| The Burgers' model of compressible fluid dynamics in one dimension is extended to include the effects of pressure back-reaction. The new system consists of two coupled equations: Burgers' equation with a pressure gradient (essentially the 1-D Navier-Stokes equation) and an advection-diffusion equation for the pressure field. It presents a minimal model of both adiabatic gas dynamics and compressible magnetohydrodynamics. From the magnetic perspective, it is the simplest possible system which allows for Alfvenization, i.e. energy transfer between the fluid and the magnetic field. For the special case of equal fluid viscosity and (magnetic) diffusivity, the system is completely integrable, reducing to two decoupled Burgers' equations in the characteristic variables v ± vsound ( v ± vAlfven). For arbitrary diffusivities, renormalized perturbation theory is used to calculate the effective transport coefficients for forced Burgerlence. It is shown that energy equi-dissipation, not equipartition, is fundamental to the turbulent state. Both energy and dissipation are localized to shock-like structures, in which wave steepening is inhibited by small-scale forcing and by pressure back-reaction. The spectral forms predicted by theory are confirmed by numerical simulations. It is shown that the velocity structures lead to an asymmetric velocity PDF, as in Burgers' turbulence. Pressure fluctuations, however, are symmetrically distributed. A Fokker-Planck calculation of these distributions is compared and contrasted with a path integral approach. The latter instanton solution suggests that the system maintains its characteristic directions in steady-state turbulence, supporting the results from perturbation theory. Implications for the spectra of turbulence and self-organization phenomena in compressible fluids and plasmas are also discussed. |
