1

Peter Stangeby. 10 August 2011.

Modified 2 Point Model of the SOL to allow for variation in Rtarget

The most basic form of the 2PM does not allow for variation in R along the length of SOL

flux tubes, say from the outside mid-plane to the outer target, and thus it does not allow for the

change of SOL

||A , the cross-sectional area of the SOL perpendicular to B, see Appendix A on

magnetic flux expansion. The most basic form of the 2PM is given for example in Sec. 5.2 of my

book:

uutt TnTn 2 (1)

oe

/t

/u

LqTT

2

72727 (2)

sttt|| ckTnq (3)

where “t” indicates target and “u” indicates upstream, here taken to be the outside mid-plane,

OMP. When R and SOL

||A vary along the flux tube, then eqn. (1) for pressure balance is

unchanged, also the expression for parallel power flux density at the target, eqn. (3), is

unchanged, provided one uses q||t in eqn. (3). However, q|| is no longer constant along the flux

tube, thus eqn. (2) for the parallel electron heat conduction needs to be modified. We continue to

make the simple 2PM assumption of no volumetric power loss/gain along the flux tube, hence

we have from eqn. (A11):

2

R

R

B

B

B

B

q

q

||

|| 0

000 (A11)

where is the poloidal location along the flux tube and “0” indicates the reference location, here

taken to be the OMP. Here we will simply write:

R

Rqq OMP

OMP|||| (4)

where the lack of a subscript indicates any general location along the flux tube. Note that we are

assuming that all of the power enters the flux tube at the upstream end, at the OMP. It only alters

results slightly to allow for the power to enter the flux tube in a spatially distributed way.

For simplicity we will assume that R changes linearly with distance s|| measured along the

flux tube, from s|| = 0 at the OMP, thus:

L

sf

R

R ||R

OMP

11 (5)

where:

OMP

etargt

OMPR

R

R

R

LRf (6)

is the specified ratio and L is the connection length from OMP to the outer target.

The electron heat conduction equation is:

||

/||

ds

dTTq 25

0 (7)

which using eqns. (4) and (5) can be integrated to give the temperature change along the SOL:

3

12

7

0

2727

R

ROMP||/t

/u

f

flnLqTT (8)

For example, for fR = 2 then

1R

R

f

fln~ 0.693, so there is little change from eqn. (2).

As usual we are interested in cases where there is non-negligible temperature drop along the

SOL, so 27 /

tT can be neglected compared with 27 /

uT , thus:

7272

0 12

7/

R

R

/OMP||

uf

flnLqT

(9)

and one can see that the effect of 1Rf is even smaller; for Rf = 2,

72

1

/

R

R

f

fln

~ 0.901. It is

therefore not important exactly how R and SOL

||A vary along the flux tube and the simple linear

assumption here is adequate.

By contrast, it is quite important to allow for the fact that the q|| in eqn. (3), i.e. q||target, is

not the same as the q|| in eqn. (9), i.e. q||OMP:

ROMP||etargt|| f/qq (10)

Thus: i

tuu

Rstuu

RstttROMP||

m

kTkTn

fckTn

fckTnfq

2

22

(11)

where we have used eqn. (1).

We now combine eqns. (9) and (11) to obtain the target plasma conditions Tt and nt as

functions of the control parameters, nu and q||OMP:

4

74

74

02222

2

12

74

2

/

R

R

/OMP||

uR

OMP||it

f

flnLq

nef

q

e

mT

(12)

76

76

02

3322

12

7

4

/

R

R

/OMP||

OMP||i

uRt

f

flnLq

qm

nefn

(13)

where T [eV], n [m-3], q [M/m2], L [m], mi [kg], e = 1.6x10-19 C and typically 0 = 2000, = 7.

Note that the only significant change compared to the expressions for the basic 2PM, eqns.

(5.9) and (5.11) my book, are the 2Rf terms, since the

1R

R

f

flnterms aren’t very important

(~1.23 in eqn. (12) and ~ 0.73 in eqn. (13), for Rf = 2). To a good approximation we have:

2

OMP

etargtt

R

RT

and

2

OMP

etargtt

R

Rn (14)

This is therefore a highly valuable effect. Using a value of Rtarget which is, for example, 2X

bigger than ROMP, will reduce the divertor temperature by ~ 4X and increase the divertor density

~ 4X. These are both very valuable since they each help to increase volumetric power loss in the

divertor. The reduction in temperature also reduces the gross sputtering rate. The higher density

also decreases net erosion relative to gross erosion by increasing the probability of prompt local

deposition of sputtered particles.

It appears that Kotschenreuter et al, “The super X divertor (SXD) and a compact fusion

neutron source (CFNS)”, Nucl. Fusion 50 (2010) 035003, have developed the same modified

2PM as above, although they don’t provide enough details in their paper to make that explicitly

clear. Consider however Fig. 5 from their paper:

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Consider the right panel and the ratio of utupdiv T/TT/T for a density of 1020 m-3, comparing

the ratio for the two values of Rdivu fB/B = 1 and 1.9: from the modified 2PM one expects

utupdiv T/TT/T to be 1.92 = 3.61X higher for Rdivu fB/B = 1 vs 1.9, and this is readily seen

to be the case. This also holds for other densities so long as utupdiv T/TT/T <<1, as required

for the approximations made here to hold. It therefore seems almost certain that the

Kotschenreuter modified 2PM is the same as the one developed here.

Kotschenreuter et al also use their modified 2PM to estimate when the SOL is in the sheath-

limited regime, i.e. ut T/T ~ 1, which is the undesirable regime, of course, but which may be

difficult to avoid for AT operation, where nu tends to be small:

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where reference [24] is my book. The modified 2PM developed above gives exactly the same

expression for S, further indicating that the Kotschenreuter modified 2PM must be the same one

as developed above.

We consider next the deposited power flux density on the target, etargt

depq [MW/m2]:

etargt

wet

totaletargtdep

A

Pq (15)

where, as in standard 2PM, we assume that the total power Ptotal [W] carried along the SOL to

the target is conserved, i.e. no volumetric power loss and etargt

wetA [m2] is the plasma-wetted area

of the target for power. We start by considering an “orthogonal target”, i.e., a target whose

surface is orthogonal to the poloidal flux surfaces at the target. Then:

etargqtetargtetargt

wet RA 2 (16)

From eqn. (A6) we have:

etargt

OMP

qOMP

etargqt

B/B

B/B

(17)

Thus:

etargt

OMPqOMPetargt

etargtwet

B/B

B/BRA

2 (18)

This is the same result given by Kotschenreuter for an orthogonal target:

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since etargtetargtetargtsintanB/B where we have added the subscript “target”.

That is eqn. (18) gives:

etargt

OMPqOMPetargt

etargtwet

sin

B/BRA

2 (19)

Note that etargt here is the same as Kotschenreuter’s .

Consider next the non-orthogonal target, Fig. 1.

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Fig.1. Non-orthogonal target.

Now: etargtetargqtetargtetargt

wet cos/RA 2 (20)

where etargt is the angle between the target surface and the orthogonal to the poloidal flux

surfaces at the target. Thus eqn. (18) is altered to:

etargtetargt

OMPqOMPetargt

etargtwet

B/Bcos

B/BRA

2 (21)

It is readily shown, see Appendix B, that etargtetargt B/Bcos is the same as

Kotschenreuter’s sin and thus eqn. (21) is the same as Kotschenreuter’s eqn. (1).

Thus from eqn. (15) we have for the deposited power flux density on the target:

9

etargtetargt

OMPqOMPetargt

totaletargtdep

B/Bcos

B/BR

Pq

2

(22)

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Appendix A: The two different definitions of SOL Magnetic Flux Expansion

It can be confusing that two different definitions of SOL “magnetic flux expansion” are used.

This note is an attempt to clarify the situation. It is important that a distinction be made between

poloidal magnetic flux expansion and total magnetic flux expansion.

The first result we need is that the toroidal magnetic field, B satifies:

RB= constant (A1)

See, e.g. Wesson [1], also Fig. 1. Eqn. (A1) assumes that toroidal << 1.

Apply Ampere’s law to the circle of radius R in the midplane of the torus for a location inside the

coils:

coilIRB 02

(A2)

where Icoil is the total current in the toroidal field coils (this current only penetrates the circle of

radius R once if the circle lies within the coils and thus there is a net current of Icoil penetrating

the circle). Eqn. (A2) then gives eqn. (A1).

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The next result we need is that the poloidal magnetic field, B , satisfies:

)(s)(R)(B = constant (A3)

where )(s is the separation between two poloidal flux surfaces 1 and 2 , see Fig. 2.

Fig. 2 Poloidal cross-section. To show that )(s)(R)(B = constant.

The total poloidal magnetic flux between 1 and 2 is conserved, thus:

z

R

2 1 0s

s

12

)(s)(R)(B 2 = constant (A4)

Recall that B is magnetic flux density and has e.g. units [webers/m2] while the total magnetic

flux, e.g. in units [webers], is constant between two magnetic flux surfaces. Eqn. (A4) then gives

eqn. (A3).

From eqn. (A3) we now obtain our first expression for flux expansion which we will call here the

poloidal magnetic flux expansion:

0

00

0

RB

RB

s

s (A5)

where is any SOL decay length. To see why we equate to the last expression in eqn. (A5),

imagine a SOL quantity, e.g. Te, which was constant at the outside midplane from the separatrix

out to some radial distance 0s then dropped to 0, and we wanted to know what the radial

extent of constant Te would be at some other poloidal location assuming no variation along the

field line: clearly it would be s , thus eqn. (A5).

We can also combine eqns. (A1) and (A5) to get another expression for the poloidal magnetic

flux expansion:

0

0000

0

B/B

B/B

B/B

B/B

s

s (A6)

This is the expression in my book, eqn. (5.52) although it was obtained there making more

approximations than were made here.

Next we consider SOL

||A , the cross-sectional area of the SOL perpendicular to B. Consider an

imaginary, flat, toroidally continuous limiter positioned so that the normal to its surface is in the

poloidal direction. The limiter can be at any poloidal location . The wetted area of the limiter

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on each side is RAwet 2 . The value of SOL

||A is then given by the projection of wetA in the

direction of B:

B/BRASOL

|| 2 (A7)

Note that each of the quantities in eqn. (A7) is a function of .

Now combining eqns. (A5) and (A7) gives:

B/ASOL

|| 1 (A8)

Suppose we want to relate the parallel flux density, say of power, q||, at one poloidal location to

another, then assuming no variation along B we get:

00 SOL

||||

SOL

|||| AqAq (A9)

Thus we obtain the total magnetic flux expansion:

00 B

B

q

q

||

||

(A10)

We could have anticipated the result of eqns. (A9) and (A10) simply on the basis of the

similarity of magnetic flux density and parallel power flux density: in both cases the flux

direction is in the direction of B and the total quantity in a flux tube is conserved.

Using eqn. (1) we obtain the useful approximation for the total magnetic flux expansion:

R

R

B

B

B

B

q

q

||

|| 0

000 (A11)

Therefore for relating SOL decay lengths at two poloidal locations we use the poloidal magnetic

flux expansion, eqn. (A6), while for relating parallel flux densities we use the total magnetic flux

expansion, eqn. (A11).

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For basic analytic work one generally needs to use both types of flux expansion. For more

advanced analysis one may use a numerical line-tracing code so that one can relate each specific

poloidal flux surface at the local location, , back to the reference poloidal location, = 0. In

that case one doesn’t need to know the local value of : the local value of ne or Te on each

specific poloidal flux surface is just the same as it was at the reference location, assuming no

variation along B. Thus one doesn’t need to use the poloidal magnetic flux expansion. On the

other hand if one wants to know the local value of flux densities, such as q||, then the total

magnetic flux expansion still has to be used. One also usually needs to know the local value of

the magnetic pitch B/B but the line-tracing analysis provides that.

Appendix B. The plasma-wetted area of non-orthogonal divertor targets

(to be completed)