Stability of ideal MHD modes of a finite pressure plasma in toroidal systems with a complex shape of the magnetic axis This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1991 Nucl. Fusion 31 1717 (http://iopscience.iop.org/0029-5515/31/9/009) Download details: IP Address: 130.102.42.98 The article was downloaded on 16/04/2013 at 06:30 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
Transcript
Stability of ideal MHD modes of a finite pressure plasma in toroidal systems with a complex
shape of the magnetic axis
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
1991 Nucl. Fusion 31 1717
(http://iopscience.iop.org/0029-5515/31/9/009)
Download details:
IP Address: 130.102.42.98
The article was downloaded on 16/04/2013 at 06:30
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
STABILITY OF IDEAL MHD MODES OF AFINITE PRESSURE PLASMA IN TOROIDAL SYSTEMSWITH A COMPLEX SHAPE OF THE MAGNETIC AXIS*
P.V. DEMCHENKO, A.Ya. OMEL'CHENKO, K.V. SAKHARInstitute of Physics and Technology,Ukrainian Academy of Sciences,Khar'kov,Union of Soviet Socialist Republics
ABSTRACT. The authors have obtained an equation for small oscillations describing the excitation of ideal smallscale MHD modes in a finite pressure plasma with allowance for inertial effects in arbitrary magnetic configurations.This equation is used to analyse the plasma stability in an ASPERATOR NP-4 type trap. It is shown that in thecurrentless regime both ideal ballooning and Mercier modes may be excited in this trap. It is established that thesemodes are absolutely unstable because of the growth with increasing plasma pressure of the magnetic 'hill' in the device.The growth rates of the excited modes are determined. In unstable regimes, a relative improvement in confinement isobserved when, with increasing pressure, a transition may occur from regimes with larger growth rates to regimeswith smaller growth rates. Such a transition is connected with finite plasma pressure effects when the destabilizingaction of finite pressure, leading to a growth of the 'magnetic hill' and to an absolute instability of Mercier modes,is partially compensated for by the stabilizing action connected with the growth of the shear value with increasingplasma pressure.
1. INTRODUCTION
For the stellarator traps with a spatial axis [1-3]which are currently being developed it is necessaryto study the plasma MHD stability with respect to theexcitation of ideal small scale modes because the growthof these instabilities can determine the maximum valuesof the plasma pressure attainable in the devices. Theanalysis of the stability of the ideal modes with theassumption of zero frequency [4-6] is limited incharacter. While such an analysis enables us to deter-mine the plasma stability boundary as a function of thevalues of the plasma parameters or the geometric para-meters of the device, it leaves open the question of themagnitudes of the growth rates of unstable oscillations.
For this reason, it is advisable to generalize theequation for small oscillations [7] which describes theexcitation of ideal MHD modes in a finite pressureplasma in arbitrary toroidal magnetic configurationswithin the zero frequency limit and allowing for iner-tial effects. Using the well known analytical methodsof solving such a generalized equation [8-10], thiswill enable us to obtain and to study numerically thedispersion equation for ideal modes in traps with aspatial shape of the magnetic axis.
2. AVERAGED EQUATIONFOR SMALL OSCILLATIONS
Let us consider a toroidal trap whose magnetic axisis a closed spatial curve of a length L characterized bycurvature k(v>) and torsion n((p). Here we use a systemof co-ordinates a, 9, <p with straightened magnetic fieldlines. We analyse the stability of ideal small scale modes,with allowance for inertial effects by the methoddescribed in Ref. [7]. In the Connor-Hastie-Taylorrepresentation [11] the three-dimensional problem ofstability of these modes is reduced to a two-dimensionalproblem with respect to the variables y and <p (d — y):
p' 4TTP' vg
(la)
47T2pg73
- L , A M - WO-L1M ' " ^ r ' j f
where
* Translated by S.K. Datta, English Translation Section,Division of Languages, IAEA, Vienna.
fLi =dy dtp
NUCLEAR FUSION, Vol.31, No.9 (1991) 1717
DEMCHENKO et al.
7 is the growth rate of unstable oscillations, p and pare the equilibrium pressure and density of the plasma,6p and f are the perturbed pressure and the radial dis-placement of a plasma element, 70 is the adiabatic index,r2 = R2q2/Cs
2, C2 = yQp/p, q = 1/M = 0 7 X ' ,
A = AM 1 2 7 r B ° ' P'l
and the prime denotes the derivative with respect tothe radius.
The quantities Ll M, Wo and AM entering the systemof equations (1) are defined by the following relations(see Ref. [12]):
L i M —1
yq'
(lb)
W 0 =0
2 ,
p'2V
<*>'2
x r ^ , nL (l) \ (Bl)
Mg22
A M =
72 7 r B Q ' l
Bg J
In relations (lb), gilc and gik are covariant andcontravariant components of the metric tensor ofthe co-ordinate system with straightened lines offorce of the steady magnetic field Bo, g = detgik,glb = g .2 gbb = g22
oto~) = (Jo'Bo)^^^^ Jo ls t n e equilibrium currentdensity, v is the oscillating part of the equilibriumcurrent density, x and <j> are the poloidal and toroidalmagnetic fluxes, V = 47r2(Vg)(0), L|~' is the operatorinverse to the operator L|. The notations of the averagedand oscillating quantities used above are as follows:
4*2 Jo Jo
r rJo Jo
Gddd(p
!
2ir p2»
0 Jo
= G - (G)(0)
The system of two-dimensional equations (1) can besimplified by averaging over fast oscillations of themetric [7] and reduced to a second order differentialequation
dy
1 df
Qfflffl •] - '• (• dy( Q X M A M ) ( 0 )
Q(0)J.M
f l M
+ Y4ic2p
(g) ( 0 ) (Lx M) ( 0 )
(0)(2a)
where f = (f)(0), |3e is the ratio of the kinetic plasmapressure to the poloidal magnetic field pressure,
r2 = °*~A
P
A = {L|A,
+ WO{[1
+ (LrVi
(2b)
1718 NUCLEAR FUSION, Vol.31, No.9 (1991)
STABILITY OF IDEAL MHD MODES
The first term in Eq. (2a) describes the stabilizing actionof the shear, the second term represents the action ofthe 'magnetic well (hill)', the terms bilinear in AM andthe quantity A describe the action of the finite plasmapressure effects that are proportional to p\ and Bj. Theterm proportional to \i is the joint action of the shearand of the current density. The last term accounts forthe effects of inertia.
Equation (2a) is a generalization of the small-oscillationequation for the case of perturbations with non-zerofrequency for sufficiently small growth rates y2r\ < 1and can be used to study the stability of ideal small scalemodes in a moderate pressure plasma, 8*e = R/4a < 1,in arbitrary toroidal magnetic configurations.
We can calculate the quantities Wo, QxM and AM
entering into Eq. (2a) by the procedure described inRef. [13]. This enables us to write the small-oscillationequation (2a) in the form
+ 2**n + n \ -
n)(n n + n, - n2)2
n — n2
n2)
(/x + n2)
- 4
+ n,)(2/* + n + n.)
- n2)(4)
- n 2
(3)
where
x " s 2
x - —
• 2 =
- 4
+ 4
R2
2K° ~ "
2
[-0*
12
16
14
r
3 12 (M + n,)(M + n2)
1 1 1+ n)0i + n,)
( n - n , ) 2
(M + n,)(M + n2)(^
[L (/x + n)4 Lv n n
l
^ 5 ,
2 (/x + n)(/x + n2)
/x + n + nj — n2
(fi + n)(2/x + n + n,)2
I+ n) I
K2
(fi + n)2
,K n 2 K n + n i _ n 2
n,)
z, = -4
+ n,
(M
2 -i
+ n)(/x
(A+ n)
+ n2)(/x
i + n + i
+ n,
-l-n +
n2
f i
1
ni
)2
m
11
- n 2 )
- n , ) 2
NUCLEAR FUSION, Vol.31. No.9 (1991) 1719
DEMCHENKO et al.
In the relationships (4) we have used the followingnotations: a is the plasma column radius, R = L/2ir,S = -a/i7/x, fi, = RJ/a<l>', t = Sy, Kn is the Fouriercomponent of K = k(<p) exp [ih(<p)] normalized to R;h(p) is the rectification function, determined with anaccuracy of up to terms of the second order in plasmapressure by the relationship h(<p) = R \ d<p[i<(<p) - K0];and K0 is
m e torsion of the magnetic axis averagedover the angular variables. In deriving the abovequantities, the distribution of plasma pressure wasassumed to be parabolic.
Henceforth, we shall be concerned with the solutionsof Eq. (3) which are localized in t space since, whenthe angular variable 6 goes over to actual space, theywill give physical solutions which determine the distri-bution of the amplitude of the perturbed radial displace-ment of the plasma element over the minor azimuth ofthe torus. In accordance with Ref. [9], the localizedsolutions of the small-oscillation equation can beexpressed in terms of oblate spheroidal functions [14]for a specific relationship between the coefficients X,Ji2 and 7*2 (within the limit of small 7* < 1):
which can be regarded as the dispersion equation. Theupper and lower signs in Eq. (5) correspond to even andodd solutions of Eq. (3) with respect to the t variable.
3. STABILITY OF IDEAL MODESOF A FINITE PRESSURE PLASMA
An analysis of the dependence of the parameter Xon /I2 and 7* shows [9] that for /I2 < 1/4 in the caseof even modes and for Ji2 < 9/4 in the case of oddmodes the eigenvalue spectrum of X has, for 7* — 0,a condensation point X = -1/4 for all harmonicsn = 1, 2, ... of localized perturbations. Note that therelation X = -1/4 leads to the well known Mercier
criterion if it is expressed, with the help of relations (4),in the form
2 " ° R ~ n
R4
and subsequently the stability boundary for localizedperturbation 7* = 0 is determined. This allows one toconsider perturbations for which the eigenvalue spectrumhas a condensation point, as is the case for Merciermodes.
For Ji2 > 1/4 in the case of even modes and Ji2 > 9/4in the case of odd modes, two types of perturbation occurin the plasma. For harmonics n < Neven = \[Ji - \]in the case of even modes and n < Nodd = j [Ji — \ ]in the case of odd modes, the eigenvalue spectrumof X for 7* — 0 has no condensation point and isdiscrete for different harmonics of localized modes:Xeven = (£ - 1 - 2n)(jZ - 2n), X ^ = (Ji - 1 - 2n)x (Ji - 2 - 2n). These relations follow from thedispersion equation (5) at 7* = 0. Because theeigenvalue X spectrum of the excited modes dependsexplicitly on the ballooning parameter Ji, they may beregarded as corresponding to ideal ballooning modes(IBM). For harmonics n > Neven in the case of evenmodes and n > Nodd in the case of odd modes, theeigenvalue spectrum of X has, for 7* — 0, a conden-sation point (perturbations of this type correspond toideal Mercier modes). The harmonics mentioned abovemay be regarded as local solutions of the Schrodinger-
like equation with the potential = 7*2 cosh2f~ (M2 ~ I) sech2 f to which Eq. (3) may be reducedby the substitution t = sinhf, f = ^(D/Vcoshf.Comparing the eigenvalues of X found from thenumerical solution of system (5) with the possiblevalues of the parameter X in the magnetic trap underdiscussion, we can establish the type of modes excitedin this trap and their growth rates.
Let us use dispersion equation (5) to analyse thestability of ideal modes of a finite pressure plasmain a closed magnetic trap with a spatial shape of themagnetic axis having parameters close to those ofASPERATOR NP-4 [3]. The trap under consideration isa helical solenoid whose magnetic axis performs N = 8turns about the circular axis of the torus. The majorand minor radii of the torus are equal to RQ = 152.4 cmand r0 = 19.05 cm, respectively. In this case, thecurvature k(̂ j) and the torsion n(<p) of the magnetic
1720 NUCLEAR FUSION, Vol.31, No.9 (1991)
STABILITY OF IDEAL MHD MODES
axis can be calculated in accordance with Ref. [15]and represented in the form:
1
rod +- x', cos Nv?
1 + x2
K(<p) = -ro(l + x2)
. , r0 2 + 5x2 + x4
X 1 + — - = COS N*>R 1 2
(6)
Ro 1 + x2
where x = Ro/Nr0.For the above values of RQ and r0 the curvature k(v?)
normalized to R = L/2TT can be represented in the form
k(<p) = Rk(<p) exp -i I d^[«(^) - K0]V J J
= 5.66 £ Jn(0.35)ei8n*n = -oo
(7)
We shall henceforth confine ourselves to the study ofthe stability of ideal modes in the currentless confine-ment regime. Then, in accordance with Ref. [13], themagnitude of shear Str can be represented in the form:
Str = (a2/3|/R2) X + n)3
n = 0
This enables us, using relationships (4) and (7), to findthe explicit form of S,r, UQ, U|, X,r and /Z2
r in a magnetictrap of the above type:
u0 = -(0.23/3^ + 0.04/3^)
u, = -(0.005/S^ + 0.47/3lStr)
- 2 = 21 2
Since in the magnetic trap under consideration/Z2
r = 21.2, in accordance with what has been statedabove, both ideal ballooning modes (with harmonicsn = 0, 1, 2 for even modes and n = 0, 1 for oddmodes) and ideal Mercier modes (with harmonicsn = 3, 4, 5, ... for even modes and n = 2, 3, 4, ...for odd modes) may be excited in the plasma.
3.1. Stability of ideal ballooning modes
For the system of equations (5) in the case of small 7*to have solutions corresponding to IBM excitation, thequantity v should be close to the poles of the T functionsentering into relationship (5): v = Jl — 1/2 — 2n foreven modes and v = /Z — 3/2 — 2n for odd modes.Condition v > 0 and the requirement following from it,Ji2 s (O.OOSjS4,+0.47-S-j32.)/S2 > (1/2 + 2n)2 (foreven harmonic), determine the region of occurrence onthe S-jSfl stability diagram of IBMs bounded by curvesS|, S2 (Fig. 1). The S-/3e diagram is constructed accordingto the results of the numerical solution of the system ofequations (5) in which all the parameters except theshear S correspond to the ASPERATOR NP-4 magnetictrap parameters. In Fig. 1, curve 3 defines the boundary7* = 0 of the IBM stability, and curves 4 and 5 deter-mine those values of shear S and plasma pressure(parameter /3e) for which in the plasma the n = 0 evenharmonic of IBMs, with growth rates 7* = 0.1 and7* = 0.8, is excited. Curve 6 determines the value ofthe shear Str versus the plasma pressure in the trapstudied. The points of intersection of curve S,r withcurves S(j30,7*) indicate those plasma pressure andshear values in the device at which the given harmonicis excited with corresponding growth rates 7*.
On the basis of numerical analysis of the dispersionequation the following conclusions can be drawn aboutIBM stability in a device with parameters close to thoseof ASPERATOR NP-4:
(1) In the plasma pressure region considered,&e < &*e — 6, only the n = 0 even harmonic of IBMscan be excited. The region of instability of this harmonicis bounded in the upper half-plane (S > 0) of the diagramby curves S! and S3 (7* = 0) for values of the parameter(le in the ranges 0 < |80 < 0.7 and 0.7 < (3e < 6,
s
20
10
FIG. 1. Curve (1): S,; curve (2): S2; curve (3): S (y' = 0);curve (4): S (y' = 0.1); curve (5): S (y' = 0.8); curve (6): Slr.
NUCLEAR FUSION, Vol.31. No.9 (1991) 1721
DEMCHENKO et al.
1 2 3 4 5 6 7150
100
40(
o!
-15
-16
-17
(2)
0.5
(D
FIG. 2. Curve (1): -X; curve (2): -\lr
respectively. In the lower half-plane (S < 0) the insta-bility region lies above curve S2. For the remainingn ?* 0 harmonics of IBMs the solutions of dispersionequation (5) lie outside the region of occurrence ofIBMs.
(2) In a device of the above type it is not possibleto achieve a plasma confinement which is stable withrespect to IBM excitation since, with rising plasmapressure, curve S,r moves away from the stabilityboundary (curve S3 (7* = 0)). The deterioration ofplasma stability in a trap with parameters close tothose of ASPERATOR NP-4 can be explained bythe fact that this trap has a magnetic 'hill' u0 < 0(see Eq. (8)) the absolute value of which increaseswith plasma pressure.
(3) The absence of points of intersection of curvesSlr and S (7* = 0.8) in Fig. 1 indicates that only IBMswith growth rates 7* > 1 can be excited in this trap.This conclusion is confirmed by the data of Fig. 2,where curve 1 determines the value of the parameter -Xdepending on y*2 according to the results of numericalsolutions of Eqs (5); curve 2, presented in (-X,j3e)co-ordinates, determines, in accordance with Eq. (8),the dependence of -X,r on the plasma pressure (theparameter /3e) in a confinement system such asASPERATOR NP-4. Curves 1 and 2 do not crossin the region 7* < 1.
3.2. Stability of Mercier modesof a finite pressure plasma
Since among the excited harmonics of Merciermodes the one with the smallest number has thehighest growth rate [9, 10], we shall henceforth confineourselves to analysing the stability of the odd and the
even mode with n = 3. Figures 3a and 3b present theresults of numerical solutions of Eqs (5) for the evenand odd n = 3 harmonics mentioned above. Curves 1determine the boundary 7* = 0 of the Mercier modestability; curves 2, 3 and 4 determine those values ofthe shear and plasma pressure (parameter (3e) at whichthe above mentioned modes are excited with growthrates 7* = 10'3, 7* = 10'1 and 7* = 0.8, respectively.Curve 5 determines the values of Slr in the magnetictrap.
In Fig. 4, curves 1 and 2 determine the eigenvalues-X versus y*2 for even and odd n = 3 harmonics ofthe Mercier mode, respectively; curve 3, presented in(-X, Pe) co-ordinates, determines the -X,r dependenceon the plasma pressure in agreement with Eq. (8). Theresults of numerical calculations shown in Figs 3a, 3band 4 allow one to draw the following conclusions:
(1) The S-(3e diagrams of the even (Fig. 3a) and theodd (Fig. 3b) n = 3 harmonics show that with a rise inplasma pressure the state of stability cannot be attained(curve S,r does not intersect the stability boundaryS| (7* = 0)). As in the case of IBMs, this is due to thegrowth of the magnetic 'hill' with increasing plasmapressure.
(D
FIG. 3a. Curve (1): S (y' = 0); curve (2): S (y' = W3);curve (3): S (y' = 0.1); curve (4): S (y' = 0.8); curve (5): Slr
FIG. 3b. Curve (1): S (y' = 0); curve (2): S (y* = 10'3);curve (3): S (y' = 0.1); curve (4): S (y' = 0.8); curve (5): Slr.
1722 NUCLEAR FUSION, Vol.31, No.9 (1991)
STABILITY OF IDEAL MHD MODES
3 4 5 6
60
40
20
0.2 0.4 0.6
FIG. 4. n = 3. Curve (J): even mode; curve (2): odd mode;curve (3): -Xlr.
(2) With a rise in pressure there may be a transitionfrom unstable regimes with larger growth rates toregimes with smaller growth rates; this is indicated bythe intersection of curves S,r and S4(7* = 0.8) in Fig. 3a.The above conclusion is confirmed by the data of Fig. 4.A comparison of the magnitudes of -X,r and -X showsthat with rising plasma pressure the growth rate 7*decreases. Such a transition is connected with finiteplasma pressure effects when the destabilizing actionof finite pressure, leading to a growth of the 'magnetichill* and to an absolute instability of Mercier modes,is partially compensated for by the stabilizing actionconnected with the growth of the shear value withincreasing plasma pressure.
4. CONCLUSIONS
In this work we have obtained an equation for smalloscillations describing the excitation of ideal small scaleMHD modes in a finite pressure plasma with allowancefor inertial effects. We have found the parameters ofthe dispersion equation, which determines the growthrates of the excited modes in an ASPERATOR NP-4type trap, and performed its numerical analysis. It hasbeen shown that in this trap both ideal ballooning modesand Mercier modes can be excited. It has been estab-lished that these modes are absolutely unstable, owingto the growth of the magnetic 'hill' in a given devicewith increasing plasma pressure. In unstable regimeswe note a relative improvement in confinement when,with rising pressure, there may occur a transition fromregimes with larger growth rates to regimes with smaller
growth rates. Such a transition is connected with finiteplasma pressure effects when the destabilizing action offinite pressure, leading to the growth of the 'magnetichill' and to the absolute instability of Mercier modes,is partially compensated for by the stabilizing actionconnected with the growth of the shear value withincreasing plasma pressure.
REFERENCES
[1] GLAGOLEV, V.M., KADOMTSEV, B.B.,SHAFRANOV, V.D., TRUBNIKOV, B.A., in ControlledFusion and Plasma Physics (Proc. 10th Eur. Conf. Moscow,1981), Vol. 1, European Physical Society (1981) E-8.
