q z
ð Þ ¼ À
Z
K z, z
0
ð
Þ
∂T e
∂z 0 dx
0 ,
ð6:41Þ
can be considered with different kernels K(z, z
0 ) which were suggested over the years
(e.g. see [33–36]). A comparison of the outcomes of these nonlocal models with the
results of numerical simulations of the decay of a small amplitude, harmonic electron
temperature perturbation is shown in Fig. 6.1. However, in spite of the reasonable
agreement of the results coming from some non-local models based on the integral
expression (6.41) and numerical simulations, in the edge fluid plasma transport
codes, a much simpler approach is usually employed. It is based on the so-called
“flux limiting” expression for the heat flux suggested in [38]. This expression simply
constrains the Spitzer-Härm heat flux, q SH (z), by the fraction, fr FS % 0.2 À 0.4, of the
free streaming electron heat flux, q FS (z) ¼ n(z)T e (z){2T e (z)/πm}
1/2 , so that
q x
ð Þ ¼
q SH z
ð Þfr FS q FS z
ð Þ
jq SH z
ð Þj þfr FS q FS z
ð Þ
:
ð6:42Þ
As we see, such expression describes the reduction of q(x) in comparison with
q SH (z). For a monotonic temperature profile T e (z), which is the case for both the
inertial fusion applications and the edge plasmas, this reduction has clear physical
meaning in the high-temperature region, where, at γ > 10
À2 , the tail of the electron
distribution function is depleted because of the runaway of the weakly collisional
electrons into the low-temperature region. However, in the low-temperature region,
these suprathermal electrons result in the increase of q(x) beyond q SH (z), but such an
effect is not captured by Eq. (6.42). Nonetheless, expression (6.42) and similar ones
for ion heat conduction and viscosity along the magnetic field are often used in edge
plasma transport codes.
10 –3
10 –4
10 –3
10 –2
10 –1
10 0
10 –2
10 –1
L
K
A
kλ T
κ
eff /
κ
SH
10 0
10 1
Fig. 6.1 Ratio of the
effective, κ eff , to the SpitzerHärm, κ SH , electron heat
conductivities as a function
of kλ T , where k is the
wavenumber of the initial
electron temperature
perturbation. The filled
circles are the results of
numerical simulations [37],
the curves A, K, and L are
from the references [33–
35]. (Reproduced with
permission from [35],
© AIP Publishing 1993)
128
6 Fluid Description of Edge Plasma Transport
ð Þ ¼ À
Z
K z, z
0
ð
Þ
∂T e
∂z 0 dx
0 ,
ð6:41Þ
can be considered with different kernels K(z, z
0 ) which were suggested over the years
(e.g. see [33–36]). A comparison of the outcomes of these nonlocal models with the
results of numerical simulations of the decay of a small amplitude, harmonic electron
temperature perturbation is shown in Fig. 6.1. However, in spite of the reasonable
agreement of the results coming from some non-local models based on the integral
expression (6.41) and numerical simulations, in the edge fluid plasma transport
codes, a much simpler approach is usually employed. It is based on the so-called
“flux limiting” expression for the heat flux suggested in [38]. This expression simply
constrains the Spitzer-Härm heat flux, q SH (z), by the fraction, fr FS % 0.2 À 0.4, of the
free streaming electron heat flux, q FS (z) ¼ n(z)T e (z){2T e (z)/πm}
1/2 , so that
q x
ð Þ ¼
q SH z
ð Þfr FS q FS z
ð Þ
jq SH z
ð Þj þfr FS q FS z
ð Þ
:
ð6:42Þ
As we see, such expression describes the reduction of q(x) in comparison with
q SH (z). For a monotonic temperature profile T e (z), which is the case for both the
inertial fusion applications and the edge plasmas, this reduction has clear physical
meaning in the high-temperature region, where, at γ > 10
À2 , the tail of the electron
distribution function is depleted because of the runaway of the weakly collisional
electrons into the low-temperature region. However, in the low-temperature region,
these suprathermal electrons result in the increase of q(x) beyond q SH (z), but such an
effect is not captured by Eq. (6.42). Nonetheless, expression (6.42) and similar ones
for ion heat conduction and viscosity along the magnetic field are often used in edge
plasma transport codes.
10 –3
10 –4
10 –3
10 –2
10 –1
10 0
10 –2
10 –1
L
K
A
kλ T
κ
eff /
κ
SH
10 0
10 1
Fig. 6.1 Ratio of the
effective, κ eff , to the SpitzerHärm, κ SH , electron heat
conductivities as a function
of kλ T , where k is the
wavenumber of the initial
electron temperature
perturbation. The filled
circles are the results of
numerical simulations [37],
the curves A, K, and L are
from the references [33–
35]. (Reproduced with
permission from [35],
© AIP Publishing 1993)
128
6 Fluid Description of Edge Plasma Transport