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

209 Chapter 8 — Wave flowsIntegrat<strong>in</strong>g with respect to x∂B∂x − U η s<strong>in</strong> kx B = C 1. (8.56)The advection-diffusion balance is therefore described by a 1st order ord<strong>in</strong>ary differentialequation. The arbitrary constant C 1 will be zero if when x=0, dBdx = 0 also2 . If this isthe case then∂B∂x = U s<strong>in</strong> kx B. (8.57)ηSeparat<strong>in</strong>g the variables<strong>and</strong> then <strong>in</strong>tegrat<strong>in</strong>g∫ dBB = ∫ Uηs<strong>in</strong> kx dx, (8.58)ln B = − U kη cos kx + C 2, (8.59)so thatB(x, t) t→∞ = C 2 e − U kη cos kx . (8.60)To determ<strong>in</strong>e the amplitude of this steady state magnetic field, the constant C 2 must beevaluated <strong>and</strong> the <strong>in</strong>itial <strong>and</strong> steady state field strengths must be l<strong>in</strong>ked. This can beachieved follow<strong>in</strong>g the approach adopted by Clark (1965) <strong>and</strong> consider<strong>in</strong>g the <strong>in</strong>tegratedmagnetic field over one wavelength of the wave flowddtπ∫k− π kB(x, t) dx. (8.61)S<strong>in</strong>ce the velocity field u is periodic <strong>in</strong> x (wavelength 2π k ), if the <strong>in</strong>itial field B 0 is uniform,then the steady state field B <strong>and</strong> its derivatives with respect to x(∂B∂x<strong>and</strong> ∂2 B∂x 2 )also always be periodic with the same period as u <strong>and</strong> will <strong>in</strong>tegrate to zero over the<strong>in</strong>terval ( −πk , π k). Therefore, return<strong>in</strong>g to the time dependent <strong>in</strong>duction equationwill∂B∂t+ U s<strong>in</strong> kx∂B∂x + Uk cos kxB = η ∂2 B∂x 2 , (8.62)<strong>and</strong> <strong>in</strong>tegrat<strong>in</strong>g from x = −πkto x = π k leavesddtπ∫k− π kB(x, t)dx = 0. (8.63)Physically this says that the amount vertical magnetic flux with<strong>in</strong> one wavelength ofthe stationary wave pattern is conserved. This property then allows the flux with<strong>in</strong> a2 This cannot be rigorously established at this po<strong>in</strong>t. The analysis, however, proceeds on the assumptionthat C 1=0 <strong>and</strong> the assumption is justified a posteriori by the fact that the steady state solutionfound for B does <strong>in</strong>deed have this form.