Annex A - Formulas

This annex lists the formulas behind the simulation. All values are SI units unless stated.

Saturation vapour pressure (Buck 1996)

\[ e_s = 6.1121 \cdot \exp\left(\frac{(18.678 - T/234.5) \cdot T}{257.14 + T}\right) \tag{1}\]

Where \(T\) is the temperature in \(\celsius\) and \(e_s\) is in \(\hpa\).

Actual vapour pressure

\[ e = e_s \cdot \frac{RH}{100} \tag{2}\]

Virtual temperature

\[ T_v = \frac{T_K}{1 - \frac{0.37802 \cdot e}{P}} \tag{3}\]

Air density

\[ \rho = \frac{P_{Pa}}{R_d \cdot T_v}, \quad R_d = 287.05287 \, \frac{J}{kg \cdot K} \tag{4}\]

The density \(\rho\) from Eq. 4 drives the ballistics model: a colder, drier, higher-pressure atmosphere is denser and slows the round more.

Wet-bulb temperature (Stull 2011)

\[ T_w = T \cdot \text{atan}(0.151977 \cdot \sqrt{RH + 8.313659}) + \text{atan}(T + RH) - \text{atan}(RH - 1.676331) + 0.00391838 \cdot RH^{1.5} \cdot \text{atan}(0.023101 \cdot RH) - 4.686035 \tag{5}\]

The atan terms are in radians. SQF atan returns degrees, so each term is multiplied by the factor 0.0174532925.

WBGT

\[ WBGT = 0.7 \cdot T_w + 0.2 \cdot T_g + 0.1 \cdot T \tag{6}\]

The globe temperature \(T_g\) is \(T + 15\) in full sun and \(T\) in full shade, interpolated by overcast. See Eq. 5 for \(T_w\).

Wind chill (JAG/TTI)

\[ T_{wc} = 13.12 + 0.6215 \cdot T - 11.37 \cdot V^{0.16} + 0.3965 \cdot T \cdot V^{0.16} \tag{7}\]

Where \(V\) is wind speed in \(\kmh\). Valid below 10 \(\celsius\) and above 4.8 \(\kmh\).

Solar declination

\[ \delta = 23.45 \cdot \sin\left(\frac{360}{365} \cdot (doy + 284)\right) \tag{8}\]

Solar elevation

\[ \sin(\alpha) = \sin(\phi) \cdot \sin(\delta) + \cos(\phi) \cdot \cos(\delta) \cdot \cos(h) \tag{9}\]

Where \(\phi\) is latitude and \(h\) is the hour angle (local_hour - 12) * 15 degrees. Radiation is max(0, sin(alpha)), scaled by cloud transmission 1 - 0.75 * overcast. The declination comes from Eq. 8.

Lapse rate

\[ \Delta T = -0.0065 \cdot \Delta z \tag{10}\]

Deterministic random (LCG)

\[ x_{n+1} = (x_n \cdot 1103515245 + 12345) \bmod 2^{31} \tag{11}\]

The seed combines mission time and an event salt, so the value is identical on every machine.

Determinism guarantee

The LCG in Eq. 11 is the reason AEE gives identical results on every client: the same seed produces the same value, so stochastic effects (scent dispersion, turbulence) resolve identically regardless of machine.

Rain extinction (Atlas 1954)

\[ \sigma = 0.21 \cdot R^{0.74} \qquad V = \frac{3.912}{\sigma} \tag{12}\]

Where \(\sigma\) is the extinction coefficient in \(\mathrm{km^{-1}}\), \(R\) is the rain rate in \(\mathrm{mm\,h^{-1}}\) (Arma rain maps to mm/h via \(R = \text{rain} \cdot 25\)), and \(V\) is the visual range in kilometres (Koschmieder). The visibility modifier is \(V/20\), clamped to 0.05..1.

Evaporation (FAO-56 Penman-Monteith)

\[ ET_0 = \frac{0.408 \cdot \Delta \cdot R_n + \gamma \cdot \frac{37}{T + 273.15} \cdot u_2 \cdot (e_s - e_a)}{\Delta + \gamma \cdot (1 + 0.34 \cdot u_2)} \tag{13}\]

Where \(\Delta\) is the slope of the saturation vapour-pressure curve (\(\mathrm{kPa\,^\circ C^{-1}}\)), \(R_n\) is the net radiation (game-scaled), \(\gamma\) is the psychrometric constant, \(u_2\) is the wind speed at 2 m in \(\ms\), and \(e_s - e_a\) is the vapour-pressure deficit.

Atmospheric seeing (Cn² boundary layer)

\[ C_n^2 = C_T^2 \cdot \left(\frac{79 \times 10^{-6} \cdot P}{T^2}\right)^2 \qquad s = 0.1 + 0.9 \cdot \frac{\log_{10}(C_n^2) + 17}{5} \tag{14}\]

Where \(C_T^2\) is the temperature structure parameter (from the ground-air delta and absolute temperature), \(P\) is pressure in hPa, \(T\) is temperature in K, and \(s\) is the seeing index (0.1..1). Night convection drops \(C_T^2\) by a factor of 100.

Smoke dispersal (physical chain)

\[ M = \frac{1}{1 + 0.12 v} \cdot \frac{1}{1 + 0.5 \tau} \cdot F_{K\ddot{o}hler} \cdot F_{buoyancy} \cdot \frac{1}{1 + 3 R} \tag{15}\]

Where \(v\) is wind speed, \(\tau\) is turbulence, \(F_{K\ddot{o}hler}\) is the hygroscopic growth factor (1 above 50 %RH), \(F_{buoyancy}\) is the buoyancy/inversion factor, and \(R\) is the rain rate. The chain combines advection, Taylor diffusion, particle growth, buoyancy, and rain washout.

Fire spread (Rothermel)

\[ ROS = 0.03 \cdot f \cdot (1 + 0.2 v) \cdot (1 + 0.1 \cdot slope) \cdot e^{-0.05 \cdot m} \tag{16}\]

Where \(f\) is the fuel factor (grass 1.0, forest 0.6), \(v\) is wind speed, \(slope\) is the slope in percent, and \(m\) is the fuel moisture. Area grows as a circle: perimeter \(= 2\pi\sqrt{A/\pi}\).

Freeze and thaw (Stefan)

\[ d = 0.05 \cdot \sqrt{FDD} \qquad d = 0.05 \cdot \sqrt{TDD} \tag{17}\]

Where \(FDD\) and \(TDD\) are the accumulated freezing and thawing degree-days. Depth grows with the square root, so a warm spell melts fast initially then slows — the physically correct non-linear response.

CBRN persistence (Arrhenius Q10)

\[ P(T) = \frac{P_{15}}{2^{(T-15)/10}} \cdot \frac{1}{1 + 0.5 \cdot \frac{RH - 50}{50}} \cdot \frac{1}{1 + 0.05 v} \tag{18}\]

Hydrolysis roughly doubles per 10 °C rise (Q10 rule of thumb). \(P_{15}\) is the reference persistence at 15 °C, \(RH\) is relative humidity in percent, and \(v\) is wind speed.

Avalanche slope risk

\[ R_{slope} = 1 - \frac{|slope - 35|}{20} \tag{19}\]

Clamped to 0..1: risk peaks at 35°, falls to zero at 15° and 55° — the published release-zone distribution (Schweizer et al., 2003).

Harmonic tides

\[ h(t) = \sum_i A_i \sin(\omega_i t + \phi_i) \tag{20}\]

With the four principal constituents: M2 (28.9841 °/h, 1.00 m), S2 (30.0000 °/h, 0.47 m), K1 (15.0411 °/h, 0.58 m), O1 (13.9430 °/h, 0.42 m). The M2-S2 beat produces the spring/neap cycle (about 14.8 days). \(t\) is hours since a fixed epoch, so the tide is a deterministic function of the in-game date.

Sea state (WMO Beaufort)

\[ B = \left(\frac{v}{0.836}\right)^{2/3} \qquad H_s = 0.0246 \cdot v^2 \tag{21}\]

Where \(v\) is wind speed in \(\ms\) and 0.836 m/s is one knot. \(B\) is the Beaufort number (0..12) and \(H_s\) is the significant wave height in metres (Pierson-Moskowitz fully-developed sea).

Turbulence (ICAO EDR)

\[ EDR = 0.7 \cdot \tau \qquad \text{classes: } <0.1, 0.1\text{-}0.3, 0.3\text{-}0.5, >0.5 \tag{22}\]

Where \(\tau\) is the raw turbulence value (0..1) and \(EDR\) is the eddy dissipation rate in \(\mathrm{m^{2/3}\,s^{-1}}\). The classes map to LIGHT, MODERATE, SEVERE, EXTREME per ICAO Annex 3.

Radio path loss (Friis)

\[ FSPL = 20\log_{10}(d) + 20\log_{10}(f) - 147.55 \tag{23}\]

Where \(d\) is distance in metres and \(f\) is frequency in Hz. The propagation index is \(0.3 + 1.7\sqrt{signal}\), where signal is the link budget ratio. Ducting adds a dB bonus; hot-dry air and terrain add loss.

Ionospheric absorption (ITU-R P.531)

\[ A = 0.5 \cdot \left(\frac{3 \times 10^6}{f}\right)^2 \cdot D \cdot \left(1 + \frac{SSN}{100}\right) + flare \tag{24}\]

Where \(f\) is frequency in Hz, \(D\) is the day/night factor (1 day, 0.05 night), and \(SSN\) is the sunspot number. Absorption falls as \(1/f^2\) and follows the solar zenith angle. Flares add a sudden ionospheric disturbance term (Dellinger effect).

Hypoxia (FAA time of useful consciousness)

\[ z_{eq} = 44330 \cdot \left(1 - \left(\frac{P}{1013.25}\right)^{0.1903}\right) \qquad \text{risk} = \frac{exposure}{TUC} \tag{25}\]

The equivalent oxygen altitude \(z_{eq}\) comes from the ISA hypsometric relation. The TUC table interpolates 6000 m → 1800 s, 7000 m → 240 s, 8000 m → 120 s, 9000 m → 45 s, 10000 m → 15 s. Below 6000 m the TUC is effectively infinite. Risk clamps to 0..1.

UV index (ISO 17166 / CIE)

\[ UVI = 12.5 \cdot \sin(\alpha) \cdot e^{-0.0013 \cdot O_3 / \sin(\alpha)} \cdot F_{alt} \cdot F_{cloud} \cdot F_{month} \tag{26}\]

Where \(\alpha\) is the solar elevation, \(O_3\) is the ozone column in Dobson units (300 default), \(F_{alt}\) adds 10 % per 1000 m, \(F_{cloud}\) cuts 70 % at full overcast, and \(F_{month}\) is the northern-hemisphere summer boost. Clamped 0..13.

Engine power (SAE J1349 density correction)

\[ \frac{P}{P_0} = \left(\frac{\rho}{1.225}\right)^{1.2} \tag{27}\]

The naturally-aspirated power derates with the air-density ratio to the power 1.2. The turbocharged model adds an altitude term: 35 % loss by 9000 m.

Traction (slip curve)

\[ F = \mu \cdot W \cdot (1 - e^{-k \cdot s}) \tag{28}\]

Where \(\mu\) is the surface friction coefficient, \(W\) is the wheel load, \(s\) is the slip (0..1), and \(k\) is the slip coefficient (8..12 dry). Traction degrades smoothly at high slip instead of dropping abruptly.

River water level (Nash cascade)

A cascade of three serial linear reservoirs smooths rainfall into a lagged, peaked hydrograph response instead of an instantaneous one.

Route degradation (NRMM cone index)

Every vehicle passage reduces the soil cone index in proportion to ground pressure; the soil recovers exponentially. Passability is the ratio of the current cone index to the required threshold, clamped 0..1.