which justifies our assumption b k z ∼ λ
À1
D . For the case of a monoenergetic ion
distribution function along the magnetic field, the expression (4.11) is reduced to
the Chodura inequality (4.1) [2].
Once the ion flux onto the target is defined by f
À1
ð
Þ
i
v
!
, the potential drop
between the plasma away from the target and the target can be found by equilibrating
the electric currents in the plasma and through the sheath. For the case of ambipolar
plasma flow, assuming that the ion flow to the target along the magnetic field is
%n sh C s , ignoring other effects that could alter the ion flux to the target (e.g. drifts),
and adopting the Maxwellian electron distribution function, we find e φ sh % Λ sh T e ,
where Λ sh ~ ℓn(M/m) ~ 3 Ä 4 [3].
In [5] the constrains (4.5) and (4.11) were criticized, in particular, from the point
of view of “unphysical” v
À2
z
v
À2
k
moment of the ion distribution function and the
omission of the impact of the collision operator for small v z (v k ) in the course of the
derivation of Eqs. (4.5) and (4.11) from the corresponding Vlasov equations. A
replacement of Eq. (4.5) based on positive powers of the velocity moments of the
distribution function, which do not diverge at v ¼ 0, was suggested (see [5] for
details). But in [11] it was argued that the inequality (4.5) holds also in the presence
of collisions provided that Eq. (4.5) is applied at the entrance to the sheath and the
sheath is considered in the limit of λ D ! 0. Although this discussion is important
from an academic point of view, for the practical application of the Bohm constraint
as the boundary condition for edge plasma flow to the target it becomes rather
meaningless. This is because in the most relevant, from the point of view of the
reduction of the power loading of the target, high recycling regimes of divertor
operation, strong plasma-neutral interactions become ubiquitous. As a result, in this
case, all the models suggest virtually the same constraint on the plasma flow velocity
hv z i % C s (for the case of normal incidence of the magnetic field lines onto the target)
with some correction for the finite ion temperature. The difference between various
models is within the error bar imposed by the boundary conditions used for the ion
and electron heat fluxes to the target, the calculation of which should take into
account spatial variation of both the electrostatic potential and the electron and ion
distribution functions at the entrance to the sheath (e.g. see [12]).
In our considerations, we assume that the electron distribution function along the
magnetic field lines for v k > 0 (in the direction towards the target) is Maxwellian,
whereas for v k < 0 the tail of the distribution function is cut at velocities beyond
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
eφ sh =m
p
. For the case of the ambipolar plasma flow, φ sh can be found by
equilibrating the electron and ion fluxes. However, such approximation is only
applicable for the case where electron collisionality in the SOL plasma is relatively
high and the tail can be replenished by Coulomb collisions before the electrons reach
the target. In the opposite case, depletion of the tail of the electron distribution
function can drive the whistler waves, which are capable of scattering the electrons
and effectively populating the “gap” in the electron distribution function (see [13]
and the references therein). However, in practice, in the SOL plasmas, the electrons
with the energies ~ e φ sh % Λ sh T e are weakly collisional, which makes it difficult to
4 Sheath Physics
79
À1
D . For the case of a monoenergetic ion
distribution function along the magnetic field, the expression (4.11) is reduced to
the Chodura inequality (4.1) [2].
Once the ion flux onto the target is defined by f
À1
ð
Þ
i
v
!
, the potential drop
between the plasma away from the target and the target can be found by equilibrating
the electric currents in the plasma and through the sheath. For the case of ambipolar
plasma flow, assuming that the ion flow to the target along the magnetic field is
%n sh C s , ignoring other effects that could alter the ion flux to the target (e.g. drifts),
and adopting the Maxwellian electron distribution function, we find e φ sh % Λ sh T e ,
where Λ sh ~ ℓn(M/m) ~ 3 Ä 4 [3].
In [5] the constrains (4.5) and (4.11) were criticized, in particular, from the point
of view of “unphysical” v
À2
z
v
À2
k
moment of the ion distribution function and the
omission of the impact of the collision operator for small v z (v k ) in the course of the
derivation of Eqs. (4.5) and (4.11) from the corresponding Vlasov equations. A
replacement of Eq. (4.5) based on positive powers of the velocity moments of the
distribution function, which do not diverge at v ¼ 0, was suggested (see [5] for
details). But in [11] it was argued that the inequality (4.5) holds also in the presence
of collisions provided that Eq. (4.5) is applied at the entrance to the sheath and the
sheath is considered in the limit of λ D ! 0. Although this discussion is important
from an academic point of view, for the practical application of the Bohm constraint
as the boundary condition for edge plasma flow to the target it becomes rather
meaningless. This is because in the most relevant, from the point of view of the
reduction of the power loading of the target, high recycling regimes of divertor
operation, strong plasma-neutral interactions become ubiquitous. As a result, in this
case, all the models suggest virtually the same constraint on the plasma flow velocity
hv z i % C s (for the case of normal incidence of the magnetic field lines onto the target)
with some correction for the finite ion temperature. The difference between various
models is within the error bar imposed by the boundary conditions used for the ion
and electron heat fluxes to the target, the calculation of which should take into
account spatial variation of both the electrostatic potential and the electron and ion
distribution functions at the entrance to the sheath (e.g. see [12]).
In our considerations, we assume that the electron distribution function along the
magnetic field lines for v k > 0 (in the direction towards the target) is Maxwellian,
whereas for v k < 0 the tail of the distribution function is cut at velocities beyond
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
eφ sh =m
p
. For the case of the ambipolar plasma flow, φ sh can be found by
equilibrating the electron and ion fluxes. However, such approximation is only
applicable for the case where electron collisionality in the SOL plasma is relatively
high and the tail can be replenished by Coulomb collisions before the electrons reach
the target. In the opposite case, depletion of the tail of the electron distribution
function can drive the whistler waves, which are capable of scattering the electrons
and effectively populating the “gap” in the electron distribution function (see [13]
and the references therein). However, in practice, in the SOL plasmas, the electrons
with the energies ~ e φ sh % Λ sh T e are weakly collisional, which makes it difficult to
4 Sheath Physics
79