A scale-separated approach for studying coupled ion and electron scale turbulence

Multiple space and time scales arise in plasma turbulence in magnetic confinement fusion devices because of the smallness of the square root of the electron-to-ion mass ratio and the consequent disparity of the ion and electron thermal gyroradii and thermal speeds. Direct simulations of this turbulence that include both ion and electron space-time scales indicate that there can be significant interactions between the two scales. The extreme computational expense and complexity of these direct simulations motivates the desire for reduced treatment. By exploiting the scale-separation between ion scales (IS) and electron scales (ES), and expanding the gyrokinetic equations for the turbulence in, we derive such a reduced system of gyrokinetic equations that describes cross-scale interactions. The coupled gyrokinetic equations contain novel terms which provide candidate mechanisms for the observed cross-scale interaction. The ES turbulence experiences a modified drive due to gradients in the IS distribution function, and is advected by the IS drift, which varies in the direction parallel to the magnetic field line. The largest possible cross-scale term in the IS equations is sub-dominant in our expansion. Hence, in our model the IS turbulence evolves independently of the ES turbulence. To complete the scale-separated approach, we provide and justify a parallel boundary condition for the coupled gyrokinetic equations in axisymmetric equilibria based on the standard 'twist-and-shift' boundary condition. This approach allows one to simulate multi-scale turbulence using ES flux tubes nested within an IS flux tube.