Transport Enhanced by Saturated Magnetic Islands

Magnetic islands enhance transport by locally short-circuiting the
plasma. Since parallel diffusivity is much larger than
perpendicular diffusivity, the temperature and density profiles
are flattened within each island [10].
This assumption breaks down for small magnetic islands where the
pressure flattening is incomplete. The critical island width, below
which pressure flattening is incomplete, is given as
,
where is the localized inverse aspect ratio and is
the localized magnetic shear [10].
For islands that are wider than , however,
the perturbation on the background pressure and temperature profiles
near the magnetic island is given by Eq. (33),
which can be written in the form

where is the plasma pressure, is the diffusivity and is the diffusive heat flux, it follows that the diffusivity is inversely proportional to the radial pressure gradient and

Consider a finite difference scheme in which the pressure
is computed on zone centers , while the
thermal diffusivity and heat flux are computed
on zone boundaries . Then, a simple finite difference
approximation for Eq. (37) is

at the zone boundary. To determine the finite difference form for , integrate Eq. (37) from one zone center to the next

Hence, an approximation for the thermal diffusivity at zone boundary is

>From Eq. (38), it is concluded that the diffusivity is enhanced near a magnetic islands with widths located at mode rational surfaces by the following factor

Eq. (41) provides a reasonable approximation to the enhancement of the diffusivities caused by each magnetic island even if the width of the magnetic island is much smaller than the grid spacing or even if part of an island lies in one grid zone while the other parts of the same island lie in one or more adjacent zones. Even though the underlying physics that results in this flattening of the profiles makes use of the fact that the diffusivity along magnetic field lines is much larger than the diffusivity across magnetic field lines, the magnitudes of the parallel and perpendicular diffusivities are not needed in this derivation. When the enhanced diffusivity given by Eq. (41) is used in the transport equations in an integrated modeling code, the effect is to flatten the pressure profile to a form that approximates Eq. (36). As the electron temperature profile is flattened by each island, the resulting resistivity profile will be flattened and the magnetic diffusion equation will produce a corresponding flattening in the current density profile. Hence, the use of the enhanced diffusivities in an integrated modeling code produces an approximately self-consistent treatment of the axisymmetric effects of the magnetic islands.