Finite-time singularities and flow regularization in a hydromagnetic shell model at extreme magnetic Prandtl numbers

Abstract

Conventional surveys on the existence of singularities in fluid systems for vanishing dissipation have hitherto tried to infer some insight by searching for spatial features developing in asymptotic regimes. This approach has not yet produced a conclusive answer. One of the difficulties preventing us from getting a definitive answer is the limitations of direct numerical simulations which do not yet have a high enough resolution so far as to properly describe spatial fine structures in asymptotic regimes. In this paper, instead of searching for spatial details, we suggest seeking a principle, that would be able to discriminate between singular or not-singular behavior, among the integral and purely dynamical properties of a fluid system. We investigate the singularities developed by a hydromagnetic shell model during the magnetohydrodynamic turbulent cascade. Our results show that when the viscosity is equal to the magnetic diffusivity (unit magnetic Prandtl number) singularities appear in a finite time. A complex behavior is observed at extreme magnetic Prandtl numbers. In particular, the singularities persist in the limit of vanishing viscosity, while a complete regularization is observed in the limit of vanishing diffusivity. This dynamics is related to differences between the magnetic and the kinetic energy cascades towards small scales. Finally a comparison between the three-dimensional and the two-dimensional cases leads to conjecture that the existence of singularities may be related to the conservation of different ideal invariants.

Study of the fractality in an MHD Shell model forced by solar wind fluctuations

“…In this work, we use the MHD GOY shell model, which has been shown to be adequate to describe the dynamics of the energy cascade in MHD turbulence (Lepreti et al, 2004), dynamo effect (Nigro and Carbone, 2010;Nigro and Veltri, 2011), statistics of solar flares (Boffetta et al, 1999;Lepreti et al, 2004;Nigro et al, 2004), finite-time singularities in turbulent cascades (Nigro and Carbone, 2015) and to model the fractal features of a magnetized plasma (Domínguez et al, 2017(Domínguez et al, , 2018).…”

“…In previous work (Domínguez et al, 2017;Lepreti et al, 2004;Nigro et al, 2004;Nigro and Carbone, 2010;Nigro and Veltri, 2011;Nigro, 2013;Nigro and Carbone, 2015) the forcing terms were obtained from the Langevin equation…”

An Argument in Favor of Magnetic Polarity Reversals Due to Heat Flux Variations in Fully Convective Stars and Planets

“…Among simplified models are shell models (Gloaguen et al 1985;Plunian et al 2013;Biferale 2003). Despite fewer degrees of freedom being considered compared to the corresponding original fluid equations, shell models are able to successfully reproduce the main characteristics of the smallscale statistics of the fields (Biferale 2003;Plunian et al 2013;Nigro & Carbone 2015), fractality in geomagnetic activity (Domínguez et al 2017(Domínguez et al , 2018Muñoz et al 2018), and dynamo mechanism (Frick & Sokoloff 1998;Nigro & Carbone 2010;Nigro & Veltri 2011;Nigro 2013). Shell models have been adopted for thermal convection (Jensen et al 1992;Mingshun & Shida 1997) and magnetoconvection (Brandenburg 1992), where dynamical real scalar variables are considered.…”

Turbulence in a Coronal Loop Excited by Photospheric Motions

“…However, due to numerical limitations, the wide range of spatial scales of coronal turbulence is not adequately described. This problem can be circumvented using "reduced" models of turbulence, like the so-called shell models [30,31,48], in which the dynamics of turbulence is represented in a simplified Fourier space. In a coronal context, hybrid shell models have been developed [33,37], based on Reduced MHD (RMHD) [49,50].…”