Exercises

1. Derive Equations (9.11)–(9.14) from Equations (9.3)–(9.10).

2. Consider the linear tearing stability of the following field configuration,
   $$F(\hat{x}) = \left\{\begin{array}{lcl} F'(0)\,\hat{x}&\mbox{\hspa... ... [0.5ex] F'(0)\,{\rm sgn}(\hat{x})&&\vert\hat{x}\vert\geq 1\end{array}\right..$$
   This configuration is generated by a uniform, $z$-directed current sheet of thickness $a$, centered at $x=0$.
   1. Solve the tearing mode equation, (9.31), subject to the constraints $\hat{\psi}(-\hat{x})=\hat{\psi}(\hat{x})$, and $\hat{\psi}(\hat{x})\rightarrow 0$ as $\vert\hat{x}\vert\rightarrow\infty$. Hence, deduce that the tearing stability index for this configuration is
      $${\mit\Delta}' = \frac{2\,\hat{k}}{\tanh\hat{k}}\left[\frac{\hat{k} + \hat{k}\,\tanh\hat{k}-1}{1-\hat{k}/\tanh\hat{k}-\hat{k}}\right].$$

   2. Show that
      $${\mit\Delta}' \rightarrow \frac{2}{\hat{k}} -\frac{8}{3}+{\cal O}(\hat{k})$$
      as $\hat{k}\rightarrow 0$, and
      $${\mit\Delta}'\rightarrow -2\,\hat{k} + 2\left[1+\frac{1}{2\,\hat{k}}+{\cal O}\left(\frac{1}{\hat{k}^{2}}\right)\right]\exp(-2\,\hat{k})$$
      as $\hat{k}\rightarrow\infty$.
   3. Demonstrate that the field configuration is tearing unstable (i.e., ${\mit\Delta}'> 0$) provided that $\hat{k}<\hat{k}_c$, where
      $$\hat{k}_c\,(1+\tanh\hat{k}_c )= 1.$$
      Show that $\hat{k}_c=0.639$.

3. We can incorporate plasma viscosity into the reduced-MHD equations, (9.11)–(9.14), by modifying Equation (9.4) to read
   $$\rho\left[\frac{\partial {\bf V}}{\partial t} + ({\bf V}\cdot\nabla) {\bf V}\right]+\nabla p-{\bf j}\times {\bf B} -\mu\,\nabla^2{\bf V}=0,$$
   where $\mu$ is the viscosity.
   1. Show that the reduced-MHD equations generalize to give

      $$\frac{\partial\psi}{\partial t} = [\phi,\psi]+\frac{\eta}{\mu_0}\,(J-J_0),$$

      $$\rho\,\frac{\partial U}{\partial t} = \rho\,[\phi,U] + \mu_0^{-1}\,[J,\psi]+\mu\,\nabla^2 U,$$

      $$J =\nabla^2\psi,$$

      $$U =\nabla^2\phi.$$

   2. Show that Equations (9.22) and (9.23) generalize to give

      $$\gamma\,\psi_1 = {\rm i}\,k\,B_0\,F\,\phi_1 + \frac{\eta}{\mu_0}\left(\frac{d^2}{dx^2}-k^2\right)\psi_1,$$

      $$\gamma\,\rho\left(\frac{d^2}{dx^2}-k^2\right)\phi_1 = {\rm i}\,\mu_0^{-1}\,k\,B_0\,F\left( \frac{d^2}{dx^2}-k^2 - \frac{d^2F/dx^2}{F}\right)\psi_1+\mu\left(\frac{d^2}{dx^2}-k^2\right)^2\phi_1.$$

   3. Show that Equations (9.27) and (9.28) generalize to give

      $$S\,\hat{\gamma}\left(\hat{\psi}-F\,\skew{3}\hat{\phi}\right) = \left(\frac{d^2}{d\hat{x}^2}-\hat{k}^2\right)\hat{\psi},$$

      $$\hat{\gamma}^2\left(\frac{d^2}{d\hat{x}^2}-\hat{k}^2\right)\skew{3}\hat{\phi} = - F\left(\frac{d^2}{d\hat{x}^2} -\hat{k}^2-\frac{F''}{F}\right)... ...S^{-1}\,P\left(\frac{d^2}{d\hat{x}^{2}}-\hat{k}^{2}\right)^2\skew{3}\hat{\phi},$$

      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.

   4. Show that the resistive layer equations, (9.32) and (9.33), generalize to give

      $$S\,\hat{\gamma}\,(\hat{\psi}-\hat{x}\,\skew{3}\hat{\phi}) = \frac{d^2\hat{\psi}}{d\hat{x}^2},$$

      $$\hat{\gamma}^{2}\,\frac{d^2\skew{3}\hat{\phi}}{d\hat{x}^{2}} = -\hat{x}\,\frac{d^2\hat{\psi}}{d\hat{x}^2}+ \hat{\gamma}\,S^{-1}\,P\,\frac{d^4\skew{3}\hat{\phi}}{d\hat{x}^{4}}.$$

   5. Show that the Fourier transformed resistive layer equation, (9.45), generalizes to give
      $$\frac{d}{dp}\!\left(\frac{p^{2}}{Q+p^{2}}\frac{d\skew{3}\bar{\phi}}{dp}\right) -\left(Q\,p^{2}+ P\,p^{4}\right)\skew{3}\bar{\phi} = 0.$$

   6. 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
      $${\mit\Delta} = 2\pi\,\frac{\Gamma(3/4)}{\Gamma(1/4)}\,S^{1/3}\,Q^{5/4},$$
      whereas if $Q\ll P^{-1/3}, P^{2/3}$ then
      $${\mit\Delta} = 6^{2/3}\pi\,\frac{\Gamma(5/6)}{\Gamma(1/6)}\,S^{1/3}\,Q\,P^{1/6}.$$

4. 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 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

   $$\hat{v}_\ast \sim \frac{\hat{B}_\ast\,S}{\hat{k}\,P^{1/2}},$$

   $$\hat{v}_0 \sim \left(\frac{\hat{B}_\ast\,S}{\pi\,P^{1/2}}\right)^{1/2},$$

   $$\hat{\delta}_\ast \sim \left(\frac{\pi\,P^{1/2}}{\hat{B}_\ast\,S}\right)^{1/2},$$

   $$\frac{d\hat{\mit\Psi}}{d\hat{t}} = \frac{2^{3/2}}{\sqrt{\pi}}\,\frac{S^{1/2}}{P^{1/4}}\,\hat{\mit\Psi}^{3/4}.$$