MHD, drift, kinetic-Alfvén and resonant wavefields

High frequency $\omega$ magneto-hydrodynamic (MHD) perturbations are susceptible to undergo resonant interactions with fast (f) super-Alfvénic particles $v_f>v_A=B_0/\sqrt{4\pi n_i m_i}$ for frequencies that range roughly from the electron (e) or ion (i) drift to the Alfvén frequency: $\omega^*=(\ln nT)^\prime k_\theta T/(m\Omega) \leq \omega \leq \omega_A=v_A/R \ll \Omega_i$, where $\Omega_i=q_i B/m_i$ stands for the cyclotron frequency of the ions. From a local dispersion analysis of electromagnetic waves, keeping the pressure gradients and the finite Larmor radius (FLR) of the ions, it is known that the drift-waves, the kinetic-Alfvén wave (KAW) and the global MHD wavefield get coupled and yield a relation of the form

$$\omega^2 -\omega\omega_i^* -v_A^2 k_\Vert^2 \left\{1 + k_\... ...\omega_i^*}{\omega-\omega_e^*} \right) \right] \right\} = 0$$ (1)

Taking the homogeneous limit $\{\omega^*_e,\omega^*_i\} \rightarrow 0$ to recover the standard kinetic-Alfvén wave [10], one immediately sees with Faraday's law how the electrostatic component of a magnetic perturbation $k_\perp \times B \sim E_\Vert$ is related to the ion Larmor radius dispersion term $k_\perp^2\rho_i^2$ and yields resonant Landau interactions with passing electrons that move along the magnetic field lines

$$\frac{\gamma}{\omega} = -\sqrt{\frac{\pi}{4}}v_s \frac{v_A... ... \stackrel{v_i \ll v_A \ll v_e}{\propto} -k_\perp^2 \sqrt{A}.$$ (2)

In the regime typical of tokamaks nowadays, the Alfvén velocity lies between the ion v_i and electron thermal velocities v_e, so that for a fixed mode structure $k_\perp$ and similar plasma conditions, the electron Landau damping is proportional to the square root of the isotope mass $\sqrt{A}$; this argument has recently been tested against the experiment [11]. Damping occurs also via the collisions between passing or trapped electrons [12,13], but this collisional damping is negligible compared with the electron Landau damping for fusion relevant regimes - as will be substantiated below in answer to [1].

If the inhomogeneity drifts become sufficiently large $\omega^*>\omega$, the kinetic energy of the particles may be transferred to the wave and provides a finite drive that may overcome the total damping. This can under circumstance occur for the bulk species, but is most easily achieved by fast (energetic) ions. Substituting $i\rightarrow f$ in (eq.1) when the fast particle pressure gradient dominates the ions response $\beta_f^\prime>\beta_i^\prime$, it is clear that the wavefield acquires a resonant or energetic character when the normalized fast particle pressure $\beta_f=8\pi P_\mathrm{kin}/B^2$ is comparable with the bulk $\beta$.