Flux-tube axial field virial identity ($⟨B_z^2⟩=B_p^2-2{\mu }_0⟨p⟩$)

For a regular cylindrical flux tube with $\mathbf{B}=(0,B_{\varphi }(s),B_z(s))$, vanishing field beyond radius $s_0$, and exterior pressure $p_e$, define $⟨h⟩=2s_0^{-2}{\int }_0^{s_0}sh(s)ds$ and $B_p^2=2{\mu }_0{p}_e$. Integrating cylindrical magnetostatic pressure balance against $s^2$ gives the displayed identity. The toroidal magnetic pressure and hoop magnetic tension cancel in this cross-sectional axial-field average. Nonnegative gas pressure gives $⟨B_z^2⟩\le B_p^2$, with strict inequality precisely when $⟨p⟩>0$.

For a flux tube with $B_{\varphi }=\gamma (s/s_0)B_z$, regularity at the axis and cylindrical magnetostatic pressure balance give $B_z(0{)}^2=B_p^2-2{\mu }_{0}p(0)+{\gamma }^2⟨B_z^2⟩$. The flux-tube axial field virial identity and $p\ge 0$ give the bound. A twisted toroidal magnetic field can support a central axial field above the untwisted magnetic pressure limit even though its cross-sectional mean obeys that limit.