The Ideal MHD Equations
The single-fluid, perfect-conductor model of a magnetized plasma used for equilibrium and gross stability.
The closed set
Ideal magnetohydrodynamics treats the plasma as one conducting fluid with negligible resistivity. It couples fluid conservation laws to Maxwell's equations through the Lorentz force and a perfect Ohm's law.
- Continuity: drho/dt + div(rho v) = 0
- Momentum: rho dv/dt = -grad p + J x B
- Ideal Ohm's law: E + v x B = 0
- Induction: dB/dt = curl(v x B)
- Ampere (low frequency): mu0 J = curl B
- Adiabatic energy: d/dt (p rho^-gamma) = 0
The frozen-in condition
Ideal Ohm's law implies the magnetic flux through any fluid loop is conserved: field lines are frozen into the fluid and move with it. This single property explains why plasmas can be confined at all and why ideal instabilities are so violent when they occur.
What it is good for
Ideal MHD is remarkably accurate for the largest, fastest plasma motions even though it ignores kinetic detail. It predicts equilibrium (via Grad-Shafranov), Alfven and magnetosonic waves, and the ideal instabilities (kink, interchange, ballooning) that set hard operational limits such as the Troyon beta limit.
How it is solved numerically
Time-dependent nonlinear ideal MHD is advanced with conservative finite-volume or finite-element schemes; shock-capturing (Godunov-type) methods are common because the equations are hyperbolic. The divergence-free constraint div B = 0 is enforced by constrained transport, projection, or divergence-cleaning.
For stability, the linearized equations are recast as an eigenvalue problem through the energy principle, and codes minimize the potential energy functional dW over trial displacements. A negative minimum means instability. This is how growth rates and marginal beta limits are computed for machines like the Hyperion breeder.