global magnetorotational instability with inflokersale/comm/talk-gdr.pdf · • wall modes...

Global Magnetorotational Instability with Inflow

Evy Kersale

PPARC Postdoctoral Research Associate

Dept. of Appl. Maths — University of Leeds

Collaboration:D. Hughes & S. Tobias (Appl. Maths, Leeds)

N. Weiss & G. Ogilvie (DAMTP, Cambridge)

Journees GDR Dynamo, Nice — 3rd & 4th May 2004

Introduction 1

Accretion Discs

Accretion of material occurs in different

galactic or extra-galatic environments like

AGNs or YSOs

Which mechanisms to explain:

• the turbulent transport of angular

momentum

• the existence of the well-collimated jets

Magnetic field ≡ link between accretion and ejection

• MRI drives turbulence ⇒ outward angular momentum transport

• Magneto-centrifugal ejection and Bϕ-collimation

Origin of the large scale magnetic field 〈B〉 ?

Journees GDR Dynamo — Nice 3rd & 4th May 2004

Introduction 2

Magnetorotational Instability in Accretion Discs

Velikhov 1959, Chandrasekhar 1960, Balbus & Hawley 1991

Local linear analysis

• Weakly magnetized (β 1) and differential rotating flows unstable if dΩ/dr < 0

• Free energy ≡ differential rotation ⇒ MRI extremely powerful γ ∼ r|dΩ/dr|

herent well into the nonlinear regime and look nothinglike turbulence. On the other hand, when the averagevalue of the poloidal field over the computational box iszero, the flow quickly breaks down into turbulence. The

turbulence does not persist, however; it decays over aperiod of many orbits (Fig. 21).

Let us consider the latter result first, as it is moreeasily understood. It is in fact a consequence of the‘‘anti-dynamo theorem’’ (Moffatt, 1978; the theorem isdue to Cowling). Simply stated, any sort of sustainedmagnetic-field amplification by axisymmetric turbulencein an isolated dissipative system is impossible. To see

FIG. 19. Contours of angular momentum perturbations in asimulation with an initial radial field, viewed in (R ,z) crosssection. The axes are oriented as in Fig. 18. The long radialwavelength and short vertical wavelength are characteristic ofthe most unstable modes of a radial background magneticfield. From Hawley and Balbus 1992.

FIG. 20. Magnetic-field lines (solid curves) and velocity vec-tors (arrows) in a simulation of a uniform initial vertical field,viewed in (R ,z) cross section. The axes are oriented as in Fig.18. The flow evolves to two rapidly flowing channels. In crosssection, they appear as oppositely moving radial streams. FromHawley and Balbus, 1992.

FIG. 21. Two dimensional MHD turbulence simulation: (a)Grey-scale plot of angular momentum perturbations in an axi-symmetric simulation of an initial vertical field with ^BZ&50,viewed in (R ,z) cross section. The axes are oriented as in Fig.18. This field configuration does not lead to streams (cf. Fig.20); instead, the flow becomes turbulent. (b) The time evolu-tion of poloidal magnetic-field energy in axisymmetric simula-tions of an initial vertical field with ^BZ&50. Labels corre-spond to number of grid zones. After an initial period ofgrowth, the magnetic field declines with time at a rate deter-mined by the numerical resolution. This behavior accords withCowling’s anti-dynamo theorem.

38 S. A. Balbus and J. F. Hawley: Instability and turbulence in accretion disks

Rev. Mod. Phys., Vol. 70, No. 1, January 1998

Small vertical length-scales & large radial length-scales

2-D studies

• 〈Bz〉 6= 0: channel flow unstable in 3D

• 〈Bz〉 = 0: turbulence which decays over a

resolution dependent time scale

3-D Studies

• 〈Bz〉 or 〈Bϕ〉 6= 0 determine the saturation

level of turbulence

• No dependence on initial Bz if its average value

is 0; 〈B2〉 relies on the level of numerical

dissipation


Introduction 3

Magnetorotational Instability & Dynamo

Shear flow hydrodynamically nonlinearly unstable but stable to the MRI: 〈U2〉 saturates

but 〈B2〉 decreases.

Hawley et al. 1996

1996ApJ...464..690H

Brandenburg et al. 1995: 〈B〉 6= 0

1995ApJ...446..741B

Nonlinear dynamo

Flux

Current

Field B

Flow


Set-up of the model 4

Global Dissipative Study

Bz

(r)ϕU

rU (r)

NL evolution equations:

(∂t + U·∇) U = −∇Φ−∇Π + B·∇B + ν∆U

(∂t + U·∇) B = B·∇U + η∆B

∇ · B = ∇ ·U = 0

Boundary conditions:

• Two BCs only for ideal discs

• Ten BCs for dissipative discs

Generalized pressure:

P = Φ + Π

(• Does not evolve explicitly

• Can vanish ⇒ rotation not supported

Differential rotation sustained by the BCs which may

either enforce the rotation or constrain the pressure

variations



Basic State & Boundary Conditions

Disc equilibrium: Axisymmetric and Z-invariant, Uz = 0, Br = Bϕ = 0

Uro = −3 ν

2 r

Uϕo =ζ

r1/2

Πo = δ −9

8

ν2

r2+

GM∗ − ζ2

r

BZo = B0

9>>>>>>>>>>=>>>>>>>>>>;

ζ2 = GM∗ (Keplerian), or

ζ2 = GM∗ − ν2|α| (Sub-Keplerian)

No BCs on Ur or Br

˛˛˛

∂rUϕ = −Uϕo/2r

∂rUz = 0

∂r(rBϕ) = 0

∂rBz = 0

and

˛˛ Uϕ = Uϕo

or

Π = Πo

to drive the shearing flow



Linearized Problem

Linear evolution equations:

• Normal modes:

K(r, t) = κ(r) exp(σt + im ϕ + ik z)

• 10th order linear system:

σ I(r) κ(r) = L(r) κ(r)

• Π evolves on much shorter time scales:

∇ ·U = 0 ⇐⇒ Iπ = 0

˛˛ =⇒

Solved numerically:

• inverse iteration

• shooting (double checking)

Linear boundary conditions:˛˛ druϕ = 0

druz = 0

dr(rbϕ) = 0

drbz = 0

and

˛˛ uϕ = 0

or

π = 0

butforcing Uϕ itself seems more reliable

at first to sustain differential rotation


Cylindrical MRI Modes 7

Ideal MRI Modes

No inflow in the basic state

• m = 0 is the most unstable global mode

• Quenching by the magnetic tension

• Saturation: γmax → r1/2 |dΩ/dr|r1


Cylindrical MRI Modes 8

Dissipative MRI Modes

• Modes slightly modified by inflow & dissipation

• Growth rates globally reduced

• Damping of the small-scale modes


Cylindrical Wall Modes 9

Wall Modes

• Wall modes solutions of the linear system too

• γ too large for the free energy available and large range of k unstable

• Ideal case: γ scales linearly with k and increases rapidly with B0

• Significant flux of energy through the boundaries to feed these modes

• Inflow, curvature and Coriolis force non crucial


Simplified Problem 10

Cartesian Linear Shearing Flow

o

oz

y x

B

U (z)

Incompressible, non dissipative basic state:

ρ0 = 1

U0 = z ex, z ∈ [−z0, +z0]

B0 = B0 ey

2nd order system of linear ODEs:

χHux = −U′0Huz − ikx π

χHuy = −iky π

χHuz = −π′

0 = ikx ux + iky uy + u′z

where,

K(x, t) = κ(z) exp(σt + ikx x + iky y)

ωa = kB0

χ = σ + ikxU0

H =“1 + ω

2a/χ

2”



Cartesian Wall Modes: HD Limit

Hydrodynamic limit: ωa = 0 and H = 1

HD modes are solutions of χ

»u′′z −

„k

2+

χ′′

χ

«uz

–= 0

Linear shear ⇒ χ′′ = 0 and uz = c− exp(−kz) + c+ exp(kz)

• No discrete mode with the BCs uz =

0, only a continuum of stable modes

• Neutral wall modes solutions with BCs

ux = 0



Cartesian Wall Modes in MHD

Magnetic field destabilizes the wall modes

u′′z − 2

ω2a

χ2 + ω2a

χ′

χu′z −

"k

2+

χ′′

χ− 2

ω2a

χ2 + ω2a

„χ′

χ

«2#

uz = 0

Singularity when γ = 0 and ω = −kxU0 ± ωa



Origin of the Instability

Analyse on the boundaries:

γ2=

(S2 U ′0)

2

(S2 U ′0)

2 + ω2a k2 k2

x

"( U

′0)

2 K2

k2x

+ ω2a k

2L2 − ω2a

#

where S2 = |π′′|/|π|, K = |π|′/|π|, L2 = |π| |π|′′/|π′′|2 and kx = ky

B0 and U ′0 both non zero at either of the boundaries ⇒ wall modes unstable

Mechanism:

uz

ky B0−−−→ bz

U ′0−→ bx

ky B0−−−→ Tx = ky B0 bx

In the vicinity of the boundary ux ∼ 0 ⇒ energy from the outside required to balance Tx


New BCs Set 14

Wall Modes Treatment in Accretion Discs

• Wall modes are solutions of incompressible shearing flows when rigid BCs are relaxed

• B0 makes them linearly unstable if uϕ = 0 on the boundaries

Forcing the differential rotation of the boundaries impossible unless Ω′0 or B0 locally zero

or

BCs on the pressure to keep it low in agreement with quasi-Keplerian accretion discs

Basic State:

Πo(r1) = Πo(r2) = δ +9 ν

8 r1r2

Πo = δ +9

8

ν

r1r2

„r1 + r2

r−

r1 r2

r2

«

Uϕo =

s1

r

„GM∗ −

9 ν

8

r1 + r2

r1 r2

«BCs Π = Πo:

external pressure does not work


New BCs Set 15

Boundary Conditions on the Total Pressure

• Body modes still unstable and wall modes properties now consistent with

MRI

• Larger range of unstable m for the wall modes


New BCs Set 16

Modes in Slim Discs

Disc thickness constrains min(k) ⇒ low ν & η

H/R =

1/4

1/10⇒ kmin =

12.6

31.4


New BCs Set 17


Nonlinear Study 18

Numerical Schemes Implemented

Spectral decomposition: X(r, ϕ, z) =P

l,m,kXlmkTl(s) eimϕ cos, sin(kz),

Tl(s) = cos(l cos−1 s) & 2r = [(r1 − r2) s + r1 + r2], s ∈ [−1, +1]

Spatial differentiations are performed using the properties of spectral decompositions (Fourier:

multiplication & Chebyshev: recurrence relation) & the nonlinear terms are computed in

configuration space

The advance in time uses a semi-implicit scheme Crank-Nicholson & Adams-Bashforth for the

linear & nonlinear terms respectively

[I −Θ δtL] Un+1+∇Γ

n+1= Nu

n+ [I + (1−Θ) δtL] Un

with ∇2Γ

n+1= ∇ ·N n

u

[I −Θ δtL] Bn+1+∇Ψ = Nb

n+ [I + (1−Θ) δtL] Bn

with ∇2Ψ = ∇ ·N n

b

where Θ ∈ [0, 1], Γ = δt Π, L = ν∇2 and N represents the nonlinear terms

The azimuthal and vertical boundary conditions are automatically satisfied by the trial

functions while radial ones imply some linear relations between the expansion coefficients such

thatP

l αlXlmk = const. The BCs used to compute Ur and Br come from the divergence

free constrains.


Nonlinear Study 19

Basic State


Nonlinear Study 20

2-D Results


Conclusions & Perspectives 21

Conclusions & Future Work

Linear study

• Slightly sub-keplerian basic state with inflow

• No assumption of the behavior of Ur and Br on the boundaries

• Accurate description of the global dissipative MRI modes

Fully nonlinear study

• Will rely on high performance computations

• Parallel, 2D & 3D

• Systematic parameter space survey

• Toward a comparison of dynamical properties with a reduced MRI approach


global magnetorotational instability with inflokersale/comm/talk-gdr.pdf · • wall modes...

Documents