In fluid mechanics, meteorology (weather) and oceanography, a trajectory traces the motion of a single point, often called a parcel, in the flow.
Trajectories are useful for tracking atmospheric contaminants, such as smoke plumes, and as constituents to Lagrangian simulations, such as contour advection or semi-Lagrangian schemes.
If the velocities are gridded in space and time, then bilinear, trilinear or higher-dimensional linear interpolation is appropriate.
Bicubic, tricubic, etc., interpolation is used as well, but is probably not worth the extra computational overhead.
The differential equations for tracing a two-dimensional, atmospheric trajectory in longitude-latitude coordinates are as follows: where,
One problem with this formulation is the polar singularity: notice how the denominator in the first equation goes to zero when the latitude is 90 degrees—plus or minus.
One means of overcoming this is to use a locally Cartesian coordinate system close to the poles.