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 Growth

spectral analysis

ni tnt

f 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 2

2

Bt

Bc

jBBdt

Bd

Bjc

rc

dt

d s

tzbbzB ,,v,v,

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z

R0

H0

Thin Disk Geometry

Vertical stretching

z

0

0

H

R

1

z

Supersonic rotation:

1sM

r

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2 2

2

dP

dt n

Vj B

Vertically Isothermal Disks

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

*2*

4 P

B

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 rr

B r

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

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2

( ) 12 ( ) 0

( ) ( )

1 ( ) 10

2 ( ) ( )

( ) 0

ln( ) 0

ln

r r

r

r r

r

V Brr V

t r n

V BrV

t r n

B Vr

t

B V dr B

t d r

The In-Plane Coriolis-Alfvén System

2 /2( ) n e

2 2

2

( ) ( ) ( )( )

( )

z

N r H r rr

B 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

variabletanh( ) b

1

2

Spitzer, (1942)

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The Coriolis-Alfvén Spectrum

,ˆ( )( ) 0 rK K V L L

2[(1 ) ]

d d

d d L

Legendre operator

2 22

( )3 2 9 16

2

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 16

2

rK

b ( 1)k k

2 4 2 ( )ˆ ˆ ˆ ˆ ˆ( 6) 9(1 ) 0 , cr

r

2

( 1)3

crb

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 nr

t n

nr n V

t

ˆ ˆ( ) ( 1) 0n n

The Vertical Magnetosonic System

2 22

2 2

ˆˆ(1 ) 2 0

1

d nn

d

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

2

11

( ) ( )1

f

2

21

( ) ( )1

f

2 21

Define: 2( ) 1 ( )n g

2 2 22

2 2 2

1(1 ) 2 2 0

1

d g d gg

d 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 t

H r H r

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Two main regimes of perturbations dynamics

I. Decoupled inertia-coriolis

and MS modes

II. Coupled inertia-coriolis

and 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 10

( ) ( )

[ ( ) ]10

( )

z

z

z

bV n

n

n Vn

n

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 nb

n d nd

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WKB solution for the eigenfunctions

Turning points at *

The Bohr-Sommerfeld

condition*

2 2

*

1( ) ( 1) , ( ) ( * )

2 4d 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

VV L n V b L n V b

VV 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 ik

d

1

( )r

rdv

bdS

ln( )

dD

d

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II – Coupled regime

Need to go to high axial and azimuthal wave numbers

1, zk k

22

2

12 2 ( ) ( ) {[ ( ) 1] ( ) }r z r

r r r r r zd b db db

k k k k b k k bd dd

22

2

1[(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. 2008

Rebusco et al. 2010

Shtemler et al. 2011

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II – Nonlinear theory

),()1((4

1)1(

22

1

Ov

vv

b

bb

v

vL

v

v

tr

zr

zr

Ar

),()1( 22

Ob

bv

v

vb

b

bL

b

b

tr

zr

zr

Ar

),(1

)1(8

11

22

22

O

v

bbvL

v

t z

rzs

z

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Weakly nonlinear asymptotic model

.)(,,3

21427

cc i

~A

Ginzburg-Landau equation

3AAt

A

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

.)(,,3

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

27222cc i

)].()([)(

)()(),(

1

)()( 20

218

74

32

176

yPyPtAb

byPtA

v

v

c

crcr

)561)((

)1()(42

2

tC

tBvz

).(10

),(9

4

),(35

8

3

32

4

AOB

dt

dC

AOACdt

dB

AOABAdt

dA

nmmn ndPP

1

1 12

2)()(

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Nonlinear theory, range II

)()()()(

),()()()(

4422

3311

PtCPtC

PtBPtBvz

).(7

20

),(5

6

5

1

),(13

35

52

245

104

315

),(17

49

52

105

104

385

),()175

8

75

8(

33

4

331

2

3242

3

3242

1

321

AOBdt

dC

AOBBdt

dC

AOACCdt

dB

AOACCdt

dB

AOBBAAdt

dA

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Generation of toroidal magnetic field

)]()()[( 20 PPtAb

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

2)2(

1)1( yPtbyPtbyPtbytb

rtdr

dbb r

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Conclusions

It is shown that the spectrum in a stratified disk (the finite thickness) contains a discrete part and a continuous one for AC and MS waves respectively. In the linear approximation, the AC and MS waves are decoupled. There is a special critical value of parameter plasma beta for which the system becomes unstable if

Both main regimes of perturbations dynamics, namely decoupled (coupled) AC and MS modes, are discussed in detail and compared with hydrodynamic limits.

It was found from numerical computations that the generalization to include nonlinear dynamics leads to saturation of the instability.

The nonlinear behavior is modeled by Ginzburg-Landau equation in case of weakly nonlinear regime (small supercriticality) and by truncated system of equations in the case of fully nonlinear regime. The latter is obtained by appropriate representation of the solutions of the nonlinear problem using a set of eigenfunctions for linear operator corresponding to AC waves

c0c