Cancellation exponent and multifractal structure in two-dimensional magnetohydrodynamics: Direct numerical simulations and Lagrangian averaged modeling

Abstract

We present direct numerical simulations and Lagrangian averaged (also known as α-model) simulations of forced and free decaying magnetohydrodynamic turbulence in two dimensions. The statistics of sign cancellations of the current at small scales is studied using both the cancellation exponent and the fractal dimension of the structures. The α-model is found to have the same scaling behavior between positive and negative contributions as the direct numerical simulations. The α model is also able to reproduce the time evolution of these quantities in free decaying turbulence. At large Reynolds numbers, an independence of the cancellation exponent with the Reynolds numbers is observed. PACS numbers: 47.27.Eq; 47.27.Gs; 47.11.+j The magnetohydrodynamic (MHD) approximation is often used to model plasmas or conducting fluids in astrophysical and geophysical environments. However, given the huge amount of temporal and spatial scales involved in the dynamics of these objects, simulations are always carried out in a region of parameter space far from the observed values. Lagrangian averaged magnetohydrodynamics (LAMHD), also called the MHD alpha-model [1, 2] (or the Camassa-Holm equations in early papers studying the hydrodynamic case [3]), has been recently introduced as a way to reduce the number of degrees of freedom of the system, while keeping accurate evolution for the large scales. This approach (as well as large eddy simulations, or LES, for MHD; see e.g. [4]) is intended to model astrophysical or geophysical flows at high Reynolds numbers using available computational resources. Several aspects of the MHD alpha-model have already been tested in two and three dimensions at moderate Reynolds numbers, against direct numerical simulations of the MHD equations [2]. These studies were focused on comparisons of the evolution of global quantities and the dynamics of the large scale components of the energy spectrum [2,5].All these models introduce changes in the small scales in order to preserve the evolution of the large scales. In several cases, it is of interest to know the statistics of the small scales. It is also important to model properly the small scales because they have an effect on large scales, as for example in the case of eddy noise: the beating of two small scales eddies produces energy at the large scale, and this may affect the global longtime evolution of the flow, an issue that arises in global climate evolution or in solar-terrestrial interactions. Moreover, plasmas and conducting fluids generate thin and intense current sheets where magnetic reconnection takes place. In these regions, the magnetic field and the current rapidly change sign, and after reconnection the magnetic energy is turned into mechanical and thermal energy. These events are known to take place in the magnetopause [6], the magnetotail [7], the solar atmosphere [8], and the interplanetary medium [9].Current sheets are strongly localized and intermittent. To preserve reliable statistics of these events in models of MHD turbulenc...

Lagrangian-averaged model for magnetohydrodynamic turbulence and the absence of bottlenecks

“…As in the case of LES, the actual dissipation process is not as important as the fact that large-scale dynamics should be reproduced with minimal contamination by the sub-grid model. We believe the results presented here (and in earlier work [27,28,29,30,31]) show this is the case, and allow the use of the LAMHD equations as a subgrid model of MHD turbulence. However, considering the differences observed between LANS and LAMHD, we discuss the dissipation processes in LAMHD.…”

“…Recall that differences at the largest scales, stem from the differences in initial conditions as stated in Section III A, and from time evolution of the flow. Finally, noting that the computer saving here is 6 3 in memory and 6 4 in running time, we conclude that the LAMHD continues to behave satisfactorily, as already shown both in two space dimensions [27,28,29] and in 3D [30], in particular in the context of the dynamo problem of generation of magnetic fields by velocity gradients; thus, LAMHD may prove to be a useful tool in many astrophysical contexts where magnetic fields are dynamically important, such as in the solar and terrestrial environments, or in the interstellar and intergalactic media. ; labels are as in Fig.…”

“…It has been shown to reproduce a number of features of DNS. In two dimensions (2D) for Taylor Reynolds numbers (R λ ) up to ≈ 5000 it has been shown to reproduce selective decay, the inverse cascade of mean-square vector potential, and dynamic alignment between the velocity and magnetic fields [27] as well as the statistics of small-scale cancellation [28] and inter-mittency [29]. In three dimensions (3D) at Reynolds numbers (Re) of ≈ 500, LAMHD reproduced the inverse cascade of magnetic helicity (associated with the development of force-free magnetic field) and the helical dynamo effect [30].…”

“…This limitation is not apparent in low and moderate Reynolds number (resolution) simulations (e.g., 64 3 LANS compared with 256 3 DNS) as the scale separation is not enough for the above-mentioned phenomenon of contamination of small-scale spectra because of rigid body regions in the flow to appear. The bottleneck (and superfilter-scale contamination) was not studied as such but neither was it observed in 2D LAMHD for high Reynolds number [27,28,29]. 3D LAMHD has only been tested at more moderate Reynolds number [30] (see also [34] for a recent review).…”

Sign cancellation and scaling in the vertical component of velocity and vorticity in rotating turbulence

“…The exponent gives information about the rapid changes in the sign of scalar quantities, and has been used before to characterize fluctuations of velocity and magnetic-field components in hydrodynamic turbulence and magnetohydrodynamic dynamos [23], of the current density in 2D magnetohydrodynamic turbulence [25,26], of magnetic helicity in solar wind observations [27], and of helicity in isotropic and homogeneous hydrodynamic turbulence [28]. We analyze data from high-resolution direct numerical simulations (up to 1536 3 grid points) and compute the cancellation exponent for the Cartesian components of the velocity and vorticity fields and for the helicity.…”