Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part 3, Chapter 3

MHD Equilibrium

Force balance and magnetic confinement configurations

3.1 Static Equilibrium Condition

In MHD equilibrium, all time derivatives vanish and flow velocity u = 0. The momentum equation reduces to:

$$\mathbf{J} \times \mathbf{B} = \nabla p$$

Force Balance: Lorentz force balances pressure gradient

Consequence 1

$$\mathbf{B} \cdot \nabla p = 0$$

Pressure is constant along field lines

Consequence 2

$$\mathbf{J} \cdot \nabla p = 0$$

Pressure is constant along current lines

3.2 Magnetic Surfaces

Since both B and J are perpendicular to ∇p, they lie on surfaces of constant pressure called magnetic surfaces or flux surfaces.

Properties of Magnetic Surfaces

  • Nested toroidal surfaces in toroidal geometry
  • Field lines wrap helically around torus
  • Characterized by rotational transform ι or safety factor q
  • Rational surfaces where field lines close on themselves

Safety Factor

$$q = \frac{\text{toroidal turns}}{\text{poloidal turns}} = \frac{r B_\phi}{R B_\theta}$$

Measures pitch of field line helix; q > 1 typically needed for stability

3.3 Grad-Shafranov Equation

For axisymmetric toroidal equilibria, the force balance reduces to a single scalar equation:

$$R^2 \nabla \cdot \left(\frac{\nabla \psi}{R^2}\right) = -\mu_0 R^2 \frac{dp}{d\psi} - F\frac{dF}{d\psi}$$

Grad-Shafranov Equation

Poloidal Flux

$$\psi = \text{flux function}$$

Labels magnetic surfaces

Poloidal Current

$$F(\psi) = RB_\phi$$

Toroidal field function

3.4 Plasma Beta

The ratio of plasma pressure to magnetic pressure is a key dimensionless parameter:

$$\beta = \frac{p}{B^2/(2\mu_0)} = \frac{\text{thermal pressure}}{\text{magnetic pressure}}$$

Low β

β ≪ 1: Magnetic pressure dominates (tokamaks typically β ~ few %)

High β

β ~ 1: Comparable pressures (spheromaks, RFPs)

β Limit

Maximum β before instabilities (Troyon limit)

3.5 Simple Equilibria

θ-Pinch

$$p + \frac{B_z^2}{2\mu_0} = \frac{B_0^2}{2\mu_0}$$

Axial field confines plasma radially; no end confinement

Z-Pinch

$$\frac{d}{dr}\left(p + \frac{B_\theta^2}{2\mu_0}\right) = -\frac{B_\theta^2}{\mu_0 r}$$

Azimuthal field from axial current; inherently unstable

Screw Pinch

$$\frac{d}{dr}\left(p + \frac{B_\theta^2 + B_z^2}{2\mu_0}\right) = -\frac{B_\theta^2}{\mu_0 r}$$

Combined θ and Z-pinch; basic tokamak geometry

3.6 Bennett Relation

For a uniform pressure Z-pinch in equilibrium:

$$I^2 = \frac{8\pi N k_B (T_e + T_i)}{\mu_0}$$

Bennett Pinch Condition

Where N is the line density (particles per unit length) and I is the total current. This gives the minimum current needed to confine the plasma.

Key Takeaways

  • ✓ Static equilibrium: J × B = ∇p
  • ✓ B and J lie on magnetic flux surfaces of constant p
  • ✓ Grad-Shafranov equation describes axisymmetric equilibria
  • ✓ β = p/(B²/2μ₀) measures confinement efficiency
  • ✓ Pinch configurations: θ-pinch, Z-pinch, screw pinch
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