Global Magnetorotational Instability with Inflow

Transcription

1 Global Magnetorotational Instability with Inflow Evy Kersalé 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) Journées GDR Dynamo, Nice 3 rd & 4 th May 2004

2 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 B ϕ -collimation Origin of the large scale magnetic field B?

3 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 38 S. A. Balbus and J. F. Hawley: Instability and turbulence in accretion disks Small vertical length-scales & large radial length-scales 2-D studies B z = 0: channel flow unstable in 3D B z = 0: turbulence which decays over a resolution dependent time scale 3-D Studies B z or B ϕ 0 determine the saturation level of turbulence No dependence on initial B z if its average value is 0; B 2 relies on the level of numerical dissipation FIG. 19. Contours of angular momentum perturbations in a simulation with an initial radial field, viewed in (R,z) cross section. Journées TheGDR axes Dynamo are oriented Nice as in Fig. 18. The long radial 3 rd & 4 th May 2004 wavelength and short vertical wavelength are characteristic of the most unstable modes of a radial background magnetic

4 Introduction 3 Magnetorotational Instability & Dynamo Shear flow hydrodynamically nonlinearly unstable but stable to the MRI: U 2 saturates but B 2 decreases. Hawley et al ApJ B Brandenburg et al. 1995: B 0 Nonlinear dynamo Flux Flow Current Field B

5 Set-up of the model 4 Global Dissipative Study B z NL evolution equations: U (r) r U ϕ (r) ( 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: ( Does not evolve explicitly P = Φ + Π Can vanish rotation not supported Differential rotation sustained by the BCs which may either enforce the rotation or constrain the pressure variations

6 Set-up of the model 5 Basic State & Boundary Conditions Disc equilibrium: Axisymmetric and Z-invariant, U z = 0, B r = B ϕ = 0 U ro = 3 ν 2 r ζ U ϕo = r 1/2 Π o = δ 9 8 B Zo = B 0 ν 2 r + GM ζ 2 2 r 9 >= >; ζ 2 = GM (Keplerian), or ζ 2 = GM ν 2 α (Sub-Keplerian) No BCs on U r or B r r U ϕ = U ϕo /2r r U z = 0 r (rb ϕ ) = 0 r B z = 0 and U ϕ = U ϕo or Π = Π o to drive the shearing flow

7 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 I π = 0 Solved numerically: inverse iteration shooting (double checking) Linear boundary conditions: d r u ϕ = 0 d r u z = 0 u ϕ = 0 and d r (rb ϕ ) = 0 or π = 0 d r b z = 0 but forcing U ϕ itself seems more reliable at first to sustain differential rotation

8 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: γ max r 1 /2 dω/dr r1

9 Cylindrical MRI Modes 8 Dissipative MRI Modes Modes slightly modified by inflow & dissipation Growth rates globally reduced Damping of the small-scale modes

10 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 B 0 Significant flux of energy through the boundaries to feed these modes Inflow, curvature and Coriolis force non crucial

11 Simplified Problem 10 Cartesian Linear Shearing Flow z B o U (z) o Incompressible, non dissipative basic state: ρ 0 = 1 U 0 = z e x, z [ z 0, +z 0 ] y x 2nd order system of linear ODEs: χh u x = U 0 H u z ik x π χh u y = ik y π χh u z = π 0 = ik x u x + ik y u y + u z B 0 = B 0 e y where, K(x, t) = κ(z) exp(σt + ik x x + ik y y) ω a = kb 0 χ = σ + ik x U 0 H = 1 + ω 2 a /χ2

12 Simplified Problem 11 Cartesian Wall Modes: HD Limit Hydrodynamic limit: ω a = 0 and H = 1 HD modes are solutions of χ» «u z k 2 + χ χ u z = 0 Linear shear χ = 0 and u z = c exp( kz) + c + exp(kz) No discrete mode with the BCs u z = 0, only a continuum of stable modes Neutral wall modes solutions with BCs u x = 0

13 Simplified Problem 12 Cartesian Wall Modes in MHD Magnetic field destabilizes the wall modes u ω 2 a z 2 χ 2 + ωa 2 χ χ u z " ω 2 a k 2 + χ χ 2 χ 2 + ωa 2 χ «2 # χ u z = 0 Singularity when γ = 0 and ω = k x U 0 ± ω a

14 Simplified Problem 13 Origin of the Instability Analyse on the boundaries: γ 2 (S 2 U 0 = )2 (S 2 U 0 )2 + ωa 2 k2 kx 2 " ( U 0 )2 K 2 k 2 x + ω 2 a k2 L 2 ω 2 a # where S 2 = π / π, K = π / π, L 2 = π π / π 2 and k x = k y B 0 and U 0 both non zero at either of the boundaries wall modes unstable Mechanism: u z ky B 0 b z U 0 b x ky B 0 T x = k y B 0 b x In the vicinity of the boundary u x 0 energy from the outside required to balance T x

15 New BCs Set 14 Wall Modes Treatment in Accretion Discs Wall modes are solutions of incompressible shearing flows when rigid BCs are relaxed B 0 makes them linearly unstable if u ϕ = 0 on the boundaries Forcing the differential rotation of the boundaries impossible unless Ω 0 or B 0 locally zero or BCs on the pressure to keep it low in agreement with quasi-keplerian accretion discs Basic State: Π o (r 1 ) = Π o (r 2 ) = δ + 9 ν 8 r 1 r 2 Π o = δ + 9 ν r1 + r 2 8 r 1 r 2 r s 1 U ϕo = r GM 9 ν 8 r «1 r 2 r 2 «r 1 + r 2 r 1 r 2 BCs Π = Π o : external pressure does not work

16 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

17 New BCs Set 16 Modes in Slim Discs Disc thickness j constrains min(k) j low ν & η 1/ H/R = 1/10 k min = 31.4

18 New BCs Set 17

19 Nonlinear Study 18 Numerical Schemes Implemented Spectral decomposition: X(r, ϕ, z) = P l,m,k X lmkt l (s) e imϕ {cos, sin}(kz), T l (s) = cos(l cos 1 s) & 2r = [(r 1 r 2 ) s + r 1 + r 2 ], 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 δt L] U n+1 + Γ n+1 = N u n + [I + (1 Θ) δt L] U n with 2 Γ n+1 = N n u [I Θ δt L] B n+1 + Ψ = N b n + [I + (1 Θ) δt L] B n with 2 Ψ = 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 that P l α l X lmk = const. The BCs used to compute U r and B r come from the divergence free constrains.

20 Nonlinear Study 19 Basic State

21 Nonlinear Study 20 2-D Results

22 Conclusions & Perspectives 21 Conclusions & Future Work Linear study Slightly sub-keplerian basic state with inflow No assumption of the behavior of U r and B r 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