Nonlinear MRI

Jeremy Goodman

CMPD/CMSO Winter School

7-12 January 2008, UCLA

Questions for the nonlinear regime

• How does the linear instability saturate?

– i.e., What nonlinear mode-mode interactions “brake” growth?

• What are the consequences for the mean flow?

– i.e., What does the final state look like?

The final state depends on what’s

driving the mean flow

• Taylor-Couette flow is contained by pressure &

usually driven viscously by rotation of the boundaries.

– Viscous driving is weak at large Reynolds number Re=VL/

– Mean flow can change drastically

• Disks are “driven” by gravity, which is stronger than

magnetic & thermal energies

– Mean flow can’t change much

– Dissipation is balanced instead by radial accretion

Taylor-Couette flow

Nominal parameters of the

Princeton MRI experiment

(B 0)

G.I. Taylor’s (1936)

apparatus (B=0)

fluid-filled gap

between cylinders

weight & pulley drive

inner cylinder

motor drives outer

cylinder

Mechanical energy is minimized by

uniform rotation

E =pi

2

2+U(r1,

i=1

N

...,rN )

L = rii=1

N

pi

pi

E • L( ) = 0 pi = ri

Energy:

Angular momentum:

A system of minimal energy rotates uniformly with angular

velocity (= ).

Minimize E at fixed L:

TC flow can rotate uniformly

• L is constant, neglecting viscous & magnetic

interactions with the container and coils

• Centrifugal force is balanced by radial pressure

gradient:

– This pressure drop is supported by the container.

Absent external driving (torques), TC flow will relax

to uniform rotation

2r =P

r

1

cJ B( )r

, if any

...in cylindrical coordinates z,r, with = ez

P(rmax ) P(rmin )2

2rmax

2 rmin2( )

Disks cannot rotate uniformly

• Boundaries are “free” (P 0 at boundaries)

– except inner edge, sometimes (e.g. stellar surface)

• Orbital energy dominates: ( r)2 >>P/ cs2

– This means disks are thin: z<< r

• Centrifugal force is balanced mainly by gravity:

• Dissipation causes little change in , but a radial drift

toward the star (accretion), i.e. toward lower

gravitational potential

2rGM*

r2(at z << r)

r 3/2 ("keplerian")

• Since the available free energy is so large, saturation occurs not

by reduction in free energy but by secondary instabilities that

feed on the main MRI modes.

• These 2ndary may include

– Kelvin-Helmholtz

– tearing modes

– magnetic Rayleigh-Taylor & Parker instability (if sat~1)

• This has been studied mostly via analysis and simulations of a

local model for a small part of a disk called the shearing box.

• Main goal is to determine the rate of momentum transport &

dissipation at saturation ( parameter)

MRI saturation in disks

MRI linear instability primary MRI

modes

2ndary instabilities

dissipation

dynamo?

The Shearing Box: A local, corotating

Cartesian approximation

x r r0

y r0 0 (t)[ ];

0 (t) (r0 )

z z

Dimensions of the box are typically

comparable to vertical scale height

hcs r0 , cs sound speed

Shearing-box equations of motion

2A rd

dr r=r0

: shear rate

4A xx 2zz : tidal field

: kinematic viscosity dimensions : L2T 1

: magnetic diffusivity ditto[ ]

dvdt

=vt

+ v • v = 2 z v 4A xx 2zz1

P +1

cJ B +

2v

d

dt= • v

Bt

= v B( ) +2B

Standard “equilibrium” state

of the shearing box

2 z v 4A xx 2z+1 P

zz 0

v0 = 2Ax y (r0) +

d

dr r0

r r0

( )

1

0

P0

z=

2z

• Gas or plasma pressure dominates

P >> B2/8 , i.e. >>1

• Radial gradients (of pressure, etc.) are negligible

• J 0

With these assumptions,

Recap of linear theory

Consider the (most important) special case,

B0 = B0z = constant, VA

B0

4, (v, ,P, B)1 exp t + ikz( ) :

+ k2( ) + k2( ) + kVA( )2 2

+ 4 ( + A) + k2( )2

+ 4A kVA( )2

= 0

• In ideal MHD ( =0= ), weak field is always unstable: ~ |A| at k ~ /VA

— presuming d 2/dr = 4A < 0, as in disks.

—Thus growth occurs on small scales (<< h) for weak fields ( >>1)

• For 0, modes at k > (|A|/ )1/2 are stabilized.

• For =0 but 0, unstable range of k is unaffected but is reduced.

Incipient turbulence:

Parasitic instabilities

Exact Fourier modes

• On scales l << cs/ , perturbations are

incompressible:

• For a single mode , nonlinear

terms vanish (next slide).

Single Fourier modes can grow to arbitrary amplitude

– Assuming that boundary conditions allow single modes;

• The shearing box does, with shearing-periodic boundaries;

• TC flow does not.

• Though exact, sufficiently large-amplitude single

modes are not stable.

v = v0 ++ v1, • v1 0

= 0 + 1, 1 0, etc.

v,P,B( )1exp( ik • r) + c.c.

Proof of exactness

• v1

= 0 = • B1

k • v1 = 0 = k • B1;

v1 • v1

v1 • ikv1 + c.c. = ikv1( ) • v1 + c.c. = 0.

Similarly, v1 • B1

= 0 = B1 • v1 = B1 • B1.

Also, 1

P1 is linear because = 0.

More generally, v1(r,t) = an (t)exp(ikn • r)n

+ c.c.

so v1 • v1 ikm • an( ) am 0 in general unless km kn = 0.

Modes at parallel wavenumbers can be linearly superposed

• provided >>1, i.e. in the incompressible limit

• Also true in non-ideal MHD, since 2v1 & 2v1 are linear

2D (x,z) simulations in “ideal” MHDHawley & Balbus 1992, ApJ 400, 595

vy, 1 at t=2.4, 3.1, 3.6, & 4.1 “orbits”.

=4000, box size=0.25cs/

Cartoon of evolution

of the magnetic field.

Above: growth of magnetic

energy in several runs.

Below: power versus (kx,kz)z

x or r

Parasitic modesGoodman & Xu 1994, ApJ 432, 213

• Consider stability of a single large-amplitude mode

v1(r,t) = Vh exp(St)sin(Kz) v0 = 2Axx,

B1(r,t) = Bh exp(St)cos(Kz) B0 = B0z.

Vh , Bh are constant, horizontal (xy) vectors; Vh • Bh = 0 in ideal MHD.

(1)

• Linearize about (1) as basic state, neglecting (v0,B0)

– Also neglect because we seek 2ndary (=“parasitic”) growth rates ~ Kv1 >> S

• Describe 2ndary mode by its lagrangian displacement : v1, B1

v = v0 + v1 + v /2 , B = B0 + B1 + B2

v2 =d 2

dt t+ v1 • 2 , and in ideal MHD, B2 = 2 B1( )

• Since basic state is independent of x, y, and (by assumption) t, and

periodic in z, seek a mode of “Bloch” form (like an electron wavefunction in a crystal):

2 X(z)exp( i t + ikxx + ikyy), where X is quasiperiodic:

X(z + ) = X(z)exp(ikz ) with 2

K and kz

K

2,

K

2

Parasitic modes (continued)

1

zz

zkh

2z = 0, where

kh (kx ,ky ), the horizontal part of k, and

kh • v1( )2 (kh • B1)2

4= (z)

It can be shown that in ideal MHD, the vertical displacement satisfies

• Growth rates Im( )=O(b ), where b=B1/B0

• Fastest growth in 3D (ky 0) with kh•B1=0–These are Kelvin-Helmholtz modes

–No perturbed magnetic tension since kh•B1=0

• slower modes with kh•B1 0 that are

almost discontinuous at z where B1(z) 0

–Proto-tearing modes(?) But =0 in this analysis

• Non-ideal ( 0, 0) is feasible but has

not been done!

(2)

Im( )/b vs. kz/K for kh•B1=0

z vs. zK for a kh•B1 0 mode

Fully developed MRI turbulence

• Distinguish net-flux (i.e., <Bz> 0) cases

– shearing-box boundary conditions preserve <Bz>

persistent instability & turbulence are guaranteed

— but at what level, and how does this depend on <Bz>, , ,...?

• ...from zero-net-flux (<Bz>=0) cases

– turbulence must act as a magnetic (fluctuation?) dynamo to

sustain the field against reconnection & Ohmic dissipation

– impossible in 2D: Cowling anti-dynamo theorem

– even in 3D ideal MHD, persistent turbulence isn’t guaranteed

Turbulent momentum transport:

The parameter

Let ij=

ji turbulent stress tensor: flux of i th component of momentum

in the j th direction (discounting transport by the mean flow).

Dimensions: ij= force area[ ] = energy volume[ ].

Associated dissipation rate = ijiv j energy/volume/time[ ]

In disks, this is dominated by r r r 2A xy in shearing box.

xy = vxvy +BxBy

4 where vi vi vi .

Reynolds stress Maxwell stress

Shakura & Sunyaev (1973, before MRI was understood) postulated

r xy( ) = P

dimensionless “viscosity” parameter

“Measured” values of

?AGN ( QSOs, quasars, Seyferts...)

direct observational constraints are few to none

10-3-100Cataclysmic variables

based on models of “dwarf nova” outbursts

10-2-10-3Protostellar disks

based on disk masses, temperatures, accretion

rates, and lifetimes

10-3-10-1Numerical simulations of MRI

varies with large-scale field, dissipation terms

What we used to think we knew about

MRI turbulent transport

• If <Bz> 0 (“nonzero net flux”),

then 0.1 iff Rem>1, and

Lu >1*, and VA2< cs

2 (i.e.,

plasma>1)

– Necessary only that <Bz> 0within the computational domain,

usually having size ~ h << r

• If <Bz> =0 (“zero net flux”),

then 0.1 iff Rem>104:

magnetic [energy] dynamo

– no external support requiredlo

g <

B2 >

time in orbits

Fleming, Stone, & Hawley 2000

*Rem=VL/ ; Lu = VAL/

Zero net flux (closeup view)

Fleming, Stone, & Hawley 2000

orbits

Pm

ag /P

~

10-1

10-4

Figure above is taken from Pessah, Chan, & Psaltis (2007), but the scalingwith <Bz> was noted already by Hawley, Gammie, & Balbus (1995).

scaled <Bz>

scal

ed

<Bz> in net-flux cases

Nx=64

Nx=128

Nx=256

t [2 / ]log k

log

PB

• Level & dominant lengthscale of the turbulence decrease grid scale,not domain size.

- Not what one expects from an inertial cascade.• Calculations above (Fromang & Papaloizou 2007) were carried outwithout explicit or , hence dissipation is purely numerical.

Saturation depends on grid scale

when <Bz> = 0

With sufficiently large explicit viscosity ( ) and diffusivity ( ),saturation should be independent of grid resolution.

increases with Pm insimulations with <Bz> 0(Lesur & Longaretti 2007)

=0 when <B> = 0unless Pm > 1(Fromang, Papaloizou, Lesur,& Heinemann 2007)

Pm < 10-10 in protostellar disks

increases with magnetic Prandtl

number Pm = /

Pm << 1 in most astrophysical &

laboratory systems

• Protostellar disks: Pm ~ 10-12

– is very uncertain due to nonthermal ionization

• Cataclysmic-variable (white-dwarf) disks: Pm ~ 1

• Quasar disks: Pm ~ 10-10 [broad-line region]

• Liquid metals (e.g. Na, Ga): Pm ~ 10-5-10-6