**Accurate
neoclassical resistivity, bootstrap current and other
transport coefficients**

(Fortran 90 subroutines and matlab functions**)**

The Fokker-Planck codes CQL3D
[1] and CQLP [2] have been used to calculate the neoclassical transport coefficients
valid for arbitrary tokamak geometry and collisionality.
These results provide better and more complete coefficients than the previously
used Hirshman, Hinton, Chang, Taguchi, etc models.

We have fitted the exact
Fokker-Planck results to a very good accuracy while maintaining relatively
simple formulas.

This was possible because we
found that the equilibrium effects can be very well encapsulated through a
trapped fraction ft (which definition depends on the related transport
coefficients [3: C. Angioni
et al + errata]).

On the other hand, the collisionality effects can be well encapsulated through
modifications of the trapped fraction, leading to effective trapped fractions.

The neoclassical conductivity sigma_{neo} and bootstrap current <j_{BS} B> are given in the paper [4] O. Sauter et al
and its errata (+0.315
instead of -0.315 in alpha coefficient)

The other neoclassical
transport coefficients, like chi_i for example, are
given in [3] C. Angioni
et al (+ Errata, refs corrected),
also valid for arbitrary geometry, aspect ratio and collisionality.

The input data are the
profiles and the trapped fraction. The trapped fraction for conductivity and
bootstrap, ft, can be obtained with either of:

1) from
the full formula as a double integral (as done in CHEASE), Eq.
(12) of [4]:

2) From
the approximate formula by Lin-Liu using simple flux surface average of B's,
thus only single integrals: Ref. [5]

3) For
most tokamak cases the following formula works relatively well (Eq. (4) of Ref [6]):

Modules have been developed
and are included in the CHEASE
code for example. They are available here below:

**F90
and matlab routines/modules for neoclassical resistivity and bootstrap current: NEOCLASSICAL.tar**

·
**How to use the formulas with
experimental data and the questions of Z, Zeff**

The original BS paper [4a] was not precise enough about the
pressure, Ti, Z and Z_{eff}. Additional information
is given in the Errata [4b].
In particular:

o
Z = Zeff in all
equations except Eq. 18(c) and Eq.
18(e) where Z=

o
Total pressure should be used for p and p_{e}/p should be used where stated and not some
approximations with Te and Ti for example

o
The formula for ln L should be used as
defined in the paper (Eqs 18(d,e))

The above are due to the fact
that the fits to the exact code results have been performed using these
definitions. The above statements are particularly true if a code benchmark is
being performed.

The code results for various Z_{eff} have been obtained using a single ion
species with Z_{eff} and therefore also
assuming T_{I}=T_{i}. This is a limitation which needs to be
tested against experimental results. There are several comparisons with
reconstructed profiles. The latest detailed comparison has been performed on
AUG [7]. In that
case they assumed a single impurity and Z_{eff}
to define a total ni=ni_{main} + ni_{imp} and to define total pressure. When ion
temperature and/or impurity temperature were not available, T_{i}=T_{e},
T_{I}=T_{i} were used. It is shown in Ref. [7] that the edge
bootstrap current is well predicted.

In smaller inverse aspect
ration in particular, a correction is necessary near the edge as proposed
recently in Ref. [8] and
NSTX. This correction should be added in all cases, in particular in H-mode for
the pedestal. A useful discussion of Z, Zeff is also
given in Sec. V.

As a general statement and in
the absence of specific more precise calculations, additional effects should be
added in total pressure and equivalent Z_{eff},
inspired by MHD effects. For example, fast particles contribution should be
included in total pressure and therefore total p_{i} be used as well.

**References**:

[2] O. Sauter, R. W. Harvey, and F. L. Hinton, Contrib. Plasma
Phys. **34** (1994) 169.

[3a]
C. Angioni and O. Sauter,
Phys. Plasmas **7** (2000) 1224

[3b] C. Angioni and
O. Sauter, **Errata**, Phys. Plasmas **7** (2002) 3122.

[4a] O. Sauter, C. Angioni, and Y. R. Lin-Liu, Phys. Plasmas **6** (1999)
2834

[4b] O. Sauter, C. Angioni, and Y. R. Lin-Liu, **Errata**, Phys Plasmas **9** (2002) 5140.

[5] Y. R. Lin-Liu
and R. L. Miller, Phys. Plasmas **2** (1995) 1666.

[6] O. Sauter et al, Plasma Phys. Control. Fusion
**44 **(2002) 1999.

[7] M. G. Dunne et al, Nucl. Fusion **52 **(2012) 123014

[8] S. Koh
et al, Phys. Plasmas **19** (2012) 072505