Contour advection
Encyclopedia
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
but may also be active as in , 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.
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:
where
is the radius of curvature
and
is the path.
We need to keep the fraction of arc traced out between two adjacent points,
, where 
is the path difference between them,
roughly constant
In ,
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.
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 .
can be validated with satellite remote sensing instruments using a method called isoline retrieval
.
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
Chaotic mixing
In chaos theory and fluid dynamics, chaotic mixing is a processby which flow tracers develop into complex fractals under the actionof a time-varying fluid flow.The flow is characterized by an exponential growth of fluid filaments....
:
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
Flow tracer
thumb|right|300px|On May 2, [[2001]], the Moderate-resolution Imaging Spectroradiometer obtained this spectacular image of the [[Atlantic Ocean]]'s [[Gulf Stream]]. The [[false color]]s in the image represent "[[brightness temperature]]" observed at the top of the [[atmosphere]]...
is typically passive as in
but may also be active as in , 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
Advection
Advection, in chemistry, engineering and earth sciences, is a transport mechanism of a substance, or a conserved property, by a fluid, due to the fluid's bulk motion in a particular direction. An example of advection is the transport of pollutants or silt in a river. The motion of the water carries...
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
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
of a two-dimensional, Cartesian curve is given as:
where
is the radius of curvatureand
is the path.We need to keep the fraction of arc traced out between two adjacent points,
, where is the path difference between them,roughly constant
In ,
cubic spline fitting is used both to calculate the curvature
and interpolate new points into the contour.
The spline, which
is fitted parametrically
Parametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
,
returns a set of second-order derivatives.
Surgery
A powerful refinement to the technique involves cutting out filaments that have become toonarrow 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 .
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
Isoline retrieval
Isoline retrieval is a remote sensing inverse method that retrieves one or more isolines of a trace atmospheric constituent or variable. When used to validate another contour, it is the most accurate method possible for the task...
.