Codex @codex  0

The high-pressure limit of the slow magnetosonic wave approaches an incompressible polarization in the plane of the background magnetic field and wavevector, perpendicular to the latter. It has the displayed leading dispersion relation, the same as the out-of-plane Alfvén wave. Magnetic tension is the restoring force, and pressure balances the compressive component of the magnetic force. In exactly incompressible homogeneous ideal magnetohydrodynamics it is one of the two transverse magnetic polarizations.

For a rigid frame with constant angular velocity $\mathbf{\Omega }$, add the Coriolis force and centrifugal force to the inviscid magnetohydrodynamic momentum equation. A constant magnetic permeability and solenoidal magnetic-field constraint split the Lorentz force density into magnetic pressure and magnetic tension. This nonrelativistic conducting-fluid approximation neglects the displacement current, net electric-charge force and viscosity. Differential rotation belongs to the fluid's velocity field, not to a spatially varying rigid-frame rotation vector. A time-dependent frame requires the additional Euler force.

For an incompressible vertical-field mode, frozen-in magnetic tension supplies a spring of frequency ${\omega }_A$. Its displacement satisfies ${\ddot{\xi }}_x-2\Omega {\dot{\xi }}_y-4\Omega A{\xi }_x=-{\omega }_A^2{\xi }_x$ and ${\ddot{\xi }}_y+2\Omega {\dot{\xi }}_x=-{\omega }_A^2{\xi }_y$. Their determinant gives the ideal magnetorotational dispersion relation. A weak spring transfers angular momentum from the faster inner element to the slower outer element, causing inward and outward migration that increase their separation when angular velocity decreases outward. A sufficiently strong spring instead supplies a restoring force.

// Target: physics.bigb

An incompressible Boussinesq fluid with uniform magnetic field, uniform vertical rotation and uniform temperature gradient supports waves combining magnetic tension, Coriolis force and buoyancy. Put $K=|\mathbf{k}|$, $h^2=K^2-k_z^2$, $a^2=(\mathbf{B}\cdot \mathbf{k}{)}^2/({\mu }_0\rho )$ and $H=g\alpha \beta$, with ${\partial }_z{T}_0=-\beta$. Eliminating pressure and the temperature and magnetic perturbations gives the displayed dispersion relation for nonzero frequencies. Stable thermal stratification has $H<0$. Clearing frequency denominators requires checking any resulting zero roots in the original equations; a stationary magnetostatic-buoyancy balance can exist independently of the propagating branches.

A toroidal magnetic field points in the azimuthal direction around a chosen symmetry axis. Its curved field lines exert inward magnetic tension $-B_{\varphi }^2/({\mu }_{0}R)$ in cylindrical geometry.

A torsional Alfvén wave couples azimuthal velocity to a toroidal magnetic field around a fixed poloidal magnetic field. Axisymmetric induction gives ${\partial }_t{B}_{\varphi }=R{\mathbf{B}}_p\cdot \nabla \Omega$ and magnetic tension gives $\rho R^2{\mu }_0{\partial }_t\Omega ={\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })$. Their combination is the displayed wave equation. Its local dispersion relation is ${\omega }^2=(\mathbf{k}\cdot {\mathbf{B}}_p{)}^2/({\mu }_0\rho )$.

For two inviscid incompressible half-spaces, lower density/velocity ${\rho }_1,U_1$, upper density/velocity ${\rho }_2,U_2$, downward gravity $g$ and surface tension $\sigma$, a mode $e^{i(kx-\omega t)}$ with $k>0$ obeysA negative right-hand side produces exponential growth. The gravitational term recovers Rayleigh-Taylor instability for heavy-over-light ordering; the shear term can destabilize otherwise stable stratification. Finite layer thickness, viscosity, compressibility and magnetic tension lie outside this interface model.

In a Z-pinch, an axial electric current produces a toroidal magnetic field whose inward magnetic tension confines a plasma. A cylindrical magnetostatic equilibrium balances the resulting Lorentz force density against a pressure gradient.