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Equilibrium, Formation and
Performances of the
PROTO-SPHERA Experiment
Presented by
P. Micozzi
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Outline of the Talk
1) PROTO-SPHERA Equilibria
2) PROTO-SPHERA Formation Sequence:
i) ST current ramp-up
ii) formation time-scale & expected plasma performaces
iii) eddy currents and their influence
3) Chandrasekhar-Kendall-Furth configurations inside PROTO-SPHERA
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PROTO-SPHERA configuration
Screw Pinch (SP, open field lines ending upon the electrodes) surrounded by a Spherical Torus
(ST, closed field lines): SP and ST have a common magnetic separatrix
Coupled equilibrium calculations performed in free boundary:
•
Spherical co-ordinates (r=radius, =colatitude, =azimuth)
•
Scalar flux function   2RA    r    
•
Force-Free Screw Pinch with p() & f() continuous at the ST-SP interface
 j may have jumps at the interface
•
ST-SP interface defined by the common separatrix
•
Ip and p inside the ST and the longitudinal Pinch current Ie are input, I Pinch
inside the

SP is an output
M in (r) r -n  M en (r) r n+1  sin  P1n (cos )

n1

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Input parameters for the PROTO-SPHERA Reference Equilibrium
- Ie = 60 kA (i.e. BT0 = 0.08 T at R = 0.15 m) ; Ip = 180 kA (Ie/Ip = 1/3) ; p=0.22
2
2 2
2
- p() = pe = constant ;
f () = (0Ie/2 ( /x )
inside SP
1.1
2
2
1.1
- p()pe+Cp(x) ;
f () = (0Ie/2 +Cf(x)
inside ST
ST=22%
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Two important requirements to be fulfilled by the PROTO-SPHERA equilibria:
1)
Tokamak-like safety factor profile of ST: q0~0.94, q95~2.6 at the edge
2)
Strong jump of the surface averaged relaxation parameter <>=0< j • B /B2>
between SP and ST  Helicity Injection (~35 m-1 inside SP, ~10 m-1 inside ST)
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Simulation of a TS-3 equilibrium
Ie = 40 kA; Ip = 50 kA (Ie/Ip = 4/5) ; p=0.6 (to match the measured plasma shape)
2
Same functional form of p() & f () as in PROTO-SPHERA

of the SP (larger radial extension)
-

In comparison with the PROTO-SPHERA equilibria, TS-3 shows:
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1)
Spheromak-like safety factor profile of ST: q0~0.62 and q95~0.95 at the edge
 essentially due to the lower elongation and to the higher aspect ratio
2) Almost flat profile of the surface averaged relaxation parameter <>=0< j • B /B2>
 essentially due to the weaker compression which lowers <> inside the SP
Furthermore the presence of the SP plasma disks in PROTO-SPHERA
increases the rigid stability of the configuration towards tilt & vertical shift
PROTO-SPHERA

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Design philosophy of the poloidal coils
Only 2 groups to simplify the power supplies:
Group 'A'  fast varying compression coils
with thin Inconel case (≈200 s)
Group 'B'  fixed current coils for the
shaping of the SP with thick
metal case (≈2 ms) to stabilize
the plasma disks near the
electrodes
Current density in the PF coils limited to
jPF≤2 kA/cm2 (dT/dt < 2 ˚C/sec)
 simple water cooling system
(due to the presence of hot electrodes)
and high duty-cycle (1 shot every 5')
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Screw Pinch Formation
•
Filling pressure pH~10-3÷10-2 mbar
•
Cathode filaments heated to 2600 °C,
current in group 'B' reaches I'B'=1875 A
(no current in group 'A')
•
Breakdown of the Hydrogen gas with
Ve~100 V on anode
•
Longitudinal arc current limited to
Ie~8.5 kA
•
Pinch discharge kink stable
(qPinch≥2, Pinch~2.6 m-1)
•
Low power required (<1 MW),
plasma can be sustained for 1 s
 SP density control through gas
puffing from the anode
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Spherical Torus Formation
The formation of the ST is obtained by the
kink destabilization of a screw pinch, through
an increase of the longitudinal arc current, as
demonstrated on the TS-3 experiment
(University of Tokyo)
Linear and non-linear phase of a kink
unstable screw pinch, with longitudinal field
BZ and 'toroidal' field B
 qPinch = 2 Pinch BZ/ LPinch B
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ST Formation Procedure
Field geometry at the ST formation is the
same as in TS-3 (formation 100% successful)
SP current pushed up to Ie=60 kA in ~500 s
 Screw Pinch goes kink unstable (qPinch«1)
After ~100 s the group 'A' PF coils current
starts to increase (up to I'A'=0.7 kA in ~1 ms)
After a further delay of 100 s the ST starts to
form (as in the TS-3 experiment), reaching
Ip=120 kA in ~1 ms
Success of the TS-3 formation scheme helped
by the compression coils flux swing
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TS-3
PROTO-SPHERA
PROTO-SPHERA provides a loop voltage
Vloop~10 V, for about 1 ms
 enough to push up Ip to 120 kA in ~1 ms
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Formation sequence starts at 250 s
after the ST formation
(Ip=30 kA=Ie/2 )
Equilibrium parameter are:
aspect ratio A= 1.80,
elongation = 2.17,
safety factor at the edge q95= 3.4,
paramagnetic effect BT/BT0= 1.20,
toroidal pinch current IPinch = 179 kA
Relaxation parameter in the SP is PinchRsph=6
and its volume average in the ST is
STRsph>Vol~2.45(Rsph=0.35 m)
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Fast phase of the formation sequence lasts
1000 s after the ST formation
(Ip= 120 kA = 2•Ie)
Equilibrium parameter are:
aspect ratio A= 1.32,
elongation = 2.16,
safety factor at the edge q95= 2.8,
paramagnetic effect BT/BT0= 2.10,
toroidal pinch current IPinch = 310 kA
Relax. parameter in the SP is PinchRsph=10.5
and its volume average in the ST is
STRsph>Vol~3.85
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Formation ends
(Ip= 240 kA = 4•Ie)
after the ST formation
Helicity Injection is required
Equilibrium parameter are:
aspect ratio A= 1.21,
an elongation = 2.35,
safety factor at the edge q95= 2.8,
paramagnetic effect BT/BT0= 3.10,
toroidal pinch current IPinch = 407 kA
Relax. parameter in the SP is PinchRsph=14
and its volume average in the ST is
STRsph>Vol~4.2
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PROTO-SPHERA performances at Ip=180 kA
•
The case chosen for evaluating the plasma performances of the ST of PROTO-SPHERA is
intermediate between the time slice at 1.0 ms (Ip=120 kA) and the one at 10 ms (Ip=240 kA)
•
This choice is due to the fact that it is a high current case (Ip=180 kA), but still ideal MHD
stable with a large beta value (ST=22%)
•
Density of the ST chosen in order to match ST (below the Greenwald limit of
PROTO-SPHERA, <nG> = 3.0•1020 m-3)
•
E evaluated from the semi-empirical Lackner-Gottardi L-mode plateau-scaling
•
Total power for helicity injection taken as PHI=4•PSToh, (PSToh=IpVloop = equivalent ohmic
power to sustaining the ST and comes from the Spitzer conductivity). It is assumed that
half of this power is dissipated inside the ST, therefore P =2•PSToh is used in the E scaling
•
E and <Te> calculated iteratively by adjusting Vloop in PSToh
With <ne>=0.5•1020 m-3 & Zeff=2  <T>=140 eV (Vloop=0.8 V, PHI=580 kW, ELG~1.6 ms)
Alfvén time is Aaxis=Raxis/vA~0.55 s; resistive time
R =0a2/~69 ms

The Lundquist number of PROTO-SPHERA is S=R /A~1.2•105

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TS-3 Lundquist number
•
The time-scale connecting the sequence of PROTO-SPHERA equilibrium calculation
is derived scaling up the time-scale obtained in the TS-3 flux-core spheromak experiment
•
The analyzed TS-3 equilibrium (Ie=40 kA, Ip=50 kA) is the same shown previously
With <ne>=0.55•1020 m-3 & Zeff=2 (density limit <nG>=1.8•1020 m-3)
<T>=55 eV (Vloop=2.5 V, PHI=500 kW, ELG~0.27 ms)
Alfvén time is Aaxis=Raxis/vA~0.75 s; resistive time
R =0a2/~6.5 ms

The Lundquist number of TS-3 is S=R /A~9•103
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Formation time-scale
•
Magnetic reconnections are required to form the ST from the screw pinch, therefore the
time for the formation of PROTO-SPHERA must be extrapolated from the experimental
results of the TS-3 flux-core spheromak by using the reconnection time-scale
•
TS-3 needed 80 s to reach a ratio Ip/Ie=50 kA/40 kA.
The Sweet-Parker reconnection theory predicts that the reconnection rate scales like S1/2
•
This prescription for the formation time form=1.12•S1/2A, applied to TS-3,
gives the measured time form=80 s
•
The time-scale for the formation of PROTO-SPHERA can be calculated applying the
prescription form=1.12•S1/2A =210 s in order to reach the same ratio Ip/Ie=5/4 (Ip=75 kA)
•
Therefore a minimum resistive MHD time-scale of 350 s is required to reach Ip=120 kA
•
However the effect of the eddy currents over all the passive components inside the vacuum
vessel introduces a further 650 s delay, therefore an overall time-scale of 1 ms is required
for achieving Ip=120 kA
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Eddy Currents Effect
on the Formation Scenario
• Effect of the eddy currents on the
PROTO-SPHERA formation scenario
evaluated by the code ANSYS
• Active/passive elements included in the
model are axisymmetric and continuous
• ST and SP plasma shapes fixed for all the
formation sequence
• All plasma current values at the various
time slices provided by the equilibrium code
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•
Waveforms of the PF coil currents imposed (ideal power supplies)
•
Induced currents determined iteratively (current waveforms of the PF coils kept constant).
As a first guess, the time sequence described by the "ideal" equilibria has been used:
Ip=30 kA (t=t0+250 s); Ip=60 kA (t=t0+500 s); Ip=120 kA (t=t0+1 ms); Ip=240 kA (t=t0+10 ms)
•
Resulting eddy currents fed back into equilibrium code:
 Process iterated until plasma currents & passive element currents convergence (few %)
•
The result of this procedure is that the formation scenario is compatible with the eddy
current of the PROTO-SPHERA device:
Ip=15 kA (t=t0+250 s); Ip=50 kA (t=t0+500 s); Ip=120 kA (t=t0+1 ms); Ip=240 kA (t=t0+10 ms)
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•
At t=t0+500 s a comparison between the "ideal" and the "effective" equilibria (with the
same currents in the poloidal field coils) shows negligible differences
•
At the end of the fast rise (t=t0+1 ms, Ip=120 kA) a similar comparison shows that the eddy
current effects are weak enough to allow for obtaining the same plasma current as in the
"ideal" case
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In conclusion, the eddy current analysis provides the following results:
1)
The rise of the ST toroidal current is slightly delayed.
This effect could be probably minimized with an earlier discharge of the capacitor bank
feeding the PF coils of "Group A".
2)
It is not possible to rise Ip up to the maximum value of 240 kA in a time shorter than 10 ms.
This is essentially due to the eddy currents induced in the PF coils with thick metal cases.
If a faster rise is applied, the screw pinch would not fit the electrodes. This effect implies
that, after the first ms, the ST current must raise up to Ip=240 kA quite slowly (~10 ms),
relying upon an effective helicity injection.
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Chandrasekhar-Kendall-Furth
Configuration
inside PROTO-SPHERA?
The load assembly of PROTO-SPHERA
could host a CKF configuration with
limited changes inside the machine
The aim of this experiment is to prove
that such magnetic configuration can be
obtained through the destabilization of a
screw pinch and then it can survive as
long as the inductive drive of the
"compression coils" is present
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Required modifications of the PF coils
• removal of the PF2 & PF3.1 "divertor coils"
• removal of the PF6.2 "focalizing coils"
• displacement of the PF6.1 "focalizing coils"
near the PF1 "compression coils"
• each poloidal field coil fed independently
(6 feeders instead of 2).
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Breakdown requires some
modifications of the electrodes
• slightly shorter distance between the
electrodes: 1.54 m instead of 1.63 m
• larger vertical thickness of the electrodes:
Z=12 cm instead of Z=5.4 cm
The total current flowing at the start-up inside
the PF coils would be IPF=  I =234 kA.
Comparing the signs and magnitudes of the
currents flowing inside the PF coils at the
start-up with those of the final configuration,
it turns out that the sudden variation of these
currents could provide approximately enough
magnetic flux to the plasma as to induce the
full toroidal plasma current of the final
configuration.
N PF
i 1
i
PF
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Final CKF equilibrium
• Free-boundary equilibrium calculations
assuming for the poloidal spheromak current
the value Ie=60 kA (same of the longitudinal
SP current of PROTO-SPHERA):
the total toroidal current in the main
spherical torus would be Ip=327 kA (larger
than in the case of PROTO-SPHERA, which
has Ip=120÷240 kA flowing inside the ST)
• The configuration is calculated to be ideal
MHD stable at ST=50%
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Conclusions
1) PROTO-SPHERA Equilibria:
•
•
The flux-core spheromak magnetic configuration is designed in order to obtain a
tokamak-like q profile inside ST (spherical torus-like transport & stability) and
a gradient of the <> parameter between SP and ST (helicity injection)
A comparison with TS-3 equilibria (simulated with the same assumptions)
shows that these two requirements were not fulfilled in TS-3
2) PROTO-SPHERA Formation Sequence:
•
•
•
•
The PF coils design is as simple as possible (only two feeders)
The formation time-scale has been extrapolated from TS-3 in the framework of the
resistive MHD (Sweet-Parker reconnection theory)
The expected performances are coherent with an hot plasma regime (S~1•105)
The formation sequence is compatible with the eddy currents (only small delay)
3) Chandrasekhar-Kendall-Furth configurations inside PROTO-SPHERA:
•
Minor changes of the load assembly could allow for a "proof of principle"
PROTO-SPHERA Workshop
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