Note that the ratio of the left- to the right-hand sides of Eq. (9.5) for T d ¼ T Ã and
P
tot
ft ¼ P
tot
ft
À Á
max
gives us virtually parameter Π 1 , which, as we discussed before, sets
the limit, imposed by the available power, on the rate of plasma recycling.
To find the relation between N
tot
ft and T d , we take into account the plasma outside
the narrow, (L ft , recycling region, where there is no ionization source and
the plasma flow is stagnant. Therefore, outside the recycling region, the only
available mechanism to provide the energy flux q ft is heat conduction. So we have
q ft ¼ κ e (T)dT/dℓ, where κ e (T) / T
5/2 is the electron heat conductivity and ℓ is the
coordinate along the magnetic field line (we take ℓ ¼ 0 at the target). Then we find
the following expression for the electron temperature: T ℓ
ð Þ ¼ T
7=2
d þ q ft ℓ=b κ
2=7
,
where b κ ¼ 2=7
ð
Þκ e T
ð ÞT
À5=2
¼ const: As a result, assuming T(L ft ) ) T d , using the
expression for the plasma density n ℓ
ð Þ ¼ P
tot
ft =T ℓ
ð Þ and taking into account
Eq. (9.6), we find the following expression for the upstream plasma density
n up n(L ft ):
n up T d
ð Þ ¼ P
tot
ft T d
ð Þ
b κ
q ft L ft
2=7
:
ð9:8Þ
Note that the expression virtually identical to Eq. (9.7), (9.8) was used in [32] as
the justification for the SOL plasma density limit.
Similarly to the derivation of Eq. (9.8) we find the dependence N
tot
ft T d
ð Þ:
N
tot
ft T d
ð Þ ¼ L
À1
ft
Z L ft
0
n ℓ
ð Þdℓ %
7
5
P
tot
ft T d
ð Þ
b κ
q ft L ft
2=7
,
ð9:9Þ
and finally, we have
N
tot
ft T d
ð Þ ¼
2q ft
γT d þ E
H
ion
ffiffiffiffiffi
T d
M
r
7
5
b κ
q ft L ft
2=7
:
ð9:10Þ
Examining the expression (9.10) one finds that in accordance with our dimensionless analysis it can be re-written in terms of the dimensionless parameters (9.3)
and the functions (9.5).
As we see from Eqs. (9.6), (9.8), and (9.9), the functions n up (T d ), N
tot
ft T d
ð Þ, and
P
tot
ft T d
ð Þhave a similar dependence on T d , which seems to suggest that like P
tot
ft , both
n up (T d ) and N
tot
ft T d
ð Þ have some maximum values, n up T d
ð Þ / N
tot
ft
À
Á
max
/ q ft
ð Þ
5=7 .
However, more detailed analysis and numerical simulations [29] show that the
model we consider here is too crude to describe properly the recycling region for
small T d . In practice, the dependence T d N
tot
ft
À Á
for some cases can be described by an
N-shaped curve shown schematically in Fig. 9.8. Moreover, as it often happens,
some part of the N-shaped curve T d N
tot
ft
À
Á
is unstable (see Fig. 9.8), and there is a
bifurcation of T d N
tot
ft
À Á
dependence around N
tot
ft % N
tot
ft
À
Á
max
[29, 30, 33]. With
238
9 Physics of Some Edge Plasma Phenomena
P
tot
ft ¼ P
tot
ft
À Á
max
gives us virtually parameter Π 1 , which, as we discussed before, sets
the limit, imposed by the available power, on the rate of plasma recycling.
To find the relation between N
tot
ft and T d , we take into account the plasma outside
the narrow, (L ft , recycling region, where there is no ionization source and
the plasma flow is stagnant. Therefore, outside the recycling region, the only
available mechanism to provide the energy flux q ft is heat conduction. So we have
q ft ¼ κ e (T)dT/dℓ, where κ e (T) / T
5/2 is the electron heat conductivity and ℓ is the
coordinate along the magnetic field line (we take ℓ ¼ 0 at the target). Then we find
the following expression for the electron temperature: T ℓ
ð Þ ¼ T
7=2
d þ q ft ℓ=b κ
2=7
,
where b κ ¼ 2=7
ð
Þκ e T
ð ÞT
À5=2
¼ const: As a result, assuming T(L ft ) ) T d , using the
expression for the plasma density n ℓ
ð Þ ¼ P
tot
ft =T ℓ
ð Þ and taking into account
Eq. (9.6), we find the following expression for the upstream plasma density
n up n(L ft ):
n up T d
ð Þ ¼ P
tot
ft T d
ð Þ
b κ
q ft L ft
2=7
:
ð9:8Þ
Note that the expression virtually identical to Eq. (9.7), (9.8) was used in [32] as
the justification for the SOL plasma density limit.
Similarly to the derivation of Eq. (9.8) we find the dependence N
tot
ft T d
ð Þ:
N
tot
ft T d
ð Þ ¼ L
À1
ft
Z L ft
0
n ℓ
ð Þdℓ %
7
5
P
tot
ft T d
ð Þ
b κ
q ft L ft
2=7
,
ð9:9Þ
and finally, we have
N
tot
ft T d
ð Þ ¼
2q ft
γT d þ E
H
ion
ffiffiffiffiffi
T d
M
r
7
5
b κ
q ft L ft
2=7
:
ð9:10Þ
Examining the expression (9.10) one finds that in accordance with our dimensionless analysis it can be re-written in terms of the dimensionless parameters (9.3)
and the functions (9.5).
As we see from Eqs. (9.6), (9.8), and (9.9), the functions n up (T d ), N
tot
ft T d
ð Þ, and
P
tot
ft T d
ð Þhave a similar dependence on T d , which seems to suggest that like P
tot
ft , both
n up (T d ) and N
tot
ft T d
ð Þ have some maximum values, n up T d
ð Þ / N
tot
ft
À
Á
max
/ q ft
ð Þ
5=7 .
However, more detailed analysis and numerical simulations [29] show that the
model we consider here is too crude to describe properly the recycling region for
small T d . In practice, the dependence T d N
tot
ft
À Á
for some cases can be described by an
N-shaped curve shown schematically in Fig. 9.8. Moreover, as it often happens,
some part of the N-shaped curve T d N
tot
ft
À
Á
is unstable (see Fig. 9.8), and there is a
bifurcation of T d N
tot
ft
À Á
dependence around N
tot
ft % N
tot
ft
À
Á
max
[29, 30, 33]. With
238
9 Physics of Some Edge Plasma Phenomena