On the Conservation of Turbulence Energy in Turbulence Transport Models
2021
Zank et al. developed models describing the transport of low frequency
incompressible and nearly incompressible turbulence in inhomogeneous flows. The
formalism was based on expressing the fluctuating variables in terms of the
Els\"assar variables and then taking "moments" subject to various closure
hypotheses. The turbulence transport models are different according to whether
the plasma beta regime is large or of order 1 or smaller. Here, we show
explicitly that the two sets of turbulence transport models admit a
conservation representation that resembles the well-known WKB transport
equation for Alfv\'en wave energy density after introducing appropriate
definitions of the "pressure" associated with the turbulent fluctuations. This
includes introducing a distinct turbulent pressure tensor for 3D incompressible
turbulence (the large plasma beta limit) and pressure tensors for quasi-2D and
slab turbulence (the plasma beta order 1 or small regimes) that generalize the
form of the WKB pressure tensor. Various limits of the different turbulent
pressure tensors are discussed. However, the analogy between the conservation
form of the turbulence transport models and the WKB model is not close for
multiple reasons, including that the turbulence models express fully nonlinear
physical processes unlike the strictly linear WKB description. The analysis
presented here serves both as a check on the validity and correctness of the
turbulence transport models and provides greater transparency of the energy
dissipation term and the "turbulent pressure" in our models, which is important
for many practical applications.
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