Electromagnetic gyrokinetics (GK)

By now, it should be clear that a consistent description of the coupling between a fluid AE and the kinetic-Alfvén wave requires a gyrokinetic description of the passing bulk ions with FLR corrections at least to $\mathcal{O}(k_\perp^2\rho_i^2)$. This has first been derived in Ref.[37] and implemented in the PENN code [8]. The perturbed current contributes mainly along the magnetic field and can be written symbolically using the dispersion function¹

$$\begin{split} j_\Vert &= -\frac{i\omega}{4\pi} \left\{ \fr... ...ight\} E_b +\mathrm{neoclassical} +\mathrm{drifts} \end{split}$$ (7)

The second order term $v_i^2/\Omega_i^{-2}\nabla^\dagger_\perp\nabla_\perp \sim -\rho_i^2 k_\perp^2$, reproduces the FLR induced kinetic-Alfvén wave dispersion (eq.1) with the real part of the second dispersion function. In addition, a number of terms are not shown here and contribute in a non-evident manner to account for the proper amount of mode conversion in toroidal geometry. The expression shows explicitly how the resistive damping (collision frequency $\nu_e$ in the imaginary part of the first dispersion function) differs from the selective Landau damping ($\omega$ in the imaginary part of the second dispersion function). Most important compared with the previous models, is that the Landau damping is here proportional to the electrostatic component $E_\Vert$: solving Maxwell's equations for an Alfvénic perturbation $\nabla_\perp \times B = \nabla_\perp \times \nabla \times E \sim E_\Vert$with the spatial scale self-consistently described with FLR effects, this reproduces the selective damping $\sim k_\perp^2$ from (eq.2) that affects mainly the shorter wavelength kinetic-Alfvén wave. To perform analytically the integration over the resonant denominators $(\omega-k_\Vert v_\Vert)^{-1}$ and formulate a differential problem in both the radial and poloidal directions, the wave-particle resonance has been approximated assuming passing particles and using a functional dependence for the parallel wave vector, e.g. $k_\Vert\simeq k_\varphi^\mathrm{TAE}(s,\theta)=1/(2qR)$. This approximation can be tested a posteriori in two manners:

• by monitoring the sensitivity of the global damping on different choices that are plausible $k_\varphi\in\{(2qR)^{-1},(qR)^{-1},n/R\}$,
• comparing the complex eigenvalue calculated directly from the model [37] with the damping evaluated perturbatively from the gyrokinetic wavefields $\gamma/\omega=P_{tot}/(\omega W)$. To give a quantitative answer to the controversy in Ref.[1], the drift-kinetic power for passing electrons P_DKe [3] is here extended to account for what turns out to be a really negligible damping from trapped electrons [39]
  

  $$(1-\alpha_t) P_{DKe} +\alpha_t P_{col} \quad\mathrm{with}\quad \alpha_t=\sqrt{\frac{B}{B_{max}}}$$ P_e = (8)

  $$-\frac{(q\omega)^2}{2}\int dV \left(\frac{\nu}{\omega^2+\nu^2}\right) \frac{n_e}{T_e} \vert\Phi\vert^2$$ P_col = (9)

  
  
  where $\nu=(\nu_{ee}+\nu_{ei})R/\rho$ and $\Phi$ is the electrostatic potential calculated here consistently with the gyrokinetic model. The power transfer with passing fast- and bulk ions is evaluated with the non-local expression from Ref.[40] valid to all orders in the Larmor radius $k_\perp\rho_i>1$.

Except for DKAE modes where the mode conversion takes place to an electromagnetic drift wave [15,38] and depends rather sensitively on the choice of $k_\varphi$, these self-consistency checks largely justify the approximations made. They show that the global damping of AEs depends mainly on the location where the conversion occurs, which determines how much power is converted and finally deposited by the kinetic Alfvén wave. In other words, the global damping of AEs depends only weakly on the local Landau damping that affects only the distance the wave covers before it is ultimately damped.

The strength of our approach is that it does not a priori rely on any particular mode conversion mechanism that has been mocked-up from an informed guess; following the dispersion and damping of both the fast and slow global wavefields, mode conversion spontaneously occurs where $k^2_{\perp,\mathrm{fast}}\approx k^2_{\perp,\mathrm{slow}}$. Five conversion mechanisms have been found so far, of which four occur to the kinetic Alfvén wave and only two had been expected from heuristic arguments. They are illustrated in the sketch of Fig.3:

  

Figure 3: Sketch of four mode conversion mechanisms between the fluid (MHD) and kinetic Alfvén (KAW) wavefields.

1.

Near the plasma center, where the aspect ratio is large and the shear is sufficiently weak, the kinetic-Alfvén wave expands radially until it matches the global fluid scale [31]. This mechanism reproduces the global AE dampings measured in a set of similar JET discharges, showing with (eq.2) that the gyrokinetic mode structure correctly changes with the isotope mass [11].

2.

Within gaps, mode conversion sometimes reduces to radiative damping if the kinetic-Alfvén wave is damped in the vicinity of this gap, so that both models reproduce the global damping measurements from JET to a good degree of accuracy [31,35]. Global effects within a single gap can however also split a weakly damped AE into KAEs [14,41], couple adjacent gaps to form high-n global KAE [38] and the fast particles may even drive the kinetic Alfvén wave [38].

3.

At Alfvén resonances, mode conversion is generally not as efficient as continuum damping. The wavefield and damping predicted by the PENN and CASTOR-K codes have been compared with measurements from JET in Ref.[30,31] showing clear contradictions with continuum damping model.

4.

In the plasma edge, the large magnetic shear associated with the plasma shaping (X-point) squeezes the wavefield radially and triggers a mode conversion. The strong global damping rate that results has been tested in the time evolution of Ref.[31].

5.

Mode conversion to electromagnetic drift waves becomes possible in the neighborhood of rational surfaces where $k_\Vert\simeq 0$ if $\omega_*/\omega_{TAE} \simeq 2 n q^2 (\rho/a)^2 (R\omega_{pi}/c)$ approaches unity, so that drift-kinetic AEs (DKAEs) become unstable. They provide for a plausible mechanism for the instabilities observed in the TAE frequency range of DIII-D [42,15]; their modeling is however likely to suffer from the approximative treatment of the parallel dynamics with $k_\varphi=1/(2qR)$.

So far, the gyrokinetic PENN model with approximate parallel dynamics is the only tool that deals properly with the mode conversion of global wavefields in a tokamak; most toroidal predictions remain therefore to be confirmed using other models.

The KIN-2DEM code [9] from PPPL/Princeton does not presently retain the dynamics of passing ions and therefore misses the mode conversion to kinetic-Alfvén wave that is essential for AEs and perhaps also important for kinetic ballooning modes (KBM). A gyrokinetic model including the main ingredients has recently been formulated in Ref.[43] and might be implemented in a future version of the NOVA code. A model for electromagnetic micro-instabilities is currently under development at CRPP/Lausanne for low $\beta$ plasmas in simple toroidal geometry and includes the parallel dynamics of passing and trapped bulk particles to all orders in $k_\perp\rho_i$ [46]; the model retains the mode conversion both to the kinetic-Alfvén and the drift waves and will soon be used to confirm and extend the present knowledge beyond the FLR approximation. In parallel, a new electromagnetic model for macro-instabilities has been derived for a future version of the PENN code, keeping the complete parallel dynamics of passing bulk- and energetic ions together with large $\beta$ and arbitrary shape in the limit of small Larmor radii. A complete derivation for the current perturbation along the magnetic field will be given elsewhere, but finally results in an integro-differential expression in configuration space that can be written symbolically as

$$\begin{split} j_\Vert = \frac{i\omega}{4\pi}&\left\{-\frac{c... ...B}B_\Vert\right] \right) +\mathrm{drifts} \right\} \end{split}$$ (10)

The kernel $K(\theta,\theta^\prime)$ clearly plays an essential role here, since it contains the wave-particle resonances that are integrated numerically over $dv_\Vert$ and couples different poloidal locations through the integral operator. Finite Larmor radius corrections appear explicitly with the second order operators $\Delta_\perp$ and inhomogeneities with the drift frequency $\omega^*$. Resonant effect from energetic particles are not shown here, but are also kept in the small Larmor radius approximation. The physics we hope to study with this new model ranges down in frequency from the energetic-particle-kinetic AEs, drift-kinetic AEs, beta-induced AEs and finally the non-ideal counterparts of MHD instabilities such as KBMs, internal kink and resistive-wall modes.