knosos: a fast orbit-averaged neoclassical code for arbitrary...

KNOSOS: a fast orbit-averaged neoclassicalcode for arbitrary stellarator geometry

Jose Luis Velasco

Laboratorio Nacional de Fusion, CIEMAT

Developed in collaboration with Ivan Calvo, Felix Parra (U. Oxford,CCFE) Jose Manuel Garcıa-Regana and J Arturo Alonso

Kinetic Theory Working Group Meeting 2018, Madrid

Laboratorio Nacional de Fusión

Aknowledgements: A. Mollen, C. Nuehrenberg (IPP),S. Satake, S. Morita, XL Huang (NIFS)...

Motivation and goal

In stellarators, neoclassical transport:

Tends to dominate radial particle and energy fluxes in the coreof reactor-relevant plasmas [Dinklage, NF (2013)].

Explains impurity accumulation ob-served in many stellarator plasmas[Burhenn, NF (2009)].

Energy transport simulations Equations Algorithms First results Ongoing work 1/26

Motivation and goal

Goal of this work: write a neoclassical code (i.e., a code thatsolves the Drift Kinetic Equation together with the quasineutralityequation) that is, at the same time, accurate and fast:

Accurate so that:

it gets the simulation quantitatively closer to the experiment;

Fast so that:

it can be included in an optimization loop;it allows to perform comprehensive parameter scans;

i.e., generally speaking, so that it improves confidence in ourpredictions and that these predictive capabilities can be fullyexploited.


Outline

Energy transport simulations (short overview).

Equations.

Algorithms.

First simulations for arbitrary geometry.

Examples of other applications.


Stellarator-specific regimes, εeff

At low collisionality, stellarator neoclassical transport is verydifferent from (and larger than in) tokamaks [Helander, PPCF (2012)].

Several strategies for characterizing magnetic configurations w.r.t.neoclassical energy transport:

1 Calculation of the level of 1/ν transport.

One scalar, effective ripple, gives level of 1/ν transport.

Fast calculation of εeff provided by NEO [Nemov, PoP (1999)].

Fails to explain configuration dependence of the experimentalτE stellarator scaling [Yamada, NF (2005)].


Neoclassical τE [Pedersen, NF (2015)]

2 Calculation of τE solving the energybalance for every species b:

3

2

〈∂nbTb〉∂t

+1

V ′∂

∂rV ′〈Qb〉 = 〈Pb〉 .

Qb calculated with DKES [Hirshman, PoF (1986)]1.

τE ∼∫dV

∑nbTb∫

dV∑Pb

, gives level of transport for given configuration and

sources [Turkin, PoP (2011), Geiger, PPCF (2015)].

Monoenergetic approach by DKES (and others [Beidler, NF (2011)])saves some time by calculating a database of transport coefficientsinstead of solving the DKE multiple times.

Still, for each new magnetic configuration, creating databasetakes time (∼ 1− 2 days [Beidler, W7-X Workshop (2016)]).See also [Landreman, APS (2017)].

It oversimplifies the DKE (next slides).1

The net energy source is provided by a suite of codes, and there is an ad-hoc anomalous model at the edge.


Improved calculations of neoclassical energy flux

3 Accurate calculation of Qb for fixed kinetic profiles.

Recently, there have been refinements of some aspects of neo-classical theory that have lead to more complete neoclassicalcodes (EUTERPE, SFINCS, FORTEC-3D...):

electric field tangent to magnetic surface (aka ϕ1);magnetic drift tangent to magnetic surface;radially global effects;complete collision operator.

Calculations with all the ingredients can get extremely time-consuming and memory-demanding, specially at low collisionalities.

⇒ Comprehensive parameter scans may become impossible.


Specific goal

We have outlined three strategies for characterizing the energytransport of a magnetic configuration. Our goal is to create a codethat can contribute to the three strategies by:

1 Giving figures of merit for neoclassical transport of collisionalityregimes other than the 1/ν (i.e.

√ν and sb-p).

2 Employing a different ”database of coefficients” that allows3

to be very fast at low collisionalities and, at the same time,to retain all the necessary ingredients of neoclassical theory.

KNOSOS solves the bounce-averaged drift kinetic and quasineutralityequations discussed in [Calvo, PPCF (2017)] for arbitrary geometry.


Coordinates on phase space

We are interested in calculating the ion2 distribution functionF (ψ(x), α(x), l(x), E , µ, σ), where:

ψ is the radial coordinate.We choose ψ = ψt, where 2πψt is the toroidal magnetic flux.

α is an angle that labels magnetic field lines on the surface.We choose α ≡ θ − ιζ, where θ and ζ are Boozer angles and ιis the rotational transform.

l is the arc length along the magnetic field line.

B = ∇ψ ×∇α ,b = B/B = B/|B| .

E = v2 + Ze/mϕ is the total energy per mass unit, v = |v|,and ϕ(ψ, α, l) is the electrostatic potential.

µ is the magnetic moment.

σ = v‖/v is the sign of the parallel velocity, v‖ = v · b.2

We will take a mass ratio expansion√me/m � 1; all quantities will refer to ions and Ze, m, etc, will

have the usual meaning.


Drift kinetic equation

In these coordinates, the equation for the ion distribution func-tion F can be written as

(v‖b + vd) · ∇F = C[F, F ] ,

where vd = vM +vE is the sum of magnetic and E×B drifts,and C[F, F ] is the (ion-ion) collision operator.

We know that the motion of particles along the magnetic fieldis much faster than the motion perpendicular to it.

At low collisionalities (ν∗ ≡ νiiR0/vT � ε3/2), the parallelstreaming term also dominates over the collision term for bothpassing and trapped trajectories.

In an expansion in the normalized Larmor radius, the first termis:

v‖∂lF = 0 ,

and, to next order, F is determined by integrals over l of thedrift-kinetic equation.


Bounce-averaged drift kinetic equation

We have:

Passing particles :

∫ 2π

0dα

∫ lmax(ψ,α)

0

1

|v‖|C[F, F ]dl = 0 .

Trapped particles : −∂ψJ∂αF+∂αJ∂ψF =∑σ

Ze

m

∫ lb2

lb1

1

|v‖|C[F, F ]dl .

Everything has become expressed in terms ofderivatives of the second adiabatic invariant:

J(ψ, α, E , µ) ≡ 2

∫ lb2

lb1

|v‖|dl .

At the bounce points, lb1 and lb2 , v‖ = 0.


Bounce-averaged drift kinetic equation (II)

−∂ψJ∂αF + ∂αJ∂ψF =∑σ

Ze

m

∫ lb2

lb1

1

|v‖|C[F, F ]dl .

These derivatives of the second adiabatic invariant are connectedto the drifts perpendicular to the magnetic field line:

∂αJ =2Ze

m

∫ lb2

lb1

1

|v‖|vd·∇ψ dl , ∂ψJ = −2Ze

m

∫ lb2

lb1

1

|v‖|vd·∇α dl .

∂αJ contains the radial magnetic and E × B drift caused bythe inhomogeneity of the magnetic field strength and of theelectrostatic potential on the flux surface respectively.

Can model 1/ν, global effects and transport caused by ϕ1.

∂ψJ contains the precession tangential to the flux surface causedby the radial variation of the magnetic field strength and of theelectrostatic potential (i.e. Er) respectively.

Eqs. contain the√ν and superbanana-plateau regimes.


Local bounce-averaged drift kinetic equation

We would like to solve a drift-kinetic equation that

is linear and radially local and,

rigorously includes the tangential magnetic drift.

It can be proven that [Calvo, PPCF (2017)] this can be done rigorouslyif the magnetic configuration is optimized enough. Then3

|∂αJ∂ψg| � |∂ψJ∂αg| ,where g(ψ, α, v, λ) is the dominant piece of the non-adiabaticcomponent of the deviation of the ion distribution function from aMaxwellian FM :

g = F − FM [1− (Zeϕ1/T )] ,

and we have changed coordinates to:

v = 2√E − Zeϕ0/m) and λ = µ/(E − Zeϕ0/m) ,

and used ϕ(ψ, α, l) = ϕ0(ψ) + ϕ1(ψ, α, l) with |ϕ1| � |ϕ0| .3

This also happens in large aspect-ratio stellarator for large values of the radial electric field.


Local bounce-averaged drift kinetic equation (II)

The equations that determine g(ψ, α, v, λ) are∫ lb2

lb1

dl

|v‖|vd · ∇α ∂αg +

∫ lb2

lb1

dl

|v‖|vd · ∇ψΥFM =

∫ lb2

lb1

dl

|v‖|C lin[g]

with

∫ 2π

0

g dα = 0 for trapped particles and g = 0 for passing

and

(Z

T+

1

Te

)ϕ1 =

2π

en

∫ ∞

0

dv

∫ B−1

B−1max

dλv3B

|v‖|g

Υ =∂ψnn +

∂ψTT

(mv2

2T −32

)+

Ze∂ψϕ0

T .

C lin[g] is the linearized collision operator. We will use pitch-angle-scattering for the moment.

vE · ∇α ≈ ∂ψϕ0, so the system is linear in ϕ1.

This is the set of equations solved by KNOSOS4.4

We do not use explicitly the expansion of [Calvo, PPCF (2017)] of the magnetic configuration around aperfectly optimized stellarator as in [Velasco, PPCF (2018)] but, for small |∂ψϕ0|, if the stellarator is not optimizedenough, it is not justified to use the local DKE.


Explicit expressions

Using the expressions of the drift in Boozer coordinates5,everything gets written in terms of a few bounce integrals:(IvM,α(α, λ) +

1

vdIvE ,α(α, λ)

)∂αg +

(IvM,ψ (α, λ) +

1

vdIvE,ψ (α, λ)

)FMΥ +

=vν

vd∂λIν(α, λ)∂λg ,

with vd ≡ mv2

Ze and

IvE,α = ∂ψϕ0

∫ lb2

lb1

dl√1− λB

, IvM,α =1− λ

2

∫ lb2

lb1

dl√1− λB

∂ψB

B,

IvE,ψ = −∫ lb2

lb1

dl√1− λB

∂θϕ1 , IvM,α = −1− λ2

∫ lb2

lb1

dl√1− λB

∂θB

B,

Iν =

∫ lb2

lb1

dl

√1− λBB

.

5For the sake of simplicity, in this talk we assume vaccum magnetic field, B = Bζ∇ζ.


Coefficients

Integrals in l done using extended midpoint rule:

Number of points tripled until the result converges.Note that integrals like:

I =

∫ l2

l1

dlf(l)√

1− λB(l)

diverge logarithmically close to the passing/trapped boundary. Onecan ease the convergence (and thus speed up the calculation) byremoving the divergence and solving it analytically:

I = I0 +∑j

Ij ,

I0 =

∫ l2

l1

dl

f(l)√1− λB(l)

−∑j

f(lj)√−λ(l− lj)[∂lB|lj + 1

2∂2lB|lj (l− lj)]

,

Ij =

√√√√ −2

λ∂2lB|lj

ln

2λ

√√√√ ∂2lB|lj−2

√−∂lB|lj x−

1

2∂2lB|lj x

2 − λ(∂2l B|l1x + ∂lB|lj )

l2−l1

0

.

The same can be done at other boundaries (e.g. from trapped inone period to trapped in two periods).


Coefficients (II)

When integrating along the field line using fixed step ∆ζ,∆θ,instead of evaluating

B(θ, ζ) =∑m,n

Bm,n cos[mθ + nζ] ,

B(θ + ∆θ, ζ + ∆ζ) =∑m,n

Bm,n cos[m(θ + ∆θ) + n(ζ + ∆ζ)] ,

B(θ + 2∆θ, ζ + 2∆ζ) =∑m,n

Bm,n cos[m(θ + 2∆θ) + n(ζ + 2∆ζ)] ,

...

we can precalculate a few sines and cosines cos(mθ + nζ),sin(mθ + nζ), cos(m∆θ + n∆ζ) and sin(m∆θ + n∆ζ) and usetrigonometric identities to iterate:

B(θ + ∆θ, ζ + ∆ζ) =∑m,n

Bm,n[cos(mθ + nζ) cos(m∆θ + n∆ζ)−

sin(mθ + nζ) sin(m∆θ + n∆ζ)]


Discretization

We are left with two variables:

Bounce-averages in l.

Pitch-angle scattering collision operator ⇒ v is a parameter.Calculate ∂vQi for 28 values of v and integrate Gauss-Laguerre.

Radially local equations ⇒ ψ is a parameter.

(IvM,α(α, λ) +

1

vdIvE ,α(α, λ)

)∂αg +

(IvM,ψ (α, λ) +

1

vdIvE,ψ (α, λ)

)FMΥ +

=vν

vd∂λIν(α, λ)∂λg .

Discretize gi,j ≡ g(αi, λj), with uniform grid in λ and α.Second order central differences.

g = 0 at boundary between passing and trapped.

v‖∂λg = 0 at well bottoms.

Subtlety: at bifurcations, g is continuous in α and λ, but ∂λg is notcontinuous in λ.


Discretization (II) gi,j ≡ g(αi, λj)(IvM,α(α, λ) +

1

vdIvE ,α(α, λ)

)∂αg +

(IvM,ψ (α, λ) +

1

vdIvE,ψ (α, λ)

)FMΥ +

=vν

vd∂λIν(α, λ)∂λg ,

Top: gi,1 = 0 .

Arbitrary point:

∂αg|i,j =gi+1,j − gi−1,j

2∆α, ∂λg|i,j =

gi,j+1 − gi,j−1

2∆λ,

∂2λg|i,j =

gi,j+1 + gi,j−1 − 2gi,j(∆λ)2

.

Bifurcations (W = I, II...):

∂αg|i,j =gi+1,j,W − gi−1,j

2∆α,

rhs =∑w

Iν ,i,j+1,w−gi,j+3,w + 2gi,j+2,w − 3gi,j+1,w

4(∆λ)2

− Iν ,i,j−1gi,j − gi,j−2

4(∆λ)2.

Bottom (j = N + 1): rhs = −Iν ,i,N−1gi,N−gi,N−2

4(∆λ)2.


Linear problem

(IvM,α(α, λ) + 1

vdIvE,α(α, λ)

)∂α

[IvM,ψ(α, λ) +

+vνvd∂λν(α, λ)∂λ g 1

vdIvE,ψ(α, λ)

]FMΥ

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

·. . .. . .. . .. . .

=

. . .. . .. . .. . .

Square matrix, Nλ ×Nα elements per row.

Sparse, ∼ 5 non-zero elements per row.Non-zero elements are always at the same position for a givenflux-surface, but relative weight varies with v, Er...

Direct solver, based on LU factorization.

Will be useful for solving quasineutrality.


Benchmark with DKES (vM · ∇α = 0, ϕ1 = 0)

W7-X (standard configuration, EIM,low ripple) at:

ψ/ψLCMS = 0.3 (top right).

ψ/ψLCMS = 0.5 (bottom left).

ψ/ψLCMS = 0.7 (bottom right).

10-4

10-3

10-2

10-1

100

10-6 10-5 10-4 10-3 10-2 10-1

Γ11

[A.U

.]

ν/v [m-1]

Er/v=0.0Er/v=1x10-5

Er/v=3x10-5

Er/v=1x10-4

Er/v=3x10-4

Er/v=1x10-3

Er/v=3x10-3

10-4

10-3

10-2

10-1

100

10-6 10-5 10-4 10-3 10-2 10-1

Γ11

[A.U

.]

ν/v [m-1]

Er/v=0.0Er/v=1x10-5

Er/v=3x10-5

Er/v=1x10-4

Er/v=3x10-4

Er/v=1x10-3

Er/v=3x10-3

10-4

10-3

10-2

10-1

100

10-6 10-5 10-4 10-3 10-2 10-1

Γ11

[A.U

.]

ν/v [m-1]

Er/v=0.0Er/v=1x10-5

Er/v=3x10-5

Er/v=1x10-4

Er/v=3x10-4

Er/v=1x10-3

Er/v=3x10-3



LHD (R = 3.75) at ψ/ψLCMS = 0.5(top right).

QIPC at:

ψ/ψLCMS = 0.1 (bottom left).

ψ/ψLCMS = 0.25 (bott. right).

10-6

10-5

10-4

10-3

10-2

10-1

100

101

10-6 10-5 10-4 10-3 10-2 10-1

Γ11

[A

.U.]

ν/v [m-1]

Er/v=0.0Er/v=1x10-5

Er/v=3x10-5

Er/v=1x10-4

Er/v=3x10-4

Er/v=1x10-3

Er/v=3x10-3

10-3

10-2

10-1

100

101

102

10-6 10-5 10-4 10-3 10-2 10-1

Γ11

[A

.U.]

ν/v [m-1]

Er/v=0.0Er/v=1x10-5

Er/v=3x10-5

Er/v=1x10-4

Er/v=3x10-4

Er/v=1x10-3

Er/v=3x10-3

10-5

10-4

10-3

10-2

10-1

100

101

10-6 10-5 10-4 10-3 10-2 10-1

Γ11

[A

.U.]

ν/v [m-1]

Er/v=0.0Er/v=1x10-5

Er/v=3x10-5

Er/v=1x10-4

Er/v=3x10-4

Er/v=1x10-3

Er/v=3x10-3



49 simulations at low collisionalities

Nα=32, Nλ=256.

Total time: 2.4 s (0.05 s/point)at one single core of Dual Xeonquad-core 3.0 GHz (∼ severaltimes slower than e.g. Marconi).

Relatively large overhead:

Grid in α and λ: 0.2 s.Bounce integrals 0.9 s.Integral of g in λ: 0.2 s.

Easy to parallelize: v, ψ, bounceintegrals, ∂ψϕ0, ϕ1 (next slide)...

10-4

10-3

10-2

10-1

100

10-6 10-5 10-4 10-3 10-2 10-1

Γ11

[A.U

.]

ν/v [m-1]

Er/v=0.0Er/v=1x10-5

Er/v=3x10-5

Er/v=1x10-4

Er/v=3x10-4

Er/v=1x10-3

Er/v=3x10-3

* One could think that anything below (say) 1 s is equally fine, butnote that solving ambipolarity, quasineutrality, transport balance...requires evaluating the drift kinetic equation MANY times.


Solving quasineutrality (matrix response approach)

[IvM,α(α, λ) +

1

vdIvE,α(α, λ)

]∂αg(α, λ)− vν

vd∂λIν(α, λ)∂λg(α, λ) =

=

(IvM,ψ (α, λ)− 1

vd

∫ lb2

lb1

dl√1− λB

∂θϕ1

)FMΥ , (1)

(Z

T+

1

Te

)ϕ1 =

2π

en

∫ ∞0

dv

∫ B−1

B−1max

dλv3B

|v‖|g . (2)

If we parametrize ϕ1 using N vectors uk and N coefficients, thenthe linear system can be esquematically written as

ϕ1 = ϕ01 +Aϕ1 (3)

where A is a N ×N matrix:

ϕ01 obtained by solving (1) with ϕ1 = 0 and then using (2).

The kth row of A is obtained by solving (1) with ϕ1 = uk,

solving (2) and substracting ϕ01.

DKE is solved N + 1 times, but LU factorization is done once(e.g., in prev. case, for N = 1000, CPU time is 40 times larger).

Once ϕ01 and A are known, the linear system (3) can be solved.


Example: effect of plasma profiles on transport ofcollisional impurities [Velasco et al., in preparation]

Besides, depending on the plasma parameters, affecting Qi, weknow that ϕ1 caused by the bulk species may have an importanteffect on radial transport of collisional impurities.

Yet, only a total of ∼ 100 calculations of ϕ1 (and of ΓZ) havebeen published in papers [Garcıa-Regana, NF (2013), Garcıa-Regana,

NF (2017), Garcıa-Regana, NF (2018), Mollen, PPCF (2018), Velasco,

PPCF (2018)].

Roughly speaking, this covers: ∼ 5 magnetic configurations ×5 radial positions × 5 different plasma profiles.

More comprehensive study (dependence on the configuration,colllisionality, bulk plasma profiles...) remains to be done.

We will do so by combining ϕ1 calculated with KNOSOS and an-alytical formulas derived the flux of collisional impurities [Calvo,

arXiv (2018), TTF (last week)].


Example (II): ΓW44 vs. bulk profile shape

Same plasma parameters as in [Calvo, arXiv (2018), TTF (last week)].

LHD6 (R0 = 3.67 m) at√ψ/ψLCMS = 0.8.

n = 1.2× 1020 m3, T = 1.3 keV.

Er = T2e

(∂rnn + 3.37∂rTT

)(ion root limit, 1/ν regime).

1008 calculations of ϕ1 andΓW44.

For ϕ1 = 0, outward pinch(V > 0) if ∂rT/T > 2∂rn/n[Helander, PRL (2017)].

With ϕ1 6= 0, outward pinchfor hollow density only.

Experimental evidence:[Huang, NF (2017)].

-5

-4

-3

-2

-1

0

-5 -4 -3 -2 -1 0 1 2

T’/

T [

m-1

]

ni’/ni [m-1]

Er<0 Er>0 | |

|

|||

V(ϕ1=0)<0 V(ϕ1=0)>0

Er<0 Er>0 | |

|

|||

V(ϕ1=0)<0 V(ϕ1=0)>0

-5

-4

-3

-2

-1

0

1

2

V[m

/s]

6For this plot, we used a preliminary version of KNOSOS that makes certain assumptions on the magnetic

configuration [Velasco, PPCF (2018)].


Conclusions

We have developed the code KNOSOS. It can provide very fastcalculations of:

Level of transport in several low-collisionality regimes (1/ν,√ν

and superbanana-plateau) for arbitrary geometry.DKES-like radial fluxes including the effect of vM · ∇α.ϕ1 and Er (ongoing).

Benchmarking and experimental validation:

Radial fluxes agains DKES and FORTEC-3D.ϕ1 against EUTERPE, SFINCS, FORTEC-3D and analytical ex-pressions of [Calvo, JPP (2018)].Comparison with measurements (e.g. Qi and Er in W7-X [Alonso,

EPS (2017), TTF (2018)]).

Next steps:

Transition between 1/ν and plateau.Solve DKE for electrons.Improve collision operator.


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