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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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
Set-up of the model 5
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
Set-up of the model 6
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
Cylindrical MRI Modes 8
Dissipative MRI Modes
• Modes slightly modified by inflow & dissipation
• Growth rates globally reduced
• Damping of the small-scale modes
Journees GDR Dynamo — Nice 3rd & 4th May 2004
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
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”
Journees GDR Dynamo — Nice 3rd & 4th May 2004
Simplified Problem 11
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
Simplified Problem 12
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
Simplified Problem 13
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004
New BCs Set 17
Journees GDR Dynamo — Nice 3rd & 4th May 2004
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.
Journees GDR Dynamo — Nice 3rd & 4th May 2004
Nonlinear Study 19
Basic State
Journees GDR Dynamo — Nice 3rd & 4th May 2004
Nonlinear Study 20
2-D Results
Journees GDR Dynamo — Nice 3rd & 4th May 2004
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
Journees GDR Dynamo — Nice 3rd & 4th May 2004