hold. At the values of γ > 10
À2 , which is the typical situation in the SOL plasma, the
heat is mostly transported by supra-thermal particles having a longer mean-free path
and the parallel energy transport becomes non-local – the flux is determined by the
whole temperature profile, not by the local values of the plasma parameters and their
gradients. Physically, this means that the hot tails of the distribution functions are
depleted in the hot SOL and enhanced in the colder divertor region, therefore
reducing the heat flux upstream (see Sect. 6.4 for details). The empirical way of
taking this reduction of the conducted energy flux into account is the introduction of
the so-called flux-limit factors that effectively limit the conducted flux to some
fraction of the free-streaming one, see Eq. (5.42). As noted in Chap. 6, this approach
describes reasonably well the reduction of the heat flux upstream because of the
depletion of the hot tails of the distribution functions there but does not take into
account the appearance of the supra-thermal particles in the cold plasma. This may
be not so important since, in the divertor regions, where intensive particle recycling
occurs, the heat fluxes are dominated by convective transport [12]. In addition, low
electron temperature and high plasma density in the recycling region cause strong
dissipation of the electron tail so that it does not affect the ionization rates strongly
[13]. The heat flux correction with the flux limiting factors is implemented in
virtually all major 2D modeling codes together with similar treatment of the parallel
viscosity coefficient.
In principle, non-local transport requires a full kinetic description that allows a
significant deviation of the velocity distribution functions of the charged particles
from the Maxwellian ones. However, full kinetic treatment of a problem including
many different interactions occurring on different time scales with adequate spatial
resolution does not look realistic at the present state of computer development.
Given the complexity of the full kinetic treatment, the attempts have been made to
combine a simplified kinetic model with a fluid model of the edge plasma, treating
the Coulomb collisions in kinetics and leaving the slower processes for the fluid
description. In [12], a simplified kinetic model in the BGK approximation [14] was
combined with a 2D fluid model. The solution was obtained in iterations where the
plasma parameters from the fluid model were taken as the background for the kinetic
calculations for electrons and ions, then the effective heat conductivities evaluated
from the kinetic fluxes and fluid gradients were applied in the fluid code – and so on,
until convergence. In a recent study [13], a Fokker-Planck kinetic model for electrons was coupled to a 1D fluid model in a similar way. However, these attempts are
rather exotic and are not being used in the massive calculations with the 2D models.
8.1.3 Cross-Field Transport
Presently, there is no concise, credible theory that would describe anomalous plasma
transport across the magnetic field (see Chap. 7 for the present state of the theory in
this area). In this situation, the most common approach is using the diffusive ansatz
that meets at least the first law of thermodynamics. The “classical” cross-field
diffusion of particles, parallel momentum and energy, which is related to the binary
8.1 Transport Modeling of the Plasma
205
À2 , which is the typical situation in the SOL plasma, the
heat is mostly transported by supra-thermal particles having a longer mean-free path
and the parallel energy transport becomes non-local – the flux is determined by the
whole temperature profile, not by the local values of the plasma parameters and their
gradients. Physically, this means that the hot tails of the distribution functions are
depleted in the hot SOL and enhanced in the colder divertor region, therefore
reducing the heat flux upstream (see Sect. 6.4 for details). The empirical way of
taking this reduction of the conducted energy flux into account is the introduction of
the so-called flux-limit factors that effectively limit the conducted flux to some
fraction of the free-streaming one, see Eq. (5.42). As noted in Chap. 6, this approach
describes reasonably well the reduction of the heat flux upstream because of the
depletion of the hot tails of the distribution functions there but does not take into
account the appearance of the supra-thermal particles in the cold plasma. This may
be not so important since, in the divertor regions, where intensive particle recycling
occurs, the heat fluxes are dominated by convective transport [12]. In addition, low
electron temperature and high plasma density in the recycling region cause strong
dissipation of the electron tail so that it does not affect the ionization rates strongly
[13]. The heat flux correction with the flux limiting factors is implemented in
virtually all major 2D modeling codes together with similar treatment of the parallel
viscosity coefficient.
In principle, non-local transport requires a full kinetic description that allows a
significant deviation of the velocity distribution functions of the charged particles
from the Maxwellian ones. However, full kinetic treatment of a problem including
many different interactions occurring on different time scales with adequate spatial
resolution does not look realistic at the present state of computer development.
Given the complexity of the full kinetic treatment, the attempts have been made to
combine a simplified kinetic model with a fluid model of the edge plasma, treating
the Coulomb collisions in kinetics and leaving the slower processes for the fluid
description. In [12], a simplified kinetic model in the BGK approximation [14] was
combined with a 2D fluid model. The solution was obtained in iterations where the
plasma parameters from the fluid model were taken as the background for the kinetic
calculations for electrons and ions, then the effective heat conductivities evaluated
from the kinetic fluxes and fluid gradients were applied in the fluid code – and so on,
until convergence. In a recent study [13], a Fokker-Planck kinetic model for electrons was coupled to a 1D fluid model in a similar way. However, these attempts are
rather exotic and are not being used in the massive calculations with the 2D models.
8.1.3 Cross-Field Transport
Presently, there is no concise, credible theory that would describe anomalous plasma
transport across the magnetic field (see Chap. 7 for the present state of the theory in
this area). In this situation, the most common approach is using the diffusive ansatz
that meets at least the first law of thermodynamics. The “classical” cross-field
diffusion of particles, parallel momentum and energy, which is related to the binary
8.1 Transport Modeling of the Plasma
205