PHYS/OCEA 4411/5411 A Atmospheric Dynamics I - Key formulas and constants

Nov 2, 2026

Uploaded by Aurelia

Compiled reference sheet for Atmospheric Dynamics, listing core physical constants and standard fluid-dynamics quantities used in the course. Includes dry adiabatic lapse rate, thermodynamic parameters, Earth rotation and Coriolis terms, ideal-gas and potential-temperature relations, hydrostatic and momentum balances, continuity in Z and pressure coordinates, geostrophic and thermal-wind relationships, and circulation/vorticity with potential vorticity formulations, including associated tendencies.

LanguageEnglish

CategoryTechnology

FormatPDF

Pages6

Reading Time15 min

AccessFree to download

Constants and thermodynamic definitions

Wind, forces, and momentum equations

Hydrostatic balance and derivatives

Stability, buoyancy, and thickness relations

Geostrophic wind, prognostic forms, and continuity

Pressure coordinates and thermal wind

Circulation and vorticity

Potential vorticity and vorticity tendency

Formulas/Constants: PHYS/OCEA 4411/5411 A Atmospheric Dynamics I 􀀀 d = 9 .8 oC/km (dry adiabatic lapse rate) 􀀍 = cp /cv = 1 .4 R = 287.05 J/(kg K) cv = 717.5 J/(kg K) cp = 1004.5 J/(kg K) g0 = 9 .81 m/s2 f = 2􀀊sin􀀞 􀀊 = 7 .292 × 10 􀀀5 rad/sec 􀀚0 = 1 .225 kg/m3 p0 = 1013 .25 hPa Md = 28 .94 g/mole (Mean molecular weight of dry air) 􀀗 = 1 .46 × 10 􀀀5m2 /sec (sea level) 􀀀 d = 9 .8 oC/km a = 6370 km (radius of the earth) speci􀀌c volume: 􀀋 = 1/􀀚 Rossby Number: R0 = U/fL Viscous Force per unit area A between two plates with relative velocity u0 and separation l: 􀀖Au 0 F = ~~ ~~ l Viscous acceleration within a 􀀍uid in the x direction due to a shear in u in the z direction (􀀗 is the kinematic viscosity): 􀀖 ∂2 u ∂ 2 u a = = 􀀗 􀀚 ∂z 2 ∂z 2 Angular momentum: L = Iω, where the moment of inertia of a point mass m is I = mR2 (R is the distance from the axis of rotation), ω = u/R is the angular velocity, and u is the tangential (zonal) velocity. Momentum/unit mass about earths axis(with zonal speed u) : M = L/m = (􀀊acos􀀞 + u)cos􀀞 Relationship between local and total derivative: ∂ d ~~ ~~ = ~~ ~~ − U · ∇ ∂t dt ∂ ∂ ∂ ∇ = i + j + k ∂x ∂y ∂z ∇2 F = F2 + 2 + 2 Relative vector vorticity : ω = ∇ × U ∂v ∂u Relative scalar vorticity : 􀀘 = − ∂x ∂y Total vs Local Derivative : ddt = + U · ∇ Pressure Gradient Acceleration : Fm = − 1􀀚∇p = −∇p 􀀈 Total Gravity : g (􀀞) = g􀀃 (􀀞) + 􀀊2 R De􀀌nition of Geopotential : d􀀞 = gdz = g0 dZ 􀀞 (z) = Z0 z gdz 􀀞 (z) Z = ~~ ~~ g0 Hydrostatic Relationship : dpdz = −􀀚g dU 1 Momentum Equation (Z surface) : dt = −2􀀊 × U − 􀀚 ∇p + g + Fr Primitive equations: du uv tan􀀞 uw 1 ∂p dt − a + a = − 􀀚 ∂x + 2􀀊v sin􀀞 − 2􀀊w cos􀀞 + Frx dvdt + u2~~ ~~taan􀀞 + vwa = − 1􀀚 py + −2􀀊usin􀀞 + Fry dw u2 + v 2 1 ∂p dt − a = − 􀀚 ∂z − g + 2􀀊u cos􀀞 + Frz Ideal Gas Law : p = 􀀚RT Potential Temperature: 􀀒 = T (p0 /p)0.286 = T (p0 /p)R/cp , where p0 = 105 Pa T d􀀒 dT g dT = + = + 􀀀 d 􀀒 dz dz cp dz Buoyancy acceleration : B = dwdt = g ~~ ~~(T~~ ~~−T0T0~~ ~~) = g 􀀒~~ ~~0􀀒0 = −N2 􀀎z (small 􀀎z) 2 d ln􀀒 g d􀀒 Buoyancy frequency : N = g dz = 􀀒 dz N is an angular frequency so the period of the oscillation is T = 2􀀙/N Thickness Relationship : 􀀁Z = R~~ ~~ln (p1 /p2 ) 1 Geostrophic wind (Z surface) : Vg = k × 􀀚f ∇p 1 ∂pug = − 􀀚f ∂y vg = 1􀀚f px Approximate Prognostic equations (coordinate form on a Z surface): du 1 ∂p dt = fv − 􀀚 ∂x = f (v − vg) dv 1 ∂p dt = −fu − 􀀚 ∂y = −f(u − ug) Approximate Prognostic equation (vector form on a Z surface): dV = −fk × Vag dt Continuity equation (Z surface) : 􀀚t + ∇ · (􀀚U) = 0 1 d􀀚 Continuity equation (Z surface) : + ∇ · U = 0 􀀚 dt Thermodynamic equation : cv dTdt + p dd􀀋t = Q˙ Thermodynamic equation : cp dTdt − 􀀋 dpdt = Q˙ Mechanical Energy Equation : Q˙ = ddt 􀀔 ~~ ~~(u2~~ ~~+~~ ~~v~~ ~~+~~ ~~w2~~ ~~) + 􀀈 + cvT + p􀀋􀀕 − 􀀋 pt − V · F Bernouilli Equation : (u2~~ ~~+~~ ~~v~~ ~~+~~ ~~w2~~ ~~) + 􀀈 + cvT + p􀀋 = constant d􀀒 􀀒 Q˙= dt T cp T􀀒 􀀒z = Tz + gcp = −􀀀 + 􀀀d dV Momentum Equation p surface : dt + fk × V = −∇p 􀀞 De􀀌nition of pressure velocity : ω = ddpt 1 Vector Geostrophic wind on pressure surface :Vg = k × f ∇p 􀀞 ug = − ~~ ~~1f 􀀞y 1 ∂􀀞 vg = f ∂x ∇p · Vg ≈ 0 Continuity equation in pressure coordinates : 􀀒 ux + vy􀀓 p + ωp = 0 Continuity equation in pressure coordinates : ∇ · Vh = − ωp Thermodynamic equation in pressure coordinates : 􀀒 Tt + u Tx + v Ty􀀓 − ω􀀛p = Qcp˙ (−∇T · Vh) Approximate expression on a p surface : ω ≈ ~~ ~~ −􀀛p Approximate expression on a PT surface : ω ≈ Vh · ∇p T ∂􀀒 ∂T RT 􀀋 ∂T 􀀀 d − 􀀀 Static stability : 􀀛p = − = − + = − = 􀀒 ∂p ∂p cpp cp ∂p 􀀚g If pressure levels have 􀀌xed z, or ∂p/∂t = 0: ω = −􀀚gw Surface Pressure Tendency : s ≈ − Z0ps (∇ · V)dp Kinematic Estimate of ω : ω(p) = ω(ps) − Zp 􀀒 ux + vy􀀓 dp Natural coordinates : dVdt = − 􀀞s V 2 ∂􀀞 Natural coordinates : = − −

FAQs

What core atmospheric constants and gas parameters are listed?

The sheet provides values such as the dry adiabatic lapse rate, the specific heat ratio, gas constant for dry air, and standard gravity, along with Earth-rotation and sea-level reference density and pressure.

How are geostrophic and wind relationships expressed?

It includes geostrophic wind relations on Z and pressure surfaces, plus approximate prognostic equations that connect ageostrophic components to pressure-gradient forcing.

Which stability and vertical-motion concepts are included?

It covers potential temperature, buoyancy acceleration and buoyancy frequency, static stability, thickness relations, and links between vertical velocity and continuity in pressure coordinates.

Nov 2, 2026

Uploaded by Aurelia