<s>
In	O
fluid	O
mechanics	O
,	O
meteorology	O
and	O
oceanography	O
,	O
a	O
trajectory	B-Application
traces	O
the	O
motion	O
of	O
a	O
single	O
point	O
,	O
often	O
called	O
a	O
parcel	O
,	O
in	O
the	O
flow	O
.	O
</s>
<s>
Trajectories	B-Application
are	O
useful	O
for	O
tracking	O
atmospheric	O
contaminants	O
,	O
such	O
as	O
smoke	O
plumes	O
,	O
and	O
as	O
constituents	O
to	O
Lagrangian	O
simulations	O
,	O
such	O
as	O
contour	O
advection	O
or	O
semi-Lagrangian	B-Application
schemes	I-Application
.	O
</s>
<s>
The	O
motion	O
of	O
a	O
fluid	O
parcel	O
,	O
or	O
trajectory	B-Application
,	O
is	O
given	O
by	O
the	O
following	O
system	O
of	O
ordinary	O
differential	O
equations	O
:	O
</s>
<s>
While	O
the	O
equation	O
looks	O
simple	O
,	O
there	O
are	O
at	O
least	O
three	O
concerns	O
when	O
attempting	O
to	O
solve	O
it	O
numerically	B-General_Concept
.	O
</s>
<s>
The	O
first	O
is	O
the	O
integration	B-Algorithm
scheme	I-Algorithm
.	O
</s>
<s>
This	O
is	O
typically	O
a	O
Runge-Kutta	B-Algorithm
,	O
although	O
others	O
can	O
be	O
useful	O
as	O
well	O
,	O
such	O
as	O
a	O
leapfrog	B-Algorithm
.	O
</s>
<s>
The	O
second	O
is	O
the	O
method	O
of	O
determining	O
the	O
velocity	O
vector	O
,	O
at	O
a	O
given	O
position	O
,	O
,	O
and	O
time	O
,	O
t	O
.	O
Normally	O
,	O
it	O
is	O
not	O
known	O
at	O
all	O
positions	O
and	O
times	O
,	O
therefore	O
some	O
method	O
of	O
interpolation	B-Algorithm
is	O
required	O
.	O
</s>
<s>
If	O
the	O
velocities	O
are	O
gridded	O
in	O
space	O
and	O
time	O
,	O
then	O
bilinear	O
,	O
trilinear	O
or	O
higher-dimensional	O
linear	O
interpolation	B-Algorithm
is	O
appropriate	O
.	O
</s>
<s>
Bicubic	B-Algorithm
,	O
tricubic	O
,	O
etc.	O
,	O
interpolation	B-Algorithm
is	O
used	O
as	O
well	O
,	O
but	O
is	O
probably	O
not	O
worth	O
the	O
extra	O
computational	O
overhead	O
.	O
</s>
<s>
The	O
differential	O
equations	O
for	O
tracing	O
a	O
two-dimensional	O
,	O
atmospheric	O
trajectory	B-Application
in	O
longitude-latitude	O
coordinates	O
are	O
as	O
follows	O
:	O
</s>
<s>
Trajectories	B-Application
can	O
be	O
validated	O
by	O
balloons	O
in	O
the	O
atmosphere	O
and	O
buoys	O
in	O
the	O
ocean	O
.	O
</s>
