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
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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Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
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]
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
3/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
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 .
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
4/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
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
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
5/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
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!
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
6/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
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!
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
7/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
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
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
8/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
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.
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
9/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
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)
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
10/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
Recurrence - TG (X) flow
Kinetic Isosurfaces Magnetic Isosurfaces
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
11/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
NO Recurrence - ABC (7) flow
Kinetic Isosurfaces Magnetic Isosurfaces
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
12/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
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]
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
13/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
Summary & Discussions
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.
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
14/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
Summary & Discussions
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
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
14/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
Summary & Discussions
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
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
15/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
TG Flow & ABC Flow - Kinetic & Magnetic energy isosurfaces
Recurrence - TG (X) & ABC (7) flow
TG
ABC
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
16/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
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
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
17/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
GPU Performance of G-MHD3D (With NVIDIA)
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
18/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
GPU Performance of G-MHD3D with Passive Tracers (With NVIDIA)
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
19/14
Motivation G-MHD3D: DNS GPU code Nonlinear Alfven wave Recurrence Summary Extra
Results from DGX (With NVIDIA)
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