Numerical schemes for multifluid magnetohydrodynamics Sam Falle, - PowerPoint PPT Presentation

Numerical schemes for multifluid magnetohydrodynamics Sam Falle, Department of Applied Mathematics, University of Leeds.

Giant Molecular Clouds (GMC) e.g. Rosette Molecular Cloud ≃ 35 pc Size ≃ 10 5 M ⊙ Mass ≃ 10 − 22 gm cm − 3 Mean Density ≃ 10 K ⇒ sound speed ≃ 0 . 2 km s − 1 Temperature ≃ 2 km s − 1 ⇒ magnetic pressure dominates Alfv´ en speed ≃ 10 km s − 1 Velocity dispersion

Translucent Clumps Rosette GMC not Homogeneous: CO maps show that it consists of ≃ 70 clumps with ≃ 3 . 5 – 8 . 0 pc Sizes ≃ 10 2 – 2 10 3 M ⊙ Masses 10 − 21 gm cm − 3 Densities ≃ 10 K ⇒ Sound speed ≃ 0 . 2 km s − 1 Temperature ≃ 2 km s − 1 ⇒ magnetic pressure dominates (Crutcher 1999) Alfv´ en speed ≃ 1 km s − 1 Velocity dispersion 3 10 3 M ⊙ (based on velocity dispersion) ⇒ Jeans Mass

Dense Cores These clumps also have substructure. Contain dense cores with < 1 pc Sizes ≃ 10 – 100 M ⊙ Masses ≃ 10 − 19 gm cm − 3 Densities ≃ 10 K ⇒ Sound speed ≃ 0 . 2 km s − 1 Temperature ≃ 2 km s − 1 ⇒ magnetic pressure dominates Alfv´ en speed ≃ 0 . 3 km s − 1 Velocity dispersion ⇒ Jeans Mass 10 M ⊙ (based on velocity dispersion)

Ambipolar Diffusion (Ion–Neutral Drift) Low ionization fraction X i ( < 10 − 4 ) → ambipolar diffusion. Magnetic Reynolds No = 1 for � B � 3 / 2 � � 10 − 6 � � 10 3 Length scale = 0 . 04 1 ( M A is Alfv´ en Mach No ≃ 1 ) pc 10 − 5 M A X i n ⇒ Magnetic Reynolds number < 100 in Translucent Clumps and Dense cores ⇒ Ambipolar Diffusion important on scales smaller than GMC. Viscosity In neutral gas, Reynolds No = 1 for � 1 � Length scale = 3 . 2 10 − 4 pc ( M is Mach No) Mn

Multifluid Equations N fluids with equations ( i = 1 · · · N ) ∂ρ i ∂t + ∂ρ i v ix � = S ij ∂x j � = i S ij – rate of conversion of i to j ∂ρ i v i + ∂ � ∂x ( ρ i v ix v i + p i ˆ ı) = α i ρ i ( E + v i ∧ B ) + f ij ∂t j � =1 f ij – force exerted on i by j , α i – charge to mass ratio ∂e i ∂t + ∂ ∂x [ v ix (1 � 2 ρ i v 2 i + p i )] = H i + G ij j � = i H i – energy loss rate for i , G ij – energy transfer rate from j to i ∂ B � ∂t = −∇ ∧ E , ∇ ∧ B = J = α i ρ i v i i Species 1 - neutral ( α 1 = 0 ), Species 2 · · · N charged.

Force is of the form f ij = K ij ρ i ρ j ( v j − v i ) Define Hall parameter β i = α i B ρ 1 K i 1 β i ≫ 1 ⇒ Species i tied to field lines β i ≪ 1 ⇒ Species i tied to neutrals in ISM β ≫ 1 for ions and electrons, but not for grains

Time Dependent Numerical Scheme Two Fluid β i ≫ 1 for all i > 1 ⇒ single conducting fluid. Upwind (Godunov Type) scheme for each fluid. Add source terms. Subshocks captured in usual way. But Must have all Hall parameters β i ≫ 1 – true for ions and electrons, but not for grains. If density of conducting fluid ≪ total density ⇒ conducting fluid wavespeeds ≫ equilibrium wavespeeds ⇒ small timestep with explicit scheme Can increase mass of ions to increase timestep (Li, McKee & Klein 2006). But only works for single conducting fluid.

Multi-Fluid Some species with β i ≃ 1 Total density of charged species ≪ total density ⇒ neglect inertia of charged species (otherwise equations are stiff) f ij = ∂ρ i v i + ∂ � α i ρ i ( E + v i ∧ B ) + ∂x ( ρ i v ix v i + p i ˆ ı) ≃ 0 ∂t j � =1

Get single fluid with induction equation ∂ B ∂t = −∇ ∧ E = ∇ ∧ ( 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 Here v is neutral velocity.

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 i ) B B B 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 If not isothermal, must include Lorentz force, J ∧ B as source term in momentum and energy equations to get correct relations across subshock. Hall term dispersive with ω 2 = ν 2 1 cos 2 θk 4 ( θ is angle between field and x axis) i.e. phase and group velocity → ∞ as wavelength → 0 (whistler waves). Might suppose that group velocity, 2 ν 1 cos θk , is effective wavespeed and ∆ x is smallest wavelength ∆ x 2 ⇒ stable timestep for explicit scheme ∆ t = 4 πν 1 cos θ . But

Obvious explicit scheme unconditionally unstable for pure Hall effect ⇒ either implicit scheme for resistive terms or differencing in O’Sullivan & Downes 2006 and super-time-stepping Algorithm 1) Calculate solution at half time using a first order scheme which is explicit for hyper- bolic terms, implicit for resistive terms. 2) Use this to calculate explicit, second order accurate fluxes for both hyperbolic and resistive terms. 3) Advance solution by complete timestep using these fluxes. ⇒ scheme is second order and stability limited by hyperbolic timestep, not resistive timestep, even if Hall term is dominant.

Shock Structure with Large Hall Parameters Two charged species: β 2 = − 5 . 8 10 6 (electrons), β 3 = 5 . 8 10 3 (ions) Preshock state: B x = 1 . 0 , B y = 0 . 6 , Fast shock with Fast Mach No = 1.5 ν 0 = 1 . 7 10 − 12 , ν 1 = 10 − 5 , ν 2 = − 0 . 058 (Hall effect negligible) Isothermal – neutral pressure negligible.

High Resolution U 1 B y -1.0 1.6 Transverse field Fluid 1 x velocity -1.2 1.4 1.2 -1.4 1.0 -1.6 0.8 0.6 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 X X ∆ x = 5 10 − 3 . Line – Integration of steady equations, markers – Numerical scheme No rotation – Z component of field ≃ 10 − 4

Low Resolution U 1 -1.0 Fluid 1 x velocity -1.2 -1.4 -1.6 0.0 0.2 0.4 0.6 0.8 X ∆ x = 2 . 5 10 − 2 . Line – Integration of steady equations, markers – Numerical scheme

Shock Structure with Strong Hall Effect Two charged species: β 2 = − 5 . 8 10 6 (electrons), β 3 = 0 . 233 (grains). Preshock: B x = 1 . 0 , B y = 0 . 6 , Fast shock with Fast Mach No = 1.5 Preshock ν 0 = 1 . 7 10 − 9 , ν 1 = 0 . 01 , ν 2 = 0 . 0023 (Significant Hall effect) Isothermal – neutral pressure negligible.

U 1 High Resolution -1.0 B y 1.5 y component of field -1.2 Fluid 1 x velocity -1.4 1.0 -1.6 0.5 -1.8 0.0 0.1 0.2 0.0 0.1 0.2 X X B z 0.2 0.0 -0.2 z component of field -0.4 0.0 0.1 0.2 X ∆ x = 2 10 − 3 . Line – Integration of steady equations, markers – Numerical scheme

Low Resolution U 1 -1.0 Fluid 1 x velocity -1.2 -1.4 -1.6 -1.8 0.0 0.1 0.2 X ∆ x = 5 10 − 3 Line – Integration of steady equations, markers – Numerical scheme

Shock Structure with Neutral Subshock Two charged species: β 2 = − 5 . 8 10 6 (electrons), β 3 = 5 . 8 10 3 (ions) Preshock state: B x = 1 . 0 , B y = 0 . 6 , Fast shock with Fast Mach No = 5 ν 0 = 1 . 7 10 − 12 , ν 1 = 10 − 5 , ν 2 = − 0 . 058 (Hall effect negligible) Isothermal – neutral sound speed a = 1 .

B y U 1 8.0 High Resolution -2.0 transverse field Fluid 1 x velocity 6.0 4.0 -4.0 2.0 -6.0 0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20 X X U 2 -2.0 Fluid 2 x velocity -4.0 -6.0 0.05 0.10 0.15 0.20 X ∆ x = 10 − 3 . Line – Integration of steady equations, markers – Numerical scheme

Low Resolution U 1 -2.0 Fluid 1 x velocity -4.0 -6.0 0.05 0.10 0.15 0.20 X ∆ x = 5 10 − 3 . Line – Integration of steady equations, markers – Numerical scheme

Multidimensions Resistive terms contain cross-derivatives ⇒ fully implicit scheme messy. But Can treat cross-derivatives explicitly and only use implicit approximation for diagonal terms: ∂ 2 B y ∂x 2 , ∂ 2 B x etc ∂y 2 Scheme then has same stability properties as in one dimension. Cheap because just have tridiagonal matrices to invert.

Can use scheme for: 1. Stability of multifluid shocks (Wardle instability) 2. Ambipolar diffusion in star forming regions. 3. Ambipolar diffusion and Hall effect in accretion discs 4. etc