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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


Transition to turbulence: Pipe flow

Osborne Reynolds 1883

Linearly stable.

Re = VL/ν.

Requirement for transitionShear + high Re −→ 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.


Part 1: Non-Rotating MHD shear turbulence

with Eric Blackman.


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, ...


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


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


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


Hawley+ 1996:

Ideal MHD + small resolution→ small effective Re,Rm!


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.



Averagingxy ≡ 〈〉

〈V 2〉 ∼ 〈B2〉

Velocity fields: smooth.

Magnetic fields: ∼ smooth.



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.


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.


Part 2: Rotating MHD shear turbulence

with Martin Pessah.


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


Rotating MHD shear turbulence

Re = SL2/ν

Rm = SL2/η

S = L = 1

Lx = 1,Ly = 2,Lz = 4

64× 64× (64 ∗ Lz)



Re = 10,000,Lx = 1,Ly = 2

Low Pm turbulenceLarge Lz −→ Pm < 1 turbulence possible.



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.



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.


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.


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.


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.


Thank you!


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


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.


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.


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.
