Bounds for inertialess dynamo — Draconitic Research

Abstract

We derive necessary conditions for instantaneous dynamo action for rotating convection. A magnetohydrodynamic model is considered in two settings: the rapidly rotating plane layer where inertia and viscosity are absent, and at an arbitrary rotation rate where viscosity is finite. In contrast to kinematic dynamo bounds, the evolution of the magnetic field is coupled via an inertialess force balance. The buoyancy-driven part of the flow u^A in the event of dynamo action must in fact satisfy, for 3≤ p ≤ ∞ Rm A_p| u^A|_L^p ≥ 1 where A_p is an explicit constant, and Rm is the magnetic Reynolds number. In the inviscid model, u^A depends only on the horizontal gradients of the vertical primitive of temperature. A refinement via the poloidal-toroidal decomposition allows us to replace L^p in our constraint with an anisotropic norm for L^∞_z Ḣ¹_x,y. For the viscous model, we also derive necessary conditions for the growth of magnetic enstrophy and a combined thermo-magnetic energy. One branch of our constraints implies that the scaling Ra_ν≳ Ek^−3/2 is necessary for dynamo action, where Ra_ν is the classical Rayleigh number and Ek is the Ekman number.

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BibTeX

@misc{arslan2026bounds,
  title = {{Bounds for inertialess dynamo}},
  author = {Ali Arslan and Hezekiah Grayer II},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28584},
  eprint = {2608.28584},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28584},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Bounds for inertialess dynamo
AU  - Ali Arslan
AU  - Hezekiah Grayer II
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28584
UR  - https://arxiv.org/abs/2608.28584
AB  - We derive necessary conditions for instantaneous dynamo action for rotating convection. A magnetohydrodynamic model is considered in two settings: the rapidly rotating plane layer where inertia and viscosity are absent, and at an arbitrary rotation rate where viscosity is finite. In contrast to kinematic dynamo bounds, the evolution of the magnetic field is coupled via an inertialess force balance. The buoyancy-driven part of the flow u^A in the event of dynamo action must in fact satisfy, for 3≤ p ≤ ∞ Rm A_p| u^A|_L^p ≥ 1 where A_p is an explicit constant, and Rm is the magnetic Reynolds number. In the inviscid model, u^A depends only on the horizontal gradients of the vertical primitive of temperature. A refinement via the poloidal-toroidal decomposition allows us to replace L^p in our constraint with an anisotropic norm for L^∞_z Ḣ^1_x,y. For the viscous model, we also derive necessary conditions for the growth of magnetic enstrophy and a combined thermo-magnetic energy. One branch of our constraints implies that the scaling Ra_ν≳ Ek^−3/2 is necessary for dynamo action, where Ra_ν is the classical Rayleigh number and Ek is the Ekman number.
ER  -

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