# Coarse-grained incompressible magnetohydrodynamics: Analyzing the turbulent cascades

## Abstract

Here, we formulate a coarse-graining approach to the dynamics of magnetohydrodynamic (MHD) fluids at a continuum of length-scales. In this methodology, effective equations are derived for the observable velocity and magnetic fields spatially-averaged at an arbitrary scale of resolution. The microscopic equations for the bare velocity and magnetic fields are renormalized by coarse-graining to yield macroscopic effective equations that contain both a subscale stress and a subscale electromotive force (EMF) generated by nonlinear interaction of eliminated fields and plasma motions. At large coarse-graining length-scales, the direct dissipation of invariants by microscopic mechanisms (such as molecular viscosity and Spitzer resistivity) is shown to be negligible. The balance at large scales is dominated instead by the subscale nonlinear terms, which can transfer invariants across scales, and are interpreted in terms of work concepts for energy and in terms of topological flux-linkage for the two helicities. An important application of this approach is to MHD turbulence, where the coarse-graining length ℓ lies in the inertial cascade range. We show that in the case of sufficiently rough velocity and/or magnetic fields, the nonlinear inter-scale transfer need not vanish and can persist to arbitrarily small scales. Although closed expressions are not available for subscale stressmore »

- Authors:

- Univ. of Rochester, Rochester, NY (United States)

- Publication Date:

- Research Org.:
- Univ. of Rochester, Rochester, NY (United States); Univ. of Rochester, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24); USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); USDOE National Nuclear Security Administration (NNSA), Office of Defense Nuclear Security (NA-70)

- OSTI Identifier:
- 1437700

- Alternate Identifier(s):
- OSTI ID: 1356076; OSTI ID: 1358358

- Grant/Contract Number:
- SC0014318; NA0001944; 20150568ER

- Resource Type:
- Journal Article: Published Article

- Journal Name:
- New Journal of Physics

- Additional Journal Information:
- Journal Volume: 19; Journal Issue: 2; Journal ID: ISSN 1367-2630

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 22 GENERAL STUDIES OF NUCLEAR REACTORS; energy-dissipation rate; large-eddy simulation; free magnetic-fields; isotropic turbulence; hydromagnetic turbulence; ideal hydrodynamics; inverse cascade; inertial-range; helicity; flows; magnetohydrodynamics; turbulence; coarse-graining; cascade

### Citation Formats

```
Aluie, Hussein.
```*Coarse-grained incompressible magnetohydrodynamics: Analyzing the turbulent cascades*. United States: N. p., 2017.
Web. doi:10.1088/1367-2630/aa5d2f.

```
Aluie, Hussein.
```*Coarse-grained incompressible magnetohydrodynamics: Analyzing the turbulent cascades*. United States. doi:10.1088/1367-2630/aa5d2f.

```
Aluie, Hussein. Tue .
"Coarse-grained incompressible magnetohydrodynamics: Analyzing the turbulent cascades". United States.
doi:10.1088/1367-2630/aa5d2f.
```

```
@article{osti_1437700,
```

title = {Coarse-grained incompressible magnetohydrodynamics: Analyzing the turbulent cascades},

author = {Aluie, Hussein},

abstractNote = {Here, we formulate a coarse-graining approach to the dynamics of magnetohydrodynamic (MHD) fluids at a continuum of length-scales. In this methodology, effective equations are derived for the observable velocity and magnetic fields spatially-averaged at an arbitrary scale of resolution. The microscopic equations for the bare velocity and magnetic fields are renormalized by coarse-graining to yield macroscopic effective equations that contain both a subscale stress and a subscale electromotive force (EMF) generated by nonlinear interaction of eliminated fields and plasma motions. At large coarse-graining length-scales, the direct dissipation of invariants by microscopic mechanisms (such as molecular viscosity and Spitzer resistivity) is shown to be negligible. The balance at large scales is dominated instead by the subscale nonlinear terms, which can transfer invariants across scales, and are interpreted in terms of work concepts for energy and in terms of topological flux-linkage for the two helicities. An important application of this approach is to MHD turbulence, where the coarse-graining length ℓ lies in the inertial cascade range. We show that in the case of sufficiently rough velocity and/or magnetic fields, the nonlinear inter-scale transfer need not vanish and can persist to arbitrarily small scales. Although closed expressions are not available for subscale stress and subscale EMF, we derive rigorous upper bounds on the effective dissipation they produce in terms of scaling exponents of the velocity and magnetic fields. These bounds provide exact constraints on phenomenological theories of MHD turbulence in order to allow the nonlinear cascade of energy and cross-helicity. On the other hand, we show that the forward cascade of magnetic helicity to asymptotically small scales is impossible unless 3rd-order moments of either velocity or magnetic field become infinite.},

doi = {10.1088/1367-2630/aa5d2f},

journal = {New Journal of Physics},

number = 2,

volume = 19,

place = {United States},

year = {Tue Feb 21 00:00:00 EST 2017},

month = {Tue Feb 21 00:00:00 EST 2017}

}

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