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MHD Equilibrium

Source lecture(s): pc368_lec14_mhd_equilibrium

Intuition

A magnetic confinement device (tokamak, stellarator, Z-pinch) can only confine plasma if the outward pressure gradient is balanced by the inward magnetic tension. This static balance is MHD equilibrium. Finding the self-consistent \(\mathbf{B}(\mathbf{r})\) and \(p(\mathbf{r})\) that satisfy \(\mathbf{J}\times\mathbf{B} = \nabla p\) is a central problem in fusion energy.

Formal Definition

An equilibrium solution of the MHD force-balance equation satisfies:

\[\frac{\mathbf{J}\times\mathbf{B}}{\mu_0} = \nabla p\]

with \(\nabla\cdot\mathbf{B} = 0\) and \(\mathbf{J} = (1/\mu_0)\nabla\times\mathbf{B}\).

Mathematical Formulation

For axisymmetric systems (tokamaks), the Grad–Shafranov equation describes flux surfaces:

\[\Delta^* \psi + \mu_0 R^2 \frac{dp}{d\psi} + F \frac{dF}{d\psi} = 0\]

where \(\psi(R,Z)\) is the poloidal flux stream function and \(F(R,\psi) = RB_\phi\).

Derivation

  1. Start from \(\mathbf{J}\times\mathbf{B} = \nabla p\).
  2. Assume static equilibrium (\(\partial/\partial t = 0\)) and no flow (\(\mathbf{U}=0\)).
  3. Take the curl of both sides and use \(\nabla\times\mathbf{B} = \mu_0\mathbf{J}\).
  4. Substitute \(\nabla\times(\mathbf{J}\times\mathbf{B}) = \nabla(\mathbf{J}\cdot\mathbf{B}) - \mathbf{J}(\nabla\cdot\mathbf{B}) - (\mathbf{B}\cdot\nabla)\mathbf{J} + (\mathbf{J}\cdot\nabla)\mathbf{B}\). With \(\nabla\cdot\mathbf{B}=0\) and constant \(p\)-surfaces, simplify.
  5. In axisymmetry (\(\partial/\partial \phi = 0\)), the Grad–Shafranov equation follows.

Worked Example

Circular tokamak: Assume \(\psi = \psi(r)\) with \(r\) the minor radius. The poloidal field is \(B_\theta = (1/R)\partial\psi/\partial r\). Pressure balance \(d p/dr = B_\theta B_\phi/\mu_0\) gives a natural profile \(p(r) = p_0 (1 - r^2/a^2)^2\) compatible with a parabolic current density.

Common Mistakes

  • Ignoring the J×B force. Pressure gradients alone cannot confine a plasma.
  • Treating all equilibria as 1D. Shafranov shift and triangularity are inherently 2D effects.
  • Forgetting the bootstrap current. Neoclassical physics contributes a self-generated current important for steady-state tokamaks.

Quiz Questions

  1. Conceptual: Why can’t a straight Z-pinch achieve high-β equilibrium?
  2. Computational: For a force-free field \(\mathbf{J} = \alpha\mathbf{B}\), show that \(\nabla^2\psi + \alpha^2\psi = 0\) in cylindrical geometry.
  3. MCQ: The Grad–Shafranov equation requires:
  4. A) Time dependence
  5. B) Axisymmetry
  6. C) \(\nabla\cdot\mathbf{J} \neq 0\)
  7. D) Zero resistivity

Further Reading

  • J. P. Freidberg, Plasma Physics and Fusion Energy.