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[[File:wv cadv ecmwf small.gif|frame|right|alt=Contour advection animation|Contour advection simulation]]
 
'''Contour advection''' is a [[Lagrangian coordinates|Lagrangian method]]
of simulating the evolution of one or more contours or isolines of
a tracer as it is stirred by a moving fluid.
Consider a blob of dye injected into a river or stream: to first order it could be modelled by tracking only the motion of its outlines.
It is an excellent method for studying [[chaotic mixing]]:
even when advected by smooth or finitely-resolved velocity fields,
through a continuous process of stretching and folding,
these contours often develop into intricate [[fractals]].
The [[flow tracer|tracer]] is typically passive as in <ref name="Waugh_Plumb1994">{{Cite journal
| author1 = D. W. Waugh
| author2 = R. A. Plumb
| title = Contour advection with surgery: a technique for investigating the fine scale structure in tracer transport
| journal = [[Journal of the Atmospheric Sciences]]
| volume = 51
| issue = 4
| pages = 415&ndash;422
| year = 1994
}}
</ref>
but may also be active as in,<ref name="Dritschel1988">{{Cite journal
| author = D. G. Dritschel
| title = Contour surgery: A topological reconnection scheme
| journal = [[Journal of Computational Physics]]
| volume = 77
| pages = 240&ndash;266
| year = 1988
}}
</ref> representing a dynamical property of the fluid such as [[vorticity]].
At present, advection of contours is limited to two dimensions,
but generalizations to three dimensions are possible.
 
==Method==
 
First we need a set of points that accurately define the contour.
These points are [[advection|advected]] forward using a [[trajectory (meteorology)|trajectory]]
integration technique.
To maintain its integrity,
points must be added to or removed from the curve
at regular intervals based on some criterion or metric.
The most obvious criterion is to maintain the distance between adjacent points
within a certain interval.
A better method is to use curvature since fewer points are required for
the same level of precision.
The [[curvature]] of a two-dimensional, Cartesian curve is given as:
 
:<math>
\frac{1}{r} = \sqrt{\left ( \frac{\mathrm d^2 x}{\mathrm d s^2}\right )^2 +
              \left ( \frac{\mathrm d^2 y}{\mathrm d s^2} \right )^2}
</math>
 
where <math>r</math> is the radius of curvature
and <math>s</math> is the path.
We need to keep the fraction of arc traced out between two adjacent points,
<math>
r \Delta s
</math>, where <math>\Delta s</math> is the path difference between them,
roughly constant
 
In,<ref name="Mills2009">{{Cite journal
| author = Peter Mills
| title = Isoline retrieval: An optimal method for validation of advected contours
| journal = Computers & Geosciences
| volume = 35
| number = 11
| pages = 2020&ndash;2031
| year = 2009
| doi = 10.1016/j.cageo.2008.12.015
| url = http://peteysoft.users.sourceforge.net/Mills2009.pdf
}}
</ref>
[[cubic spline]] fitting is used both to calculate the curvature
and interpolate new points into the contour.
The spline, which
is fitted [[parametric equation|parametrically]],
returns a set of second-order derivatives.
 
===Surgery===
 
A powerful refinement to the technique involves cutting out filaments that have become too
narrow to be significant.
If the distance method of adding/removing points is used,  
then it is relatively straight forward
to check the distances between all combinations of points.
If a distance between non-adjacent points is too small,  
then the two points are separated from their neighbours,
joined together and their neighbours joined also.
Points may then be removed if necessary. 
Once we allow surgery, we allow [[multiply connected]] domains inside the same contour.
A piece of the contour only one point in length would be removed from the simulation.
The most challenging part of the exercise is keeping track of all the points in order
to reduce the number of distance calculations---see [[nearest neighbour search]].
If the curvature method is used,  
then it may be difficult to recognize when two sections of the contour
are close enough to apply the surgery because of differing spacing
in strongly curved versus relatively straight sections.<ref name="Dritschel1988"/>
 
==Validation==
 
Advected contours, e.g. of trace gases (such as ozone) in the stratosphere,
can be validated with satellite remote sensing instruments using a method called [[isoline retrieval]].
<ref name="Mills2009"/>
 
==External links==
* [http://ctraj.sourceforge.net ctraj]: A library for Lagrangian advection simulations.
 
==References==
 
<references/>
 
[[Category:Fluid dynamics]]
[[Category:Meteorology]]

Revision as of 15:55, 7 September 2013

Contour advection animation
Contour advection simulation

Contour advection is a Lagrangian method of simulating the evolution of one or more contours or isolines of a tracer as it is stirred by a moving fluid. Consider a blob of dye injected into a river or stream: to first order it could be modelled by tracking only the motion of its outlines. It is an excellent method for studying chaotic mixing: even when advected by smooth or finitely-resolved velocity fields, through a continuous process of stretching and folding, these contours often develop into intricate fractals. The tracer is typically passive as in [1] but may also be active as in,[2] representing a dynamical property of the fluid such as vorticity. At present, advection of contours is limited to two dimensions, but generalizations to three dimensions are possible.

Method

First we need a set of points that accurately define the contour. These points are advected forward using a trajectory integration technique. To maintain its integrity, points must be added to or removed from the curve at regular intervals based on some criterion or metric. The most obvious criterion is to maintain the distance between adjacent points within a certain interval. A better method is to use curvature since fewer points are required for the same level of precision. The curvature of a two-dimensional, Cartesian curve is given as:

1r=(d2xds2)2+(d2yds2)2

where r is the radius of curvature and s is the path. We need to keep the fraction of arc traced out between two adjacent points, rΔs, where Δs is the path difference between them, roughly constant

In,[3] cubic spline fitting is used both to calculate the curvature and interpolate new points into the contour. The spline, which is fitted parametrically, returns a set of second-order derivatives.

Surgery

A powerful refinement to the technique involves cutting out filaments that have become too narrow to be significant. If the distance method of adding/removing points is used, then it is relatively straight forward to check the distances between all combinations of points. If a distance between non-adjacent points is too small, then the two points are separated from their neighbours, joined together and their neighbours joined also. Points may then be removed if necessary. Once we allow surgery, we allow multiply connected domains inside the same contour. A piece of the contour only one point in length would be removed from the simulation. The most challenging part of the exercise is keeping track of all the points in order to reduce the number of distance calculations---see nearest neighbour search. If the curvature method is used, then it may be difficult to recognize when two sections of the contour are close enough to apply the surgery because of differing spacing in strongly curved versus relatively straight sections.[2]

Validation

Advected contours, e.g. of trace gases (such as ozone) in the stratosphere, can be validated with satellite remote sensing instruments using a method called isoline retrieval. [3]

External links

  • ctraj: A library for Lagrangian advection simulations.

References

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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  3. ↑ 3.0 3.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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