Wave-number spectrum of dissipative drift waves and a transition scale

We study the steady state spectrum of the Hasegawa-Wakatani (HW) equations that describe drift wave turbulence. Beyond a critical scale k_{c}, which appears as a balance between the nonlinear time and the parallel conduction time, the adiabatic electron response breaks down nonlinearly and an internal energy density spectrum of the form F(k_{⊥})∝k_{⊥}^{-3}, associated with the background gradient, is established. More generally a dual power law spectrum, approximately of the form F(k_{⊥})∝k_{⊥}^{-3}(k_{c}^{-2}+k_{⊥}^{-2}) is obtained, which captures this transition. Using dimensional analysis, an expression of the form k_{c}∝C/κ is derived for the transition scale, where C and κ are normalized parameters of the HW equations signifying the electron adiabaticity and the density gradient, respectively. The results are numerically confirmed using a shell model developed and used for the Hasegawa-Wakatani system.