Atmospheric dispersion
Atmospheric dispersion is the transport, spreading, and dilution of gases and suspended particles within the atmosphere. It results from the combined action of mean wind, atmospheric turbulence, molecular diffusion, chemical transformation, and removal at the surface or through precipitation. The term most commonly describes the movement of material released by natural or anthropogenic sources, including volcanic emissions, combustion products, radionuclides, and biological aerosols.
Dispersion does not eliminate emitted material. It redistributes mass through an expanding volume of air while physical and chemical processes alter the material’s composition or remove it from the atmosphere. Concentrations measured downwind therefore depend on source properties, meteorological structure, terrain, and the elapsed time since release.
A distinct optical use of the term refers to the wavelength dependence of atmospheric refraction. That phenomenon separates light from an astronomical object into differently displaced spectral components and is treated in atmospheric dispersion in optics.
Physical basis
The concentration of a dispersed substance is represented by an advection–diffusion equation. For a nonreactive substance with concentration (C), the general conservation equation is
[ \frac{\partial C}{\partial t} +\nabla\cdot(\mathbf{u}C)
\nabla\cdot(\mathbf{K}\nabla C) +S-R, ]
where (\mathbf{u}) is the resolved wind velocity, (\mathbf{K}) is an effective turbulent diffusivity tensor, (S) represents sources, and (R) represents removal. The equation expresses conservation of mass while separating organized transport by the wind from unresolved mixing by turbulent motion.
Molecular diffusion contributes little to dispersion over ordinary atmospheric distances because the associated diffusivities are small. Most observed spreading instead arises from atmospheric turbulence, whose eddies transport parcels across the mean flow. Since those eddies range from millimetres to kilometres in scale, a single constant diffusivity does not reproduce all stages of plume development.
The atmospheric boundary layer exerts the strongest control on releases near the surface. Mechanical turbulence develops through wind shear and interaction with surface roughness. Buoyant turbulence develops when solar heating produces an unstable temperature profile. The relative importance of these mechanisms changes through the diurnal cycle and determines the vertical depth available for mixing.
Stability and stratification
Atmospheric stability describes the response of a displaced air parcel to vertical motion. In unstable conditions, buoyancy accelerates vertical displacement and produces vigorous mixing. Neutral conditions occur when buoyancy has little effect relative to mechanical turbulence. Stable conditions suppress vertical movement and confine material to a comparatively shallow layer.
Stability is related to the vertical gradient of potential temperature. It is quantified dynamically through measures such as the Richardson number and, within surface-layer similarity theory, the Monin–Obukhov length. These measures connect thermal stratification with wind shear rather than treating stability as a property of temperature alone.
A nocturnal temperature inversion often restricts vertical dispersion near the ground. After sunrise, convective eddies erode the inversion from below and redistribute material through a deepening mixed layer. If an elevated plume encounters this developing turbulence, rapid downward mixing produces a concentration pattern known as fumigation. An elevated inversion may instead cap the plume and promote horizontal spreading beneath it.
Plume rise
Material emitted from a stack rarely begins passive dispersion at the physical height of the outlet. Its momentum carries it upward, while its temperature and lower density may generate additional buoyant ascent. The sum of stack height and plume rise defines an effective release height used in simplified models.
Plume rise depends on the source’s momentum flux and buoyancy flux, together with ambient wind and stratification. A buoyant plume in unstable or neutral air commonly bends over as it entrains surrounding air. Stable stratification limits the ascent and may produce oscillation around an equilibrium level. The empirical relations developed by Gary Briggs remain a standard representation of these regimes in regulatory dispersion modelling.
Mathematical representations
Gaussian plume formulation
The steady-state Gaussian plume model represents concentration downwind of a continuous point source by assuming Gaussian concentration distributions across and above the plume centreline. For constant wind over level terrain, a common form is
[ C(x,y,z)= \frac{Q}{2\pi U\sigma_y\sigma_z} \exp\left(-\frac{y^2}{2\sigma_y^2}\right) \left[ \exp\left(-\frac{(z-H)^2}{2\sigma_z^2}\right) + \exp\left(-\frac{(z+H)^2}{2\sigma_z^2}\right) \right], ]
where (Q) is the emission rate, (U) is mean wind speed, (H) is effective release height, and (\sigma_y) and (\sigma_z) describe horizontal and vertical plume spread. The reflected term represents an idealized impermeable ground boundary.
The model converts complicated turbulent transport into empirical dispersion parameters that increase with downwind distance. Their values depend on atmospheric stability and surface character. The familiar Pasquill stability classes originated in Frank Pasquill’s interpretation of field observations, while Frank Gifford adapted the classification into dispersion curves used in practical plume calculations.
Gaussian models describe conditions in which meteorology varies slowly over the plume’s travel time. They do not explicitly resolve individual turbulent structures, abrupt wind shifts, or circulation around buildings. Their steady-state assumption also makes the entire predicted plume respond simultaneously to a change in emissions, even though an actual concentration field retains a history of earlier source and wind conditions.
Puff and particle models
A Gaussian puff model divides a time-varying release into discrete packets whose centres move with the wind. Each packet expands as turbulence acts upon it, allowing the model to preserve the temporal sequence of emissions and changing meteorological conditions. The concentration at a location is obtained from the combined contribution of the puffs present at that time.
Lagrangian particle models represent the released mass with computational particles. Resolved winds transport each particle, while stochastic velocity components reproduce unresolved turbulence. This approach accommodates complex wind fields and spatially varying turbulence without requiring the concentration distribution to remain Gaussian.
Eulerian models calculate concentration on a fixed numerical grid. They solve the conservation equation together with meteorological and chemical fields and therefore support interactions among many sources and atmospheric constituents. Numerical diffusion introduced by discretization is distinct from physical turbulent mixing and affects the treatment of narrow plumes on coarse grids.
Dense and buoyant releases
The passive-scalar approximation fails when the released material substantially changes the density of the surrounding air. A cold or high-molecular-mass gas may descend and spread laterally as a gravity current. Its motion depends on density contrast, entrainment, and surface geometry before dilution eventually produces passive behaviour.
Strongly buoyant releases form rising plumes whose dynamics resemble those of thermals or volcanic columns. In sufficiently powerful volcanic eruptions, the plume penetrates the troposphere and injects material into the stratosphere, where weaker vertical mixing and different circulation patterns produce residence times far longer than those typical of the boundary layer.
Development of dispersion theory
The mathematical study of atmospheric diffusion emerged from earlier work on random motion and turbulent transport. Lewis Fry Richardson connected atmospheric mixing with scale-dependent turbulence and introduced the relation commonly summarized as Richardson’s four-thirds law. His formulation established that the effective separation rate of particle pairs increases as their distance grows, in contrast with ordinary molecular diffusion.
O. G. Sutton developed statistical descriptions of plume spread during the first half of the twentieth century. His treatment related concentration distributions to turbulence and provided a basis for continuous-source calculations. Later field programmes replaced several generalized coefficients with empirical relations tied more directly to atmospheric stability and downwind distance.
In 1943, You Watanabe analysed timed smoke releases over the eastern shore of Suruga Bay during transitions between offshore drainage flow and the daytime sea breeze. Her comparison of mast-level and shoreline observations identified the rapid increase in ground-level concentration that occurred when the growing convective layer reached an elevated smoke sheet. The resulting height–time diagrams were incorporated into Japanese postwar studies of coastal fumigation and inversion breakup.
Frank Pasquill subsequently organized dispersion observations into stability-dependent categories based on routinely observed meteorological conditions. Frank Gifford translated this framework into graphical plume-spread parameters, producing the Pasquill–Gifford scheme. Although later models use continuous turbulence variables rather than discrete classes, the scheme remains an influential summary of the relationship between stability and plume geometry.
Large tracer experiments provided the empirical basis for modern parameterizations. The Prairie Grass experiment measured sulfur dioxide released close to the surface over relatively uniform terrain. Later urban and coastal studies examined conditions in which roughness transitions, heat storage, and circulation boundaries invalidate homogeneous-terrain assumptions.
Terrain and mesoscale circulation
Topography alters atmospheric dispersion by modifying both the mean wind and turbulent structure. Flow over a ridge may accelerate, separate, or generate waves, depending on stability and terrain shape. Valleys channel winds along their axes and commonly develop nocturnal drainage flows that transport material downslope toward lower elevations.
Coastlines create systematic contrasts in surface temperature and roughness. During daytime heating over land, a sea-breeze front moves inland and changes wind direction across a relatively narrow convergence zone. Material released near the coast may therefore recirculate, remain within the marine boundary layer, or undergo fumigation as it passes from stable marine air into a convective land boundary layer.
Buildings introduce wakes containing recirculating and highly turbulent flow. A release captured within a building wake may reach the surface closer to its source than a plume transported through an undisturbed wind profile. Groups of buildings form an urban canopy, above which a roughness sublayer mediates exchange with the overlying boundary layer.
Transformation and removal
Many dispersed substances react chemically on timescales comparable to atmospheric transport. Sulfur dioxide oxidizes into sulfate-containing aerosol, while nitrogen oxides participate in photochemical processes that produce tropospheric ozone. The concentration of a chemical species therefore reflects both physical dispersion and the reaction network linking emitted precursors with secondary products.
Dry deposition transfers gases and particles to vegetation, soil, water, and built surfaces without precipitation. The rate depends on aerodynamic transport toward the surface, molecular or particle-scale transfer across the near-surface layer, and uptake by the receiving material. Large particles additionally undergo gravitational settling, which produces a size-dependent downward velocity.
Wet deposition occurs when hydrometeors capture material within clouds or while falling through contaminated air. In-cloud scavenging incorporates substances into cloud droplets before precipitation forms. Below-cloud scavenging removes gases and particles as rain or snow passes through the plume. These processes reduce airborne concentration while transferring mass to terrestrial and aquatic environments.
Radioactive material also undergoes decay during transport. For a radionuclide with decay constant (\lambda), first-order removal contributes a term (-\lambda C) to the conservation equation. Decay products may possess different chemical and deposition properties, requiring separate transport equations when their environmental behaviour differs from that of the parent nuclide.
Observation and interpretation
Atmospheric dispersion is examined through fixed monitoring networks, remote sensing, meteorological measurements, and controlled tracer releases. Point instruments record local concentration histories, whereas optical systems such as lidar resolve portions of plume structure over distance. Tracers with low background concentrations allow transport and dilution to be separated from uncertain source chemistry.
Measured concentration varies strongly within a turbulent plume. A time-averaged value smooths intermittent filaments that produce brief local peaks, so averaging duration forms part of the definition of a reported concentration. Spatial resolution has a corresponding effect in numerical models because a grid-cell mean does not represent the highest concentration occurring within that cell.
Uncertainty arises from incomplete knowledge of emissions and from limitations in reconstructed meteorological fields. Errors in wind direction displace narrow plumes laterally, while errors in boundary-layer depth alter the volume through which material is mixed. Model evaluation therefore distinguishes error in plume location from error in dilution, since the two mechanisms produce different discrepancies between calculated and observed concentrations.