2. Theory |

2.1 Contour Dynamics

2.2 Equations of Cotour Dynamics

2.3 Theory of an Elliptical Vortex

2.4 Finite-Time Lyapunov Analysis on Fluid Deformation

## 2.2 Equations of Cotour Dynamics

Basics of contour dynamics are summarized following Dritschel(1989).The streamfunction Y( x,y,t ) is related to the vorticity Q( x,y,t ) through Poisson's equation,

(2.3)

General solution can be obtained using the Green function G(r):

(2.4)

where

(2.5)

As for the two-dimensional infinite plane, G(r) = ( log r )/ 2p :

(2.6)

The velocity (u,v) is given by

(2.7)

By integrating (2.7) partially by h,

(2.8)

The h integral of the second part of the right-hand side is equal to the vorticity jump q^{( j )}, since ¶Q/¶h≠0 only on the contour. Letting q^{( j )}be the vorticity jump crossing inward, and taking the line integral anti-clockwise, we obtain

(2.9)

Calculating v in the same way yields

(2.10)

These two equations are bound up to

(2.11)

which means that the velocity at an arbitrary point can be calculated from the contour position {x^{( j )}} . Since the velocity on the contour can be calculated. Time evolution of the contour can be computed by advecting the contour using that velocity.

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