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Large-scale dynamos at low magnetic Prandtl numbers

Explore the dynamics of large-scale and small-scale dynamos at low Prandtl numbers, above, below, and inside the lab, revealing insights into dynamo excitation and energy dissipation. Key studies by Axel Brandenburg, Novikov, Rogachevskii, Boldyrev, Ponty, Schekochihin, and more highlight critical relationships between Rm, PrM, and dynamo behavior. Investigate the impact on stars, discs, turbulence, and energy spectra in various dynamo regimes. Gain insights into the distinctions between large-scale and small-scale dynamos and their implications for astrophysical and laboratory systems.

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Large-scale dynamos at low magnetic Prandtl numbers

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  1. Large-scale dynamos at low magnetic Prandtl numbers above, below, and inside the lab: PrM=n/h~10-5 • Small-scale dynamos • Progressively harder to excite at low PrM • But may level off … • Large-scale dynamos • Independent of PrM • Low PrM can be used to “filter” out SS dynamo • Most of energy dissipated Ohmically • Can decrease n even further Axel Brandenburg (Nordita, Stockholm)

  2. Winter School 11-22 January

  3. Small-scale vs large-scale dynamo

  4. Low PrM results • Small-scale dynamo: Rm.crit=35-70 for PrM=1 (Novikov, Ruzmaikin, Sokoloff 1983) • Leorat et al (1981): independent of PrM (EDQNM) • Rogachevskii & Kleeorin (1997): Rm,crit=412 • Boldyrev & Cattaneo (2004): relation to roughness • Ponty et al.: (2005): levels off at PrM=0.2

  5. Maybe no small scale “surface” dynamo? Small PrM=n/h: stars and discs around NSs and YSOs Schekochihin et al (2005) k Boldyrev & Cattaneo (2004)

  6. Levels off for Taylor-Green flow • Confirmation for finite Rm for SS dynamo? • Or effect of LS dynamo?

  7. Hyperviscous, Smagorinsky, normal height of bottleneck increased Haugen & Brandenburg (PRE, astro-ph/0402301) onset of bottleneck at same position Inertial range unaffected by artificial diffusion

  8. Re-appearence at low PrM Gap between 0.05 and 0.2 ? Iskakov et al (2005)

  9. Fully helical turbulence Here: Rm=urmsl/h Brandenburg (2001, ApJ)

  10. ABC flow dynamo • Rm,crit varies still by factor 2 • Spectral magnetic energy peaks at k=1 Mininni et al. (2007, PRE)

  11. Cartesian box MHD equations Magn. Vector potential Induction Equation: Momentum and Continuity eqns Viscous force forcing function (eigenfunction of curl)

  12. Growth rate • Growth rate scaling for large Rm as for SS dynamo • Helical dynamo still excited for low Rm

  13. Kinematic regime

  14. Kinematic vs saturated regime

  15. Spectra in kinematic regime • Kazantsev scaling for PrM=1 • Progressively more energy at large scale

  16. Compensated spectra kinematic saturated

  17. Low PrM dynamoswith helicity do work • Energy dissipation via Joule • Viscous dissipation weak • Can increase Re substantially!

  18. PrM=1, saturated case

  19. U and B fields: minor changes

  20. Conclusions 1) low PrM helps to distinguish LS and SS dynamos • LS dynamo must be excited • SS dynamo too dominant, swamps LS field • Dominant SS dynamo: artifact of large PrM=n/h Brun, Miesch, & Toomre (2004, ApJ 614, 1073) 2) Important also for accretion disc dynamos

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