Key Formulas & Equations
Hydrostatic Equation: ∂P/∂z = -ρg
Pressure-Height: P = P₀ × e(-z/H) (H = scale height ≈ 8.5 km)
Geopotential Height: Z = (r × z)/(r + z)
Coriolis Parameter: f = 2Ω sin(φ) (Ω = 7.292×10⁻⁵ rad/s)
Geostrophic Wind: Vg = (1/ρf) × (∂P/∂n)
Gradient Wind: Balance of pressure gradient, Coriolis, and centrifugal forces
Saturation Vapor Pressure: es = 6.11 × 10(7.5T/(237.3+T)) hPa
Relative Humidity: RH = e/es × 100%
Dew Point: Td = (237.3 × ln(e/6.11))/(7.5 - ln(e/6.11))
Lapse Rate: Γ = -dT/dz; Dry adiabatic: Γd = 9.8°C/km
Saturated Adiabatic: Γs ≈ 4-7°C/km
Static Stability: Stable if Γ < Γs; Unstable if Γ > Γd
Wind Chill: WC = 13.12 + 0.6215T - 11.37v0.16 + 0.3965Tv0.16
Heat Index: Complex function of T and RH
Hypsometric Equation: Z₂ - Z₁ = (RdTv/g) × ln(P₁/P₂)
Core Concepts
Atmospheric Layers
Troposphere (0-12 km), stratosphere (12-50 km), mesosphere (50-80 km), thermosphere (80-600 km). Temperature and composition vary with altitude.
Air Masses and Fronts
Large bodies of air with uniform temperature and humidity. Fronts are boundaries between air masses: cold, warm, stationary, and occluded.
Cloud Formation
Clouds form when air rises, cools to saturation, and water vapor condenses. Types: cumulus, stratus, cirrus, and nimbus based on altitude and shape.
Severe Weather
Thunderstorms, tornadoes, hurricanes, and blizzards. Formation requires specific atmospheric conditions: instability, moisture, and lift.
Climate Systems
Long-term weather patterns: El Niño/La Niña, monsoons, and jet streams. Influenced by ocean circulation, solar radiation, and greenhouse gases.
Weather Forecasting
Prediction of future weather conditions using numerical weather prediction (NWP) models, satellite data, and radar. Accuracy decreases with time.
Atmospheric Optics
Optical phenomena caused by atmospheric conditions: rainbows, halos, mirages, and blue sky. Caused by refraction, reflection, and scattering of light.
Climate Change
Long-term changes in global temperature and weather patterns. Driven by greenhouse gas emissions, with impacts on sea level, ecosystems, and extreme weather.
Worked Examples
Example 1: Relative Humidity
Problem: Air at 20°C has vapor pressure 12 hPa. Saturation vapor pressure at 20°C = 23.4 hPa. Calculate RH.
Solution:
RH = 12/23.4 × 100% = 51.3%
Example 2: Dew Point
Problem: Air has vapor pressure 10 hPa. Estimate dew point.
Solution:
Td ≈ (237.3 × ln(10/6.11))/(7.5 - ln(10/6.11)) = 7.3°C
Example 3: Potential Temperature
Problem: Air at 850 hPa has temperature 15°C. Calculate potential temperature (P₀ = 1000 hPa, R/cp = 0.286).
Solution:
θ = 288.15 × (1000/850)0.286 = 288.15 × 1.048 = 302 K = 29°C
Practice Problems
Problem 1
Calculate RH if e = 8 hPa and es = 16 hPa.
Answer: RH = 8/16 × 100% = 50%
Problem 2
Calculate Coriolis parameter at 45°N (f = 2Ω sin(45°)).
Answer: f = 2 × 7.292×10⁻⁵ × 0.707 = 1.03×10⁻⁴ s⁻¹
Problem 3
Calculate potential temperature for T = 10°C at 900 hPa.
Answer: θ = 283.15 × (1000/900)0.286 = 283.15 × 1.029 = 291.4 K
Problem 4
Identify the dry adiabatic lapse rate.
Answer: Γd = 9.8°C/km
Problem 5
Calculate geostrophic wind if ∂P/∂n = 2 hPa/100 km, ρ = 1.2 kg/m³, f = 10⁻⁴ s⁻¹.
Answer: Vg = (1/(1.2 × 10⁻⁴)) × (200/100,000) = 16.7 m/s
Problem 6
Calculate dew point for vapor pressure 15 hPa.
Answer: Td ≈ (237.3 × ln(15/6.11))/(7.5 - ln(15/6.11)) = 13.1°C
Problem 7
Calculate height difference if P₁ = 1000 hPa, P₂ = 850 hPa, Tv = 15°C (Rd = 287, g = 9.81).
Answer: ΔZ = (287 × 288.15/9.81) × ln(1000/850) = 8435 × 0.1625 = 1,370 m