Exercises

1. We can add viscous effects to the MHD momentum equation by including a term $\mu\,\nabla^2 {\bf V}$, where $\mu$ is the dynamic viscosity, so that
   $$\rho\,\frac{d{\bf V}}{dt} = {\bf j}\times {\bf b} -\nabla p + \mu\,\nabla^2{\bf V}.$$
   Likewise, we can add finite conductivity effects to the Ohm's law by including the term $(1/\mu_0\,\sigma)\,\nabla^2{\bf B}$, to give
   $$\frac{\partial {\bf B}}{\partial t} = \nabla\times ({\bf V}\times {\bf B}) + \frac{1}{\mu_0\,\sigma}\,\nabla^2{\bf B},$$
   Show that the modified dispersion relation for Alfvén waves can be obtained from the standard one by multiplying both $\omega^{2}$ and $V_S^{2}$ by a factor
   $$[1+{\rm i}\,k^{2}/(\mu_0\,\sigma\,\omega)],$$
   and $\omega^{2}$ by an additional factor
   $$[1+{\rm i}\,\mu\,k^{2}/(\rho_0\,\omega)].$$

   If the finite conductivity and viscous corrections are small (i.e., $\sigma\rightarrow\infty$ and $\mu\rightarrow 0$), show that, for parallel ($\theta=0$) propagation, the dispersion relation for the shear-Alfvén wave reduces to

   $$k\simeq \frac{\omega}{V_A} + {\rm i}\,\frac{\omega^2}{2\,V_A^{3}}\left(\frac{1}{\mu_0\,\sigma} + \frac{\mu}{\rho_0}\right).$$

2. Demonstrate that $V_+> V_S\,\cos\theta$, and $V_-<V_S\,\cos\theta$, where $V_+$ and $V_-$ are defined in Equation (8.45).

3. Demonstrate that Equation (8.65) can be rearranged to give
   $$\frac{du^2}{dr}\left(1-\frac{u_c^{2}}{u^2}\right) = \frac{4\,u_c^{2}}{r} \left(1-\frac{r_c}{r}\right),$$
   Show that this expression can be integrated to give
   $$\left(\frac{u}{u_c}\right)^2 -\ln\left(\frac{u}{u_c}\right)^2 = 4\,\ln\left(\frac{r}{r_c}\right) + 4\,\frac{r_c}{r} + C,$$
   where $C$ is a constant.

   Let $r/r_c =1 +x$. Demonstrate that, in the limit $\vert x\vert\ll 1$, the previous expression yields either

   $$u^2= u_c^{2}\left[1\pm 2\,x + {\cal O}(x^2)\right]$$
   or
   $$u^2 = u_0^{2}\left[1+\frac{2\,u_c^{2}\,x^2}{u_0^{2}-u_c^{2}} + {\cal O}(x^3)\right],$$
   where $u_0\neq u_c$ is an arbitrary constant. Deduce that the former solution with the plus sign is such that $u$ is a monotonically increasing function of $r$ with $u\lessgtr u_c$ as $r\lessgtr r_c$ (this is a Class 2 solution); that the former solution with the minus sign is such that $u$ is a monotonically decreasing function of $r$ with $u\gtrless u_c$ as $r\lessgtr r_c$ (this is a Class 3 solution); that the latter solution with $u_0 < u_c$ is such that $u<u_c$ for all $r$ (this is a Class 1 solution); and that the latter solution with $u_0>u_c$ is such that $u>u_c$ for all $r$ (this is a Class 4 solution).

4. Derive expression (8.111) from Equations (8.107)–(8.110).

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

   $${\bf B} = \nabla\psi\times{\bf e}_z + B_z\,{\bf e}_z,$$

   $${\bf V} =\nabla\phi\times {\bf e}_z+ V_z\,{\bf e}_z,$$

   respectively, where $\psi=\psi(x,y)$ and $\phi=\phi(x,y)$. Here, $(x,\,y,\,z)$ are standard Cartesian coordinates. Demonstrate from the MHD Ohm's law and Maxwell's equations that
   $$\frac{d}{dt}\!\int \psi^{2}\,dx\,dy=-\frac{2\,\eta}{\mu_0}\int\!\int \vert\nabla\psi\vert^{2}\,dx\,dy,$$
   where $\eta$ is the (spatially uniform) plasma resistivity. Hence, deduce that a two-dimensional “poloidal” magnetic field, ${\bf B}_p=\nabla\psi\times {\bf e}_z$, cannot be maintained against ohmic dissipation by dynamo action.

   Given that ${\bf B}_p={\bf0}$, show that

   $$\frac{d}{dt}\!\int B_z^{2}\,dx\,dy = -\frac{2\,\eta}{\mu_0}\int\!\int \vert\nabla B_z\vert^{2}\,dx\,dy.$$
   Hence, deduce that a two-dimensional “axial” magnetic field, ${\bf B}_t=B_z\,{\bf e}_z$, cannot be maintained against ohmic dissipation by dynamo action.

6. Derive Equations (8.142) and (8.143) from Equations (8.139)–(8.141).

7. Derive Equations (8.149) and (8.150) from Equations (8.142)–(8.148).

8. Derive Equation (8.156) from Equations (8.151)–(8.155).

9. Derive Equation (8.161) from Equations (8.156)–(8.159).

10. Derive Equation (8.163) from Equation (8.161).

11. Derive Equations (8.169)–(8.176) from the MHD equations, (8.1)–(8.4), and Maxwell's equations.

12. Derive Equations (8.188)–(8.191) from the MHD Rankine-Hugoniot relations.

13. Demonstrate that for a parallel MHD shock the downstream Mach number has the following relation to the upstream Mach number:
    $$M_2 = \left[\frac{2+({\mit\Gamma}-1)\,M_1^{2}}{2\,{\mit\Gamma}\,M_1^{2} - ({\mit\Gamma}-1)}\right]^{1/2}.$$
    Hence, deduce that if $M_1>1$ then $M_2 < 1$.

14. Derive Equations (8.201)–(8.204) from the MHD Rankine-Hugoniot relations.

15. Demonstrate that Equation (8.204) is equivalent to
    $$- {\mit\Gamma}\,({\mit\Gamma}+1)\,\beta_2\,M_2^{2}\,r^{2} + {\mit... ...\beta_2)+ ({\mit\Gamma}-1)\,\beta_2\,M_2^{2}\right] r+2\,(2-{\mit\Gamma})\}=0.$$
    Hence, deduce that if the second law of thermodynamics requires the positive root of this equation to be such that $r>1$ then
    $$M_2^{2} < 1+ \frac{2}{{\mit\Gamma}\,\beta_2}:$$
    that is,
    $$V_2 < V_{+\,2},$$
    where $V_{+\,2}=(V_{S\,2}^{2}+V_{A\,2}^{2})^{1/2}$ is the downstream fast wave velocity.

16. Derive Equations (8.210)–(8.216) from the MHD Rankine-Hugoniot relations combined with Equations (8.208) and (8.209).