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 PublicationsManager@ccfe.ac.uk.

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.