Influence of rotation in unforced MHD shearing box turbulence Farrukh Nauman Niels Bohr International Academy Niels Bohr Institute August 17th, 2016 Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Influence of rotation in unforced MHDshearing box turbulence
Farrukh Nauman
Niels Bohr International AcademyNiels Bohr Institute
August 17th, 2016
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Unforced?
Most work:MHD + Linear shear + Forcing in Navier Stokes
This talk:MHD + Linear shear + Finite amplitude perturbations at t = 0
BUT shear ∼ forcing on all scales
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Transition to turbulence: Pipe flow
Osborne Reynolds 1883
Linearly stable.
Re = VL/ν.
Requirement for transitionShear + high Re −→ turbulence.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Transition to turbulence: pipe flow
Re < Recrit :Laminar.
Re ∼ Recrit :Coherent structures.
Re > Recrit :Featureless turbulence.
From: Willis+ 2008
Turbulence lifetimeLifetime ∼ exp |Re − Recrit|. Hof+ 2006
Recrit ∼ 2000.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Part 1: Non-Rotating MHD shear turbulence
with Eric Blackman.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Non-rotating MHD shear turbulence: Background
Questions:
Unforced: Is sustained growth in magnetic energy possible?(No: Hawley+ 1996)
2D, Yes: Mamatsashvili+ 2014
Forced: Can shear + non-helical forcing lead to large scale dynamo?(Yes: Yousef+ 2008)
See also work by: Bhattacharjee, Brandenburg, Cattaneo, Sridhar, Subramanian, Tobias, ...
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Non-rotating MHD shear turbulence: Background
Questions:
Unforced: Is sustained growth in magnetic energy possible?(No: Hawley+ 1996)
2D, Yes: Mamatsashvili+ 2014
Forced: Can shear + non-helical forcing lead to large scale dynamo?(Yes: Yousef+ 2008)
See also work by: Bhattacharjee, Brandenburg, Cattaneo, Sridhar, Subramanian, Tobias, ...
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Non-rotating hydrodynamic shear turbulence
ShearV = −Sxey + v
∂vx
∂t= ν∇2vx
∂vy
∂t= Svx + ν∇2vy
Stable up to infinite Re.(Romanov 1972)
Nonlinearly unstableRecrit ∼ 350.
x
y
Vy = −Sx
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Non-rotating hydrodynamic shear turbulence
Shear + Magnetic FieldsB =ZZB0 + b
∂vx
∂t=ZZZZ
B0∂bx
∂z+ ν∇2vx
∂vy
∂t=ZZZZ
B0∂by
∂z+ Svx + ν∇2vy
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.δv ∼ SL
Rm = SL2/η,Re = SL2/ν.Vy = −Sx , S = L = 1
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
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.δv ∼ SL
Rm = SL2/η,Re = SL2/ν.Vy = −Sx , S = L = 1
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Hawley+ 1996:
Ideal MHD + small resolution→ small effective Re,Rm!
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
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
Averagingxy ≡ 〈〉
〈V 2〉 〈B2〉 (except one run, next slide)
Velocity fields: smooth.
Magnetic fields: Bx rough, By smoother.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Non-rotating MHD shear turbulence
Averagingxy ≡ 〈〉
〈V 2〉 ∼ 〈B2〉
Velocity fields: smooth.
Magnetic fields: ∼ smooth.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Non-rotating MHD shear turbulence
x: Velocity and Magnetic field
0.1 1.0 10.0 100.0kz
10−8
10−6
10−4
10−2
100
Pow
er
Spectr
um
Vx 1x2x1
Bx 1x2x1
Vx 1x2x4
Bx 1x2x4
Vx 4x8x4
Bx 4x8x4
y: Velocity and Magnetic field
0.1 1.0 10.0 100.0kz
10−8
10−6
10−4
10−2
100
102
Pow
er
Spectr
um
Vy 1x2x1
By 1x2x1
Vy 1x2x4
By 1x2x4
Vy 4x8x4
By 4x8x4
Re = Rm = 10,000.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Non-rotating MHD shear turbulence: Conclusions
Rotation is NOT required to sustain 〈B2〉 if Rm > Rmcrit.
SYSTEM scale dynamo.
No clear forcing scale in the system.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Part 2: Rotating MHD shear turbulence
with Martin Pessah.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Rotating MHD shear turbulence: Background
Keplerian shear with zero net magnetic flux
(Pm = Rm/Re = ν/η)
Protoplanetary and CV disks: 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) Rotation in MHD shear turbulence
Rotating MHD shear turbulence
Re = SL2/ν
Rm = SL2/η
S = L = 1
Lx = 1,Ly = 2,Lz = 4
64× 64× (64 ∗ Lz)
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Rotating MHD shear turbulence
Re = 10,000,Lx = 1,Ly = 2
Low Pm turbulenceLarge Lz −→ Pm < 1 turbulence possible.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Rotating MHD shear turbulence
Comparison with Walker+ 2016:
0 1000 2000 3000 4000 5000
10-8
10-6
10-4
10-2
100
0 1000 2000 3000 4000 5000Time ( S
-1)
10-8
10-6
10-4
10-2
100
Str
ess
ReynoldsMaxwell
2x4x12x4x22x4x42x4x8
Re = Rm = 10,000
Low Pm turbulenceLarge Lz −→ Pm < 1 turbulence possible.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Rotating MHD shear turbulence
x: Velocity and Magnetic field
0.1 1.0 10.0 100.0kz
10−10
10−8
10−6
10−4
10−2
Pow
er
Spectr
um
Vx Lz=4
Bx Lz=4
Vx Lz=8
Bx Lz=8
Vx Lz=12
Bx Lz=12
y: Velocity and Magnetic field
0.1 1.0 10.0 100.0kz
10−10
10−8
10−6
10−4
10−2
Pow
er
Spectr
um
Vy Lz=4
By Lz=4
Vy Lz=8
By Lz=8
Vy Lz=12
By Lz=12
Re = Rm = 10,000, Lx = 1,Ly = 2.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Rotating MHD shear turbulence: Conclusions
Turbulence is sustained with Pm 1 if Lz ≥ 4.
Bx very rough and intermediate scale - incoherent αΩ?
By oscillates on much longer time scales compared to V .
Stresses not very sensitive to Pm for Pm 1,consistent with net flux results by Meheut+ 2015.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Comparison: Influence of Rotation?
Non-Rotating: Nauman, Blackman (submitted)
Rotating: Nauman, Pessah (submitted)
V : Spatially smooth for both, but rotation introduces oscillations.Non-rotating: V ∼ sin kz, Rotating: V ∼ sin(κt + sin kz + phase)
Bx : rough in both cases, box scale in non-rotating - only intermediatescale in rotating.
By : smooth and oscillating in both cases.
Rotating: Vrms < Brms.
Non-rotating: Vrms > Brms.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Some speculations
Incoherence due to lack of scale separation in shearing boxes.High resolution global simulations might lead to different results.
Box scale velocity structures transitional,might disappear at higher Re and/or L.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Thank you!
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
Numerical Simulations: Limitations
AGN disks:
Re ∼ 1015
Experiments: 104 − 106 PROMISE, Maryland, Princeton
Simulations: 4.5× 104 Meheut+ 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) Rotation in MHD shear turbulence
Ω = Ωez
vφ = Ωr
r0
φ r0 −→∞ x
y
vy = −qΩx
Shear parameter:
q = −d ln Ω/d ln r
x = r − r0y = r0(φ− φ0)
Figure: Ω ∼ r−q. Centrifugal force + radial gravity = linear shear.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
The need for large amplitude in simulations
Re = L2S/ν, Modes = 1000
k ∂tv v · ∇v Re−1 ∇2v
Ωv v2k Re−1 k2 v
1 1 10−2 1 10−4 10−4 10−6
10 1 10−2 10 10−3 10−2 10−4
100 1 10−2 100 10−2 1 10−2
1000 1 10−2 1000 10−1 102 1
Ω = 1 = L, k = n/L (ignore 2π) Perturbations v in units of LΩ.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence
The need for large amplitude in simulations
Re = L2S/ν, Modes = 1000
Re =104 Re =104
v = 1 v = 10−2
k ∂tv v · ∇v Re−1 ∇2v
Ωv v2k Re−1 k2 v
1 1 10−2 1 10−4 10−4 10−6
10 1 10−2 10 10−3 10−2 10−4
100 1 10−2 100 10−2 1 10−2
1000 1 10−2 1000 10−1 102 1
Ω = 1 = L, k = n/L (ignore 2π) Perturbations v in units of LΩ.
Farrukh Nauman (Niels Bohr Institute) Rotation in MHD shear turbulence