Small-scale hydromagnetic flow in the Earth's core: Rise of a vertical buoyant plume

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Abstract
The steady and transient flow induced by a vertical cylinder of buoyant electrically conducting fluid immersed in an infinite extent of slightly denser fluid in the presence of a horizontal magnetic field is investigated, with the aim of elucidating the small-scale flow within the Earth's core. The evolution from a state of rest may be divided into three regimes. For short times [t < O(L 2μ[sgrave]) where L is the horizontal scale of the plume] Alfvén waves propagate a distance V A t along the magnetic field lines, accelerating the fluid to a speed of order (Δρ)gL/ρV A where V A = B/(ρμ)½ is the Alfvén speed and Δρ is the density deficit of the buoyant plume. For intermediate times [O(L 2μ[sgrave])t O(L 2/v)] lateral viscous diffusion also becomes important and a quasi-steady state is reached having rise speed of order (Δρ)gL/B(ρ[sgrave]v)½ and lateral extent of order L 2 B([sgrave]/ρv)½. For values of parameters thought to be relevant to the core, the short-time solution lasts roughly an hour and the intermediate-time solution lasts several decades. The long-time solution may not be relevant to the core as the fluid can rise to the top during the intermediate-time regime. The rise speeds associated with this plume flow may exceed those estimated from secular variation, but this result is sensitive to the size of the density deficit, which is poorly known, and to the particular orientation of the plume that has been chosen.

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