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An accretion column carries an inward accretion flow onto a localized region of a star. In the strong-field slender approximation, the magnetic field controls the flux-tube area while longitudinal pressure and gravity determine acceleration. The signed outward-coordinate velocity is negative; the inward speed and mass accretion rate are usually defined positive.

The Alfvén frequency is the frequency of an Alfvén wave with wavevector $\mathbf{k}$ in a uniform magnetic field $\mathbf{B}$ and mass density $\rho$. For a vertical field and vertical wavenumber $k$, its square is ${\omega }_a^2=k^2{B}_z^2/({\mu }_0\rho )$. Its competition with orbital shear determines the ideal magnetorotational dispersion relation.

The Alfvén velocity is the signed vector ${\mathbf{v}}_a=\mathbf{B}/\sqrt{{\mu }_0\rho }$, whose magnitude is the Alfvén speed. It follows the orientation of the magnetic field, unlike the nonnegative scalar speed.

The alpha effect is the part of a mean-field electromotive force linear in the mean magnetic field itself, $\mathcal{E}=\mathbf{\alpha }\overline{\mathbf{B}}+\cdots$. The alpha tensor can be anisotropic; it need not be a scalar multiple of the identity. Its curl can couple transverse mean-field components and produce an alpha-squared dynamo.

An alpha-squared dynamo generates a mean magnetic field using an alpha effect without requiring an additional large-scale shear coupling. For $\mathbf{\alpha }={\alpha }_0\mathrm{diag}(1,1,0)$ and a transverse mean field varying as $e^{iKz}$, the growth rates are $-\eta K^2\pm {\alpha }_{0}K$. A nonzero ${\alpha }_0$ therefore allows growth for sufficiently long wavelengths, provided the domain and scale separation permit them.

The alpha tensor is the linear map from a uniform test magnetic field to the corresponding mean-field electromotive force. Its components satisfy ${\mathcal{E}}_i={\alpha }_{ij}{B}_{0j}$. Anisotropic helical flows can produce a real symmetric rank-two response, such as ${\alpha }_0\mathrm{diag}(1,1,0)$.

A regular level surface of the axisymmetric poloidal magnetic flux function contains both the poloidal and toroidal magnetic field directions. Conservation along field lines can be expressed locally as a function of the flux label on connected regular surfaces. Degenerate points with no poloidal magnetic field and disconnected branches with the same label require separate treatment.

For a steady current density $J$, the magnetic field is

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 $D\psi /Dt=0$.

Let $h_c=\mathbf{u}\cdot \mathbf{B}$ be the cross-helicity density. The ideal magnetohydrodynamic induction equation implies $D\mathbf{B}/Dt=(\mathbf{B}\cdot \nabla )\mathbf{u}-\mathbf{B}\nabla \cdot \mathbf{u}$. Dotting the momentum equation with $\mathbf{B}$ eliminates the Lorentz force density, since it is perpendicular to the magnetic field. The material derivative of $h_c$ is thereforeUsing $dh=Tds+dp/\rho$ for specific enthalpy $h$ and specific entropy $s$, together with Gauss's law for magnetism, converts this toThe source vanishes when magnetic field lines lie in constant-specific entropy surfaces, in particular in a homentropic flow. Integrating over a volume conserves its total cross-helicity only if the boundary flux also vanishes. This local law is derived in Ogilvie's astrophysical fluid dynamics notes.

A current sheet idealizes electric current as supported on a surface. It produces a jump in the tangential magnetic field.

Dynamo action is growth or sustained maintenance of a magnetic field by conducting-fluid motion despite magnetic diffusion. In the kinematic problem the velocity is prescribed and the resistive induction equation is linear in the magnetic field. Magnetic feedback through the Lorentz force density matters once the field becomes dynamically important.

In uniform perpendicular electric and magnetic fields, the guiding centre of a charged particle drifts with ${\mathbf{v}}_D=\mathbf{E}\times \mathbf{B}/|\mathbf{B}{|}^2$. This velocity is independent of the charge and mass; cyclotron motion is superposed on it.

The combined electric field and magnetic field that determine the Lorentz force on a charged particle.

Electromagnetic radiation consists of propagating oscillations of the electric field and magnetic field that carry energy and momentum.

For smooth ideal magnetohydrodynamics of constant mass density, define $I_{\pm }={\int }_V|{\mathbf{z}}^{\pm }{|}^{2}dV$. Taking the dot product of the Elsässer variable equation with $2{\mathbf{z}}^{\pm }$ givesTherefore each integral is constant when its outward boundary flux vanishes, for example when both velocity and magnetic field are tangent to the fixed boundary, or for periodic boundary conditions. The kinetic energy plus magnetic energy is $\rho (I_{+}+I_{-})/4$, while the cross-helicity is $\sqrt{{\mu }_0\rho }(I_{+}-I_{-})/4$.

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.

In steady axisymmetric ideal magnetohydrodynamics, induction makes $\omega$ constant on regular connected poloidal flux surfaces. It describes field-line rotation. The material angular velocity is $u_{\varphi }/R=\omega +k{B}_{\varphi }/(\rho R)$ and differs when the toroidal magnetic field contributes. If the poloidal magnetic field vanishes, the flux-surface integration requires separate treatment.

If the horizontal magnetic field perturbations vanish at $z=\pm H$, their allowed vertical wavenumbers are $k_m=m\pi /(2H)$, $m=1,2,\dots$. The lowest wavenumber is unstable exactly whenIf that magnetic mode is stable, all higher magnetic modes are stable as well. A weak field admits longer unstable wavelengths that fit inside the slab; sufficiently strong magnetic tension stabilizes every allowed wavelength.