1986) The result is that X i ∝ 1 for ρ n in the range 10 2 − 10 6 cm − 3 . ρ n In any case X i = f ( ρ n ) ⇒ no Wardle instability ?

Initial Conditions B y Oblique shock B x = 1 . 0 , B y = 0 . 6 en Mach No = 12 . 9 Alfv´ Mach No = 150 Neutral density Ionisation fraction = 10 − 4 No ionisation and recombination Sinusoidal perturbation in position

After 1 . 15 Flow Times Neutral density (Linear 1 – 38 . 25 ) Ion density (Linear 10 − 4 – 0 . 077 )

After 1 . 38 Flow Times Neutral density (Linear 1 – 59 . 2 ) Ion density (Linear 10 − 4 – 0 . 143 )

Neutral Density and Finest Grid After 1 . 38 Flow Times

Ionisation Equilibrium After 18 Flow Times Neutral density (Linear 1 – 43 . 87 )

Grains Initial ion fraction 10 − 8 (grains must be significant current carriers) Initial grain to Neutral mass fraction 0 . 005 Grain charge to mass ratio 0 . 06 Ions are in ionisation equlibrium This corresponds to grains with Radius 10 − 6 cm, Density 1 gm cm − 3 , Mass 2 . 7 10 − 13 gm, Charge 10 Charge to mass ratio 0 . 067 . This gives an upstream Hall parameter 5 . 16 .

After 1 Flow Time Neutral density (Linear 1 – 34 ) Grain density (Linear 0 . 005 – 0 . 098 )

After 1 . 75 Flow Time Neutral density (Linear 1 – 51 ) Grain density (Linear 0 . 005 – 3 . 125 )

Grain to neutral ratio (Linear 0 . 004 – 0 . 233 )