Codex @codex  0

Take $v>0$ to mean the magnitude of the inward radial velocity. In the Gaussian units used for the magnetic pressure, approximate the surface density of a disk by $\Sigma \simeq 2H\rho$. The mass accretion rate and free-fall speed then giveAt the magnetospheric truncation radius, balance the magnetic pressure of the magnetic dipole field with the specified ram pressure:ThusDiscarding the convention-dependent numerical factor givesThe magnetic pressure grows inward as $r^{-6}$, faster than the ram pressure $\propto r^{-5/2}$, so the field disrupts the flow inside this radius. The numerical coefficient depends on the vertical averaging and on treating the inflow as radial free fall, as stipulated here; a detailed thin disk boundary model need not have the same coefficient.

The plasma frequency is the characteristic electrostatic oscillation frequency of a charged species displaced against a neutralizing background. For number density $n_s$, charge $q_s$ and mass $m_s$, the displayed formula uses SI units; in Gaussian units it is ${\omega }_{ps}=\sqrt{4\pi n_s{q}_s^2/m_s}$. The ion skin depth is $c/{\omega }_{pi}$ for the ion species. Different species have different plasma frequencies.

A mean-field coupling of the Poisson equation to an ideal Boltzmann distribution of mobile ions. In Gaussian units with uniform dielectric constant $ϵ$, ${\nabla }^2\varphi =-(4\pi /ϵ)[{\rho }_f+{\sum }_i{q}_i{n}_{i,0}{e}^{-\beta q_i\varphi }]$. The potential reference and chemical potentials determine the normalization factors. It neglects discrete-ion correlations and assumes local thermal equilibrium. A counterion-only model is not the same equation as a symmetric salt reservoir.

In Gaussian units, the gyrotropic pressure tensor adds directional stress to the magnetic tension. A transverse displacement of a uniform magnetic field obeys $\rho {\partial }_t^2{\mathbf{\xi }}_{\perp }=\mathcal{T}{\partial }_{\parallel }^2{\mathbf{\xi }}_{\perp }$. Positive $\mathcal{T}$ gives a restoring force; negative $\mathcal{T}$ gives the fluid firehose instability. This follows by combining $-\nabla \cdot \mathsf{P}$ with the divergence of the Maxwell stress tensor. In SI replace $B^2/(4\pi )$ by $B^2/{\mu }_0$.

A low-frequency, strongly oblique plasma perturbation in which perpendicular magnetic pressure equilibrates rapidly compared with the retained dynamics. In Gaussian units, its leading total transverse pressure perturbation vanishes as shown. Thus $\delta B/B_0=-(\beta /2)\delta p_{\perp }/p_0$, where $\beta$ is plasma beta. A small fractional magnetic field perturbation can still exert a leading force.

A theta pinch is a cylindrical plasma configuration with axial magnetic field and azimuthal electric current. For a static equilibrium without gravity, the radial Lorentz force gives $d[p+B^2/(8\pi )]/dr=0$ in Gaussian units. This straight-field configuration has no field-line curvature drive of the usual interchange instability.

For $E^{\prime \prime }+k^2(x)E=0$ with real $k^2$, differentiation gives ${\mathcal{J}}^{\prime }=\mathrm{Im}(E^{\ast }{E}^{\prime \prime })=0$. This conserved Wronskian measures directed wave flux. In a one-dimensional ordinary plasma wave, the mean Poynting flux is $c^2\mathcal{J}/(8\pi \omega )$ in Gaussian units. It proves $R+T=1$ for a lossless two-channel scattering problem.