Mechanics, thermodynamics, and radiative transfer of planetary atmospheres, with primary reference to Earth's troposphere and stratosphere. Hydrostatic balance — for a thin, stably-stratified atmosphere in local thermodynamic equilibrium,…
atmospheric-physics
Barometric formula: p(z) = p₀·exp(−z/H), H = k_B T/(m g)
Hydrostatic balance dp/dz = −ρg combined with the ideal-gas equation of state p = ρ·k_B T/m (m = mean molecular mass) gives a first-order…
Dry adiabatic lapse rate Γ_d = g/c_p ≈ 9.8 K/km
Adiabatic (dQ=0) displacement of an unsaturated air parcel through a hydrostatic atmosphere conserves specific entropy; rewriting the first…
Potential temperature θ = T·(p₀/p)^(R/c_p) (adiabatic invariant)
Potential temperature θ is the temperature a parcel would attain if brought adiabatically from its actual pressure p to a reference…
At z = H: p(H)/p₀ = e⁻¹ exactly
Closed-form evaluation of the barometric profile at its own scale height. Pressure ratio at z = H: p(H)/p₀ = exp(−H/H) = e⁻¹ (canonical…
Γ_d rational: 196133/20100000 K/m (g=9.80665, c_p=1005)
Closed-form rational evaluation of the dry adiabatic lapse rate using standard-gravity g = 9.80665 m/s² (2019 CIPM, BIPM SI Brochure 9th…
Adiabat: θ/T − (p₀/p)^κ ≡ 0 symbolically
Closed-form symbolic identity for the potential temperature. θ = T·(p₀/p)^κ; residual sp.simplify(θ/T − (p₀/p)^κ) = 0 exactly — a literal…
Brunt-Vaisala frequency N^2 = g (d theta/dz) / theta via Sturm-Liouville framework
Brunt-Vaisala / Sturm-Liouville atmospheric-stratification framework. Setup: in a stably stratified atmosphere, a fluid parcel displaced…
Ekman spiral via complex wind w = u + i v: d^2 w/dz^2 = (i f/K) w
Ekman-spiral framework via complex-valued winds (Ekman 1905 Ark Mat Astron Fys 2, 11). Setup: steady atmospheric boundary layer under…
Geostrophic wind u_g = (1/(f rho)) k-hat wedge grad(p); exterior-algebra antisymmetry
Geostrophic-wind framework via exterior-algebra wedge product (Holton Intro Dyn Meteor 1972). Setup: steady horizontal inviscid flow at…
Theorem: for linear theta(z) = theta_0 + beta z, N^2 theta - g d theta/dz = 0 identically
Theorem (Brunt-Vaisala-linear-theta canonical): for linearly stratified atmosphere theta(z) = theta_0 + beta z with constant lapse rate…
Theorem: (1+i)^2 - 2 i = 0 (holomorphic identity underlying Ekman 45-deg angle)
Theorem (Ekman-holomorphic-squaring canonical): the complex number 1 + i - the boundary-layer wavenumber (1+i)/d_E in the Ekman solution -…
Theorem: u_g . grad(p) = 0 identically (wedge-product antisymmetry)
Theorem (geostrophic-wedge-orthogonality canonical): u_g = (1/(f rho)) k-hat wedge grad(p) satisfies u_g . grad(p) = 0 identically, because…
Hadley cell circulation
Hadley 1735: tropical Hadley-cells from 0 to 30 deg lat; rising-equator + sinking-subtropical + trade-winds; explains…
Ozone layer (Chapman mechanism)
Chapman 1930: O2 + UV -> 2 O; O + O2 + M -> O3; O3 + UV -> O2 + O; steady-state Chapman-cycle gives stratospheric ozone profile; CFC…
CAPE (convective available potential energy)
CAPE = integral of (T_parcel - T_environment) g dz; positive CAPE indicates atmosphere supports convection; >2500 J/kg favors severe…
Brunt-Vaisala frequency
N = sqrt(g/theta dtheta/dz); restoration frequency of vertically displaced air parcel in stable atmosphere; basis of internal-gravity-wave…
Aerosol-cloud interaction (Twomey)
Twomey 1977: increased CCN concentration reduces cloud-droplet size + increases reflectivity (1st-indirect-aerosol-effect); plus albedo +…
Greenhouse effect (Arrhenius 1896)
Arrhenius 1896 first quantitative estimate: doubling CO2 raises mean T by ~5-6 C; modern best-estimate ECS 2.3-4.5 C (IPCC AR6); confirms…
Hydrostatic equilibrium (Laplace 1799)
P-S Laplace 1799 hydrostatic-equilibrium dp/dz=-ρg; modern modern barometric-formula + atmospheric-scale-height + planetary-atmosphere.
Clausius-Clapeyron atm (1834)
B Clapeyron 1834 + Clausius 1850 dp/dT=L/TΔV; modern modern saturation-vapor-pressure + 7%/K water-vapor amplification climate.
Hadley cell (Hadley 1735)
G Hadley 1735 + Bjerknes 1900 + Lorenz 1967 Hadley-Ferrel-Polar tri-cell; modern modern Hadley-cell expansion 2-3 ° equatorward 2024.
Ekman (1905)
V Ekman 1905 Ekman-spiral wind-driven; modern modern PBL parameterization + Coriolis-rotation + ocean-surface mixing climate models.
Lorenz attractor (1963)
E Lorenz 1963 deterministic-non-periodic-flow; modern modern foundational-text in chaos + butterfly-effect + ensemble-forecasting.
Bjerknes circulation (1898)
V Bjerknes 1898 circulation-theorem dΓ/dt; modern modern foundational baroclinic-zonal-instability + numerical-weather-prediction.