Next: Waves in Warm Plasmas Up: Magnetohydrodynamic Fluids Previous: Oblique MHD Shocks

# Exercises

1. We can add viscous effects to the MHD momentum equation by including a term , where is the dynamic viscosity, so that

Likewise, we can add finite conductivity effects to the Ohm's law by including the term , to give

Show that the modified dispersion relation for Alfvén waves can be obtained from the standard one by multiplying both and by a factor

If the finite conductivity and viscous corrections are small (i.e., and ), show that, for parallel ( ) propagation, the dispersion relation for the shear-Alfvén wave reduces to

2. Demonstrate that , and , where and are defined in Equation (7.45).

3. Demonstrate that Equation (7.65) can be rearranged to give

Show that this expression can be integrated to give

where is a constant.

Let . Demonstrate that, in the limit , the previous expression yields either

or

where is an arbitrary constant. Deduce that the former solution with the plus sign is such that is a monotonically increasing function of with as (this is a Class 2 solution); that the former solution with the minus sign is such that is a monotonically decreasing function of with as (this is a Class 3 solution); that the latter solution with is such that for all (this is a Class 1 solution); and that the latter solution with is such that for all (this is a Class 4 solution).

4. Derive expression (7.111) from Equations (7.107)-(7.110).

5. Consider a two-dimensional'' MHD fluid whose magnetic and velocity fields take the divergence-free forms

respectively, where and . Here, are standard Cartesian coordinates. Demonstrate from the MHD Ohm's law and Maxwell's equations that

where is the (spatially uniform) plasma resistivity. Hence, deduce that a two-dimensional poloidal'' magnetic field, , cannot be maintained against ohmic dissipation by dynamo action.

Given that , show that

Hence, deduce that a two-dimensional axial'' magnetic field, , cannot be maintained against ohmic dissipation by dynamo action.

6. Derive Equations (7.142) and (7.143) from Equations (7.139)-(7.141).

7. Derive Equations (7.149) and (7.150) from Equations (7.142)-(7.148).

8. Derive Equation (7.156) from Equations (7.151)-(7.155).

9. Derive Equation (7.161) from Equations (7.156)-(7.159).

10. Derive Equation (7.163) from Equation (7.161).

11. Derive Equations (7.177) and (7.178) from Equations (7.171)-(7.176).

12. Consider the linear tearing stability of the following field configuration,

This configuration is generated by a uniform, -directed current sheet of thickness , centered at . Solve the ideal-MHD equation, (7.186), subject to the constraints , and as . Here, . Hence, deduce that the tearing stability index for this configuration is

Show that

as , and

as . Demonstrate that the field configuration is tearing unstable (i.e., ) provided that , where

Show that .

13. We can incorporate plasma viscosity into the linearized resistive-MHD equations, (7.172)-(7.175), by modifying Equation (7.173) to read

where is the dynamic viscosity.

1. Show that, in this case, Equations (7.177) and (7.178) generalize to give

respectively.

2. Show that Equations (7.182) and (7.183) generalize to give

where

is the magnetic Prandtl number, and

is the viscous diffusion time.

3. Show that the resistive layer equations, (7.187) and (7.188), generalize to give

4. Show that the Fourier transformed resistive layer equation, (7.196), generalizes to give

5. Finally, solve the Fourier transformed resistive layer equation to determine the layer matching parameter, . Demonstrate that if then

whereas if then

14. Consider the effect of plasma viscosity on the Sweet-Parker reconnection scenario. The viscosity is conveniently parameterized in terms of the magnetic Prandtl number

where is the dynamic viscosity. Demonstrate that if then the conventional Sweet-Parker reconnection scenario remains valid, but that if then the scenario is modified such that

15. Derive Equations (7.239)-(7.246) from the MHD equations, (7.1)-(7.4), and Maxwell's equations.

16. Derive Equations (7.258)-(7.261) from the MHD Rankine-Hugoniot relations.

17. Demonstrate that for a parallel MHD shock the downstream Mach number has the following relation to the upstream Mach number:

Hence, deduce that if then .

18. Derive Equations (7.271)-(7.274) from the MHD Rankine-Hugoniot relations.

19. Demonstrate that Equation (7.274) is equivalent to

Hence, deduce that if the second law of thermodynamics requires the positive root of this equation to be such that then

that is,

where is the downstream fast wave velocity.

20. Derive Equations (7.280)-(7.286) from the MHD Rankine-Hugoniot relations combined with Equations (7.278) and (7.279).

Next: Waves in Warm Plasmas Up: Magnetohydrodynamic Fluids Previous: Oblique MHD Shocks
Richard Fitzpatrick 2016-01-23