Magnetohydrodynamic equilibria

Set bulk acceleration to zero in the scalar-pressure momentum equation and neglect gravity, viscosity, and electrostatic charge-density force at the retained quasineutral order. The result is $\nabla p=\mathbf j\times\mathbf B$. Dotting with the magnetic field shows that pressure is constant along each field line in this model. It need not be constant between distinct lines. Dotting with current likewise constrains pressure variation along current paths where the fields are regular.

The force equation fixes only the component of current perpendicular to the field. A parallel current supplies no magnetic force locally and must instead be determined consistently by field geometry, current continuity, and boundaries. A force-free configuration has current parallel to magnetic field and can have nonzero current at constant scalar pressure. Thus finding a nonzero current is not enough to infer a pressure gradient, and finding constant pressure is not enough to infer a vacuum field.

Using the magnetic divergence constraint, the Lorentz force separates into a magnetic-pressure gradient and a field-line tension term. The pressure part pushes away from larger field magnitude; the tension part depends on variation of the field vector along itself. A straight field varying transversely can balance gas pressure through magnetic pressure alone. Curved fields can supply another force component, so replacing magnetic stress everywhere by one scalar pressure discards relevant geometry.

For the special slab $\mathbf B=B_z(x)\hat{\mathbf z}$, field-line tension vanishes and Ampère's law gives $j_y=-dB_z/dx\,/\mu_0$. Force balance then integrates to constant gas plus magnetic pressure across x. An increasing positive axial field requires negative y current and decreasing gas pressure. These signs are worth checking directly with the cross product rather than trusting an unsigned magnetic-pressure estimate. The slab is a genuine divergence-free geometry despite its transverse variation.

The constant total slab pressure is fixed by boundary data, not by the differential force equation alone. A proposed magnetic profile can demand negative gas pressure somewhere, revealing incompatible boundary values rather than an exotic physical equilibrium. At an interface a field jump can require a surface current, and the normal stress must match the external medium. A vacuum region has no gas pressure but still exerts magnetic stress through its field.

Plasma beta, $\beta=2\mu_0p/B^2$, compares gas pressure with magnetic energy density locally. It does not by itself quantify curvature forces, solve the boundary problem, or prove stability. A pinch can satisfy static force balance while admitting kink or interchange perturbations. Stability requires an additional dynamical or energy analysis with specified admissible displacements. Resistive evolution is another separate timescale: a nearly stationary force balance can slowly change even when it is stable to ideal perturbations.