Kinetic Eulerian Simulation of Electrostatic Phase Space Vortices … · 2020-07-30 · Eulerian...
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3 rd Asia-Pacific Conference on Plasma Physics, Hefei, China Kinetic Eulerian Simulation of Electrostatic Phase Space Vortices (PSVs) In A Driven-Dissipative Vlasov-Poisson System Pallavi Trivedi †* , Rajaraman Ganesh * † P rincetonP lasmaP hysicsLaboratory, P rinceton, U SA * InstituteF orP lasmaResearch, HBN I, Gandhinagar, Gujarat, India November 6, 2019 [email protected]
Transcript of Kinetic Eulerian Simulation of Electrostatic Phase Space Vortices … · 2020-07-30 · Eulerian...
3rd Asia-Pacific Conference on Plasma Physics, Hefei, China
Kinetic Eulerian Simulation ofElectrostatic Phase Space Vortices
(PSVs) In A Driven-DissipativeVlasov-Poisson System
Pallavi Trivedi†∗,Rajaraman Ganesh∗
†PrincetonPlasmaPhysicsLaboratory, Princeton, USA
∗InstituteForP lasmaResearch,HBNI,Gandhinagar,Gujarat, India
November 6, 2019
Motivation
Collisionless plasmas are ubiquitious in nature
Vlasov equation give accurate description of weakly correlatedcollisionless plasmas and have a wide range of applications:-
from interplanetary environment to laboratory plasmas
to understand kinetic effects of plasmas such as wave particleresonant interactions,
to understand damping effects, instabilities, nonlinear particletrapping, several nonlinear coherent structures, double layers inlaboratory plasmas and more.
Motivation
Energetic particles produced in fusion experiments, solar wind andmagnetospheric plasmas etc can excite various modes and leads tovarious frequency bursts over the spatial and temporal scales.
Associated nonlinear wave-particle interactions can generatesignificantly enhanced levels of energetic particle transport which canhappen both along and across the magnetic field lines. [For example,increased energetic particle transport by Alfven eigenmodes has beencorrelated with a fast frequency oscillation (chirping) with asubmillisecond period that has been observed in many experiments].[Zhang et. al., PRL 109, 025001 (2012)]
Several investigations aim to understand the features of dynamics ofwave-particles interaction such as excitation of electrostatic modes andphase space structures, at ion scales and electron scales in spaceplasmas by analyzing both spacecraft data, solar wind observationsand numerical results from kinetic or phase spacesimulations.[Valentini et.al., Fajans et.al., Berk et.al.]
Eulerian Simulations (PSVs)
In systems governed by kinetic processes, limit of lowcollisionality is not the same as the limit of zero collisionality.
Particle collisions work to restore thermal equilibrium, which caneventually change the features of the kinetic dynamics of aplasma, even in situations where collisionality can be consideredvery weak.
In these conditions,
Kinetic processes works to produce deformations of the particledistribution function away from a Maxwellian
Collisionality tends to restore the Maxwellian configuration.
The evolution of the plasma is, therefore, a result of complexcombination of these two effects.
Inclusion of Collisional Effects
Nearly collisionless regimes are important to a number ofphysical processes, including:-
runaway electrons in magnetically confined fusion plasmas
magnetic reconnection in weakly collisional regimes
low density edge in a tokamak plasma
the solar plasma near sunspots, and non-neutral plasmas etc.
Broadly speaking, two types of collisions:1 Boltzmann collisions where the colliding particles can be treated
as isolated pairs
2 Fokker-Planck (FP) collisions where many weak collisions lead toparticle diffusion in velocity space.
Daniel H. E. Dubin, PHYSICS OF PLASMAS 21, 052108 (2014).
Eulerian algorithms for Vlasov simulations
A simplest approach is to model the unbounded or periodicdirection (eg toroidal direction in Tokamaks or along the B-fieldin Astroplasmas) using a 1D-1V Vlasov-Poisson model where anexternal electric field is used to produce kinetic species.
In the limit of zero correlations and weak collisions, plasmas arewell described in their electrostatic limit by Vlasov-Poisson (VP)system of equations.
Vlasov Equation-1D
∂fj∂t
+−→vj .∂fj∂−→x
+qjmj
(−→E +−→v ×
−→B ).
∂fj∂−→vj
= 0
Along the B-field or in absence of B-field : −→v ×−→B = 0.
In a Cartesian system, further simplifies:∂fj∂t
+ vj∂fj∂x
+qjmjE∂fj∂vj
= 0.
Inclusion of Collisional Effects
∂tf + v∂xf + E∂vf = (∂tf)c = C(f).
Bhatnagar-Gross-Krook (Krook) Operator
Bhatnagar-Gross-Krook (Krook)- C(f)=ν(f − f0).
Approximates by a single relaxation process.
Dissipative operator
Zakharov-Karpman (ZK) Operator
Zakharov-Karpman (ZK) - C(f)=ν ∂∂v
( ∂f∂t
+ vf)
Also known as Lenard-Bernstein collisional operator.
Dissipative operator
Fokker-Planck form which preserves:-
conservation the number of electrons;represent diffusion in velocity space;
P.L. Bhatnagar; E.P. Gross; M. Krook, Physical Review. 94 (3) 511525,(1954).V. E. Zakharov and V. I. Karpman, Sov. Phys. JETP 16, 351 (1963).A. Lenard and I. B. Bernstein, Phys. Rev. 112, 1456 (1958).
Numerical Scheme - VPPM 2.0 Solver
Phase Space and Time Discretization
∆x = Lx/Nx
∆v = 2vmax/Nv
tn = n∆t, n = 0, nstep
∆t :- CFL
∣∣∣∣v∆t
∆x
∣∣∣∣ ≤ 1 CFL
Time-Splitting Scheme
1D Vlasov-Poisson system =⇒ 1D advection equations & Poissonequation, [Time-Stepping method, Cheng & Knorr].
∂fe∂t
+ ve∂fe∂x
+ E∂fe∂ve
= 0,∂E
∂x=∫fedv −
∫fidv
↙ ↘
∂f
∂t+ v
∂f
∂x= 0
∂f
∂t+ E
∂f
∂v= 0
Numerical Scheme - VPPM 2.0 Solver
1D advection eqn. → Piecewise Parabolic Method (PPM),[Collela & Woodward]. PPM method
VPPM - Vlasov Poisson Solver with PPM VPPM
Time stepping method for one time step ∆t:
PPM routine∂f
∂t+ v
∂f
∂x= 0 in x domain, for ∆t/2.
Poisson FFT routine −→ E .
PPM routine∂f
∂t+ E
∂f
∂v= 0 in v-domain, for ∆t.
PPM routine∂f
∂t+ v
∂f
∂x= 0 in x domain, for ∆t/2.
Periodic boundary conditions → both spatial and velocitydomain.
Numerical Scheme - VPPM 2.0 Solver
∂tf + v∂xf + E∂vf = (∂tf)c = C(f).
Numerical splitting scheme for a single time step ∆t
1 ∆t/2 transport step:- ∂tf + v∂xf + E∂vf = 0.
2 ∆t Collision step:- ∂tf = C(f).
3 ∆t/2 transport step:- ∂tf + v∂xf + E∂vf = 0.
Transport Step for ∆t′ = ∆t/2
PPM routine ∂tf + v∂xf = 0 in x domain, for ∆t′/2.
Poisson FFT routine −→ E.
PPM routine ∂tf + E∂vf = 0 in v-domain, for ∆t′.
PPM routine ∂tf + v∂xf = 0 in x domain, for ∆t′/2.
F. Filbet and L. Pareschi, J. Comput. Phys. 179, 1 (2002).
Chirp Driven Phase Space Vortices
A common characteristic of an evolving nonlinear system is thatthe mode frequency also evolves in time. Such behavior, referredto as frequency chirping/sweeping, is normally a relaxationprocess.
It can be found in nonlinear optics, developing turbulent systems,and unsaturated nonlinear wave-wave and/or wave-particleinteraction, in particular, beam driven activities in tokamakplasmas.
Previously, a homogeneous plasma with Maxwellian velocitydistribution is driven with an external drive of time dependentfrequency ω(t) for time interval ∆td → PSVs.
Pallavi Trivedi and R. Ganesh, Phys. of Plasmas 23, 062112 (2016)
Pallavi Trivedi and R. Ganesh, Phys. of Plasmas 24, 032107 (2017)
Pallavi Trivedi and R. Ganesh, Manuscript in communication (2019)
Chirp Driven Phase Space Vortices
A homogeneous plasma with Maxwellian velocity distribution isdriven with an external drive of time dependent frequency ω(t)for time interval ∆td.
Chirp With downward
frequency
ω=α t+ βω2
ω1
ω( t)
0 tTimet1
Chirp Driven Phase Space Vortices
Chirp With downward
frequency
ω=α t+ βω2
ω1
ω( t)
0 tTimet1
f0e = 1√2πexp(−v
2e
2 )
∆td = 250, k = 0.4, E0 = 0.025
ωhigh = 0.8 to ωlow = 0.4
Nx = 512, Nv = 4000
α = −1.6× 10−03, β = 0.8.
−6 −4 −2 0 2 4 6v
−6
−5
−4
−3
−2
−1
0
1
2
log 1
0f̂(v
,t)
t=0,Chirp on
t=50,Chirp on
t=100,Chirp on
t=250,Chirp off
t=500,Chirp off
t=1000,Chirp off
t=2000,Chirp off
Effects of Collisions on Chirp Driven PSVs
What will happen to the plasma response for the externalfrequency chirp in the presence of collisions?
When collisions are turned on after attaining steady state?
When collisions are on from the start of the simulation i.e. fromt = 0?
Weak Collisional effects on chirp driven PSVs -
∆td = 250, ν = 10−5, f(x, v, t = 5000)
Collisions turned on at t = 2000
Collisionless case ν = 0 Collisions turned on at t = 2000
Collisions
On
Drive
On
0 t t1 2
t3
Dri
ve/
Dis
sipat
ion
Krook Operator, ν = 10−5 ZK operator, ν = 10−5
Weak Collisional effects on chirp driven PSVs -
∆td = 250, ν = 10−5, f(x, v, t = 5000)
Collisions turned on at t = 2000.
v-5 0 5
log10f̂e(v,t)
-8
-6
-4
-2
0
ν=0, t=0
ν=0, t=5000
ν=10-5
, t=5000, Krook
ν=10-5
, t=5000, ZK
t0 1000 2000 3000 4000 5000
δn(x
=π/L
,t)/n0
-0.6
-0.4
-0.2
0
0.2
0.4
ν=0
ν=10-5
, Krook
ν=10-5
, ZK
4000 4500 5000
0.08
0.09
0.1
Drive Off
Collision Off
Drive Off
Collision On
t
0 1000 2000 3000 4000 5000
δW
0
2
4
6
8
10
ν=0
ν=10-5
, Krook
ν=10-5
, ZK
2000 3000 4000 50008
8.5
9
Drive Off
Collisions Off
Drive Off, Collisions On
t0 1000 2000 3000 4000 5000
log(S
rel)
-40
-30
-20
-10
0
ν=0
ν=10-5
, Krook
ν=10-5
, ZK
2000 3000 4000 5000-2.2
-2
-1.8
Drive Off
Collisions Off
Drive Off
Collision On
Weak Collisional effects on chirp driven PSVs -
∆td = 250, ν = 10−5, f(x, v, t = 5000)
Collisions are on since t = 0, throughout the simulation B
Collisionless case ν = 0
t0 1000 2000 3000 4000 5000
δn(x
=π/L
,t)/n0
-0.4
-0.2
0
0.2
0.4ν=10
-5, Krook
ν=10-5
, ZK
ν=0
Drive Off
Collisions are on throught the simulation
Krook Operator, ν = 10−5 ZK operator, ν = 10−5
Weak Collisional effects on chirp driven PSVs -
∆td = 250, ν = 10−5, f(x, v, t = 5000)
Collisions are on since t = 0, throughout the simulation. B
t
0 1000 2000 3000 4000 5000
δW
0
5
10
ν=0
ν=10-5
, Krook
ν=10-5
, ZK
Drive
Off
Collisions areon throughoutthe simulation
t
0 1000 2000 3000 4000 5000
δK
0
5
10
ν=0
ν=10-5
, Krook
ν=10-5
, ZK
Drive Off
Collisions are on throughout the simulation
t
0 1000 2000 3000 4000 5000
δP
0
0.5
1
1.5
2ν=10
-5, Krook
ν=10-5
, ZK
ν=0
DriveOff
Collisions are on throughout the simulation
t0 1000 2000 3000 4000 5000
log(S
rel)
-1.5
-1
-0.5
0
0.5
ν=0
ν=10-5
, Krook
ν=10-5
, ZK2000 40000.03
0.04
0.05
Summary
Electrostatic PSVs : On applying a small (linear-like) amplitude,external drive, when chirped downwards, it is shown to coupleeffectively to the plasma and increase both streaming of“untrapped” and “trapped” particle fraction.
To understand dissipative effect of weak collisions on drivenPSVs, two operators have been applied:-
Bhatnagar-Gross-Krook (Krook) : Boltzmann collisions(particle-particle collisions)Zakharov-Karpman (ZK) : Fokker-Planck type (FP) collisions(velocity diffusion)
Using both collisional operators, it is shown that for weakcollisions (eg. 10−5), the giant PSVs smoothen out, yet retainlarge excess density fractions.
However...
The physical results are strongly dependent on the way onechoose to model collisions.
Next step:-nonlinear form of Fokker plank operator.
Inclusion of ion-ion & electron-ion collisions.
Velocity space dependent collision frequency.
Acknowledgement
Prof. A. Sen, IPR, India
Prof. R. Ganesh (PhD Thesis Supervisor), IPR, India
UDAY cluster, IPR, India
Financial Assistance
AAPPS-DPP 2019
PPPL, Princeton, USA
