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 (7.45).

3. Demonstrate that Equation (7.65) 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 (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

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

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

   
   
   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 (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,
    $$F(\bar{x}) = \left\{\begin{array}{lcl} F'(0)\,\bar{x}&\mbox{\hspa... ... [0.5ex] F'(0)\,{\rm sgn}(\bar{x})&&\vert\bar{x}\vert\geq 1\end{array}\right..$$
    This configuration is generated by a uniform, $z$ -directed current sheet of thickness $a$ , centered at $x=0$ . Solve the ideal-MHD equation, (7.186), subject to the constraints $\psi(-\bar{x})=\psi(\bar{x})$ , and $\psi(\bar{x})\rightarrow 0$ as $\vert\bar{x}\vert\rightarrow\infty$ . Here, $\bar{x}=x/a$ . Hence, deduce that the tearing stability index for this configuration is
    $${\mit\Delta}' = \frac{2\,\bar{k}}{\tanh(\bar{k})}\left[\frac{\bar{k} + \bar{k}\,\tanh(\bar{k})-1}{1-\bar{k}/\tanh(\bar{k})-\bar{k}}\right].$$
    Show that
    $${\mit\Delta}' \rightarrow \frac{2}{\bar{k}} -\frac{8}{3}+{\cal O}(\bar{k})$$
    as $\bar{k}\rightarrow 0$ , and
    $${\mit\Delta}'\rightarrow -2\,\bar{k} + 2\left[1+\frac{1}{2\,\bar{k}}+{\cal O}\left(\frac{1}{\bar{k}^{\,2}}\right)\right]\exp(-2\,\bar{k})$$
    as $\bar{k}\rightarrow\infty$ . Demonstrate that the field configuration is tearing unstable (i.e., ${\mit\Delta}'>0$ ) provided that $\bar{k}<\bar{k}_c$ , where
    $$\bar{k}_c\,[1+\tanh(\bar{k}_c) ]= 1.$$
    Show that $\bar{k}_c=0.639$ .

13. We can incorporate plasma viscosity into the linearized resistive-MHD equations, (7.172)-(7.175), by modifying Equation (7.173) to read
    $$\rho_0\,\frac{\partial{\bf V}}{\partial t} = -\nabla p + \frac{(... ...0} + \frac{(\nabla\times{\bf B}_0)\times{\bf B}}{\mu_0}+\mu\,\nabla^2{\bf V},$$
    where $\mu$ is the dynamic viscosity.
    1. Show that, in this case, Equations (7.177) and (7.178) generalize to give

       $$\gamma\,B_x = {\rm i}\,k\,B_{0\,y} \,V_x + \frac{\eta}{\mu_0} \left(\frac{d^2}{dx^2} - k^2\right) B_x,$$

       $$\gamma\,\rho_0\,\left(\frac{d^2}{dx^2}-k^2\right) V_x = \frac{{\rm i}\,k\,B_{0\,y}}{\mu_0}\left( \frac{d^2}{dx^2} - k^2... ...ac{B_{0\,y}''}{B_{0\,y}}\right) B_x+\mu\left(\frac{d^2}{dx^2}-k^2\right)^2 V_x,$$

       
       
       respectively.

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

       $$\bar{\gamma}\,(\psi-F\,\phi) = S^{-1}\left(\frac{d^2}{d\bar{x}^{\,2}}-\bar{k}^{\,2}\right) \psi,$$

       $$\bar{\gamma}^{\,2}\left(\frac{d^2}{d\bar{x}^{\,2}} -\bar{k}^{\,2}\right)\phi = -\bar{k}^{\,2}\, F \left( \frac{d^2}{d\bar{x}^{\,2}} -\bar{k}^{... ...{\gamma}\,S^{-1}\,P\left(\frac{d^2}{d\bar{x}^{\,2}}-\bar{k}^{\,2}\right)^2\phi,$$

       
       
       where
       $$P = \frac{\tau_R}{\tau_M}$$
       is the magnetic Prandtl number, and
       $$\tau_M = \frac{\rho_0\,a^2}{\mu}$$
       is the viscous diffusion time.

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

       $$\bar{\gamma}\,(\psi - \bar{x}\,\phi) = S^{-1}\frac{d^2\psi}{d\bar{x}^{\,2}},$$

       $$\bar{\gamma}^{\,2}\,\frac{d^2\phi}{d\bar{x}^{\,2}} = - \bar{x}\,\frac{d^2\psi}{d\bar{x}^{\,2}}+ \bar{\gamma}\,S^{-1}\,P\,\frac{d^4\phi}{d\bar{x}^{\,4}}.$$

       
       

    4. Show that the Fourier transformed resistive layer equation, (7.196), generalizes to give
       $$\frac{d}{dt}\!\left(\frac{t^{\,2}}{Q+t^{\,2}}\frac{d\skew{3}\hat{\phi}}{dt}\right) -(Q\,t^{\,2}+ P\,t^{\,4})\,\skew{3}\hat{\phi} = 0.$$

    5. Finally, solve the Fourier transformed resistive layer equation to determine the layer matching parameter, ${\mit\Delta}$ . Demonstrate that if $1\gg Q\gg P^{\,2/3}$ then
       $${\mit\Delta} = 2\pi\,\frac{{\mit\Gamma}(3/4)}{{\mit\Gamma}(1/4)}\,S^{1/3}\,Q^{\,5/4},$$
       whereas if $Q\ll P^{\,-1/3}, P^{\,2/3}$ then
       $${\mit\Delta} = 6^{2/3}\,\pi\,\frac{{\mit\Gamma}(5/6)}{{\mit\Gamma}(1/6)}\,S^{1/3}\,Q\,P^{\,1/6}.$$

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
    $$P = \frac{\mu_0\,\mu}{\eta\,\rho},$$
    where $\mu$ is the dynamic viscosity. Demonstrate that if $P\ll 1$ then the conventional Sweet-Parker reconnection scenario remains valid, but that if $P\gg 1$ then the scenario is modified such that

    $$\frac{v_\ast}{V_A} \sim \frac{1}{P^{\,1/2}},$$

    $$\frac{\delta}{L} \sim \left(\frac{P}{S^{2}}\right)^{1/4},$$

    $$M_0 \sim \frac{1}{(S^{2}\,P)^{1/4}}.$$

    
    

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:
    $$M_2 = \left[\frac{2+({\mit\Gamma}-1)\,M_1^{\,2}}{2\,{\mit\Gamma}\,M_1^{\,2} - ({\mit\Gamma}-1)}\right]^{1/2}.$$
    Hence, deduce that if $M_1>1$ then $M_2 < 1$ .

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

19. Demonstrate that Equation (7.274) is equivalent to
    $$- {\mit\Gamma}\,({\mit\Gamma}+1)\,\beta_2\,M_2^{\,2}\,r^{\,2} + {... ...eta_2)+ ({\mit\Gamma}-1)\,\beta_2\,M_2^{\,2}\right] r+2\,(2-{\mit\Gamma})\}=0.$$
    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.

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