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Turbulence in MHD Shear Flows
Farrukh Nauman
Niels Bohr International AcademyNiels Bohr Institute
August 4th, 2016
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 1 / 37
“... This model will be a simplification and anidealization, and consequently a falsification. It is tobe hoped that the features retained for discussion arethose of greatest importance in the present state ofknowledge.”
— Alan Turing, 1952
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 2 / 37
Two themes:
Nonlinear non-rotating and rotating shear MHDflows.
Turbulence in accretion disks (if time permits).
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 3 / 37
Transition to turbulence: Pipe flow
Osborne Reynolds 1883
Linearly stable.
Re = VL/ν.
Requirement for transitionShear + high Re −→ turbulence.
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 4 / 37
Transition to turbulence: pipe flow
Not a plasma.
No radiation, dust.
Incompressible.
Turbulence lifetimeLifetime ∼ exp |Re − Recrit|. Hof+ 2006
Recrit ∼ 2000.
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 5 / 37
How do you study stability?
Infinitesimal (. 1%)
Ignore nonlinear termsO(v2,b2)
v ,b ∼ exp[i(kx − ωt)]
StabilityUnstable: ω2 < 0Stable: ω2 > 0
Finite amplitude(∼ 10− 100%)
Cannot ignore v · ∇v , b · ∇bv · ∇b, b · ∇v
Critical AmplitudeAcrit ∼ 1/Re
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 6 / 37
The need for large amplitude in simulations
Re = L2Ω/ν, Modes = 1000,
Ω = 1 = L, k = n/L (ignore 2π),
Perturbations v in units of 1/ΩL.
k ∂tv v · ∇v Re−1 ∇2v
Ωv v2k Re−1 k2 v
1 v v2 Re−1 v
10 v 10 v2 Re−1 102 v
100 v 100 v2 Re−1 104 v
1000 v 1000 v2 Re−1 106 v
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 7 / 37
The need for large amplitude in simulations
Re = 104 Re = 104
v = 1 v = 10−2
k ∂tv v · ∇v Re−1 ∇2v
Ωv v2k Re−1 k2 v
1 1 1 10−4
10 1 10 10−2
100 1 100 1
1000 1 1000 102
k ∂tv v · ∇v Re−1 ∇2v
Ωv v2k Re−1 k2 v
1 10−2 10−4 10−6
10 10−2 10−3 10−4
100 10−2 10−2 10−2
1000 10−2 10−1 1
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 8 / 37
Theme 1
MHD Shear Turbulence
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 9 / 37
Linear StabilityShearV = −Sxey + vS = qΩ
∂vx
∂t= ν∇2vx
∂vy
∂t= Svx + ν∇2vy
Stable up to infinite Re.(Romanov 1972)
Nonlinearly unstableRecrit ∼ 350.
x
y
vy = −qΩx
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 10 / 37
Linear Stability
Shear + Magnetic FieldsB =@@B0 + b
∂vx
∂t=HH
HHH
1µ0
B0∂bx
∂z+ ν∇2vx
∂vy
∂t=
HH
HHH
1µ0
B0∂by
∂z+ Svx + ν∇2vy
Linearly stable!
Nauman, Blackman (submitted)
0 500 1000 1500
10-10
10-8
10-6
10-4
10-2
0 500 1000 1500Time
10-10
10-8
10-6
10-4
10-2
Energy
KineticMagnetic
Rm=1000Rm=1200Rm=1500Rm=2000
Finite amplitude.Rm = SL2/η,Re = SL2/ν.
Vy = −Sx , S = L = 1
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 11 / 37
Linear Stability
Shear + Magnetic Fields
Hawley+ 1996:B fields cannot sustain growth
Finite amplitude.
Nauman, Blackman (submitted)
0 500 1000 1500
10-10
10-8
10-6
10-4
10-2
0 500 1000 1500Time
10-10
10-8
10-6
10-4
10-2
Energy
KineticMagnetic
Rm=1000Rm=1200Rm=1500Rm=2000
Finite amplitude.
Rm = SL2/η,Re = SL2/ν.Vy = −Sx , S = L = 1
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 12 / 37
Hawley+ 1996:
Ideal MHD + small resolution→ small effective Re,Rm!
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 13 / 37
Non-rotating MHD shear turbulence
0 25 50 75 100
-0.5
0.0
0.5
0 25 50 75 100
-0.5
0.0
0.5
0 25 50 75 100
-0.5
0.0
0.5
-0.16
0.011
0.18 Vx
0 25 50 75 100
-0.5
0.0
0.5
0.00.0
0 25 50 75 100
-0.5
0.0
0.5
0.00.0
-0.29
0.014
0.32 Vy
0 25 50 75 100
-0.5
0.0
0.5
0.00.00.00.0
0 25 50 75 100
-0.5
0.0
0.5
0.00.00.00.0
-0.038
-0.0021
0.034 Bx
0 25 50 75 100
-0.5
0.0
0.5
0.00.00.00.0
0 25 50 75 100
-0.5
0.0
0.5
0.00.00.00.0
-0.045
-0.0018
0.041 By
1x2x1
Time (in shear times)
zz
x − y averaged.
< V 2 >< B2 > (except one run)
Velocity fields: very smooth.
Magnetic fields: Bx dominated by small scale, By smooth.
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 14 / 37
Let’s add ROTATION!
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 15 / 37
Some background
Keplerian shear with zero net magnetic flux
(Pm = Rm/Re = ν/η)
Protoplanetary and CV disks have Pm 1.
Lab. experiments Pm 1.
Previous work (Lz = 1): Pm < 1 turbulence not observed.Fromang+ 2007, Rempel+ 2010, Rincon+ 2011-16, Walker+ 2016
Lz = 4: Pm = 1 very short lived turbulence. Shi+ 2016
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 16 / 37
Nauman, Pessah (in prep)
Re = SL2/ν
Rm = SL2/η
S = L = 1
Lx = 1,Ly = 2,Lz = 4
64× 64× (64 ∗ Lz)
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 17 / 37
Influence of Rotation?
Rotating: Nauman, Blackman (submitted)
Non-rotating: Nauman, Pessah (in prep)
V : Spatially smooth for both, but rotation introduces oscillations.
B: Bx rough in both cases, By smooth.
Rotating: Vrms < Brms.
Non-rotating: Vrms > Brms (except one simulation).
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 18 / 37
Theme 2
Accretion disk turbulence
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 19 / 37
Linear Stability
Shear + Rotation + Magnetic FieldsPerturb: exp(i(kzz − ωt)):
∂vx
∂t= 2Ωvy +
1µ0
B0∂bx
∂z+ ν∇2vx
∂vy
∂t= qΩvx − 2Ωvx +
1µ0
B0∂by
∂z+ ν∇2vy
Aside: MRI linearly stable for B0 = 0 for q < 2 but found to be unstable in simulations!
Magnetorotational Instability (MRI)Unstable: 0 < q < 2
ω2 = −(
q2−q
)k2
z v2A for k2
z v2A 1
where v2A = B2
0/µ0.
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 20 / 37
Shear + Rotation + Magnetic Fields: Sims
Nauman, Blackman arXiv2015
Hydrodynamics
0 500 1000 1500 2000
10-20
10-15
10-10
10-5
100
0 500 1000 1500 2000Time
10-20
10-15
10-10
10-5
100
En
erg
y a
nd
str
ess
Kinetic energyReynolds stress
q=1.5q=2.1q=4.2
Magnetohydrodynamics
0 20 40 60 80 100
0.01
0.10
1.00
10.00
0 20 40 60 80 100Time
0.01
0.10
1.00
10.00
En
erg
y
Kinetic energyMagnetic energy
Both 0 < q < 2 and q > 2 unstable with magnetic fields!
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 21 / 37
Rotating shear stability
Nauman, Blackman arXiv2015 Consistent with linear stabilityanalysis. No surprises!
Hydro: Keplerian flow immediately decays exponentially.
MHD:
I q = 1.5: Magnetically dominated.I q > 2: Kinetically dominated.
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 22 / 37
Keplerian flow: q = 1.5 is unstable to MRI!
Velikhov 1959, Chandrasekhar 1960, Balbus Hawley 1991.
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 23 / 37
Shakura Sunyaev MRI simulations
Winds and jets
Radiation
Dust
Temperature variations
G. Relativity?
Magnetic fields
Vortices
(Partially) ionized
Density variations
BH spin
Global, thin
Hydrodynamics
∂t = 0, ∂φ = 0
No outflow
νT = αcsH
Local, thin
MagnetoHD
3D + time
Corona?
〈ρvxvy − bxby〉
α =〈ρvxvy−bxby〉
ρ0c2s
Figure : Comparison between Shakura Sunyaev and MRI.Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 24 / 37
Numerical Simulations: LimitationsAGN disks:
Re ∼ 1015
Experiments: 104 − 106PROMISE, Maryland, Princeton
Simulations: 4.5× 104Meheut+ 2015, Walker+ 2016
Physical ResolutionAssume H = 1AU ∼ 1.5× 1013cm, and H/R ∼ 0.1.
Shearing Box: H/1000 ∼ 1010cmGlobal Disk: R/1000 ∼ H/100 ∼ 1011cm
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 25 / 37
Accretion disks: Problems
Angular momentum transport:
Viscous time scale R2/ν ∼ 1011 years
PPD disk lifetime: ∼ 1 million years
AGN disk lifetime: ∼ 108 years
Brightness: Source of AGN radiation?
Jets: Origin of large scale fields?
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 26 / 37
Disk Theory: Shakura Sunyaev (SS) Model
(citations: ∼ 7500)
For Lz = ρvφr ,
∂〈Lz〉∂t
+∇ · 〈Lzv〉 = ∇ ·⟨ρr2ν∇vφ
r
⟩Replace ν with turbulent viscosity:
SS α prescriptionνT = αcsH.
cs = sound speed, H = scale height (density ρ = exp(−z2/H2)).
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 27 / 37
Other models
Linear/Quasi-linear: (Pessah+ 2006, Ebrahimi+ 2016)
Including jets: α + large B. (Kuncic+ 2004, Salmeron+ 2007)
Nonlinear closure: (Kato+ 1990s, Ogilvie+ 2003, Lingam+ 2016)
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 28 / 37
SS vs MRI
Question 1:
Are MRI generated stresses local?
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 29 / 37
MRI stresses are non-local
Nauman, Blackman MNRAS 2014
Mag. Energy: |B|2Kin. Energy: |√ρv |2xy Maxwell: |BxBy |2yz Maxwell: |ByBz |2
x-axis:k = 1.0→ L = Hy-axis:1.0 means 100% power!
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 30 / 37
Figure : Stratified: 〈B〉 = By0.
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 31 / 37
SS vs MRI
Question 2:
Is Trφ ∼ Txy = qαc2s ?
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 32 / 37
Shear dependence of stressesNauman, Blackman MNRAS 2015
Figure : Total stress vs shear.
Txy ∝ q/(2− q) unlike SS prediction!
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 33 / 37
Conclusions
MRI: No analytical model yet that captures large scale features.
Large (coherent) magnetic fields: a generic property ofmagnetized shear turbulence.
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 34 / 37
Thank you!
Farrukh Nauman (Niels Bohr Institute) MHD Shear Turbulence August 4th, 2016 35 / 37