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Extended fluid transport theory in the tokamak plasma edge

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2017 Nucl. Fusion 57 066034

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1. Introduction

The high pressure pedestal over roughly the outer 5% of the confined magnetic flux [1] that is characteristic of the High Confinement regime [2] is the subject of intense interest because the properties of the edge pedestal seem to have a strong effect on the performance properties of the entire plasma (e.g. [3]). Recent research has been primarily in two areas: (i) establishment of the stability boundary in edge physics parameters for the onset of the magneto- hydrodynamic peeling-ballooning mode instability (edge localized mode-ELM) that destroys the edge pedestal [4] and (ii) understanding the transport in the edge pedestal that determines the evolution of these edge physic parameters [5] (pressure, electric field, etc). This paper is concerned with the second area and collects several recent advances into an extended fluid transport theory of the plasma edge. The paper is organized into brief discussions of ion orbit loss and intrinsic co-rotation; determination of the radial particle and energy fluxes; determination of the radial pressure and temperature distributions; determination of the radial elec-tric field and rotation velocities; interpretation of diffusive transport coefficients from measurements; solution of the transport equations; flux surface geometry representation of poloidal asymmetries; and finally the poloidal distribution of particle and energy fluxes out of the edge region into the scrape-off layer (SOL).

2. Ion orbit loss

Kinetic ion orbit loss from the thermalized ion distribu-tion directly affects the ion particle, pressure, energy and momentum distributions in the plasma edge.

The fluid transport calculation is usually carried out for the distribution of ‘thermalized’ ions that are flowing radially out-ward from a central or recycling ion source in the presence of the distributions of other ion species and electrons. The interactions of these thermalized ion species and their effect on the outward flows is traditionally described by the fluid transport theory, making use of ‘transport coefficients’ which are determined from kinetic theory or from turbulence theory, in various approximations. However, there is a further kinetic theory effect associated with thermalized ions that can access an orbit which carries them across the last closed flux sur-face (LCFS) and out of the confined plasma. Although such ion orbit loss (IOL) was identified 20 years ago by Miyamoto [6], IOL has only recently been recognized as an important phenom enon in plasma edge physics [7–13].

Following Miyamoto [6], we have used the constraints of conservation of magnetic moment, energy and canonical angular momentum to develop an equation for the determi-nation of the minimum energy for which an ion at a given location on an internal flux surface going in a given direc-tion can reach a given point on the LCFS [9–13], and from this have developed a numerical procedure for calculating

Nuclear Fusion

Extended fluid transport theory in the tokamak plasma edge

W.M. Stacey

Georgia Institute of Technology, Atlanta, GA 30332 United States of America

E-mail: weston.stacey@nre.gatech.edu

Received 15 November 2016, revised 20 February 2017Accepted for publication 4 April 2017Published 3 May 2017

AbstractFluid theory expressions for the radial particle and energy fluxes and the radial distributions of pressure and temperature in the edge plasma are derived from fundamental conservation (particle, energy, momentum) relations, taking into account kinetic corrections arising from ion orbit loss, and integrated to illustrate the dependence of the observed edge pedestal profile structure on fueling, heating, and electromagnetic and thermodynamic forces. Solution procedures for the fluid plasma and associated neutral transport equations are discussed.

Keywords: transport, edge plasma, tokamaks

(Some figures may appear in colour only in the online journal)

W.M. Stacey

Printed in the UK

066034

NUFUAU

© 2017 IAEA, Vienna

57

Nucl. Fusion

NF

10.1088/1741-4326/aa6b34

Paper

6

Nuclear Fusion

IOP

International Atomic Energy Agency

2017

1741-4326

1741-4326/17/066034+7$33.00

https://doi.org/10.1088/1741-4326/aa6b34Nucl. Fusion 57 (2017) 066034 (7pp)

W.M. Stacey

2

the cumulative loss fraction for outflowing thermalized ions

( )F rjiol , for ion energy ( )E rj

iol and for ion parallel momentum ( )M rj

iol across the LCFS [9–13]. We estimate [13]from calcul-ation that a fraction ≈R 0.5jloss

iol of these escaping particles, energy and momentum do not return into the plasma on the same orbit and thus constitute a kinetic loss of the outflowing fluxes of thermalized ions. We note that some of the particles lost in the SOL could have interactions that cause them to re-enter the plasma as kinetic sources at other plasma locations, but this effect is not treated in the calculation of this paper.

A representative calculation of the IOL loss fractions for a DIII-D H-mode shot is shown in figure 1, calculated for the exact flux surface geometry (empty symbols) and for an effec-tive circular model flux surface geometry (solid symbols),

both with =R 1.0jlossiol . A very large fraction of both the parti-

cles and energy in the outflowing thermalized ion distribution actually crosses the LCFS on IOL orbits which deposit them in the scrape-off layer (SOL) preferentially near the outboard midplane [11]. Thus, taking IOL into account in fluid calcul-ations will greatly affect the calculated distribution both of the radial pressure and temperature profiles of thermalized ions within the edge plasma and the poloidal profiles of thermal-ized ions, energy and momentum from the edge plasma into the SOL.

The calculation of the ion orbit loss fraction is discussed in [9–13], and this paper is concerned more with how to include these ion orbit loss fractions in a fluid model calculation. We have in mind low collisionality edge plasmas and did not include the additional ion orbit loss that would be associated with confined ions scattering into the loss cone.

There is also a similar loss of fast ions produced by neutral beam injection (nbi), which must be taken into account in calcu-lating the source of thermalized ions [13] (and in calculating the alpha particles remaining in the plasma in future fusion plasmas).

3. Intrinsic co-rotation

The ion orbit loss of particles directed opposite to the cur-rent direction (ctr-I) is significantly larger than the loss of co-I

particles, resulting in ( )>M r 0jiol in figure 1 and a preferential

retention of co-I particles, which constitutes an intrinsic co-

rotation [14–16] ( ) ( )∥( )∆ =

πV r M rj

kT r

m

2 iol 2 j

j1

2. The peaking in

( )M rjiol in figure 1 corresponds to the radial location at which

most of the ctr-I particles have been lost and the co-I parti-cles are beginning to be lost. Identification of measured co-I toroidal rotation peaking with IOL predictions has now been made in DIII-D [14–17] and EAST [18], but measured co-I intrinsic poloidal rotation has not yet been reported, although there is also a predicted co-I poloidal component of ∥∆V . Intrinsic co-rotation due to IOL is a kinetic effect that must be added to the fluid calculation of rotation.

4. Ion radial particle flux distribution

Particle conservation determines the radial distribution of the thermalized ion flux. The flux surface average (FSA) radial particle continuity equation for the main ion species ‘j’ (in the cylindrical approximation) is

( ( ))( )( ( )) ( ) ( )

( )( ) ( )

( )( ) ( )

α ν

α β

α β

∂ Γ

∂= − +

−∂

∂Γ −

∂∂Γ

≡ −∂

∂Γ −

∂∂Γ

r

r r

rN r f r n r r

F r

rr z

F

rr

SF r

rr z

F

rr

11

jj j j

jj k

kk

jj

j kk

k

rnb nb

iole ion

iol

r

iol

r

n

iol

r

iol

r

(1)where N jnb and f jnb

iol are the fast neutral beam source rate and the differential ion-orbit loss fraction for fast beam ions of species ‘j’, ne is the electron density, ν συ= nj o jion ion (with noj being the density of neutral atoms of species j) is the

ionization frequency of neutrals of species ‘j’, ( )F rjiol is the

cumulative ion-orbit loss of thermalized particles of species ‘j’ over < <′r r0 and Γ jr is the outward radial particle flux of thermalized main ions of species ‘j’. The parameters α and β depend on how charge neutrality is assumed to be maintained: i) if neutrality is maintained by a compensating electron loss by unspecified means ( )α β= =1, 0 ; ii) if neutrality is main-tained by a return inward current of thermalized ions from the SOL with a divergence equal to the local IOL ion orbit charge loss rate ( )α β= =2, 1 ; and (iii) if there is no ion orbit loss ( )α β= =0, 0 . It is thought that ( )α β= =2, 1 is the most physically reasonable case. Equation (1) may be integrated to obtain

( ) ( ) ( )

( )

[ ( ) ( )]

[ ( ) ( )]

∫

∫

α ν βΓ = − + −∂∂Γ

×

≡

′ ′

′ ′ ′

α

α

− −

− −

′

′

⎡

⎣⎢

⎤

⎦⎥r r N f n z

F

rr

r r

S r r r

1

e d

e d .

j

r

j j j kk

k

F r F r

r

jF r F r

r 0 nb nbiol

e ion

iol

r

0n

j j

j j

iol iol

iol iol

(2)

The radial particle flux of thermalized ions at any radius r depends on the integral over ion particle flux sources minus sinks at all radii < <′r r0 but attenuated by the ion orbit loss of a certain fraction of these source particles. Since the

ion orbit loss particle loss fraction, ( )F rjiol , increases sharply

Figure 1. Ion orbit loss fractions calculated for a DIII-D H-mode discharge #123302 (i) using measured flux surface geometry and (ii)

using an effective circular flux surface representation ( =R 1.0jlossiol ).

(Reprinted from [13], with the permission of AIP Publishing.)

Nucl. Fusion 57 (2017) 066034

W.M. Stacey

3

with radius just inside the separatrix, this effect would at least partially offset the effect of the increasing ionization source just inside the separatrix, as shown in figure 2 for a DIII-D H-mode discharge. Note that no conductive transport coeffi-cient (neoclassical or turbulent) enters the determination of the ion radial particle flux from the particle continuity equa-tion, so the radial particle flux depends on the sources and sinks, not the transport coefficient. (The transport coefficient will of course affect the particle distribution.)

5. Radial ion pressure distribution

Momentum conservation determines the radial ion pressure distribution. The flux surface average (FSA) of the radial momentum balance for species ‘j’ is

( )( )

∂

∂= + −θ ϕ ϕ θ

r

rp

rn e E n e V B V B

1 jj j j j j jr (3)

The replacement ∫ϕ ϕ ϕ∇ ∇ Π ≡ ∇ ∇ Ππ

R R· · d · ·p

20

2 2∮ [ ]

∫ ϕ νπ

ϕ�B B Rnm Vdℓ d dℓpp p 0

2p p visc∮/ / / in which the radial

gradient of the velocity is replaced by a radial gradient scale length of the velocity in the neoclassical νvisc is rigorous for the rate-of-strain tensor representation of viscosity [22] (in which the parallel viscosity component vanishes identically for axisymmetric plasmas), and a similar replacement can be made in the inertial term. Here, the subscript ‘p’ refers to the poloidal component and ϕ refers to toroidal. For a toroidally axisymmetric tokamak the FSA of the toroidal momentum balance equation is then

( )ν ν ν+ − = + Γ +ϕ ϕ ϕ θ ϕn m V n m V n e E e B Mj j jk dj j j j jk k j j j j jr (4)

and similar equations obtain for the other impurity ion spe-cies ‘k’. The νjk is an interspecies collision frequency, the ν ν ν ν ν≡ + + +ϕdj j j j jvisc in cx ion is a composite momentum loss frequency due to viscosity plus inertia plus charge-exchange plus ionization, ϕM j is the toroidal momentum source rate, and

E B, and V are the electric and magnetic fields and the fluid velocity. ϕE includes the inductive transformer field ϕEA plus any toroidal electric field produced by turbulence. Making the assumption of a single impurity species ‘k’ with the same density distribution and temperature as the main ions ‘j’, equations (3) and (4) and similar equations for the impurity species ‘k’ may be solved for the FSA of the main ion radial pressure gradient [22]

( ) ( )( ( / ) )

( )⎛

⎝⎜

⎞

⎠⎟

ν ν≡ −

∂

∂=

− +Γ − Γθ−p L

r

rp

r

e B

m e e

1

1j pjj j

j jk j k djrj j

12

pinch

(5)where the ( ( / ))− e e1 j k term approximately accounts for the presence of the impurity species ‘k’, and the electromagnetic pinch particle flux given by [22]

( )ν ν

ν

Γ = − − ++

+

−

ϕ

θ

ϕ

θ θ θ

ϕ

θθ

ϕ

⎛⎝⎜

⎞⎠⎟

n E

B

M

e B

n m

e B

E

B

B

BV

n m V

jj

Aj

j

j j jk dj

j

rj

j j jk k

rpinch

(6)

is a FSA radial ‘pinch’ particle flux associated with electro-magnetic, external, viscous and friction forces. Note that the momentum balance requirements of equations (3) and (4) dic-tate that the radial particle flux must be of the pinch-diffusive form of equation (5).

Equation (5) can be integrated from some interior edge location ‘r’ outward to the separatrix at rsep to obtain the FSA pressure distribution of the main ion species ‘j’

∫ ν ν− =

Γ − Γ

− +′ ′

θrp r r p

e B

m e er r

1d .j j

r

r j rj j

j jk j k djsep

sep2 pinch

sep

( )( ) ( )( ( / ) )

(7)

Equation (7) indicates that the radial ion pressure profile in the plasma edge depends on particle sources through the Γ jr given by equation  (2) and on the electromagnetic

and external momentum sources through the Γ jrpinch term

given by equation  (6). This has interesting implications for the control of the radial ion pressure distribution in the edge by injection of neutral beam momentum in the edge to control edge rotation and the radial electric field (see section 8).

Similar equations, but with the ‘j’ and ‘k’ interchanged, obtain for the pressure of the impurity ions.

6. Ion and electron total radial energy fluxes

The ion and electron radial energy fluxes are determined by ion and electron energy conservation.

The FSA radial energy balance equation for the main ion species is

α

συ

∂

∂= − − −

− − −∂

∂

�r

rQ r

rq r r q r q r

n r n r T r TE r

rQ r

11 e

3

2

jj

ii j jk

j oj j ojj

l

j

r nbnbiol

e

cx

io

r

( ( ))( )( ( )) ( ) ( )

( ) ( ) ( ( ) )( )

( )

(8)

Figure 2. Radial particle flux calculated from equation (2) with (α β= =2, 0 empty symbols) and without (α β= =0, 0 solid symbols) ion orbit loss.

Nucl. Fusion 57 (2017) 066034

W.M. Stacey

4

with solution

∫

∫

α ν= − − − − −

×

≡

′ ′

′ ′

− −

− −

′

′

�⎜ ⎟⎛⎝

⎞⎠

rQ r

q q q n T T

r r

S e r r

1 e3

2

e d

d

jr

ji

i j jk j j j oj

E r E r

r

EjE r E r

r

0nb

nbiol

e cx

0

j j

j j

iol iol

iol iol

( )

( ) ( )

[ ( ) ( )]

[ ( ) ( )]

(9)

where q jinb is the neutral beam (or other) heating rate of ion

species ‘j’, /=T p nj j j is the temperature, qje and qjk are the energy loss rates due to scattering with electrons and other

ion species ‘k’, E jiol and e j

iol are the ion orbit energy loss

fractions for thermalized (E jiol cumulative) and fast beam

(e jiol differential) ions of species ‘j’, and the subscript ‘oj‘

refers to the neutral atoms of species ‘j’. A calculation for MAST found ��α 0 for co-current beam injection, due to heating effects associated with the compensating fast ion current, and �α< <0.5 1.0 for counter-current beam injection [19]. Similar equations obtain for the other ion species ‘k’. Note that the radial ion energy flux at radius r is determined by sources and sinks of energy and energy ion orbit loss over < <′r r0 , and not by any conductive or other transport coef-

ficient. (The temper ature distribution will be determined by the diffusive transport coefficient, of course.)

Equation (9) indicates that the radial ion heat flux at radius r is determined by the net of heat sources and sinks integrated over < <′r r0 but attenuated by ion orbit energy loss.

For the electrons the FSA radial energy balance is

( ( )) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

συ

συ

∂∂

= + + −

− − −r

rQ r

rq r q r q n r n r E

n r n r E n n L r n n L r

1 ij k oj j j

ok k k j j k k

reenb

e e e ion ion

e ion ion e e

(10)

where Eion is the ionization energy and L is the radiation emis-sivity. The last four terms are the ionization energy loss of neutral main and impurity atoms and the radiation loss due to interactions with the main ions and with partially stripped impurity ions, respectively. The solution is

( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )∫

συ

συ=

+ + −

− − −′

′ ′ ′ ′ ′

′ ′ ′ ′′

⎡

⎣⎢⎢

⎤

⎦⎥⎥

rQ r

rq r q r q r n r n r E

n r n r E n n L r n n L rrd .

r ij k oj j j

ok k k j j k k

re

0

enb

e e e ion ion

e ion ion e e

(11)

7. Ion and electron temperatures

The quantity Qj is the sum of the conductive and convec-tive heat fluxes and perhaps other heating mechanisms (e.g. viscous). Since equations (1), (3), (4) and (8) are the zeroth, first and second velocity moments of the Boltzmann trans-port equation, it would be logical to use the third velocity moment of the Boltzmann equation to obtain an expression

relating the conductive heat flux to the temperature and density, but this quickly becomes computationally intrac-table and we join everyone else in using the Fourier radial heat conduction relation χ χ= − ∇ ≡ −q n T nT LTr r

1 as a sur-rogate for this purpose to obtain an equation  for the ion temperature

( ( ))( ) ( ) ( ) ( )

( )

( ) ( )

[ ( ) ( )]

[ ( ) ( )]

∫

∫

χ χ−∂∂

≡ = = − Γ

=

−

′ ′ ′

′ ′ ′

−

− −

− −

′

′

⎛⎝⎜

⎞⎠⎟n

r

rT r

rn T L q r Q r T r r

rS r r r

T rr

S r r r

1 5

2

1e d

5

2

1e d

j jj

j j j Tj j j j j

r

EjE r E r

j

r

jF r F r

1r

0

0 n

j j

j j

iol iol

iol iol

(12)

and for the electron temperature

∫

χ χ

συ

συ

−∂∂

≡ = = − Γ

=+ + −

− − −

− Γ + Γ

′′ ′ ′ ′ ′

′ ′ ′ ′′

−⎜ ⎟⎛⎝

⎞⎠

⎡

⎣⎢⎢

⎤

⎦⎥⎥

nr

rT r

rn T L q r Q r T r r

rr

q r q r q r n r n r E

n r n r E n n L r n n L rr

T r z r z r

1 5

2

1d

5

2

T

r ij k oj j j

ok k k j j k k

j j k k

e ee

e e e e1

e e e re

0

enb

e e e ion ion

e ion ion e e

e r r

( ( )) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) [ ( ) ( )]

(13)

where zk is the charge state of the impurity ion. The determi-nation of the total ion or electron radial temperature distribu-tions requires knowledge of the thermal diffusivities due to turbulent or other classical processes, as well as the determi-nation of radial energy flux, which latter is sensitive to the ion orbit loss effects emphasized in this paper.

8. Radial electric field

The radial electric field is governed, via momentum conser-vation constraints, by the ion rotation velocities and pressure gradients. Multiplying the radial momentum balance equa-tion (3) by /e m and summing over main ions ‘j’, impurity ions ‘k’ and electrons, then using the leading order force balance × = ∇ +∇ +∇p p pj B  j k e leads to an Ohm’s Law [20] for

the determination of

( )( / )

( )( / )

( )( )

η= − −−

+−

−

+

−+

+

θ ϕ ϕ θ θ ϕ ϕ θ

− −

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪⎧⎨⎩

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

⎫⎬⎭

⎧⎨⎩

⎫⎬⎭

E jV B V B

n m n m

V B V B

n m n m

p L p L

e n z n

1 1

.

j j

k k j j

k k

j j k k

j pj k pk

j k k

r r

1 1

(14)

Here η is the Spitzer resistivity and the first term is generally small compared to the other two terms. The second, motional electric field, term and the pressure gradient term were domi-nant in several DIII-D discharges that have been examined [20]. This expression, when evaluated with rotation velocities determined from experiment, agrees very well [20] with the conventional ‘experimental’ electric field determined by using measured carbon density, temperature and rotation velocity in the radial momentum balance of equation (3) for carbon.

Nucl. Fusion 57 (2017) 066034

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5

9. Toroidal and poloidal ion rotation velocities

The toroidal and poloidal fluid rotation velocities are deter-mined largely by the toroidal and poloidal momentum bal-ance equations. The major issue here is the evaluation of the FSA of the toroidal and poloidal components of the viscous

stress tensor ∇ ΠϕRn · · and ∇ Πθn · · , where Π is the viscous stress associated with flows parallel, perpendicular and encir-cling the magnetic field, giving rise to parallel, perpendicular and gyro viscous forces, respectively. The parallel viscosity

component of ∇ ΠϕRn · · vanishes identically for an axisym-metric tokamak with zero radial magnetic field [21] and the remaining gyro and perpendicular viscosity components have

the form [21] � ν∇ Πϕ ϕ ϕR nmR Vn · · visc .The toroidal component of the FSA of the inertial force

( ) ( )� ν ν∇ ∇ +ϕ ϕR nm Rnm Vn V V· · in ion where νin is an ‘inertial’ momentum transport frequency and νion is the ioniz-ation frequency [22]. A similar term probably exists for the poloidal component of the inertial force, but we have not investigated this.

Using these representations of the viscous and inertial terms and including also the charge-exchange momentum exchange term, we obtain a composite momentum exchange frequency ν ν ν ν ν= + + +ϕd visc in ion cx in terms of which the toroidal momentum balance equation for each ion species can be written in the form of equation (4) to obtain a coupled set of n equations in n unknowns, where n is the number of ion spe-cies. We note that such a representation is strictly valid only for a toroidally axisymmetric system with no radial magnetic field components, and that the presence of non-axisymmetric radial magnetic field components changes the structure of the viscosity representation [21, 23], as well as introduces a host of new viscosity mechanisms [24].

Employing a similar representation for the poloidal comp-onent of viscosity � ν∇ Π −θ θ θnm Vn · · visc yields

[ ]ν ν ν ν ν+ + − = −Γ

θ θ θ θϕ ϕ

−

V VB

B

K L

e

e B

n mj jk j j jk k j

jTj

j

j j

jvisc atom visc 2

1r

(15)

and a similar equation  with ‘j’ and ‘k’ interchanged [19]. Here, ν ν ν= +j j jatom cx ion , /ν ν µ=θ θB Bj jjvisc 00 and the Kj and µ00 are the Hirshman–Sigmar coefficients [25, 26].

The fluid velocities calculated from equations  (4) and (15) do not include the effects of IOL momentum loss. This can be remedied by including IOL momentum loss terms in equations (4) and (15) or, equivalently, by considering equa-tions (4) and (15) without IOL momentum loss terms, which are based on fluid momentum balances on the thermalized plasma ions in the absence of IOL to define fluid velocities for the thermalized ions. In the latter case the effect of IOL can be taken into account by calculating an intrinsic rotation of these thermalized ions due to the preferential ion orbit loss of counter-current ions. This is discussed in [12]. We find the latter procedure more straightforward.

If the solutions to the pair of equations  (4) and (15) are designated with a fluid superscript, then the total rotation velocities observed experimentally should be

( )∥π

= + = +ϕ ϕ ϕ ϕ ϕV V V V MkT

mn n

2 2·j j j j j

j

j

tot fluid intrin fluid1

2

iol

(16)and

π= + = +θ θ θ θ θV V V V M

kT

mn n

2 2· .j j j j j

j

j

tot fluid intrin fluid1

2

iol ( )∥

(17)

10. Interpretation of diffusive/conductive transport coefficients from experimental data

The basic diffusive/conductive transport coefficients that have been introduced are the conductive radial heat transport coef-ficients, χ, and the viscous momentum exchange frequencies ν ϕvisc and ν θvisc .

The Fourier heat conduction relation may be inverted

χ = =− Γ

− −

q

nT L

Q T

nT LT T1

5

2 r

1 (18)

for the determination of ion and electron thermal diffusivities from measured temperatures and densities of the respective species. As indicated in equations (2) and (9), ion orbit loss must be taken into account in calculating the total radial par-ticle and heat fluxes. It is clear from figures 2 and 3 that IOL will have a major impact on the interpretation of ion thermal diffusivity in the plasma edge.

The diffusive/conductive transport coefficients impor-tant for ion particle transport are the ion viscous momentum exchange frequencies, ν ϕvisc and ν θvisc (along with the collision, charge exchange and ionization frequencies). If the toroidal velocities are measured for main and impurity ions, then these measurements may be used as input to equations (4) for both species in the calculation of ν ϕjvisc and ν ϕkvisc . If the velocity of one species is not measured, then there are perturbation methods [27] that can be used. Similarly, if the poloidal veloc-ities are measured for main and impurity ions, these may be used as input to equations (15) for both species in the calcul-ation of ν θjvisc and ν θkvisc .

11. Solution of the edge transport equations

Assuming that the χ are known, equations  (12) and (13) may be integrated to determine ( )T rj and ( )T re . Equation (7) may be solved for ( )p rj and the ion density determined from

( ) ( )/ ( )=n r p r T rj j j . The electron density can be determined from charge neutrality ( ) ( )= +n z n r z n rj j k ke , and the elec-tron pressure can be determined from =p n Te e e. This pro-cess must be carried out iteratively. Evaluation of the atomic physics terms requires also the solution of neutral particle transport equations, which is usually done with Monte Carlo, but may be done more efficiently with the TEP integral trans-port method developed specifically for plasma edge transport calcul ations [28–32].

In existing large 1D and 2D (e.g. [33, 34]) edge trans-port codes, a diffusion theory is used for the calculation of densities and temperatures, and the electromagnetic pinch

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6

forces in equation (6) are taken into account by adding drift flows into the total poloidal and radial convective velocitie [33,  34]. We have suggested a systematic procedure for developing a momentum-conserving pinch-diffusion theory by substituting the momentum-conserving radial particle flux determined from equation  (5) into the particle conserving continuity equation  (1), resulting in a generalized diffusion equation  [22](written here in the slab approximation for clarity)

( )

( )( ) ( )

⎜ ⎟⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

⎛

⎝⎜

⎞

⎠⎟

⎛⎝⎜

⎞⎠⎟

ν ν ν

−∂∂

∂

∂−∂∂

∂∂

−∂∂

∂

∂

−∂∂

∂∂

+∂ Γ

∂=

≡+

≡θ θ

rD

n

r rD

n

r rD

n

T

T

r

rD

n

T

T

r rS

Dm T

e BD

m T

e e B,

jjj

jkk

jjj

j

j

jkj

k

k jj

jjj j dj jk

jjk

j j jk

j k

rpinch

2 2

(19)

and a similar equation with ‘j’ and ‘k’ interchanged for the impurity species. This equation can be simplified somewhat by assuming a single impurity species distributed as the main ions with the same temperature. Substituting the resulting expression for Γ jr into the continuity equation  results in a somewhat simpler diffusion equation  (written now in the cylindrical approximation)

ν ν

−∂∂

∂∂

+∂∂Γ

+∂

∂Γ −

∂∂

=

≡ − + θ

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

r rD

rrn T

r rr

F

rD

r rrn T S

D m e e e B

1 1

21

1 .

j j j j

jj j j j j

j j jk j k dj j

rpinch

riol

rpinch

n

2

( ) ( )

( )

( ( / ) )/( )

(20)

It is not clear that this diffusion theory formalism has any computational advantage over direct numerical solution of the previous equations of this paper.

12. 2D poloidally asymmetric flux surface geometry

Although the foregoing equations  are based on cylindrical flux surface geometry to minimize the complexity of the for-malism, the fluid equations  can also be developed on more realistic flux surface geometries, such as the analytical flux surface geometry known as the Miller model [35] and its var-ious extensions [36–38], or on exact flux surface geometries constructed from experimental measurements of the magnetic field [39]. Comparison calculations [38]indicate that the ana-lytical Miller model flux surfaces reproduce all features of the exact EFIT [39]flux surfaces.

The ion orbit loss fractions calculated using the exact EFIT [38] flux surface geometry and using an effective circular geometry that conserved flux surface area are compared [13] in figure 1 for a DIII-D discharge. Clearly, it is important to represent the geometric details in making the IOL calculation.

A fully 2D calculation could be developed in principle, but this remains a task for the future.

13. Poloidal asymmetry of particle and energy fluxes into SOL

The above calculations of FSA radial particle and energy fluxes are of course poloidally symmetric. However, the ana-lytical Miller model flux surface geometry takes into account poloidally asymmetric flux surface expansion and compres-sion in Euclidian space, hence enables the calculation of poloidally-dependent radial gradients in Euclidian space [40], which predict the poloidally dependent conductive heat flux distribution shown in figure  3 for an H-mode DIII-D dis-charge [11]. This type of poloidal distribution compares well [40] with that predicted by the explicitly 2D UEDGE [33] and SOLPS [34] codes.

Furthermore, as seen from figure 1, the cumulative (with radius) fraction of thermalized ions and their energy that is being ion orbit lost becomes quite large in the very edge. The poloidal distribution of IOL particles and energy is strongly peaked about the outboard midplane [11], as shown for the IOL energy flux distribution in figure 3 for an H-mode discharge. The implication is that ion orbit loss produces a stronger out-board peaking of particle and energy fluxes into the SOL than would be predicted by codes that calculate only the diffusive and convective fluid radial particle and heat fluxes.

14. Relation to previous work

Braginskii [41] and subsequent researchers have taken velocity moments of the Boltzmann transport equation  (or approximate versions thereof) to derive fluid particle, momentum and energy balance equations involving various integrals over the distribution function—constitutive rela-tions identified as viscosity, thermal conductivity, etc. Yet other workers have carried out a similar procedure involving fluctuations to arrive at similar fluid balance equations with

Figure 3. Normalized poloidal distribution of energy flux into SOL (outboard midplane θ π= 0, 2 , inboard midplane θ π= ). (Reprinted from [11], with the permission of AIP Publishing.)

Nucl. Fusion 57 (2017) 066034

W.M. Stacey

7

turbulent constitutive relations. These fluid balance equa-tions have been combined with the assumptions that particle transport can be described by a diffusion equation and heat conduction is given by the Fourier heat conduction relation to get the modeling equations used in 2D SOL fluid codes such as SOLPS, UEDGE, EDGE2D-EIRENE, etc. The par-ticle (1), momentum (3 & 4) and energy (8 & 10) balance equations of this paper are similar to the ones from which the UEDGE etc modeling equations  were derived, except that ion orbit loss is included in the equations of this paper. The Fourier heat conduction relation is also used in this paper. Furthermore, instead of assuming that particle transport is determined by a diffusion equation, in this paper the particle transport equation is derived by solving the momentum bal-ance for the radial particle flux which is then substituted into the particle balance (continuity) equation to obtain a ‘pinch-diffusion’ particle transport equation (19) or (20) that incor-porates electromagnetic as well as thermodynamic forces.

15. Summary

The particle, energy and momentum balance equations in the edge plasma have been reformulated to include ion orbit loss and integrated to obtain expressions that display explicitly the quantities upon which the pressure, the temperature gradients and the radial particle and energy fluxes in the edge plasma depend. The numerical solution of these equations and coupled equations for recycling neutral atoms are discussed. Inclusion in existing codes of ion orbit loss of thermalized ions will rather dramatically change both (i) the predicted radial pro-files of density, temperature, velocity and radial electric field in the edge pedestal inside the separatrix and (ii) the predicted poloidal distribution of ions, energy and momentum from the plasma edge into the scrape-off layer.

Acknowledgment

The calculations shown in figures  1–3 were based on mea-sured data from DIII-D H-mode shot 123302 @ 2600 ms. The author is grateful to T. M. Wilks for preparing figures 1 and 2, to M. T. Schumann for preparing figure 3 and to R.J. Groeb-ner for fruitful discussions over the years about plasma edge physics.

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