Numerical schemes for multifluid
magnetohydrodynamics
Sam Falle,
Department of Applied Mathematics,
University of Leeds.
Giant Molecular Clouds
(GMC)
e.g. Rosette Molecular Cloud
Size
' 35 pc
Mass
' 105 M
Mean Density
' 10−22 gm cm−3
Temperature
' 10 K ⇒ sound speed ' 0.2 km s−1
Alfvén speed
' 2 km s−1 ⇒ magnetic pressure dominates
Velocity dispersion ' 10 km s−1
Translucent Clumps
Rosette GMC not Homogeneous: CO maps show that it consists of ' 70 clumps with
Sizes
' 3.5 – 8.0 pc
Masses
' 102 – 2 103 M
Densities
10−21 gm cm−3
Temperature
' 10 K ⇒ Sound speed ' 0.2 km s−1
Alfvén speed
' 2 km s−1 ⇒ magnetic pressure dominates (Crutcher 1999)
Velocity dispersion ' 1 km s−1
⇒ Jeans Mass
3 103 M (based on velocity dispersion)
Dense Cores
These clumps also have substructure. Contain dense cores with
Sizes
< 1 pc
Masses
' 10 – 100 M
Densities
' 10−19 gm cm−3
Temperature
' 10 K ⇒ Sound speed ' 0.2 km s−1
Alfvén speed
' 2 km s−1 ⇒ magnetic pressure dominates
Velocity dispersion ' 0.3 km s−1
⇒ Jeans Mass
10 M (based on velocity dispersion)
Ambipolar Diffusion (Ion–Neutral Drift)
Low ionization fraction Xi (< 10−4 ) → ambipolar diffusion.
Magnetic Reynolds No = 1 for
1
Length scale = 0.04
MA
B
10−5
10−6
Xi
103
n
3/2
pc (MA is Alfvén Mach No ' 1)
⇒ 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
Length scale = 3.2 10−4
1
Mn
pc
(M is Mach No)
Multifluid Equations
N fluids with equations (i = 1 · · · N )
∂ρi ∂ρi vix X
+
=
Sij
∂t
∂x
j6=i
Sij – rate of conversion of i to j
X
∂
∂ρi vi
+
(ρi vix vi + pi ı̂) = αi ρi (E + vi ∧ B) +
fij
∂t
∂x
j6=1
fij – force exerted on i by j, αi – charge to mass ratio
X
∂ei
∂
1
+
[vix ( ρi vi2 + pi )] = Hi +
Gij
∂t
∂x
2
j6=i
Hi – energy loss rate for i, Gij – energy transfer rate from j to i
∂B
= −∇ ∧ E,
∂t
∇∧B=J=
Species 1 - neutral (α1 = 0), Species 2 · · · N charged.
X
i
αi ρi v i
Force is of the form
fij = Kij ρi ρj (vj − vi )
Define Hall parameter
βi =
αi B
ρ1 Ki1
βi 1 ⇒ Species i tied to field lines
βi 1 ⇒ Species i tied to neutrals
in ISM β 1 for ions and electrons, but not for grains
Time Dependent Numerical Scheme
Two Fluid
βi 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 βi 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 βi ' 1
Total density of charged species total density
⇒ neglect inertia of charged species (otherwise equations are stiff)
αi ρi (E + vi ∧ B) +
X
j6=1
fij =
∂
∂ρi vi
+
(ρi vix vi + pi ı̂) ' 0
∂t
∂x
Get single fluid with induction equation
∂B
= −∇ ∧ E =
∂t
∇ ∧ (v ∧ B)
(J · B)
B]
B2
conduction parallel to field
(J ∧ B)
]
B
Hall effect
−
∇ ∧ [ν0
−
∇ ∧ [ν1
− ∇ ∧ [ν2
Here v is neutral velocity.
hyperbolic
(J ∧ B)
∧ B] ambipolar diffusion
B2
Resistivities
Conductivities are
σ0 =
1 X
1 X αi ρ i
1 X αi ρi βi
,
σ
=
−
αi ρi βi , σ1 =
2
B i
B i (1 + βi2 )
B i (1 + βi2 )
Resistivities are
ν0 =
σ2
σ1
1
ν1 = − 2
ν2 = − 2
2
σ0
(σ1 + σ2 )
(σ1 + σ22 )
Note |ν1 | 1 if all βi 1 i.e. no Hall effect
To compute these need charged species densities, ρi .
Momentum equations for charged species reduce to
βi
(E + vi ∧ B) + (v1 − vi ) = 0
B
i = 2···N
(Neglecting inertia and collisions between charged species)
Also have
J=∇∧B=
X
αi ρi vi
i
These N equations determine E and the vi for i = 2 · · · N .
Given the vi , determine the ρi 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 = ν12 cos2 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, 2ν1 cos θk, is effective wavespeed and ∆x is smallest
wavelength
∆x2
⇒ stable timestep for explicit scheme ∆t =
.
4πν1 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 hyperbolic 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:
β2 = −5.8 106 (electrons), β3 = 5.8 103 (ions)
Preshock state:
Bx = 1.0, By = 0.6, Fast shock with Fast Mach No = 1.5
ν0 = 1.7 10−12 , ν1 = 10−5 , ν2 = −0.058 (Hall effect negligible)
Isothermal – neutral pressure negligible.
High Resolution
U1
By
-1.0
1.6
Fluid 1 x velocity
-1.2
Transverse field
1.4
1.2
-1.4
1.0
-1.6
0.8
0.6
0.0
0.2
0.4
0.6
0.8
X
0.0
0.2
0.4
0.6
0.8
X
∆x = 5 10−3 .
Line – Integration of steady equations, markers – Numerical scheme
No rotation – Z component of field ' 10−4
Low Resolution
U1
-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:
β2 = −5.8 106 (electrons), β3 = 0.233 (grains).
Preshock:
Bx = 1.0, By = 0.6, Fast shock with Fast Mach No = 1.5
Preshock ν0 = 1.7 10−9 , ν1 = 0.01, ν2 = 0.0023 (Significant Hall effect)
Isothermal – neutral pressure negligible.
U1
High Resolution
-1.0
By
1.5
-1.2
y component of field
Fluid 1 x velocity
-1.4
1.0
-1.6
0.5
-1.8
0.0
0.1
0.0
0.2
0.1
Bz
0.2
0.0
-0.2
z component of field
-0.4
0.0
∆x = 2 10−3 .
0.2
X
X
0.1
0.2
X
Line – Integration of steady equations, markers – Numerical scheme
Low Resolution
U1
-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:
β2 = −5.8 106 (electrons), β3 = 5.8 103 (ions)
Preshock state:
Bx = 1.0, By = 0.6, Fast shock with Fast Mach No = 5
ν0 = 1.7 10−12 , ν1 = 10−5 , ν2 = −0.058 (Hall effect negligible)
Isothermal – neutral sound speed a = 1.
By
8.0
U1
High Resolution
-2.0
Fluid 1 x velocity
transverse field
6.0
4.0
-4.0
2.0
-6.0
0.05
0.10
0.15
X
0.05
0.20
0.10
0.15
X
U2
-2.0
Fluid 2 x velocity
-4.0
-6.0
0.05
0.10
0.15
X
0.20
∆x = 10−3 .
Line – Integration of steady equations, markers – Numerical scheme
0.20
Low Resolution
U1
-2.0
Fluid 1 x velocity
-4.0
-6.0
0.05
0.10
0.15
X
0.20
∆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 By ∂ 2 Bx
,
etc
∂x2
∂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