Astrophysical fluid dynamics

Astrophysical fluid dynamics studies gases, plasmas, stars, discs, and outflows under gravity, rotation, compressibility, and magnetic fields.

An affine stellar model represents a fluid body by a time-dependent ellipsoid whose internal velocity is linear in position. Its axis lengths become finitely many dynamical degrees of freedom.

The affine breathing mode changes all three principal axes by the same fraction and therefore changes the stellar volume and density.

An affine quadrupole mode has axis perturbations whose fractional sum vanishes. It changes the ellipsoidal shape without changing volume to first order.

A polytropic atmosphere obeys $p=K{\rho }^{1+1/m}$ for polytropic index $m$. Under uniform vertical gravity and a free surface, its density and pressure are powers of depth below that surface.

A polytropic atmosphere is neutrally stratified when $1+1/m=\gamma$, so its equilibrium pressure-density relation matches the adiabatic relation and its buoyancy frequency vanishes.

A surface gravito-inertial wave combines free-surface gravity restoration with Coriolis restoration. In a uniformly rotating deep atmosphere its incompressible dispersion relation can be written

In a neutrally stratified polytropic atmosphere, vertically trapped polynomial solutions of degree $n\ge 1$ describe acoustic pressure modes. The degree-zero member is the surface-gravity mode.

Magnetohydrodynamics treats an electrically conducting fluid coupled to a magnetic field through the Lorentz force and electromagnetic induction.

Ideal magnetohydrodynamics assumes vanishing resistivity, so magnetic flux is frozen into the fluid and the electric field in the fluid rest frame vanishes.

The Alfvén speed associated with magnetic-field magnitude $B$ and density $\rho$ is $v_A=B/\sqrt{{\mu }_0\rho }$.

An Alfvén number is the ratio of a flow speed to the corresponding Alfvén speed. For a poloidal flow parallel to a poloidal magnetic field, $A^2={\mu }_0\rho u_p^2/B_p^2$.

A magnetohydrodynamic shock is a discontinuity satisfying conservation of mass, momentum, energy, normal magnetic field, and tangential electric field.

The de Hoffmann-Teller frame is a tangentially boosted shock frame in which the electric field vanishes. Ideal MHD then makes the fluid velocity parallel to the magnetic field on both sides.

A rotational discontinuity changes the tangential direction of the magnetic field and velocity while leaving density, pressure, and normal components continuous. The normal flow is Alfvénic.

A steady axisymmetric magnetohydrodynamic wind follows nested magnetic-flux surfaces and has conserved mass loading, field-line angular velocity, angular momentum, entropy, and Bernoulli quantities along each surface.

For ${\mathbf{B}}_p=\nabla \psi \times \nabla \varphi$, surfaces $\psi =\text{constant}$ are poloidal magnetic surfaces and $2\pi \psi$ is magnetic flux up to the choice of axial reference value.

When $\rho {\mathbf{u}}_p=k{\mathbf{B}}_p$, the flux-surface function $k$ is the mass flux per unit poloidal magnetic flux.

Ideal induction in a steady axisymmetric flow makes each magnetic surface rotate with a constant field-line angular velocity $\omega (\psi )$.

The total specific angular momentum transported by matter and magnetic stress isand is constant on each magnetic surface.

An Alfvén surface is where the poloidal flow speed equals the poloidal Alfvén speed. Smooth passage through it imposes a regularity condition relating field-line angular velocity and angular momentum.

Magnetocentrifugal acceleration occurs when matter tied to a rotating inclined magnetic field moves outward under the effective centrifugal force. For a Keplerian disc, cold launching is possible when the field tilts by more than $30^{\circ }$ from the vertical.