Fundamentals of Plasma Physics 2025, 34 - Static & Dynamic Equilibria
For static equilibrium, the MHD equation of motion reduces to ∇P = J × B, whereas the dynamic one reduces to ρU∇U + ∇P = J × B + νρ∇2U where the last term is the viscous damping. ν is the kinematic viscosity. Static equilibria are specifically relevant to plasma confinement.
A simplest static equilibrium is the Bennett pinch, consisting of an infinitely long axisymmetric cylindrical plasma with some axial current density Jz(r) and no other currents. The axial current flowing within a circle with radius r is I(r) = ∫0r 2πr'Jz(r')dr', By Ampere's law,
Confinement using currents that flow in the azimuthal direction is also possible, but the resulting pinch is transient. The radial confining force is a radial inward force. It will confine finite pressure plasma. This can't be extended to 3 dimensions, using MHD, unfortunately. Instead, first assume a pressure profile P(r) and then determine a corresponding B(r) with associated current J(r). The arbitrary magnetic field can't be specified, as ∇ × ∇P = 0. This is always true, but physically problematic. Equilibria then only exist for certain situations, requiring symmetry in some direction. Generally,
MHD-driven flows have ∇×(J×B) ≠ 0. This makes J×B non-conservative, and its finite curl acts as torque / hydrodynamic vorticity. When the MHD force is non-conservative, a closed line integral over the quantity is finite, while that of the pressure gradient vanishes. The modified MHD equation comes out to ρ(∂tU + U∇U) = J×B-∇P + ρv∇2U where v is the kinematic viscosity. Assume that the motion is incompressible, meaning ρ is constant; the system is cylindrically symmetric; and the flow velocities and current density are in poloidal direction. For such currents, the magnetic field must be purely toroidal. There are forms for this, that are consistent with the integral form of Ampere's law. The full magnetic force is
There is no viscosity in this construction, as ∂zI2 = 0. A generalized case with pre-existing poloidal magnetic field works with local radial pressure to determine the toroidal component of the magnetic field.
