Theoretical study of long-lived mode in tokamaks

Long-lived mode (LLM) in rotating plasmas is studied by using a alternative dispersion relation. This work offers a theoretical interpretation of LLM observed in experiments [Heidbrink, Phys. Rev. Lett. 57, 835 (1986); Chapman, Nucl. Fusion 50, 045007 (2010)]. It is found that in rotating plasmas, LLM can transform from the fishbone mode. The real frequency of LLM, independent of the characteristic frequency of energetic ions (EIs), is proportional to the on-axis rotation frequency of bulk plasma. The key conditions to cause transition between the fishbone mode and LLM are weak magnetic shear (for both monotonic and nonmonotonic safety factor profile) and plasma rotation. The fishbone mode can evolve into LLM only when rotating speed is bigger than a critical value in low shear plasmas. The critical beta of EIs to induce LLM is very low. LLM is nonresonantly excited by EIs, and the driving sources mainly come from plasma inertia.

Figure 1

Temporal evolution of mode frequency of LLM (solid line) and central toroidal rotation frequency (dashed line) in HL-2A discharge 21787 (a) and in MAST shot 21781 (b).

Figure 2

(a) Safety factor profile inside $q=1$ ($q_{\min }$) surface, the dashed (solid) line denotes positive (negative) magnetic shear. (b) The profiles of EI density (blue lines) and rotation (red line). The dashed line is experimental EI density profile given by ONETWO and the solid blue one is plotted by $n_{0h}\left(r\right)\sim \exp \left[-{\sigma }^2{\left(r-r_h\right)}^2\right]$ with $\sigma =2.8, {\stackrel{⌢}{r}}_h=0.16$. The dotted line denotes the position of $q=1$ ($q_{\min }$) flux surface.

Figure 3

Real frequencies (a) and growth rates (b) of LLM (red lines) and the FB mode (blue ones) as functions of ${\beta }_h$ for different beam energy ($E_b$) with ${\mathrm{\Omega }}_0=2\times {10}^4\mathrm{rad}/\mathrm{s}, {\stackrel{⌢}{r}}_{\mathrm{\Omega }}=0, {\sigma }_{\mathrm{\Omega }}=7, {\sigma }_{\rho }=2, \sigma =2.8$ and ${\stackrel{⌢}{r}}_h=0.16$. The magnetic shear $\widehat{s}=0.072$ for LLM and $\widehat{s}=0.81$ for FB. The circles are theory results while other lines are the numerical results by solving Eq. (2). The solid red and blue lines denote beam energy $E_b=40\mathrm{keV}$ whereas the dashed red and blue ones represent $E_b=60\mathrm{keV}$.

Figure 4

Real frequencies (a) and growth rates (b) of the mode as functions of toroidal rotation frequency with $\widehat{s}=0.072, {\stackrel{⌢}{r}}_{\mathrm{\Omega }}=0, {\sigma }_{\mathrm{\Omega }}=7, {\sigma }_{\rho }=2, {\beta }_h=0.004, E_b=40\mathrm{keV}$ for solid lines and $E_b=60\mathrm{keV}$ for dashed lines. The dotted lines denote theory calculation with the same parameters as the red lines while other lines are the numerical results by solving Eq. (2). The solid red and dotted lines denote beam energy $E_b=40\mathrm{keV}$ whereas the dashed blue ones represent $E_b=60\mathrm{keV}$.

Figure 5

Real frequencies (a) and growth rates (b) of the mode as functions of local magnetic shear with ${\sigma }_{\mathrm{\Omega }}=7, {\stackrel{⌢}{r}}_{\mathrm{\Omega }}=0, {\beta }_h=0.004, {\sigma }_{\rho }=2, {\mathrm{\Omega }}_0=2\times {10}^4\mathrm{rad}/\mathrm{s}$. The dashed, dash-dotted, and solid lines denote the cases of $E_b=40$, 60 and 80 (keV), respectively.

Figure 6

Real frequencies (a) and growth rates (b) of LLM (red lines) and the FB mode (blue ones) as functions of ${\beta }_h$ for different beam energy with ${\mathrm{\Omega }}_0=2\times {10}^4\mathrm{rad}/\mathrm{s}, {\stackrel{⌢}{r}}_{\mathrm{\Omega }}=0, {\sigma }_{\mathrm{\Omega }}=7, {\sigma }_{\rho }=2$ and $q_{\min }=1.03$. The magnetic shear $\widehat{s}=0.056$ for LLM and $\widehat{s}=1.28$ for the FB mode. The solid red and blue lines denote beam energy $E_b=40\mathrm{keV}$ whereas the dashed red and blue ones represent $E_b=80\mathrm{keV}$.

Figure 7

Real frequencies (a) and growth rates (b) of the modes as functions of ${\mathrm{\Omega }}_0$ with ${\stackrel{⌢}{r}}_{\mathrm{\Omega }}=0, {\sigma }_{\mathrm{\Omega }}=7, q_{\min } q_{\min }=1.03, {\beta }_h=0.004$, the red and blue lines represent $\widehat{s}=0.056$ and $\widehat{s}=1.28$, respectively. The solid red and solid blue lines denote beam energy $E_b=40\mathrm{keV}$ whereas the dashed red and dashed blue ones represent $E_b=80\mathrm{keV}$.

Figure 8

Real frequencies (a) and growth rates (b) of the mode as functions of local magnetic shear with ${\sigma }_{\mathrm{\Omega }}=7, {\stackrel{⌢}{r}}_{\mathrm{\Omega }}=0, {\beta }_h=0.004, q_{\min }=1.03, {\sigma }_{\rho }=2, {\mathrm{\Omega }}_0=2\times {10}^4\mathrm{rad}/\mathrm{s}$. The solid and dashed lines denote beam energy $E_b=40$ and 80 (keV), respectively.