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

156 Chapter 6 — Theory(∇ 2 − P r ∂ )Θ = − ∂χ∂t ∂y , (6.56)()∇ 2 ∂− P r m j = − ∂∂t ∂y ∇2 χ. (6.57)Plane wave solutions for χ, j <strong>and</strong> Θ proportional to e i(kx+ky−ωt) (<strong>in</strong>variant <strong>in</strong> the ẑdirection <strong>and</strong> tak<strong>in</strong>g k to be an estimate of the wavenumber <strong>in</strong> both the x <strong>and</strong> y directionsfor simplicity) can then be substituted <strong>in</strong>to equations (6.55) to (6.57). This yield relationsbetween j <strong>and</strong> χ, <strong>and</strong> between Θ <strong>and</strong> χΘ =−ikχ−k 2 + iP rω<strong>and</strong> j =ik 3 χ−k 2 + iP r m ω , (6.58)which when substituted <strong>in</strong>to equation (6.55) give a dispersion relation <strong>in</strong> the complexfrequency ω,E(−iω + k 2 ) + iβ∗k = ERa−iP r ω + k 2 − Λk 2−iP r m ω + k 2 . (6.59)The first term comes from the <strong>in</strong>ertial <strong>and</strong> viscous forces, the second term comes from thevariable Coriolis force, the third term represents thermal (convective) forc<strong>in</strong>g, <strong>and</strong> thefourth term the <strong>in</strong>fluence of the uniform magnetic field. This dispersion relation is key tounderst<strong>and</strong><strong>in</strong>g <strong>and</strong> predict<strong>in</strong>g the properties of hydromagnetic wave driven by convection<strong>and</strong> is also capable of recover<strong>in</strong>g the dispersion relations for free <strong>waves</strong> deduced by Hide(1966) if convective forc<strong>in</strong>g <strong>and</strong> diffusion are ignored. By neglect<strong>in</strong>g terms <strong>in</strong> equation(6.59), simple explicit solutions can be found for ω which give <strong>in</strong>sight <strong>in</strong>to the types ofwave motion possible <strong>in</strong> this system:(i) Rossby <strong>waves</strong>.Ignor<strong>in</strong>g thermal driv<strong>in</strong>g <strong>and</strong> the <strong>in</strong>fluence of magnetic fields leav<strong>in</strong>g <strong>in</strong>ertia, viscosity<strong>and</strong> the β-effect <strong>in</strong> balance,E(−iω + k 2 ) = − iβ∗k=> ω = β∗Ek − ik2 E . (6.60)This is the viscous non-dimensionalisation dispersion relation expected for freeRossby <strong>waves</strong>, damped by viscous diffusion effects <strong>and</strong> travell<strong>in</strong>g <strong>in</strong> the prograde(eastward) direction <strong>in</strong> the annulus geometry which mimics the scenario outsidethe tangent cyl<strong>in</strong>der <strong>in</strong> a thick spherical shell.(ii) Hide’s MC-Rossby <strong>waves</strong>.Ignor<strong>in</strong>g viscosity, <strong>in</strong>ertia, thermal driv<strong>in</strong>g <strong>and</strong> magnetic diffusion, the β-effect isbalanced by the effect of changes <strong>in</strong> the magnetic field,iβ ∗k = Λk2−iP r m ω=> ω = − Λk3β ∗ P r m. (6.61)This is the dispersion relation expected for Hide’s unforced MC-Rossby <strong>waves</strong>, travell<strong>in</strong>g<strong>in</strong> the retrograde (westward) direction with frequency <strong>in</strong>versely proportionalto the β-effect <strong>and</strong> proportional to the square of the magnetic field strength.