Up-down asymmetric tokamaks

Justin Ball Prof. Felix Parra and Prof. Michael Barnes

Oxford University and CCFE 9th Gyrokinetic Working Group Meeting

29 July 2016

The problem

• Toroidal plasma rotation has been used in experiments to significantly increase the plasma pressure

(on-axis)

2

*multiply by ~10 to get

MS ⌘ R⌦⇣

vsound

*MA ⌘ R⌦⇣

vAlfven⇡ 0.5� 5%

Liu et al. Nucl. Fusion (2004).

⌦⇣• The usual methods to drive rotation do not appear to scale well to larger

devices such as ITER

• Numerical modeling suggests that, to see a benefit, ITER requires rotation with

The solution?

• Use plasma turbulence to transport momentum and spontaneously generate “intrinsic” rotation from a stationary plasma

• For future large devices, we need intrinsic rotation that scales with size

• As we will see, this severely restricts our options: up-down asymmetry in the magnetic equilibrium

3

Camenen et al. PPCF (2010).

r aTo

roid

al v

eloc

ity [k

m/s

]

Outline

4

Non-mirror symmetric Mirror symmetric

Up-

dow

n as

ym.

enve

lope

Up-

dow

n sy

m.

enve

lope

Up-down symmetric

Generalization of Miller local equilibrium

• Works well with GS2, a local δf gyrokinetic code

• Specify the flux surface of interest as a Fourier decomposition:

• Specify how it changes with minor radius:

5

✓r

⇣@r0@r

����✓

= 1�X

m

C 0m cos (m (✓ + ✓0tm))

Miller et al. Phys. Plasmas (1998).

r0 (✓) = r 0

1�

X

m

Cm cos (m (✓ + ✓tm))

!

Gyrokinetics

• Governs turbulence in tokamaks:

• Allows us to calculate the turbulent fluxes, such as

• Calculate the eight geometric coefficients from MHD equilibrium

7

k? =

rk2

���~r ���2+ 2k k↵~r · ~r↵+ k2↵

���~r↵���2

where

⇧ = 2⇡iIX

k ,k↵

k↵

⌧� (k , k↵)

Zdv||dµ v||J0 (k?⇢s)hs (�k ,�k↵)

�

@hs

@t+ v||b · ~r✓

@hs

@✓

����v||

+ i (k vds + k↵vds↵)hs + a||s@hs

@v||�

X

s0

hC(l)ss0i' + {J0 (k?⇢s)�, hs}

=ZseFMs

Ts

@

@t(J0 (k?⇢s)�)� v�s FMs

1

ns

dns

d +

✓msv2

2Ts� 3

2

◆1

Ts

dTs

d

�

Estimating the Alfvén Mach number

8

Ball et al. PPCF (2014).

ignore

• Ignoring pinch is conservative, may enhance rotation by a factor of 3

Peeters et al. PRL (2007).

Pr ⌘ D⇧

DQ⇡ 1 ⇡ constant

h⇧ (0, 0)it � P⇧⌦⇣ �D⇧d⌦⇣dr

= 0

⌧⇧

✓⌦⇣ ,

d⌦⇣dr

◆�

t

= 0 hQiit = �DQdTi

dr

MA ⇡p2�T

Pr

h⇧ithQiit

Up-down symmetry argumentPeeters et al. PoP (2005). & Parra et al. PoP (2011).

• Second solution has a canceling momentum flux:

• Constrains to lowest order in

• Negating kψ, θ, and v|| leads to a second solution of the gyrokinetic eq.

⇢⇤ ⌘ ⇢i/a ⌧ 1

9

Sugama et al. PPCF (2011).

!⇢B, b · ~r✓,�vds , vds↵,�a||s,

���~r ���2,�~r · ~r↵,

���~r↵���2�

hs

�k , k↵, ✓, v||, µ, t

�! �hs

��k , k↵,�✓,�v||, µ, t

�

h⇧it ! �h⇧ith⇧it = 0

MA = 0

Qud

geo

2⇢B, b · ~r✓, vds , vds↵, a||s,

���~r ���2

, ~r · ~r↵,���~r↵

���2

�

Outline

11

Non-mirror symmetric Mirror symmetric

Up-

dow

n as

ym.

enve

lope

Up-

dow

n sy

m.

enve

lope

Up-down symmetric

Small in ⇢⇤ ⌧ 1

MHD equilibrium argument

m=4 modem=2 mode

Low m penetrates best

• Grad-Shafranov equation for a constant toroidal current profile:

• To lowest order in aspect ratio, solutions are cylindrical harmonics:m=3 mode

13

Rodrigues et al. Nucl. Fusion (2014).Ball et al. PPCF (2015).

R2~r · ~r R2

!= �µ0R

2 dp

d � I

dI

d = const

amount of shaping / rm�2

Outline

15

Non-mirror symmetric Mirror symmetric

Up-

dow

n as

ym.

enve

lope

Up-

dow

n sy

m.

enve

lope

Exp. bad MHD

Up-down symmetric

Exp. bad MHD

Exp. bad MHD

Small in ⇢⇤ ⌧ 1

Screw pinch argument

• Screw pinches have no toroidicity, so up-down symmetry has no meaning

• Mirror symmetric flux surfaces generate no rotation

• Rotation can be generated by breaking mirror symmetry (i.e. the direct interaction of two different shaping effects)

• This can occur in tokamaks

16

MA 6= 0

MA = 0

Outline

18

Non-mirror symmetric Mirror symmetric

Up-

dow

n as

ym.

enve

lope

Up-

dow

n sy

m.

enve

lope

Exp. bad MHD

Up-down symmetric

Screw pinch limit

Screw pinch limit Exp. bad MHD

Exp. bad MHD

Small in ⇢⇤ ⌧ 1

Poloidal tilting symmetry argument

• Rewrite geometry specification to distinguish (the fast poloidal scale) from (the connection length scale):

• Convert to the form of a 2-D Fourier series using

• Define according to the physics of the scale separation (defines any mode as “fast”)

19

Ball, et al. PPCF 58 045023 (2016).

✓z ⌘ mc✓

r0 (✓) = r 0

1�

X

m

Cm cos (m (✓ + ✓tm))

!

k ⌘ m� lmc

l ⌘ bm/mccm � mc

r0 (✓, z) = r 0

1�

1X

l=0

mc�1X

k=0

Ck+lmc

⇥⇥cos (l (z +mc✓tm)) cos (k (✓ + ✓tm))� sin (l (z +mc✓tm)) sin (k (✓ + ✓tm))

⇤!

z ✓

mc = 5

Poloidal tilting symmetry argument

• But remember we expanded in

• Specify

25

Ball, et al. PPCF 58 045023 (2016).

Qud

geo

(✓, z) 2⇢B, b · ~r✓, vds , vds↵, a||s,

���~r ���2

, ~r · ~r↵,���~r↵

���2

�

Qtilt

geo

= Qud

geo

(✓, z + ztilt

)

htilts (✓, z) = hud

s (✓, z + ztilt)⌦⌦⇧tilt

↵t

↵z=

⌦⌦⇧ud

↵t

↵z= 0

ztilt

⇡tilt⇣ (✓, z) = ⇡ud

⇣ (✓, z + ztilt)

• Verify by looking for

rtilt0 (✓, z) = rud0 (✓, z + ztilt)

Up-down sym.

Tiltedmc � 1⌦⌦⇧

tilt↵t

↵z⇠ MA ⇠ exp (�mc)

28

m=7 geometry: Simulation (up-down sym.)

Simulation (tilted)

Theory prediction for tilted

-π πθ

-0.06

0.06⇡⇣

-π π

-0.06

0.06

Ball, et al. PPCF 58 045023 (2016).Verify ⇡tilt

⇣ (✓, z) = ⇡ud⇣ (✓, z + ztilt)

29

m=5:

-π πθ

-0.06

0.06

m=7:

m=6:

m=8:

-π πθ

-0.06

0.06 ⇡⇣⇡⇣

⇡⇣⇡⇣

-π π

-0.06

0.06

-π π

-0.06

0.06

-π π

-0.06

0.06

-π π

-0.06

0.06

-π πθ

-0.06

0.06

-π πθ

-0.06

0.06

Ball, et al. PPCF 58 045023 (2016).Verify ⇡tilt

⇣ (✓, z) = ⇡ud⇣ (✓, z + ztilt)

Outline

31

Non-mirror symmetric Mirror symmetric

Up-

dow

n as

ym.

enve

lope

Up-

dow

n sy

m.

enve

lope

Exp. bad MHD

Up-down symmetric

Screw pinch limit & Exp. small in

Screw pinch limit Exp. bad MHD

Exp. bad MHDExp. small in m � 1

m � 1

Small in ⇢⇤ ⌧ 1

32

• Created using two modes, and , with distinct tilt angles, and

• Calculate geometric coefficients order-by-order in

• Look for beating between fast shaping effects (creates an envelope on the connection length)

m � 1

Ball, et al. PPCF 58 055016 (2016).Envelope argument: Expansion in m � 1

R0 R0 R0 R0 R0R

m = 2 m = 3 m = 4 m = 5 m = 6

m✓tm ✓tn

n = m+ 1

37

0

Ball, et al. PPCF 58 055016 (2016).

vds↵ =

B0

R0⌦s

dr d

(cos (✓) + s✓sin (✓))

+

B0

2R0⌦s

dr d

⇥�m3C2

m + n3C2n

�✓sin (✓)

� s✓cos (✓) (mCmsin (m (✓ � ✓tm)) + nCnsin (n (✓ � ✓tn)))

+ s✓sin (✓) (mCmcos (m (✓ � ✓tm)) + nCncos (n (✓ � ✓tn)))

+mnm+ n

n�mCmCnsin (✓)

⇥ (sin ((n�m) ✓) cos (n✓tn �m✓tm) + cos ((n�m) ✓) sin (n✓tn �m✓tm))]

Envelope argument: Expansion in

vds↵• Calculate magnetic drift coefficient within flux surface,

m � 1

breaks symmetry⇠ m�1

h⇧ithQiit

⇠ m�1MA / m�1

Envelope argument: Numerical scaling with

38

m � 1Ball, et al. PPCF 58 055016 (2016).

2 3 4 5 6 7 8

m

−0.02

0

0.02

0.04

v thi⟨Π

⟩ t/R

0⟨Q

i⟩ t

Mirror Sym.

exp(-m)

Non-mirror Sym.

1/m

2 3 4 5 6 7 8

mc

−0.02

0

0.02

0.04

(vth

i/R

c0)⟨Π

ζi⟩

t/⟨Q

i⟩t

Mirror sym. Exp. scaling

Non-mirror sym., up-down asym. envelopeNon-mirror sym., up-down sym. envelope

h⇧i t/hQ

iit

Outline

40

Non-mirror symmetric Mirror symmetric

Up-

dow

n as

ym.

enve

lope

Up-

dow

n sy

m.

enve

lope

Exp. bad MHD

Up-down symmetric

Screw pinch limit & Exp. small in

Screw pinch limit & Poly. small in

Exp. bad MHD

Exp. bad MHDExp. small in m � 1

m � 1

Poly. small in m � 1 m � 1

Small in ⇢⇤ ⌧ 1

Two options to maximize rotation

42

Elongation & triangularity

• Use lowest possible to break mirror symmetry and create up-down asymmetric envelopes

• Prefer modes to be controllable by external shaping magnets

Shafranov shift & elongation(i.e. and ) (i.e. and )

m

m = 1 m = 2 m = 2 m = 3

1 2 3 4 5- 2

- 1

0

1

2

R

Z

44

Momentum flux from Shafranov shift

• Reduced by including effect of a constant , which affects the magnetic equilibrium through the Grad-Shafranov equation

dp/dr

0 0.25 0.5 0.75 1

ρ0

0

0.02

0.04

0.06

0.08

0.1

(vth

i/R

c0)⟨Π

ζi⟩

t/⟨Q

i⟩t

0 0.25 0.5 0.75 1

ρ0

0

0.02

0.04

0.06

0.08

0.1

(vth

i/R

c0)⟨Π

ζi⟩

t/⟨Q

i⟩t

0 0.25 0.5 0.75 1

ρ0

0

0.02

0.04

0.06

0.08

0.1

(vth

i/R

c0)⟨Π

ζi⟩

t/⟨Q

i⟩t

/ includesdp

dr

Ball, et al. arXiv:1607.06387 (2016).

r a

Local Shafranov shift increases with minor radius

1

0

2

MA

(%)

Two options to maximize rotation

45

Elongation & triangularity

• Use lowest possible to break mirror symmetry and create up-down asymmetric envelopes

• Prefer modes to be controllable by external shaping magnets

Shafranov shift & elongation(i.e. and ) (i.e. and )

m

m = 1 m = 2 m = 2 m = 3

1 2 3 4 5- 2

- 1

0

1

2

R

Z

Roughly no net enhancement of momentum flux

47

Momentum flux from elongation and triangularity

Up-down sym. envelopeMirror sym.

MA (%)

48

Momentum flux from elongation and triangularityhQiit

Up-down sym. envelopeMirror sym.

ITER

Conclusions

• Intrinsic rotation generated by up-down asymmetry scales well to larger machines (ITER, DEMO, etc.), unlike other mechanisms

• Tilting the elongation of flux surfaces is a simple way to generate significant rotation

• The magnitude of rotation is roughly what is needed to permit increasing the plasma pressure in ITER

• Breaking ALL the symmetries, especially with external shaping, can boost the rotation significantly

49

} in ITER

= 1.7

� = 0.35

1 2 3 4 5- 2

- 1

0

1

2

R

Z

✓ = ⇡/4

✓� = ⇡/6

MA ⇡ 1.5%

The “optimal” geometry

Thank you!

Summary

51

Non-mirror symmetric Mirror symmetric

Up-

dow

n as

ym.

enve

lope

Up-

dow

n sy

m.

enve

lope

Exp. bad MHD

Up-down symmetric

Screw pinch limit & Exp. small in

Screw pinch limit & Poly. small in

Exp. bad MHD

Exp. bad MHDExp. small in m � 1

m � 1

Poly. small in m � 1 m � 1

Small in ⇢⇤ ⌧ 1

Extra Slides

62

Expansion fails at m=4:

-π πθ

-0.06

0.06⇡⇣

-π π

-0.06

0.06

Ball, et al. PPCF 58 045023 (2016).

Simulation (up-down sym.)

Simulation (tilted)

Theory prediction for tilted

Verify ⇡tilt⇣ (✓, z) = ⇡ud

⇣ (✓, z + ztilt)

Breakdown in poloidal tilting symmetry

63

64

Global Shafranov shift in tilted elliptical geometry

• Calculate from Grad-Shafranov eq. (to next order in aspect ratio) for ITER-like parameters

• Verify with the equilibrium code ECOM

✓axis

raxis

○○

○

○○ ○

□□

□

□□ □

△ △△

△△ △

π8

π4

3 π8

π2

θκb

0.2

0.4

0.6

raxisa

○

○○

○

○

○□

□□

□

□

□△

△△

△

△

△π8

π4

3 π8

π2

θκb

π8

π4

θaxis

Ball, et al. arXiv:1607.06387 (2016).

• Insensitive to shape of current profile (holding fixed)Ip

65

Global Shafranov shift in tilted elliptical geometry

• Calculate from Grad-Shafranov eq. (to next order in aspect ratio) for ITER-like parameters

• Verify with the equilibrium code ECOM

○○

○

○○ ○

□□

□

□□ □

△ △△

△△ △

π8

π4

3 π8

π2

θκb

0.2

0.4

0.6

raxisa

○

○○

○

○

○□

□□

□

□

□△

△△

△

△

△π8

π4

3 π8

π2

θκb

π8

π4

θaxis

Ball, et al. arXiv:1607.06387 (2016).

• Insensitive to shape of current profile (holding fixed)

✓axis

raxis

paxis

/ b

Global to local Shafranov shift

•

66

• GS2 requires the local change in the flux surface center, anddRc

dr

dZc

dr

dRc

dr =

dRc

d

d

dr = �2

r a

raxis

asin (✓

axis

)dRc

d = const

Ball, et al. arXiv:1607.06387 (2016).

■■ ■■

■■

■■

■■

▲▲ ▲▲

▲▲

▲▲ ▲▲

○○ ○○

○○

○○ ○○

□□ □□

□□

□□ □□

△△ △△

△△

△△ △△0 0.25 0.5 0.75 1

ψψb

0

0.05

0.1

0.15

Rc -R0a

70

• I think so, but there is a catch

• The shape of the first wall is fixed

Reduced plasma volume

• Each shaping coil has a current limit

Reduced plasma current

• For , it’s worth a shot�N = 3

Can something like this be done in ITER?

X

X