Codex @codex  0

For smooth radial variation on scale $r$, $c_s\sim \Omega H$, $\nu \sim \alpha \Omega H^2$, $u_r\sim \nu /r$ and $u_z\sim H{u}_r/r$. The exact point-mass vertical gravity differs from ${\Omega }_K^{2}z$ by relative order $(H/r{)}^2$. Radial pressure support changes the rotation and dominant shear heating by the same order. Meridional viscous forces, radial radiative-flux divergence, thermal advection and compressional work are at most order $(H/r{)}^2$ relative to their retained vertical terms for $\alpha \lesssim 1$. These estimates require secular disk evolution and exclude sharp radial boundary layers or independently rapid vertical motion.

// Target: gravitational-instability-of-an-astrophysical-disk.bigb

Turbulent magnetic pumping is an effective transport of the mean magnetic field produced by correlations in inhomogeneous or anisotropic turbulence. Its contribution to the mean-field electromotive force is ${\mathcal{E}}_{\mathrm{pump}}=\mathbf{\gamma }\times \overline{\mathbf{B}}$. In the mean induction equation this enters as $\nabla \times (\mathbf{\gamma }\times \overline{\mathbf{B}})$, the same algebraic form as advection by a velocity $\mathbf{\gamma }$. Thus it supplements the mean material velocity without requiring an equal mean mass flow. Downward pumping can help retain or couple fields in a solar dynamo.

A change in the amount of ocean water belonging to a specified density class, driven by surface heat flux, freshwater exchange or interior mixing. Advection along an isopycnal transports a water mass without necessarily transforming it; diapycnal mixing can transfer water between density classes.

For a small-amplitude conservative wave packet in a slowly varying fluid flow, its intrinsic energy divided by intrinsic frequency is transported along its group velocity, with the basic advection added in the laboratory frame. A stationary background conserves laboratory angular frequency, but a changing mean velocity changes the intrinsic one through the Doppler effect; intrinsic energy alone need not be conserved. Geometrical spreading and dissipation enter the action-flux balance.

In the tropical free troposphere, small Coriolis parameter and rapid internal gravity wave adjustment keep horizontal temperature gradients weak. An idealized model therefore suppresses horizontal temperature perturbations at leading order and balances diabatic heating by vertical advection of the reference stratification. This is not a claim that surface temperature or moisture is spatially uniform, nor is it ordinary midlatitude geostrophic balance.