The Elbert range of magnetostrophic convection. I. Linear theory
Susanne Horn, J. M. Aurnou
Abstract
In magnetostrophic rotating magnetoconvection, a fluid layer heated from below and cooled from above is equidominantly influenced by the Lorentz and the Coriolis forces. Strong rotation and magnetism each act separately to suppress thermal convective instability. However, when they act in concert and are near in strength, convective onset occurs at less extreme Rayleigh numbers ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> <mml:mi>a</mml:mi> </mml:math> , thermal forcing) in the form of a stationary, large-scale, inertia-less, inviscid magnetostrophic mode. Estimates suggest that planetary interiors are in magnetostrophic balance, fostering the idea that magnetostrophic flow optimizes dynamo generation. However, it is unclear if such a mono-modal theory is realistic in turbulent geophysical settings. Donna Elbert first discovered that there is a range of Ekman ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:math> , rotation) and Chandrasekhar ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> <mml:mi>h</mml:mi> </mml:math> , magnetism) numbers, in which stationary large-scale magnetostrophic and small-scale geostrophic modes coexist. We extend her work by differentiating five regimes of linear stationary rotating magnetoconvection and by deriving asymptotic solutions for the critical wavenumbers and Rayleigh numbers. Coexistence is permitted if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> <mml:mo><</mml:mo> <mml:mn>16</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>27</mml:mn> <mml:mi>π</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> <mml:mi>h</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>27</mml:mn> <mml:msup> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . The most geophysically relevant regime, the Elbert range , is bounded by the Elsasser numbers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mn>4</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mstyle> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>4</mml:mn> <mml:mn>4</mml:mn> </mml:msup> <mml:msup> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> </mml:mo> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>≤</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>≤</mml:mo> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>3</mml:mn> <mml:mn>4</mml:mn> </mml:msup> <mml:msup> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> . Laboratory and Earth’s core predictions both exhibit stationary, oscillatory, and wall-attached multi-modality within the Elbert range.