An energy estimate for a perturbed Hasegawa-Mima equation

It is commonly believed that drift waves and drift-wave turbulence play a major role in the understanding of anomalous transport at the plasma edge of a tokamak fusion reactor. A one-field equation describing the electrostatic potential fluctuations in this regime is the so-called Hasegawa-Mima equation. If this equation is driven by some instability and damped by some hyperviscous term, the energy grows exponentially in time which is not consistent with the approximations made in the derivation of the equation. Numerical simulations of a perturbed Hasegawa-Mima equation which includes in addition a so-called E x B nonlinearity showed that the energy saturates at a finite level. Ir. this paper this numerical observation is proven analytically.