Nonlinear Dispersion Relation of Geodesic Acoustic Modes

The energy input and frequency shift of geodesic acoustic modes (GAMs) due to turbulence in tokamak edge plasmas are investigated in numerical two-fluid turbulence studies. Surprisingly, the turbulent GAM dispersion relation is qualitatively equivalent to the linear GAM dispersion but can have drastically enhanced group velocities. In up-down asymmetric geometry the energy input due to turbulent transport may favor the excitation of GAMs with one particular sign of the radial phase velocity relative to the magnetic drifts and may lead to pulsed GAM activity.

Figure 1

Results of a nonlocal NLET turbulence study with slow parameter variation, $\lambda \approx 200$. (a) Flux-surface averaged poloidal $\mathbf{E}\times \mathbf{B}$ velocity $⟨v_{E,\theta }⟩$. (b) Temporal Fourier transform (linear color scale, truncated at 60% of the intensity maximum) of (a) with the local GAM frequency indicated by the dashed line. (c) Bandpass filtered flow profile at $\omega ={\omega }_{\mathrm{GAM},0}(r_0)=1.03$ with fits to curves of constant phase according to Eq. (2) with $\beta =4$ (dash-dotted), $\beta =2$ (dashed), and $\beta =1$ (dotted). Reference radius $r_0$ indicated by horizontal dashed line.

Figure 2

(a) Spectrum of the flux-surface averaged poloidal $\mathbf{E}\times \mathbf{B}$ velocity $|⟨v_{E,\theta }⟩|$ (logarithmic color scale) of a local NLET turbulence study otherwise equivalent to the one shown in Fig. 1 (using the parameters at $r=0$) and dispersion relations ${\omega }_{\mathrm{GAM}}(r,k_r)={\omega }_{\mathrm{GAM},0}(r)(1+\alpha |k_r{|}^2)$ with $\alpha ={\rho }_s^2/2$ (linear dispersion) and $\alpha =39{\rho }_s^2$. (b) Nonlinear frequency shifts due to the couplings between turbulence and GAMs for the Fourier mode $\omega =1.03$ of Fig. 1a: frequency shift necessary for the radial mode structure in Fig. 1c (dash-dotted), shift due to the up-down asymmetric component of the turbulent transport (solid), the Reynolds stress (dotted), and the diamagnetic velocity (dashed).

Figure 3

Flux-surface averaged poloidal $\mathbf{E}\times \mathbf{B}$ flows $⟨v_{E,\theta }⟩$ of NLET turbulence studies using the same parameters as in Fig. 1 but with a single-null geometry. (a) Local run. Left: Ion magnetic inhomogeneity drift ${\mathbf{v}}_d$ directed away from the $X$ point. Right: ${\mathbf{v}}_d$ directed towards the $X$ point. (b) Nonlocal run with $\lambda \approx 200$ and ${\mathbf{v}}_d$ towards the $X$ point. Color coded: short-time rms of $⟨v_{E,\theta }⟩$. Contours: turbulence intensity. Within the shaded areas, the turbulence intensity is above 70% of its rms average. (c) Like (b) but with opposite ${\mathbf{v}}_d$.