Models of Turbulent Angular Momentum Transport Beyond the Parameterization

Martin PessahInstitute for Advanced Study

Workshop on Saturation and Transport Properties of MRI-Driven Turbulence - IAS- June 16, 2008

C.K. Chan - ITC, HarvardD. Psaltis - University. of Arizona

Timescales and Lengthscales in Accretion Disks

1Gyr

1yr

1sec

0.1AU 1AU 1000AU

Mean Field Models

BLRInside Horizon

NumericalSimulations

Viscous

Orbital Saturation

BHGrowth

BHVariability

Hubble Time 107Mo

Disk Spectrum

Standard Accretion Disk Model

€

∂l

∂t+ ∇ ⋅(lv) = −

1

r

∂

∂r(r2T rφ )

€

Σ, P,T, H, vr , ...This prescription dictates the structure of the accretion disk

Angular momentum transport in turbulent media

€

T rφ = αP −d lnΩ

d ln r

⎛

⎝ ⎜

⎞

⎠ ⎟

Shakura & Sunyaev (‘70s)Angular momentum transport: Turbulence and magnetic fields

Standard Disk MRI Disk

€

€

Trφ

€

T

€

P

€

H

€

Pmag

€

H

€

Bz

€

Trφ

€

T

€

P

MRI

The need to go beyond -viscosity

Issues…

*Dynamics: is not constant

*Causality: Information propagates across sonic points (Stress reacts instantly to changes in the flow)

*Stability criterion: Standard model works always (Stresses only for MRI-unstable disks)

*”Viscosity”: MRI does not behave like Newtonian viscosity

Long-Term Evolution of MRI

MRI drives MHD turbulence at a well defined scale which depends only on the magnetic field and the shear

small scales

large scales

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

MRI Accretion Disks in Fourier Space

k

“Stress Power”

MRI DissipationParker

Mode interactions

Parasitic Instabilities

small scales

large scales

(Talk by C.K. Chan later today)

The Closure Problem in MHD Turbulence

We can derive an equation for angular momentum conservation

Need for a CLOSURE model beyond alpha!!

which involves 2nd order correlations…

… but they involve 3rd order correlations…

We can derive equations for 2nd order correlations…

We can derive equations for 3nd order correlations…

€

∂l

∂t+ ∇ ⋅(lv) = −

1

r

∂

∂r(r2T rφ )

€

∂l

∂t+ ∇ ⋅(lv) = −

1

r

∂

∂r[r2(Rrφ − M rφ )]

€

∂l

∂t+ ∇ ⋅(lv) = −

1

r

∂

∂rr2 ρδvrδvφ −

δBrδBφ

4π

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

(Closure models with no mean fields; papers by Kato et al 90’s; Ogilvie 2003)

Stress Modeling with No Mean Fields

Kato & Yoshizawa, 1995

Pressure-strain tensor,tends to isotropize the

turbulence

Stress Modeling with No Mean Fields

Ogilvie 2003

C2 return to isotropy in decaying turbulence

C3, C4 energy transfer

C1, C5 energy dissip.

New correlation drives the growthof Reynolds & Maxwell stresses

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W ik = δviδjk

A Model for Turbulent MRI-driven Stresses

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M 0 = ξρ 0HΩ0 vAz

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k ij = ζ kmax

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q = −d lnΩ

d ln r

W in the Turbulent State

W tensor dynamically important!

-Wr-Maxwell Reynolds

The model produces initial exponential growth and leads to stresses with ratios in agreement with the saturated regime

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k = ζ kmax

The parameter controls the ratio between Reynolds

& Maxwell stresses

Pessah, Chan, & Psaltis, 2006a

=0.3

Calibration of MRI Saturation with Simulations

Pessah, Chan, & Psaltis, 2006b

Hawley et al ‘95

Calibration of MRI Saturation with Simulations

A physically motivated model for angular momentum transportKeplerian disks that incorporates the MRI

The parameter controls the level

at which the magnetic energy saturates

€

M 0 = ξρ 0HΩ0 vAz=11.3

€

Rrφ − M rφ = αP −d lnΩ

d ln r

⎛

⎝ ⎜

⎞

⎠ ⎟

Shakura & Sunyaev, 1973Pessah, Chan, & Psaltis, 2008Shakura & Sunyaev, 1973

Pessah, Chan, & Psaltis, 2006bPessah, Chan, & Psaltis, 2008Shakura & Sunyaev, 1973

Pessah, Chan, & Psaltis, 2006bPessah, Chan, & Psaltis, 2008

A Local Model for Angular Momentum Transport in Turbulent Magnetized Disks

A model for MRI-driven angular momentum transport in agreement

with numerical simulations

Pessah, Chan, & Psaltis, 2006b

Pessah, Chan, & Psaltis, 2006b

Simulations by Hawley et al. 1995

Simulations by Hawley et al. 1995

Scaling in MRI Simulations

Pessah, Chan, & Psaltis, 2007; Sano et al. 2004

€

~ Bz

Deeper understanding

of scalings?

Non-zero Bz

(few % Beq)

Magnetic pressure must be important

in setting H

Spectral hardening(Blaes et al. 2006)

(Talk by S. Fromang later today)

Viscous, Resistive MRI

Pes

sah

& C

han,

200

8

€

γmax =1

η

4 −κ

4κ 2

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kmax =1

η

4 −κ

4κ 2

Pm<<1 Pm<<1

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γmax =4 −κ 2

2κν

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kmax =κ

ν

Pm>>1 Pm>>1

(Various limits studied by Sano et al., Lesaffre &Balbus, Lesur & Longaretti, … )

Viscous, Resistive MRI: Stresses & Energy

* Lowest ratio magnetic-to-kinetic: Ideal MHD* Linear MRI effective at low/high Pm as long as Re large* Need higher Re to drive MHD turbulence at low Pm(Lesur & Longaretti, 2007; Fromang et al., 2007)

Pm<<1

Pm>>1

Pm<<1Pm>>1

Viscous, Resistive MRI Modes

€

δv(z, t) = v0eσt (cosθv sin kz,sinθv sin kz)

δb(z, t) = b0eσt (cosθB coskz,sinθB coskz)

€

θv,B = θv,B (ν ,η )

Pm=1

Viscous, Resistive MRI Modes

Ideal MHD

kh

Parasitic Instabilities sensitive to viscosity and resistivity

(Goodman & Xu, 1994)

Viscous, Resistive MRI ModesPessah & Chan, 2008

Pm<<1

Pm>>1

Viscous, Resistive Parasitic Instabilities

PRELIMINARY

(with J. Goodman, in

prep.)

Parasitic Instabilities seem to be weaker at high PmHint for difficulties to drive MRI at low Pm?

Can the destruction of primary MRI modes by parasitic instabilities explain (Re, Pm) dependencies?

1ow Pm

high Pm