Ad-hoc complex resistivity

It was apparent in (eq.6) how it is possible to form a 4th order equation with a small change to the resistive MHD by adding an imaginary part to the plasma resistivity [36]:

$$j_\Vert = -\frac{i\omega}{4\pi}\rho_s^2\left(\frac{2\omega}... ..._e}{T_i}\right)}_% {\mathrm{KAW mock-up}} \right] E_\Vert .$$ (6)

The kinetic-Alfvén wave dispersion and the mode conversion is mocked-up, keeping a tiny collisional dissipation $\sim\nu_e$ to reproduce weakly damped KTAE modes that were predicted analytically by Mett & Mahajan [16]. Implemented in the CASTOR-K code for global wavefields, the results from the complex resistivity model are however often in contradiction with the gyrokinetic calculations from the PENN code. This not surprising, since the dispersion is here not consistent with the FLR correction $\mathcal{O}(k_\perp^2\rho_i^2)$ characteristic of the kinetic-Alfvén wave. The 4th order complex resistivity equation is very different from a consistent gyrokinetic ordering, the amount of power converted where the spatial scales match $k^2_{\perp,\mathrm{TAE}} \approx k^2_{\perp,\mathrm{KAW}}$is altered and the Landau damping, which should be a selective 2nd order differential operator to reproduce the $\vert\gamma/\omega\vert\sim k_\perp^2$dependence of (eq.2), is entirely absent. This explains the large qualitative differences between the complex resistivity spectrum calculated by the CASTOR-K code and the gyrokinetic spectrum. To our knowledge, it was never possible to show a quantitative agreement between damping rates predicted using the complex resistivity model and the measurements from the JET tokamak.