The Effect of Plasma Beta on High-n Ballooning Stability … concepts, namely the effects on the...
Transcript of The Effect of Plasma Beta on High-n Ballooning Stability … concepts, namely the effects on the...
1
The Effect of Plasma Beta on High-n Ballooning Stability
at Low Magnetic Shear
J W Connor1, 2
, C J Ham1 and R J Hastie
1
1
CCFE, Culham Science Centre, Abingdon, Oxon, UK, OX14 3DB
2Imperial College of Science and Technology and Medicine, London SW7 2BZ
Abstract
An explanation of the observed improvement in H-mode pedestal characteristics with increasing core plasma
pressure or poloidal beta, pol , as observed in MAST and JET, is sought in terms of the impact of the
Shafranov shift, , on ideal ballooning MHD stability. To illustrate this succinctly, a self-consistent treatment
of the low magnetic shear region of the '' s stability diagram is presented using the large aspect ratio
Shafranov equilibrium, but enhancing both and so that they compete with each other. The method of
averaging, valid at low s, is used to simplify the calculation and demonstrates how , , plasma shaping and
‘average favourable curvature’ all contribute to stability.
1. Introduction
Tokamak performance in H-mode is strongly dependent on the characteristics of the edge pedestal
namely its steepness and width, since these determine the effective edge temperature which provides
the boundary condition for core transport models [1]. The EPED model [2], which is based on the
stability of the edge plasma to ideal MHD peeling-ballooning modes [3, 4], can be used to determine
these quantities. There is experimental evidence, e.g. from {DIII-D [5], JT-60U [6], ASDEX Upgrade
[7], JET [8] and MAST [9] and, that the pedestal characteristics improve as the core plasma pressure,
or poloidal beta, pol , increases. This appears to be related to improvements in the ideal MHD
stability [10, 11]. The purpose of this note is to explain the origin of this in terms of basic tokamak
equilibrium concepts, namely the effects on the familiar '' s stability diagram of the Shafranov
shift, )(r , plasma shaping and ‘favourable average curvature’.
The stability against high-n ballooning modes is usually investigated in full toroidal geometry, but
with the stability boundaries being described in terms of a normalised pressure gradient parameter,
'' , and plasma current density, j , or equivalently the magnetic shear, s , local to the magnetic
surface being analysed. However this description hides other potential equilibrium dependencies such
as on the Shafranov shift, )(r , which is a consequence of the global pressure profile, i.e. the
plasma beta, , and the surface shape (e.g. ellipticity, ). A role for the Shafranov shift has been
proposed [11, 121], but it is not readily separable from other potential causes. In order to understand
these aspects it is useful to develop a model ballooning equation that explicitly contains such
2
parameters, so we need to consider an analytic, tokamak equilibrium. We take the large aspect ratio
1/ Rr , Shafranov tokamak equilibrium with 2~ and approximately circular magnetic
surfaces, constantr , albeit displaced by the Shafranov shift, )(r with some weak shaping, in
particular ellipticity, parameterised by a quantity rE , where the ellipticity
rrErrEr / [13]. The familiar s ballooning equation [14] corresponds to
taking the limit 0,0 , i.e. 0)( r of this equilibrium, while assuming a steep pressure
gradient with 1~)/( drnprd exists in a narrow region, of width r , in the vicinity of the
surface under consideration, so that the parameter )1(0~//2 22 BdrdpRq , where q is the
safety factor. Since the pressure gradient is only large in a narrow radial region, rr , one can
consistently assume concentric circular magnetic surfaces [15].
In order to investigate the effect of on ballooning stability we consider a somewhat different
modification of the Shafranov equilibrium in which the pressure gradient is enhanced globally, rather
than locally. We also consider the limit of small magnetic shear, which simplifies the analysis by
allowing a two-scale approach [16], but clearly shows the impact of the effects of finite on
ballooning stability. An optimal ordering is adopted that allows competition between and and
also with the effects of mild plasma shaping and favourable average curvature due to retaining terms
of 0/0 Rr :
3
0
2 ~/;~~~;~~ RrEEs , (1)
This allows the surfaces to remain circular in leading order, but to be self-consistent they must have
some ellipticity at a level that is driven by . We also allow for the possibility of an imposed
ellipticity at the plasma boundary, parameterised by E(a), at a comparable magnitude. Higher
harmonic shaping such as triangularity is found not to contribute at low magnetic shear. The modified
ballooning equation contains the parameters ,and,/ 0 ERa andtoadditionin s , with
and E involving the effects of so that we can explicitly examine the effect of on ballooning
stability.
2. MHD Ballooning Stability at Low Shear
The high-n, ideal MHD ballooning equation in general geometry is [17, 18]
4
2
22
2
2/where,0
.2..
B
Bp
B
Sp
B
S BBκ
κBBB
(2)
is the curvature, the poloidal flux and S is the eikonal, kdrqS , with k a radial wave-
number [19]. We use non-orthogonal straight field line co-ordinates ,,r with Jacobian
3
0
2 / RrRJ [8] and express the equilibrium magnetic field as rrfrgBR 00B ,
so that fRrgq 0/ , with r the magnetic surface label. In this co-ordinate system eqn. (2) becomes
.02/
211
4
23
0
2
2
B
SBp
grB
q
r
p
JB
S
J
B. (3)
Here
,.21
0
22
0
22
2
2
2
2 rrsrsr
r
q
RS (4)
with drdqqrs // the magnetic shear and where we have introduced the ballooning angle,
0qk , in place of the radial wave-number k. We can express
,2/
022
0
4
2
sBrR
qIR
B
SBpr
B. (5)
where
2/1
,2/1 2
2
2
2Bp
rBBp
rBr
(6)
With the aid of eqns. (4)-(6), eqn. (3) can be written in the form
0
G
d
dF
d
d (7)
where
000
2
0
2
2
2
; RsRgR
RGS
q
rF r
(8)
At this point we introduce the ordering (1), the two scales, su , [16], so that
us // 2 and expand
......;....; 4
4
3
3
2
2
103
3
2
2
102
2
10 GGGGGFFFF
(9)
In )0(λ0 we find
000
F (10)
4
so that )(00 u . In next order
0010
GF
, (11)
while in )0(λ2
0110001120
GGu
FsFF
. (12)
In next order
.02112001
1010122130
GGGu
Fs
Fu
su
FsFFF
(13)
Finally, in the )0(λ4equation, we annihilate the term in 4 by the operation 2/...... d , to
obtain
03122130
00
2
1120
GGGG
uF
usF
usF
us
(14)
Since we shall see that )(00 uFF , the first term on the left hand side vanishes and we only require
F to order 2 ; furthermore, although we need to expand G to 30 , we only need retain
that part of 3G that is independent of periodic terms in . Similarly, since we shall find that 1
contains only the sinandcos harmonics, we only need retain the same harmonics when
calculating 2G .
3. Equilibrium Quantities for the Ballooning Equation
It remains to evaluate the geometrical quantities in eqn. (7). To do this we use the Shafranov
equilibrium, expressed in co-ordinates ,,r through the representation [13]
.sin)(2sinsin)(sin
,cos)(2coscos)()(cos
7653
76543
0
rPrTrErZ
rPrTrErrRR
(15)
We note that we have measured the angle from the inboard side of the tokamak. Although the
surfaces are taken to be circular one finds that the equilibrium pressure forces a small amount of
5
ellipticity, parameterised by )(rE , and triangularity, rT , which can be removed at a given surface
by applying external shaping but necessitates a local gradient and curvature of )(rE and rT .
The function )(rP merely allows one to re-label surfaces; for the moment we have differentiated
between the radial co-ordinates rr and but we shall choose the function )(rP later such that they
can be identified and will from now on ignore the distinction.
We write ....)(1)(....;)()( 4
4
4
4 rgrgrprp , since 4
04 ~/~ Rrp . Substitution
of expansion (15) into the Grad-Shafranov equation yields equations for )(and)(),(4 rErrp [13]:
)18(02
3
2
3
23
312
)17(022
12
)16(,01
2222
4
2
0
0
2
0
2
2
2
0
42
0
4
rr
q
q
rE
rE
r
q
q
rEr
gqRR
r
q
r
R
q
r
q
q
rr
q
r
qRg
B
p
where we ignore terms small in . We can integrate eqn. (17) to obtain )(r . Since our ballooning
analysis is applied near the plasma edge we effectively require )(a which is given by
2
)()(
0
al
R
aa i
pol (19)
where pol is the poloidal of the plasma and il is the internal inductance per unit length of the
plasma column. Thus we see that is a parameter representing the global plasma and is distinct
from the parameter , which only represents the local pressure gradient. For more precise
calculations we should use
r
i
r
poli
pol rdrBBr
rlrpprdrrB
rrl
R
rr
0
2
22
0
22
0
2,)(
22;
2)(
, (20)
but il is smaller than pol with our ordering (1). From eqn. (15) we can compute the Jacobian for this
co-ordinate system:
Z
r
R
r
ZRJ (21)
The straight field line angle is then obtained from [19]
6
0
00ˆ//ˆ/2 dJRRdJRR . (22)
The radial co-ordinate r is defined as [19]:
RRJdrdr
r
/ˆ20
0
2
(23)
Inserting the expansion into JRR ˆ/0 and expanding in this serves to determine )(rP as:
r
E
R
r
R
rrP
2
0
2
0
3
2
1
2
1
8
1)(
(24)
Similarly, with the aid of eqn. (22), we can calculate from eqns. (15), (21) and (22) and, on
inverting the expression, we obtain
3sin324
3
sin524
sin2sin2
1sin
2
2
0
2
TE
r
E
Er
E
R
rE
r
E
(25)
To compute 222and., rrrr we substitute the expansion (25) into eqn. (15) and form
ZR and in terms of rr and . Inverting these expressions one can readily form
222and., rrrr since ZR and are orthogonal unit vectors. After some lengthy
algebra one obtains, to )(0 2 ,
2cos22
5
2cos21 2
22
Er (26)
2cos22
22
cos21
2
22
222
E
rrr
rr (27)
2sin32
1sin. 2
ErE
r
Errrr . (28)
It remains to replace r and Er from eqns. (16) - (18). These are given by:
r
EEEr
R
rsr
223;32
2
0
. (29)
7
to )(0 3 , where we have substituted for 4g in favour of 4p from eqn. (16) and expressed 4p in
terms of . Thus .and22 rrr are modified and when substituted in eqn. (4) we obtain
2cos22
5
2cos21
2sin34
32sin22
2cos22
22
5
22
2
7cos21
22
2
0
2
2
0
22
222
2
2
Es
Er
Es
ESq
rF
(30)
We now evaluate andr as defined in eqn. (6). Thus
22
42
0
402
0
2
2
0
2
2
00
21
2r
fg
B
pR
R
R
B
pR
rR
RR r , (31)
on using eqns. (16) and (26), so that
2
0
2
0
2
002
0
2
2 qR
r
R
R
r
RR
R
Rr
(32)
as required for forming G . Likewise
2
0
2
002
0
2
2 R
R
r
RR
R
R
(33)
Expressing 2R as a function of andr using eqns. (15) and (25), we obtain
.......2
cos3
2
3
22cos
2cos
21
2
0
22
00000
2
0
R
rE
r
E
R
r
R
r
R
r
RR
r
R
R (34)
where, as mentioned earlier, we only retain sinandcos harmonics in 50 and constant terms
in 60 as these produce the required contributions to 32 and GG . It is interesting to note that neither
TP nor (i.e. triangularity) contribute to 3G . Calculating G using eqns. (32), (33) and (34) and
recalling eqn. (29) for and E , we obtain
8
(35)
11
3
2
3sin
4cos
3
2
3
2
9
4
1
2sin2cos222
1
2sincos
2
0
202
00
sqR
rE
r
Es
r
EE
sssG
4. The ‘Averaged’ Ballooning Equation
We are now in a position to develop eqn. (14) for 0 which determines ideal MHD ballooning stability
at low magnetic shear. With the substitution us 0 in eqns. (30) and (35) we can identify the
quantities 321210 and,,,, GGGFFF . In particular, )1( 2
0 uF , which is indeed independent of . It
is also convenient to introduce the notation
..........sincos2sin2cos2
sincos)(
...2sin2cossincos)(
3
3
11
2
11
222
2
110
guggugguG
ufffuffFF
scsc
scsc
(36)
We can readily evaluate )(03 uG :
303 )( guG (37)
Equation (11) then yields
sincos)(
0
01 u
F
u , (38)
from which we can evaluate /11Fs and 12G
)(2
011
0
11 uffu
F
sFs cs
, )(
202
2
2
0
12 ugugF
G sc
(39)
From eqn. (12) we learn
)(2cos2sin2cos22sin122
)(11
2
0
000
11
20 uCugguu
F
u
usFFF sc
(40)
Periodicity of 2
9
.)(
)( 00
11
u
usFFuC
(41)
2cos2sin2cos22sin122
)(
2cos2sin2
)(
11
2
0
0
111
2
1
0
020
sc
scsc
ugguuF
u
ffufufF
uF
(42)
By integrating by parts we can then evaluate 21G :
)(2128
0
2
1
2
101
2
1
2
111
2
1
2
112
0
2
21 uuggFguguffgufufgF
G scsccssscc
(43)
Finally, we can evaluate 30G by integrating twice by parts and using eqn. (13): this generates a
plethora of terms:
(44)021120
0
0
0110101221
0
030
GGGF
G
uFsF
us
uFsFF
F
GG
These are evaluated in the appendix and given in eqn. (A.9).
The results (37), (39), (43) and (A.9) provide all the information needed to complete eqn. (14) for
)(0 u . The development of this equation is also described in more detail in the appendix. The result is
02
0
2
0
1000
2
F
A
F
AA
du
dF
du
ds (45)
where
E
r
E
qR
rA
4
3
8
911 2
2
0
0 (46)
sE
r
EA 43
2
9
8
9 22
1 (47)
423
2
9
4
9
8
3 223
2 sEr
EA (48)
10
The form of this equation has been confirmed by using computer algebra [20].
It is interesting to consider the large u limit:
0
2
2
0
00
2
4
3
8
911
E
r
E
qR
r
du
dF
du
ds , (49)
yielding the Mercier criterion [21]:
,04
3
8
911
4
2
2
0
2
E
r
E
qR
rs (50)
which we can write in the standard form:
E
r
E
qR
r
sDD MM
4
3
8
911;0
4
1 2
2
0
2. (51)
5. Ideal MHD Ballooning Stability
We now investigate the marginal stability curves corresponding to solutions of eqn. (45) that vanish
as u , determining the impact of (through its impact on and E) and 0/ Ra on the s
diagram at low magnetic shear. Several influences can affect the stability with respect to the s
diagram: the role of finite aspect ratio (the first term in 0A , 2
0 1)/( qRrd ); the role of the
Shafranov shift as increases; and finally the effect of ellipticity through E , which has a direct
effect through the shaping but also through Ewhich
itself responds to . A fully self-consistent treatment
requires all these effects to be included, but it is
instructive to consider their effects sequentially.
Fig. 1: The effect of finite aspect ratio on the
s stability diagram. The solid line
shows 0/ 0 Ra , the dash-dot line
025.0/ 0 Ra , the dotted line 05.0/ 0 Ra ,
and the dashed line 1.0/ 0 Ra when q = 3.
11
In Fig. 1 we show the effect of 0/ Ra through d in
isolation, for several values of 0/ Ra with a typical
value for q 3i.e. q ; thus stability is seen to
increase rapidly with increasing 0/ Ra , relative to the
standard s diagram. Note that, strictly, the
analysis is only valid for the region s << 1. Having
established the effect of d , we can effectively remove
it as a parameter by scaling eqn. (45) so that it involves
just four independent parameters, namely:
./and/,/,/ 3/23/23/13/1 dsdEdd The
corresponding scaled quantity for is 3/4
0 /ˆ d ,
where 2
00 /02 Bp , which we need when we
calculate Eand . We can thus plot stability curves
parameterised by through its impact on E and
(for given values of )(aE ) in a ‘normalised s diagram’ labelled by axes
3/23/1 /ˆand,/ˆ dssd . However, before considering the complete problem we examine the
effect of increasing the parameter 3/1/ˆ d alone (one of the two outcomes of increasing ),
the results being shown in Fig. 2: again we observe a stabilising effect.
To address the complete problem we must first determine EE and , which involves a global
solution for these quantities. In this ordering these are obtained from solutions of the equation
(a) (b)
Fig. 3: The scaled ellipticity parameter, E , as a function of normalised radius, r/a, calculated using Eq. (52)
with 0ˆ (solid line), 1.0ˆ (dash-dot line), 2.0ˆ (dotted line), and 3.0ˆ (dashed line). (a) is
the circular boundary case; (b) )3/1,3/1(69.0ˆ dEE .
Fig. 2: The effect of increasing on the
scaled s stability diagram. The solid
line shows 0ˆ , the dash-dot line shows
3.0ˆ , the dotted line shows 6.0ˆ , and
the dashed line shows 9.0ˆ .
12
)(4
3
2
3
2
3312 222
2
22
2
2rH
dr
d
rq
r
r
qE
rE
rq
r
r
qE
(52)
satisfying a boundary condition on )(aE , where
.2
0
2
2
0
3
2
0
r
dr
dprdr
Br
qRr (53)
In our illustrative calculations we take simple global pressure and safety factor profiles of the form
42
2
3
11
0;10
a
r
a
r
qrq
a
rprp
(54)
with 10 q , so that 3aq . In Fig.3 we show radial profiles of 3/2/ˆ dEE for several values
of : in Fig. 3(a) we choose 0ˆ aE , the circular boundary case as a reference, while in Fig. 3(b)
we set 69.0ˆ aE , corresponding to a JET-like situation with .3/1 and3/1 d aE The
magnitude, and even the sign, of aEˆ is seen to vary with . While we observe that Eˆ is negative
near the edge, we note that the ellipticity, , is nevertheless an increasing function.
Using this information on aE and aEˆ and calculating aˆ from eqn. (53) as input, we solve
the averaged ballooning equation (45). We must emphasize that we can treat ands as free,
independent parameters on a given flux surface as the global equilibrium changes with increasing ,
(a) (b)
Fig. 4 : Stability in the scaled s diagram as global pressure is increased; in each plot = 0 (solid
line), = 0.1 (dash-dot line), = 0.2 (dotted line), and = 0.3 (dashed line). (a) is the case with a
circular plasma boundary. (b) the case with a shaped plasma boundary: 69.0ˆ E .
13
since the necessary sharp gradients in these quantities can
be considered to exist only over a localised region,
without affecting the magnetic geometry of the underlying
equilibrium [14, 15]. The curves of marginal stability in
the ˆ s diagram corresponding to ar are shown in
Fig. 4: Fig. 4(a) shows the impact of increasing for the
‘cylindrical’ case 0ˆ aE , while Fig. 4(b) repeats this
for the finite ellipticity case 69.0ˆ aE . We see that the
stable regions increase as increases for both cases and
that a finite value of aE provides further stabilisation.
Finally, in Fig. 5 we show the dependence of MD , related
to Mercier stability, as a function of . Clearly it
becomes yet more positive, precluding any question of Mercier instability.
6. Conclusions
The confinement properties of tokamaks depend on the H-mode pedestal characteristics and
there is experimental evidence that these improve with plasma pressure. High-n ideal MHD
ballooning modes, which may serve as an indicator for the effect of the more relevant kinetic
ballooning modes, are believed to play a role in defining the pedestal properties. However,
simple analysis based on the s stability diagram assumes that only the local pressure
gradient, not the global pressure, or , matters.
In this paper we have explored the impact of plasma , as mediated through the Shafranov
shift, , for example, on stability. We employ a Shafranov equilibrium, but one in which the
pressure is enhanced globally to allow to compete with , whereas in the s
equilibrium calculation one only enhances locally. This means that we also have to consider
the effect of plasma ellipticity, parameterised by E, since it too responds to . We also include the
stabilising effects of favourable average curvature, proportional to 2
0 1/ qRad . In order to
extract the essence of such effects we have considered the limit of low magnetic shear, s, (the region
of the s diagram where the relative impact of and d is greatest), allowing access to the
second stability region for example. Furthermore the presence of the bootstrap current in the pedestal
region tends to produce low s, making the calculation even more relevant. An added advantage of this
region of parameter space is that a two-scale averaging process can be invoked to reduce the ideal
MHD ballooning equation to a simpler form devoid of the usual poloidally periodic terms. An
optimal ordering scheme )~/;~~;~~( 3
0
2 RrEs for the quantities
ERas and/,,, 0 has been introduced which ensures that all these effects on stability
compete equally. This equilibrium information has been fed into the general high-n MHD ballooning
Fig. 5: Plot of the Mercier index, MD ,
against , solid line shows the circular
plasma boundary case and the dash-dot
line shows the shaped plasma boundary.
14
equation, which has then been processed order by order to generate the averaged equation (eqn.(45)),
which appears in 40 . The equilibrium information required is reduced as a result of the averaging
process: had we attempted to calculate the complete modified s diagram we would have needed
to account for more poloidal harmonic structure in the higher orders of , including the presence of
triangularity driven by .
The averaged marginal MHD ballooning equation has been solved and the effects of E, and d on
the s diagram explored. There are thus six parameters to set: Eds ,,,, and E . Since E
plays a role we have to solve the global equilibrium equation for rE to calculate this quantity.
Examples of these solutions for given values of the imposed edge ellipticity parameter, aE = 0 (the
circular boundary case) and aE =1/3 (a JET-like case) for several values of are shown in Fig. 4.
Figure 1 demonstrates the effect of just including the effect of d , setting ;0 EE clearly
second stability access is rapidly opened up with increasing 0/ Ra . In order to explore the effects of
through its impact on and E it is convenient to reduce the number of independent parameters
by a suitable scaling, introducing 3/23/23/13/1 /ˆand/ˆ,/ˆ,/ˆ dssdEEdd (we also
define 3/4/ˆ d , needed when calculating the radial profiles of )(ˆ r and E(r)). In Fig.2 we
examine the effect of just including the normalized Shafranov shift parameter, ˆ , alone, although
this is not a consistent procedure; increasing ˆ is seen to have a stabilizing effect. Finally we
examine the full effect of through its self-consistent impact on E, and E . Figure 4 shows this
to be stabilizing for the two cases considered: 3/1,0aE . Finally, we note that increasing
leads to increasingly Mercier stability, as shown in Fig. 5.
In summary we find that increasing has a stabilising effect on the s diagram through its
impact on the Shafranov shift, , and ellipticity parameter, E, providing a potential explanation for
the experimental observations that the pedestal characteristics pertaining to tokamak confinement
improve with increasing plasma pressure, even if the local pressure gradient is unaffected, and
complementing studies with full MHD stability codes.
Acknowledgments
This work has received funding from the RCUK Energy Programme [grant EP/I501045]. For further
information on this paper contact [email protected].
References
[1] J Kinsey et al, Nucl. Fusion 51 083001 (2011)
[2] P B Snyder et al, Phys. Plasmas 19 056115 (2012)
15
[3] H R Wilson et al, Phys. Plasmas 6 1925 (1999)
[4] P B Snyder et al, Nucl. Fusion 44 320 (2004)
[5] A W Leonard et al, Phys. Plasmas 15 056114 (2008)
[6] Y Kamada et al, Plasma Phys. Control. Fusion 48 A419 (2006)
[7] C Maggi, Nucl. Fusion 50 066001 (2010)
[8] C D Challis, J Garcia, M Beurskens, P Buratti et al, Nucl. Fusion 55 053031 (2015)
[9] I Chapman, J Simpson, S Saarelma, A Kirk et al, Nucl. Fusion 55 013004 (2015)
[10] J Simpson, S Saarelma, I T Chapman, C D Challis et al, to be submitted to Plasma Phys. Control.
Fusion
[11] N Aiba and H Urano, Nucl. Fusion 54 114007 (2014)
[12] H Urano, S Saarelma, L Frassinetti, C F Maggi et al, ‘Origin of Edge Transport Barrier
Expansion with Edge Poloidal Beta at Large Shafranov Shift in ELMy H-mode’, EUROFUSION
WPJET1-PR(15)42, submitted to Phys. Rev. Lett. (2015)
[13] J M Greene, J L Johnson and K E Weimer, Phys. Fluids 14 671 (1971)
[14] J W Connor, R J Hastie and J B Taylor, Phys. Rev. Lett. 40 396 (1978)
[15] J M Greene and M S Chance, Nucl. Fusion, 21 453 (1981)
[16] D Lortz and J Nuhrenberg, Nucl. Fusion, 19 1207 (1979)
[17] J W Connor, R J Hastie and J B Taylor, Proc. R. Soc. Lond. A 365 (1979)
[18] R L Dewar and A H Glasser, Phys. Fluids 26 3038 (1983)
[19] M N Bussac, R Pellat, D Edery and J L Soule, Phys. Rev. Lett. 35 1638 (1975)
[20] Wolfram Research Inc. Mathematica Version 3.0, Champaign IL (1996)
[21] C Mercier, Nucl. Fusion 1 47 (1960)
Appendix: Details of the Derivation of the Averaged Ballooning Equation. (45).
In this appendix we provide more detail on the derivation of eqn. (45). First we evaluate the quantity
30G , given by eqn. (44):
(A.1)021120
0
0
0110101221
0
030
GGGF
G
uFsF
us
uFsFF
F
GG
16
term by term. For the first term on the right hand side, we obtain
0111
2
11
2
11
2
1
0
2
1
2
12
0
21
0
0
28
scscscscsc gffugfuffufF
uffF
FF
G
(A.2)
The second results in
02
2
2
2
202
0
12
0
0 2124
sc fufufFF
FF
G
(A.3)
The third and fourth are equal:
0
0
10
0
010
0
0
2
F
sF
us
F
G
uFs
F
G
(A.4)
while for the fifth,
011
0
01
0
0
2
uffu
F
s
uFs
F
G cs
(A.5)
Turning to the terms involving G , we have
00
1
2
1
2
1
2
12
0
2
2
0
2
0
221
16
Fgugufuf
FF
G scsc , (A.6)
01
2
1
2
2
0
2
0
0
3
11
0
0 2144
sc guguFF
GF
G (A.7)
and
02
2
2
0
02
0
0
2
sc gug
FG
F
G . (A.8)
Assembling these results, we finally have
17
)(2
)()(2
)(32
7)(
2
)(2116
5)(
8
)(2128
011
0
0
0
02
0
0
0
2
02
2
2
0
01
2
1
2
2
0
2
01
2
1111
2
12
0
01
2
1
2
2
2
2
22
1
2
12
0
30
uu
ffF
suu
F
suf
Fu
Fugug
F
uguguF
uguffgfufF
ufuffufuuffF
G
cssc
scscscsc
scscsc
(A.9)
Now we turn to the derivation of eqn. (45). The results (37), (39), (43) and (A.9) provide all the
information needed to complete eqn. (14) for )(0 u :
.213
48
2212
8
4232
7
0301
2
1
2
0
2
2
2
2
1
2
1
0
2
011
0
02
2
2
2
1
2
1
2
1
2
12
0
2
01111
2
112
0
2
02
3
0
00
2
gguguF
guguggF
ffF
u
du
dsfufufufufuf
F
ffugfufgF
sfFdu
dF
du
ds
scscsc
scscscsc
scsscc
(A.10)
It is convenient to isolate the u dependence of the coefficients above by substituting:
2
2
222
2
221
2
11
ˆandˆ;ˆ fufffufffuff cccccc (A.11)
Then eqn. (A.10) can be rewritten as
02
0
2
0
1000
2
F
A
F
AA
du
dF
du
ds (A.12)
Where
2
2
11
22
1
2
2
2
12
22
1
2
30
ˆ
2ˆ
4ˆ
8ˆ
4ˆ
228ffgfff
sgggA cscccss
, (A.13)
18
,ˆˆ28
ˆ8
ˆ324
ˆ2
ˆ
2
1
32
7
ˆˆ428
28
3
2
1111
2
11
3
222
2
11122
22
1111111
2
22
22
1
2
1
2
11
3
1
scccsc
ccsccs
ccsscscscsccs
ffffff
ffffffs
sff
ffgffggggggggA
(A.14)
.ˆ8
ˆ4
ˆ8
ˆ2
ˆ28
3
2
1
2
11
2
11111
2
111
3
222
2
11111
3
2
sccsccsc
scccsccscsc
ffffffgg
fffffffffsggA
(A.15)
Recalling the definitions
sqR
rg
E
r
Eg
E
r
Eggg
Er
EfuEEfuff
ufufffufuff
scsc
sccc
sccc
2
0
3
2
2
2
211
2
2
222
2
2
2
22
222
2
2
2
221
2
1
2
11
11,
44
3
8
3,
44
3
8
3
8
9,,
2
;262
34,2
2
52
22
2
5ˆ
,22
22
7ˆ,22,12ˆ
(A.16)
we finally obtain
E
r
E
qR
rA
4
3
8
911 2
2
0
0 (A.17)
sE
r
EA 43
2
9
8
9 22
1 (A.18)
423
2
9
4
9
8
3 223
2 sEr
EA (A.19)
Equation (A.12), now with the coefficients (A.17) – (A.19), is the required equation for u0 , given
as eqn. (45) - (49) in the main text.