# Astrophysical Fluid Dynamics – Problem Set 8

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Astrophysical Fluid Dynamics Problem Set 8 Readings Selections from Sturrock in the Course Reader Problem 1 Rigidly Rotating Magnetospheres A magnetized star is set spinning The immediate environoment of the star its magnetosphere where the magnetic field of the star dominates all other sources of energy density is filled with plasma This problem considers how magnetospheric plasma moves Work in the inertial lab frame in which the star is spinning and in cylindrical coordinates d z where d measures distance from the rotation axis of the star a Take the plasma to have effectively infinite electrical conductivity Write down an expression in terms of the velocity of the plasma v and the magnetic field B for the electric field E b Take the magnetosphere to be in steady state Use Faraday s law of induction to prove that E 0 c Combine a and b to deduce that the plasma s poloidal non velocity is parallel to the poloidal magnetic field d Use the induction equation for B t the absence of magnetic monopoles and the assumption that the flow is incompressible to find that B v v B B t 1 e Assume the plasma s poloidal velocity is zero as would be the case if the plasma is in hydrostatic equilibrium along a poloidal field line That is take v to be purely azimuthal r r r z r v v r 2 where r is the displacement vector from the origin and z is the local angular velocity of plasma At this stage there is no reason to believe that cannot vary arbitrarily with position i e the plasma might be differentially rotating in an arbitrary way In fact though part f will show that is not free to vary arbitrarily Prove from 1 the condition of steady state and the absence of magnetic monopoles that B Bd Bz 0 1 3 Note this is not the same as B 0 since unit vectors in cylindrical coordinates have non zero phi derivatives 0 That is the angular velocity of plasma is constant along a f Show that e implies B field line This result is Ferraro s 1937 Law of Iso Rotation Every field line is rooted on the star If the star is rigidly rotating then Ferraro s Law implies that magnetospheric plasma is also rigidly rotating at the same angular velocity of the star The plasma is definitely not moving on Kepler orbits Problem 2 Drift This problem affords another perspective on Ferraro s Law and magnetic B fields each oriented perpendicular to the other Consider uniform electric E a Show that the motion of a charged particle can be decomposed into a fast gyromotion about B plus a slow drift at velocity vD c B E B2 4 This is a standard result that can be found in many textbooks Please provide a complete derivation and also sketch the motions of both an electron and a proton annotating your sketch with directions and magnitudes of motion b Verify the validity of 4 in the context of problem 1 Problem 3 In the Beginning the Biermann Battery In class in deriving Ohm s Law the difference equation between the ion and electron momentum equations we dropped a term that looked like ene hwer wes i xs 5 Restore this term but keep only the diagonal components of the electron stress tensor i e set ne me hwer wes i Pe rs where me is the mass of the electron Pe is the electron partial pressure and rs is the Kronecker delta In other words ignore the viscous contribution to the stress tensor and keep only the pressure contribution 2 Derive the slightly more general form of the induction equation 2 B c 2 B c Pe ne v B t 4 n2e e 6 The last term on the right hand side is the term introduced by Ludwig Biermann to generate magnetic fields from material that is initially field free It relies upon the contours of constant electron pressure being misaligned with the contours of constant electron density to pressureaccelerate electrons away from ions in such a way as to produce current which in turn produces field The action is similar to that of a battery which relies on chemical reactions to accelerate electrons away from ions and for this reason we call this term the Biermann battery term It has been invoked to generate truly primordial magnetic field in proto galactic plasma see Kulsrud s 1997 article in Critical Dialogues in Cosmology 3