200C, Winter 2009, First Homework Set1. Polarization drift (time-varying electric field). Consider an electric field that varies in time, starting from zero and increasing linearly, but very slowly compared to the gyroperiod.First sketch the ion and electron orbits and show qualitatively why the particles are slowly drifting in opposite directions due to the varying field, while they are moving in the same direction due to the instantaneous electric field. Show that the polarization drift due to the time-varying electric field is perpendicular to the ExB drift.Derive the polarization drift in a simple manner by conservation of energy: As the particle drifts slowly along the electric field, it gains kinetic energy ½ mvE2. The rate of energy gain is equal to the force applied times the particle drift in the direction of the force. Use this equation to compute the polarization drift.2. Bounce-average motion in constant external E, B fields. Consider a particle drifting in a modified (distorted) but planetary field B, which is constant in time, subject to an externally imposed planetary electric field E (e.g., corotation, convection). The particle motion is tracked by following the equatorial crossing of the gyrocenter, r, drifting at a “bounce-averaged” motion of its equatorial guiding center, vgc. The particle equatorial kinetic energy is a function of positionand the two adiabatic invariants: W(r, , J) and the potential energy is q, E=-. Show that for the bounce-averaged sum of the electric and gradient/curvature motion, the total guiding center motion is vgc = vE + vGC =(B( q+W))/qB2, i.e., the particles follow equatorial contours of constant total energy Wtot (kinetic + potential) as follows:First show that in an arbitrary, static E, B configuration the gradient/curvature bounce-averaged drifts are given by: vGC = (zW)/qB, i.e. the particle moves in an “effective” potential W/q. If no electric field is present the bounce-averaged guiding center motion is along contours of constant W, preserving the particle kinetic energy. To do that consider conservation of the particle energy in the equatorial plane: (vgc )Wtot= (vE + vGC) (q+W)=0, and use the fact that since this must be satisfied for arbitrary potentials, including =0, it must be: vGC =c (zW), where c is a constant – then determine the constant. Next show that in the electrostatic potential and W/q add up as in vgc above. 3. Quasi-neutrality is an important property of space plasmas, where the difference in density between the ions and electrons responsible for any space charge is much less than the number density. As an example top show this, consider the electric field associated with a corotating plasma. For simplicity assume the magnetic field is represented by a dipole and the dipole is oriented along the spin axis. First sketch the dipole field lines, and show that for the Earth, the corotation electric field requires a negative space charge near the equatorial plane, and a positive space charge above the magnetic pole.Second, estimate the electric field in the equatorial plane, and show that for the Earth the electric field is ~ 14(RE/R)2 mV/m, assuming an equatorial magnetic field of 30,000 nT at the surface of the Earth [Here RE is the radius of the earth, and R is the radial distance[.Third, assuming that the scale size for variation of the electric field is of order the radial distance show that the corotation electric field requires a density of ~ 2 x10–7 electrons/cm3. Compare this to typical magnetospheric plasma densities.Bonus question: Determine the electron density exactly for an axisymmetric dipole, and comparethis to the estimate given in the third part of the question.4. Derive a general form for the MHD equations, based on integrals over velocity space of moments of the Vlasov equation. Discuss the open-ended nature of the resultant equations, and specifically discuss how the energy equation can be closed [Hint, only the heat flux cannot be expressed in terms of lower order moments]. What assumption is required for the energy equation to reduce to the adiabatic form P– = constant?5. Consider a plasma that contains a uniform magnetic field in which the plasma is flowing radially outward (i.e., cylindrically symmetric flow). Show that in this simple geometry B dA/dt + A dB/dt = 0, where B is the magnetic field strength and A is the area enclosed by circle expanding with the flow. Discuss why this is equivalent to the field lines being frozen to the