Ideal magnetohydrodynamic induction equation (ideal MHD induction equation)

The ideal magnetohydrodynamic induction equation is ${\partial }_t\mathbf{B}=\nabla \times (\mathbf{u}\times \mathbf{B})$. Together with $\nabla \cdot \mathbf{B}=0$, it implies magnetic-flux freezing.

For an axisymmetric velocity $\mathbf{u}=(0,rf(r),g(r))$ independent of $z$, write $\mathbf{B}=B_z\hat{\mathbf{z}}+\nabla \times (A\hat{\mathbf{z}})$. Ideal induction becomesThe first equation expresses magnetic flux freezing of the transverse flux function, while the second is axial stretching $B_r{g}^{\prime }$. Starting from a uniform field $B_0\hat{\mathbf{x}}$ gives $A(r,\varphi ,t)=B_{0}r\sin (\varphi -f(r)t)$ and $B_z(r,\varphi ,t)=B_{0}t{g}^{\prime }(r)\cos (\varphi -f(r)t)$. In complex azimuthal-mode notation, $\hat{A}=B_{0}r{e}^{-ift}$ and ${\hat{B}}_z=i{B}_{0}t{g}^{\prime }{e}^{-ift}$, obtained by elementary integrating factors. The transverse field has $B_r=r^{-1}{A}_{\varphi }$ and $B_{\varphi }=-A_r$, preserving zero divergence.

For a $y$-independent solenoidal magnetic field, the transverse components can be written $B_x=-{\psi }_z$ and $B_z={\psi }_x$. Contours of $\psi$ are transverse magnetic field lines. The independent axial component $B_y$ can twist the flux tube. Under the MHD induction equation, an additive time-dependent gauge can be chosen so that $D\psi /Dt=0$.

A magnetic island in a planar magnetic field is a region of closed magnetic field lines surrounding an extremum of the Cartesian magnetic flux function. A surrounding separatrix can pass through a saddle of that function and separate the island from other flux regions. In ideal incompressible flow, the flux-function value carried by each fluid particle and the area enclosed by each material contour are conserved.

In a $y$-independent magnetostatic equilibrium with no axial pressure or gravitational force, $\nabla \psi \times \nabla B_y=0$. Thus $B_y=b(\psi )$ on a regular connected flux region. The remaining balance is $[{\nabla }^2\psi +b{b}^{\prime }]\nabla \psi /{\mu }_0+\nabla p+\rho \nabla \Phi =0$. Constancy on separate components of a level set does not automatically give one global single-valued $b$.

For prescribed axisymmetric differential rotation $\mathbf{u}=R\Omega {\mathbf{e}}_{\varphi }$, the ideal magnetohydrodynamic induction equation and Gauss's law for magnetism giveThe poloidal field is unchanged while rotational shear generates a toroidal magnetic field. For time-independent $\Omega$, the toroidal component grows linearly until neglected dynamical feedback becomes relevant.

For Keplerian rotation, initial $B_R\propto R^{-1}$ and thermal pressure $p\propto R^{-2}$, axisymmetric magnetic winding gives $B_{\varphi }=-(3/2)\Omega t{B}_R$. If the initial plasma beta is ${\beta }_p>1$, the formal equality of magnetic pressure and gas pressure occurs atThe magnetic-to-thermal pressure ratio decreases with radius, so magnetic pressure dominates inside this radius. In a disk with a finite radial range, an equality radius exists in the disk only when this formal radius lies within that range.

A stationary axisymmetric magnetic field in ideal purely rotational flow requires constant angular velocity along the magnetic field lines of its poloidal magnetic field. Otherwise axisymmetric magnetic winding generates a time-dependent toroidal component.