Ideal MHD stability calculations in presence of magnetic separatrices

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Ideal MHD stability calculations in presence of magnetic separatrices

Franco Alladio, Alessandro Mancuso, Paolo Micozzi, François Rogier

To cite this version:

Franco Alladio, Alessandro Mancuso, Paolo Micozzi, François Rogier. Ideal MHD stability calculations

in presence of magnetic separatrices. 2004. �hal-00001811v2�

Ideal MHD stability calculations in presence of magnetic separatrices

F. Alladio ^a , A. Mancuso ^a , P. Micozzi ^a and F. Rogier ^b

a Associazione EURATOM-ENEA, CR-ENEA Frascati, Rome, Italy

b ONERA-CERT, Toulouse, France (Dated: October 21,2004)

The use of magnetic coordinates, in which field lines are straight, defines the simplest approach to the numerical study of the ideal MHD stability of magnetoplasma equilibria: it allows the use of 1-dimensional radial Finite Elements, while adopting a Fourier decomposition in the poloidal angle. However the magnetic coordinates become singular in presence of magnetic separatrices. This difficulty becomes particularly relevant in the stability analysis of a simply connected axisymmetric plasmas (Flux-Core-Spheromak or Chandrasekhar-Kendall-Furth configurations), which embed a magnetic separatrix with regular X-points (B≠0) and are bounded by an edge magnetic separatrix that includes a part of the symmetry axis (R=0) and is limited by two singular X − points (B=0). The approach taken in the present paper is that of maintaining the 1- dimensional radial Finite Elements, while using an asymptotic analysis of the perturbed plasma displacement near the separatrices. The permissible asymptotic limits for the perturbed displacement are derived: they have simple analytical expressions in Boozer magnetic coordinates.

An intensified numerical radial mesh following Boozer magnetic coordinates is set up near the magnetic separatrices; it requires: a logarithmic fit to the rotational transform near the embedded magnetic separatrix; a minimum distance between the radial mesh and both separatrices; finally an extended spectrum of poloidal mode numbers in the Boozer angle. The numerical results are compared “a posteriori” with the permissible asymptotic limits for the perturbed displacement. The radial displacement variable (radial contravariant component of the perturbed displacement) is found to be always near to its most unstable asymptotic limit, while the full range of permissible asymptotic behaviours can be obtained for the binormal and parallel displacement variables (which are combinations of the poloidal and toroidal contravariant components of the perturbed displacement).

I. INTRODUCTION

A very few examples of numerical results for the ideal MHD stability analysis of (even axisymmetric) plasma equilibria endowed with magnetic separatrices have been published in the literature [1]. The reason is that numerical analysis of the ideal MHD stability of plasma equilibria is usually undertaken expanding the perturbed plasma displacement ^ξ in magnetic coordinates and then solving the normal mode equation [2]

^{δ W p} ( ^{ξ * , ξ} ) = ω ² δW _k ( ξ ^* , ξ ) ^, (1) where ^{δ W p and δ W k} are the perturbed potential and kinetic plasma energies and ω is the eigenvalue, through a 1D radial finite element method that is based upon a Fourier expansion in the poloidal angle of the magnetic coordinates.

The magnetic coordinates [3] are however plagued by divergent metrics coefficients if magnetic separatrices are

present within (or at the boundary of) the plasma; in particular the choice of the poloidal angle becomes singular at the

ordinary X-point of a magnetic separatrix, leading to the use of a large number of Fourier harmonics in the poloidal

angle, which implies difficulties in the numerical convergence. The problem with magnetic separatrices is doubled if

the aim is to evaluate the ideal MHD stability of simply connected axisymmetric MHD plasma equilibria, which embed

a first magnetic separatrix with regular X − points ( B -0 ) and vanishing rotational transform, while being bounded by a

second edge magnetic separatrix, which includes a part of the symmetry axis (R=0) and is limited by two singular

Fig. 1. Comparison between: a) a CKF configuration; b) the PROTO−SPHERA Flux-Core-Spheromak.

X − points ( ^{B = 0} ). Such general axisymmetric toroidal MHD plasma equilibria, with poloidal flux ^{ψ = ∫ B ⋅ d S p}

( ψ = 0 on the symmetry axis), normalized toroidal flux ψ _Τ = ∫ B ⋅ d S _Τ 2 π ( ψ _Τ = 0 on the main magnetic axis), rotational transform ^{ι ψ /} ( ) ^{Τ = −} ( ^{d ψ d ψ} _Τ ) ^{2 π} , normalized toroidal current, I ( ) ψ _Τ = ∫ ∇ ∧ B ⋅ d S _Τ 2 π and normalized poloidal current ^f ( ) ^{ψ Τ} = ∫ ∇ ∧ B ⋅ d S _p 2 π ), can be expressed by non − orthogonal Boozer magnetic coordinates [4] ( ψ _Τ =radial coordinate, θ =poloidal angle, φ =toroidal angle, with Jacobian

^{g =} [ ^f ( ) ^{ψ Τ + / ι ψ} ( ) Τ ^I ( ) ^ψ Τ ] ^{B 2} and non-orthogonality coefficient β _* ( ψ _Τ , θ ) , defined by B = β _* ∇ ψ

_Τ

+ I ∇ θ + f ∇ φ ).

The first example of such MHD equilibrium is a Chandrasekhar-Kendall-Furth (CKF) configuration [5], shown in Fig. 1a. The embedded magnetic separatrix ( ψ _Τ = ψ _Τ ^X ), while enclosing a main spherical torus (ST, carrying a total current with toroidal component I ST ) along with two secondary tori (SC, each carrying a total current with toroidal component I SC ) on top and bottom, is on its turn enclosed within a toroidal shell surrounding plasma (SP, carrying a total current with poloidal component I e ), which is bounded by an edge magnetic separatrix ( ψ _Τ = ψ _Τ ^max ).

A second example (see Fig. 1b) is a plasma equilibrium, where the embedded separatrix ( ) divides a spherical torus (ST, with and closed flux surfaces, carrying a current with toroidal component I

ψ _Τ = ψ _Τ ^X

0 ≤ ψ _Τ ≤ ψ _Τ ^X ST ) from a

screw pinch discharge (SP, and open flux surfaces ending upon ring electrodes, carrying a plasma electrode current I

0 ≤ ψ _Τ ≤ ψ _Τ ^max

e , which is mainly poloidal). Such a configuration has been devised theoretically under the name of

"Flux − Core − Spheromak" (FCS) [6] and an example is the PROTO − SPHERA system [7], proposed at CR − ENEA Frascati.

II. DIVERGENCES OF MAGNETIC COORDINATES NEAR SEPARATRICES At the embedded magnetic separatrix ( ψ _Τ = ψ _Τ ^X ) the rotational transform ι / vanishes like:

lim

ψ

_T

→ ψ

_T^X

/ ι ψ ( ) _Τ ^{∝ 1}

ln ψ _Τ ^X − ψ _Τ , (2)

while its derivative with respect to the radial Boozer coordinate diverges. For flux surfaces approaching the separatrix, the component of ^{∇ θ} upon the flux surfaces diverges near the regular X − point ( ^{B -0 ) like 1 ψ Τ}

X − ψ _Τ ln ψ _Τ ^X − ψ _Τ , implying that, although the Boozer poloidal angles θ have to converge there, each

Fig. 2. Boozer coordinate mesh for a Flux-Core-Spheromak configuration ( ψ T – radial coordinate, θ - poloidal angle), shown in the poloidal plane.

regular X-point can be roughly indicated by its “central angle” θ X (see Fig.2). The “integrated residual shear”

^{γ * = ∇ ψ Τ ⋅} ( ^{∇ θ − / ι ∇ φ} ) ∇ ψ _Τ ² diverges near the X − point like 1 ψ _Τ ^X − ψ _Τ ln ψ _Τ ^X − ψ _Τ , while vanishing far from the X − point like ^{1 ln ψ Τ X − ψ Τ}

2

. Correspondingly ∇ ψ _Τ ² vanishes near the X − point like ψ _Τ ^X − ψ _Τ ln ψ _Τ ^X − ψ _Τ and diverges far from the X − point like ^{ln ψ Τ X − ψ Τ}

2

.

At the edge magnetic separatrix ( ^{ψ Τ = ψ Τ max} ) the rotational transform ι / is finite:

^ψ

^T

^{lim → ψ}

^Τmax

^{/ ι ψ Τ}

( ) max ^{= ι /} symm -0 . (3) Near the singular X-points, using local spherical coordinates (r, ϑ , ϕ ), the magnetic field vanishes as ^{B ∝ r , ∇ ψ} Τ

vanishes like r ² , the integrated residual shear diverges like 1/r (while being regular far from the singular X − points) and the component of upon the flux surfaces vanishes like r; this explains why no convergence of poloidal Boozer angles θ occurs into the lower singular X − point, which is therefore uniquely labeled by the poloidal angle

γ _*

∇ θ

θ _{B= 0} (the upper one being 2 π − θ _B=0 , see Fig. 2).

A number of metric coefficients, defined as dot products between the covariant basis vectors: ^{g ij = e i ⋅ e j , or}

some among their radial and poloidal derivatives furthermore diverge on either of the two magnetic separatrices.

III. PERMISSIBLE ASYMPTOTIC LIMITS FOR THE PERTURBED DISPLACEMENT In order to avoid the singularities connected with the magnetic separatrices, some ideal MHD stability codes [1]

have adopted 2-dimensional Finite Elements, still aligned with the flux surfaces, either all over the mesh or just near

the magnetic separatrix. The approach taken in this paper is instead that of maintaining the 1-dimensional radial Finite

Elements and of using an asymptotic analysis of the perturbed plasma displacement near the separatrices. As a matter

of fact the logarithmical dependence (2) of the rotational transform near the embedded separatrix creates asymptotic limits to the perturbed displacement ^ξ for low-n global MHD perturbations, while the vanishing of plays the same role for the edge separatrix. The actual vector components of

B

ξ are most easily expressed through the three (normal, binormal and parallel) orthonormal vectors:

ξ = ^{ξ ⋅ ∇ ψ T}

∇ ψ _T

⎡

⎣ ⎢ ⎤

⎦ ⎥ ^{∇ ψ} _{∇ ψ} ^T

T

+ ^ξ ⋅ ( B ∧ ∇ ψ _T )

∇ ψ _T B

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥

B ∧ ∇ ψ _T

∇ ψ _T B + ^ξ ⋅ B B

⎡

⎣ ⎢ ⎤

⎦ ⎥ ^B _B ^. (4) The perturbed plasma displacement can be better expressed substituting the covariant radial basis vector

e _ψ = ^{∇ ψ Τ}

∇ ψ _Τ ² + ^{β *} B ² B − ^{γ *}

B ² ( B ∧ ∇ ψ _Τ ) , into (4):

ξ = ^ξ

ψ

∇ ψ

_T

⎛

⎝ ⎜ ⎞

⎠ ⎟ ^{∇ ψ}

^T

∇ ψ

_T

+ ^{∇ ψ}

^T

B ^η

ψ

− γ

_*

ξ

^ψ

( )

⎛ ⎝ ⎞

⎠ ^B _{∇ ψ} ^{∧ ∇ ψ}

^T

T

B + ^β

^*

^ξ

ψ

+ I η

^ψ

− µ B

²

B

⎛

⎝ ⎜ ⎞

⎠ ⎟ ^B

B . (5) At the embedded separatrix, the ideal MHD stability analysis requires [8] the continuity of the radial plasma perturbed displacement variable, ^{ξ ψ = ξ ⋅ ∇ ψ Τ} ; the other two variables, η ^ψ = ξ ⋅ ( ∇ θ − / ι ∇ φ ) and ^{µ = − g ξ ⋅ ∇ φ can}

instead be discontinuous. In general at an edge magnetic separatrix (irrespectively from whether it has regular or singular X-points) it can be shown that the radial derivatives of the perturbed potential plasma energy and of the free − boundary vacuum perturbed magnetic energy compensate each other, if no surface equilibrium plasma current flows at the plasma − vacuum boundary:

δ W _p δ W _v

∂

∂ψ _T ( δ W _v )

ψ

_T^edge

= − ^∂

∂ψ _T ( ) δ W _p

ψ

_T^edge

. (6)

Taking into account all the sources of divergence connected with the asymptotic behaviours of ^{∇ ψ} Τ ^, ∇ θ / ^, ι , γ _* ,

^{d / ι d ψ Τ} , of the metric coefficients ^{g ij = e i ⋅ e j} and of some among their radial and poloidal derivatives, the permissible asymptotic limits for the perturbed displacement variables ( ξ ^ψ , η ^ψ , µ ), and therefore for the vector displacement components (5) ( [ ^{ξ ⋅ ∇ ψ T ∇ ψ T} ] ⁼ [ ^{ξ ψ ∇ ψ} T ] ^, [ ^{ξ ⋅} ( ^{B ∧ ∇ ψ} T ) B ∇ ψ _T ] ⁼ [ ( ^{η ψ − γ} * ξ ^ψ ) ∇ ψ T B ] ^,

[ ^{ξ ⋅ B B} ] ⁼ [ ( ^β * ξ ^ψ + I η ^ψ − µ B ² ) ^B ] ), can be calculated analytically at both separatrices, imposing:

1) The regularity of all plasma perturbed energies δ W _p ( ξ ^* , ξ ) ^and δW _k ( ξ ^* , ξ )

2) The regularity of the perturbed displacement ξ .

At the embedded magnetic separatrix ( ψ _Τ = ψ _Τ ^X ) the results for the displacement variables are:

lim

ψ

_T

→ ψ

_T^X

ξ ^ψ < ο ^{ψ T}

X − ψ _T ln ψ _T ^X − ψ _T

⎛

⎝ ⎜ ⎞

⎠ ⎟ ^; (7)

ο ¹

ln ψ _T ^X − ψ _T

⎛

⎝ ⎜ ⎞

⎠ ⎟ ≤ lim

ψ

_T

→ ψ

_T^X

η ^ψ < Ο ¹

ψ _T ^X − ψ _T ln ψ _T ^X − ψ _T ^3/2

⎛

⎝ ⎜ ⎞

⎠ ⎟ ^; (8)

lim

ψ

_T

→ ψ

_T^X

µ ≤ Ο ¹

ψ _T ^X − ψ _T ln ψ _T ^X − ψ _T ^3/2

⎛

⎝ ⎜ ⎞

⎠ ⎟ ^; (9)

but, in case of divergence, the parallel and the binormal displacement variables compensate each other:

^ψ

^T

^{lim → ψ}

^TX

^{µ − η}

ψ I B ² ≤ Ο ( ) 1 . (10)

At the edge magnetic separatrix, respectively near the symmetry axis ( ψ _Τ = ψ _Τ ^max , R=0) or the singular X-points ( ψ _Τ = ψ _Τ ^{max ,} θ = θ _B=0 ), the results for the displacement variables are:

;

^ψ

^T

^{→ ψ lim}

^Tmax

^{, R → 0 ξ}

ψ ≤ ο ( ) R lim

ψ

_T

→ ψ

_T^max

, θ→θ

_B=0

ξ ^ψ ≤ ο ( ) r ² ; (11)

^ψ

^T

^{→ ψ lim}

^Tmax

^{, R → 0 η}

ψ < Ο ¹ R

⎛ ⎝ ⎞

⎠ ^; ψ

_T

→ ψ ^lim

_T^max

, θ→ θ

_B=0

η ^ψ ≤ ο ( ) r ; (12)

^ψ

^T

^{→ ψ lim}

^Tmax

^{, R → 0 µ < Ο}

1 R

⎛ ⎝ ⎞

⎠ ^; ψ

_T

→ ψ

_T^max

^lim , θ→θ

_B=0

µ < Ο ¹ r ^{3/ 2}

⎛ ⎝ ⎞

⎠ ; (13) but, in case of divergence, the parallel and the binormal displacement variables compensate each other:

^ψ

^T

^{→ ψ lim}

^Tmax

^{, R → 0 µ − η}

ψ I B ² ≤ Ο ( ) 1 ; lim

ψ

_T

→ ψ

_T^max

, θ→ θ

_B=0

µ − η ^ψ I B ² ≤ Ο ¹ r

⎛ ⎝ ⎞

⎠ . (14)

IV. METHOD FOR COMPUTING THE STABILITY IN PRESENCE OF MAGNETIC SEPARATRICES

To solve the normal − mode equation (1) for the ideal MHD stability of an axisymmetric plasma equilibrium [8], the perturbed displacement variables ( ξ ^ψ , η ^ψ , µ ) away from the equilibrium are expanded in a trigonometric Fourier series of modes; each mode is labeled by an index l, which corresponds to a poloidal number m

l

and a (fixed as separable) toroidal number n:

^ξ

ψ = ξ _l

l

∑ ( ) ψ _Τ sin m ( _l θ − n φ ) ,

η ^ψ = η _l

l

∑ ( ) ψ _Τ cos m ( _l θ − n φ ) , µ = µ _l

l

∑ ( ) _{ψ Τ} cos m ( _l θ − n φ ) . (15) The reduction to a sine component for ξ ^ψ and to a cosine component for η ^ψ and µ is permitted if up − down symmetric equilibria are assumed.

Numerical accuracy requires that stability calculations always use an inhomogeneous (intensified) numerical radial mesh near the magnetic separatrices; an example is illustrated in Fig. 3a. As the field lines, labeled by , are in correspondance at the ST-SP interface along the embedded magnetic separatrix, the continuity of the radial variable ξ

^{θ 0 = θ − ι ψ / ( ) φ = constant} _ψ

implies the continuity of the angles θ and φ and forces the matching condition:

^{ι / SP} ( ^ψ

^X

^{− ε}

^SP

) = / ι ^ST ( ψ

_X

+ ε

_ST

) = / ι _X , (16) between the two adjacent radial mesh points on the two sides of the separatrix. For any MHD axisymmetric equilibria, in terms of the poloidal flux ψ , the rotational transform near the separatrix ( ψ = ψ _X ) exhibits the logarithmic behaviour

^{ψ→ ψ lim}

^X

^{ι ψ / ( ) = − Α ln C} ( ^{ψ − ψ X} ) . The two coefficients A and C can be found by a fitting procedure down to the minimum distance from the magnetic separatrix where ι ψ / ( ) can still be calculated numerically, typically

^{ψ − ψ X ≈} ( ^{1 ⋅ 10 −3 − 5 ⋅ 10 −3} ) ^ψ X (see Fig. 3b and 3c). The values of the coefficient C, calculated from the fitting procedure, indicate that at the minimum values of ψ − ψ _X where the rotational transform can be calculated numerically, the constant |lnC| still amounts to 17-40% of ln ψ − ψ _X .

A characteristic of FCS examples is that the numerically calculated rotational transform on the ST side is about the

same as the rotational transform on the SP side: ι / ^SP _FCS ( ψ _X − ε _SP ) ≈ / ι _FCS ^ST ( ψ _X + ε _ST ) , for ε SP ≈ε ST ; this allows for a simple

solution of the matching condition (16), where ε SP is derived as a function of ε ST .

Fig. 3. a) Refined mesh for the ideal MHD stability analysis of a CKF configuration, shown in the poloidal plane. Fit (line) to numerical rotational transform (dots) near the embedded magnetic separatrix: b) on the SP side; c) on the ST side.

On the other hand for CKF configurations, the rotational transform on the ST side is about twice as much as the rotational transform on the SP side (see Fig. 3b and 3c): ι / _CKF ^ST ( ψ _X + ε _ST ) ≈ 2 ⋅ ι / _CKF ^SP ( ψ _X − ε _SP ) for ε SP ≈ε ST and therefore, given ε SP , the matching condition (16) is used to extrapolate ε ST to quite smaller values: ε ST ≈  ^ ε SP .

Figure 4a shows the effect of the distance between the radial mesh and the separatrix upon the most unstable eigenvalue calculated by the numerical code; the results are expressed as the ratio between the square of the growth rate of the instability ω ² and the squared Alfvén rate: ω _A ² = B ₀ ² / µ ₀ ρ

₀

R ₀ ² , where R 0 is the position of the magnetic axis, upon which the toroidal field strength is B 0 and ρ 0 is the plasma density (assumed to be constant).

Fig. 4. a) Behaviour of ω ² / ω _A ² , for an unstable FCS example with n=1, as a function of the radial distance from the embedded

magnetic separatrix of the last ST radial mesh point, expressed through its safety factor q( ψ X + ε ST ). b) Behaviour of ω ² / ω _A ² , for a

different FCS example, as a function of the poloidal mode number spectrum.

The radial distance ε ST between the radial mesh and the separatrix has been decreased, in the FCS example of Fig. 4a, from ε ST =2 • 10 ^-2 ψ max (which corresponds to a safety factor q X =1/ ι / X =4.4) down to ε ST =2 • 10 ^-3 ψ max (which corresponds to a safety factor q X =1/ ι / X =5.5).

An acceptable convergence of the most unstable eigenvalue is obtained only when ε ST ≤1 • 10 ^-2 ψ max , which typically corresponds to ^{ψ T X − ψ T ≈ 2.5 ⋅ 10 −2 ψ T X} . The study of the eigenvalue convergence as a function of the poloidal mode numbers spectrum is illustrated in Fig. 4b with another FCS example. The choice of for the poloidal spectrum represents a reasonable convergence. ^{m l = -5,15} [ ]

V. RESULTS

The numerical results are compared “a posteriori” with the permissible asymptotic limits for the perturbed displacement. Figure 5 shows the numerical perturbed displacement arrow plot in the poloidal plane around the embedded separatrix, for a CKF configuration unstable to the n=1 toroidal mode number. Figure 5 confirms that:

• The normal vector component of (4) vanishes far from and near the X − point:

lim

ψ

_T

→ ψ

_T^X

, θ−θ

_X

→ π / 2

ξ ⋅ ∇ ψ _T

∇ ψ

_T

< ο ^{ψ T}

X − ψ _T

ln ψ _T ^X − ψ _T ^{3/ 2}

⎛

⎝ ⎜ ⎞

⎠ ⎟ ^, _ψ ^lim

T

→ ψ

_T^X

, θ→θ

_X

ξ ⋅ ∇ ψ _T

∇ ψ _T < ο ¹ ln ψ _T ^X − ψ _T

⎛

⎝ ⎜ ⎞

⎠ ⎟ . (17)

• The binormal vector component of (4) can be regular far from the X-point but must vanish near the X-point:

lim

ψ

_T

→ ψ

_T^X

, θ−θ

_X

→ π / 2

ξ ⋅ B ∧ ∇ ψ _T

∇ ψ _Τ B = Ο ( ) 1 , lim

ψ

_T

→ ψ

_T^X

, θ→θ

_X

ξ ⋅ B ∧ ∇ ψ _T

∇ ψ _Τ B ≤ ο ¹ ln ψ _T ^X − ψ _T

⎛

⎝ ⎜ ⎞

⎠ ⎟ . (18)

Fig. 5. Perturbed displacement arrow plot for an n=1 unstable CKF configuration, shown in the poloidal plane.

Fig. 6. Fourier components of the perturbed displacement variables ( ^{ξ l s} , ^{η l c} , ^{µ l c} with m

^l

=0,1,2) inside the ST for an n=1 unstable mode of a CKF configuration.

The only exception to the vanishing asymptotic limits at regular X-points is the perturbed displacement of the (n=0) axisymmetric mode ^{ξ n= 0} , which can exhibit finite normal and binormal vector components:

lim

ψ

_T

→ ψ

_T^X

, θ→θ

_X

ξ _{n =0} ⋅ ∇ ψ _T

∇ ψ _T , ξ _{n= 0} ⋅ B ∧ ∇ ψ _T

∇ ψ _Τ B = Ο ( ) 1 . (19)

• Although not visible from the poloidal plane arrow plot of Fig. 5, the parallel vector component of (4) is regular along the whole embedded magnetic separatrix:

lim

ψ

_T

→ ψ

_T^X

ξ ⋅ B B ≤ Ο ( ) 1 _. (20)

The numerical results for the main Fourier components of the displacement variables ( ξ ^ψ , η ^ψ , µ ) are shown in Fig. 6 on the ST side of the embedded separatrix, for a CKF configuration unstable to the n=1 toroidal mode number.

The vanishing of the normal variable ξ ^ψ obtained from the numerical computations is always quite near to the most unstable vanishing asymptotic limit (7), lim

ψ

_T

→ ψ

_T^X

ξ ^ψ ~ ο ( ψ

_T^X

− ψ _T ln ψ

_T^X

− ψ _T ) , whereas the numerical binormal component η ^ψ can span all its asymptotic permissible range (8), and the numerical parallel component µ , if diverging, compensates the divergence of η ^ψ , like in (10).

Similar results are obtained near the symmetry axis upon the edge magnetic separatrix, where the radial component of the numerical plasma displacement eigenvector turns out to be always near to its most unstable vanishing asymptotic limit, , whereas the numerical binormal component η

^ψ

^T

^{→ ψ lim}

^Tmax

^{, R → 0 ξ}

ψ ~ ο ( ) R ^ψ spans the broader range (12)

and the numerical parallel component µ , if diverging, compensates the divergence of η ^ψ , like in (14). At the edge

magnetic separatrix, near the singular X-points, the vanishing of the binormal variable of the numerical plasma

displacement eigenvector (12) is observed even in the cases where η ^ψ diverges near the symmetry axis.

VI. CONCLUSIONS

The problem of calculating the numerical ideal MHD stability of (even) axisymmetric plasma equilibria in presence of magnetic separatrices, internal or at the edge of the plasma, remains beyond the scope of most of the existing ideal MHD stability codes. The approach taken in the present paper is that of adopting Boozer coordinates that, among other magnetic coordinates, optimize the Fourier decomposition in poloidal angle, while setting up an intensified numerical radial mesh near the magnetic separatrices. The nume ical results are compared “a posteriori”

with the permissible asymptotic limits for the perturbed displacement r

ξ , which have again simple analytical expressions if Boozer coordinates are adopted.

The numerical unstable eigenvectors are in good agreement, respectively within distances

^{ψ T X − ψ T ≤ 2.5 ⋅ 10 −2 ψ T X and ψ T max − ψ T ≤ 2.5 ⋅ 10 −2 ψ T max − ψ T X} from the two magnetic separatrices, with the permissible asymptotic limits, under the two conditions that at least all poloidal numbers in the range ^{m l = -5,15} [ ]

are used and that the minimum distance between the radial mesh and the two separatrices is at least within

^{ψ T X − ψ T ≤} ( ^{3 − 9} ) ^{⋅ 10 −3 ψ} T

X and ^{ψ T max − ψ T ≤ ( 3 − 9 ) ⋅ 10 −3 ψ T max − ψ T X} , respectively.

While the radial numerical displacement variable ξ ^ψ is found to be always near to its most unstable asymptotic limit at both magnetic separatrices, the full range of permissible asymptotic behaviours can be obtained for the numerical binormal and parallel displacement variables η ^ψ and µ . This conclusion agrees with the dominant role that the radial displacement variable plays in the stability calculation for global ideal perturbations with low toroidal mode number (n=0 to 3).

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