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

- Demonstrate that
, and
, where
and
are defined in Equation (7.45).
- Demonstrate that Equation (7.65) can be rearranged to give
Let . Demonstrate that, in the limit , the previous expression yields either

- Derive expression (7.111) from Equations (7.107)-(7.110).
- 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 thatGiven that , show that

- Derive Equations (7.142) and (7.143) from Equations (7.139)-(7.141).
- Derive Equations (7.149) and (7.150) from Equations (7.142)-(7.148).
- Derive Equation (7.156) from Equations (7.151)-(7.155).
- Derive Equation (7.161) from Equations (7.156)-(7.159).
- Derive Equation (7.163) from Equation (7.161).
- Derive Equations (7.177) and (7.178) from Equations (7.171)-(7.176).
- Consider the linear tearing stability of the following field configuration,
- We can incorporate plasma viscosity into the linearized resistive-MHD equations, (7.172)-(7.175), by modifying Equation (7.173)
to read
- Show that, in this case, Equations (7.177) and (7.178) generalize to give

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

where*magnetic Prandtl number*, and - Show that the resistive layer equations, (7.187) and (7.188), generalize to give

- Show that the Fourier transformed resistive layer equation, (7.196), generalizes to give
- Finally, solve the Fourier transformed resistive layer equation to determine the layer matching
parameter,
. Demonstrate that if
then

- Show that, in this case, Equations (7.177) and (7.178) generalize to give
- Consider the effect of plasma viscosity on the Sweet-Parker reconnection scenario. The viscosity is conveniently parameterized in terms
of the magnetic Prandtl number

- Derive Equations (7.239)-(7.246) from the MHD equations, (7.1)-(7.4), and Maxwell's equations.
- Derive Equations (7.258)-(7.261) from the MHD Rankine-Hugoniot relations.
- Demonstrate that for a parallel MHD shock the downstream Mach number has the following relation to the upstream Mach number:
- Derive Equations (7.271)-(7.274) from the MHD Rankine-Hugoniot relations.
- Demonstrate that Equation (7.274) is equivalent to
- Derive Equations (7.280)-(7.286) from the MHD Rankine-Hugoniot relations combined with Equations (7.278) and (7.279).