The Ideal MHD Equations

The single-fluid model of a perfectly conducting magnetized plasma: mass, momentum, energy, and induction.

The Closed Set

Ideal magnetohydrodynamics treats the plasma as one conducting fluid with negligible resistivity. The equations are mass continuity drho/dt + div(rho u) = 0; momentum rho du/dt = J x B - grad p; an adiabatic energy law d/dt (p/rho^gamma) = 0; and the ideal induction equation dB/dt = curl(u x B). Ampere's law J = curl B / mu0 and div B = 0 complete the set. The electric field is E = -u x B, the ideal Ohm's law with zero resistivity.

The Lorentz Force

The J x B force can be split into magnetic pressure and magnetic tension: J x B = -grad(B^2/2mu0) + (B . grad)B / mu0. The first term acts like an isotropic pressure of the field; the second is a tension along curved field lines. This decomposition explains equilibrium (pressure balanced by field) and the Alfven wave (tension as restoring force).

Validity

Ideal MHD is valid when the plasma is highly conducting (large Lundquist number), collisional enough to behave as a fluid, and observed on scales larger than the ion gyroradius and skin depth. It fails in thin reconnection layers, in low-collisionality kinetic regimes, and wherever two-fluid or finite-Larmor-radius effects matter. Despite these limits it captures the fastest and largest-scale plasma dynamics remarkably well.

Relevance

Equilibrium (Grad-Shafranov) and gross stability (kink, ballooning, interchange) both derive from ideal MHD, making it the first screen any confinement design must pass. For the Hyperion breeder concept, ideal-MHD equilibrium and stability analysis frame the design point; results are computational for a machine in simulation.