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# NIFS PROC 88

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```The transport equation of particles is
\u2202
\u2202t
n(r, t) +\u2207 · (n(r, t)V (r, t)) = 0 (7.1)
provided processes of the ionization of neutrals and the recombination of ions are negligible (see
ch.5.1). The particle \ufb02ux \u393 = nV is given by
n(r, t)V (r, t) = \u2212D(r, t)\u2207n(r, t)
in many cases, where D is di\ufb00usion coe\ufb03cient. (Additional terms may be necessary in more general
cases.)
Di\ufb00usion coe\ufb03cient D and particle con\ufb01nement time \u3c4p are related by the di\ufb00usion equation of
the plasma density n as follows:
\u2207 · (D\u2207n(r, t)) = \u2202
\u2202t
n(r, t).
Substitution of n(r, t) = n(r) exp(\u2212t/\u3c4p) in di\ufb00usion equation yields
\u2207 · (D\u2207n(r)) = \u2212 1
\u3c4p
n(r).
When D is constant and the plasma column is a cylinder of radius a, the di\ufb00usion equation is
reduced to
1
r
\u2202
\u2202r
(
r
\u2202n
\u2202r
)
+
1
D\u3c4p
n = 0.
The solution satisfying the boundary condition n(a) = 0 is
n = n0J0
(
2.4r
a
)
exp
(
\u2212 t
\u3c4p
)
and the particle con\ufb01nement time is
\u3c4p =
a2
2.42D
=
a2
5.8D
, (7.2)
where J0 is the zeroth-order Bessel function. The relationship (7.2) between the particle con\ufb01ne-
ment time \u3c4p and D holds generally, with only a slight modi\ufb01cation of the numerical factor. This
formula is frequently used to obtain the di\ufb00usion coe\ufb03cient from the observed values of the plasma
70
7 Di\ufb00usion of Plasma, Con\ufb01nement Time 71
The equation of energy balance is given by (A.19), which will be derived in appendix A, as
follows:
\u2202
\u2202t
(
3
2
n\u3baT
)
+\u2207 ·
(
3
2
\u3baTnv
)
+\u2207 · q = Q\u2212 p\u2207 · v \u2212
\u2211
ij
\u3a0ij
\u2202vi
\u2202xj
. (7.3)
The \ufb01rst term in the right-hand side is the heat generation due to particle collisions per unit
volume per unit time, the second term is the work done by pressure and the third term is viscous
heating. The \ufb01rst term in the left-hand side is the time derivative of the thermal energy per
unit volume, the second term is convective energy loss and the third term is conductive energy
loss. Denoting the thermal conductivity by \u3baT, the thermal \ufb02ux due to heat conduction may be
expressed by
q = \u2212\u3baT\u2207(\u3baT ).
If the convective loss is neglected and the heat sources in the right-hand side of (7.3) is zero, we
\ufb01nd that
\u2202
\u2202t
(
3
2
n\u3baT
)
\u2212\u2207 · \u3baT\u2207(\u3baT ) = 0.
In the case of n = const., this equation reduces to
\u2202
\u2202t
(
3
2
\u3baT
)
= \u2207 ·
(
\u3baT
n
\u2207(\u3baT )
)
.
When the thermal di\ufb00usion coe\ufb03cient \u3c7T is de\ufb01ned by
\u3c7T =
\u3baT
n
,
the same equation on \u3baT is obtained as (7.1). In the case of \u3c7T = const., the solution is
\u3baT = \u3baT0J0
(
2.4
a
r
)
exp
(
\u2212 t
\u3c4E
)
, \u3c4E =
a2
5.8(2/3)\u3c7T
. (7.4)
The term \u3c4E is called energy con\ufb01nement time.
7.1 Collisional Di\ufb00usion (Classical Di\ufb00usion)
7.1a Magnetohydrodynamic Treatment
A magnetohydrodynamic treatment is applicable to di\ufb00usion phenomena when the electron-to-
ion collision frequency is large and the mean free path is shorter than the connection length of the
inside regions of good curvature and the outside region of bad curvature of the torus; i.e.,
vTe
\u3bdei
<\u223c
2\u3c0R
\u3b9
,
\u3bdei >\u223c \u3bdp \u2261
1
R
\u3b9
2\u3c0
vTe =
1
R
\u3b9
2\u3c0
(
\u3baTe
me
)1/2
where vTe is electron thermal velocity and \u3bdei is electron to ion collision frequency. From Ohm\u2019s
law (5.28)
E + v ×B \u2212 1
en
\u2207pi = \u3b7j,
the motion of plasma across the lines of magnetic force is expressed by
nv\u22a5 =
1
B
((
nE \u2212 \u3baTi
e
\u2207n
)
× b
)
\u2212 me\u3bdei
e2
\u2207p
B2
71
72 7 Di\ufb00usion of Plasma, Con\ufb01nement Time
Fig.7.1 Electric \ufb01eld in a plasma con\ufb01ned in a toroidal \ufb01eld. The symbols \u2297 and \ufffd here show the
direction of the P\ufb01rsch-Schlu¨ter current.
=
1
B
((
nE \u2212 \u3baTi
e
\u2207n
)
× b
)
\u2212 (\u3c1\u3a9e)2\u3bdei
(
1 +
Ti
Te
)
\u2207n (7.5)
where \u3c1\u3a9e = vTe/\u3a9e, vTe = (\u3baTe/me)1/2 and \u3b7 = me\u3bdei/e2ne (see sec.2.8).
If the \ufb01rst term in the right-hand side can be neglected, the particle di\ufb00usion coe\ufb03cient D is
given by
D = (\u3c1\u3a9e)2\u3bdei
(
1 +
Ti
Te
)
. (7.6)
The classical di\ufb00usion coe\ufb03cient Dei is de\ufb01ned by
Dei \u2261 (\u3c1\u3a9e)2\u3bdei = n\u3baTe
\u3c3\u22a5B2
=
\u3b2e\u3b7\u2016
\u3bc0
, (7.7)
where \u3c3\u22a5 = nee2/(me\u3bdei), \u3b7\u2016 = 1/2\u3c3\u22a5.
However the \ufb01rst term of the right-hand side of (7.5) is not always negligible. In toroidal con-
\ufb01guration, the charge separation due to the toroidal drift is not completely cancelled along the
magnetic \ufb01eld lines due to the \ufb01nite resistivity and an electric \ufb01eld E arises (see \ufb01g.7.1). Therefore
the E×b term in (7.5) contributes to the di\ufb00usion. Let us consider this term. From the equilibrium
equation, the diamagnetic current
j\u22a5 =
b
B
×\u2207p, j\u22a5 =
\u2223\u2223\u2223\u2223 1B \u2202p\u2202r
\u2223\u2223\u2223\u2223
\ufb02ows in the plasma. From \u2207 · j = 0, we \ufb01nd \u2207 · j\u2016 = \u2212\u2207 · j\u22a5. By means of the equation
B = B0(1\u2212 (r/R) cos \u3b8), the P\ufb01rsch-Schlu¨ter current j\u2016 is given by (refer to (6.43))
j\u2016 = 2
2\u3c0
\u3b9
1
B0
\u2202p
\u2202r
cos \u3b8. (7.8)
If the electric conductivity along the magnetic lines of force is \u3c3\u2016, the parallel electric \ufb01eld is
E\u2016 = j\u2016/\u3c3\u2016. As is clear from \ufb01g.7.1, the relation
E\u3b8
E\u2016
\u2248 B0
B\u3b8
holds. From B\u3b8/B0 \u2248 (r/R)(\u3b9/2\u3c0), the \u3b8 component of the electric \ufb01eld is given by
E\u3b8 =
B0
B\u3b8
E\u2016 =
R
r
2\u3c0
\u3b9
1
\u3c3\u2016
j\u2016 =
2
\u3c3\u2016
R
r
(
2\u3c0
\u3b9
)2 1
B0
\u2202p
\u2202r
cos \u3b8. (7.9)
Accordingly (7.5) is reduced to
nVr = \u2212nE\u3b8
B
\u2212 (\u3c1\u3a9e)2\u3bdei
(
1 +
Ti
Te
)
\u2202n
\u2202r
72
7.1 Collisional Di\ufb00usion (Classical Di\ufb00usion) 73
Fig.7.2 Magnetic surface (dotted line) and drift surfaces (solid lines).
= \u2212
(
R
r
· 2
(
2\u3c0
\u3b9
)2 n\u3baTe
\u3c3\u2016B20
cos \u3b8
(
1 +
r
R
cos \u3b8
)
+
n\u3baTe
\u3c3\u22a5B20
(
1 +
r
R
cos \u3b8
)2)
×
(
1 +
Ti
Te
)
\u2202n
\u2202r
.
Noting that the area of a surface element is dependent of \u3b8, and taking the average of nVr over \u3b8,
we \ufb01nd that
\u3008nVr\u3009 = 12\u3c0
\u222b 2\u3c0
0
nVr
(
1 +
r
R
cos \u3b8
)
d\u3b8
= \u2212 n\u3baTe
\u3c3\u22a5B20
(
1 +
Ti
Te
)(
1 +
2\u3c3\u22a5
\u3c3\u2016
(
2\u3c0
\u3b9
)2) \u2202n
\u2202r
. (7.10)
Using the relation \u3c3\u22a5 = \u3c3\u2016/2, we obtain the di\ufb00usion coe\ufb03cient of a toroidal plasma:
DP.S. =
nTe
\u3c3\u22a5B20
(
1 +
Ti
Te
)(
1 +
(
2\u3c0
\u3b9
)2)
. (7.11)
This di\ufb00usion coe\ufb03cient is (1 + (2\u3c0/\u3b9)2) times as large as the di\ufb00usion coe\ufb03cient of (7.2). This
value is called P\ufb01rsch-Schlu¨ter factor (ref.). When the rotational tranform angle \u3b9/2\u3c0 is about
0.3, P\ufb01rsch-Schlu¨ter factor is about 10.
7.1b A Particle Model
The classical di\ufb00usion coe\ufb03cient of electrons
Dei = (\u3c1\u3a9e)2\u3bdei
is that for electrons which move in a random walk with a step length equal to the Larmor radius.
Let us consider a toroidal plasma. For rotational transform angle \u3b9, the displacement \u394 of the
electron drift surface from the magnetic surface is (see \ufb01g.7.2)
\u394 \u2248 ±\u3c1\u3a9e 2\u3c0
\u3b9
. (7.12)
The ± signs depend on that the direction of electron motion is parallel or antiparallel to the
magnetic \ufb01eld (see sec.3.5). As an electron can be transferred from one drift surface to the other
by collision, the step length across the magnetic \ufb01eld is
\u394 =
(
2\u3c0
\u3b9
)
\u3c1\u3a9e. (7.13)
73
74 7 Di\ufb00usion of Plasma, Con\ufb01nement Time
Consequently, the di\ufb00usion coe\ufb03cient is given by
DP.S. = \u3942\u3bdei =
(
2\u3c0
\u3b9
)2
(\u3c1\u3a9e)2\u3bdei, (7.14)
thus the P\ufb01rsch-Schlu¨ter factor has been reduced (|2\u3c0/\u3b9| \ufffd 1 is assumed).
7.2 Neoclassical Di\ufb00usion of Electrons in Tokamak
The magnitude B of the magnetic \ufb01eld of a tokamak is given by
B =
RB0
R(1 + \ufffdt cos \u3b8)
= B0(1\u2212 \ufffdt cos \u3b8), (7.15)
where
\ufffdt =
r
R
. (7.16)
Consequently, when the perpendicular component v\u22a5 of a electron velocity is much larger than
the parallel component v\u2016, i.e., when(
v\u22a5
v
)2
>
R
R+ r
,
that is,
v\u22a5
v\u2016
>
1
\ufffd
1/2
t
, (7.17)
the electron is trapped outside of the torus, where the magnetic \ufb01eld is weak. Such an electron
drifts in a banana orbit (see \ufb01g.3.9). In order to complete a circuit of the banana orbit, the e\ufb00ective
collision time \u3c4e\ufb00 = 1/\u3bde\ufb00 of the trapped electron must```