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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

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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 TX 

• Coordinates are defined through the

SP (up to the symmetry axis R=0)

using the force-free equilibrium

equation:

df/d+()dI/d=0

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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:

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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)

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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

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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 = 30240 kA, i.e. Ip/Ie = 0.54, A=R/a= 1.81.2

•Three value of ST 20<P>Vol/<B2Vol considered: 10%, 20% and 30%

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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

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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

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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)

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•Boozer coordinates (T,,)

joined at interfaces

Inside Tori:

Surrounding coupled mode:

Surrounding Internal mode:

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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

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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

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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