Tutorial: Incorporating kinetic aspects of RF current drive in MHD simulation

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Tutorial: Incorporating kinetic aspects of RF current drive in MHD simulation J. Pratt, E. Westerhof Background Models of Current drive RF physics from reduced-MHD simulation

Tutorial: Incorporating kinetic aspects

of RF current drive in MHD simulation

with a focus on ECCD stabilization of tearing modes

J. Pratt, E. Westerhof

Lorentz Workshop:

Modeling Kinetic Aspects of Global MHD Modes

4 Dec 2013, Leiden, Netherlands

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Outline

I

Background

I

radio frequency (RF) heating

I

electron cyclotron current drive (ECCD)

I

suppression of tearing modes

I

Review of models for ECCD in MHD

I

flux function model of driven current (Yu, G¨

unter)

I

anisotropic diffusive model (Giruzzi, Yu, Gianakon)

I

RF force model (Kruger, Jenkins)

I

convective model (Pratt, Westerhof)

I

basics of EC current drive

I

two equation convective model

I

single equation convective model

I

results from the reduced MHD simulation JOREK

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Tearing Modes

I

Magnetic reconnection breaks up the nested flux surfaces in a

tokamak.

I

This creates

regions of closed

magnetic field

lines, called

magnetic islands

.

I

The tearing

instability

produces

magnetic islands

that grow in size.

I

Large islands

increase radial

transport, cause

loss of

confinement.

I

A neoclassical tearing mode, in contrast to a classical tearing

mode, is driven by reduction of bootstrap current.

(4)

Tearing Mode Suppression

Maraschek 2012 Nuc. Fus.

I

Suppression can be

accomplished by

replacing current inside

the islands using:

electron cyclotron

resonance

heating/current drive,

lower hybrid resonance

heating/current drive.

I

Microwave power is

injected at the electron

cyclotron resonance

frequency.

I

ECCD targets the island

center, at a surface of

rational safety factor

q

around which the island

forms.

(5)

Electron cyclotron current drive

I

A small group of electrons resonates with the RF waves.

I

The current produced is localized – tight control over current

profile.

I

ECCD produces a steady-state, non-Maxwellian distribution of

electrons.

Nice short review:

La Haye. Phys.

Plasmas 2006.

Neoclassical tearing

modes and their

control.

Figure from:

Pletzer and Perkins. Phys. Plasmas, 1999. Stabilization of neoclassical tearing modes using a continuous localized current drive.

(6)

Flux Function Model

Yu and G¨

unter (PPCF 1998) model the contribution to the

helical magnetic flux

ψ

Dψ

Dt

=

E

0

+

η(

j

−

j

BS

−

j

EC

),

η

and

j

EC

: functions of

ψ

, centered at O-point.

(b), (c), and (d) curves use different width and intensity of EC current

Result: width of

magnetic island

abruptly drops

when simple ECCD

is applied, partial

suppression of

NTM!

(7)

Anisotropic Diffusive Model

Sometimes referred to as the Giruzzi model, this model

derives from the bounce-averaged kinetic equation for

electrons:

collisions EC waves radial diffusion

∂f

e

∂t

=

C(f

e

) +

Q

EC

(f

e

)

+

1

ρ

∂

∂ρ

ρD

r

∂f

e

∂ρ

+

eE

||

∂f

e

∂p

||

Expressed as an evolution equation for the electron

cyclotron current, this model includes parallel and

perpendicular diffusive terms:

Dψ

Dt

=

E

+

η(

j

−

j

BS

−

j

EC

)

∂

j

EC

∂t

=

ν(

j

−

j

EC

) +

∇ ·

(χ

k

∇

k

j

EC

) +

∇ ·

(χ

⊥

∇

⊥

j

EC

)

Giruzzi, G. PPCF 1993. Modelling of RF current drive in the presence of radial diffusion.

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Anisotropic Diffusive Current Drive Results

dotted – source at O-point dashed – rotating source

Yu, et al. Phys Plasmas, 2000 & 2004. (TM code)

Gianakon. Phys Plasmas 2001. (NIMROD code)

“Even with the

complicated equations

mentioned above, the

time evolution of RF

current has not been

completely described by

our model....The major

purpose of the present

paper is on the

stabilization of the

NTM’s by the RF

current drive rather than

on the RF current drive

physics itself.”

(9)

RF force model

ρ

D

u

∂t

=

−∇

p

+

j

×

B

− ∇ ·

Π

E

+

u

×

B

=

η

j

+

F

rf

e

/n

|

q

e

|

3

2

n

DT

Dt

=

−

p

∇ ·

v

− ∇ ·

q

+

Π

:

∇

u

+

Q

+

S

rf

RF force RF energy

Jenkins et al. Phys. Plasmas 2010.

Kruger et al. in Proceedings of the 5th IAEA Technical Meeting on the Theory of Plasma Instabilities, Austin, Texas, USA, 2011

(1) ignore the electron stress tensor (2) use Braginskii closure for

parallel heat flux (3) ignore heat flux contributions to resistivity.

Hybrid code: NIMROD (extended MHD) + ray-tracing for RF force:

w

4

∼

B

(

n

= 1)

2

– results

show reduction of

magnetic island size.

(10)

Outline

I

Background

I

radio frequency (RF) heating

I

electron cyclotron current drive (ECCD)

I

suppression of tearing modes

I

Review of models for ECCD in MHD

I

flux function model of driven current (Yu, G¨

unter)

I

anisotropic diffusive model (Giruzzi, Yu, Gianakon)

I

RF force model (Kruger, Jenkins)

I

convective model

(Pratt, Westerhof)

I

basics of EC current drive

I

two equation convective model

I

single equation convective model

I

results from the reduced MHD simulation JOREK

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Description of EC current drive

Gyrophase-averaged kinetic equation for electrons:

collisions

∂f

e

∂t

=

C(f

e

) +

Q

EC

(f

e

)

−

v

k

∇

k

f

e

I

averaged EC wave-effect

Q

EC: quasi-linear diffusion model

I

electron distribution is permitted to convect along the

magnetic field.

Hegna and Callen. Phys. Plasmas 2009. Two-fluid MHD

equations are produced by taking moments:

collisional friction RF force RF energy collisional energy exchange heat flux stress

m

s

n

s

∂

∂t

+

u

s

· ∇

u

s

=

−∇

p

+

n

s

q

s

(

E

s

+

u

s

×

B

)

− ∇ ·

Π

s

+

R

s

+

F

rf

s

3

2

n

s

∂

∂t

+

u

s

· ∇

T

s

=

−

n

s

T

s

∇ ·

u

s

− ∇ ·

q

s

−

Π

s

:

∇

u

s

+

Q

s

+

S

rf

s

(12)

Choice of operators

For the collision operator we use a simple Krook operator:

C(f

e

) =

−

ν(v)(f

e

−

f

M

).

We assume that the quasi-linear diffusion is non-relativistic

and dominantly in the perpendicular direction, reasonable

for EC resonance. Thus

Q

EC

(f

e

)

=

∂

v

·

D

EC

·

∂

v

f

e

,

D

EC

≈

Dδ(v

k

−

v

res

k

)ˆ

v

⊥

ˆ

v

⊥

,

where

D

is a constant, and

v

res

k

= (ω

−

nΩ

e

)/k

k

is the

parallel velocity of the resonant electrons.

(13)

Effect of ECCD on electrons

ECCD produces a perturbation

δf

e

in the gyrophase-averaged electron

distribution function. This perturbation:

I

creates zero net parallel momentum.

I

is localized at the electron cyclotron resonant

parallel velocity

.

I

in perpendicular velocity takes the form of a velocity space

“hole” for

v

⊥

<

√

2

v

th

and a “bump” for

v

⊥

>

√

2

v

th.

I

is convected along the magnetic field lines out of the deposition

region.

v

⊥

v1

2 vth

v2

0

hole

bump

δf

e

(v

par

=v

EC

,v

⊥

)

(14)

Asymmetric Resistivity

I

Collisions would eventually return the distribution function

to the equilibrium Maxwellian state:

δf

e

→

0

.

I

But the asymmetric energy exchange between waves and

electrons (heating electrons moving in one toroidal

direction) produces an asymmetric collision rate, and thus

an asymmetric resistivity.

I

The hole is filled in more quickly than the bump is eroded,

because of the velocity dependence of the collision

frequency.

I

This is the “Fisch-Boozer” current mechanism, the

dominant mechanism for ECCD.

I

Net current decays at the slower collision rate of the high

velocity electrons in the bump.

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Developing a fluid model that uses asymmetric

resistivity

δf

e

can be reasonably represented by two delta functions at

perpendicular velocities

v

1

and

v

2

(representing the hole and bump

respectively) with different collision rates.

We take a moment of the kinetic equation to get the EC current

evolution:

∂

j

EC

∂t

=

e

m

e

R

δf

e

−

v

k

∇

k

j

EC

,

j

EC

(x, t)

=

−

e

Z

d

3

v

k

δf

e

,

R

δf

e

=

−

Z

d

3

v

m

e

v

ν(v)δf

e

.

Here

R

δf

e

is the

transient electron-ion friction

associated with

the EC-driven quasi-linear modification of the distribution

function.

(16)

Two equation convective model

We model

R

δf

e

in the standard way as the sum of a current

generation and collisional decay. This produces a convective

model with two equations for the EC current:

Dψ

Dt

=

E

0

+

η(

j

−

j

BS

−

(

j

EC2

+

j

EC1

)

∂

j

EC1

∂t

=

−

S

EC

−

ν

1

j

EC1

+

v

res

k

∇

k

j

EC1

∂

j

EC2

∂t

=

+

S

EC

−

ν

2

j

EC2

+

v

res

k

∇

k

j

EC2

4 parameters to be tuned:

I

current source:

S

EC

I

parallel velocity of the resonant electrons :

v

res

k

I

collision frequency of the resonant electrons that

(17)

Single equation convective model

Dψ

Dt

=

E

0

+

η(

j

−

j

BS

−

j

EC

)

∂

j

EC

∂t

=

S

∗

−

ν

2

j

EC

+

v

res

k

∇

k

j

EC

I

j

EC

is the sum of

j

EC1

and

j

EC2

.

I

current source

S

∗

is the flux surface average of

S

EC

in

the limit where the collision time of the slower

electrons is sufficiently long that they travel around

the entire flux surface.

I

parallel velocity of the resonant electrons :

v

res

k

A more detailed discussion of the single equation model is presented

in: E. Westerhof and J. Pratt. Expression of electron cyclotron

current drive in plasma fluid models. Proceedings of the 40th EPS

Conference on Plasma Physics. Espoo, Finland, July 1st – 5th 2013.

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Evolution of the EC driven current density along a

field line

x/mean free path

j

EC

current

2 equation model

1 equation model

0

50

100

150

200

0.0

0.2

0.4

0.6

0.8

1.0

1.2

The EC driven current is generated as the perturbation

δf

e

flows out of the EC power deposition region,

0

≤

x

≤

10

−

3

.

Time is normalized to the collision time and length is normalized to a electron-thermal-mean-free-path. Parameters: vresk= 2 v1= 0 ν1= 1/8 v2= 2.7 ν2= 1/38

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About JOREK

I

Over the last decade, the 3D nonlinear reduced-MHD

simulation JOREK has been developed by an

international developers group centered at

ITER/Cadarache.

I

simulation work performed with JOREK: edge localized

modes, resonant magnetic perturbations, pellet pacing,

resistive wall modes, disruptions

I

poloidal plane treated with 2D Bezier finite elements

(based on bicubic Bezier surfaces – a generalization of

cubic Hermite elements, elements are aligned with

magnetic flux surfaces)

I

toroidal direction treated with Fourier decomposition

I

fully-implicit time-stepping (choice of Crank-Nicholson,

BDF1 Implicit Euler, or BDF2 Gears method)

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Results: classical tearing mode suppression in

JOREK

t(s)

w/a

ECCD application

without ECCD

with ECCD

0.07

0.09

0.11

0.13

0.15

0.00

0.05

0.10

0.15

JOREK

reduced MHD

simulation:

high resistivity,

low viscosity,

low collision

frequency, 8

toroidal

harmonics

(21)

Results: RF current drive physics

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Results: RF current drive physics

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Results: RF current drive physics

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Results: RF current drive physics

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Results: RF current drive physics

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Results: RF current drive physics

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Results: RF current drive physics

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Results: RF current drive physics

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How do these models compare ...

... and what physics is important for practical results?

convection model

anisotropic diffusion model

∂

j

EC

∂t

=

S

∗

−

ν

2j

EC

+

v

res

k

∇

k

j

EC

∂

j

EC

∂t

=

ν

(j

−

j

EC

) +

∇ ·

(

χ

k

∇

k

j

EC

) +

∇ ·

(

χ

⊥

∇

⊥

j

EC

)

j

EC

=

C

exp

"

−

2

ψ

−

ψ

(

ro

)

ψ

(

r

o

)

−

ψ

(

r

o

−

d/

2)

2

#

flux function model

H & C two-fluid eqs

RF force model

ρ

s

D

u

s

Dt

=

−∇

p

+

n

s

q

s

(

E

s

+

u

s

×

B

)

− ∇ ·

Π

s

+

R

s

+

F

rf

s

ρ

D

u

∂t

=

−∇

p

+

j

×

B

− ∇ ·

Π

E

+

u

×

B

=

η

j

+

F

rf

e

/n

|

qe

|

(44)

Discussion

I

What are the limits of validity for the given

approximations/assumptions?

I

How accurate does the physics of the RF current need to

be to capture relevant features of NTM supression?

I

What is the role of self-induction of the current produced

by RF current drive?

I

How do we decide that the time evolution of the RF current

is sufficiently described? ... to predict power required in a

tokamak, time required to reduce the island, minimum

island width possible with a realistic RF source?

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Thanks!

Many thanks to G.T.A. Huysmans, Marina Bécoulet, Matthias Hölzl, Wolf-Chrisian Müller and the participants of the ASTER project and the JOREK collaboration. This work was performed on the Helios system at the system at Computational Situational Centre, International Fusion Energy Research Centre (IFERC-CSC), Rokasho-Japan and the Cartesius system, the Dutch national supercomputer, at SURFsara, Amsterdam, Netherlands.

The work in this tutorial talk has been performed in the framework of the NWO-RFBR Centre of Excellence (grant 047.018.002) on Fusion Physics and Technology. This work, supported by the European Communities under the contract of Association between EURATOM/FOM, was carried out within the framework of the European Fusion Programme. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

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References

I

La Haye. Phys. Plasmas 2006. Neoclassical tearing modes and

their control.

I

Giruzzi, G. Modelling of RF current drive in the presence of radial

diffusion. PPCF 1993.

I

Giruzzi, G., et al. Nuc. Fus. 1999. Dynamical modelling of

tearing mode stabilization by RF current drive.

I

Gianakon, T. A. Limitations on the stabilization of resistive

tearing modes. Physics of plasmas 8 (2001): 4105.

I

Hegna and Callen. Phys. Plasmas 2009. A closure scheme for

modeling RF modifications to the fluid equations.

I

Yu, G¨

unter, Giruzzi, et al. Phys Plasmas, 2000. Modeling of the

stabilization of neoclassical tearing modes by localized radio

frequency current drive.

I

Yu, Zhang, and G¨

unter. Phys Plasmas 2004. Numerical studies

on the stabilization of neoclassical tearing modes by radio

frequency current drive.

I

E. Westerhof and J. Pratt. Expression of electron cyclotron

current drive in plasma fluid models. Proceedings of the 40th

EPS Conference on Plasma Physics. July 1st – 5th 2013.

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Reduced MHD Formulation in JOREK

Vector fields are represented in terms of

u

(

velocity

stream function

) and

ψ

(

poloidal magnetic flux

):

B

=

−

ˆ

e

φ

×

1

R

∇

ψ

+

F

0

/Rˆ

e

φ

v

=

e

ˆ

φ

×

R

∇

u

+

v

||

B

JOREK solves for 6 scalar variables: also toroidal

current

density

j

, toroidal

vorticity

ω

,

density

ρ

,

temperature

T

.

(48)

Reduced MHD Equations

∂ψ

∂t

+

R

[

u, ψ

]

=

η

(

j

−

j

BS

,

0

−

E

0

/η

−

j

EC

)

−

F

0

∂u

∂φ

.

eφ

· ∇ ×

[

ρ

∂

v

∂t

=

−

ρ

(

v

· ∇)

v

− ∇

p

+

j

×

B

+

µ

4

v

]

jφ

=

R

2

∇ ·

(

R

−

2

∇

ψ

)

ω

=

∇

2

pol

u

∂ρ

∂t

=

−∇ ·

(

ρ

v

) +

∇ ·

(

D⊥

∇

⊥ρ

) +

Sρ

ρ

∂T

∂t

=

−

ρ

v

· ∇

T

−

(

κ

−

1)

p

∇ ·

v

+

∇ ·

(

K⊥

∇

⊥T

+

K

||

∇

||

T

) +

S

T

K

||,⊥

are the parallel and perpendicular heat diffusivity, and

κ

= 5

/

3

is the ratio of specific heats.

S

ρ

is a particle sources and

S

T

is a heat source.

Poisson bracket