The complete mathematical formulation behind RADOSE — from source term definition to atmospheric transport and radiological dose assessment.
Overview
RADOSE implements a steady-state Gaussian plume model to estimate the atmospheric transport of radionuclides released from a point source, and converts the resulting air concentrations and ground deposition into radiological dose. The computation proceeds in three stages, evaluated independently at every receptor of a fixed downwind grid.
Source Term
The released activity Q is derived from the total inventory using the five-factor formula. Each factor scales down the material at risk to the fraction that actually becomes airborne and escapes to the environment in respirable form.
Atmospheric Dispersion
The time-integrated air concentration at a receptor located at downwind distance x, crosswind offset y and height z follows the Gaussian plume solution of the advection–diffusion equation with total reflection at the ground:
The vertical term Fz(x) superimposes the direct plume and its ground-reflected image, so no activity is lost through the surface:
Dispersion Coefficients
Turbulent spread grows with distance at a rate set by the Pasquill–Gifford atmospheric stability class. RADOSE uses the standard-terrain parameterisations below, with x in metres:
| Class | Stability | σy(x) [m] | σz(x) [m] |
|---|---|---|---|
| A | Very unstable | 0.22·x/√(1+0.0001x) | 0.20·x |
| B | Unstable | 0.16·x/√(1+0.0001x) | 0.12·x |
| C | Slightly unstable | 0.11·x/√(1+0.0001x) | 0.08·x/√(1+0.0002x) |
| D | Neutral | 0.08·x/√(1+0.0001x) | 0.06·x/√(1+0.0015x) |
| E | Slightly stable | 0.06·x/√(1+0.0001x) | 0.03·x/(1+0.0003x) |
| F | Stable | 0.04·x/√(1+0.0001x) | 0.016·x/(1+0.0003x) |
Class A–C: daytime convective conditions · Class D: overcast / high wind · Class E–F: clear, calm nights with strong temperature inversions.
Wind Profile
Wind speed is measured at a reference height (typically 10 m) and extrapolated to the release height with a power-law profile whose exponent p depends on stability class and terrain roughness:
| Class | p — Standard terrain | p — Urban terrain |
|---|---|---|
| A | 0.07 | 0.15 |
| B | 0.07 | 0.15 |
| C | 0.10 | 0.20 |
| D | 0.15 | 0.25 |
| E | 0.35 | 0.30 |
| F | 0.55 | 0.30 |
Radioactive Decay
Activity decays during plume travel. With a transport time of x/u to the receptor, the surviving fraction is:
For short-range assessments this term is significant only for short-lived nuclides; for Cs-137 (T½ = 30.17 y) it is effectively unity within the 80 km grid.
Plume Depletion
Dry deposition continuously removes activity from the plume as it travels. Conservation of mass is enforced with a source-depletion factor, evaluated by numerical integration along the plume path:
For noble gases vd = 0, so DF(x) = 1 and the plume is not depleted.
Ground Deposition
The activity deposited per unit ground area is proportional to the ground-level time-integrated concentration (z = 0):
Dg is reported in kBq/m² and drives the ground-shine dose pathway below.
Dose Pathways
Three exposure pathways are evaluated at each receptor and summed into the Total Effective Dose (TED). Dose conversion factors couple the physical fields to effective dose.
Dose Conversion Factors
Nuclide-specific factors are taken from the ICRP 107 decay database with effective-dose coefficients per ICRP 68/72 methodology (inhalation: 1 μm AMAD, 50-year committed effective dose). Cs-137 values include the Ba-137m daughter in secular equilibrium.
| Nuclide | T½ | DCF inh [Sv/Bq] | DCF sub [(Sv·m³)/(Bq·s)] | DCF gnd [(Sv·m²)/(Bq·s)] |
|---|---|---|---|---|
| I-131 | 8.04 d | 7.39E-09 | 1.70E-14 | 3.65E-16 |
| Cs-137 (+Ba-137m) | 30.17 y | 4.67E-09 | 2.55E-14 | 5.49E-16 |
| Xe-133 | 5.24 d | — (noble gas) | 1.37E-15 | 4.06E-17 |
Noble gases have no inhalation or ingestion pathway and do not deposit (v_d = 0); their dose is dominated by cloud submersion.
Receptor Grid & Assumptions
All quantities are evaluated on a fixed logarithmic-like mesh of 20 downwind receptors from 0.03 km to 80 km, at a default receptor height of 1.5 m (breathing zone). Plume arrival time at each receptor is estimated as x/u.
- Steady-state release into a homogeneous wind field over flat terrain.
- Total ground reflection — no activity is absorbed at the surface except via the explicit deposition term.
- Dry deposition only; wet scavenging (rain-out / wash-out) is not modelled.
- H is the effective release height — buoyant plume rise must be included in H by the user.
- Valid in the near-to-mid field (≲ 80 km); long-range transport requires segmented or puff models.
- Doses are computed for a stationary adult receptor with constant breathing rate.