Burgers' turbulence with self-consistently evolved pressure

The Burgers' model of compressible fluid dynamics in one dimension is extended to include the effects of pressure back-reaction. The system consists of two coupled equations: Burgers' equation with a pressure gradient (essentially the one-dimensional 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 magnetic field excitations. 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+/-v(sound) (v+/-v(Alfven)). For arbitrary diffusivities, renormalized perturbation theory is used to calculate the effective transport coefficients for forced "Burgerlence." It is shown that energy equidissipation, not equipartition, is fundamental to the turbulent state. Both energy and dissipation are localized to shocklike 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.