Contour advection

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.

To maintain its integrity, points must be added to or removed from the curve at regular intervals based on some criterion or metric.

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 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.

[2] 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.