Magnetic Fields in Stars - Math Tripos - Past Exam, Exams of Mathematics

Download Magnetic Fields in Stars - Math Tripos - Past Exam and more Exams Mathematics in PDF only on Docsity! MATHEMATICAL TRIPOS Part III Thursday 30 May 2002 9 to 12 PAPER 43 MAGNETIC FIELDS IN STARS Attempt THREE questions There are four questions in total The questions carry equal weight Candidates may bring their own notebooks into the examination You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 Explain the distinction between lack of equilibrium and instabilities associated with magnetic buoyancy. A plane stratified atmosphere contains a perfect gas with density ρ(z), pressure p(z) and temperature T (z), and a magnetic field B = B0(z)ŷ, referred to cartesian axes with the z-axis pointing vertically upward. Consider a bodily displacement of a horizontal flux tube in the xz-plane. Given that heat transfer is so efficient that the displaced tube always remains in thermal equilibrium with its surroundings, find the condition for the atmosphere to be unstable to interchanges driven by magnetic buoyancy. 2 A turbulent conducting fluid occupies an infinite conducting slab of depth d in the presence of a sheared horizontal velocity u = (πV z/d)ŷ, referred to cartesian co-ordinates with the z-axis vertical. The mean magnetic field does not vary in the y-direction and can be decomposed into a poloidal component in the xz-plane, described by a flux function A(x, z, t), and a toroidal component B(x, z, t)ŷ. The equations describing a kinematic αω-dynamo are ∂A ∂t = αB + η∇2A, ∂B ∂t = ( πV d ) ∂A ∂x + η∇2B. What is the significance of the various terms in these equations? Find the minimum value of the dynamo number D = αV d 2 π2η2 for which there exists an exponentially growing dynamo wave solution of the form A = Ã(t) exp[i(kx+ πz/d)], B = B̃(t) exp[i(kx+ πz/d)]. What is the corresponding critical value of the wave number k? Paper 43