Foundations

Reactive transport couples species fluxes to kinetic/equilibrium reactions while enforcing elemental and charge balance.

Richards flow conserves water in a variably saturated porous medium while Darcy flux uses saturation-dependent permeability.

Coupled groundwater and deformation models account for pore pressure, body forces, elastic response and boundary conditions.

Shared stream-power models combine detachment and transport limits instead of identifying sediment flux with erosion.

Ice thermal models combine advection, diffusion and deformation heating; enthalpy tracks sensible heat and liquid fraction at the pressure-melting boundary.

Mode-I stress intensity depends on load and crack geometry.

Rate-and-state friction distinguishes instantaneous velocity response from state evolution.

Mantle reconstruction solves mass, momentum and energy equations with rheology and thermal boundaries, then uses observations to constrain a past state.

Glacial isostatic adjustment jointly depends on ice/ocean loading history and Earth's elastic-viscous response.

Rotating dynamo models couple magnetic induction, momentum, heat/composition transport and solenoidal constraints.

A statistically steady dynamo with no net mechanical or Poynting power through its boundary balances supplied buoyancy power against ohmic and viscous dissipation.

Earth-system simulations couple atmosphere, ocean, land, cryosphere and biogeochemistry.

Paleoclimate reconstruction combines proxy calibration/age support with a climate-model prior.

Deep-time full-plate reconstructions combine paleomagnetism, geological ties and evolving topological boundaries.

Scott et al. extended G-ADOPT to elastic-compressible, laterally variable and non-Maxwell viscoelastic models.

The TERRA-NG v1.0 preprint reports a matrix-free GPU Stokes–energy solver specialized for spherical wedges and tests against a spherical-shell convection benchmark.

An elliptic two-body ephemeris solves Kepler's equation for eccentric anomaly at a declared time origin, then transforms orbital-plane coordinates to the observing frame.

For an elliptic orbit, d(E-e sin E)/dE is positive, so the root is unique for a chosen unwrapped mean anomaly.

A far-field source produces a geometric delay b·s/c between two antennas.

An interferometer measures a finite set of Fourier samples, leaving unmeasured sky modes.

A collisionless distribution follows Hamiltonian trajectories in a mean gravitational potential.

The stationary spherical nonrotating Jeans moment equation connects tracer density and radial velocity dispersion to enclosed mass through velocity anisotropy.

A projected dispersion averages line-of-sight moments through the tracer system.

Discretize stellar structure, transport and composition into a residual vector F(y).

A stellar or planetary EOS can blend differently valid microscopic models.

Angle-dependent radiative transfer yields moments J,H,K; flux is 4πH.

NonLTE level populations solve coupled radiative and collisional rate equations with particle/charge constraints and the transfer solution.

Shellular stellar evolution approximates angular velocity as nearly constant on isobars and models vertical transport and mass loss with prescriptions.

Ideal conducting-fluid motion evolves magnetic flux by induction and preserves an initially solenoidal field.

Magnetic pressure and tension contribute a stress tensor whose divergence gives the Lorentz force in the stated quasistatic limit.