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Broken force-free
electrodynamics, current
sheets, and your favourite
magnetosphere problems
Kyle Parfrey
with
Anatoly Spitkovsky
Dimitrios Giannios
Andrei Beloborodov
Force-free electrodynamics
�
µν
∇µ T(Matter)
+
µν
T(EM)
� + J� × B
� =0
ρE
�
� = −∇ × E
�
∂t B
=0
� ·B
� =0
E
B2 > E2
� =∇×B
� − J�
∂t E
�
� B
�
� ×B
�
E
� · (∇ × B)
� −E
� · (∇ × E)
�
�
J� = B
+
(∇
·
E)
B2
B2
Uchida 97, Thompson & Blaes 98, Gruzinov 99, Komissarov 02, Blandford 02,...
Advantages
Only need BCs for E & B — “plasma” maintained as needed
Infinite magnetization limit — no density floors
‣ No shocks, since vA → c
‣
‣
Problems
‣ How do you add dissipation? No fluid frame...
‣ Current sheets — tangential discontinuities in B
“Broken” force-free electrodynamics
�
� B
�
� ×B
�
E
� · (∇ × B)
� −E
� · (∇ × E)
�
�
J�ideal = B
+
(∇
·
E)
Ideal current:
B2
B2
�
� �
�
�
�
� ·B
� = ∇×B
� ·B
� − J� · B
� −E
� · ∇×E
� =0
Parallel current found from: ∂t E
� · B|
� target = η J� · B
�
Instead, set target value: E
�
�
�
� ·B
� =γ E
� · B|
� target − E
� ·B
�
Now drive to target value: ∂t E
J�||resist
�
� ·∇×B
� −E
� ·∇×E
� + γE
� ·B
�
B
�
=
B
(1 + γη) B 2
� ·B
� �= 0
E
� · J� �= 0
E
�
�
� B,
� E,
� ...
Can have η = η J,
Current sheet capturing
1. Identify current sheets: Bi changes sign & B2<E2 , even if between grid points
2. Find ΔE required at that point to set E2→B2 & leave E⋅B unchanged
3. Reconstruct smooth distribution of ΔE in nearby region
4. Set ΔE → ΔE - [ΔE]⋅B/B at each point
5. E → E - ΔE around current sheet
Simulates particle acceleration,
pair cascades, and E dissipation
1D moving
current sheet
By
Ez
Demonstrations
‣
“Waving” aligned rotator (axisymmetric pulsar)
‣
Magnetospheric Wald problem
Komissarov 04, 05
Nathanail & Contopoulos 14
Using pseudospectral GRFFE code PHAEDRA (Parfrey+ 12)
Colour: Bϕ
positive
negative
η=0
η = 2 × 10
−4
�
�
J�ideal · B
1+ 2
B + E2
�
Colour: Bϕ
positive
negative
η=0
η = 2 × 10
−4
�
�
J�ideal · B
1+ 2
B + E2
�
poloidal
current
function
Hφ ≡ B T
a/M = 0.98
4
η = 5 × 10
light surface
ergosphere
Application: jets from small-scale field
2
1
3
risco
rotating
black hole
‣
Field loops created by MRI in disc, on scale ~ H
‣
Rise into corona via Parker instability
‣
Can open up when first footpoint is accreted
‣
Swallowed or ejected when second footpoint is accreted
Black hole-disc coupling
no field lines
to infinity
open field
is potential
Steady-state solution of
Grad-Shafranov eqn.
region of closed field
lines connecting the
inner disc to the hole
Uzdensky 05
Quasi-steady BH-disc magnetospheres
Prograde
Retrograde
risco
risco
a/M = 0.7
positive
Colour: Hϕ negative
Quasi-steady BH-disc magnetospheres
Prograde
risco
Retrograde
rcrit
closed BH-disc field lines
E2 > B2 inside current sheet
risco
a/M = 0.7
positive
Colour: Hϕ negative
all field lines open
E2 < B2 inside current sheet
Prograde disc
loop width = 2 M
loop direction
clockwise
anti-clockwise
positive
Colour: Hϕ negative
vaccrete = c/100
a/M = 0.98
Retrograde disc
loop width = 2 M
loop direction
clockwise
anti-clockwise
positive
Colour: Hϕ negative
vaccrete = c/100
a/M = 0.98
3C 120
rin ≈ 8.6 M
Leahy & Perley 95
Kataoka+ 07
3C 390.3
Walker 94
rin � 20 M
Radio Galaxies
Sambruna+ 09
Seyferts
a/M > 0.8
Leahy & Perley 91
3C 382
MCG 6-30-15
Miniutti+ 07
rin ≈ 10 M
NGC 3783
Brenneman+ 11
NGC 1365
Risaliti+ 13
Sambruna+ 11
Summary
‣
Resistive method for magnetospheric dynamics
— stiff in the diffusive rather than ideal limit
— suitable for both nearly ideal and very diffusive regimes
‣
Current sheet capturing method
— keeps sheets well-behaved and marginally resolved
— simulates effect of cross-field conductivity when E2 > B2
‣
Jets from small-scale loops grown in discs
— prograde: minimal poloidal scale for jet
differential-rotation powered magnetic wind
— retrograde: get BH-powered jet even for small loops
— possible relevance to (1) radio-loud vs. -quiet AGN?
(2) jet quenching in XRB soft states?
(3) retrograde microquasers?
1/--pages
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