Intrinsic rotation in
up-down asymmetric tokamaks
Justin Ball, Felix Parra,
and Michael Barnes
!
Oxford University
!
Madrid Working Group Meeting
8 July 2014
Outline
• Argument limiting intrinsic rotation in up-down symmetric devices
• Identify attractive up-down asymmetric configurations
• Heat and momentum flux found in the tilted elliptical tokamak
• Reasonable agreement with experimental results
• Poloidal distribution of momentum flux
Momentum transport symmetry argument
Parra et al. (2011)
• Constrains intrinsic rotation (rotation observed in the absence of external
momentum injection) to be small
gs k , k↵ , , ✓, ⇣, v|| , µ !
gs
k , k↵ , , ✓, ⇣, v|| , µ
• Negating θ, v||, and kψ leads to a
second solution of the
gyrokinetic equation with exactly
canceling momentum flux
• Contributions to momentum flux
from two mirror symmetric
particles cancel
Symmetry breaking mechanisms
Camenen et al. (2011)
0.3
We want to creating nonzero rotation from an initially stationary plasma:
0.2
1. Rotation
2. Gradient in rotation
0.1

vM
0.0
-0.1
3. Up-down asymmetry
• Order ⇢⇤ ⌘ ⇢i /a mechanisms
(radial profile variation, neoclassic
flows, …) as well as options 1 and 2
likely weaken significantly in future
larger machines
-0.2
-0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
MHD equilibria of up-down asymmetry
Grad-Shafranov Eq. to
0th
1 @
order in aspect ratio: r @r
Let the toroidal current be constant
✓
@ 0
r
@r
µ 0 j ⇣ R0 =
◆
1 @2 0
+ 2
=
r @✓2
µ0 R02
dp2
d 0
µ0 R02
dp2
d 0
dI2
I0
= const
d 0
Solutions are cylindrical harmonics:
m=2 mode
Only elongation penetrates
Contours of ψ0
m=3 mode
dI2
I0
d 0
m=4 mode
MHD equilibria of up-down asymmetry
Grad-Shafranov Eq. to
0th
1 @
order in aspect ratio: r @r
Let the toroidal current be constant
✓
@ 0
r
@r
µ 0 j ⇣ R0 =
◆
1 @2 0
+ 2
=
r @✓2
µ0 R02
dp2
d 0
µ0 R02
dp2
d 0
dI2
I0
= const
d 0
Solutions are cylindrical harmonics:
m=2 mode
Contours of ψ0
m=3 mode
Only elongation penetrates
θκ
θδ
dI2
I0
d 0
m=4 mode
Intuition for shaping penetration - poloidal shaping
Look at cases with strong shaping ( R0 Bp |b = 0 ):
b (a)
(a) ⌘
a
db
a
b (a)
d
da
=
da
a2
I ⇡
Z r
1
⌘
d⇣
drRBp
2⇡
⇡
0
✓
◆
d
1 RBp |a
=
1
da
a
RBp |b
m=2
b
!1
a
b
m=3
=2
a
b
m=4
a
=
Low m shaping effects penetrate to the magnetic axis better!
p
2
Intuition for shaping penetration - current profile
Write the Grad-Shafranov equation as
1 @
2
(R0 Bp ) + R0 Bp p = µ0 j⇣ R0 where ~
p ⌘
2 @ #,⇣
~ b̂p
b̂p · r
For m=2, separate effects of magnetic pressure and curvature
s
Z
b
R0 B p | a =
2
d
0µ
µ 0 j ⇣ R0
R0 B p | b =
p | b
0 j ⇣ R0
0
d
= 0 for the constant current case to find p |b
Use
da
0s R
1
µ
j
R
0µ j R
0
⇣
0
d
d
Hence, da
=
a
@
0
0 ⇣
µ 0 j ⇣ R0
0
1A
More hollow current profiles allow shaping to penetrate radially!
a
Tilted ellipse simulation geometry in modified GS2
0
= 0 22
= /8
= /4 11
= 3 /4
= /2
00
-1
-1
-1
-2
-2
-2
2
=
=3
=
ZZN
Z
1
=
1
2
R0
4
5
RminN
2
R
3
3
4
4
RN
R
5
5
Constant during rotation
R0 , , rmin ,
Rmin , , rmin ,
LT (rmin ) ,
LT (Rmax ) ,
Ln (rmin )
Ln (Rmax )
6
6
Instability threshold and stiffness
γN
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
Linear Growth Rate, (vth,i/a)
Linear Growth Rate, (vth,i/a)
0.4
=0
=
0
= /4
== /4
/2
= /2
0.3
R0 /a = 3.0
 = 2.0
rmin /a = 0.54
0.2
a/Ln (rmin ) = 0.733
θκ = 0
θκ = π/4
θκ = π/2
0.1
0
3
3
3
66
6
99
RR
/LTNs
0N/L
R00/LTT
9
12
12
12
Nonlinear heat flux in rotated cross sections
R0 /a = 3.0
30
 = 2.0
<QNtot>t
rmin /a = 0.54
a/LT (rmin ) = 3.45
20
a/Ln (rmin ) = 0.733
R0 /a = 3.0
10
 = 2.0
rmin /a = 0.54
a/LT (rmin ) = 2.3
0
-π/2
-π/4
0
θκ
π/4
π/2 a/Ln (rmin ) = 0.733
Momentum flux from a tilted ellipse
Rmin /a = {2.0}
 = {2.0}
rmin /a = {1.0}
a/LT (Rmax ) = {2.5}
a/Ln (Rmax ) = {0.733}
R0 /a = {3.0, 3.0, 6.0}
 = {2.0, 2.0, 2.0}
<Πud
ζNi>t/(R0N<QNi>t)
0.06
0.03
0
-0.03
-0.06
-π/2
a/LT (rmin ) = {3.45, 2.3, 2.3}
a/Ln (rmin ) = {0.733, 0.733, 0.733}
rmin /a = {0.54, 0.54, 0.54}
@u⇣i
@r
@Ti
@r
⇡
✓
1
mi P ri
◆
⇧ud
⇣i
R0 Q s
-π/4
0
θκ
π/4
π/2
Ti du⇣i /dr
⇡ 5%
vthi dTi /dr
Up-down asymmetry and rotation in TCV
0 D
B
@
⇧ud
⇣N i
⇧ud
⇣N i
D
t
R0N hQN i it
0 D
B
@
E
E
✓ =⇡/8
R0N hQN i it
✓ =⇡/8
⇧ud
⇣N i
E
1
t
R0N hQN i it
D
t
⇧ud
⇣N i
E
Camenen et al. (2010)
✓ = ⇡/8
1
t
R0N hQN i it
C
A
✓ = ⇡/8
C
A
⇡ 0.03
exp
sim
⇡ 0.06
2
22
1
1
11
0
0
ZZN
2
Z
Z
Tilted triangle simulation geometry in modified GS2
00
-1
-1
-1
-1
-2
-2
-2
-2
1
2
R0
4
R
5
2
3
4
R
5
6
1
2
RR0N
0
RRN
4
5
Constant during rotation
R0 , , rmin ,
Rmin , , rmin ,
R0 ,
, rmin ,
LT (rmin ) ,
LT (Rmax ) ,
LT (rmin ) ,
Ln (rmin )
Ln (Rmax )
Ln (rmin )
Effect of triangularity
R0 /a = {3.0}
= {0.7}
ud
<ΠζNi>t/(R0N<QNi>t)
0.06
0.03
0
-0.03
-0.06
rmin /a = {1.0}
-π/2
a/LT (rmin ) = {2.3}
a/Ln (rmin ) = {0.733}
-π/4
0
θδ or θκ
π/4
π/2
Up-down
asymmetric
Effect of Shafranov shift
ud
<ΠζNi>t/(R0N<QNi>t)
0.06
Add
shift
0.03
Add
shift
& β’
Pure
vertical
shift 0
0
Add
β’
π/4
θκ
π/2
Poloidal distribution of momentum flux
θκ = 0
Vertical Shift
β’ Only
0.02
0.02
0.02
0.01
0.01
0.01
0
0
0
-0.01
-0.01
-0.01
-0.02
-0.02
-0.02
-π
-π/2
0
θ
π/2
π
-π
-π/2
0
θ
π/2
π
0.02
0.02
0.01
0.01
0.01
0
0
0
-0.01
-0.01
-0.01
-0.02
-0.02
-0.02
0
θ
π/2
π
-π
-π/2
0
θ
0
θ
π/2
π
π/2
π
Shift Only
0.02
-π/2
-π/2
θκ = π/8
Shift and β’
-π
-π
π/2
π
-π
-π/2
0
θ
Relationship between heat and momentum flux
The w|| moment of the gyrokinetic eq. is roughly
✓
◆
0
1 ⇧⇣i
2 ~
@Qi
1 @B
1 ⌧nl R ⇡ 3 b̂ · r✓ @✓ + B @✓ Qi + 3rd order moments
θκ = 0
Vertical Shift
β’ Only
0.03
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0
0
0
-0.01
-0.01
-0.01
-0.02
-0.02
-0.02
-0.03
-0.03
-0.03
-π
-π/2
0
Shift θand β’
π/2
π
-π
-π/2
0
θκ =θ π/8
π/2
π
0.03
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0
0
0
-0.01
-0.01
-0.01
-0.02
-0.02
-0.02
-0.03
-0.03
-0.03
-π
-π/2
0
θ
π/2
π
-π
-π/2
0
θ
π/2
π
-π
-π/2
0
ShiftθOnly
π/2
π
-π
-π/2
0
θ
π/2
π
Conclusions
• Elongated flux surfaces with a π/8 tilt appear optimal for achieving rotation
• Tilt of vertical elongation seems to increase the critical gradients, but it
makes the heat flux more stiff
• The sustainable momentum gradient is about 5% that of the sustained
temperature gradient
• In ITER, this corresponds to an Alfvén Mach number of a couple
percent
• Momentum flux seems roughly proportional to the poloidal derivative of
the heat flux
Future work
• Determine the relationship between the geometric coefficients and the
heat/momentum fluxes
• Rigorously prove that the momentum flux must have two zeroes
• Understand the dramatic effect of β’ on momentum transport
• Further investigate the effect of tilt on energy transport
Questions?