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 weight & pulley drive inner cylinder fluid-filled gap between cylinders motor drives outer cylinder G.I. Taylor s (1936) apparatus (B=0) Nominal parameters of the Princeton MRI experiment (B0)
Mechanical energy is minimized by uniform rotation Energy: Angular momentum: E = L = N p i 2 2 + U(r 1,...,r N ) i=1 N r i p i i=1 Minimize E at fixed L: ( E L)= 0 p i = r i p i A system of minimal energy rotates uniformly with angular velocity (=).
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: 2 r = P 1 ( r c J B), if any r...in cylindrical coordinates z,r, with = e z P(r max ) P(r min ) 2 ( 2 ) 2 r max 2 r min This pressure drop is supported by the container. Absent external driving (torques), TC flow will relax to uniform rotation
Disks cannot rotate uniformly Boundaries are free (P0 at boundaries) except inner edge, sometimes (e.g. stellar surface) Orbital energy dominates: (r) 2 >>P/ c s 2 This means disks are thin: z<< r Centrifugal force is balanced mainly by gravity: 2 r GM * r 2 (at z << r) r 3/2 ("keplerian") Dissipation causes little change in, but a radial drift toward the star (accretion), i.e. toward lower gravitational potential
MRI saturation in disks 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. dynamo? MRI linear instability primary MRI modes 2 ndary instabilities These 2 ndary may include Kelvin-Helmholtz dissipation 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)
The Shearing Box: A local, corotating Cartesian approximation x r r 0 y r [ 0 0 (t)]; 0 (t) (r 0 ) z z Dimensions of the box are typically comparable to vertical scale height h c s r 0, c s sound speed
Shearing-box equations of motion dv dt = v t + v v = 2ẑ v 4Ax ˆx 2 zẑ 1 P + 1 c J B + 2 v d dt = v B t = ( v B )+ 2 B 2A r d dr r=r0 : shear rate 4Ax ˆx 2 zẑ : tidal field : kinematic viscosity dimensions : L 2 T 1 [ ] : magnetic diffusivity ditto
Standard equilibrium state of the shearing box Gas or plasma pressure dominates P >> B 2 /8, i.e. >>1 Radial gradients (of pressure, etc.) are negligible J 0 With these assumptions, 2ẑ v 4Ax ˆx v 0 = 2Axŷ 1 0 P 0 z = 2 z P 2 z+ 1 z ẑ 0 (r 0 ) + d ( ) dr r0 r r 0
Recap of linear theory Consider the (most important) special case, B 0 = B 0 ẑ = constant, V A B 0 4, (v,,p, B) 1 exp( t + ikz) : ( +k 2 )( +k 2 )+ kv A ( ) 2 2 + 4( + A) +k 2 ( ) 2 + 4A( kv A ) 2 = 0 In ideal MHD (=0=), weak field is always unstable: ~ A at k ~/V A 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 << c s /, perturbations are incompressible: v = v 0 + v 1, v 1 0 = 0 + 1, 1 0, etc. For a single mode ( v,p,b) 1 exp(ik r) + c.c., 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.
Proof of exactness v 1 = 0 = B 1 k v 1 = 0 = k B 1 ; ( ) v 1 + c.c. = 0. v 1 v 1 v 1 ikv 1 + c.c. = ikv 1 Similarly, v 1 B 1 = 0 = B 1 v 1 = B 1 B 1. Also, 1 P 1 is linear because = 0. More generally, v 1 (r,t) = a n (t)exp(ik n r) n so v 1 v 1 ik m a n + c.c. ( ) a m 0 in general unless k m k n = 0. Modes at parallel wavenumbers can be linearly superposed provided >>1, i.e. in the incompressible limit Also true in non-ideal MHD, since 2 v 1 & 2 v 1 are linear
2D (x,z) simulations in ideal MHD Hawley & Balbus 1992, ApJ 400, 595 z Above: growth of magnetic energy in several runs. Below: power versus (k x,k z ) x or r v y, 1 at t=2.4, 3.1, 3.6, & 4.1 orbits. =4000, box size=0.25c s / Cartoon of evolution of the magnetic field.
Parasitic modes Goodman & Xu 1994, ApJ 432, 213 Consider stability of a single large-amplitude mode v 1 (r,t) = V h exp(st)sin(kz) v 0 = 2Ax ˆx, B 1 (r,t) = B h exp(st)cos(kz) B 0 = B 0 ẑ. V h, B h are constant, horizontal (xy) vectors; V h B h = 0 in ideal MHD. (1) Linearize about (1) as basic state, neglecting (v 0,B 0 ) Also neglect v because we seek 2 nd 1, B 1 ary (= parasitic ) growth rates ~ Kv 1 >> S Describe 2 nd ary mode by its lagrangian displacement : v = v 0 + v 1 + v /2, B = B 0 + B 1 + B 2 v 2 = d 2 dt t + v 1 2, and in ideal MHD, B 2 = 2 B 1 ( ) 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(it + ik x x + ik y y), where X is quasiperiodic: X(z + ) = X(z)exp(ik z ) with 2 and K k K z 2, K 2
Parasitic modes (continued) It can be shown that in ideal MHD, the vertical displacement satisfies 1 z z z k 2 h z = 0, where k h (k x,k y ), the horizontal part of k, and (2) ( k h v 1 ) 2 (k h B 1 )2 4 = (z) Growth rates Im()=O(b), where b=b 1 /B 0 Fastest growth in 3D (k y 0) with k h B 1 =0 These are Kelvin-Helmholtz modes No perturbed magnetic tension since k h B 1 =0 slower modes with k h B 1 0 that are almost discontinuous at z where B 1 (z)0 Proto-tearing modes(?) But =0 in this analysis Non-ideal (0, 0) is feasible but has not been done! Im()/b vs. k z /K for k h B 1 =0 z vs. zk for a k h B 1 0 mode
Fully developed MRI turbulence Distinguish net-flux (i.e., <B z > 0) cases shearing-box boundary conditions preserve <B z > persistent instability & turbulence are guaranteed but at what level, and how does this depend on <B z >,,,...?...from zero-net-flux (<B z >=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 = ij i v j energy/volume/time [ ] In disks, this is dominated by r r r 2A xy in shearing box. xy = v x v y + B xb y 4 where v i v i v i. Reynolds stress Maxwell stress Shakura & Sunyaev (1973, before MRI was understood) postulated r ( xy )= P dimensionless viscosity parameter
Measured values of Numerical simulations of MRI varies with large-scale field, dissipation terms 10-3 -10-1 Protostellar disks based on disk masses, temperatures, accretion rates, and lifetimes Cataclysmic variables based on models of dwarf nova outbursts 10-2 -10-3 10-3 -10 0 AGN ( QSOs, quasars, Seyferts...) direct observational constraints are few to none?
What we used to think we knew about MRI turbulent transport If <B z >0 ( nonzero net flux ), then 0.1 iff Re m >1, and Lu >1 *, and V A2 < c 2 s (i.e., plasma >1) Necessary only that <B z >0 within the computational domain, usually having size ~ h << r log <B 2 > time in orbits If <B z > =0 ( zero net flux ), then 0.1 iff Re m >10 4 : magnetic [energy] dynamo no external support required * Re m =VL/; Lu = V A L/ Fleming, Stone, & Hawley 2000
10-1 Zero net flux (closeup view) P mag /P ~ 10-4 orbits Fleming, Stone, & Hawley 2000
<B z > in net-flux cases scaled scaled <B z > Figure above is taken from Pessah, Chan, & Psaltis (2007), but the scaling with <B z > was noted already by Hawley, Gammie, & Balbus (1995).
Saturation depends on grid scale when <B z > = 0 N x =64 N x =128 N x =256 log P B t [2/] log k 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 out without explicit or, hence dissipation is purely numerical.
increases with magnetic Prandtl number Pm = / With sufficiently large explicit viscosity () and diffusivity (), saturation should be independent of grid resolution. increases with Pm in simulations with <B z > 0 (Lesur & Longaretti 2007) Pm < 10-10 in protostellar disks =0 when <B> = 0 unless Pm > 1 (Fromang, Papaloizou, Lesur, & Heinemann 2007)
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