The dynamics of two interacting compositional plumes in the presence of a magnetic field

Abstract

The dynamics of two compositionally buoyant columns of fluid rising in an infinite less buoyant fluid is studied in the presence of a uniform magnetic field, B ₀. The fluid is thermally stably stratified and has a viscosity, ν, a thermal diffusivity, κ and magnetic diffusivity, η. The stability of the mean state to infinitesimal disturbances is governed by the seven dimensionless parameters: the Reynolds number, R (=UL/ν, where U, L are characteristic velocity and length respectively) which measures the strength of the compositional buoyancy; the dimensionless measures x ₀, x ₁, d of the thickness of the two plumes and the distance between them, respectively; the ratio Γ of the strengths of the two plumes (as measured by their basic concentration of light material); the Chandrasekhar number, Qc (=, in which μ is the magnetic permeability, ρ₀ the fluid density and B ₀ a characteristic unit of magnetic field), is a measure of the magnitude of the magnetic field and the normalized horizontal projection of the magnetic field, where θ measures the inclination of the magnetic field to the vertical. The stability is examined for small values of R. The preferred mode of instability is studied in the parameter space (x ₀, x ₁, d, ). It is shown that the influence of the magnetic field does not change the order of the magnitude of the growth rate from O(R ⁰) of the two non-magnetic interacting plumes and it does not introduce any new modes to the stability problem. However, the presence of the magnetic field introduces novel features to the stability problem. For any fixed set x ₀, x ₁, d, Γ, Qc, the growth rate can either increase with or initially decrease reaching a minimum before it increases again. As Qc increases, with fixed, the growth rate can assume one of four different behaviours: (i) it maintains the same value of the non-magnetic case with the disturbance propagating along field lines; (ii) it decreases steadily with Qc; (iii) it maintains the same value as in the absence of the field until a value is reached when it starts to increase to a maximum before it decreases to zero for large values of Qc and (iv) it increases from its value for Qc = 0 reaching a maximum before it decreases steadily to zero at some value of Qc dependent on the other parameters. The helicity and α-effect have also been studied to find that the unstable motions can produce mean helicity and α-effect.

Keywords:

• Plumes
• Instability
• Helicity
• α-effect
• Dynamo action
• Earth's core

Acknowledgement

We thank the anonymous referees for their constructive suggestions on an earlier version of this article.