The Trajectory Equation

The advection of a particle or puff is computed from the average of the three-dimensional velocity vectors at the initial-position P(t) and the first-guess position P'(t+Δt). The velocity vectors are linearly interpolated in both space and time. The first guess position is

• P'(t+Δt) = P(t) + V(P,t) Δt

• P(t+Δt) = P(t) + 0.5 [ V(P,t) + V(P',t+Δt) ] Δt.

The integration time step (Δt) can vary during the simulation. It is computed from the requirement that the advection distance per time-step should be less than the grid spacing. The maximum transport velocity is determined from the maximum transport speed during the previous hour. Time steps can vary from 1 minute to 1 hour and are computed from the relation,

• U_max(grid-units min^-1) Δt (min) < 0.75 (grid-units).

The integration method is very common (e.g. Kreyszig, 1968) and has been used for trajectory analysis (Petterssen, 1940) for quite some time. Higher order integration methods will not yield greater precision because the data observations are linearly interpolated from the grid to the integration point. Trajectories are terminated if they exit the meteorological data grid, but advection continues along the surface if trajectories intersect the ground.