Turbulent Cascades in 3d Incompressible Plasmas

This report surveys mathematical results related to plasma turbulence obtained while supported by the VSGC Graduate Research Fellowship. The solar-wind and interstellar medium are highly turbulent plasmas exhibiting certain features ubiquitous across present numerical and phenomenological theories. The objective of the present research is to provide a rigorous validation of such aspects of these theories directly from the governing equations. The most significant of the included results establishes the existence of a cascade of the total energy in non-forced 3D MHD turbulence over an inertial range the lower limit of which is a modified Taylor length scale. Additional cascades are established for the total fluid energy, the magnetic energy, the energy exchanged between the fluid and magnetic mediums, and the current and vorticity enstrophies. Scale locality of the various fluxes as well as a scenario in which the inter-medium energy transfer is predominantly from kinetic to magnetic are also considered.