Reeb stability theorem: Difference between revisions

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In [[fluid mechanics]], [[meteorology]] and [[oceanography]], a '''trajectory''' traces the motion of a single point, often called a parcel, in the flow.
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Trajectories are useful for tracking atmospheric contaminants, such as smoke plumes,
and as constituents to [[Lagrangian coordinates|Lagrangian]] simulations,
such as [[contour advection]] or [[semi-Lagrangian scheme]]s.
Suppose we have a time-varying flow field, <math>\vec v(\vec x,~t)</math>.
The motion of a fluid parcel, or trajectory, is given by the following
system of [[ordinary differential equations]]:
 
:<math>
\frac{d \vec x}{dt} = \vec v(\vec x, ~t)
</math>
 
While the equation looks simple, there are at least three concerns
when attempting to solve it [[numerically]]. The first is the  
[[numerical ordinary differential equations|integration scheme]].
This is typically a [[Runge-Kutta]], although others can be useful as
well, such as a [[leapfrog integration|leapfrog]]. The second is the
method of determining the velocity vector, <math>\vec v</math>
at a given position, <math>\vec x</math>, and time, ''t''.
Normally, it is not known at all positions and times, therefore
some method of [[interpolation]] is required. If the velocities
are gridded in space and time, then [[bilinear interpolation|bilinear]],
[[trilinear interpolation|trilinear]] or higher dimensional linear
interpolation is appropriate.
[[Bicubic interpolation|Bicubic]], [[Tricubic interpolation|tricubic]], etc.,
interpolation is used as well, but is probably not worth the extra [[overhead (computing)|computational overhead]].
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. [[data assimilation|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:
 
:<math>
\frac{d \theta}{dt} = \frac{u}{r \cos \phi}
</math>
 
:<math>
\frac{d \phi}{dt} = \frac{v}{r}
</math>
 
where, <math>\theta</math> and <math>\phi</math> are, respectively,
the longitude and latitude in [[radian]]s, ''r'' is the [[radius of the Earth]],
''u'' is the zonal wind and ''v'' is the meridional wind.
 
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 projection]]s—one for the N. Hemisphere
and one for the S. Hemisphere.
<ref name="Mills2012">{{cite techreport
|author=Peter Mills
|year=2012
|title=Principal component proxy tracer analysis
|number=arXiv:1202.2194v1
|url=http://arxiv.org/abs/1202.1999
}}</ref>
 
Trajectories can be validated by [[balloons]] in the [[atmosphere]] and [[buoys]] in the
[[ocean]].
 
== External links ==
* [http://ctraj.sourceforge.net ctraj]: A trajectory integrator written in C++.
 
== References ==
 
* {{cite book
|title=Numerical Recipes in C: the Art of Scientific Computing
|author1=William H. Press
|author2=Brian P. Flannery
|author3=Saul A. Teukolsky
|author4=William T. Vetterling
|year=1992
|publisher=Cambridge University Press
|edition=2nd
}}
 
{{Reflist}}
 
[[Category:Fluid dynamics]]
[[Category:Meteorology]]
[[Category:Numerical analysis]]

Latest revision as of 22:10, 5 August 2014

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Decide how and where you want to ride, then choose a mountain bike that fits your personal style, rather than that of the sales staff. The more bikes you can test drive, you better you'll understand what works and what doesn't. It is a quality piece available at affordable price. Instead of having fun playing with him (like a child with a favorite playmate), they evaluate his potential as a father. 7,934,739) CVA is an award-winning cancellation style and now it is recognized with a US Apparent.

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