However, the simple model of the magnetic field (7.83) can more or less
adequately describe the effect of the magnetic shear for the case where the shear is
not varying much on a magnetic flux surface. This is not the case for the magnetic
flux surfaces close to the separatrix which contains at least one X-point where the
magnitude of the poloidal magnetic field, B p , is zero. In the vicinity of the X-point,
the magnitude of the poloidal magnetic field is proportional to the distance from the
X-point, ℓ X (for the case of the first-order X-point), so B p (ℓ X ) / ℓ X . We notice that
the similarity of such dependence of B p (ℓ X ) with the model (7.89) is illusory. The
z-direction in Eq. (7.89) is the direction of the total magnetic field at some effective
“magnetic flux surface” corresponding to x ¼ 0, whereas for the case of the magnetic
field in the vicinity of the X-point, we are dealing with the “exact” poloidal and
toroidal magnetic fields.
Such a structure of the magnetic field near the X-point results in a very strong
magnetic shear localized there. One can see this by considering how the poloidal
projection of a magnetic flux tube cross-section evolves along the tube, Fig. 7.12.
The magnetic flux in the tube is mainly determined by the large toroidal magnetic
field that does not vary much, so we can consider the area of the poloidal crosssection of the tube to be almost constant if the pitch angle of the magnetic field line
B p /B ( 1, which is always the case near the X-point. Assume that the magnetic flux
tube has a circular poloidal cross-section closer to the mid-plane (see the contour
(1) in Fig. 7.12). The shape of this cross-section evolves along the tube, becoming a
thin oval close to the X-point (contours 2 and 3) since the magnetic surfaces that
determine the radial extent of the cross-section diverge there. The variation of the
poloidal width of the tube cross-section, δℓ p , can be estimated taking into account
that B p (ℓ X ) / ℓ X [50]:
δℓ p
δℓ p 0
ð Þ
%
ℓ X
ℓ X 0
ð Þ
:
ð7:90Þ
Fig. 7.12 Poloidal
projection of magnetic flux
tube, having circular crosssection at the “mid-plane”,
in different toroidal
locations
170
7 Anomalous Cross-Field Transport in Edge Plasma
adequately describe the effect of the magnetic shear for the case where the shear is
not varying much on a magnetic flux surface. This is not the case for the magnetic
flux surfaces close to the separatrix which contains at least one X-point where the
magnitude of the poloidal magnetic field, B p , is zero. In the vicinity of the X-point,
the magnitude of the poloidal magnetic field is proportional to the distance from the
X-point, ℓ X (for the case of the first-order X-point), so B p (ℓ X ) / ℓ X . We notice that
the similarity of such dependence of B p (ℓ X ) with the model (7.89) is illusory. The
z-direction in Eq. (7.89) is the direction of the total magnetic field at some effective
“magnetic flux surface” corresponding to x ¼ 0, whereas for the case of the magnetic
field in the vicinity of the X-point, we are dealing with the “exact” poloidal and
toroidal magnetic fields.
Such a structure of the magnetic field near the X-point results in a very strong
magnetic shear localized there. One can see this by considering how the poloidal
projection of a magnetic flux tube cross-section evolves along the tube, Fig. 7.12.
The magnetic flux in the tube is mainly determined by the large toroidal magnetic
field that does not vary much, so we can consider the area of the poloidal crosssection of the tube to be almost constant if the pitch angle of the magnetic field line
B p /B ( 1, which is always the case near the X-point. Assume that the magnetic flux
tube has a circular poloidal cross-section closer to the mid-plane (see the contour
(1) in Fig. 7.12). The shape of this cross-section evolves along the tube, becoming a
thin oval close to the X-point (contours 2 and 3) since the magnetic surfaces that
determine the radial extent of the cross-section diverge there. The variation of the
poloidal width of the tube cross-section, δℓ p , can be estimated taking into account
that B p (ℓ X ) / ℓ X [50]:
δℓ p
δℓ p 0
ð Þ
%
ℓ X
ℓ X 0
ð Þ
:
ð7:90Þ
Fig. 7.12 Poloidal
projection of magnetic flux
tube, having circular crosssection at the “mid-plane”,
in different toroidal
locations
170
7 Anomalous Cross-Field Transport in Edge Plasma