V x
de x
dt
¼ À
c
B 0
∇e φ Â e
!
z :
ð7:86Þ
As a result, from Eqs. (7.85 and 7.86) we find
de x
dt
¼
k y
k z
cE
B 0
dℓn σ
ð Þ
dx
e x,
ð7:87Þ
and for the proper sign of the k y /k z ratio (which defines the inclination angle of the
slab with respect to the direction of the magnetic field), we have the so-called
current-convective instability with a characteristic growth rate
γ ¼
cE
B 0
dℓn σ
ð Þ
dx
k y
k z
:
ð7:88Þ
We should recall that the electric conductivity of plasma is σ / T
3=2
e , therefore the
perturbation of conductivity, which drives the current-convective instability, is
associated with inhomogeneity of the electron temperature along the magnetic
field lines. However, electron temperature perturbations along the magnetic field
can be washed away by very fast parallel electron heat conduction, κ e / T
5=2
e , and no
instability will be possible. However, recently it was shown [43] that in asymmetric
“detached divertor” regimes, where the plasma temperature in the inner divertor falls
to ~eV range but the outer divertor is still relatively hot, T hot ~ 10 eV), currentconvective instability can be very “active” and important in plasma transport in the
inner divertor. There are two reasons for this: i) the electron thermal conductivity
effects are suppressed in the inner divertor and ii) the asymmetry of the electron
temperatures in the inner and outer divertors results in onset of a large electrostatic
potential drop, U ~ few  T hot , through the inner divertor leg, which boosts the
growth rate of the current-convective instability there, see Eq. (7.88).
7.2.9 Impact of Magnetic Shear
So far, considering the slab approximation of a magnetic confinement device, we
assumed that all magnetic field lines are parallel to each other. However, in practice,
this is not the case and the direction of the vector b
! ¼ B
!
=B turns around the minor
radius r
! with increasing r, similar to that shown for the slab geometry in Fig. 7.10a
where the x-coordinate plays the role of the minor radius. Therefore, if we specify
the poloidal (y-direction) component of the wavenumber (k y ), the effective parallel
component of the wave vector will depend on the minor radius. One can see this
from Fig. 7.10b, where the blue and orange stripes along the toroidal (z) coordinate
correspond to different phases of the perturbed plasma parameters with given k y and
the vector b
!
is shown for different radial (x-coordinate) locations.
168
7 Anomalous Cross-Field Transport in Edge Plasma
de x
dt
¼ À
c
B 0
∇e φ Â e
!
z :
ð7:86Þ
As a result, from Eqs. (7.85 and 7.86) we find
de x
dt
¼
k y
k z
cE
B 0
dℓn σ
ð Þ
dx
e x,
ð7:87Þ
and for the proper sign of the k y /k z ratio (which defines the inclination angle of the
slab with respect to the direction of the magnetic field), we have the so-called
current-convective instability with a characteristic growth rate
γ ¼
cE
B 0
dℓn σ
ð Þ
dx
k y
k z
:
ð7:88Þ
We should recall that the electric conductivity of plasma is σ / T
3=2
e , therefore the
perturbation of conductivity, which drives the current-convective instability, is
associated with inhomogeneity of the electron temperature along the magnetic
field lines. However, electron temperature perturbations along the magnetic field
can be washed away by very fast parallel electron heat conduction, κ e / T
5=2
e , and no
instability will be possible. However, recently it was shown [43] that in asymmetric
“detached divertor” regimes, where the plasma temperature in the inner divertor falls
to ~eV range but the outer divertor is still relatively hot, T hot ~ 10 eV), currentconvective instability can be very “active” and important in plasma transport in the
inner divertor. There are two reasons for this: i) the electron thermal conductivity
effects are suppressed in the inner divertor and ii) the asymmetry of the electron
temperatures in the inner and outer divertors results in onset of a large electrostatic
potential drop, U ~ few  T hot , through the inner divertor leg, which boosts the
growth rate of the current-convective instability there, see Eq. (7.88).
7.2.9 Impact of Magnetic Shear
So far, considering the slab approximation of a magnetic confinement device, we
assumed that all magnetic field lines are parallel to each other. However, in practice,
this is not the case and the direction of the vector b
! ¼ B
!
=B turns around the minor
radius r
! with increasing r, similar to that shown for the slab geometry in Fig. 7.10a
where the x-coordinate plays the role of the minor radius. Therefore, if we specify
the poloidal (y-direction) component of the wavenumber (k y ), the effective parallel
component of the wave vector will depend on the minor radius. One can see this
from Fig. 7.10b, where the blue and orange stripes along the toroidal (z) coordinate
correspond to different phases of the perturbed plasma parameters with given k y and
the vector b
!
is shown for different radial (x-coordinate) locations.
168
7 Anomalous Cross-Field Transport in Edge Plasma