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Next: Waves in Warm Plasmas Up: Magnetohydrodynamic Fluids Previous: Oblique MHD Shocks

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

    $\displaystyle \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

    $\displaystyle \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

    $\displaystyle [1+{\rm i}\,k^{\,2}/(\mu_0\,\sigma\,\omega)],$

    and $ \omega^{\,2}$ by an additional factor

    $\displaystyle [1+{\rm i}\,\mu\,k^{\,2}/(\rho_0\,\omega)].$

    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

    $\displaystyle 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

    $\displaystyle \frac{du^2}{dr}\left(1-\frac{u_c^{\,2}}{u^2}\right) = \frac{4\,u_c^{\,2}}{r} \left(1-\frac{r_c}{r}\right),$

    Show that this expression can be integrated to give

    $\displaystyle \left(\frac{u}{u_c}\right)^2 -\ln\left(\frac{u}{u_c}\right)^2 = 4\,\ln\left(\frac{r}{r_c}\right) + 4\,\frac{r_c}{r} + C,$

    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

    $\displaystyle u^2= u_c^{\,2}\left[1\pm 2\,x + {\cal O}(x^2)\right]
$

    or

    $\displaystyle 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

    $\displaystyle {\bf B}$ $\displaystyle = \nabla\psi\times{\bf e}_z + B_z\,{\bf e}_z,$    
    $\displaystyle {\bf V}$ $\displaystyle =\nabla\phi\times {\bf e}_z+ V_z\,{\bf e}_z,$    

    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

    $\displaystyle \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

    $\displaystyle \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,

    $\displaystyle 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

    $\displaystyle {\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

    $\displaystyle {\mit\Delta}' \rightarrow \frac{2}{\bar{k}} -\frac{8}{3}+{\cal O}(\bar{k})
$

    as $ \bar{k}\rightarrow 0$ , and

    $\displaystyle {\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

    $\displaystyle \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

    $\displaystyle \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

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

      respectively.

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

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

      where

      $\displaystyle P = \frac{\tau_R}{\tau_M}
$

      is the magnetic Prandtl number, and

      $\displaystyle \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

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

    4. Show that the Fourier transformed resistive layer equation, (7.196), generalizes to give

      $\displaystyle \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

      $\displaystyle {\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

      $\displaystyle {\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

    $\displaystyle 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

    $\displaystyle \frac{v_\ast}{V_A}$ $\displaystyle \sim \frac{1}{P^{\,1/2}},$    
    $\displaystyle \frac{\delta}{L}$ $\displaystyle \sim \left(\frac{P}{S^{2}}\right)^{1/4},$    
    $\displaystyle M_0$ $\displaystyle \sim \frac{1}{(S^{2}\,P)^{1/4}}.$    

  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:

    $\displaystyle 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

    $\displaystyle - {\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

    $\displaystyle M_2^{\,2} < 1+ \frac{2}{{\mit\Gamma}\,\beta_2}:
$

    that is,

    $\displaystyle 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).


next up previous
Next: Waves in Warm Plasmas Up: Magnetohydrodynamic Fluids Previous: Oblique MHD Shocks
Richard Fitzpatrick 2016-01-23