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A flux tube is a bundle of magnetic field lines bounded by a surface everywhere tangent to the magnetic field. Gauss's law for magnetism makes the magnetic flux through any transverse cross-section constant along the tube. For a narrow tube with nearly uniform field magnitude $B$ and transverse area $A$, this gives $BA=\mathrm{constant}$.

The cross-sectional area of a narrow flux tube, measured normal to its magnetic field. Constancy of the magnetic flux gives $A|\mathbf{B}|=\mathrm{constant}$ along the tube. This area controls the mass transport when flow follows the field.

Hartmann flow is a fully developed viscous conducting-fluid flow driven along a planar channel while a uniform magnetic field crosses the walls. Induction and the Lorentz force density couple its velocity and tangential induced field. For no-slip, normal-field walls separated by $2L$, the velocity is a hyperbolic cosine profile that approaches plane Poiseuille flow at zero imposed field and develops a uniform core with Hartmann layers at large Hartmann number.

For a streamwise pressure gradient $-\rho G$ and walls at $z=\pm L$ with zero tangential magnetic field, the volumetric flow rate per unit span isIts zero-field value is $2G{L}^3/(3\nu )$; it decreases strictly for $H>0$ and is asymptotic to $2G{L}^3/(\nu H)$. The partial-fraction expansion of the hyperbolic cotangent proves monotonicity directly.

A Hartmann layer is the thin viscous-magnetic boundary layer next to a wall crossed by the applied magnetic field. For channel half-width $L$ and large Hartmann number $H$, its thickness is $\delta \sim L/H=\sqrt{{\mu }_0\rho \nu \eta }/|B_0|$. It restores the no-slip boundary condition and the specified magnetic wall condition outside the nearly uniform Hartmann flow core.

In a uniform magnetic field, the Lorentz force leaves the component of velocity parallel to the field constant and rotates the perpendicular component at the nonrelativistic cyclotron frequency $|q|B/m$. The resulting path is a helix around a field line.

For ideal magnetohydrodynamics at constant mass density $\rho$, a shearing sheet with velocity $-q\Omega x{\mathbf{e}}_y+{\mathbf{v}}_h(z,t)$ and magnetic field ${\mathbf{B}}_h(z,t)+B_z{\mathbf{e}}_z$ obeysHorizontal invariance and vanishing vertical velocity eliminate nonlinear advection. These equations retain Coriolis acceleration, orbital shear and magnetic tension. The vertical pressure adjusts to balance gravity and magnetic pressure.

For a vertical magnetic field in a homogeneous shearing sheet, a horizontal normal mode with growth rate $s$ and vertical Alfvén frequency ${\omega }_a=k{v}_{Az}$ satisfiesThe two roots for $s^2$ are real. For $q>0$, a growing magnetic mode exists precisely when $0<{\omega }_a^2<2q{\Omega }^2$. Magnetic tension is strong enough to couple radially displaced elements, yet weak enough to allow their separation to increase. Ogilvie's lecture on magnetorotational instability derives this relation and its finite-thickness boundary conditions.

In aligned steady ideal flow with $p=c_s^2\rho$ and constant isothermal sound speed, the Lorentz force density has no field-parallel component. Projecting momentum along the magnetic field makes $u^2/2+\Phi +c_s^2\ln (\rho /{\rho }_0)$ constant on each connected magnetic surface. The arbitrary reference density changes only the additive constant.

The $n=0$ level of the transverse harmonic oscillator in a uniform magnetic field, with energy $\hslash |qB|/(2m)$ before longitudinal kinetic energy or spin contributions.

In magnetically channelled accretion, a strong magnetic field guides an accretion flow along flux tubes. The component of the Lorentz force density parallel to the magnetic field vanishes, so the longitudinal flow can be treated hydrodynamically while the field controls the transverse geometry. A magnetic dipole field near a neutron star channels matter toward small polar caps.

For a particle obeying $m\dot{\mathbf{v}}=q\mathbf{v}\times \mathbf{B}$ in a constant uniform magnetic field, the displayed combination is conserved. The derivative of mechanical angular momentum projected onto the field is canceled by the quadratic position term. This is the rotational analogue of the distinction between mechanical and canonical momentum in an electromagnetic field.

A magnetic buoyancy instability is a gravitational instability of a magnetized stratified fluid in which displacements of the gas and magnetic field can release Newtonian gravitational potential energy. Magnetic pressure supports part of the weight. Both interchange displacements and undular displacements can enter this broad category; Parker instability is the undular mechanism involving drainage along bent magnetic field lines.

Finite electrical conductivity permits resistive smoothing of a magnetic field. With uniform coefficients and no motion the field satisfies a diffusion equation; with motion the diffusion term supplements the curl of $\mathbf{u}\times \mathbf{B}$. The magnetic Reynolds number compares these effects.

A magnetic field line is an integral curve tangent to the magnetic field. In coordinates $x^i$, it obeys $d{x}^i/B^i=d{x}^j/B^j$ wherever the relevant components are nonzero.

For a smoothly moving circuit and a divergence-free magnetic field, the magnetic flux derivative is $d\Phi /dt={\int }_S{\partial }_t\mathbf{B}\cdot d\mathbf{S}-{\oint }_C(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{r}$. The minus sign follows by orienting the swept side surface compatibly with the two spanning surfaces. Combining this transport identity with Faraday's law yields the moving-loop electromotive-force law.

In a simply connected current-free region of constant magnetic permeability, Ampère's circuital law makes the magnetic field curl-free, so it admits a scalar potential. The absence of magnetic monopoles then gives Laplace's equation for that potential. Some conventions define the potential for $\mathbf{H}$ instead of $\mathbf{B}$, differing by the constant permeability. Boundary conditions, including imposed fields at infinity, determine the potential up to an additive constant.

A plane-parallel magnetostatic equilibrium with constant downward gravitational acceleration $g$, horizontal magnetic field, constant isothermal sound speed $c_s$, and constant Alfvén speed $v_a$ hasIndeed the magnetohydrodynamic total pressure is $(c_s^2+v_a^2/2)\rho$, and its vertical derivative is $-g\rho$. The magnetic pressure therefore increases the scale height above that of an unmagnetized atmosphere.

Magnetoconvection is thermal convection coupled to the evolution and forces of a magnetic field. Conserved magnetic flux can supply a slow field that must be retained along with the convective complex amplitude, rather than being eliminated into a single local amplitude equation.