Helical and nonhelical (magneto-)Burgers turbulence: I. Compressibility reduction and beyond

We compare the helical and nonhelical (magneto-)Burgers turbulence for the \textit{helicity fastening effect}. Theoretical arguments and heuristic mathematical analysis are offered for the latter notion in the new system loosing some ``nice'' properties as previously used in addressing the Navier-Stokes and various plasma fluids. Miscellaneous discussions are also offered, including the inferences of several consequences on the transports of passive scalars for both the density and tracer, particularly, the opposite consequences of the helicity fastening effect for the latter two scalars in appropriate situations (with the caveat of the possibility of the inverse cascade of the tracer energy). Basic numerical results of the fractions of the parallel-mode spectra, with maximally-helical random forcing on some small-wavenumber modes, present a benefit of about $0.2$ over those with nonhelical forcing, indicating regularization (to some degree) of the solutions. Such helicity ``fastening'' effect of Burgers turbulence is much more marked than that for low-Mach-number Navier-Stokes turbulence. The magnetic helicity in magneto-Burgers dynamics can present an even stronger benefit, of around $0.5+$.