AOD Formula (First part)

Ideal gas law for moist air (pressure in Pa)

P = rho * Rd * Tv

Hydrostatic pressure at height z for a thin layer (in hPa)

P(z) = P0 - (rho * g * z) / 100

Geometric thickness between two pressure levels P1 > P2 (Hypsometric equation)

dz = (Rd * Tv_mean / g) * ln(P1 / P2)

Mean virtual temperature of a layer from thickness dz and pressures P1, P2

Tv_mean = (g * dz) / (Rd * ln(P1 / P2))

Potential temperature theta from T (in K) and P (in hPa)

theta = T * (1000 / P)^0.286

Actual temperature T (in K) from potential temperature theta and pressure P

T = theta * (P / 1000)^0.286

Exact virtual temperature Tv from T (in K) and mixing ratio w (in kg/kg)

Tv = T * (1 + 0.61 * w)

Fast approximation of virtual temperature Tv from T and w (in g/kg)

Tv = T + w_g_kg / 6

Magnus formula for saturation vapor pressure es (in hPa) using T in Celsius

es(T) = 6.11 * exp((17.27 * T_C) / (T_C + 237.3))

Inverse Magnus formula for dewpoint Td (in Celsius) from actual vapor pressure e

Td = (237.3 * ln(e / 6.11)) / (17.27 - ln(e / 6.11))

Relative humidity RH (%) from vapor pressure or mixing ratio

RH = (e / es) * 100 = (w / ws) * 100

Exact and approximate mixing ratio w (in kg/kg) from e and P

w = 0.622 * e / (P - e) ≈ 0.622 * e / P

Actual vapor pressure e from mixing ratio w (in kg/kg) and total pressure P

e = (w * P) / (0.622 + w)

Condensed liquid water mixing ratio above the LCL

w_L = w_initial - ws(P)

Approximate LCL height (in meters) from surface T and Td

z_LCL ≈ 125 * (T - Td)

Equivalent potential temperature theta_e from theta, w (kg/kg), and T_LCL (K)

theta_e ≈ theta * exp((Lv * w) / (cpd * T_LCL))

Brunt-Vaisala frequency squared N^2 from potential temperature gradient

N^2 = (g / theta_mean) * (d_theta / dz)

Brunt-Vaisala frequency squared N^2 from environmental lapse rate

N^2 = (g / T_mean) * (Gamma_d - Gamma_env)

Lifted Index (LI) formula and meaning of T'_surface

LI = T500 - T'_surface (parcel lifted from surface to 500 hPa)

Showalter Stability Index (SSI) formula and meaning of T'_850

SSI = T500 - T'_850 (parcel lifted from 850 hPa to 500 hPa)

Stability thresholds for Lifted Index (LI) and Showalter Index (SSI)

0: Stable | 0 to -3: Weakly unstable | -3 to -6: Moderately unstable | <= -6: Severe thunderstorms

Total Totals Index (TTI) formula

TTI = T850 + Td850 - 2 * T500

Convective thresholds for Total Totals Index (TTI)

< 44: Unlikely | 44-50: Isolated thunderstorms | 51-54: Severe thunderstorms | >= 55: Tornadoes likely

K-Index (KI) formula

KI = (T850 - T500) + Td850 - (T700 - Td700)

Convective thresholds for K-Index (KI)

< 20: No thunderstorms | 20-30: Isolated thunderstorms | > 30: Severe thunderstorms

Approximate total positive area on Skew-T (in K*m) between LFC and EL

Tot_Area ≈ (Delta_T_max * dz) / 2

CAPE (in J/kg) from total positive area and mean layer temperature (in K)

CAPE ≈ (g / ) * Tot_Area

Instability thresholds for CAPE (in J/kg)

0: Stable | 0-1000: Weak | 1000-2500: Moderate | > 2500: Extreme / Severe storms