boozer

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boozer

Associazione Euratom-ENEA sulla Fusione
Presented by
P. Micozzi
PROTO-SPHERA Workshop
Frascati, 18-19/03/2002
2
1) Ideal MHD Code for "Flux-Core Spheromak"
Configurations
2) PROTO-SPHERA Stability Analysis
3) Stability Analysis of the
Chandrasekhar-Kendall-Furth Configurations
Frascati, 18-19/03/2002
3
In PROTO-SPHERA resistive MHD instabilities are required
to inject magnetic helicity from Screw Pinch (SP) into Spherical Torus (ST),
but the combined configuration must be stable in ideal MHD
New ideal MHD stability codes*, built in collaboration with
François Rogier (ONERA de Toulouse, France)
*
Validated upon the well-known stability results of analytic Solovev equilibria with fixed and free
boundary conditions in presence of vacuum regions surrounding the plasma
The codes contain a number of new features:
• Boozer coordinates on open field lines are defined and joined to the closed
field lines Boozer coordinates at the ST-SP interface
• Boundary conditions at the ST-SP interface
• Vacuum magnetic energy in presence of multiple plasma boundary
• 2D finite element method for accounting the perturbed vacuum energy
• Presence of plasma on the symmetry axis
Frascati, 18-19/03/2002
4
MAGNETIC COORDINATES WITH OPEN FIELD LINES
Ideal MHD stability code treats configurations with closed and open field lines
New feature: Boozer coordinates
joined at SP-ST interface
Boozer Coordinates (yT,q,f)
(yT radial, q poloidal,
fG geometric)
r
r f toroidal≠
I(y T ) = (1 2p )ò Ñ Ù B × dST = m0Ip/2p
r r
f (y T ) = (1 2 p)ò Ñ Ù B × dSp = RBT
yT=tor. flux/2p in ST, i/ (yT)=rotat.tran.
Jacobian g = èæ f + i/Iøö /B2
r ö
r 2 r
æ r
ç
Nonorthogonal: g * Ñy T = Ñ y T ×çç Ñq-i/ Ñf÷÷÷
è
r
r
r
rø
b* from B = b*Ñy T + IÑq + fÑf
Combined equilibrium calculation
•
•
Spherical Torus (ST), closed lines
Screw Pinch (SP), open line
Frascati, 18-19/03/2002
5
ENERGY PRINCIPLE
y
STABLE rcode:
r displacement (normal x , binormal h and parallel m)
r
B Ù Ñy T æ Ih
y r r
ör
yr
x = x ey + h
+ 2 - m B : x = x • Ñy T ,
2
èB
ø
B
r r
r r
r
æ
ö
ç
=h= x • è Ñq - /iÑfø , m
g x • Ñf
Fourier expansion of displacement
x = å x l (y T ) sin(m l q - n l f )
Ü
nl=n is a pure toroidal number
h = å hl (y T ) cos(m l q - n l f)
Ü
up/down symmetry
Ü
ml is a spectrum of poloidal harmonics
y
l
l
(
m = å m l (y T ) cos m l q - n l f
l
)
Boundary conditions at ST-SP interface in ideal MHD:
y
1) Constraint of continuous normal displacement x
Frascati, 18-19/03/2002
6
2) Tangential displacements h, m jump (no constraint)
Frascati, 18-19/03/2002
7
Boozer coordinates can be chosen
almost arbitrary inside the Pinch
• Coordinates join "smoothly" at
ST-SP interface [yT= ;qX≤q≤2p-qX]
imposing
q
q T qX


 
• Coordinates are defined through the
SP (up to the symmetry axis R=0)
using the force-free equilibrium
equation:
df/dy+ (y)dI/dy=0
Radial coordinate yT inside the SP:
( <yT<
)
Frascati, 18-19/03/2002
8
Frascati, 18-19/03/2002
9
PERTURBED VACUUM MAGNETIC ENERGY
Using the perturbed scalar magnetic potential F, the vacuum contribution
r
r
m0
is expressed as an integral over the plasma surface: dWv = òò F ÑF × dSy
2
Sy
with multiple plasma surfaces
The vacuum contribution is present on
three plasma surfaces:
(i= N ST
y ),
v2
X
y T = y T + eSP 2 p/i X
(i= N ST
y +1)
v3
max
ST
SP
y T = y T - e symm 2 p /isymm (i= N y + N y )
X
y v1
/X
T = y T - eST 2 p i
Perturbed vacuum magnetic energy
Frascati, 18-19/03/2002
10
In vacuum the 2D scalar potentials
˜ nc cos(nf ) + F
˜ nssin (nf )) obeys:
F=F
n cs
æ
n{sc ö
˜
2
˜
1 ¶ ç R ¶F ÷ + ¶ F - n 2 F˜ n{cs = 0
R ¶R çè ¶R ÷ø ¶Z 2 R2
G
G
ì
í
î
with B.C.
¶F
( i)
y
S
˜ ns
¶F
¶n
{
¶n
( i)
y
)
[
( )]
(i)
xk y(i)
T (i/ mk - nk )
[
( )]
S
=å
k
=å
k
= 0 , on conductors
Sc
)
i
/i( mk - nk
( )(
xk y(i)
T
˜ nc
¶n
˜
¶F
n sc
(i )
(i)
cosm
q - nk n q
r
k
m0 g(i) Ñy(Ti)
( )
(i)
(i)
sinm
q - nkn q ,
r
k
m0 g(i) Ñy(Ti)
on all the three y v(i)
surfaces
T
nq ffG (geometrical-Boozer) azimuth
Frascati, 18-19/03/2002
11
2D finite element method to solve the equation for the perturbed scalar potential
Frascati, 18-19/03/2002
12
AXIS (R=0)
rTHE PROBLEM
r r OF THE SYMMETRY
r
r
Ñ yT æ
B Ù Ñy T æç b*xy -Ih ö
y
y
ö
x = x r 2 + è h- g *x ø
+ç
- m÷÷ B inadequate for plasmas at R=0
2
B2
B
ø
è
Ñy T
r
r
y
1) Ñy T ® 0 like Ñy T » R on symmetry axis, so x ®0 to avoid divergences
1/2+e
2) h( y T ® y max
on the degenerate X-point (B=0), so h®0 to avoid
T )≈r
divergences
y
1) x =0 at the symmetry axis yT=
(after degenerate X-point, yT=
2) h=0 at yT=
easy to impose, but questionable!
does not coincide with symmetry axis)
impossible to impose, as no (¶h/¶yT) in energy principle
Frascati, 18-19/03/2002
13
SOLUTION OF THE R=0 PROBLEM (STABLEC code)
Only way to solve the symmetry
axis problem is a change of variables:
y
x
x˜ = r
Ñy T
˜=
h
h
B
r
˜
˜
In terms of the new variables ( x , h,m) the perturbed displacement x becomes:
r
r
æ
æ
r
r
ö r
ö
r
r
Ñ
y
Ñ
y
T
T
ç
ç
Ñy T ˜
÷ B Ù Ñy T
÷
I
˜
˜
˜
˜ -m B
+ çh- g *
+ çb*
x=x r
x÷
x+ h
÷
B
B ÷ B
B
ç
÷
Ñy T ç
ø
ø
è
è
All the divergences on the symmetry axis are avoided
Prices to pay:
•
•
expression of the perturbed potential energy dWp much more complicated
slower convergence of w2 by varying range of poloidal numbers [mmin,mmax]
Frascati, 18-19/03/2002
14
STABILITY RESULTS FOR PROTO-SPHERA
•
•
•
•
Formation sequence of PROTO-SPHERA:
ST toroidal current Ip = 30®240 kA, i.e. Ip/Ie = 0.5®4, A=R/a= 1.8®1.2
Three value of ST b=2m0<P>Vol/<B2>Vol considered: 10%, 20% and 30%
At b≈10% PROTO-SPHERA stable
up to:
y
max
Ip/Ie=4 (Ip=240 kA), A=1.2 if x ( y T )=0 is imposed in STABLE code
y
Ip/Ie=2 (Ip=120 kA), A=1.3 if x ( y max
T )≠0 (with both STABLE & STABLEC)
At b≈20% PROTO-SPHERA stable up to:
y
T )=0 is imposed in STABLE code
y
T )≠0 (with both STABLE & STABLEC)
•
At b≈30% PROTO-SPHERA stable only up to:
Ip/Ie=1 (Ip=60 kA), A=1.5, but at higher Ip the ST alone is fixed-boundary
unstable
Frascati, 18-19/03/2002
15
PROTO-SPHERA (Ip=120 kA, Ie=60 kA, b≈20%, A=1.3)
toroidal number n=1, poloidal harmonics mÎ [-5,15]
y
max
x ( y T )=0
y
max
x ( y T )≠0
Stable oscillatory motions on resonant q surfaces
Frascati, 18-19/03/2002
16
Upper/lower limiters needed on SP
y
max
x ( y T )=0
Stable motions
y
max
x ( y T )≠0
Kink of the SP, Tilt of the ST
Frascati, 18-19/03/2002
17
y
max
x ( y T )=0
y
max
x ( y T )≠0
Frascati, 18-19/03/2002
18
COMPARISON WITH THE TS-3 EXPERIMENT
TS-3 results extremely important since:
1)
2)
The only experiment with similar formation scheme and without close fitting shell,
that has sustained a "Flux-Core Spheromak" for tens of Alfvén times
Strong analogies between TS-3 and PROTO-SPHERA, but also differences:
i)
ST Þ the rotational transform is quite different in the two experiments
PROTO-SPHERA
TS-3
ii) SP Þ the plasma disk near the electrodes is absent in TS-3
Frascati, 18-19/03/2002
19
TS-3 (Ip=50 kA, Ie=40 kA, b≈12%, A≈1.7)
y
max
x ( y T )=0
y
max
x ( y T )≠0
Stable oscillatory motions on resonant q surfaces
Frascati, 18-19/03/2002
20
TS-3 (Ip=100 kA, Ie=40 kA, b≈14%, A≈1.5)
y
max
x ( y T )=0
y
max
x ( y T )≠0
Frascati, 18-19/03/2002
21
SUMMARY for PROTO-SPHERA
1) With moderate b (≤20%) in the ST, the Pinch dominates the stability:
compression A≥1.3, Ip/Ie≤2
2) Degenerate X-point on symmetry axis and "plasma disk" improve stability:
in TS-3 compression A≈1.6, Ip/Ie≈1
y
3) If x ( y max
T )=0 in presence of degenerate X-point at R=0 is imposed,
compression A≈1.2, Ip/Ie≈4 (with b≈10% in the ST) are obtainable
Þ upper/lower conducting shells close-fitting the pinch plasma (limiters)
4) Pressure profile shape (relatively peaked) not changed in the analysis
Þ CKF results will show that flat pressure profiles improve stability
Frascati, 18-19/03/2002
22
Frascati, 18-19/03/2002
23
Stability of the Chandrasekhar-Kendall-Furth Configurations
r
r
Unrelaxed ( Ñm ≠0, Ñp≠0) CKF Equilibria
of the plasma boundary & the full toroidal
current of the configuration
r
r
Equilibrium profiles such that Ñp & Ñm are
concentrated in a region 0<y≤yc,
where yc=yx+[1- a •(yaxis-yx)] with 0<a≤1
Fixed pressure jump between plasma edge
and ST magnetic axis (paxis/pedge=5),
variable jump of m between edge and axis
ST
Dm controls the ratio IST/Ie (µ q 95 )
yc controls the value of qST
0
r r 2
B.C.Þ m=m0 j · B B =const. only at the edge
Analysis performed keeping fixed the shape
Two bST values considered: 1/3 & 1
Investigated toroidal numbers n=1,2,3
(n=0 vertical stability not yet investigated)
Frascati, 18-19/03/2002
24
•
Boozer coordinates (yT,q,f)
joined at interfaces
Inside Tori:
xy = å x l (y T ) sin (m lq - n l f )
l
h = å hl (y T ) cos(ml q - n l f )
l
m = å m l (y T ) cos(m lq - nl f )
l
Surrounding coupled mode:
xy = å x l (y T ) sin (3m l q - n l f)
l
h = å hl (y T ) cos(3m l q - nl f)
l
m = å m l (y T ) cos(3m l q - n l f)
l
Surrounding Internal mode:
xy = å x l (y T ) sin (m lq - n l f )
l
h = å hl (y T ) cos(ml q - n l f )
l
m = å m l (y T ) cos(m lq - nl f )
l
Frascati, 18-19/03/2002
25
•
The problem to avoid divergences at the symmetry axis (R=0)
is the same as in PROTO-SPHERA
•
The solution is still a change of variable, but, for the CKF ideal MHD
analysis, the choice of the new variables has been improved:
xy
x˜ = N
R
˜=h
h
B
•
In fact the regularity of the perturbed magnetic energy
at the symmetry axis suggest to use N=2 both for dWp and dWv
•
The representation adopted for PROTO-SPHERA is equivalent to the
r
r
choice N=1 (since Ñy T ® 0 like Ñy T » R on symmetry axis), so there are some
evidences that the results obtained for PROTO-SPHERA could be pessimistic
Frascati, 18-19/03/2002
26
Ideal MHD Stability Results (wall at ∞) for CKF: b =1
Frascati, 18-19/03/2002
27
Ideal MHD Stability Results (wall at ∞) for CKF: b =1/3
Frascati, 18-19/03/2002
28
Stability behaviour Vs. q0 at bST=1, IST/Ie=3 (q95~2.8)
Stable Motion
Stable Motion
Resonance on SP
Resonance on ST
Stable Motion
Resonance on Sec. Tori
Frascati, 18-19/03/2002
29
Stability behaviour Vs. q0 at bST=1, IST/Ie=5 (q95~2.2)
Unstable Motion
Global mode on SP
Stable Motion
Resonance on ST
Unstable Motion
Global mode on ST
Frascati, 18-19/03/2002
30
Stability behaviour at bST=1/3, low IST/Ie
Unstable Motion
Internal Global mode on SP
Unstable Motion
Internal Global mode on SP
Frascati, 18-19/03/2002
31
Conclusions for the CKF Ideal MHD Stability
•
CKF configuration shows large stability region at unitary b even without close fitting walls
Þ the surrounding "spheromak" plasma has a strong stabilizing effect on the ST
•
With bST=1 only flat pressure profiles ( qST
~1) are allowed if IST/Ie>4;
0
ST
if 1.5< IST/Ie<4 (i.e. 2.7<q ST
95 <4) even peaked pressure profiles (high q 0 ) show stability
•
With bST =1/3 the region showing stability with peaked pressure profiles is extended to
1.2< IST/Ie <5.5 (i.e. 2<q ST
95 <5) and the stability region with flat pressure profiles is enlarged
•
The stability region found for the CKF configurations strongly supports the aim of the
PROTO-SPHERA experiment:
2< IST/Ie<4 ,
q ST
95 ~2.8 ,
bST ~20%
Frascati, 18-19/03/2002

boozer

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