Can drift-ordered fluid models be made fully consistent?

Although drift-ordered fluid models based on the Braginskii equations are widely used in plasma turbulence codes, no single implementation of these models conserves energy – except in extremely limiting cases where density and magnetic gradients are completely neglected. The main reason for this is the difficulty of obtaining a closed-form expression for the polarization flux and its relation to the vorticity. Our objective is to provide a simple and practical expression for the polarization flux, applicable to drift-Braginskii models. The lowest order form of the model, which truncates the perpendicular kinetic energy flux, results in a closed form for the polarization velocity as a sum of drifts. This result mainly stems from considering a generalized vorticity function related to the curl of the momentum. The resulting vorticity equation is formulated in divergence form, which ensures charge conservation and can be adapted to different numerical schemes. However, the correct kinetic energy flux can only be reconstructed by defining the polarization flux recursively – which yields a challenging 3D vector equation to solve. This poses the question of whether drift-ordered fluid models can ever be made energy-consistent in a practical manner.