Nonmodal Growth and the Unstratified MRI Dynamo

MPPC general meeting, Berlin, June 24 Nonmodal Growth and the Unstratified MRI Dynamo Jonathan Squire!! and!! Amitava Bhattacharjee

MRI turbulence Observed accretion rate in disks in astrophysical disks is far too large to be explained by molecular dissipation Likely explanation is that disks are usually turbulent Central object We focus on the case where the disk is ionized and use MHD! Turbulent disk? From nationalgeographic.com Keplerian! rotation The MRI can develop into strong turbulence in Keplerian disks, with sufficient momentum transport to explain observed accretion.

Understanding the properties of the turbulence is important in formulating observationally useful models! But direct numerical simulation encounters its standard problem:!!! The Reynolds numbers in disks can be O( 4 ).! This is unimaginable with current computing resources An understanding of the basic character of the turbulence is needed to apply certain current results to a real astrophysical disk.

Preliminary local MRI Global domain Stratified shearing box Unstratified shearing box Modified from Jacob Simon s webpage Compressible MHD! equations Compressible local! MHD equations z r z Incompressible local! MHD equations y x The simplest relevant system exhibiting MRI turbulence is the local incompressible MHD equations remove global curvature! In the shearing box, boundary conditions are periodic in y (azimuthal) and z (vertical), and shearing periodic in x (radial).

OUTLINE We are approaching the MRI turbulence/dynamo problem from the most basic possible standpoint Linear MRI nonmodal methods! The importance of shearing waves! What is the role of linear physics in turbulence?! Addition of nonlinearity! An overview of shearing box turbulence! Quasi-linear theory/second order cumulants! The shearing box dynamo instability! Conclusions and future work

Nonmodal stability For a non self-adjoint system, eigenvalue growth rates do not tell the full story! Non-orthogonality of eigenmodes allows for transient growth faster than the least stable eigenmode.! Method: choose a norm and maximize the solution at a chosen time.! In general, nonmodal effects important when the eigenmodes are very sensitive to changes to the system.

Advection equation @ t u + @ x u = d @ 2 xu d =/2 with hard wall boundaries -. -.5.5. x Least stable! eigenmode λ = -6.3488 -.5 -. -.5

6 5 What do the nonmodal structures look like for the MRI?! 6 t= 5 4 4 3 3 2 2 For non-axisymmetric modes, very different to the eigenmodes.! -. Shearing wave structures are always present, despite the incompatibility of the boundary conditions.! 4 4 y3 3 2 2 This is also true for global modes. Log HGL 6 -.5..5 -. t= -.5 5..5 x 5 4 4 y. 5 2 -.5..5. Eigenmodes 3 2 Least stable! eigenmode 5.5 6 5. t=5 -.. Nonmodal pseudo-mode 2 -.5 6 3 [Squire & Bhattacharjee, accepted for PRL, arxiv:46.6582] -.. 6 5 t=5 t -. -.5..5 x ky =, kz = 4, Bz = /, By = Re = Rm = 4 Hard wall BC.

In a turbulent situation we most likely care most about growth over relatively short time-scales.! Because of this, the presence of a spectral instability is not necessarily important.! Log HEL 5 4 3 2 5 5 2 t Growth from random noise realizations The prevalence of shear waves indicates shearing boundary conditions are sensible (at least linearly).! In fact, over very short time-scales the MRI linear growth rate is the same for all modes (k y,k z ): de dt t= + = q The same as the maximum MRI eigenmode growth rate! for axisymmetric modes! (k x =k y =) at k z =.968/B z

Shearing box MRI turbulence The net-flux shearing box is linearly unstable to the axisymmetric MRI. This evolves nonlinearly into strong turbulence.! A zero net-flux shearing box is spectrally stable!!! Nonetheless, at sufficiently high Reynolds numbers sustained zero net-flux turbulence is possible The turbulence is subcritical, arising from nonmodal growth (similar to hydrodynamic pipe flow)

But shearing box turbulence is poorly understood Around 27 it was realized that many shearing box results were not converged (Fromang et al.)! Correlation length resolution 2 resolution An increase resolution could lead to a decrease in transport this is troubling!! 4 resolution From Fromang & Papaloizou 27 This seems to be due to a strong dependence of the turbulence on Pm=Re/Rm (Fromang et al. 27b)! It is still unclear whether this is a real effect or just an artifact of the shearing box

The MRI dynamo A dynamo generated mean azimuthal magnetic field seems to be a key component of self-sustaining MRI turbulence! How does the dynamo scale with Rm and/or Pm? This is likely related to the shearing box convergence problems! Interestingly, this dynamo can never be kinematic.9 B y. Nonlinear shearing box turbulence at Re=25, Rm=8 z.8.7.6.5.4.3.2..5.5. 2 3 4 5 6 7 8 9 t

A possible turbulence/dynamo cycle By analogy with work on hydrodynamic pipe flow, Rincon et al. (27) suggest the following mechanism for self-sustaining turbulence and dynamo: Axisymmetric azimuthal! magnetic field " effect Non-axisymmetric! MRI Weak axisymmetric! poloidal magnetic field Non-axisymmetric! perturbations Nonlinear feedback! (Reynolds and Maxwell stress)

A possible turbulence/dynamo cycle By analogy with work on hydrodynamic pipe flow, Rincon et al. (27) suggest the following mechanism for self-sustaining turbulence and dynamo: LINEAR (nonmodal) " effect Axisymmetric azimuthal! magnetic field Weak axisymmetric! poloidal magnetic field Non-axisymmetric! MRI Non-axisymmetric! perturbations LINEAR (nonmodal) Nonlinear feedback! (Reynolds and Maxwell stress) NONLINEAR

Nonlinear feedback The linear phases of the cycle are best studied using nonmodal methods.! We have seen that shearing box boundary conditions are linearly well suited to the MRI.! We wish to study the nonlinearity in the simplest way possible Technique: remove as much nonlinearity as possible while retaining the important physics

CE2/S3T CE2 (Second-order cumulant expansion, Marsden & Tobias) or S3T (Stochastic structural stability theory, Farrell & Ioannou) has been successful in geophysical modeling:!!!! Neglect all nonlinear interactions except those between the mean fields and noise-driven fluctuations. Usually, evolve the statistics of the fluctuations.! For MRI, the mean field is the k x =k y = mode.

CE2 is nothing but a driven quasi-linear theory Linear fluctuations interact with the mean fields b (k x,k y ) The Reynolds and Maxwell stresses feedback on the mean fields hbi b Neglected b b hbi b (k x,k y ) b ( k x, k y ) b (k x,k y ) Solve for statistics of b C(x, y, z,z 2,t)= hui u j i hb i u j i hu i b j i hb i b j i or for a single realization

With an ergodicity assumption, equations are closed! There is no turbulent cascade! The mean field equations are very simple (aside from nonlinear stresses) @U X @T =2U Y + @ 2 ZU X + b rb X u ru X @U Y @T =(q 2) U X + @ 2 ZU Y + b rb Y u ru Y @B X @T = @2 ZB X + b ru X u rb X @B Y @T = qb X + @ 2 ZB Y + b ru Y u rb Y @ Z U Z = @ Z B Z = Basic CE2: only the mean B y interacts with fluctuations.! Full CE2: all mean fields interact with fluctuations.

We are throwing away a lot of physics, but still get rather good agreement.2.8 z Nonlinear..6.4..2.2 2 3 4 5 t 6 7 8 9 Direct numerical! simulation.5..8.5.6 z.4 Quasi-linear.5.2..5 2 3 4 5 t 6 7 8 9 Ux.4.8 Rm=8, Pm=3.2 z Nonlinear.2.6.4.2 z.2.4.6 5 5 2 Uy 25 t 3 35 t.8.2.6.8..2..4.6. z.4.2 Quasi-linear.2 5 2 5 t 3 2 4 5 25 t 6 3 7.2..3.2 35 8 9.5

The quasi-linear system also exhibits the same scaling behavior as the nonlinear system: At fixed Rm and noise, a decrease in Re causes the transport to increase Momentum transport 8 x 3 Rm=5!! Aspect ratio:! (,",) 7 6 5 4 3 2 Pm =.5 Pm = Pm = 2 Pm = 4 Quasi-linear Quasi-linear with Pm 5 5 2 25 3 Driving noise amplitude This is not a linear effect, but is due to the dynamo Momentum transport 4 x 3 3.5 3 2.5 2.5 Pm =.5 Pm = Pm = 2 Pm = 4 Linear.5 with Pm 5 5 2 25 3 Driving noise amplitude

The quasi-linear system also exhibits the same scaling behavior as the nonlinear system: At fixed Rm and noise, a decrease in Re causes the transport to increase Rm=5!! Aspect ratio:! (,",) Fromang et al. 27 Pm= Momentum transport Pm=8 8 x 3 Pm=6 7 6 5 4 3 2 Pm =.5 Pm = Pm = 2 Pm = 4 4 5 5 2 25 3 3 2 Pm=4 This is not a linear effect, but is due to the dynamo Pm= Momentum transport 3.5 3 2.5 2.5 4 x 3 Pm=2 Pm =.5 Pm = Pm = 2 Pm = 4 Rm = 25 Quasi-linear Quasi-linear Driving noise amplitude Linear 9 8 7 6 5 x 3 Pm=6 Quasi-linear with Pm Pm=4 Pm=8 5 5 2 25 3 t Pm=2.5 with Pm 5 5 2 25 3 Driving noise amplitude

It is very convenient to evolve the statistics The turbulent state is time-independent, we can study its stability! Computationally intensive grid size N x N y (4N z ) 2 but easily parallelized! All interesting information can often be obtained in a single run without time averaging B Y Rm=5, Pm=2 (driving noise: 5).9.8. 2 B Y z.7.6.5.4.3.2..5.5. Log(B) 4 6 8 B X 5 5 2 25 3 35 4 45 t 25 5 t

At some critical parameter, the homogeneous turbulence equilibrium becomes unstable to a mean-field! dynamo instability The instability is modal and real (no oscillation). It saturates at values consistent with nonlinear simulations.! Control parameter can be Rm or the driving noise Mean field saturation! amplitude Equilibrium with! strong B Y is stable Transitions to a time-! periodic saturated state Homogenous! turbulence! is stable Homogenous! turbulence! is unstable Control parameter! (e.g., noise or Rm) Described by a real Ginzburg-Landau! equation around this point?

The instability growth rate and saturation depends on Pm stronger instability at higher Pm.2 Growth rate Rm=5!! Aspect ratio:! (,",) Dynamo growth rate..8.6.4.2 Pm =.5 Pm = Pm = 2 Pm = 4.2.4 5 5 2 25 3 Driving noise amplitude

The instability growth rate and saturation depends on Pm stronger instability at higher Pm Growth rate Dynamo growth rate.5.4.3.2 Rm=5!!. Aspect ratio:! (,",) Pm = Pm = 2 Pm = 3 Pm = 4 Dynamo growth rate.2..8.6.4.2 Pm =.5 Pm = Pm = 2 Pm = 4 Noise: 3..2.2.4,5 225 3 375 5 45 5 525 2 6 25 675 3 75 Driving Rm noise amplitude

.25 Time-independent! Saturation! amplitude Rm=5!! Aspect ratio:! (,",) Magnetic field saturation amplitude.2.5..5 Pm =.5 Pm = Pm = 2 Pm = 4 5 5 2 25 3 Driving noise amplitude

At higher Reynolds numbers, the saturation can become more complicated B y (z,t) 6.4 5.2 4 Pm= z 3 2.2 Rm = 75! Noise = 5 2 4 6 8 2 4 6 5 t.4.5. Pm=4 z 4 3 2.5.5..5 2 4 6 8 2 4 t.2 Resembles nonlinear cycles, but this requires more study

Conclusions Nonmodal growth is fundamental to the MRI!!! For non-axisymmetric modes, the fastest growing structures are invariably shearing waves Since short time growth rates may be much more important in a turbulent situation, perhaps net-flux and zero net-flux turbulence are not so different? We will examine net-flux turbulence (both linearly and nonlinearly) in the near future.

Despite quantitative differences, the quasi-linear (CE2) system exhibits similar trends to the nonlinear shearing box:!!! The transport increases as Re is decreased Possible quantitative comparison driven nonlinear simulation Above a critical value of Rm or the driving noise, homogenous turbulence is unstable. Both the growth rate and saturation amplitude increase with Pm More work is needed to better understand both the growth and saturation of the dynamo instability. We have recently finished an improved CE2 code, and hope to also make analytic progress. Stratified dynamo can also be studied in this way

Self-sustaining turbulence? Our quasi-linear method must die out if noise is turned off not self sustaining Can probably self-sustain with 2-D mean fields; e.g., global run (Farrell et al. 22) Shearing of B x is the dominant positive forcing on B y. EMF acts to damp field in quasi steady-state. Dynamo generated! field is important! for the turbulence Shearing waves are! important. Short! time growth quite! different to eigenmode! predictions. The time-periodic saturated! state seems most similar to! real MRI turbulence Homogenous turbulence! is more stable at low Pm.! Breaks the cycle?

The driving terms in the dynamo @ r r z E y Diss x @B X @T = @2 ZB X + b ru X u rb X @B Y @T = qb X + @ 2 ZB Y + b ru Y u rb Y Shear Diss y @ z E x MF dynamo theory E = hv bi = ij B j + ijk B j,k We see that the B y dynamo is primarily driven by the shear term, and nonlinearly saturated by z ε x Comparing terms for k z = 2π Rm = 75! Noise = 5.2.5..5.5..5 Pm= Shear @ z " x Diss y @ z " y Diss x.2 2 4 6 8 t.5.5 Pm=4 Shear @ z " x Diss y @ z " y Diss x.5 5 5 t