In this paper, we derive estimates for size of the small scales and the attractor dimension in low $Rm$ magnetohydrodynamic turbulence by deriving a rigorous upper bound of the dimension of the attractor representing this flow. To this end, we find an upper bound for the maximum growth rate of any $n$-dimensional volume of the phase space by the evolution operator associated to the Navier-Stokes equations. As explained by Constantin et al. (J. Fluid Mech., 1985), The value of $n$ for which this maximum is zero is an upper bound for the attractor dimension. In order to use this property in the more precise case of a 3D periodical domain, we are led to calculate the distribution of $n$ modes which minimises the total (viscous and Joule) dissipation. This set of modes turns out to exhibit most of the well known properties of MHD turbulence, previously obtained by heuristic considerations such as the existence of the Joule cone under strong magnetic field. The sought estimates for the small scales and attractor dimension are then obtained under no physical assumption as functions of the Hartmann and the Reynolds numbers and match the Hartmann number dependency of heuristic results. A necessary condition for the flow to be tridimensional and anisotropic (as opposed to purely two-dimensional) is also built.
As a reduced representation of the nonlinear spectral fluxes of ideal invariants in incompressible magnetohydrodynamics, we construct a gradient-diffusion network model that combines phenomenological considerations and geometrical analysis of the exact nonlinear energy transfer function. The reduced-order representation of the conservative spectral transport of energy and cross-helicity is of port-Hamiltonian form, which highlights the flexibility and modularity of this approach. Numerical experiments with Reynolds numbers up to $~10^6$ yield clear power-law signatures of inertial-range energy…
We present pseudospectral direct-numerical-simulation (DNS) studies of the three-dimensional magnetohydrodynamic (MHD) equations (3DRFMHD) with a stochastic force that has zero mean and a variance $\sim k^{-3}$, where $k$ is the wavenumber, because 3DRFMHD is used in field-theoretic studies of the scaling of energy spectra in MHD turbulence. We obtain velocity and magnetic-field spectra and structure functions and, from these, the multiscaling exponent ratios $ζ_p/ζ_3$, by using the extended self similarity (ESS) procedure. These exponent ratios lie within error bars of their counterparts for …
The four-fifths law for third-order longitudinal moments is examined, by the use of direct numerical simulation data on three-dimensional forced incompressible magnetohydrodynamic (MHD) turbulence without a uniformly imposed magnetic field in a periodic box. The magnetic Prandtl number is set to one, and the number of grid points is $512^3$. A generalized Kármán-Howarth-Kolmogorov equation for second-order velocity moments in isotropic MHD turbulence is extended to anisotropic MHD turbulence by means of a spherical average over the direction of $\textbf{r}$. Here, $\textbf{r}$ is a separation …
A new sub-grid-scale model is developed for studying influences of the Hall term on macroscopic aspects of magnetohydrodynamic turbulence. Although the Hall term makes numerical simulations extremely expensive by exciting high-wave-number coefficients and makes magnetohydrodynamic equations stiff, studying macroscopic aspects of magnetohydrodynamic turbulence together with the Hall term is meaningful since this term often influences not only sub-ion-scales but also macroscopic scales. A new sub-ion-scale sub-grid-scale model for large eddy simulations of Hall magnetohydrodynamic turbulence is …
In the period of 1948-1955, Chandrasekhar wrote four papers on magnetohydrodynamic (MHD) turbulence, which are the first set of papers in the area. The field moved on following these pioneering efforts. In this paper, I will briefly describe important works of MHD turbulence, starting from those by Chandrasekhar.
We explore the suitability of deep learning to capture the physics of subgrid-scale ideal magnetohydrodynamics turbulence of 2-D simulations of the magnetized Kelvin-Helmholtz instability. We produce simulations at different resolutions to systematically quantify the performance of neural network models to reproduce the physics of these complex simulations. We compare the performance of our neural networks with gradient models, which are extensively used in the extensively in the magnetohydrodynamic literature. Our findings indicate that neural networks significantly outperform gradient models…
The effect of precession in a rotating sphere filled with fluid was studied with direct numerical simulations, both in the incompressible hydrodynamics (HD) and magnetohydrodynamics (MHD) scenarios. In both cases the asymptotic state and its dependence with both rotating and precession frequency was analyzed. For the MHD case no self-sustaining dynamos were found for the prograde precession case, whereas on the other hand a critical retrograde precession frequency was found above which dynamo action is self-sustained. It was also found that these correspond to small-scale dynamos with a develo…
In this review, we describe turbulent drag reduction in a variety of flows using a universal framework of energy flux. In a turbulent flow with dilute polymers and magnetic field, the kinetic energy injected at large scales cascades to the velocity field at intermediate scales, as well as to the polymers and magnetic field at all scales. Consequently, the kinetic energy flux, $ Π_u(k) $, is suppressed in comparison to the pure hydrodynamic turbulence. We argue that the suppression of $Π_u(k)$ is an important factor in the reduction of the inertial force $\langle {\bf u \cdot \nabla u} \rangle$…
