Pankin - ICTP

Computer Modeling of 

the Accretion Disk Corona 

Alexei Y Pankin 

Institute for Nuclear Research, Kyiv, Ukraine 

Lehigh University, Bethlehem, PA, USA 

Z. Mikic, S. Titov 

San Diego, CA, USA 

J. Goodman, D.A. Uzdensky 

Princeton University, Princeton, NJ, USA 

D.D. Schnack 

University of Wisconsin, Madison, WI, USA 

International Conference on Turbulence Mixing and Beyond 

August 17-26, 2007, Trieste, Italy

Outline 

Introduction 

Model for coronal accretion disk 

Initial state, equilibrium, assumptions 

Coupling of accretion disk and corona 

Simulation results 

Summary 

ICTMB, Trieste, Italy, August 2007

Introduction 

Accretion disks are unique phenomena that observed 

for very different astrophysical objects such as 

Protostars 

Galactic X-ray binaries 

Active galactic nuclei 

The problem of formation and dynamics of accretion 

disks is closely related to the problem of jet formation 

The accretion of matter should be associated with the 

release of energy and angular momentum 

Energy losses for most astrophysical objects can be 

explained by emission of radiation 

Angular momentum losses remain an open problem in 

astrophysics 


Models for angular momentum dissipation 

There are several models for angular 

momentum dissipation in accretion disks: 

Dissipation due to viscosity 


Cataclysmic Variable 

T accretion =l 2 / v ≈ 108 2 [m 2 ] 

10 [m 2 /sec] 

Companion star 

=10 15 [sec]≈10 9 [years] 

Accretion disk 

White dwarf

Models for angular momentum dissipation 

There are several models for angular 

momentum dissipation in accretion disks: 

Dissipation due to viscosity 

anisotropic viscosity 

Dissipation due to magnetized winds (Blandford & 

Payne'82): 

ejection of matter that spun up to the velocities that are 

grater than the escape velocity 

occurs when the projection of gravitational force is not 

sufficient to balance the centrifugal force. 

Dissipation due to magnetic stresses 

Balbus-Hawley ('91) instability (following Velikhov'59 

derivations) 


Conservation of angular momentum 

Angular momentum equation 

∂ 

∂t ∫ V r×v⋅z dV =−∫ ∂ V n⋅⋅r * ⋅z dS−∫ v v ∇×v⋅z dV 

Stress tensor: =v⊗v− 1 

In cylindrical coordinates (r, φ, z) 

(disk rotation is around z-axis) 

• the microscopic viscosity is too small to sustain the accretion 

• a turbulent viscosity ? 

• magnetic field helps to make differential-rotation flows unstable 

MRI in the disk 

coronal MRI 

ICTMB, Trieste, Italy, August 2007 

4 

B⊗B pB2 

8 I − ∇ v 

v

Coronal Magneto-Rotational Instability (MRI) 

Side view 


Top view 

The flux tube stretches 

• leading/lower spot loses 

to the trailing/upper one, 

• the lower spot sinks, 

while the upper spot rises; 

• the difference in their increases

cont. eq. 

induct. eq. 


Disk 

In cylindrical coordinates (r, φ, z), integrated over z 

Eqs. of motion 

input (from corona) 

output (for corona)

ρ 


Corona 

Resistive MHD eqs. 

input (from disk) 

∇ ×B= 4π 

c 

J , ∇ ×E=− 1 

c 

at z = 0 

∂ B 

∂t 

, 5 

E 1 

v×B= η J , 6 

c 

∂ ρ 

∇⋅ ρv =0, 7 

∂t 

∂ p 

∇⋅pv =−γ−1 p ∇⋅v , 8 

∂t 

v 

∂ v 

1 

v⋅∇ = J×B−∇ pρg∇⋅νρ ∇ v , 9 

∂ t c 

output (for disk)

Initial Equilibrium State 

Axisymmetric disk and corona 

Kepler differential flow profile: Ω=Ω 0 (r/r 0 ) –3/2 

Self-consistent equilibrium with finite pressure and g appropriate to a central object 

(modified to remove singularity) 

Isothermal equation of state: γ = 1, p=c s 2 ρ with cs constant 

Barotropic equation of state: p = p(ρ) 

Bipolar potential magnetic field: J×B=0 

The initial B z is chosen to minimize force imbalance when the coupling between the 

disk and corona is included 

Equilibrium state is found as a balance between pressure, gravity and centrifugal 

force 

For p = constant at z = 0, 

max(M a )=1, max(M b )≡v 0 /v A

Initial Equilibrium State 

Linear and angular 

velocities 

Gravitational 

potential 


Magnetic field bipole with poles at r = 

3a and r = 6a 

Kepler Flow with Initial Contours of B z

MHD resistive code MAB 

3D Cartesian (x,y,z) resistive MHD code 

Finite difference mesh, structured, nonuniform 

Implicit (semi-implicit waves, fully implicit diffusion) 

OpenMP support 

Used for solar coronal modeling (magnetic and thermal 

structure of active regions) 

MAB code has first-order upwind diffusion 

Estimated viscosity and resistivity due to upwinding: 


ηuw ,ν uw ~vΔ x /2

Magnetic torque density 

We are interested in the evolution of the 

magnetic torque density on the disk: 

Eqs. of motion on the disk 


K =R B B z 

2


Evolution of B z at z = 0 

Magnetic field lines 

Launch point at B z =.5 B zmax 



Launch point at B z =.75 B zmax 


Launch point at B z =.25 B zmax

Evolution of magnetic torque 

Projection of this simulation results indicates that there a transport of 

angular momentum inward for the inner spot and outward for the outer 

spot 


Deviation from force free conditions 

∣J⋅B∣ 

∣J× B∣ 


∣J⋅B∣ and ∣J× B∣ 

time

New initial conditions for magnetic perturbations 

Disk magnetic flux function: 

=sina RtanhbR 4 tanhR 2 tanhR out −R , 

Ψ 

where a=4/R out tanhbR out 4 


B z

New initial conditions for magnetic dipoles 

Disk flux function: 

=sina RtanhbR 4 tanhR 2 tanhR out −R , 

Ψ 

where a=4/R out tanhbR out 4 


B z

Results for multispot case 

Bz 

Σ 


Evolution of magnetic torque density

Results for multispot case 

Evolution of 

magnetic field 

lines 


Summary 

Model for accretion disk in solar corona is developed 

and implemented in resistive MHD code 

Evolution of single and multiple magnetic loops in the 

corona driven by Keplerian shear flow in the disk is 

reported 

There are indications that the evolution of magnetic 

loop is associated with reconnection of magnetic 

field lines in corona and 

produces angular momentum transport towards 

the central object for inner spot

Pankin - ICTP

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