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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
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