# Magnetic Prandtl number dependence of the kinetic-to-magnetic dissipation ratio

## Abstract

Using direct numerical simulations of three-dimensional hydromagnetic turbulence, either with helical or non-helical forcing, we show that the kinetic-to-magnetic energy dissipation ratio always increases with the magnetic Prandtl number, i.e., the ratio of kinematic viscosity to magnetic diffusivity. This dependence can always be approximated by a power law, but the exponent is not the same in all cases. For non-helical turbulence, the exponent is around 1/3, while for helical turbulence it is between 0.6 and 2/3. In the statistically steady state, the rate of energy conversion from kinetic into magnetic by the dynamo must be equal to the Joule dissipation rate. We emphasize that for both small-scale and large-scale dynamos, the efficiency of the energy conversion depends sensitively on the magnetic Prandtl number, and thus on the microphysical dissipation process. To understand this behavior, we also study shell models of turbulence and one-dimensional passive and active scalar models. We conclude that the magnetic Prandtl number dependence is qualitatively best reproduced in the one-dimensional model as a result of dissipation via localized Alfvén kinks.

- Authors:

- Nordita, KTH Royal Institute of Technology and Stockholm University, SE-10691 Stockholm (Sweden)
- (Sweden)

- Publication Date:

- OSTI Identifier:
- 22365420

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Astrophysical Journal; Journal Volume: 791; Journal Issue: 1; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ACCRETION DISKS; ALFVEN WAVES; APPROXIMATIONS; COMPUTERIZED SIMULATION; EFFICIENCY; ENERGY CONVERSION; ENERGY LOSSES; MAGNETOHYDRODYNAMICS; PRANDTL NUMBER; SHELL MODELS; SHOCK WAVES; STEADY-STATE CONDITIONS; THREE-DIMENSIONAL CALCULATIONS; TURBULENCE; VISCOSITY

### Citation Formats

```
Brandenburg, Axel, and Department of Astronomy, Stockholm University, SE-10691 Stockholm.
```*Magnetic Prandtl number dependence of the kinetic-to-magnetic dissipation ratio*. United States: N. p., 2014.
Web. doi:10.1088/0004-637X/791/1/12.

```
Brandenburg, Axel, & Department of Astronomy, Stockholm University, SE-10691 Stockholm.
```*Magnetic Prandtl number dependence of the kinetic-to-magnetic dissipation ratio*. United States. doi:10.1088/0004-637X/791/1/12.

```
Brandenburg, Axel, and Department of Astronomy, Stockholm University, SE-10691 Stockholm. Sun .
"Magnetic Prandtl number dependence of the kinetic-to-magnetic dissipation ratio". United States.
doi:10.1088/0004-637X/791/1/12.
```

```
@article{osti_22365420,
```

title = {Magnetic Prandtl number dependence of the kinetic-to-magnetic dissipation ratio},

author = {Brandenburg, Axel and Department of Astronomy, Stockholm University, SE-10691 Stockholm},

abstractNote = {Using direct numerical simulations of three-dimensional hydromagnetic turbulence, either with helical or non-helical forcing, we show that the kinetic-to-magnetic energy dissipation ratio always increases with the magnetic Prandtl number, i.e., the ratio of kinematic viscosity to magnetic diffusivity. This dependence can always be approximated by a power law, but the exponent is not the same in all cases. For non-helical turbulence, the exponent is around 1/3, while for helical turbulence it is between 0.6 and 2/3. In the statistically steady state, the rate of energy conversion from kinetic into magnetic by the dynamo must be equal to the Joule dissipation rate. We emphasize that for both small-scale and large-scale dynamos, the efficiency of the energy conversion depends sensitively on the magnetic Prandtl number, and thus on the microphysical dissipation process. To understand this behavior, we also study shell models of turbulence and one-dimensional passive and active scalar models. We conclude that the magnetic Prandtl number dependence is qualitatively best reproduced in the one-dimensional model as a result of dissipation via localized Alfvén kinks.},

doi = {10.1088/0004-637X/791/1/12},

journal = {Astrophysical Journal},

number = 1,

volume = 791,

place = {United States},

year = {Sun Aug 10 00:00:00 EDT 2014},

month = {Sun Aug 10 00:00:00 EDT 2014}

}