N.Tronko 1, T.Goerler 2 , A.Bottino 2 , B.D.Scott 2, E.Sonnendrücker 1

Verification of Gyrokinetic codes: Theoretical background & Numerical implementations

NumKin 2016, IRMA, Strasbourg, France

1 NMPP, Max Planck Institute für Plasmaphysik 2 TOK, Max Planck Institute für Plasmaphysik VeriGyro Project Participants

Enabling Research Project on Verification of Gyrokinetic codes

• Germany Max Planck Insitute for Plasma Physics (Garching and Greifswald) • Switzerland SPC-EPFL, Lausanne • Finland Aalto Univeristy • Great Britain University of Warwick • France CEA Cadarache, Université Paris IV, Rennes, Toulouse, Lorraine, Bretagne IRMA, Maison de la Simulation (Saclay) • USA Saint Michel’s College, VT

NumKin 2016 VeriGyro Project @ IPP Garching

Interdisciplinary Project in IPP Max Planck (Garching)

NMPP Division (Numerical Methods for Plasma Physics) • Development and implementation of new algorithms for fusion and astrophysical plasma modeling • Development of libraries (i.e. Selalib) • Verification of existing codes • Participants: N. Tronko, E. Sonnendrücker

Enabling Research Project VeriGyro TOK (Tokamak Theory) • Development of major codes • Plasma modeling and comparison with experimental results (ASDEX Upgrade) • Participants: T. Görler, A. Bottino, B. D. Scott

NumKin 2016 VeriGyro Project: Motivation

• Verification of Global (Electromagnetic) Gyrokinetic codes: Why?

• Most popular tools for magnetised plasmas simulations:

• Significant Development since last 10 years • Electrostatic gyrokinetic implementations : well established [Dimits Phys. Pl. 2000 , Bottino&Sonnendrücker J.Pl.Phys. 2015]

• Global Electromagnetic gyrokinetic implementations:

• Another level of complexity with respect to electrostatic codes • A lot of freedom for approximations (Poisson and Ampère equations) • Different codes use different version of GK equations)

• What was missing in the beginning of VeriGyro (2014) ?

• General hierarchy of the GK equations implemented numerically • Global Electromagnetic Intercode benchmark

NumKin 2016 VeriGyro: Strategy & Accomplishments

• Theory: Establishing Hierarchy of models for existing codes (Task leader Natalia Tronko)

• Systematic derivation from the Variational GK framework [Tronko, Bottino, Sonnendrücker, Phys.Plasmas 2016]

• Verification of approximations consistency and regimes of applicability

• Intercode Benchmark: challenging electromagnetic problems (Task leader Tobias Görler)

• Implicit numerical schemes verification • Microinstability characterization and exploration of required grid • Microturbulence: one of the next steps

Linear intercode benchmark [Görler, Tronko, Hornsby et al Phys.Pl. 2016]

NumKin 2016 VeriGyro: Intercode Benchmark Participating codes

• PIC codes: • ORB5 [Jolliet Comp. Phys. Comm. 2007; Bottino PPCF, 2011; Tronko Phys. Pl., 2016] N. Tronko (IPP Garching) • EUTERPE [Kornilov Phys. Pl. 2004] R. Kleiber (IPP Greifswald)

• Eulerian codes: • GENE [Jenko Phys. Pl.,2000; Görler, J.Comp. Phys. 2011] T. Görler (IPP Garching) • GKW [Peeters Comp. Phys. Comm. 2009; Buchholz Phys. Pl. 2014] W. Hornsby (IPP Garching)

• Semi-Lagrangian: • GYSELA[Grandgirard J. Comput. Phys. 2006; Grandgirard Comp. Phys. Comm. 2016] V. Grandgirard, C. Norcini (CEA Cadarache)

NumKin 2016 VeriGyro: Numerical resources

• Supercomputer Helios (Japan) Hydra (IPP Garching)

• Since Oktober 2017 Marconi Fusion (Italy)

• Using between 512 and 2048 processors (for ORB5)

• CPU time pro simulation: 2h (adiab) towards several days (electromagnetic linear)

NumKin 2016 PART I: Theoretical foundations

• Gyrokinetic theory for numerical implementations: • Systematic derivation from the first principle of dynamics

• Second order Gyrokinetic Maxwell-Vlasov Parent model • Focus on the ORB5 code model

• Continuous descriptions • Gyrokinetic dynamical reduction on single particle phase space • Systematic coupling of reduced particle dynamics with fields dynamics • Conservation laws and code diagnostics (ORB5)

• Discretized description (ORB5) • Finite Element Monte-Carlo Particle-In-Cell Lagrangian • Discrete equations of motion

NumKin 2016 PART II : Intercode Benchmark

• Establishing common test case:

• Adiabatic electrons simulations • Electrostatic (2 species) • Electromagnetic linear Benchmark

• Identifying differences in simulations: in function of the implemented model and numerical scheme

NumKin 2016 GK Orderings

⇢ ln B ✏ • Guiding- center: background quantities | ir | ⇠ B

f c E B • Gyrocenter: fluctuating fields | ?| | | ✏ F ⇠ Bv ⇠ B ⇠ th e ✏ =(k ⇢i) • Gyrokinetics ? Ti • Drift-kinetics

k ⇢i 1 k ⇢i 1 ? ⇠ ? ⌧

• Maximal ordering • Code ordering

✏ ✏ ✏B ✏ B ⇠ ⌧ [Brizard, Hahm Rev.Mod.Phys., 2007]

NumKin 2016 Local Particle coordinates

• Non-canonical local particle coordinates 6D:

Z↵ =(x, v) x,v ,µ,✓ ! k ? ⇢ • Rotating particle basis: b ✓ = b sin ✓ b cos ✓ b ? 1 2 ⇢ = b cos ✓ b sin ✓ b b1 b 2 • Magnetic • Velocity vector bdecompositionb b Momentum: 2 mv 1/2 adiabatic invariant µ = ? v = v b = v = v b +(2µmB) 2B k ? k ? b b b • Goal of the dynamical reduction: build up a change of variables such that µ˙ =0 fast gyromotion is uncoupled

• Relevant dynamics on the reduced (4+1)D phase space: X,v ; µ k NumKin 2016 Two step dynamical reduction

• Two polarization displacements x X + ⇢ (X,µ,✓)+⇢ (X,µ,✓) ⌘ 0 1 • Guiding-center displacement • Gyrocenter displacement 2 mc 2µ mc pz ⇢0 ⇢ ⇢0⇢ ⇢1 = 1(X) A1 (X) ⌘ e mB ⌘ B2 r? mc k r 0 ⇣ ⌘ (✏B) (✏) b⇠ O b ⇠ O

• First order electromagnetic potential e 1 = 1 pzA1 mc k • Canonical gyrocenter momentum: e pz = mv + A1 (X + ⇢0 + ⇢1) k c k

NumKin 2016 Second order Gyrocenter Lagrangian

• Particle Lagrangian on the reduced phase space (X,pz,µ,✓)

e mc GK theory does L = A + p b X˙ + µ ✓˙ H p c z · e not stop here!!! ⇣ ⌘ b p2 • Non-perturbed dynamics (guiding-center) H = z + µB 0 2m

ep • First order dynamics H = e gc ( (X + ⇢ )) z gc (A (X + ⇢ )) 1 J0 1 0 mc J0 1 0

The gyroaveraging operator: with respect to the guiding-center displacement 1 2⇡ gc [ (X + ⇢ (✓))] = d✓ (X + ⇢ (✓)) J0 1 0 2⇡ 1 0 Z0

NumKin 2016 Second order Hamiltonian hierarchy

• Full FLR 2nd order electromagnetic Hamiltonian equivalent to Hahm 1988 electrostatic model 1 e 2 e2 @ Hfull = gc ( (X + ⇢ )) gc (X + ⇢ ) (X + ⇢ ) 2 2m c J0 1 0 2mcJ0 @µ 1 0 1 0 ⇣ ⌘ ✓ ◆ Electromagnetice couplinge between GK Poisson and • Truncated: All 2nd FLR terms Ampère equations

2 FLR e 2 µ 2 1 µ 2 H2 = A1 (X) + A1 (X) + A1 (X) A1 (X) 2mc2 k 2B r? k 2 B k r? k 2 2 mc pz 1(X) A1 (X) 2B2 r? mc k nd • ORB5 2 Order : Second order FLR magnetic part and first order FLR in Electrostatic Polarization part 2 2 ORB5 e 2 1 µ 2 mc 2 H2 = A1 (X) + A1 (X) A1 (X) 1(X) 2mc2 k 2 B k r? k 2B2 |r? | Uncoupled GK Poisson and Ampère equations NumKin 2016 Gyrokinetic field theory Which components we do really need for building up a self- consistent model for a code? [Sugama Phys. Pl. 2000, Brizard PRL 2000] 2 2 E1 B1 L = dV dW F (Z ,t ) L Z[Z ,t ; t], Z˙ [Z ,t ; t]; t + dV | | | ?| 0 0 p 0 0 0 0 8⇡ sp X Z particles⇣ ⌘ Z fields 2⇡ 2 Z =(X,pz,µ,✓); dW = 2 B⇤dpzdµ; B1 = A1 m k ? r? k Goal: Coupling reduced particle dynamics with fields within the common mathematical structure

Getting consistently reduced set of Maxwell-Vlasov equations

F (Z0,t0) Distribution function of species “sp” at arbitrary initial time t0 Lp Gyrocenter Lagrangian: reduced motion of a single particle

NumKin 2016 How to build up a gyrokinetic model consistently?

• All approximations should be performed on the Lagrangian L before deriving the equations of motion

• Approach guarantees energetic consistency of final equations

• Symmetry properties of Lagrangian are automatically transferred to the equations

• Set up polarization effects into the system (hierarchy of models):

• Choice of the second order gyrocenter Hamiltonian defines reduced field- particles dynamics

NumKin 2016 Quasi-neutrality approximation

• Electrostatic contributions from fields and gyrocenter polarization 2 2 2 E1 mc 2 1 ⇢s 2 dV | | + dW dV F 1 = dV 1+ 1 8⇡ 2B2 |r? | 8⇡ 2 |r? | Z Z Z ✓ d ◆ Polarization from the second order particle dynamics: ORB5 model

2 2 ORB5 e 2 1 µ 2 mc 2 H2 = 2 A1 (X)+ A1 (X) A1 (X) 2 1 (X) 2mc k 2 B k r? k 2B |r? |

k T k T mc2 2 B e ⇢2 B e d ⌘ 4⇡ne2 s ⌘ e2B2 Debye length Sound ion Larmor radius

2 2 2 ⇢s 4⇡nmc c 2 = 2 = 2 1 d B vA NumKin 2016 Linearised polarisation approximation

e mc B2 L = dW dV A + p b X˙ + µ✓˙ H F dV ? c z · e 8⇡ sp X Z ⇣⇣ ⌘ 2 ⌘ Z Up to (✏ ) b ⇠ O • Initially: Hamiltonian decomposition H = H0 + H1 + H2

• Dynamical part of distribution function (H0 + H1)F

• Background part of distribution function H2F0 e mc L = dW dV A + p b X˙ + µ✓˙ H H F c z · e 0 1 sp X Z ⇣⇣ ⌘ ⌘ b B2 dW dV H F dV ? 2 0 8⇡ sp X Z Z This approximation leads to linearised field equations NumKin 2016 GK Field equations Phase space volume d⌦ dV dW ⌘ • Polarization equation in a weak form Test function L 1 =0 1 = 1 1 2 mc 2 gc ✏ d⌦ F0 1 = d⌦ b0 (1)(F0 + b✏F1) B2 |r? | J sp sp X Z X Z

L • Ampere’s equation in a weak form A1 =0 Test function A1 k k A1 = A1 b k k 2 1 2 e gc 2 pz gc dV A1 + ✏ d⌦ 0 A1 F0 = d⌦ 0 A1 (F0 + ✏F1) 4⇡ r? k mc2 J k mcJ bk Z sp Z sp Z X X Weak form of the equations of motion is suitable for the finite elements discretisation!!!! NumKin 2016 Vlasov equation

Vlasov equation is reconstructed from the characteristics

L ˙ @ (H0 + H1) B⇤ c =0 X = b (H0 + H1) Z @pz B⇤ eB⇤ ⇥ r k k b B⇤ p˙z = (H0 + H1) Z =(X,pz,µ) B⇤ · r k

d @ dZ @ f(Z [Z ,t ,t];t)= f(Z,t)+ f(Z,t) dt 0 0 @t dt · @Z

Linearized characteristics: only H0 contributions

NumKin 2016 PIC code ORB5 “J.E” diagnostics

PIC code: separation in calculation of fields and particles dynamics:

Particles: characteristics Fields: Grid evaluation

Controlling quality of the simulation: contributions from particles energy and fields energy should be calculated independently.

Calculation of the instability growth rate via two different methods

1 d 1 d = Ek = EF 2 dt 2 dt EF EF

Calculation from the linear d k d F Direct evaluation on the grid at E characteristics using the Noether E dt each time step method for energy conservation dt J.E Diagnostics NumKin 2016 ORB5 diagnostics derivation: main steps • Expression for second order energy density ORB5 = dW dV H (F + ✏ F )+ dW dV H (F + ✏ F ) E 0 0 1 1 0 1 sp sp X Z X Z 2 A1 + dW dV HORB5 F + dV |r? k| 2 0 8⇡ sp X Z Z • Use Equations of motion to simplify 2 pz gc A1 ORB5 ? = dV dW H0 ✏ e 0 A1 (F0 + ✏F1)+ dV r k E m J k 8⇡ sp sp X Z ⇣ ⌘ X Z 2 2 e 2 mc 2 + dV dW 2 A1 + 2 1 F0 k + F 2mc k 2B |r? | ⌘ E E sp X Z ✓ ◆ • Separate kinetic and field parts. Use power Balance equation for simulations diagnostics 1 d 1 d = Ek = EF 2 F dt 2 F dt E E NumKin 2016 Electromagnetic “J.E”

1 gc ˙ 1 gc p˙z = dV dW (F0 + ✏F1) e 0 ( 1gc) X 0 A1 gc 2 F rJ · 0 c J k m 0 E sp Z h ✓ ◆ X d Ek dt (X ,p ) Linearized characteristics |0 z|0

Energy diagnostics does not require knowledge of full particles trajectories

Diagnostics of linear simulations

Detailed diagnostics: identification of instabilities stabilizing and destabilizing mechanisms (Part II) NumKin 2016 PART I: Conclusions

Continuous Dynamics:

• GK dynamical reduction starts with reduced particle Lagrangian • Reduced particle dynamics defines hierarchy of GK Maxwell-Vlasov models

• Common mathematical structure for coupling reduced particle dynamics with those of electromagnetic fields

• Derivation of simulation diagnostics via conservation laws, exactly corresponding to the fields –gyrocenters Lagrangian on the infinite dimensional phase space: key element for models verification

• ORB5 model: inside the hierarchy

• Perspective: include GENE code: hierarchy model & diagnostics

NumKin 2016 PART I: Theoretical foundations

First principle dynamics discretisation:

• Monte-Carlo Lagrangian

• Discrete equations of motion

NumKin 2016 PIC Code Cycle

Start here: Initializing particle initial conditions

NumKin 2016

Integrating particle’s GK characteristics: particles pushing

Fields Particles Charge assignement: calculation of fields value at the Charge & current deposition: particles positions calculation of current and charge on grid: source terms of field equations Fields Particles Particles Fields Integrating GK fields equations : fields solver: finite elements

Particles Fields Discrete Gyrokinetic field theory

• Monte-Carlo Lagrangian density = dZ F (Z,t) L Z, Z˙ + L L sp p F sp X Z ⇣ ⌘ N 1 p = F (Z (t),t) L Z (t)Z˙ (t) + L LMC N sp k sp k k F sp p X kX=1 ⇣ ⌘ Systematic derivation of discretized equations

• Field discretization: finite-element method Ng Cubic splines h (x,t)= µ(t)⇤µ(x) µ=1 X ⇤µ(x)=⇤j(x1)⇤k(x2)⇤l(x3) • Galerkin approximation • Variations on finite-dimensional phase-space

NumKin 2016 Discrete Poisson and Ampere equations

• Discretization of particle’s density using importance sampling: this is why w_k are not simply equal to one

N 2⇡B⇤k(Xk) n = dW f k w (X X (t)) gc N ' m2 k k Z kX=1 @ MC • Discretized quasineutrality equation L =0 @1

N N g mc2 1 p µ d⌦ 2 ⇤⌫ ⇤µ = wk (e 0,k⇤⌫ (Xk)) B r? · r? Np J µ=1 sp Z sp k=1 X X FLR effects X X

@ MC • Discretized Ampère’s equation L =0 @A1 k N N g 1 e2 1 p A1 µ dV ⇤⌫ ⇤µ d⌦ 0⇤⌫ 0⇤µ = wk (e 0,k⇤⌫ (Xk)) k 4⇡ r? · r? mcJ ·J N J µ=1 sp ! sp p X Z X Z X kX=1

NumKin 2016 PART II: Models approximations: open questions

• Deriving discretised reduced model from Lagrangian framework:

• Weak form of equations issued from first principle: implementable in the PIC code

• Long wavelength approximation can be obtained inside the Lagrangian framework

• Open question: all the approximations can be derived from the Lagrangian?

NumKin 2016 PART II: Linear Intercode Benchmark

• Common framework: CYCLONE Base Test Case

• Studies of ITG/KBM transition: • Linear Electromagnetic Beta-scan • Toroidal modes (“n-scan”) with nominal beta (c.a.1%) [Görler,Tronko,Hornsby et al. Phys. Pl. 2016]

• Focus on the ORB5: identification of stabilizing/destabilizing mechanisms with electromagnetic J.E diagnostics

NumKin 2016 CYCLONE Base Case

• Common framework for benchmark: [Dimits, Phys. Pl. 2000] electrostatic simulations, adiabatic electrons • The original discharge DIII-D: H- mode shot #81499 at t=4000 ms; flux tube label r=0.5a q(r)=2.52 (r/a)2 0.16 (r/a)+0.86 • Temperature and density profiles A =(n, T )

a r r0 A/A(r0)=exp A wA tanh Lref wAa h ⇣ ⌘i 2 r r0 Lref /LA = Lref @r ln A(r)=A cosh wAa ⇣ ⌘

Lref = R0

A characteristic width

wA maximal amplitude

NumKin 2016 Linear electromagnetic Beta scan

• Looking at one of the most instable modes n = 19 • Successful comparison of 4 different codes (2xPIC and 2xEulerian) • All codes agree at the ITG/KBM transition Important for • Threshold shifted comparing to flux-tube growth rate experiment

Growth Rate scan

ORB5 code

NumKin 2016 Linear electromagnetic Beta scan

Looking at one of the most instable modes n = 19

Frequencies scan

ORB5 code

NumKin 2016 Toroidal mode scan at nominal beta

GENE/GKW: ORB5: Instability growth rate breaks down at large transition towards values of n>35 because the drift-kinetic approach Trapped Electron is outranged k ✓ ⇢ s =0.54; Modes at n=35 k ⇢ 1 ✓ ⌧

n = 19 NumKin 2016 Toroidal mode scan at nominal beta

GENE/GKW: solving the ORB5: long-wavelength Poisson equations with full approximation solver FLR corrections

n = 19 NumKin 2016 Toroidal mode scan: electrostatic

• GENE/GKW: • ORB5: Instability growth rate breaks down at large transition towards values of n>35 because the drift-kinetic approach Trapped Electron is outranged k ✓ ⇢ s =0.54; Modes at n=35

• ORB5: full FLR Poisson equation solver implementation: J.Dominski PhD thesis (SPC, EPFL)

NumKin 2016 Radial structure of modes: ORB5/GENE

1.2 GENE, global • ORB5 Novelty in Benchmark 1

0.8

• Fine structure in electrostatic θ ⟩ 2

| 0.6 φ

potential: | ⟨ • Mode-Rational Surfaces (MRS): 0.4

electrons can not be adiabatic 0.2 • Costly simulations: demanding high 0 radial resolution 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r/a 1 GENE, global 0.8 ORB5 [Dominski , Phys.Pl. 2015] 0.6 0.4 0.2 • ORB5: MRS are smoothed 0 φ because of the long- -0.2 wavelength solver -0.4 -0.6 implementation -0.8 -1 -3 -2 -1 0 1 2 3

θpoloidal NumKin 2016 Radial structure of modes: electrostatic potential

GENE Global n = 25 ORB5

NumKin 2016 Radial structure of modes: electromagnetic potential

GENE Global n = 25 ORB5

NumKin 2016 ITG/KBM Bifurcation study with ORB 5

• Use J.E diagnostics derived from the energy conservation: detailed balance of stabilizing/destabilizing mechanisms

• Low beta magnetic ITG β= 0.025% • “Gradient-driven” B ⇠ r • Stabilizing magnetic effects

• Increasing magnetic β= 0.25%: e • Driven by pz = p + A1 k k and A1 c ⇠ k Mostly destabilized by magnetic effects • Stabilizing drift effects B ⇠ r NumKin 2016 ITG/KBM Bifurcation study with ORB 5

• Bifurcation between ITG and KBM:

• Mode beating

β= 0.68%

• KBM instability : β= 0.875% • Drift-stabilising • Driven by electromagnetic mechanisms e pz = p + A1 k c k

A1 ⇠ k NumKin 2016 Conclusions and perspectives

• Achievements • Linear electromagnetic benchmark campain: toroidal modes scan for nominal β=1% value • Expensive simulations (especially at higher toroidal numbers): • High radial resolution [Goerler, Tronko, Hornsby et al, PoP 2016] • High number of particles (markers) • Small time step • Building up hierarchy of GK models for existing codes • General second order Maxwell-Vlasov model • Electromagnetic Lagrangian formulation ORB5

• Perspectives [ Tronko, Bottino, Sonnendrücker, PoP 2016] • Detailed comparison between nonlinear electromagnetic simulations between Lagrangian (ORB5) and Eulerian (GENE) Challenging!!!!

NumKin 2016 References

• Numerics • Jolliet, S., Bottino, A., Angelino, P. et al, Comp. Phys. Comm., 177, 409-425, 2007 • Bottino, A., Vernay, T., Bruce, S., PPCF, 53, 124027, 2011 • Kornilov, V., Kleiber, R., Hatzy, R., Phys. Pl., 11, 3196-3202, 2004 • Görler, T., Tronko, N., Hornsby, W. et al, Phys.Pl, 23, 072503, 2016 • Dimits A.M., Bateman, G., Beer, M.A., Phys. Pl. 7, 969-983, 2000 • Dominski J., Bruner, S., Görler, T., Phys. Pl., 22, 062303, 2015 • Jenko, F., Dorland, W., Kotschenreuther,, M. et al. Phys. Pl., 7, 1904-1910, 2000; • Görler, J., Lapillone, X., Brunner, S. et al. Comp. Phys. , 230 7053, 2011 • Buchholz, R., Camenen, Y., Casson, F.J. et al, Phys.Pl., 21, 062304, 2014 • Peeters, A.G., Camenen, Y., Casson, F.J. et al Comp.Phys.Comm., 180, 2650-2672, 2009 • Grandgirard V. Brunetti,M., Bertrand, P. et al J. Comput. Phys., 217, 395-423, 2006; • Grandgirard V. , Abiteboul, J, Bigot, J. et al, Comp. Phys. Comm. , in press, 2016

• Theory • Tronko N., Bottino A., Sonnendrucker E.,Phys.Pl. 23, 082505, 2016 • Bottino A., Sonnendrücker E., 81, 435810501, J.Pl.Phys, 2015 • Brizard A.J., Hahm T.S., Rev. Mod. Phys. 79(2), 421, 2007 • Sugama, H. Phys.Pl., 7(2), 466-480, 2000 • Brizard, A.J. Phys.Rev.Lett., 84(25), 5768, 2000

NumKin 2016 Outline (passer du temps)

• Verification of Global (Electromagnetic) Gyrokinetic codes (VeriGyro): • Electromagnetic: another level of complexity, a lot freedom for approximations(Ampere equation) we notice that different codes uses different version of GK equations) This is why we want to have a common verification being done. • Theoretical line • At the same time: Benchmark the code on challenging electromagnetic problems micro- turbulence and (MHD modes are also part of codes) it is not a stupid Benchmark at all… Why we need theory: each code slightly different theory.

• To my knowledge it was the only GK enabling research project accepted • Most popular tools for plasma turbulence investigation • Extended development over last 10 years • Variety of implemented models

• Building up hierarchy of Electromagnetic Gyrokinetic models implemented into the codes (task leader N.Tronko): • Goal: understand the limitations of existing codes • Systematic derivation from the Variational GK framework • Verification of approximations consistency • Identification of regimes of applicability

• Multi-code Benchmark: implicit numerical schemes verification (task leader T.Goerler) • Establishing a common test-case framework

NumKin 2016 J.E Components

1 gc = dV dW (F0 + ✏F1) 0 ( 1) v + v P + v B 2 rJ k r r EF sp Z X 1 gc µc B dV dW (F0 + ✏F1) 0 A1 µB b + pzb b r ⇥ B 2 rJ k r · eB ⇥ ⇥ B · r EF sp Z ⇤ ✓ ◆ ! X k b b b

p v z b • Drift velocity components k ⌘ m 2 bpz mc P v P b r 2 • Useful for analyzing r ⌘ m eB⇤ ⇥ B ⇣ ⌘ k mechanisms behind b 2 instabilities µB pz mc B v B + b r r ⌘ m m eB⇤ ⇥ B ✓ ⇣ ⌘ ◆ k b NumKin 2016 Convergence studies: number of markers

Dependencies on the number of markers in the system for relative error

Mean value Calculation from the of RelErr characteristics

Variance value of RelErr Direct calculation from finite elements method

NumKin 2016 Number of markers convergence: RelErr

How number of markers affects noise level in the simulation

NumKin 2016 Time step convergence

Separation between energy diagnostics with increasing the time step

NumKin 2016 From general second order theory to ORB5

• Quansineutrality condition • Uncoupled Quasineutrality and Ampère equations

• Reduced particle dynamics simplification via H2

• Long-wavelength approximation • Purely electrostatic polarization

NumKin 2016 1 e 2 e2 @ Hfull = (X + ⇢ ) (X + ⇢ ) (X + ⇢ ) 2 2m c h 1 0 i 2mc @µ 1 0 1 0 ⇣ ⌘ ⌧ e e

NumKin 2016