4. Magnetic energy in the corona. Leading theories indicate that magnetic processes may be responsible for heating the coronal plasma. Before exploring these processes, it is worth investigating how magnetic energy is transported up to the corona. Recall the Poynting flux, S = -EXB HO which gives the flux of electromagnetic energy in an MHD plasma. For a plane-parallel solar atmosphere, let's look at S in Cartesian coordinates. Specifically, we will evaluate its z-component, which tells us how electromagnetic energy is going upwards from the photosphere to the corona. (a) Consider a magnetic field B that is only in the 2-z plane (i.e., it has By = 0). For ideal MHD conditions (i.e., zero resistivity n), use the generalized Ohm's law to derive the expression: Sx = = (U_B} uBB:) HO where u is the MHD fluid velocity vector. (b) We will consider two geometries for B and u: MHD waves? z emerging bipoles? at apex, |B= B, | BB2 (but Bx 70) w UT First consider the left-hand panel of the above cartoon. When the field B is mostly vertical, and is being shaken back and forth by purely horizontal motions i.e., U +0), MHD waves propagate up along the field. We can make the two following two approximations about these motions: 1. Assume that the kinetic energy density of the horizontal motions is in "equilibrium" with the horizontal magnetic energy density, i.e., plug 2po 2. Assume that we know the Alfvn speed Va of the corona, where VA = VA - /B/ PMO and this is the speed at which perturbations to the magnetic field propagate up along the field. Using the above information, solve for the magnitude |S2| as a function of only densities and velocities. c) Consider the right-hand panel of the cartoon. If the solar surface is covered in emerging bipoles of magnetic field, then the velocities are all vertical (uz # 0) and the field at the apex of the loop is all horizontal. In this case, write the magnitude Szas a function of only densities and velocities. (d) Compute real numbers for (Sz| for the cases of parts (b) and (c), given the following values: The coronal density is p = 10-12 kg m-3. The coronal Alfvn speed is VA ~ 2000 km/s. In the corona, MHD waves appear to be shaken back and forth with Ug 30 km/s. Bipoles emerge very slowly. Observational estimates for their rising speed are uncertain. Some observers say uz 0.1 km/s, and others say uz 1 km/s. (e) Which of the two proposed mechanisms do you think is more important for supplying magnetic energy to the corona? 4. Magnetic energy in the corona. Leading theories indicate that magnetic processes may be responsible for heating the coronal plasma. Before exploring these processes, it is worth investigating how magnetic energy is transported up to the corona. Recall the Poynting flux, S = -EXB HO which gives the flux of electromagnetic energy in an MHD plasma. For a plane-parallel solar atmosphere, let's look at S in Cartesian coordinates. Specifically, we will evaluate its z-component, which tells us how electromagnetic energy is going upwards from the photosphere to the corona. (a) Consider a magnetic field B that is only in the 2-z plane (i.e., it has By = 0). For ideal MHD conditions (i.e., zero resistivity n), use the generalized Ohm's law to derive the expression: Sx = = (U_B} uBB:) HO where u is the MHD fluid velocity vector. (b) We will consider two geometries for B and u: MHD waves? z emerging bipoles? at apex, |B= B, | BB2 (but Bx 70) w UT First consider the left-hand panel of the above cartoon. When the field B is mostly vertical, and is being shaken back and forth by purely horizontal motions i.e., U +0), MHD waves propagate up along the field. We can make the two following two approximations about these motions: 1. Assume that the kinetic energy density of the horizontal motions is in "equilibrium" with the horizontal magnetic energy density, i.e., plug 2po 2. Assume that we know the Alfvn speed Va of the corona, where VA = VA - /B/ PMO and this is the speed at which perturbations to the magnetic field propagate up along the field. Using the above information, solve for the magnitude |S2| as a function of only densities and velocities. c) Consider the right-hand panel of the cartoon. If the solar surface is covered in emerging bipoles of magnetic field, then the velocities are all vertical (uz # 0) and the field at the apex of the loop is all horizontal. In this case, write the magnitude Szas a function of only densities and velocities. (d) Compute real numbers for (Sz| for the cases of parts (b) and (c), given the following values: The coronal density is p = 10-12 kg m-3. The coronal Alfvn speed is VA ~ 2000 km/s. In the corona, MHD waves appear to be shaken back and forth with Ug 30 km/s. Bipoles emerge very slowly. Observational estimates for their rising speed are uncertain. Some observers say uz 0.1 km/s, and others say uz 1 km/s. (e) Which of the two proposed mechanisms do you think is more important for supplying magnetic energy to the corona