Magnetic Prandtl number

The Magnetic Prandtl number (Pr_m) is a dimensionless quantity occurring in magnetohydrodynamics which approximates the ratio of momentum diffusivity (viscosity) and magnetic diffusivity. It is defined as:

$\mathrm{Pr}_\mathrm{m} = \frac{\mathrm{Re_m}}{\mathrm{Re}} = \frac{\nu}{\eta} = \frac{\mbox{viscous diffusion rate}}{\mbox{magnetic diffusion rate}}$

where:

• Re_m is the magnetic Reynolds number
• Re is the Reynolds number
• ν is the momentum diffusivity (kinematic viscosity)
• η is the magnetic diffusivity

At the base of the Sun's convection zone the Magnetic Prandtl number is approximately 10⁻², and in the interiors of planets and in liquid-metal laboratory dynamos is approximately 10⁻⁵.

As another example, a magnetic Prandtl number of order unity is a widely accepted value for the thin accretion discs. Nevertheless this complicates the problem of launching jets: strong concentrations of magnetic field are required at the inner regions of the disc in order to launch jets, but a magnetic Prandtl number of order unity causes the field to diffuse away faster than it accretes inward.

References

• Critical Magnetic Prandtl Number for Small Scale Dynamo, Schekochihin, A.A., et al. 2004, Physical Review Letters, 92, 5