The absence of bottleneck in the Lagrangian-averaged model for incompressible magnetohydrodynamics
Seminar Room 1, Newton Institute
In order to better understand the small scale dynamics of geophysical and astrophysical flows with huge Reynolds numbers, numerical modeling is an invaluable tool but it needs to be assessed against experimental and observational data as well as direct numerical simulations (DNS) at high resolution. In this context, we study the properties of the Lagrangian-averaged magnetohydrodynamics (MHD) $\alpha-$model, LAMHD hereafter; this model can be viewed as a norm-preserving filtering of the primitive MHD equations. Among its advantages is the fact that the LAMHD formulation preserves the basic properties of MHD, e.g. the Alfv\'en theorem of flux conservation, and invariants such as the total energy, the cross-correlation between the velocity and magnetic field and magnetic helicity, albeit in a modified (H_1) form.
LAMHD has been tested in two and three space dimensions and is found to behave satisfactorily, for example reproducing the threshold for dynamo action at moderately low magnetic Prandtl numbers P_M, as encountered in the liquid core of the Earth, the solar convection zone or in liquid metals in the laboratory (where P_M is the ratio of viscosity to magnetic diffusivity). Here we demonstrate that, for the case when there is initially quasi-equipartition between the velocity and the magnetic field and with a magnetic Prandtl number equal to unity, the model 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 more 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.
The LAMHD model is thus shown to be capable of leading to large reductions in required numerical degrees of freedom for a given set of kinetic and magnetic Reynolds number. Specifically, we find a reduction factor of approx 200 when compared to a direct numerical simulation on a large grid of 1536^3 points at the same Taylor Reynolds number approx 1700. The DNS having been stopped at the peak of dissipation of total energy, the run was pursued using LAMHD. We thus also report on preliminary explorations of the decaying dynamics of that high Reynolds number MHD flow at late times using the LAMHD model.