Ideal MHD Stability Results
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Ideal MHD Stability Results
for PROTO-SPHERA
and CKF Configurations
Presentedby
P. Micozzi
Outline of the Talk
1)Ideal MHD Code for "Flux-Core Spheromak"
Configurations
2)PROTO-SPHERA Stability Analysis
3)Stability Analysis of the
Chandrasekar-Kendall-Furth Configurations
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
MAGNETIC COORDINATES WITH OPEN FIELD LINES
Ideal MHD stability code treats configurations with closed and open field lines
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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Combined equilibrium calculation
New feature: Boozer coordinates
joined at SP-ST interface
Boozer coordinates (T,,)
(Tradial, poloidal, toroidal≠ Ggeometric)
= 0Ip/2
= RbT
T=tor. flux/2 in ST, (T)=rotat.tran.
Jacobian
Nonorthogonal:
* from
• Spherical Torus (ST), closed lines
•Screw Pinch (SP), open line
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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Associazione Euratom-ENEA sulla Fusione
ENERGY PRINCIPLE
STABLE code: displacement (normal, binormal and parallel )
:•••
Fourier expansion of displacement
nl=n is a pure toroidal number
up/down symmetry
ml is a spectrum of poloidal harmonics
Boundary conditions at ST-SP interface in ideal MHD:
1)Constraint of continuous normal displacement
2)Tangential displacements jump (no constraint)
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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Boozer coordinates can be chosen
almost arbitrary inside the Pinch
Radial coordinate T inside the SP:
(T)
• Coordinates join "smoothly" at
ST-SP interface [T=;X≤≤2-X]
imposing TX
• Coordinates are defined through the
SP (up to the symmetry axis R=0)
using the force-free equilibrium
equation:
df/d+()dI/d=0
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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Perturbed vacuum magnetic energy
Using the perturbed scalar magnetic potential , the vacuum contribution
is expressed as an integral over the plasma surface:
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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Perturbed vacuum magnetic energy
with multiple plasma surfaces
The vacuum contribution is present on
three plasma surfaces:
(i=),
(i=+1)
(i=)
In vacuum the 2D scalar potentials
) obeys:
with B.C. , on conductors
,
on all the three surfaces
G(geometrical-Boozer) azimuth
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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2D finite element method to solve the equation for the perturbed scalar potential
THE PROBLEM OF THE SYMMETRY AXIS (R=0)
inadequate for plasmas at R=0
1) like on symmetry axis, so 0to avoid divergences
2)()≈r1/2+ on the degenerateX-point(B=0), so 0 to avoid divergences
1)=0 at the symmetry axis T= easy to impose, but questionable!
(after degenerate X-point, T= does not coincide with symmetry axis)
=0 at T= impossible to impose, as no (/T) in energy principle
SOLUTION OF THE R=0 PROBLEM (STABLEC code)
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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Only way to solve the symmetry axis problem is a change of variables:
In terms of the new variables (,,) the perturbed displacement becomes:
All the divergences on the symmetry axis are avoided
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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Prices to pay:
•expression of the perturbed potential energy Wp much more complicated
•slower convergence of 2 by varying range of poloidal numbers [mmin,mmax]
Stability Results for PROTO-SPHERA
•Formation sequence of PROTO-SPHERA:
ST toroidal current Ip = 30240 kA, i.e. Ip/Ie = 0.54, A=R/a= 1.81.2
•Three value of ST 20<P>Vol/<B2Vol considered: 10%, 20% and 30%
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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•At ≈10% PROTO-SPHERA stable up to:
Ip/Ie=4(Ip=240 kA),A=1.2if ()=0 is imposed in STABLE code
Ip/Ie=2(Ip=120 kA),A=1.3if ()≠0 (with both STABLE & STABLEC)
•At ≈20% PROTO-SPHERA stable up to:
Ip/Ie=3 (Ip=180 kA), A=1.25 if ()=0 is imposed in STABLE code
Ip/Ie=2 (Ip=120 kA), A=1.3 if ()≠0 (with both STABLE & STABLEC)
•At ≈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
PROTO-SPHERA (Ip=120 kA, Ie=60 kA, ≈20%, A=1.3)
toroidal number n=1, poloidal harmonics m [-5,15]
()=0()≠0
Stable oscillatory motions on resonant q surfaces
PROTO-SPHERA (Ip=180 kA, Ie=60 kA, ≈20%, A=1.25)
toroidal number n=1, poloidal harmonics m [-5,15]
()=0()≠0
Stable motionsKink of the SP, Tilt of the ST
PROTO-SPHERA (Ip=210 kA, Ie=60 kA, ≈20%, A=1.25)
toroidal number n=1, poloidal harmonics m [-5,15]
()=0()≠0
Kink of the SP,Tilt of the ST
Comparison with the TS-3 Experiment
TS-3 results extremely important since:
1)The only experiment with similar formation scheme and without close fitting shell,
that has sustained a "Flux-Core Spheromak" for tens of Alfvén times
2)Strong analogiesbetween TS-3 and PROTO-SPHERA, but also differences:
i)ST the rotational transform is quite different in the two experiments
PROTO-SPHERA TS-3
ii)SP the plasma disk near the electrodes is absent in TS-3
TS-3 (Ip=50 kA, Ie=40 kA, ≈12%, A≈1.7)
toroidal number n=1, poloidal harmonics m [-5,15]
()=0()≠0
Stable oscillatory motions on resonant q surfaces
TS-3 (Ip=100 kA, Ie=40 kA, ≈14%, A≈1.5)
toroidal number n=1, poloidal harmonics m [-5,15]
()=0 ()≠0
Kink of the SP,Tilt of the ST
CONCLUSIONS for PROTO-SPHERA
•Innovative ideal MHD stability code developed for "Flux-Core Spheromak"
•Code has been validated against analytic Solovev equilibria and reproduces
satisfactorily the "Flux-Core Spheromak" experimental results of TS-3
•More pessimistic (but realistic) results obtained without constraint on R=0
(()≠0), in presence of degenerate X-point on symmetry axis
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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SUMMARY for PROTO-SPHERA
1)With moderate (≤20%) in the ST, the Pinch dominates the stability:
compression A≥1.3, Ip/Ie≤2
2)Degenerate X-point on symmetry axis improves stability:
in TS-3 compression A≈1.6, Ip/Ie≈1
3)If ()=0 in presence of degenerate X-point at R=0 is imposed,
compression A≈1.2, Ip/Ie≈4(with ≈10% in the ST) are obtainable
upper/lower conducting shells close-fitting the pinch plasma (limiters)
Stability of the Chandrasekar-Kendall-Furth Configurations
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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Unrelaxed (≠0, ≠0) CKF Equilibria
B.C.=0=const. only at the edge
Analysis performed keeping fixed the shape
of the plasma boundary & the full toroidal
current of the configuration
Equilibrium profiles such that are
concentrated in a region 0<≤c,
where c=x+[1- •(axis-x)] with 0<≤1
Fixed pressure jump between plasma edge
and ST magnetic axis (paxis/pedge=5),
variable jump of between edge and axis
controls the ratio IST/Ie ()
c controls the value of
Two ST values considered: 1/3 & 1
Investigated toroidalnumbersn=1,2,3
(n=0 vertical stability not yet investigated)
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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•Boozer coordinates (T,,)
joined at interfaces
Inside Tori:
Surrounding coupled mode:
Surrounding Internal mode:
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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•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:
•In fact the regularity of the perturbed magnetic energy
at the symmetry axis suggest to use N=2 both for Wp and Wv
•The representation adopted for PROTO-SPHERA is equivalent to the
choice N=1 (since like on symmetry axis), so there are some
evidences that the results obtained for PROTO-SPHERA could be pessimistic
Ideal MHD Stability Results (wall at ∞) for CKF: =1
Ideal MHD Stability Results (wall at ∞) for CKF: =1/3
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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Stability behaviour Vs. q0 at ST=1, IST/Ie=3
Stable Motion Stable Motion Stable Motion
Resonance on SP Resonance on ST Resonance on Sec. Tori
Stability behaviour Vs. q0 at ST=1, IST/Ie=5
Unstable Motion Stable Motion Unstable Motion
Global mode on SP Resonance on ST Global mode on ST
Stability behaviour at ST=1/3, low IST/Ie
Unstable Motion Unstable Motion
InternalGlobal mode on SP Internal Global mode on SP
PROTO-SPHERA Workshop Frascati, 18-19/03/2002
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Conclusions for the CKF Ideal MHD Stability
•CKF configuration shows large stability region at unitary even without close fitting walls
the surrounding "spheromak" plasma has a strong stabilizing effect on the ST
•With ST=1 only flat pressure profiles (~1) are allowed if IST/Ie>4
(high IST/Ie means low and small between plasma edge and ST magnetic axis);
if 1.5< IST/Ie<4 (i.e. 2.7<<4) even peaked pressure profiles (high ) show stability
•With ST =1/3 the region showing stability with peaked pressure profiles is extended to
1.2< IST/Ie <5.5 (i.e. 2<<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/Ie4 , ~2.8 , ST ~20% , relatively peaked pressure profiles
PROTO-SPHERA Workshop Frascati, 18-19/03/2002