Marco Velli

We live in an age in which high-performance computing is transforming the way we do science. Previously intractable problems are now becoming accessible by means of increasingly realistic numerical simulations. One of the most enduring and most challenging of these problems is turbulence. Yet, despite these advances, the extreme parameter regimes encountered in space physics and astrophysics (as in atmospheric and oceanic physics) still preclude direct numerical simulation. Numerical models must take a Large Eddy Simulation (LES) approach, explicitly computing only a fraction of the active dynamical scales. The success of such an approach hinges on how well the model can represent the subgrid-scales (SGS) that are not explicitly resolved. In addition to the parameter regime, heliophysical and astrophysical applications must also face an equally daunting challenge: magnetism. The presence of magnetic fields in a turbulent, electrically conducting fluid flow can dramatically alter the coupling between large and small scales, with potentially profound implications for LES/SGS modeling. In this review article, we summarize the state of the art in LES modeling of turbulent magnetohydrodynamic (MHD) flows. After discussing the nature of MHD turbulence and the small-scale processes that give rise to energy dissipation, plasma heating, and magnetic reconnection, we consider how these processes may best be captured within an LES/SGS framework. We then consider several specific applications in heliophysics and astrophysics, assessing triumphs, challenges, and future directions.

A strong indication that fast reconnection regimes exist within resistive magnetohydrodynamics was given by the proof that the Sweet-Parker current sheet, maintained by a flow field with an appropriate inflow-outflow structure, could be unstable to a reconnecting instability which grows without bound as the Lundquist number, S, tends to infinity. The requirement of a minimum value for S in order for the plasmoid instability to kick in does little to resolve the paradoxical nature of the result. Here we argue against the realizability of Sweet-Parker current sheets in astrophysical plasmas with very large S by showing that an ideal tearing mode takes over before current sheets reach such a thickness. While the Sweet-Parker current sheet thickness scales as ~S –1/2, the tearing mode becomes effectively ideal when a current sheet collapses to a thickness of the order of ~S –1/3, up to 100 times thicker than S –1/2, when (as happens in many astrophysical environments) S is as large as 1012. Such a sheet, while still diffusing over a very long time, is unstable to a tearing mode with multiple x-points: here we detail the characteristics of the instability and discuss how it may help solve the flare trigger problem and effectively initiate the turbulent disruption of the sheet.

We present a mechanism for the formation of the low-frequency 1/f magnetic spectrum based on numerical solutions of a shell-reduced MHD model of the turbulent dynamics inside the sub-Alfvénic solar wind. We assign reasonably realistic profiles to the wind speed and the density along the radial direction, and a radial magnetic field. Alfvén waves of short periodicity (600 s) are injected at the base of the chromosphere, penetrate into the corona, and are partially reflected, thus triggering a turbulent cascade. The cascade is strong for the reflected wave while it is weak for the outward propagating waves. Reflection at the transition region recycles the strong turbulent spectrum into the outward weak spectrum, which is advected beyond the Alfvénic critical point without substantial evolution. There, the magnetic field has a perpendicular power-law spectrum with slope close to the Kolmogorov –5/3. The parallel spectrum is inherited from the frequency spectrum of large (perpendicular) eddies. The shape is a double power law with slopes of – 1 and –2 at low and high frequencies, respectively, with the position of the break depending on the injected spectrum. We suggest that the double power-law spectrum measured by Helios at 0.3 AU, where the average magnetic field is not aligned with the radial (contrary to our assumptions), results from the combination of such different spectral slopes. At low frequency the parallel spectrum dominates with its characteristic 1/f shape, while at higher frequencies its steep spectral slope (–2) is masked by the more energetic perpendicular spectrum (slope –5/3).

We present a mechanism for the formation of the low frequency 1/f magnetic spectrum based on numerical solutions of a shell reduced-MHD model of the turbulent dynamics inside the sub-Alfv\'enic solar wind. We assign reasonably realistic profiles to the wind speed and the density along the radial direction, and a radial magnetic field. Alfv\'en waves of short periodicity (600 s) are injected at the base of the chromosphere, penetrate into the corona and are partially reflected, thus triggering a turbulent cascade. The cascade is strong for the reflected wave while it is weak for the outward propagating waves. Reflection at the transition region recycles the strong turbulent spectrum into the outward weak spectrum, which is advected beyond the Alfv\'enic critical point without substantial evolution. There, the magnetic field has a perpendicular power-law spectrum with slope close to the Kolmogorov -5/3. The parallel spectrum is inherited from the frequency spectrum of large (perpendicular) eddies. The shape is a double power-law with slopes of -1 and -2 at low and high frequencies respectively, the position of the break depending on the injected spectrum. We suggest that the double power-law spectrum measured by Helios at 0.3 AU, where the average magnetic field is not aligned with the radial (contrary to our assumptions) results from the combination of such different spectral slopes. At low frequency the parallel spectrum dominates with its characteristic 1/f shape, while at higher frequencies its steep spectral slope (-2) is masked by the more energetic perpendicular spectrum (slope -5/3).