In fluid mechanics, meteorology 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.
Suppose we have a time-varying flow field,
While the equation looks simple, there are at least three concerns when attempting to solve it numerically. The first is the integration scheme. This is typically a Runge-Kutta, although others can be useful as well, such as a leapfrog. The second is the method of determining the velocity vector,
Velocity fields can be determined by measurement, e.g. from weather balloons, from numerical models or especially from a combination of the two, e.g. assimilation models.
The final concern is metric corrections. These are necessary for geophysical fluid flows on a spherical Earth. 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. Another is to perform the integration on a pair of Azimuthal equidistant projections—one for the N. Hemisphere and one for the S. Hemisphere.
Trajectories can be validated by balloons in the atmosphere and buoys in the ocean.