recurrence in three dimensional magnetohydrodynamic plasma · 2020. 7. 16. · 1/14 motivation...

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Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra

Under-30 Talk

Recurrence in three dimensionalmagnetohydrodynamic plasma

Rupak Mukherjee†∗ Rajaraman Ganesh∗ Abhijit Sen∗

†Princeton Plasma Physics Laboratory, Princeton, NJ, USA∗Institute for Plasma Research, HBNI, India

[email protected]

[email protected]

[email protected]

3rd Asia-Pacific Conference on Plasma Physics

04th November, 2019

Rupak Mukherjee, PPPL, USA Fundamental Plasma Physics, Vortex Dynamics, F-O11, Willow, 14:00-16:00

Chair: Zensho Yoshida AAPPS-DPP 2019, Crowne Plaza, Hefei, China

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Aim of the Talk

Direct Numerical Simulation (DNS) study of Three dimensionalSingle fluid MagnetoHydroDynamic equations have been carried

out to explore

1 “Nonlinear Dispersion-free Alfven Waves (NDAW)” - via largeamplitude oscillations of kinetic and magnetic energy

2 “Recurrence Phenomena” - a periodic reconstruction of initialflow of fluid and magnetic field variables mediated via NDAW.

1 NDAW exists in both 2D & 3D - But Recurrence ?[Key Parameter: Initial condition]

2 Rayleigh Quotient determines criteria of Recurrence.[Key Parameter: Initial Condition]



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Governing Equations in three dimensions

Single Fluid MHD equations evaluated by G-MHD3D

∂ρ

∂t+ ~∇ · (ρ~u) = 0

∂(ρ~u)

∂t+ ~∇ ·

[ρ~u ⊗ ~u +

(P +

B2

2

)I− ~B ⊗ ~B

]= µ∇2~u + ~f

∂E

∂t+ ~∇·

[(E + P +

B2

2

)~u − ~u ·

(~B ⊗ ~B

)− η ~B ×

(~∇× ~B

)]= µ

(~∇ · ~u

)2; P = C 2

s ρ

∂ ~B

∂t+ ~∇ ·

(~u ⊗ ~B − ~B ⊗ ~u

)= η∇2 ~B

u0 = Lt0

, VA = B04π√ρ0

, Ms = u0Cs

, MA = u0VA

, Re = ρ0u0Lν , Rm = Lu0

η .



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Scope of the code

Scope and Algorithm

1 The DNS code G-MHD3D governs the time evolution of Threedimensional single fluid MagnetoHydroDynamic equations.

2 Equations are evolved in pseudo-spectrally in a periodicdomain to achieve higher accuracy and performance overfinite difference schemes. [FFTW & cuFFT library]

GPU Optimization with NVIDIA in DGX systems

1 Optimised GPU code with OpenACC and CUDA parallelisationshows two order of magnitude speedup over OpenMP code.

2 For high resolution runs multi-GPU parallalisation withNV-Link & accFFT library has been achieved to test onNVIDIA - DGX station and PSG cluster.

RM et. al., IEEE 25th International Conference on High Performance Computing Workshops (HiPCW), 2018



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Motivation

Nonlinear Dispersion-free Alfven Wave (NDAW)

Within the premise of Single fluid MHD, energy can cascadethrough both kinetic and magnetic channels simultaneously.

With weak resistivity, MHD model predicts -1). irreversible conversion of magnetic energy into fluid kineticenergy (i.e. reconnection).2). conversion of kinetic energy into mean large scalemagnetic field (i.e. dynamo).

Question: For a given fluid type and magnetic field strength,are there fluid flow profiles which do neither?

Answer: Yes. For a wide range of initial flow speeds or AlfvenMach number it is shown that the coherent nonlinearoscillation persist.

Is it a new result? - NO!



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Previous works on NDAW

Earlier works with NDAW and the take-over

The existence of such dispersionless nonlinear oscillationsare known for some time [H. Alfven, Cosmicalelectrodynamics (Clarendon Press, Oxford, 1963)]

More recently these results have been revisited by Z Yoshida[Communications in Nonlinear Science and NumericalSimulation 17, 2223 (2012)] and in another work by H MAbdelhamid & Z Yoshida [Physics of Plasmas 23, 022105(2016)] from a point of view of Hamiltonian-Casimirformulation of ideal MHD and its extended models.

H M Abdelhamid & Z Yoshida had also suggested thenecessity of special initial wave forms for sustaining suchoscillations- We started with these initial conditions!



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NDAW in two spatial dimensions at Alfven resonance (Cs = VA).

Orszag-Tang Flow

ux = −A sin(k0y)

uy = +A sin(k0x)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Sh

ifte

d K

ine

tic,

Ma

gn

etic a

nd

To

tal E

ne

rgy

Time

Kinetic Energy

Magnetic Energy

Total Energy

Cat’s Eye Flow

ux = + sin(k0x) cos(k0y)− A cos(k0x) sin(k0y)

uy = − cos(k0x) sin(k0y) + A sin(k0x) cos(k0y)

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 2 4 6 8 10

Shifte

d Kine

tic, M

agne

tic an

d Tota

l Ene

rgy

Time

Kinetic Energy

Magnetic Energy

Total Energy



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Few Observations

2D Orszag-Tang Flow withExternal Forcing

~f = α

[−A sin(kf y)+A sin(kf x)

], α = 0.1

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10

Shi

fted

Kin

etic

, Mag

netic

and

Tot

al E

nerg

y

Time

Kinetic Energy

Magnetic Energy

Total Energy

3D ABC Flow with different MA

ux = U0[A sin(k0z) + C cos(k0y)]uy = U0[B sin(k0x) + A cos(k0z)]uz = U0[C sin(k0y) + B cos(k0x)]

Frequency of energy exchangescales linearly with MA.

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0 5 10 15 20

Sh

ifte

d K

ine

tic E

ne

rgy

Time

MA = 0.1

MA = 0.2

MA = 0.3

MA = 0.4

MA = 0.5

With external forcing similar to initial flow the plasma acts asa forced-relaxed system both in two and three dimensions.



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3D DNS results of decaying chaotic flows

NDAW in 3D MHD at Alfven Resonance

Time evolution of kinetic & magnetic energy for TG, ABC flows.

Taylor-Green (TG) flow:

ux (0) = A U0 [cos(kx) sin(ky) cos(kz)]

uy (0) = −A U0 [sin(kx) cos(ky) cos(kz)]

uz (0) = 0

Arnold-Beltrami-Childress (ABC) flow:

ux (0) = U0[A sin(kf z) + C cos(kf y)]

uy (0) = U0[B sin(kf x) + A cos(kf z)]

uz (0) = U0[C sin(kf y) + B cos(kf x)]

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0 20 40 60 80 100

Shifte

d E

nerg

y

Time

Kinetic Energy

Magnetic Energy

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1 10

E(k

)

k

time = 31.2

time = 46.8

time = 62.4

time = 78.0

time = 93.6

time = 109.2

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 20 40 60 80 100

Shifte

d E

nerg

y

Time

Kinetic Energy

Magnetic Energy

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1 10

E(k

)

k

time = 31.2

time = 46.8

time = 62.4

time = 78.0

time = 93.6

time = 109.2

N L dt ρ0 U0 Re Rm Ms MA A = B = C k0

643 2π3 10−5 1 0.1 450 450 0.1 1 1 1

RM, R Ganesh, A Sen, Phys Plasmas, 26, 042121 (2019)



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Recurrence - TG (X) flow

Kinetic Isosurfaces Magnetic Isosurfaces



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NO Recurrence - ABC (7) flow

Kinetic Isosurfaces Magnetic Isosurfaces



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WHY DOES IT RECUR?

Rayleigh quotient [Q(t)] measures the number of effective‘active degrees of freedom’. [A. Thyagaraja, Physics of Fluids, 22, 2093 (1979);

AND A. Thyagaraja, Physics of Fluids, 24, 1973 (1981)]

• Q(t) =

∫V

[(~∇×~u

)2+ 1

2

(~∇×~B

)2]dV∫

V

(|~u|2+ 1

2|~B|2

)dV

=k2|ck |

2

|ck |2where, ~u & ~B are expanded in a Fourier series.

If Q(t) is bounded ⇒Recurrence can happen.

For TG flow, Q(t) isbounded.

For ABC flow Q(t)increases withoutbound.

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100

Q(t

)

Time (t)

ABC Flow

TG Flow

RM, R Ganesh, A Sen, Phys Plasmas 26, 022101 (2019) [Editor’s Pick]



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Summary & Discussions


Ideally very low probability of trapping in phase space in highdimensional systems (e.g. 3D MHD systems).

Recurrence is observed for flows involving few number ofactive degrees of freedom. [Birkhoff, Dynamical Systems, 1927, Chapter 7]

Recurrence can be helpful to make short time forecasts oncetypical initial profiles are experimentally obtained.

Energy exchange between fluid and magnetic variables opensup new cascading paths for energy amongst different scales.

Nonlinear Dispersionless Alfven Waves generated via certainspecific flow and field profiles cause Recurrence.



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Acknowledgement

A Thyagaraja, Culham Labs, UK(for several Physics discussions related to MHD)

Nagavijayalakshmi Vydyanathan, NVIDIA(for teaching GPU computing)

PPPL & AAPPS-DPP(for financial support)

Thank You



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TG Flow & ABC Flow - Kinetic & Magnetic energy isosurfaces

Recurrence - TG (X) & ABC (7) flow

TG

ABC



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Motion of Passive Tracer particles (Cloud-In-Cell Technique & Boris Algorithm) during Recurrence with Taylor-Green flow

Motion of Tracer particles during Recurrence with TG flow



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GPU Performance of G-MHD3D (With NVIDIA)



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GPU Performance of G-MHD3D with Passive Tracers (With NVIDIA)



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Results from DGX (With NVIDIA)



recurrence in three dimensional magnetohydrodynamic plasma · 2020. 7. 16. · 1/14 motivation...

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