Verification & Validation: application to the TORPEX basic plasma physics experiment

Verification & Validation: application to the TORPEX basic plasma physics experiment Paolo Ricci F. Avino, A. Bovet, A. Fasoli, I. Furno, S. Jolliet, F. Halpern, J. Loizu, A. Mosetto, F. Riva, C. Theiler, C. Wersal Centre de Recherches en Physique des Plasmas École Polytechnique Fédérale de Lausanne, Switzerland What does Verification & Validation mean? What is TORPEX? And the simulation code we use? What verification methodology did we use? and validation methodology? What have we learned?

Verification & Validation REALITY MEASUREMENT VALIDATION EXPERIMENT COMPUTATION ANALYSIS SIMULATION CODE DISCRETIZATION & CODING MODEL VERIFICATION

The TORPEX device

Key elements of the TORPEX device Plasma gradients Source (EC and UH resonance) Parallel losses Magnetic curvature

TORPEX: an ideal verification & validation testbed - Complete set of diagnostics, full plasma imaging possible - Parameter scan, N number of field line turns Example: N=2

Verification & Validation REALITY MEASUREMENT VALIDATION EXPERIMENT COMPUTATION ANALYSIS SIMULATION CODE DISCRETIZATION & CODING MODEL VERIFICATION

Properties of TORPEX turbulence n fluc ~ n eq L eq ~ L fluc L >> " i Collisional

The model Collisional Plasma Braginskii model ρ i << L, ω << Ω ci, β << 1 Electrostatic Drift-reduced Braginskii equations Convection n T e, Ω (vorticity) V e, V i! "2! = # Magnetic curvature similar equations parallel momentum balance Parallel dynamics Quasi steady state balance between: plasma source, perpendicular transport, and parallel losses Source t +[φ,n]=ĉ(nt e) nĉ(φ) (nv e )+S

Verification & Validation REALITY MEASUREMENT VALIDATION EXPERIMENT COMPUTATION ANALYSIS SIMULATION CODE DISCRETIZATION & CODING MODEL VERIFICATION

Theof GBS code, aturbulence tool to simulate line turbulenc GBS: simulation plasma in open edgefield conditions Developed by steps of increasing complexity Limited SOL Limited SOL ITER-like SOL Drift-reduced Braginskii equations Global, 3D, Flux-driven, Full-n LAPD, UCLA HelCat, UNM J. Loizu et al. 13 / 24 TORPEX, CRPP [Ricci et al PPCF 2012] The role of the sheath in magnetized plasma fluid turbulence Helimak, UTexas

3D and 2D GBS simulations Fully 3D version 2D version (k =0 hypothesis) z r z r

Verification & Validation REALITY MEASUREMENT VALIDATION EXPERIMENT COMPUTATION ANALYSIS SIMULATION CODE DISCRETIZATION & CODING MODEL VERIFICATION

Code verification, the techniques 1) Simple tests 2) Code-to-code comparisons (benchmarking) 3) Discretization error quantification 4) Convergence tests 5) Order-of-accuracy tests NOT RIGOROUS RIGOROUS, requires analytical solution Only verification ensuring convergence and correct numerical implementation

Order-of-accuracy tests, method of manufactured solution A(f) =0 Our model:, f unknown? We solve A n (f n )=0, but n = f n f = Method of manufactured solution: 1) we choose g, then For GBS: h 2 ɛ 10 5 S = A(g) 2) we solve: A n (g n ) S =0 n = g n g n T v,i v,e ω Φ 10 10 10 0 10 1 h = x/ x 0 = y/ y 0 =( t/ t 0 ) 2

Verification & Validation REALITY MEASUREMENT VALIDATION EXPERIMENT COMPUTATION ANALYSIS SIMULATION CODE DISCRETIZATION & CODING MODEL VERIFICATION

Our project, paradigm of turbulence code validation TORPEX? 3D GBS model 2D reduced model What is the agreement of experiment and simulations as a function of N? Is 3D necessary? What can we learn on TORPEX physics from the validation?

The validation methodology [Based on ideas of Terry et al., PoP 2008; Greenwald, PoP 2010] What quantities can we use for validation? The more, the better - Definition & evaluation of the validation observables What are the uncertainties affecting measured and simulation data? - Uncertainty analysis For one observable, within its uncertainties, what is the level of agreement? - Level of agreement for an individual observable How directly can an observable be extracted from simulation and experimental data? How worthy is it, i.e. what should be its weight in a composite metric? - The observable hierarchy How to evaluate the global agreement and how to interpret it - Composite metric

Definition of the validation observables? Common quantities Isat Vfloat I-V Probe model, assumptions Validation observables Probe model, assumptions to be compared - Examples: Isat t, n t, Γ,... - A validation observable should not be a function of the others - Quantities to predict should be included among the observables n Te ϕ V i V e

Evaluation of the validation observables We evaluate 11 observables: Examples n [m -3 ] low N n(r) t T e (r) t I sat (r) t δi sat /I sat k v PDF(I sat )... experiment 3D 2D δi sat /I sat high N low N high N

Experiment Uncertainty analysis x 2 = x 2 fit + x 2 prb + x 2 rep + x 2 fin Simulation I-V Fitting Probe properties, measurement uncertainties Plasma reproducibility Finite statistics y 2 = y 2 num + y 2 inp + y 2 fin Numerics Input parameters - scan in resistivity and boundary conditions Finite statistics

Agreement with respect to an individual observable Distance: Experimental measurements d = Average over all points 1 G G i=1 (x i y i ) 2 x 2 i + y2 i Simulation results Normalization to uncertainties Level of agreement: R = tanh[(d d 0)/λ]+1 2 R 1 0.8 0.6 0.4 DISAGREEMENT AGREEMENT WITHIN UNCERTAINTY d 0 =1.5 λ =0.5 0.2 0 0 1 2 3 d AGREEMENT

Observable hierarchy Not all the observables are equally worthy The hierarchy assesses the assumptions used for their deduction h exp : h sim : # of assumptions to get the observable from experimental data same for simulation results h = h exp + h sim Examples: - n : h exp = 1, h sim t = 0, h = 1 Γ Isat - : h exp = 2, h sim = 1, h = 3

Composite metric Level of agreement R j = tanh[(d j d 0 )/λ]+1 2 Sum over all the observables Hierarchy level H j =1/(h j + 1) Sensitivity Normalization: - χ = 0: perfect agreement - χ = 0.5: agreement within uncertainty - χ = 1: total disagreement 28

The validation results 1 Complete disagreement 0.8 2D simulations 0.6 0.4 3D simulations Agreement within uncertainty 0.2 0 0 5 10 15 N Perfect agreement Why 2D and 3D work equally well at low N and 2D fails at high N? What can we learn on the TORPEX physics?

Flute instabilities - ideal interchange mode k =0 n + T e eqs. Vorticity eq. 2 φ t i = 2B cm i Rn p e y p e t = c B [φ,p e] γ p e = ik y p e0c φ/b γ = γ I γ I = c s 2 L p R Compressibility stabilizes the mode at k v ρ s > 0.3γ I R/c s

Anatomy of a k =0 perturbation N =2 = L v /N L v λ v : longest possible vertical wavelength of a perturbation If k =0 then λ v = = L v N

TORPEX shows k =0 turbulence at low N! k =0 (λ v = L v /N ) Ideal interchange regime L v λ v 8 6 4 2 0 L v λ v = N -2 0 5 10 15 20 N

! For N~1-6, ideal!! k =0! interchange modes dominant! N=2!

Turbulence changes character at N>7! 8 k =0 L v λ v 6 4 2 k =0 WHY? (λ v = L v ) 0 λ v = L v -2 0 5 10 15 20 N

At high N>7, Resistive Interchange Mode turbulence Toroidally symmetric λ v L v Introducing modes k =0 k stabilization, requires high N and η =0 γ 2 = γ 2 I γ 4πV 2 A k2 η c 2 k 2 y, γ I = c s 2 RL p

Why does TORPEX transition from ideal to resistive interchange for large N? N! Resistive interchange requires high N Ideal interchange requires low N: λ v = L v N thus k v = 2πN L v stable: k v ρ s > 0.3Rγ I /c s Threshold: N~10 in TORPEX

Interpretation of the validation results 1 0.8 2D 0.6 0.4 3D 0.2 0 0 5 10 15 N k =0 - Ideal interchange turbulence - 2D model appropriate k =0 - Resistive interchange turbulence - 2D model not appropriate

Where can a verification & validation exercise help? 1. Make sure that the code works correctly Correct GBS implementation, rigorously, discretization error estimate 2. Compare codes 2D and 3D simulations agree with experimental measurements similarly at low N. Global 3D simulations are needed to describe the plasma dynamics at high N. 3. Let the physics emerge Two turbulent regimes: ideal interchange mode at low N and non-flute modes at high N. Parameter scans have a crucial role 4. Assess the predictive capabilities of a code 3D simulations predict (within uncertainty) profiles of n but not of I sat 38

What comes next? - Validation at each code refinement - Considering more observables - Involving more codes Limited SOL LAPD, UCLA HelCat, UNM TORPEX, CRPP Helimak, UTexas ITER-like SOL

What comes next? Validation on a recently achieved SOL-like configuration in TORPEX Limited SOL LAPD, UCLA HelCat, UNM TORPEX, CRPP Helimak, UTexas ITER-like SOL

Where can a verification & validation exercise help? 1. Make sure that the code works correctly Rigorously, with discretization error estimate 2. Compare codes 2D and 3D simulations agree with experimental measurements similarly at low N. Global 3D simulations are needed to describe the plasma dynamics at high N. 3. Let the physics emerge Two turbulent regimes: ideal interchange mode at low N and non-flute modes at high N. Parameter scans have a crucial role 4. Assess the predictive capabilities of a code 3D simulations predict (within uncertainty) profiles of n but not of I sat 41

Future work Missing ingredients for a complete description Use of more diagnostics: Mach probes, Triple probes or Bdot probes to compare other of plasma dynamics in TORPEX: interesting observables. Better source modeling Physics of neutrals Better boundary conditions 42

V&V A validation project requires a four step procedure: (i) Model qualification (ii) Code verification (iii) Definition and classification of observables (iv) Quantification of agreement 43