Two-dimensional mode structures on poloidal plane

Consider a harmonic with poloidal mode number $m$ and toroidal mode number $n$,

$$\delta \phi (r, \theta, \varphi, t) = A (r) \sin [m \theta + n \varphi + \omega t + \alpha (r)] .$$ (326)

Choose a radial profile of the amplitude

$$A (r) = \exp \left( - \frac{(r - r_s)^2}{\Delta^2} \right) .$$ (327)

Figure 35 plots the two-dimensional mode structures on the poloidal plane for two profiles of the radial phase variation given by

$$\alpha = 0,$$ (328)

and

$$\alpha = (r - r_s) \frac{2 \pi}{8},$$ (329)

respectively. Note that, compared with the case of $\alpha = 0$ (no radial phase variation), the radial phase variation given by Eq. (329) influence the mode structure on the poloidal plane, generating the so-called mode twist or shear[11], as shown by the left figure of Fig. 35.

Figure: Two dimensional structure (on $\varphi = 0$ plane) given by Eqs. (326) and (327) with $m = 4$, $r_s = 0.5$, $\Delta = \sqrt {0.02}$, $t = 0$. Left figure is for $\alpha = 0$ and right figure is for $\alpha$ given by Eq. (334). The mode propagates (rotates) in the clockwise direction on the poloidal plane (the zero point of $\theta$ coordinate is at the low-field-side of the midplane and the positive direction is in the anti-clockwise direction). A GIF animation of the time evolution of the mode can be found at http://theory.ipp.ac.cn/~yj/figures/mode_rotation3.gif

Consider a mode composed of two poloidal harmonics

$$\delta \phi (r, \theta, \varphi, t) = A_1 (r) \sin (m_1 \theta + ... ...\omega t + \alpha) + A_2 (r) \sin (m_2 \theta + n \varphi + \omega t + \alpha),$$ (330)

where $m_1$ and $m_2$ are the poloidal mode number of the two poloidal harmonics. Consider the case $m_2 = m_1 + 1 = 5$. Then at the high field side of the midplane $(\theta = \pi)$ of $\varphi = 0$ poloidal plane, equation (330) is written

$$\delta \phi (r, \theta, \varphi, t) = A_1 \sin (\omega t + \alpha) + A_1 \sin (\pi + \omega t + \alpha) = (A_1 - A_2) \sin (\omega t + \alpha) .$$ (331)

At the low field side of the midplane $(\theta = 0)$ of $\varphi = 0$ poloidal plane, equation (330) is written

$$\delta \phi (r, \theta, \varphi, t) = A_1 \sin (\omega t + \alpha) + A_1 \sin (\omega t + \alpha) = (A_1 + A_2) \sin (\omega t + \alpha) .$$ (332)

Equations (331) and (332) indicates the amplitude of the mode at the low field side is larger than that at the strong field side, i.e., the mode exhibits a ballooning structure.

For a radial profile given by

$$A_1 (r) = 2 A_2 (r) = \exp \left( - \frac{(r - r_s)^2}{\Delta^2} \right) .$$ (333)

and an initial phase $\alpha = 0$, Figure 36 plots the two-dimensional structure of the mode on the poloidal plane. The inital phase $\alpha$ can have radial varation and this has effects on the 2D structure of the mode. For instance, $\alpha$ is chosen to be of the form

$$\alpha = \alpha (r) = (r - r_s) \frac{2 \pi}{8} .$$ (334)

The resulting 2D mode structure is given in the right figure of Fig. 36, where the so-called mode shear can be seen[11].

Figure 36: Two dimensional structure (on $\varphi = 0$ plane) given by Eqs. (330) and (333) with $m_1 = m_2 - 1 = 4$, $r_s = 0.5$, $\Delta = \sqrt {0.02}$, $t = 0$. Left figure is for $\alpha = 0$ and right figure is for $\alpha$ given by Eq. (334). Note that the mode amplitude at the low field side is larger than that at the high field side. The mode propagates (rotates) in the clockwise direction on the poloidal plane. A GIF animation of the time evolution of the mode can be found at http://theory.ipp.ac.cn/~yj/figures/ballooning_animation.gif