Hydromagnetic waves in Earth's core and their influence on ...

167 Chapter 6 — Theorydue to the tilt of the convective columns, be<strong>in</strong>g westward near the tangent cyl<strong>in</strong>der,but eastward further from the axis. With a strong magnetic field they found that mostconvection occurs <strong>in</strong>side the tangent cyl<strong>in</strong>der, while outside there was a s<strong>in</strong>gle, large-scaleplume.Card<strong>in</strong> <strong>and</strong> Olson (1995) used the QG approximation to study fully non-l<strong>in</strong>ear magnetoconvectionfor P r = 7, P m = 0.1, E = 10 −4 with Ra = 3Ra c <strong>and</strong> 50Ra c , <strong>and</strong> Λ between10 −4 <strong>and</strong> 2. For weak forc<strong>in</strong>g, they found Reynolds stresses due to the prograde tiltof convective columns drove azimuthal flows, but that a strong magnetic field reducedthe wavenumber of convection <strong>and</strong> suppressed azimuthal flows. For strong forc<strong>in</strong>g, theReynolds stresses (rather than viscous stresses) appeared to determ<strong>in</strong>e the vortex scalewith large eddies be<strong>in</strong>g formed. Unfortunately, the imposed toroidal field chosen <strong>in</strong> thisstudy is s<strong>in</strong>gular at the orig<strong>in</strong> cast<strong>in</strong>g some doubt on the results obta<strong>in</strong>ed (Fearn, 1998).Zhang (1999) studied nonl<strong>in</strong>ear rotat<strong>in</strong>g magnetoconvection <strong>in</strong> a spherical shell, us<strong>in</strong>gthe same toroidal field as Zhang (1995), but neglect<strong>in</strong>g the <strong>in</strong>ertial terms <strong>and</strong> study<strong>in</strong>gthermally driven <strong>waves</strong> at Ra up to 3 times critical, E = 1 × 10 −3 <strong>and</strong> Λ = 10.Solutions were obta<strong>in</strong>ed by <strong>in</strong>tegrat<strong>in</strong>g the equations numerically over many magneticdiffusion times. He found two bifurcations, the first be<strong>in</strong>g the onset of steadily travell<strong>in</strong>gmagneto-convective <strong>waves</strong> with equatorial <strong>and</strong> azimuthal symmetry, the second be<strong>in</strong>gthe simultaneously break<strong>in</strong>g temporal <strong>and</strong> azimuthal symmetry lead<strong>in</strong>g to vacillat<strong>in</strong>gmagnetoconvection (for further discussion, consult Zhang <strong>and</strong> Schubert (2000)).Sakuraba <strong>and</strong> Kono (2000) studied 3D, fully non-l<strong>in</strong>ear, magnetoconvection <strong>in</strong> a rapidlyrotat<strong>in</strong>g spherical shell, with an imposed uniform magnetic field parallel to the rotationaxis <strong>and</strong> E = 2 × 10 −5 . They found a transition from a magnetically-modified thermalRossby wave to a thermal magneto-Rossby wave at Λ ∼ 1. This transition was associatedwith an <strong>in</strong>crease <strong>in</strong> the toroidal magnetic field energy, a decrease <strong>in</strong> the azimuthalwavenumber of the mode <strong>and</strong> the conf<strong>in</strong>ement of magnetic flux <strong>in</strong>to anti-cyclonic rolls,especially <strong>in</strong> high amplitude spots <strong>in</strong> the equatorial region at the outer boundary.Gillet et al. (2005) have carried out a comprehensive experimental <strong>and</strong> numerical <strong>in</strong>vestigation<strong>in</strong>to the properties of a turbulent flow produced by the nonl<strong>in</strong>ear developmentof magnetically-modified thermal Rossby <strong>waves</strong>. This study is limited to weak magneticfields (Λ