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BGUWISAP - 20111
Spectral and Algebraic Instabilities in Thin Keplerian Disks: I – Linear
Theory
Edward Liverts
Michael Mond
Yuri Shtemler
BGUWISAP - 20112
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.
BGUWISAP - 20113
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”
BGUWISAP - 20114
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.
BGUWISAP - 20115
Magneto Rotational Instability (MRI)
Governing equations
.00,v
,4
,v-v
,1v 22
Bt
BcjBBdtBd
Bjc
rcdtd s
tzbbzB ,,v,v,
BGUWISAP - 20116
z
R0
H0
Thin Disk GeometryVertical stretching
z
0
0
HR
1z
Supersonic rotation:
1sM
r
BGUWISAP - 20117
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
BGUWISAP - 2011
• 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
BGUWISAP - 20119
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
BGUWISAP - 20111010
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
BGUWISAP - 201111
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)
BGUWISAP - 201112
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
BGUWISAP - 2011
The Coriolis-Alfvén Spectrum
13
BGUWISAP - 201114
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 .
BGUWISAP - 201115
The Coriolis-Alfvén Dispersion Relation-The MRI
Im , Re Im
( 1) 0.42cr k max ˆIm for 3
Im
BGUWISAP - 2011
( ) 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
BGUWISAP - 201117
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
BGUWISAP - 201118
Continuous eigenfunctions
1( )f
2 ( )f
BGUWISAP - 201119
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
BGUWISAP - 201120
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
BGUWISAP - 201121
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
BGUWISAP - 201122
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
BGUWISAP - 201123
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
BGUWISAP - 201124
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
BGUWISAP - 201125
Instability
BGUWISAP - 201126
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
BGUWISAP - 201127
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
BGUWISAP - 201128
Weakly nonlinear asymptotic model
.)(,,32
1427
cc i
~A
Ginzburg-Landau equation3AA
tA
Landau, (1948)
Phase plane
BGUWISAP - 201129
Numerical Solution
...)()()()()()(),( 3)3(
,2)2(
,1)1(,, yPtvyPtvyPtvytv zzzz
BGUWISAP - 201130
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)()(
BGUWISAP - 201131
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
BGUWISAP - 201132
Generation of toroidal magnetic field
)]()()[( 20 PPtAb
...)()()()()()(),( 3)3(
2)2(
1)1( yPtbyPtbyPtbytb
rtdrdbb r
BGUWISAP - 201133
Conclusions Generation of toroidal magnetic field