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
and by an additional 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
Show that this expression can be integrated to give
where is a constant.
Let . Demonstrate that, in the limit , the previous expression yields either
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).
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.
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
as , and
as . Demonstrate that the field configuration is tearing unstable (i.e., ) provided that , where
Show that .
where is the dynamic viscosity.
is the magnetic Prandtl number, and
is the viscous diffusion time.
whereas if then
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
Hence, deduce that if then .
Hence, deduce that if the second law of thermodynamics requires the positive root of this equation to be such that then
where is the downstream fast wave velocity.