Effect of Magnetic Islands on the Plasma Profiles

Suppose the pressure profile across the widest part of the island, $m\theta - n\zeta = 0$, is a uniform constant $p_{0}$ within the island and is linear in the radius with slope $p'_{0}=dp_{0}/dx$, just outside the shoulder of the island,

$$p =\left\{\begin{array}{ll} p_{0}+0.5p'_{0}W_{mn}(u+1) & u<-... ...ert<1 \\ p_{0}+0.5p'_{0}W_{mn}(u-1) & u>1 \end{array} \right.$$ (29)

where

$$u \equiv 2(x-x_{mn})/W_{mn}.$$ (30)

Since ${\bf B}\cdot\nabla p=0$, the pressure is a function of $\psi$. At the widest part of the island, Eqs. (27), (28), and (30) can be used to write $u$ as a function of $\psi$

$$u = \sqrt{ \frac{ \psi - \psi^0 - \psi^1_{mn} }{ 2 \psi^1_{mn} } }$$ (31)

Then, Eq. (29) can be used to express the pressure as a function of $\psi$ in the neighborhood of a magnetic island

$$p(\psi) =\left\{\begin{array}{ll} p_{0}+0.5p'_{0}W_{mn}\left... ...1 \right) & {\rm outer\hspace{0.1in} edge} \end{array} \right.$$ (32)

Consider the poloidal (or, equivalently, the toroidal) average of the pressure

$$\langle p \rangle = \frac{1}{2\pi}\int^{2\pi}_{0}p(\psi)d\theta$$

at fixed $x$ to determine the background axisymmetric pressure profile to be used in Eq. (10). Substitution of Eq. (27) in Eq. (32) and integration over the poloidal angle leads to the following approximation to the axisymmetric poloidal averaged pressure profile in the neighborhood of a magnetic island

$$\left\langle \frac{dp^{0}}{dx}\right\rangle \approx p'_{0}\l... ...style{\frac{0.25}{u^{2}}} & \vert u\vert>1 \end{array} \right.$$ (33)

It can be seen from Eq. (33) that the presence of a magnetic island has the effect of flattening the axisymmetric background pressure profile at the corresponding mode rational surface. In particular, $\langle dp/dx \rangle$ and $\langle d^{2}p/dx^{2} \rangle$ are both equal to zero at the mode rational surface, $u=0$.

The same derivation can be applied to compute the flat spot in the current density profile caused by the magnetic island at each mode rational surface. Suppose the current density profile across the widest part of the island is

$$K\equiv\frac{\mu_{0}J^{\zeta}}{B^{\zeta}} \approx\left\{\beg... ... \vert u\vert<1 \\ K_{0}+K_{1}(u-1) & u>1 \end{array} \right.$$ (34)

with $K_{0}={mu_{0}J^{\zeta}}/{B^{\zeta}}$ at the edge of the island, $K_{1}=-{d}\left({\mu_{0}J^{\zeta}}/{B^{\zeta}}\right)/du$, at the edge of the island. The coefficient, $\gamma$, is a current peaking factor (if $\gamma > 0$, assuming $K_{1}<0$) or current suppression factor (if $\gamma < 0$) within the island, and $K_{2} \equiv \mu_{0}J^{\zeta}_{\rm bootstrap} / B^{\zeta}$ is the bootstrap current density just outside the edge of the island. Finally, $u \equiv 2(x-x_{mn})/W_{mn}$. Since the pressure profile is assumed to be flat within the magnetic island, and the bootstrap current density is roughly proportional to the pressure gradient [11], there is no bootstrap current density within the island. Hence, the abrupt transition from the presence of bootstrap current outside of the island to the complete absence of the bootstrap current within the island produces a square wave contribution to the current suppression within the island. Close to the mode rational surface, where $\vert u\vert<<1$, the axisymmetric average of the radial derivative of the toroidal current density can be approximated with the following expression

$$\left\langle\frac{d}{dx} \left(\frac{\mu_{0}J^{0\zeta}}{B^{0... ...sqrt{u^{2}+0.5}}\left[ 1 + \frac{3}{32(u^{2}+0.5)^{2}} \right]$$ (35)

$$\hspace{2cm} + \frac{2\gamma K_{1}u\cos^{-1}(2u^{2}-1)}{W_{mn}\pi} - \frac{2.52K_{2}u}{W_{mn}}.$$

Thus, near the mode rational surface, the apparently singular terms on the right side of equations (8) and (10) reduce to the following non-singular forms

$$\frac{nJ^{0\zeta}-mJ^{0\theta}}{(nq-m)B^{0\theta}} \rightarrow\frac{J^{0\zeta}}{B^{0\zeta}}$$

$$\frac{p^{1}_{mn}}{(nq-m)B^{0\theta}} \rightarrow\frac{-2.52iB^{1x}_{mn}p_{0}'}{[nB^{0\theta}W_{mn}(dq/dx)]^{2}}$$

$$\displaystyle{ \frac{\displaystyle{\frac{d}{dx}} \left(\mu_{0... ...K_{1}+4\gamma K_{1}-5.04K_{2}}{W^{2}_{mn}B^{0\theta}n(dq/dx)}}$$

Consequently the Eqs. (8) and (10) are not singular, and they can be integrated from the magnetic axis to the edge of the plasma with a numerical ordinary differential equation solver.

It should be noted that the interaction between different harmonics can produce stochastic regions in the magnetic field near the X-points of magnetic islands. The effect of the stochastic regions will be to flatten the profiles more completely near the mode rational surface than is estimated here. In addition, multiple Fourier harmonics with the same ratio of $m / n$ will produce magnetic islands with a cross-sectional shape that is different from the single harmonic given in Eq. (27). The distortion in the island shape will also modify the estimates given above for the profile flattening.