Geophysical fluid dynamics (GFD) — the unified theoretical framework for rotating, stratified fluids on a sphere, applied to atmosphere, ocean, and stellar/planetary interiors. Distinguished from general fluid dynamics by two additional…
geophysical-fluid-dynamics
QG vorticity scale: ζ_QG/f ≈ Ro
Quasi-geostrophic (QG) scaling: the relative vorticity ζ of a QG-approximated flow scales as ζ ≈ f·Ro, so the dimensionless ratio ζ/f ≈ Ro…
Rossby-wave dispersion: ω = -β·k/(k²+l²)
The dispersion relation for barotropic Rossby waves on a β-plane is ω = -β·k/(k²+l²), where β = df/dy = 2·Ω·cos(φ)/R is the latitudinal…
Baroclinic instability: σ ~ U·f/(H·N)
Baroclinic instability — the dynamical mechanism by which vertically-sheared rotating stratified flow spontaneously converts available…
QG breakdown: (ζ_QG/f)(Ro=1) = 1
Sympy-exact witness of the quasi-geostrophic breakdown anchor. Setup: ζ_QG/f = Ro as a one-symbol identity (the canonical QG scaling…
Rossby-wave limits: ω(k→0)=0; ω(l=0)=-β/k
Sympy-exact witness of the Rossby-wave dispersion-relation limiting cases. Setup: ω = -β·k/(k²+l²) as a three-symbol function. Identity 1…
Baroclinic growth doubling: σ(2U)/σ(U) = 2
Sympy-exact witness of the baroclinic-instability shear-doubling identity. Setup: σ = U·f/(H·N) as a four-symbol function. Identity:…
Rossby-wave dispersion omega = -beta k_x/(k^2); barotropic-Rossby framework
Rossby-wave dispersion framework (Rossby 1939 Mon Weather Rev 67, 281). Setup: in a beta-plane atmospheric/oceanic flow with Coriolis f =…
Quasi-geostrophic potential vorticity Dq/Dt = 0; method-of-characteristics
Quasi-geostrophic potential-vorticity (PV) conservation framework (Charney 1947 J Meteor 4, 135; Hoskins-McIntyre-Robertson 1985 Q J R…
Stream-function Poisson equation nabla^2 psi = zeta; Laplace-equation framework
Stream-function Poisson-equation framework (Helmholtz 1858 J Reine Angew Math 55, 25; Stokes 1842). Setup: 2D incompressible flow with…
Theorem: omega k_x + beta = 0 at k_y = 0, L_d -> inf (barotropic Rossby westward)
Theorem (Rossby-westward canonical): in the barotropic limit (L_d -> infinity, no stratification effect) and zonal wavenumber only (k_y =…
Theorem: dq/dt = 0 for q = const along characteristic (PV conservation)
Theorem (PV-conservation canonical): along a Lagrangian trajectory (characteristic curve) with dx/dt = u(x, t), the quasi-geostrophic PV q…
Theorem: nabla^2(x^2 + y^2) - 4 = 0 (quadratic stream-function vorticity)
Theorem (stream-quadratic-vorticity canonical): for the quadratic stream function psi = x^2 + y^2 (cylindrically-symmetric flow),…
Rossby waves (1939)
C-G Rossby 1939 planetary waves driven by beta-plane vorticity gradient; basis of large-scale atmospheric + oceanic dynamics; Rossby radius…
Ekman spiral (1905)
V W Ekman 1905 wind-driven ocean transport: balance of friction + Coriolis; spiral velocity profile; Ekman pumping basis of upwelling.
QG (Charney 1947)
J Charney 1947 quasi-geostrophic approximation Ro << 1; basis of large-scale weather prediction + first numerical-weather-prediction ENIAC…
Baroclinic instability (Eady 1949)
E Eady 1949: vertical-shear unstable mode generates synoptic eddies; basis of midlatitude weather + global energy transport.
Kolmogorov K41 (1941)
A Kolmogorov 1941 K41: -5/3 inertial-range power-law E(k) ~ epsilon^(2/3) k^(-5/3); basis of turbulence statistics + LES subgrid models.
AMOC (Stommel 1961)
H Stommel 1961 thermohaline-circulation 2-box model; bistability + tipping-point evidence; modern Rahmstorf 2002 paleoclimate confirmation.
Rossby (1939)
C-G Rossby 1939 Rossby-wave β-plane; modern modern foundational text + 2024 ML-jet-stream forecast + Rossby-deformation radius applications.
Quasi-geostrophic (Charney 1947)
J Charney 1947 + Phillips 1956 quasi-geostrophic theory; modern modern foundational text + 2024 idealized GCMs + atmospheric-baroclinic.
Eady (1949)
E Eady 1949 baroclinic-instability; modern modern foundational text + Charney-Stern + 2024 storm-track climate-model intensity.
Richardson (1922)
L F Richardson 1922 cascade + Kolmogorov 1941 K41; modern modern foundational text + 2024 ML-LES + DNS exascale geophysical-turbulence.
Smagorinsky (1963)
J Smagorinsky 1963 sub-grid-scale eddy-viscosity; modern modern foundational + Lilly 1967 + dynamic-Smagorinsky Germano 1991 + 2024 ML.
Hadley (1735)
G Hadley 1735 + Lorenz 1967 atmospheric-cells; modern modern foundational text + 2024 Hadley-cell expansion polewards 2-3° climate.