Download 6 - Enea Frascati

6. IDEAL MHD STABILITY
The purpose of this section is to analyze the ideal MHD stability of the combined pinch+spherical torus
configuration of PROTO-SPHERA. Although saturated resistive MHD instabilities are required to produce
the helicity flow from the screw pinch to the spherical torus, the combined configuration must be stable in
the ideal MHD framework. If it were unstable any resistive local saturated displacement would turn into an
exponentially growing global ideal MHD mode.
The formation of the PROTO-SPHERA configuration can be described as a tunneling from an ideal MHD
stable configuration (the screw pinch at low longitudinal current) to a different ideal MHD stable
configuration (screw pinch+spherical torus). The tunneling could occur through an ideal MHD unstable
region.
The first evaluation is that of the rigid vertical shift and tilt stability, which does not yield an rigorous
result, but provides useful indications about two of the most dangerous ideal instabilities, which could be
present during the formation of PROTO-SPHERA. One result is that the thick casings of the constant
current poloidal field coils are sufficient to stabilize the rigid vertical instability during the formation phase.
The other result is that the design of the external fields, produced by the poloidal field coils, is such that it
stabilizes the rigid tilt stability during the formation phase, even if the effect of the thick casings is
neglected.
A new finite element method ideal MHD stability code has been developed in order to analyze the
combined pinch+ST configuration of PROTO-SPHERA. As a matter of fact no existing code was able to
compute the stability of a configuration in which closed and open field lines coexist. A number of
innovative features are contained in this code, which gives a  limit for PROTO-SPHERA included in the
range
[32%-70%].
6.1 RIGID STABILITY OF THE COUPLED
ST+PINCH CONFIGURATION
The first approach to the problem of the MHD stability of the PROTO-SPHERA configurations is to
estimate its stability with respect to rigid vertical shift and to rigid tilt modes. This approach is nonrigorous,
as it assumes arbitrarily the plasma perturbed displacement and does not search for the most unstable trial
function.
Nevertheless it gives useful indications, as the rigid tilt mode approximates one of the most dangerous
instability of a spheromak equilibrium confined by poloidal field coils [80]. The MHD equilibrium requires
the magnetic dipole moment of the toroidal plasma to be opposite to the magnetic dipole moment of the
confining poloidal field coils. An instability will try to align the dipoles, but the dipole alignment is
obviously incompatible with the MHD equilibrium.
Also the rigid vertical instability is well known in the physics of tokamak configurations and makes
impossible to exceed, in particular at medium/high aspect ratio, a limit to the elongation of the plasma cross
section. In the case of PROTO-SPHERA the toroidal plasma formation begins with an aspect ratio of about
A=2 and with an elongation of about =2 (see Fig. 60). Due to the presence of only two groups of poloidal
field coils it is impossible to reduce the elongation at the beginning of the formation. Therefore the initial
phase of the formation could be plagued by a vertical instability.
The analysis is performed by operating a rigid vertical shift and a rigid tilt of all the poloidal field coils and
computing the reaction force and torque acting on the unperturbed plasma:
F =  j Plasma  B dV ,


T =  r  jPlasma  B dV
where B is the B change due to the rigid displacement of the coils.
This calculation could overestimate the stability, as the most unstable mode will not be a rigid
displacement, but it could also underestimate the stability, as the stabilizing vacuum magnetic energy
perturbation is not accounted for.
In the evaluation of the rigid stability the effect of the conducting walls surrounding the plasma can either
be completely neglected; in this case the picture of the rigid vertical shift and of the rigid tilt is the one
shown in Fig. 90a.
The effect of the thick casings (2 ms time constant) of the poloidal field coils of group "B" (PF2, PF3.1 and
PF4) can instead be accounted for, by freezing the shape of the plasma disks near the electrodes: this means
that in the two separate electrode chambers the perturbed displacement is zero, as shown in Fig. 90b. In the
case of the vertical displacements, the induced axisymmetric reacting currents in the casings produce this
effect. Also the tilt instability is counter-reacted by the almost completely closed conducting path that joins,
through the electrodes, the PF2 and PF3 to the PF4 casings. This path can allow a freezing effect due to
tilted reacting currents.
Fig. 90a. Rigid vertical shift and tilt of the PROTO-SPHERA configuration.
Fig. 90b. Vertical shift and tilt of PROTO-SPHERA with frozen plasma disks.
In the case of frozen plasma disks, the analysis is performed by operating a rigid vertical/horizontal shift
and a rigid tilt limited to the PF1, PF5 and PF3.2 poloidal field coils and computing the reaction forces and
torques acting on the unperturbed plasma. The results of both cases are shown in Tab. 6:
Shift Z=4 mm,
Fz=+38.9 N
Tilt =0.5°
Unstable

Ip= 30 kA Ty=-2.25 Nm
Stable
Fz=-9.92 N Stabilized (t<2 ms) by thick casing of PF2, PF3.1 and PF4
Fz=+19.7 N

Unstable
Ip= 60 kA Ty=-9.22 Nm
Stable
Fz=-23.5 N Stabilized (t<2 ms) by thick casing of PF2, PF3.1 and PF4
Fz=-6.9 N
Stable
Ip=120kA Ty=-12.0 Nm
Stable
Stabilization by thick casing of PF2, PF3.1 and PF4 is not needed
Fz=-68.1 N
Stable
Ip=240kA Ty=-24.5 Nm
Stable
Stabilization by thick casing of PF2, PF3.1 and PF4 is not needed
Tab. 6. Results of the rigid stability evaluation for PROTO-SPHERA.
The shift results mean that the thick casings of PF2, PF3.1 and PF4, with a time constant of 2 ms, are
sufficient to stabilize the rigid vertical instability, which would operate during the first 250 s of the
toroidal plasma formation.
The tilt results mean that the magnetic dipole moment of PF2 and PF3 provides the larger part of the disk
shaping field near the electrodes. It can stabilize the rigid tilt instability, as it is aligned with the plasma
magnetic dipole moment and dominates over the opposite (destabilizing) dipole moment of PF1, PF5 and
PF4.
6.2 RIGOROUS IDEAL MHD STABILITY
A new ideal MHD stability code suited for treating magnetic configurations with closed and open field
lines has been built in collaboration with François Rogier (ONERA, Toulouse, France), under an Euratom
mobility scheme.
The code has been validated on Solovev tokamak equilibria [87], with fixed boundary, as well as with free
boundary in presence of surrounding vacuum regions.
The code contains a number of new features:
• the Boozer coordinates on open field lines are defined and continuously joined to closed field lines
Boozer coordinates at the pinch-ST interface;
• the treatment of magnetic separatrix at the pinch-ST interface;
• the boundary conditions at the pinch-ST interface;
• the perturbed vacuum magnetic energy in presence of multiple plasma boundaries;
• a 2D finite element method for accounting the perturbed vacuum energy.
6.3 BOOZER COORDINATES
The Boozer coordinates [88] (T, , ) for closed field lines give the "simplest" expression for the MHD
stability problem [89].
A list of definitions of quantities appearing in the Boozer coordinates follows:
• radial coordinateT=(toroidal flux)/2; (T)=rotational transform=1/q;
 I  f ; nonorthogonality term *  * T,;
•
•
•
covariant field: B  * T
•
The poloidal angle  and toroidal angle  (not coincident with the geometrical azimuth G) are fixed
normalized toroidal and poloidal currents I(T)= 0Ip/2 f(T)=RB;
contravariant magnetic field:
by the Jacobian:
B   T  

 .
g  f  + 
 I   B , Fig. 91.
2
Fig. 91. Radial coordinate T and poloidal angle  for the ST of PROTO-SPHERA.
6.4 BOOZER COORDINATES
ON OPEN FIELD LINES
The configuration is analyzed in term of the flux function 2RA:
• the closed field lines inside the ST span the range (X<<max);
• the open field lines in the pinch span the range (0<<X);
The following conditions are used to extend the Boozer coordinates into the pinch:
• the continuity of  T ,  and is imposed at the ST-SP interface (=X);
•
•
•
  must remain contiguous;
at the ST-SP interface the field lines 0   
()= 
X is imposed in =X;
the continuity of the rotational transform 
the continuity of I()=IX is imposed in =X, while f() is continuous from its equilibrium definition
(see Section 3.5).
Fixing the Boozer poloidal angle to  on the equator and calculating the length parameter s on the
separatrix, with s=0 on the lower electrode and s=seq on the equatorial plane:
the Boozer poloidal angle  is evaluated on the pinch side of the interface (=X+SP);
• the length parameter s=s0 at which =0 is determined at =X+SP;
• line integral definitions for I() and (T) are provided inside the SP
I  
1
s eq
 Bpeˆ p dlp ,
s
0
  


f  
s eq
s0
the latter goes to zero (see Fig. 92) as 

 
1
R2 B
ˆe p  dlp
;
p
   X in a very narrow layer near the separatrix
|-X|≈10 •|max-X|;
-5
Fig. 92. Behavior of () near the separatrix in an ST-pinch combined configuration.
on the ST side IX and X can be calculated at =X+ST, with ST≈10-3•|max-X|;
on the pinch side IX and X can be calculated at =X+SP, with SP≠ST ;
on the separatrix at the lower electrode is calculated as EL at =X-SP.
The choice of fixing =EL on the whole lower electrode determines:
• () all over the pinch flux surfaces;
•
•
•
•
X
the radial coordinate inside the pinch as  T   T
1

2
T
1
 d 

Fig. 93 shows that this procedure makes the Boozer poloidal angle  quite accurately continuous at the STpinch interface. This interface corresponds, upon the magnetic separatrix, to the range of Boozer poloidal
angles [X<<2-X], where X represents the point where the outermost magnetic surface of the pinch
reaches its largest R value near the X-point.
Fig. 93. Boozer coordinates for PROTO-SPHERA.
6.5 ENERGY PRINCIPLE
The linearized normal-mode equation describing the ideal MHD stability can be expressed in a variational
*
form [90]. Considering  the displacement vector away from the equilibrium (  its complex conjugate)
with time dependence eit, the perturbed kinetic energy of a plasma with scalar mass density 0 is:
 
Wk * ,  


 dV    
*
0
Vp
and the perturbed potential magnetic energy of the plasma is:
    dV   F
with F  being the self-adjoint force operator. The variational principle states that any function  which
Wp * ,  
*
Vp
makes stationary the Rayleigh quotient
     ,  = 
 
W  , 
Wp * , 
2
*
2
*
k
is an eigenfunction of the normal-mode equation with eigenvalue 2.
For an arbitrary displacement  the perturbed magnetic field is
principle is written as [91]:

Wp 

with

Vp
C 2
2
dV  p    D   T
 

 


C    B   0

j    
T
 T
2
T

2
and






Q    B and the energy

j   


D 
 B  



T
2

T
T
T


2



It is convenient to decompose the displacement  in terms of the normal  , binormal and parallel
components as follows:
B  T I


 e   
  2   B .
2
B
B


The compressible displacement ( ,,) away from the equilibrium is expanded in a trigonometric
Fourier series of modes; each mode is labeled by an index l, which corresponds to a poloidal number ml and
a toroidal number nl:
 =   l  T  sinm l   n l 
 =  l  T  cosm l   nl 

l
l
 =   l  T  cosm l   n l 
l

The reduction to a sine component for  and to a cosine component for  and  is permitted if up-down
symmetric equilibria are assumed, as is the case in the combined ST+SP configurations of PROTOSPHERA.
By using the Fourier expansions, the compressible perturbed plasma kinetic energy is expressed as a
c
quadratic form of the displacements l l andl and the compressible part Wp of the perturbed plasma
potential magnetic energy is expressed as a quadratic form of the displacements l, l and l and of the
radial derivative of the normal displacement lT.
For the stability calculation the problem is discretized radially covering the T interval inside the ST
X
i
ST
ST
[0,  T ] by an equidistant mesh with the mesh points  T  i  T , for i=0,… N  , where
X
ST
X
N ST
X . The value  T of the separatrix is carefully excluded from the mesh in
 T  T  ST 2
order to avoid all the problems with the singularity of the Boozer coordinates. Also the degenerate X-points
max
sitting on the symmetry axis T =  T is excluded from the mesh (see Fig. 56), by putting the last mesh
ST
point with i= N 
max
X
max
 NSP
 symm 2
symm . Inside the SP the T interval [  T ,  T ]
 at  T   T
X
i
is covered by an equidistant mesh with the mesh points  T   T
for
where
  SP 2
X  i  NST
  1 T ,
SP
ST
X
NSP  1SPT   max
T   T  ST
i= N 
SP
 1,…, N ST
  N ,
2
X   sy mm 2
sy mm .
The radial behavior of l(T), l(T), l(T) and lTis approximated by a one dimensional Finite
ST
Element Method. For l(T) the hat functions ei(T), i=0,…, N 
ST
 NSP
 are used. For l(T), l(T) and
SP
lTthe piecewise constant functions ci-1/2(T), i=1,…, N   N are used. The finite hybrid element
 
 l i   l T i ,
 l   T   l i  l i-1  T i  Ti1.
representation
is
then:


l i  l Ti-1/2 ,

l i  l Ti-1/2

and
6.6 BOUNDARY CONDITIONS
AT THE INTERFACE
The discontinuity of the normal component ( ) of the perturbed displacement  at the ST-pinch interface
is forbidden in ideal MHD, as it would give rise to flux generation and would provide an unavoidable
divergence of the perturbed plasma potential energy. On the other hands, discontinuities of the tangential
components (,) of perturbed displacement  are allowed for at the ST-SP interface.
W x  2 K x , where the total potential magnetic energy matrix W and the
i
i
i
(positive definite) kinetic energy matrix K are symmetric and blockdiagonal, x   l , l ,  l  is the
The eigenvalue problem is
eigenvector and 2 is the eigenvalue. The system is solved by an inverse iteration method, which finds all
the lowest discrete eigenvalues and the corresponding eigenvectors.
The stability calculation outlined here is however not yet complete, as a matter of fact the (stabilizing)
perturbed vacuum magnetic energy is missing. However an incorrect stability calculation can be performed
anyway: the result is that the pinch is kink unstable and that the torus is tilt unstable, albeit the tilt
instability exhibits a 'peeling' mode character, see Fig. 94.
Fig. 94. Arrow displacement plot of the PROTO-SPHERA configuration, which is found
unstable when the perturbed vacuum magnetic energy is not accounted for.
The displacement shown is a global growing mode.
6.7 VACUUM MAGNETIC ENERGY
WITH MULTIPLE PLASMA BOUNDARIES
The treatment of the vacuum magnetic energy contribution in PROTO-SPHERA is complicated by the
presence of three plasma-vacuum surfaces (see Fig. 95):
v1
X
v3
max
v1
ST
X ;
 = 
X (i= N  ), with rotational transform 
 T =  T  ST 2
v2
X
ST
v2
with rotational transform 
X ;
 = 
X (i= N  +1),
 T =  T  SP 2 
T =T

ST
SP
v3
symm .
symm 2
symm (i= N   N ), with rotational transform 
 = 
In general couplings in the matrix elements between the three surfaces can exist.
Fig. 95. The three plasma-vacuum surfaces of PROTO-SPHERA.
n
inG
˜ (  e
In the vacuum region, the perturbed potential 
obeys in cylindrical coordinates (R,G,Z) the equation:
˜ n is the 3D scalar magnetic potential),

˜ n    
˜ n n n
1    
˜
R



 
    , with boundary conditions:
R R   R  Z
R
˜n
 
n
 0,
on the perfectly conducting axisymmetric shells Sc around the plasma;
Sc
˜ n

n S v i 

    m
vi

k
 k T
0 g
vi
v i 
k
 nk
v i 
 T
cos m vi  n  vi

k
k
 , on each   
v i
T
surfaces, where
    G is the difference between  and the geometrical azimuth G.
v3
The vacuum energy of the magnetic surface  T is an additional term on the ( l-k) components of the W
v1
v2
potential energy matrix, but it is decoupled from the surfaces  T and  T by the closed conducting path
of the Screw Pinch current Ie, which flows into the electrodes, then into the return legs and finally into the
coaxial feeder at the top(/bottom) of the machine. (see Fig. 96). This conducting path is assumed to be
axisymmetric in the MHD stability calculation.
Fig. 96. The closed conducting path of the Screw Pinch current Ie decouples the two small
vacuum regions on top and bottom of PROTO-SPHERA (shown shaded).
v1
v2
The other two surfaces  T and  T are very near, have been chosen in such a way as to have the same
rotational transform 
X and furthermore the continuity condition on the normal perturbed displacement
v1
v2
imposes l(  T )=l(  T ). These choices eliminate any coupling between the vacuum energy of the
v1
v2
magnetic surfaces  T and  T .
˜ n is therefore solved in the larger vacuum region shown in Fig. 96, on a unique plasma
The equation for 
u
v2
surface; such a surface is composed by the pinch surface (  T  T ), in the two disconnected ranges of
u
v1
poloidal Boozer angles [EL=<X] and [2-X=< =<EL], and by the ST surface (  T  T ), in
the intermediate range [Xu=<X] at the ST-SP interface.
The vacuum problem is solved both in the smaller as well as in the larger vacuum region by a 2D finite
element method that can fit any shape of the plasma and of the surrounding conductors (see Fig. 97).
Fig. 97. Example of 2D finite element mesh used to solve the vacuum problem.
The perturbed vacuum magnetic energy is then integrated in the energy principle in its final form :
 
3
Wp  * ,     Wv(i) 
  ,   
 v(i) v(i)
i1
*
 
*
Wk   ,  
 
2 *
2
   ,  =  .
v3
The vacuum energy contribution of the magnetic surface  T
components of the W potential energy matrix:
Wv3 
1


  l  Tv3  symmm l
0 l,k

are additional terms on the (l-k)

 v3

n l symmm k  n k R lk  k  v3
T 
v1
v2
The vacuum energy contributions of the two surfaces  T and  T are additional terms on the (l-k)
components of the W potential energy matrix:
Wv1 
Wv2 
1




  l  
 m l  n l 
 m k  n k R lk  k  .
l,k
11
  l Tv1 
Xm l  n l 
X m k  n k R lk  k  v1
T  ;
 0 l,k
1
 0
v2
T
where the coupling coefficients
22
X
X
v2
T
v3
11
22
R lk , R lk and R lk are calculated by the 2D finite element method.
2
In the numerical method, which solves the eigenvalue problem W x   K x , the vacuum magnetic
energy enters as additional terms in the potential magnetic energy matrix W. This terms influence only the
ST
ST
N ST
N ST 1
matrix elements which multiply the i= N  component  l  , the i= N  +1 component  l 
and the
ST
last (i= N 
ST
N
 NSP
 ) component  l
 N SP

of the eigenvector


x  l i, l i, l i .
6.8  LIMIT
The first results of the MHD stability code show that PROTO-SPHERA (Ip=240 kA, Ie=60 kA) is stable at
T=32% (Fig. 98). The wall position does not seem to be critical, as even a wall at infinity is sufficient for
stabilization.
Fig. 98. Arrow displacement plot of the PROTO-SPHERA configuration, which is found to
be stable at T=32%. The displacements shown are oscillation on resonant q surfaces.
Pushing up the beta value to T=70% (Fig. 99) PROTO-SPHERA (Ip=240 kA, Ie= 60 kA) becomes unstable
through a global mode, which is kink-like inside the pinch and tilt-like inside the torus. However neither
the stability limits to the compression, nor the effects of the shell positions and of the p( ) and q()
profiles on the ideal MHD stability have yet been explored.
Fig. 99. Arrow displacement plot of the PROTO-SPHERA configuration, which is found to
be unstable at T=70%. The displacement shown is a global growing mode.