We demonstrate that, for the case of quasi-equipartition between the velocity and the magnetic field, the Lagrangian-averaged magnetohydrodynamics alpha-model (LAMHD) reproduces well both the large-scale and small-scale properties of turbulent flows;
in particular, it displays no increased (super-filter) bottleneck effect with its ensuing enhanced energy spectrum at the onset of the sub-filter-scales. This is in contrast to the case of the neutral fluid in which the Lagrangian-averaged Navier-Stokes alpha-model is somewhat limited in its applications because of the formation of spatial regions with no internal degrees of freedom and subsequent contamination of super-filter-scale spectral properties. No such regions are found in LAMHD, making this method capable of large reductions in required numerical degrees of freedom; specifically, we find a reduction factor of 200 when compared to a direct numerical simulation on a large grid of 1536^3 points at the same Reynolds number.
We show in this paper that third- and fourth-order low storage Runge-Kutta algorithms can be built specifically for quadratic nonlinear operators, at the expense of roughly doubling the time needed for evaluating the temporal derivatives. The resulti
ng algorithms are especially well suited for computational fluid dynamics. Examples are given for the Henon-Heiles Hamiltonian system and, in one and two space dimensions, for the Burgers equation using both a pseudo-spectral code and a spectral element code, respectively. The scheme is also shown to be practical in three space solving the incompressible Euler equation using a fully parallelized pseudo-spectral code.
