spectral and algebraic instabilities in thin keplerian disks: i – linear theory

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Spectral and Algebraic Instabilities in Thin Keplerian Disks: I – Linear

Theory

Edward Liverts

Michael Mond

Yuri Shtemler

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Accretion disks – A classical astrophysical problem

Original theory is for infinite axially uniform cylinders. Recent studies indicate stabilizing effects of finite small thickness. Coppi and Keyes, 2003 ApJ., Liverts and Mond, 2008 MNRAS have considered modes contained within the thin disk.

Weakly nonlinear studies indicate strong saturation of the instability. Umurhan et al., 2007, PRL.

Criticism of the shearing box model for the numerical simulations and its adequacy to full disk dynamics description. King et al., 2007 MNRAS, Regev & Umurhan, 2008 A&A.

Magneto rotational instability (MRI) Velikhov, 1959, Chandrasekhar, 1960 – a major player in the transition to turbulence scenario, Balbus and Hawley, 1992.

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Algebraic Vs. Spectral Growthspectral analysis

ni tntf t f e ( , ) ( )

r r

Spectral stability if nIm 0

- Eigenvalues of the linear operatorn

for all s n

Short time behavior

1000eR

f

f tG t Max

f(0)

( )( )

(0)

Transient growth for Poiseuille flow.

Schmid and Henningson, “Stability and Transition in Shear Flows”

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To investigate the linear as well as nonlinear development of small perturbations in realistic thin disk geometry.

Goals of the current research

Derive a set of reduced model nonlinear equations

Follow the nonlinear evolution.

Have pure hydrodynamic activities been thrown out too hastily ?

Structure of the spectrum in stratified (finite) disk. Generalization to include short time dynamics. Serve as building blocks to nonlinear analysis.

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Magneto Rotational Instability (MRI)

Governing equations

.00,v

,4

,v-v

,1v 22

Bt

BcjBBdtBd

Bjc

rcdtd s

tzbbzB ,,v,v,

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z

R0

H0

Thin Disk GeometryVertical stretching

z

0

0

HR

1z

Supersonic rotation:

1sM

r

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2 2

2

d Pdt n

V j B

Vertically Isothermal Disks

( , ) ( , ) ( ) P r n r T r

*2*

4 PB

Vertical force balance:2 2/2 ( )( , ) ( ) H rn r N r e

Radial force balance: 3/2( ) ( ) , (r)=V r r r r

Free functions: The thickness is determined by .

( )T r( )H r ( ) ( ) / ( )H r T r r

( )N r

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• I. Dominant axial component • II. Dominant toroidal component

Two magnetic configurations

0 ( , )

( , )z zB B r

B B r

0 0

( , )

( , )

z zB B r

B B r

( )B r

Magnetic field does not influence lowest order equilibrium but makes a significant difference in perturbations’ dynamics.

Free function: ( )zB r Free function: ( )B r

8

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Linear Equations: Case Iz

zBzB2 2/2 ( )( , ) ( ) H rn r N r e

Self-similar variable

( )

H r

0B Consider

2 2

2( ) ( ) ( )( )

( )

z

N r H r rrB r

2 2 2( ) ( ) ( )sH r r C r

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2

( ) 12 ( ) 0( ) ( )

1 ( ) 1 02 ( ) ( )

( ) 0

ln( ) 0ln

r r

r

r r

r

V Brr Vt r n

V BrVt r n

B Vrt

B V dr Bt d r

The In-Plane Coriolis-Alfvén System

2 /2( ) n e

2 2

2( ) ( ) ( )( )

( )

z

N r H r rrB r

No radial derivatives Only spectral instability possible

Boundary conditions

, 0 rB

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The Coriolis-Alfvén System

n2 /2( ) n e

2( ) sec ( ) n h b

22 /2sec ( ) h b d e d

2

b

Approximate profile

Change of independent variable

tanh( ) b

1

2

Spitzer, (1942)

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The Coriolis-Alfvén Spectrum

,ˆ( )( ) 0 rK K V L L

2[(1 ) ]

d dd d

L Legendre operator

2 22

( ) 3 2 9 162 rK

b

ˆ( , , ) ( , ) , ( ) ( )f r t f r e r r i t

,ˆ ( ) r kV P ( 1) , 1,2,K k k k

3

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The Coriolis-Alfvén Spectrum

13

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The Coriolis-Alfvén Dispersion Relation

2 22

( ) 3 2 9 162 rK

b ( 1)k k

2 4 2 ( )ˆ ˆ ˆ ˆ ˆ( 6) 9(1 ) 0 , cr

r

2( 1)

3cr

b k k

Universal condition for instability:

ˆ 1 cr

Number of unstable modes increases with .

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The Coriolis-Alfvén Dispersion Relation-The MRI

Im , Re Im

( 1) 0.42cr k max ˆIm for 3

Im

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( ) 0( )

( ) ( ) 0

z

z

V nrt n

n r n Vt

ˆ ˆ( ) ( 1) 0n n

The Vertical Magnetosonic System

2 22

2 2

ˆ ˆ(1 ) 2 01

d n nd

ˆ( , , ) ( , )( ) ( )

f r t f r e

r r

i t

2( ) sec ( ) n h b

tanh( ) b

16

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Continuous acoustic spectrum

21 1 2 2( ) 1 ( ) ( )n c f c f

21

1( ) ( )1

f

22

1( ) ( )1

f

2 21

Define: 2( ) 1 ( )n g

2 2 22

2 2 21(1 ) 2 2 01

d g d g gd d

Associated Legendre

equation

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Continuous eigenfunctions

1( )f

2 ( )f

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Linear Equations: Case II z

B

( , )

( , )z zB B r

B B r

2 2/2 ( )( , ) ( ) H rn r N r e

Axisymmetric perturbations are spectrally stable.Examine non-axisymmetric perturbations.

( ) , - , , ( ) ( )

r drr tH r H r

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Two main regimes of perturbations dynamics

I. Decoupled inertia-coriolisand MS modes

II. Coupled inertia-coriolisand MS modes

A. Inertia-coriolis waves driven algebraically by acoustic modes

B. Acoustic waves driven algebraically by inertia-coriolis modes

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I.A – Inertial-coriolis waves driven by MS modes

1 1 0( ) ( )

[ ( ) ]1 0( )

z

z

z

bV nn

n Vnn

b V

2

2( ) ( )( )

( )sN c

B

The magnetosonic eigenmodes

ˆ( , , , ) ( , ) i ikf f e

22

2

ˆ ˆ( ) 1 ( ) ˆ( 2) 0( ) 1 ( ) 1

d b n db n bn d nd

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WKB solution for the eigenfunctions

Turning points at *

The Bohr-Sommerfeld condition

*2 2

*

1( ) ( 1) , ( ) ( * )2 4

d m

1MS m ˆ( , , , ) ( , ) i ikf f e

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The driven inertia-coriolis (IC) waves

1 0

0

2 { , , } { , , }

1 { , , }2

rz z

r z

V V L n V b L n V b

V V K n V b

IC operator Driving MS modes

2

ˆ[ ( ) ]1 1 1ˆˆ( ) ( ) exp[ ]( ) ( )1

MS zr MS

MS

i d vv D ik b i ikd

1( )

rr

dvbdS

ln( ) dDd

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II – Coupled regime

Need to go to high axial and azimuthal wave numbers1, zk k

22

212 2 ( ) ( ) {[ ( ) 1] ( ) }r z r

r r r r r zd b db dbk k k k b k k b

d dd

22

21 [(1 ) ( ) ]z

z z r rd b k k b k k bd

3( , ) ( )2r r rk k D k

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Instability

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Hydrodynamic limit

Both regimes have clear hydrodynamic limits. Similar algebraic growth for decoupled regime.Much reduced growth is obtained for the coupled regime.

Consistent with the following hydrodynamic calculations:Umurhan et al. 2008Rebusco et al. 2010Shtemler et al. 2011

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II – Nonlinear theory

),()1((4

1)1( 22

1

Ovv

vbb

bvv

Lvv

tr

zr

zr

Ar

),()1( 22

Obb

vvv

bbb

Lbb

tr

zr

zr

Ar

),(1

)1(81

122

22

O

vbbv

Lv

t z

rzs

z

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Weakly nonlinear asymptotic model

.)(,,32

1427

cc i

~A

Ginzburg-Landau equation3AA

tA

Landau, (1948)

Phase plane

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Numerical Solution

...)()()()()()(),( 3)3(

,2)2(

,1)1(,, yPtvyPtvyPtvytv zzzz

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Weakly nonlinear theory, range I

.)(,,32,04])3(2][2[ 14

27222cc i

)].()([)()(

)(),(1

)()( 2021874

32

176

yPyPtAbb

yPtAvv

c

crcr

)561)(()1()(

42

2

tCtBvz

).(10

),(94

),(358

3

32

4

AOBdtdC

AOACdtdB

AOABAdtdA

nmmn ndPP

1

1 122)()(

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Nonlinear theory, range II

)()()()(),()()()(

4422

3311

PtCPtCPtBPtBvz

).(720

),(56

51

),(1335

52245

104315

),(1749

52105

104385

),()175

8758(

33

4

331

2

3242

3

3242

1

321

AOBdtdC

AOBBdtdC

AOACCdtdB

AOACCdtdB

AOBBAAdtdA

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Generation of toroidal magnetic field

)]()()[( 20 PPtAb

...)()()()()()(),( 3)3(

2)2(

1)1( yPtbyPtbyPtbytb

rtdrdbb r

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Conclusions Generation of toroidal magnetic field