Trajectory (fluid mechanics)

Suppose we have a time-varying flow field, ${\displaystyle {\vec {v}}({\vec {x}},~t)}$. The motion of a fluid parcel, or trajectory, is given by the following system of

${\displaystyle {\frac {d{\vec {x}}}{dt}}={\vec {v}}({\vec {x}},~t)}$

While the equation looks simple, there are at least three concerns when attempting to solve it

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:

${\displaystyle {\frac {d\theta }{dt}}={\frac {u}{r\cos \phi }}}$

${\displaystyle {\frac {d\phi }{dt}}={\frac {v}{r}}}$

where, ${\displaystyle \theta }$ and ${\displaystyle \phi }$ are, respectively, the longitude and latitude in

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

Trajectories can be validated by

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