Curvature of Radial Electric Field Aggravates Edge Magnetohydrodynamics Mode in Toroidally Confined Plasmas

We show that the radial electric field ($E_r$) plays a dual role in edge magnetohydrodynamics (MHD) activity. While $E_r$ shear (first spatial derivative of $E_r$) dephases radial velocity and displacement, and so is stabilizing, a new finding here is that $E_r$ curvature (second spatial derivative of $E_r$) tends to synchronize the radial velocity and displacement, and so destabilizes MHD. As a highlighted result, we analytically demonstrate that $E_r$ curvature can destabilize an otherwise stable kink mode, and so form a joint vortex-kink mode. The synergetic effects of $E_r$ shear and $E_r$ curvature in edge MHD extend the familiar $E\times B$ shearing paradigm. This theory thus explains the experimental findings that a deeper $E\times B$ well may aggravate edge MHD, and so trigger the formation of the edge harmonic oscillation. A simple criterion linking $E_r$ structure and the edge MHD activity is derived.

Figure 1

(a) Current density ($j_0$) and vorticity (${\omega }_0$) distributions used in the theoretical model. (b) The eigenfunction’s amplitude of the joint mode, where $c_0=1.0328$, $c_1=0.6791$, $c_2=0.599$, $c_3=1.0$, $a_1/a_2=0.9$, and $m=5$. (c) The growth rate $\gamma =\Im (\mathrm{\Omega })$ of vortex-kink mode with $a_1/a_2=0.9$, and (d) the cross phase of the two interfacial vortex waves at $r=a_1$ and $r=a_2$. Here the growth rate is normalized to the Alfvén frequency at $a_1$ (i.e., ${\omega }_A=B_{0,\theta }/\sqrt{4\pi {\rho }_0{a}_1^2}$), and the mean vorticity is set as ${\omega }_0=1.2{\omega }_A$. The two vortex waves are decoupled when $m>7$, i.e., $k_{\theta }|a_2-a_1|>0.6223$.

Figure 2

A schematic illustration of the relationship between vortex-wave interaction and joint mode instability. The red horizontal arrows at the peaks and troughs of the black wave field represent the displacement induced by the red wave field. The black arrows at the red wave field are induced by the black wave field. The two vortex waves are phase locked, and depending on the phase difference $({\vartheta }_1-{\vartheta }_2)$, the joint mode could be (a) unstable, (b) neutrally stable, (c) stable.

Figure 3

(a) Mean current density (dashed line) and $E_r$ profiles (solid lines) in SI units. (b) Impact of $E_r$ profile on a unstable kink mode. Both $\overline{|E_r^{\prime }|}$ and $\overline{|E_r^{\prime \prime }|}$ are in dimensionless form. Red (blue) marker means the growth rate of the joint mode is larger (smaller) than that of the pure kink. (c) Growth rate of the vortex-kink mode. (d) The spatially averaged cosine of cross phase between ${\widehat{v}}_r$ and ${\widehat{\xi }}_r$. (e) The spatially averaged amplitude ratio $|{\widehat{v}}_r|/|{\widehat{\xi }}_r|$. The toroidal mode number is $n=3$.

Figure 4

(a) The equilibrium pressure (black line) and current density (red line) profiles. (b) Three typical $E_r$ profiles. (c) The linear growth rate spectrum for different $\mathcal{R}$. (d) The growth rate of the $n=7$ mode. (e) The spatially averaged cosine of cross phase (${\vartheta }_{v_r}-{\vartheta }_{{\xi }_r}$). (f) The spatially averaged amplitude ratio $|{\widehat{v}}_r|/|{\widehat{\xi }}_r|$.

Figure 5

(a) Marginal boundaries of the joint mode and the conventional PB mode. The red line corresponds to increase of $\mathcal{R}$ and the blue line to decrease of $\mathcal{R}$ with $D_0=10\mathrm{krad}/\mathrm{s}$ and $D_s=35$. (b) A schematic sketch of the $E_r$ well. $[r_L,r_R]$ is chosen as the edge region.