Main list
1. Group polar. Draw a group polar (i.e., the polar diagram of the group velocity dependence on the angle between magnetic field and the group velocity) for VA near cs.
Hint: Use the dispersion relations ω=ω(k) for the ideal MHD modes (Alfvenic, fast and slow magnetosonic) giving the expression for the group velocity.
2. MRI (magneto-rotational instability). Obtain the MRI dispersion relation (ω2=ω2(k) ) and show that the instability does exist.
Hint: Consider the Alfvenic mode only (the density remains constant). In the zero approximation there is Keplerian rotation, and magnetic field is perpendicular to the disk plane (Bz only). Disturbances of the velocity and magnetic field have the azimuthal components only, the wave vector is in the z direction.

Additional list
1. How to distinguish the gravitational red shift and the proper motion for an individual White Dwarf?
2. Model the cylindrical Sun spot within the force-free approximation.
Hint: solve the Laplace equation in the cylindrical geometry, using only one cylindrical basis function for simplicity .

## School of modern astrophysics (SOMA - 2014)

## Problems

Main list1.

Group polar.Draw a group polar (i.e., the polar diagram of the group velocity dependence on the angle between magnetic field and the group velocity) for VA near cs.Hint: Use the dispersion relations ω=ω(k) for the ideal MHD modes (Alfvenic, fast and slow magnetosonic) giving the expression for the group velocity.

2.

MRI (magneto-rotational instability).Obtain the MRI dispersion relation (ω2=ω2(k) ) and show that the instability does exist.Hint: Consider the Alfvenic mode only (the density remains constant). In the zero approximation there is Keplerian rotation, and magnetic field is perpendicular to the disk plane (Bz only). Disturbances of the velocity and magnetic field have the azimuthal components only, the wave vector is in the z direction.

Additional list1. How to distinguish the gravitational red shift and the proper motion for an individual White Dwarf?

2. Model the cylindrical Sun spot within the force-free approximation.

Hint: solve the Laplace equation in the cylindrical geometry, using only one cylindrical basis function for simplicity .