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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 $ W_{0}/r_{s} \approx 5.1(\chi_{\perp}/\chi_{\parallel})^{1/4}/
\sqrt{\epsilon s n}$, where $\epsilon$ is the localized inverse aspect ratio and $s$ is the localized magnetic shear [10]. For islands that are wider than $ W_0 $, 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

S(u) \equiv
\frac{p'_{\rm perturbed}}{p'_{\rm unperturbed}}
...frac{0.25}{u^{2}}}+\cdots & \vert u\vert>1
\end{array} \right.
\end{displaymath} (36)

The diffusive transport is modified to produce this local flattening of the profiles. If the transport equations have the following basic form
\chi\frac{\partial p}{\partial x} = F
\end{displaymath} (37)

where $ p $ is the plasma pressure, $\chi$ is the diffusivity and $F$ is the diffusive heat flux, it follows that the diffusivity is inversely proportional to the radial pressure gradient and
\frac{\chi_{\rm perturbed}}{\chi_{\rm unperturbed}}\approx 1/S(u).
\end{displaymath} (38)

Consider a finite difference scheme in which the pressure $p_{j+1/2}$ is computed on zone centers $x_{j+1/2}$, while the thermal diffusivity $\chi_{j}$ and heat flux $F_{j}$ are computed on zone boundaries $x_{j}$. Then, a simple finite difference approximation for Eq. (37) is

\begin{displaymath}\chi_{j}\frac{ p_{j+1/2} - p_{j-1/2}}{x_{j+1/2}-x_{j-1/2}} = F_{j} \end{displaymath}

at the $jth$ zone boundary. To determine the finite difference form for $\chi$, integrate Eq. (37) from one zone center to the next
p_{j+1/2} - p_{j-1/2} =
\end{displaymath} (39)

Hence, an approximation for the thermal diffusivity at zone boundary $j$ is
\displaystyle{\frac {dx}{\chi}}} }
\end{displaymath} (40)

>From Eq. (38), it is concluded that the diffusivity is enhanced near a magnetic islands with widths $ W_{mn} $ located at mode rational surfaces $ x = x_{mn} $ by the following factor
\displaystyle{\frac{\chi_{j \rm perturbed}}{\chi_{j \rm
\end{displaymath} (41)

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.

next up previous
Next: Bibliography Up: Quasi-linear Tearing Mode Equations Previous: Iterative Procedure for Determining
transp_support 2008-12-08