Magnetic Field-Line Curvature

As was mentioned in Section 1.14, the bootstrap current has a destabilizing effect on wide magnetic island chains [2,5,16]. It turns out that the mean curvature of magnetic field-lines in the inner region has a stabilizing effect on such chains that is similar in magnitude to the destabilizing effect of the bootstrap current [11,15]. Hence, it is not consistent to include the bootstrap current in our analysis without also including curvature effects. The appropriate curvature terms are derived in Reference [13]. We shall simply incorporate them into our model, which generalizes to give:

$$\frac{\partial\psi}{\partial\hat{t}} = \left[\phi,\psi\right] +\hat{d}_i\,\frac{\tau}{1+\tau}\left[\de... ...lpha_{bs}\,\frac{\partial\delta p}{\partial\hat{x}}\right) + \hat{E}_\parallel,$$ (11.52)

$$\frac{\partial\delta p}{\partial \hat{t}} = \left[\phi,\delta p\right] -c_\beta^{\,2}\left[\hat{V}_{\parall... ...el}}{\hat{\eta}_{\parallel}}\,\frac{\partial\psi}{\partial\hat{t}}, \psi\right]$$

$$\phantom{=} -c_\beta^{\,2}\,\left[\phi+\hat{d}_i\,\frac{\tau}{1+\... ...ht] + \frac{2}{3}\,(1-c_\beta^2)\,\hat{\chi}_\perp\,\hat{\nabla}^{\,2}\delta p,$$ (11.53)

$$\frac{\partial U}{\partial \hat{t}} = \left[\phi, U\right]+\frac{\hat{d}_i}{2\,(1+\tau)} \left(\hat{\... ...right]+\left[U,\delta p\right] +\left[\hat{\nabla}^2\delta p,\phi\right]\right)$$

$$\phantom{=} +\left[J,\psi\right] + [\delta p, H]+ \hat{\mit\Xi}_\... ...au}\left(1-\alpha_\theta\,\frac{\eta_i}{1+\eta_i}\right)\delta p\right] \right)$$

$$\phantom{=}+ \hat{\mit\Xi}_\perp\,\hat{\nabla}^4\left( \phi-\frac{\hat{d}_i}{1+\tau}\,\delta p\right),$$ (11.54)

$$\frac{\partial\hat{V}_\parallel}{\partial\hat{t}} =\left[\phi,\hat{V}_\parallel\right] -\left[\delta p,\psi\right]+ \frac{c_\beta^{\,2}\,\hat{d}_i}{1+\tau}\,[\hat{V}_\parallel, 2\,H-\delta p]$$

$$\phantom{=}-\hat{\mit\Xi}_\theta\,\frac{\epsilon_s}{q_s}\left(\fr... ...)\delta p\right] \right) +\hat{\mit\Xi}_\perp\,\hat{\nabla}^2\hat{V}_\parallel.$$ (11.55)

Here,

$$H = \frac{2\,\hat{x}}{\hat{L}_c},$$ (11.56)

and $\hat{L}_c=L_c/l$, where

$$L_c= \frac{R_0^{\,2}}{r_s\,(1-1/q_s^{\,2})}$$ (11.57)

is the magnetic curvature length (i.e., the mean radius of curvature of magnetic field-lines at the rational surface) [12,17]. Note that the previous expression for $L_c$ is only valid in a large-aspect ratio, low-$\beta$ plasma with circular magnetic flux-surfaces. A more general expression is given in Section A.8.