Numerical study of SOL turbulence in tokamaks using a compact finite difference scheme

High order compact finite difference scheme forsimulating interchange turbulence in the SOL

William Agnelo GraciasUniversite de Lorraine

Supervisor: F. SchwanderM2P2 - Centre National de la Recherche Scientifique

18th September 2013Bordeaux

W. A. Gracias Master Thesis Work

Outline

1. Motivations

2. Framework of study

3. Interchange instability

4. Transport model

5. Numerical model

6. Some results

7. Conclusions


Motivation...simply, why bother at all?

1. The global energy crisis Nuclear fusion2. Critical: confinement turbulence3. Particularly SOL turbulence wall fluxes target erosion4. SOL width information is very important for quantifying power

flows, target designs, etc. - ITER & Next-step devices (DEMO)

This master thesis attempts to:

I theoretically understand the interchange turbulence

I convert a FD code to Compact FD code to model interchangeturbulence

I stabilise the code for desired physical parameters

I compare the results of the code with those of other existing codes

I improve numeric scheme where possible and make code efficient


The framework of this studyAssumptions and simplifications

2D SOL turbulence study by - S. Benkadda et al, Contrib. Plasma Phys. 34 (1994),and Y. Sarazin et al, Journal of Nuclear Materials 313-316 (2003)

I We consider a simplified slab geometry for the SOLI The destabilising drive in the system will be a particle flux sourceI Fluid model - i.e. all plasma species are assumed to be in TD eqlbm

(Maxwellian distribution)

I Drift velocity ordering assumed -vEB vdia >> vpol

I 2D model to describe turbulent transport inthe poloidal plane (flute hypothesis)

I 2 symmetric toroidal limiters considered tobound the SOL at

I Sheath assumptions: Bohm sheath criterion,cold ions & adiabatic electrons; constantelectron temperature

I Electroneutrality of plasma i.e. ne = ni = n

Image: Yanick Sarazin, Thesis (1997)


Interchange instability in tokamaks

Thought to be due to electrostatic interchange turbulence produced inthe near SOL region

I Radial motion due to electric drift(B p charge separation E Bdrift), damping via parallel losses on openfield lines

I Local relaxations in the edge pressure profile bursty ejection excess particles and heatinto SOL

I role of gravity (g) curvatureImages: Seidl & Krlin (2009); Yanick Sarazin, Thesis (1997)


Main equations for SOL interchange

We start with the electron conservation equation:

tn + nv = S

and the current conservation equation:

J = 0

(where J = env = nev + nev). After some work....

tn D2n + 1B

[, n] +1

e Je = S (1)

w

t+

1

B[,w ] =

(1 + )TeB

miR0[ln n,R] +

B

n0nmi J (2)

where

w = v = 2

B(3)

All the above was done by Benkadda et al (1994) as well as Sarazin et al(2003), and verified by us too.


Normalisation and simplification

Normalisation used:

I the spatial dimensions (X radial, and Y poloidal) have beennormalised to the dimensions of the simulation box (L = 128 s)

I The rest of the normalisation is the same as that of Sarazin et al(2003):

I Viscosity & diffusion coefficients normalised by DBohm = scs ; vorticityand time by cyclotronic frequency c

I parallel current density normalised by saturation current Jsat = en0cs

Make the problem tractable: 3D 2D:

...FL = 12L+LL

...dz

Figure : Slab geometry - 3D to 2D


Final set of equations

Putting g = s(1 + )sinc()/R0, the normalised set of equations used in thenumerical model are:

t n (sL

)2D2n +

(sL

)2[, n] + ne() = S

w

t+(sL

)2[,w ] =

(sL

) g2y (ln n) +

(sL

)24 + n

(1 e()

)w =

(sL

)2 2B

The equations are of the type:dg

dt= F

and so the variation modelled during dt is

dg = F dt

In general, value at next time step is

gi+1 = gi + dg

Ensuring conservation: tn2 = 0, tw2 = 0 and v = 0W. A. Gracias Master Thesis Work

The compact finite difference (CFD) approach

Essentially, if 1st derivative is

f(x) = f

(1)i '

fi+1 fi12x

f (3)ix2

3! O(x4)

and 2nd derivative is

f

(x) = f(2)i '

fi+1 2fi + fi1x2

f (4)i2x2

4! O(x4)

then, to get the CFD-analogous of 1st derivative, we use 3-point stencil:

f(3)i =

(f

(1)i

)(2)' f

(1)i+1 2f (1)i + f (1)i1

2x2+ O(x4)

And similarly f(4)i =

(f

(2)i

)(2). So implicit CFD estimate of the 1st derivative

f(1)i1 + 4f

(1)i + f

(1)i+1

6' fi+1 fi1

2x+ O(x4)

f(2)i1 + 10f

(2)i + f

(2)i+1

12' fi+1 2fi + fi1

x2+ O(x4)

CFD: accuracy, flexibility, versatility


Implementing CFD in TOKAM2D - 1some early results

Firstly, normalised the equations implemented by TOKAM2D. Result of that:

Figure : (a)64 mesh points (b) 128 mesh points (c) 256 mesh points

Figure : (a)128 mesh points (b) 256 mesh points (c) 512 mesh points



Single-point density perturbations and PDF shows intermittency -reminiscent of blobs

Monitoring of fluctuating quantities: temporal evolution of turbulence; finallyreaching steady state



Energy spectra of fluctuating quantities in radial and poloidal directions(resp.) = energy spectrum of turbulence:I identifying the dominant mode

I study energy cascades

Figure : (a) Poloidal direction (b) Radial direction

Remark: non-anisotropy of turbulence in radial and poloidal directions


Comparing CFD-4 with other codes - 1... w.r.t. FD-2 and spectral code

Energy evolution (vorticity):

Figure : (LEFT) 256 mesh points (RIGHT) 512 mesh points

Stages of instability developmentI initial stage: CFD-4 version good concurrence with the spectral code; FD-2

version significant difference in terms of magnitudes computed; trend ofdevelopment is however close

I intermediate stage: CFD-4 higher energy (due to less accuracy of thescheme) but continues to resemble the spectral code w.r.t trend; FD-2 nolonger concurrent with spectral trend

I highly turbulent stage: loss in trend concurrence as higher wave numberperturbations are produced


Comparing CFD-4 with other codes - 2What about the energy spectrum?

Energy spectrum (poloidal direction)


I CFD-4 closer concurrence with spectral code (w.r.t.magnitude and trend);not changed much with discretisation

I deviation for higher wave numbers by both CFD-4 and FD-2I the dominant mode indicated by each version (CFD-4 & FD-2) is shifted by an

increase in discretisation


Turbulence cascadesome broad remarks w.r.t 2D turbulence

Energy transfer to higher wave numbers


I Prima facie, -3 power law of cascade is not so clearly obviousI Nonetheless, from dominant mode onward, upto to certain wavenumber, the -3

power law can be fitted with some effortI For higher discretisation the cascade is not improved greatly. However, for a

more inertial regime of parameters where the instability develops slowly (lowerdiffusion and viscosity coeffs.), this cascade is better observable [next slide]


Code sensitivity to Diffusion, viscosity coefficientssome early results

Reducing the diffusion and viscosity coefficient inertial regime ofinstability development

Figure : (a) time evolution, (b) spectra


Instability - linear growth rate...sensitivity to characteristic density gradient length

Linear growth rate of instability for different Ln

2 = [

k2 + ( + D) s

Lnk2][ || Re()

2

]1/2

I For numeric value of Ln 104 recovered from code, the maximum modenumber was ky = 8

I From spectrum, we see that energy is injected into the instability by modenumber between ky = 6 9, depending on the regime


Scope to improve CFD-4... demonstrated by Fourier analysis of schemes

Fourier analysis of schemes

I Fourier transform of derivative expression using each respectivescheme

I Modified wave number generated by each scheme versus the truewavenumber

CFD v/s Spectral: error should decrease as we go higher in order of thescheme used


Conclusions

I CFD scheme to 4th order truncation implemented in a codeformerly based on FD(2nd order) with a temporal 4th orderpredictor-corrector scheme (RK-4)

I Advcection terms specially treated - Arakawas scheme (Arakawa,J. Computation Phy., 1966) to avoid numeric instability anddoodling due to its excellent conservation properties.

I Better accuracy of results and turbulence structure details observed

I Relatively cheap - source software used, except for the PARADISOsparse system solver (& FFTW solver)

I Code has been modularised to a large extent


Way forward

I Increase accuracy of scheme to 6th (O) and compare with spectralcode

I Note: advection term based on Arakawas scheme will have to bedeveloped for this order

I Inversion of Laplacian in Poissons equation:

w = 2 = = (2)1 w . We had to use 9-point stencil to get4th(O) approximation.

I Code optimisation to increase computational efficiency of certaincalculations and parallelisation

I Implement code for more complicated geometries


Thank you for your attention!


bibliography

1. C Hirsh, Numerical Computation of Internal and External Flows: The Fundamentalsof Computational Fluid Dynamics (2nd Edition), Butterworth-HeinemannPublications, 2007

2. Joel H. Ferziger and Milovan Peric, Computational Methods for Fluid Dynamics (3rdEdition), Springer-Verlag Berlin and Heidelberg GmbH & Co. K, 2002

3. Sanjiva K. Lele, Compact Finite Difference Schemes with Spectral-like resolution,Journal of Computational Physics 103, 16-42, 1992

4. W. F. Spotz and G. F. Carey, High-order Compact Finite Difference Methods, ThirdInternational Conference on Spectral and High-order methods, Houston Journal ofMathematics, 1996

5. Yanick Sarazin, Etude de la Turbulence de Bord dans les Plasmas de Tokamaks,Doctoral Thesis, Universite Joseph Fourier - Grenoble I, 1997

6. Y. Sarazin et al, Theoretical understanding of turbulent transport in the SOL, Journalof Nuclear Materials 313-316 (2003) 796-803, 2003

7. Y. Sarazin et al, Transport due to front propagation in tokamaks, Physics of PlasmasVol 7 No 4, 2000

8. Xavier Garbet, Introduction to turbulent transport in fusion plasmas, C. R. Physique7 (2006) 573-583


Annex 1: Simulation parameters

I Simulation box dimension L = 128 sI Particle diffusivity D/DBohm = 4 103I Vorticity viscosity /DBohm = 4 103I Gravity coefficient g = 3 104I Sheath conductivity = s/(2R0q) = 2 104I Normalised magnetic field B = B/B0 = 1

I Normalised density field n = n/n0 = 1


Annex 2: Arakawas scheme


Annex 3: CFD vs spectralFourier analysis

Fourier analysis of growth rateCFD v/s spectral: error should decrease as we go higher in order

Figure : (a) Analytical (b) CFD-4 (c) FD-2


Annex 4: Runtime for CFD-4 & other codes

CFD-4 is quite competitive for a specific range of discretisation.At higher orders, this range would widen.


Annex 5: Sensitivity of code to parameters

Figure : Evolution: (a) sigma change, (b) g-term change

Figure : Spectra: (a) sigma change, (b) g-term change


Numerical study of SOL turbulence in tokamaks using a compact finite difference scheme

Documents

Transcript of Numerical study of SOL turbulence in tokamaks using a compact finite difference scheme