Hasegawa-Mima equation as limit of a magnetized pressureless Euler-Boltzmann system

We prove that the Hasegawa-Mima equation (HME) is the limit of a magnetized pressureless Euler-Boltzmann system in a special scaling regime. The proof relies on the derivation of a uniform estimate independent of the scaling parameter, using the particular structure of the singular system and the identification of a combination of the unknowns that compensates for fast oscillations in time. We also show that HME allow us to investigate a conditional property of inverse energy cascade and drift waves.