Water-bag reduced gyrokinetic model for the study of the spatial structure of ITG instability linear modes PhD work directed by N. Besse with invaluable input from P. Bertrand, E. Gravier, P. Morel,R. Klein D. Coulette - Institut Jean Lamour CNRS UMR 7198 March 2010
Outline 1 Background 2 Multi-Water-Bag : a class of reduced models 3 From physical model to numerical problem 4 Validation and results 5 conlusions, ongoing work and future perspectives
Background The big picture Turbulent transport in magnetic confinement devices steady state fusion energy production requires high energy confinement time. plasma turbulence can be thought responsible for enhanced anomalous transport need to properly characterize physical mechanism, triggering conditions and space-time structure of turbulent flows. Ion Temperature Gradient Instability ( ITG ) Features in brief among the main sources of turbulent transport driven by electrostatic v E B drift strongly anisotropic propagation k k low frequency ω 10 3 Ω CI
Background Instability threshold Instability threshold : why ITG? Triggering conditions triggered and fed by radial temperature gradient κ T = d r ln(t ) attenuated by radial density gradient κ n = d r ln(n) threshold determined by η i = d ln(t i) d ln(n i ) = κ T κ n Typical critical values Local slab fluid model : η crit = 2 Local MWB model η crit = 5 10 Density Temperature
Background Instability threshold Typical gradient profiles Normalized gradients κ Ratio η i
Background Configuration and dynamical model Plasma model cylindrical plasma column radial domain 0.1to10 20ρ s constant axial B field µ = 0 limit. adiabatic electrons. Drift kinetic equation for ions tf + v E f + v zf + q i M i E v f = 0 Quasi-neutrality with polarization drift «ni Z i n i + Z i φ = n e BΩ CI Computational cost is still high 3D space, 1D velocity,1d time, integro-differential system Are further reductions achievable?
Multi-Water-Bag : a class of reduced models Multi-Water-Bag reduction Multi-Water-Bag reduction v dependency of distribution function is approached by a sum of constant height Heaviside functions. f (r, v, t) = X A j `H `v v j (r, t) H `v v + j (r, t) j [1;N] Provided the contours stay distinct, this is an exact weak solution of the Vlasov equation.
Multi-Water-Bag : a class of reduced models MWB parameters How to set MWB parameters? Moment equivalence : constraints Minimize differences between MWB moments and reference ones. the weights A j must be positive ( to stay on the cautious side) bags contours shouldn t cross on the radial domain k th order MWB moments M MWB k = X j A j v + j k+1 v k+1 j k + 1 A non-linear functionnal optimization problem MWB moments are linear in A j and polynomials in V j : tricky non-linear optimization
Multi-Water-Bag : a class of reduced models MWB parameters Local equivalence Exact equivalence in r 0 choose N initial bags values : V j (r 0) > 0. compute even Maxwellian moments up to 2N 2 th order A j are solutions of a linear Vandermonde system. solve system to get Aj. Analytical form is known : fast and stable algorithm. How to insure A j > 0? From the Vandermonde system and Maxwellian moments one gets the condition : ( V 1 < 3T V N > p (2N 1)T
Multi-Water-Bag : a class of reduced models MWB parameters Extending equivalence while preserving η profile Radial differentiation method Vandermonde system differentiated in r 0 new Vandermonde system obtained with radial derivatives as unknowns. Compute bags derivatives extension by taylor expansion in r 0 ± h iterate to domain bounds 5 bags example
Multi-Water-Bag : a class of reduced models MWB parameters Radial differentiation method : preserved gradients Realtive errors on gradients Relative errors on moments
From physical model to numerical problem A generalized Sturm-Liouville problem Sturm-Liouville Problem 1st order linearization. Fourier transform in θ, z : φ(r, θ, z, t) = φ(r)e (mθ+nk z ωt) Radial enveloppe of potential solution of : whence and L = Lφ = 0 1 «d r (rn 0d r ) + m2 + 1 + F (ω) rn 0 r 2 Z i T e F (ω) = X j A j n 0 nk m r dr V j ω nk V j = X j Γ j ω α j with Dirichlet boundary conditions φ(r min ) = φ(r max) = 0
From physical model to numerical problem A generalized Sturm-Liouville problem Schrödinger form Setting ψ = rn 0φ one gets :» d 2 dr + Q(ω) ψ = 0 2 8 Q(ω) = B(r) + F (ω) >< with B(r) = ` m 2 + 1 + 1 r Z i T e (r) 4 (dr ln(rn0))2 + 1 d 2 2 r ln(rn 0) >: F (ω) = P Γ j ω α j j with ψ(r min ) = ψ(r max) = 0 Main features non-hermitian operator non linear spectral problem presence of poles related to equilibrium contours.
From physical model to numerical problem A generalized Sturm-Liouville problem Radial discretization scheme Discretization Uniform sampling of radial domain in N r samples of width h : i = 1... N r : r i = r min + (i 1)h Differential operators sampling 2nd order finite difference scheme d 2 X dr (r i) = X i+1 2X i + X i 1 2 2h 2
From physical model to numerical problem A generalized Sturm-Liouville problem Two resolution strategies Dispersion relation approach solve for the potential unique ODE, non linear in ω seek polynomial matrix null space determinnant yields dispersion relation find roots in complex plane get eigenmode by SVD Full eigenproblem approach solve for bags perturbations coupled ODE system, linear in ω full sprectrum and eigenspace is obtained.
Validation and results Validation - non ITG Numerical validation String modes ad-hoc form of Schrödinger potential: «2 ωπ Q(ω) = r max r min Purely imaginary eigenvalues Sine eigenvectors Similar numerical problem Collisionnal drif waves Different physical phenomenon Different poles and coefficients for F (ω) Results in adequation with previoulsy obtained ones.
Validation and results ITG in cylindrical geometry : results instability Growth rate map (m, n) Large domain (20ρ s)- wide unstable zone (6ρ s) 20 18 "solution_set.csv" using 1:2:4 4.50E-03 4.00E-03 m 16 14 12 10 8 6 4 3.50E-03 3.00E-03 2.50E-03 2.00E-03 1.50E-03 1.00E-03 (m, n) scan [1 20] [1, 10] Most unstable : (m, n) = (11, 7) Rate γ : 4.371 10 3 Ω CI 2 5.00E-04 0 1 2 3 4 5 6 7 8 9 10 n 0.00E+00
Validation and results ITG in cylindrical geometry : results Growth rate γ Frequency ω R 7-6 6-6,5 Taux de croissance 5 4 3 2 MWB Fluid Partie réelle -7-7,5-8 MWB Fluid 1-8,5 0-9 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 7 m 0 m 6-2 Taux de croissance 5 4 3 2 MWB Fluid Partie réelle -4-6 -8 MWB Fluid 1-10 0-12 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 n n
Validation and results ITG in cylindrical geometry : results Full spectrum Full spectrum - color function of m
Validation and results ITG in cylindrical geometry : results Full spectrum - ITG zone Color of points is function of m
Validation and results ITG in cylindrical geometry : results Full spectrum - ITG zone Color is function of n
Validation and results ITG in cylindrical geometry : results Radial enveloppe structure : modulus Outward shifting and narrowing when m grows
Validation and results ITG in cylindrical geometry : results Radial enveloppe structure : phase Flat phase profile in instability region : k r is small
Validation and results ITG in cylindrical geometry : results Mode (3, 7) : influence of phase? Let s check Re(phase factor) Real part k r compensated by modulus decay
Validation and results A few applications Localization of unstable zone Unstable zone Test case parameters small radial domain r max = 10ρ s narrow unstability zone ρ s Mode selection r 0(ρ s ) 2.25 4.5 6.75 n 4 4 4 m 4 10 14 ω R (10 3 ω CI ) -3.500-3.538-3.495 γ(10 3 ω CI ) 1.645 1.651 1.653 Enveloppe
Validation and results A few applications Comparison with local kinetic code : KINEZERO m 25 20 15 10 5 0 "solution_set.csv" using 1:2:4 1 2 3 4 5 6 7 8 n 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 ( C. Bourdelle - CEA - PhD Thesis 2000) MWB model - linear method 8,0E-003 8,0E-003 Most unstable mode Taux de croissance 7,0E-003 6,0E-003 5,0E-003 4,0E-003 3,0E-003 2,0E-003 1,0E-003 0,0E+000 0 5 10 15 20 25 30 m Taux de croissance 7,0E-003 6,0E-003 5,0E-003 n=1 4,0E-003 n=2 n=3 n=4 3,0E-003 2,0E-003 1,0E-003 0,0E+000 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7 n m=8 m=9 m=10 m=11 Linear MWB model : mode (m, n) = (10, 4) γ MWB = 6, 845 10 3 Ω CI KINEZERO : mode (m, n) = (11, 3) γ KINE = 6, 259 10 3 Ω CI From V.Grangirard et al, JCP (2006) Local kinetic model - KINEZERO MWB γ map
conlusions, ongoing work and future perspectives Conlusions, Ongoing work and future perspectives Conclusions typical kinetic growth rates can be obtained with a low order MWB distribution full eigenstructure can be obtained in decent time ( < 1h) Future work directions in cylindrical geometry compare linear results with N. Besse s non-linear SL code ones more parametric studies, stable modes, gyroaverage corrections. new non-linear code using discontinuous-galerkin method. Ongoing work on Toroidal geometry case coupled set of PDE instead of ODE poloidal coupling concerns with problem size...
conlusions, ongoing work and future perspectives Thanks for your attention